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--- abstract: 'This paper presents sufficient conditions for the convergence of online estimation methods and the stability of adaptive control strategies for a class of history dependent, functional differential equations. The study is motivated by the increasing interest in estimation and control techniques for robotic systems whose governing equations include history dependent nonlinearities. The functional differential equations in this paper are constructed using integral operators that depend on distributed parameters. As a consequence the resulting estimation and control equations are examples of distributed parameter systems whose states and distributed parameters evolve in finite and infinite dimensional spaces, respectively. Well-posedness, existence, and uniqueness are discussed for the class of fully actuated robotic systems with history dependent forces in their governing equation of motion. By deriving rates of approximation for the class of history dependent operators in this paper, sufficient conditions are derived that guarantee that finite dimensional approximations of the online estimation equations converge to the solution of the infinite dimensional, distributed parameter system. The convergence and stability of a sliding mode adaptive control strategy for the history dependent, functional differential equations is established using Barbalat’s lemma.' author: - 'Shirin Dadashi [^1], Parag Bobade[^2], and Andrew J. Kurdila[^3]' nocite: '[@]' title: Online Estimation and Adaptive Control for a Class of History Dependent Functional Differential Equations --- Introduction {#sec:intro} ============ It is typical in texts that introduce the fundamentals of modeling, stability, and control of robotic systems to assume that the underlying governing equations consist of a set of coupled nonlinear ordinary differential equations. This is a natural assumption when methods of analytical mechanics are used to derive the governing equations for systems composed of rigid bodies connected by ideal joints. A quick perusal of the textbooks [@shv2005], [@ssvo2010], or [@lda2004], for example, and the references therein gives a good account of the diverse collection of approaches that have been derived for this class of robotic system over the past few decades. Theses methods have been subsequently refined by numerous authors. Over roughly the same period, the technical community has shown a continued interest in systems that are governed by nonlinear, functional differential equations. These methods that helped to define the direction of initial efforts in the study of well-posedness and stability include [@m1949], [@k1956a],[@k1956b], and their subsequent development is expanded in [@driver1962], [@rudakov1974], [@rudakov1978]. More recently, specific control strategies for classes of functional differential equations have appeared in [@rs2002], [@irs2002], and [@ilr2010]. The research described in some cases above deals with quite general plant models. These can include classes of delay equations and general history dependent nonlinearities. One rich collection of history dependent models includes hysteretically nonlinear systems. General discussions of nonlinear hysteresis models can be found in [@visintin1994] or [@bs1996], and some authors have studied the convergence and stability of systems with nonlinear hysteresis. For example, a synthesis of controllers for single-input / single-output functional differential equations is presented in [@rs2002] and [@ilr2010], and these efforts include a wide class of scalar hysteresis operators. The success of adaptive control strategies in classical manipulator robotics, as exemplified by [@shv2005], [@lda2004], [@ssvo2010], can be attributed to a large degree to the highly structured form of the governing system of nonlinear ordinary differential equations. As is well-known, much of the body of work in adaptive control for robotic systems relies on traditional linear-in-parameters assumptions. The purpose of this paper is to explore the degree to which the approaches that have been so fruitful in adaptive control of robotic manipulators can be extended to robotic systems governed by certain history dependent, functional differential equations. Emulating the strategy used for robotic systems modeled by ordinary differential equations, we restrict attention to a class of hysteresis operators that satisfy a linear in [distributed]{} parameters condition. That is, the contribution to the functional differential equations takes the form of a nonlinear, history dependent operator that acts linearly on an infinite dimensional and unknown distributed parameter. Motivation: A Flapping Wing Robot for Wind Tunnel Experiments ------------------------------------------------------------- #### It has been an ongoing quest of humans to understand nature and its creations. Consequently, understanding and attempting to mimic these natural systems has helped push the boundaries of science and technology in various fields. In last few decades there has been increasing interest in area of flapping wing vehicles. The ultimate goal is to be able to mimic the flight abilities of a bird and use the merits of flapping wing to improve the agility and performance in flying. Due to the inherent nature of unsteady aerodynamics encountered in flapping wing flights designing such vehicles becomes challenging task .\ In early days of flapping wing flight research [@VT2013], [@GT2010] the wings were modeled as single rigid body without any articulations, to study the aerodynamic effects in flapping/oscillating rigid plates or membranes. This simplified approached helped understand the effect of vortex formation and shredding, on drag and lift forces during flapping flight. Later, with detailed studies in bio-mechanics of bird flight, the new flapping mechanism models incorporated the multi-body articulations to mimic the effect of bones,joints and muscles. These models were studied for their flight stability and control. The aerodynamic forces were modeled as functions of state variables i.e every combination of state variable represented a corresponding aerodynamic state there by implying quasi steady flow. This reduced the infinite dimensional coupled problem( Navier Stokes and multi-body dynamics) to finite dimensional ODEs. Since the flapping wing flight is affected by unsteady aerodynamics, these were not seen in the quasi steady model. Tobak et. al [@tobak] represented the aerodynamic forces as functional of state variables which helped incorporated the unsteady effects. This emphasizes the fact that the aerodynamic forces not just depend on instantaneous state but also on the history of state variables. Thus now the governing equation of motions is a integro-differential equation. Integro-differential equations have been widely studied in past and the existence, uniqueness and stability of such equations have also been extensively analysed . Since this makes the problem infinite dimensional it poses challenges in coming up with closed form solutions analytically. Usual approach is to represent the infinite dimensional equations to finite dimensional approximation. In this paper we assume the flapping wing mechanism through a serial kinematic chain. The governing equations involve aerodynamic forces that are modeled using history dependent integral operations. The finite dimensional approximations of these operators are used in the closed loop equation. The choice of control strategy ensures the stability and convergence of finite dimensional approximation of the operator. We also discuss the wellposedness, existence and uniqueness of solutions for such integro-differential equations. Illchman et al [@ilr2010] proved the existence and uniqueness for full state input, equations with history dependent terms and proposed universal control. Our equations do not assume the luxury of fully actuated. The proof of existence for open loop and closed loop without full state feedback is given in section \[sec:exist\]. One of the applications that motivate the theoretical considerations in this paper is a flapping wing robot that is subject to unknown and non-negligible unsteady aerodynamics. This control synthesis problem is challenging owing to the uncertain, nonlinear dynamics of the robotic system. Although the equations of motion for the mechanical system can be derived via the Euler Lagrange equations using well-documented approaches such as those found in references such as [@lda2003],[@ssvo2010],[@svh2005], the incorporation of history dependent models of aerodynamics loads results in control synthesis problems that have not been studied as extensively. We illustrate the class of models that are considered in this paper by outlining a variation on two familiar problems encountered in robotic manipulator dynamics, estimation, and control. Consider the task of developing a model and synthesizing a controller for a flapping wing, test robot that will be used to study aerodynamics in a wind tunnel. See [@bsb2013] for such a system that has been developed by researchers at Brown University over the past few years. Dynamics for a ground based flapping wing robot can be derived using analytical mechanics in a formulation that is tailored to the structure of a serial kinematic chain [@lda2004], [@shv2005], [@ssvo2010]. The equations of motion take the form $$M(q(t))\ddot{q}(t)+C(q(t),\dot{q}(t))\dot{q}(t)+\frac{\partial{V}}{\partial{q}}=Q_{a}(t,\mu) + {\tau}(t) \label{eq:robotics}$$ where $M(q)(t){\in}{\mathbb{R}}^{N{\times} N}$ is the generalized inertia or mass matrix, $C(q(t),\dot{q}(t)){\in}{\mathbb{R}}^{N{\times} N}$ is a nonlinear matrix that represents Coriolis and centripetal contributions, $V$ is the potential energy, $Q_{a}(t,\mu){\in}{\mathbb{R}}^{N}$ is a vector of generalized aerodynamic forces, and ${\tau(t)}{\in}{\mathbb{R}}^{N}$ is the actuation force or torque vector. The generalized forces $Q(t,\mu)$ due to aerodynamic loads are assumed to be expressed in terms of history dependent operators that are carefully discussed below in Section \[sec:history\], and $\mu$ is the [ distributed parameter]{} that defines the specific history dependent operator. For the current discussion, it suffices to note that the aerodynamic contributions are unknown, nonlinear, unsteady, and notoriously difficult to characterize. We consider two specific sets of equations in this paper that are derived from the robotic Equations \[eq:robotics\], both of which have similar form. We are interested in online identification problems in which we seek to find the final state and distributed parameters from observations of the states of the evolution equation. We are also interested in control synthesis where we choose the input to drive the system to some desired configuration, or to track a given input trajectory. To simplify our discussion, and following the standard practice for many control synthesis problems for robotics, we choose the original control input to be a partial feedback linearizing control that that reformats the control problem in a standard form. In the case of online identification, we choose the input $\tau=M(q)(u-G_1 \dot{q} - G_0 q)-(C(q,\dot{q})\dot{q} + \frac{\partial V}{\partial q}(q)) $ so that the governing equations take the form $$\label{eqn:2}\frac{d}{dt} \begin{bmatrix} q(t)\\ \dot{q}(t) \end{bmatrix} = \begin{bmatrix} 0&I\\ -G_0&-G_1 \end{bmatrix} \begin{bmatrix} q(t)\\ \dot{q}(t) \end{bmatrix} + \begin{bmatrix} 0 \\ I \end{bmatrix} (M^{-1}(q)Q_a(t,\mu) +u(t)).$$ in terms of a new input $u$. The goal in the online identification problem is to learn the parameters $\mu$ and limiting values $q_\infty,\dot{q}_\infty$ from knowledge of the inputs and states $(u,q,\dot{q})$. We are also interested in tracking control problems. When the desired trajectory is given by $q_d$, we choose the input $\tau=M(q)(u+\ddot{q}_d-G_1 \dot{e} - G_0 e)-(C(q,\dot{q})\dot{q} + \frac{\partial V}{\partial q}(q))$,and the equations governing the tracking error $e:=q-q_d$ take the form $$\label{eqn:3}\frac{d}{dt} \begin{bmatrix} e(t)\\ \dot{e}(t) \end{bmatrix} = \begin{bmatrix} 0&I\\ -G_0&-G_1 \end{bmatrix} \begin{bmatrix} e(t)\\ \dot{e}(t) \end{bmatrix} + \begin{bmatrix} 0 \\ I \end{bmatrix} (M^{-1}(e+q_d) Q_a(t,\mu)+u(t))$$ In either of the above two cases, we will show in the next section that the equations can be written in the general form $$\dot{X}(t)=AX(t)+B((\mathcal{H}X)(t)\circ \mu +u(t)). \label{eq:1st_order}$$ where ${A}\in \mathbb{R}^{m\times m}$ is the system matrix, ${B}\in \mathbb{R}^{m\times q}$ is the control input matrix, ${u}(t)\in \mathbb{R}^{q}$ is the corresponding input, and ${(\mathcal{H}X)}(t)$ is a history dependent operator that acts on the distributed parameter ${\mu}$. History Dependent Operators {#sec:history} =========================== There is a significant body of research to model and study the unsteady aerodynamic phenomena in flapping flight. Many different models have been presented in the last twenty years to study the aerodynamics and control of flapping flight. Numerically intensive computational fluid dynamics (CFD) presents a precise method to simulate and study the unsteady lift and drag aerodynamic forces. Generally CFD methods exploit high dimensional models that incorporate computationally expensive moving boundary techniques for the Navier-Stokes equations. They are powerful tools to explain some of the characteristics of the aerodynamic forces. One of the characteristics that has inspired the approach here is the history dependence of the aerodynamic lift and drag functions. We refer the interested reader to [@Dadashi2016] to study this phenomena in detail. Although CFD methods are advantages in several aspects, they suffer from curse of dimensionality which makes them a very unfavorable choice for online control applications. In this section, we model the unsteady aerodynamics using history dependent operators. Moreover, we present a method that provides an alternative to a high dimensional aerodynamic model that typically evolves in a much lower dimensional space. We also study the accuracy of the presented method with respect to the resolution level of the lower dimensional model. A Class of History Dependent Operators -------------------------------------- Methods for modeling history dependent nonlinearities can be formulated using a wide array of approaches. Analytical methods for the study of such systems can be based on ordinary or partial differential equations, differential inclusions, functional differential equations, delay differential equations, or operator theoretic approaches. See references [@kp89],[@visintin1994],[@shv2005]. This paper treats evolution equations that are constructed using a specific class of history dependent operators $\mathcal{H}$ that are defined in terms of integral operators constructed from history dependent kernels. These operators are studied in general in [@kp89] and [@visintin1994]. In this paper the history dependent operators are mappings $$\mathcal{H}:C([0,T),\mathbb{R}^{m}){\rightarrow} C([0,T),P^)$$ where the $T$ is the final time of an interval under consideration, $m$ is the number of input functions, $q$ is the number of output functions, $P$ is a Hilbert space of distributed parameters and its topological dual space $P^$. We limit our consideration to input$-$output relationships that take the form $$y(t)=(\mathcal{H}X)(t){\circ}{\mu} \label{eq:2.1}$$ for each $t\in [0,T)$ where $y(t) \in \mathbb{R}^q$,$(\mathcal{H}X)(t) \in P^$, and $\mu \in P $. The definition of $\mathcal{H}$ in this paper is carried out in several steps. All of our history dependent operators $\mathcal{H}$ are defined by a superposition or weighting of elementary hysteresis kernels $\kappa_i$ that are continuous as mappings $\kappa_i :\Delta \times [0,T) \times C[0,T) \rightarrow C[0,T)$ for $i=1,\ldots,\ell$. We first define the operator $h_i:C[0,T) \rightarrow C([0,T),P^)$ $$(h_if)(t) \circ \mu_i := \iint_{\Delta} \kappa_i (s,t,f) \mu_i(s) ds \label{eq:hys_int_op}$$ for $\mu_i \in P_i$ and $P=P_1 \times \cdots \times P_\ell$. When we consider problems such as in our motivating examples and numerical case studies, we must construct vectors $H$ of history dependent operators where we define the diagonal matrix $$(H X)(t) := \begin{Bmatrix} h_1(a(X))(t) & & 0 \\ &\ddots & \\0 & & h_\ell(a(X))(t) \end{Bmatrix}$$ for each $t\in [0,T)$ where $a : \mathbb{R}^m \rightarrow \mathbb{R}$ is some nonlinear smooth map. Finally, our applications to robotics require that we consider $$(\mathcal{H}X)(t) = b(X(t))(HX)(t), \label{eq:hdef1}$$ where $b:\mathbb{R}^m \rightarrow \mathbb{R}^{q\times \ell}$ is some nonlinear, smooth map. In terms of our entrywise definitions of the input–output mappings, we have $$y_i(t):= \sum_{j=1}^{l} b_{ij}(X(t))[h_j(a(X))](t)\circ \mu_j \label{eq:hdef2}$$ for $i=1,\cdots, q$. In the following discussion, let $\kappa$ be a generic representation of any of the kernels $\kappa_i$ for $i=1,\ldots, \ell$. We choose a typical kernel $\kappa(s,t,f)$ to be a special case of a generalized play operator [@visintin1994]. We suppose that $f$ is a piecewise linear function on $[0,t]$ with breakpoints $0=t_0 < t_1 < \cdots < t_N=t$. The output function $t \mapsto \kappa(s,t,f)$, for a fixed $s=(s_1,s_2) \in \Delta \subset \mathbb{R}^2$ and piecewise linear $f:[0,t] \rightarrow \mathbb{R}$, is defined by the recursion where $\kappa^{n-1}:=\kappa(s,t_{n-1},f)$ and for $t \in[t_{n-1},t_n]$ we have $$\begin{aligned} \kappa(s,t,f):= \left \{ \begin{array}{ccc} \max \left \{ \kappa^{n-1}, \gamma_{s_2}(f(t)) \right \} & & f \text{ increasing on } [t_{n-1},t_n], \\ \min \left \{ \kappa^{n-1}, \gamma_{s_1}(f(t)) \right\} & & f \text{ decreasing on } [t_{n-1},t_n]. \end{array} \right .\end{aligned}$$ The recursion above depends on the choice of the left and right bounding functions $\gamma_{s_1},\gamma_{s_2}$ that are depicted in Figure \[fig:fig\_play1\]. There are given in terms of a single ridge function $\gamma:\mathbb{R} \rightarrow \mathbb{R}$ with $$\begin{aligned} \gamma_{s_2}(\cdot)&:=\gamma(\cdot-s_2), \notag \\ \gamma_{s_1}(\cdot)&:=\gamma(\cdot-s_1). \label{eq:defgamma}\end{aligned}$$ ![Elementary hysteresis kernel $t \rightarrow \kappa(s,t,f)$ for fixed $s=(s_1,s_2)\in \mathbb{R}^2$ and piecewise continuous $f:[0,t) \rightarrow \mathbb{R}$.[]{data-label="fig:fig_play1"}](kernel_responce.pdf){width=".45\textwidth"} As noted in [@visintin1994], the definition of $\kappa$ is extended for any $f\in C[0,T)$ by a continuity and density argument. Approximation of History Dependent Operators {#subsec:approx_hist} -------------------------------------------- The integral operator introduced in Equation \[eq:hys\_int\_op\] allows for the representation of complex hysteretic response via the superposition or weighting of fundamental kernels $\kappa_i$. These fundamental kernels, each of which has simple input-output relationships, play the role of building blocks for modeling much more complex response characteristics. See [@ksw2003] for studies of history dependent active materials, [@Dadashi2016] for applications that represent nonlinear aerodynamic loading, or Section \[sec:numerical\] of this paper to see an example of richness of this class of models. In this section we emphasize another important feature of this particular class of history dependent operators. We show that relatively simple approximation methods yields bounds on the error in approximation of the history dependent operator that are uniform in time and over the class of functions $\mu\in P$. Approximation Spaces $\mathcal{A}^\alpha_2$ ------------------------------------------- The approximation framework we follow in this paper is based on a straightforward implementation of approximation spaces discussed in detail in [@dl1993] or [@devore1998], and further developed by Dahmen in [@dahmen1996]. We will see that approximation of the class of history dependent operators under consideration exploit a well-known connection between the class of Lipschitz functions and certain approximation spaces as described in [@devore1998]. Wavelets and Approximation Spaces --------------------------------- Multiresolution Analysis ( MRA ) techniques use results from wavelet theory to model multiscale phenomena. To motivate our discussion, we begin by constructing Haar wavelets in one spatial dimension and subsequently discuss how the process can be easily extended to piecewise constant functions over triangulations in two dimensions. The Haar scaling function is defined as follows: $$\begin{aligned} \phi(x) = \left \{ \begin{array}{lll} 1 & \text{if } & \text{\hspace{-.5 in}} x \in [0,1)\\ 0 & \text{otherwise.} \end{array} \right.\end{aligned}$$ The dilates and translates $\phi_{j,k}$ of $\phi$ are defined over $\mathbb{R}$ as $$\begin{aligned} \phi_{j,k}(x) = 2^{j/2}\phi(2^{j}x-k)= 2^{j/2} 1_{\Delta_{j,k}}(x)\end{aligned}$$ for $j=0, \cdots, \infty$ and $k \in \mathbb{Z}$. It is important to note that with this normalization the functions $\{\phi_{j,k}\}_{k=0}^{2^j-1}$ are $L^2[0,1]$ orthonormal so that $$\begin{aligned} \langle \phi_{j,k},\phi_{j,l} \rangle = \int_{\mathbb{R}} \phi_{j,k}(x)\phi_{j,l}(x) \mathrm{d}x = \delta_{kl}.\end{aligned}$$ In this equation $\Delta_{j,k}=\{x\|x \in [2^{-j}k,2^{-j}(k+1)]\}$ and $1_{\Delta_{j,k}}$ is the characteristic function of $\Delta_{j,k}$. For any fixed integer $j$, the $\phi_{j,k}$ form a orthonormal basis that spans the space of piecewise constants over $\{\Delta_{j,k}\}_{k=0}^{2^j-1}$. We let $V_j$ denote space of piecewise constant functions $$\begin{aligned} V_j = \underset{k=0,\cdots, 2^{j-1}}{\mathrm{span}}\{ \phi_{j,k} \}.\end{aligned}$$ Corresponding to Haar scaling function, the Haar wavelet $\psi$ is defined as $$\begin{aligned} \psi(x)=\left \{ \begin{array}{lll} 1 & x \in [0,\frac{1}{2}),\\ -1 & x \in [\frac{1}{2},1). \end{array} \right.\end{aligned}$$ Again, the translates and dilates of $\psi_{j,k}$ of $\psi$ are given by $$\begin{aligned} \psi_{j,k}(x) = 2^{j/2}\psi(2^{j}x-k),\end{aligned}$$ and the complement spaces $W_j$ are defined by $W_j= \underset{k}{\mathrm{span}}\{\psi_{j,k}\}$. It is straightforward to verify that the spaces $V_{j-1}$ and $W_{j-1}$ form an orthogonal direct sum of $V_j$. That is, we have $$\begin{aligned} V_j = V_{j-1} \bigoplus W_{j-1}.\end{aligned}$$ This process is well-known and standard in the literature as a means of constructing multiscale bases for $L^2[0,1]$. We will follow an analogous strategy to construct multiscale bases over the triangular domain depicted in Figure \[fig:refine\]. We first denote the characteristic functions over the triangular domain as shown in the Figure \[fig:refine\] $$\begin{aligned} 1_{\Delta_{s}}(x)=\left \{ \begin{array}{lll} 1 & x \in \Delta_s\\ 0 & \text{otherwise.} \end{array} \right.\end{aligned}$$ We next consider the regular refinement shown in Figure \[fig:refine\] where $\Delta_{i_1 i_2}$ is the $i_2$ child of $\Delta_{i_1}$. In general $\Delta_{i_1 i_2\hdots i_m i_{m+1}}$ is the $(m+1)^{st}$ child of $\Delta_{i_1 i_2 \hdots i_m}$. The multiscaling function $\phi_{j,k}$ is defined as $$\begin{aligned} \phi_{j,k}(x) = \frac{1_{\Delta_{i_1 i_2 \hdots i_j}}(x)}{\sqrt[]{m(\Delta_{i_1 i_2 \hdots i_j})}}\end{aligned}$$ where $j$ refers to the level of refinement in the grid. [.2]{} ![Regular refinement process for domain $\Delta$[]{data-label="fig:refine"}](j0 "fig:"){width="3cm"} [.2]{} ![Regular refinement process for domain $\Delta$[]{data-label="fig:refine"}](j1 "fig:"){width="3cm"} [.2]{} ![Regular refinement process for domain $\Delta$[]{data-label="fig:refine"}](j2 "fig:"){width="3cm"} [.2]{} ![Regular refinement process for domain $\Delta$[]{data-label="fig:refine"}](j3 "fig:"){width="3cm"} ![j level refinement[]{data-label="fig:refine"}](refine.jpg){width="50.00000%"} Since the history dependent operators $(\mathcal{H}X)(t)$ act on the infinite dimensional space $P=P_1 \times \cdots \times P_\ell$ of functions $\mu=(\mu_1,\ldots, \mu_\ell)$, we need approximations of these operators for computations and applications. In the discussion that follows we choose each function $\mu_i\in P_i:=L^2(\Delta)$ where the domain $\Delta\subset \mathbb{R}^2$ is defined as $$\Delta:=\left \{ (s_1,s_2) \in \mathbb{R}^2 \biggl \| \underline{s}\leq s_1 \leq s_2 \leq \overline{s}\right \}.$$ The modification of the construction that follows for different domains $\Delta_i$ for the functions $\mu_i \in L^2(\Delta_i)$ is trivial, but notationally tedious, and we leave the more general case to the reader. Given the domain $\Delta$ we introduce a regular refinement depicted in Figure \[fig:refine\] and disscused in more detail in Appendix A. The set $\Delta$ is subdivided into $\Delta_1,\Delta_2,\Delta_3,\Delta_4$ as shown, and each $\Delta_i$ is subdivided into $\Delta_{i1},\Delta_{i2},\Delta_{i3},\Delta_{i4}$. Further subdivision recursively introduces the sets $\Delta_{{i_1}\ldots{i_j}}$ for $i_j=1,\ldots,4$ that are the children of $\Delta_{{i_1}\ldots{i_{j-1}}}$. The characteristic functions $1_{\Delta_1},\ldots, 1_{\Delta_4}$ define a collection of multiscaling functions $\phi^1,\ldots, \phi^4$ as defined in [@keinert2004]. We define the space of piecewise constant functions $V_j$ on grid refinement level $j$ to be the span of the characteristic functions of the sets $\Delta_{i_1,\ldots,i_j}$, so that the dimension of $V_j$ is $4^j$. We denote by $\left \{ \phi_{j,k} \right \}_{k=1,\ldots, 4^j}$ the orthonormal basis obtained from these characteristic functions on a particular grid level, each normalized so that $(\phi_{j,k},\phi_{j,\ell})_{L^2(\Delta)} = \delta_{k,l}$. Each of the basis functions $\phi_{j,k}$ will be proportional to the characteristic function $\phi^\ell(2^j x + d)$ for some $\ell\in,{1,2,3,4}$ and displacement vector $d$. It is straightforward in this case [@keinert2004] to define $3$ piecewise constant multiwavelets $\psi^1, \psi^2, \psi^3$ that that are used to define functions $\psi_{j,m}$ for $m=1,\ldots,3 \times 4^j$ that span the complement spaces $W_j= \text{span}\left \{ \psi_{j,m}\right \}_{m=1,\ldots,3\cdot 4^j}$ that satisfy $$\begin{aligned} \underbrace{V_j}_{4^j functions} = \underbrace{V_{j-1}}_{4^{j-1} functions}\bigoplus \underbrace{W_{j-1}}_{3 \times 4^{j-1} functions}.\end{aligned}$$ It is a straightforward exercise to define $L^2(\Delta)-$orthonormal wavelets that span $W_j$ for each $j \in \mathbb{N}_0$, but the nomenclature is lengthly. Since we do not use the wavelets specifically in this paper, the details are omitted. Each function $\psi_{j,m}$ is proportional to one of the three scaled and translated multiwavelet functions and satisfies the orthonormality conditions $$\begin{aligned} \begin{array}{lll} (\psi_{j,k},\psi_{m,\ell}) = \delta_{j,m} \delta_{k,\ell} & & \text{for all } j,k,m,\ell,\\ (\psi_{j,k},\phi_{m,\ell}) = 0 & \quad \quad & \text{for $j\geq m$ and all $k,\ell$.} \end{array}\end{aligned}$$ In the next step, we denote the orthogonal projection onto the span of the piecewise constants defined on a grid of resolution level $j$ by $\Pi_j$ so that $$\Pi_j:P \rightarrow V_j.$$ Finally, we define the approximation space $\mathcal{A}^\alpha_2$ in terms of the projectors $\Pi_j$ as $$\mathcal{A}^\alpha_2:= \left \{ f\in P \biggl \| \\|f\\|_{\mathcal{A}^\alpha_2}:= \left ( \sum_{j=0}^\infty 2^{2\alpha j} \\| (\Pi_j-\Pi_{j-1})f \\|^2_P \right )^{1/2} \right \}.$$ Note that this is a special case of the more general analysis in [@dahmen1996]. We define our approximation method in terms of one point quadratures defined over the triangles $\Delta_{i_1\ldots i_j}$ that constitute the grid of level $j$ that defines $V_j$. For notational convenience, we collect all triangles at a fixed level $j$ in the singly indexed set $$\left \{ \Delta_{j,k} \right \}_{k\in \Lambda_j}:= \left \{ \Delta_{i_1\ldots i_j} \right \}_{i_1,\ldots,i_j \in {1,2,3,4}}$$ where $\Lambda_j:=\left \{ k\in \mathbb{N} \quad \| 1\leq k \leq 4^j \right \}$, and the quadrature points are chosen such that $\xi_{j,k}\in \Delta_{j,k}$ for $k = 1,\cdots, \Lambda_j$. We now can state our principle approximation result for the class of history dependent operators in this paper. Suppose that the function $\gamma$ that defines the history dependent kernel in Equation \[eq:defgamma\] is a bounded function in $C^{\alpha}(\mathbb{R})$, and define the approximation $h_j$ associated with the grid level $j$ of the history dependent operator $h$ to be $$(h_jf)(t)\circ \mu:= \iint_\Delta \left ( \sum_{\ell\in \Gamma_j} 1_{\Delta_{j,\ell}}(s) \kappa(\xi_{j,\ell}, t,f) \right ) \mu(s) ds .$$ Then there is a constant $C>0$ such that $$\left \| (h_jf)(t) \circ \mu - (hf)(t) \circ \mu \right \| \leq C 2^{-(\alpha+1) j} \label{eq:error_1}$$ for all $f\in C[0,T]$, $t\in [0,T]$, and $\mu\in P$. If in addition $\mu \in \mathcal{A}^{\alpha+1}_2$, there is a constant $\tilde{C}>0$ such that $$\left \| (h_jf)(t) \circ \Pi_j \mu - (hf)(t) \circ \mu \right \| \leq \tilde{C} 2^{-(\alpha+1) j} \label{eq:error_2}$$ for all $f\in C[0,T]$ and $t\in [0,T]$. We first prove the inequality in Equation \[eq:error\_1\]. By definition of the operator $h$, we can write $$\begin{aligned} \left \| (h_j f)(t) \circ \mu - (hf)(t) \circ \mu \right \| & \leq \iint_\Delta \left \| \left( \sum_{k\in \Lambda_j} 1_{\Delta_{j,k}}(s) \kappa(\xi_{j,k},t,f) -\kappa(s,t,f)\right)\mu(s) \right \| ds \\ & \leq \iint_\Delta \left \| \sum_{k\in \Lambda_j} 1_{\Delta_{j,k}}(s) \left ( \kappa(\xi_{j,k},t,f) -\kappa(s,t,f) \right ) \right \| \left \|\mu(s) \right \| ds \end{aligned}$$ Since the ridge function $\gamma$ is a bounded function in $C^{\alpha}(\mathbb{R})$, the output mapping $s \mapsto \kappa(s,t,f)$ is also a bounded function in $C^{\alpha}(\Delta)$ where the Lipschitz constant is independent of $t\in [0,T]$ and $f\in C[0,T]$. Using Proposition 2.5 of [@visintin1994], we have $$\begin{aligned} \left \| (h_j f)(t) \circ \mu - (hf)(t) \circ \mu \right \| & \leq \iint_\Delta \left \| \sum_{k\in \Lambda_j} 1_{\Delta_{j,k}}(s) L \\| \xi_{j,k} - s \\|^\alpha \right \| \left \|\mu(s) \right \| ds \\ & \leq L \sum_{k\in \Lambda_j} \left ( m(\Delta_{j,k}) \left ( \frac{\sqrt{2} (\overline{s}-\underline{s})}{2^j}\right )^\alpha \right ) \iint_{\Delta_{j,k}} \|\mu(s)\|ds \\ & \leq L \sum_{k\in \Lambda_j} \left ( m(\Delta_{j,k}) \left ( \frac{\sqrt{2} (\overline{s}-\underline{s})}{2^j}\right )^\alpha \right ) m^{1/2}(\Delta_{j,k}) \\| \mu \\|_P \\ &\leq L 2^{2j} \left ( \frac{1}{2} \frac{(\overline{s}-\underline{s})^2}{2^{2j}} \left ( \frac{\sqrt{2}(\overline{s}-\underline{s})}{ 2^j} \right )^\alpha \right ) \left(\frac{1}{2} \frac{(\overline{s}-\underline{s})^2}{2^{2j}}\right)^{1/2} \\| \mu \\|_P = C 2^{-(\alpha+1) j} \\| \mu \\|_P\end{aligned}$$ Since we have $$\left \| (h_j f)(t) \circ \Pi_j \mu - (hf)(t) \circ \mu \right \| \leq \left \| (h_j f)(t) \circ \Pi_j \mu - (h_jf)(t) \circ \mu \right \| + \left \| (h_j f)(t) \circ \mu - (hf)(t) \circ \mu \right \| ,$$ the second inequality in Equation \[eq:error\_2\] follows from the first Equation \[eq:error\_1\] provided we can show that $$\left \| ( h_jf)(t) \circ \left (\mu - \Pi_j \mu \right ) \right \| \leq C 2^{-(\alpha+1) j}$$ for some constant $C$. But it is a standard feature of the approximation spaces that if $\mu \in \mathcal{A}^{\alpha+1}_2$, then $\\|\mu-\Pi_j \mu \\|_P \leq 2^{-(\alpha+1) j} \\|\mu\\|_{\mathcal{A}^{\alpha+1}_{2}}$. To see why this is so, suppose that $\mu\in \mathcal{A}^{\alpha+1}_2$. We have $$\begin{aligned} \\|\mu-\Pi_j \mu\\|^2_P & = \sum_{k=j+1}^\infty \\| (\Pi_k -\Pi_{k-1})\mu \\|^2_P \\ & \leq \sum_{k=j+1}^\infty 2^{-2(\alpha+1) k} 2^{2(\alpha+1) k} \\| (\Pi_k -\Pi_{k-1})\mu \\|^2_P \\ & \leq 2^{-2(\alpha+1) j} \sum_{k=j+1}^\infty 2^{2(\alpha+1) k} \\| (\Pi_k -\Pi_{k-1})\mu \\|^2_P \leq 2^{-2(\alpha+1) j} \\| \mu \\|^2_{\mathcal{A}^{\alpha+1}_{2}}.\end{aligned}$$ When we apply this to our problem, the upper bound follows immediately $$\begin{aligned} \left \| ( h_jf)(t) \circ \left (\mu - \Pi_j \mu \right ) \right \| &\leq \sup_{t\in[0,T]}\left \\| (h_jf)(t)\right \\|_{P^} \\| \mu - \Pi_j \mu\\|_P \\ & \leq C2^{-(\alpha+1) j},\end{aligned}$$ since the boundedness of the ridge function $\gamma$ implies the uniform boundedness of the history dependent operators $(h_j f)(t)$ over $[0,T]$. Theorem 1 can now be used to establish error bounds for input-output maps that have the form in Equation \[eq:hdef2\]. Suppose that the hypotheses of Theorem 1 hold. Then we have $$\\|(\mathcal{H}X)(t)-(\mathcal{H}_j X)(t)\Pi_j \\| \lesssim 2^{-(\alpha+1)j}.$$ Where $\mathcal{H}$ is defined in Equations \[eq:2.1\],\[eq:hdef2\], and $\mathcal{H}_j$ is defined in Equations \[eq:Hj1\], \[eq:Hj2\] and \[eq:hj3\] below. Recall that for $i = 1 \dots q$ we had $$y_i(t) = \sum_{\substack{\ell = 1 \dots l}} b_{i\ell}(X(t))(h_\ell(a(X))(t)\circ \mu_\ell.$$ In matrix form this equation can be expressed as $$\begin{aligned} \begin{bmatrix} y_1(t)\\ \vdots \\ y_q (t) \end{bmatrix} = \underbrace{ \begin{bmatrix} b_{11}(X(t)) & \hdots &b_{1\ell}(X(t))\\ \vdots & \ddots & \vdots \\ b_{q1}(X(t)) & \hdots & b_{q\ell}(X(t)) \end{bmatrix} }_{\text{$\mathbb{R}^{q \times l}$}} \underbrace{ \begin{bmatrix} h_1(a(X))(t)\circ \mu_1\\ \vdots \\ h_\ell(a(X))(t)\circ \mu_\ell \end{bmatrix} }_{\text{$(HX)(t)\circ\mu$}}.\end{aligned}$$ It follows that, $$\begin{aligned} y(t) = \underbrace{(\mathcal{H}X)(t)\circ\mu}_{\text{$\in$ $\mathbb{R}^q$}} = \underbrace{ b(X(t))}_{\in \mathbb{R}^{q\times\ell}} \underbrace{(HX)(t) \circ \mu}_{\mathbb{R}^{\ell}}.\end{aligned}$$ By assumption $X \in C([0,T],\mathbb{R}^m)$. The construction of $H$ and $\mathcal{H}$ guarantees that $ (HX)(t) \in \mathcal{L}(P,\mathbb{R}^l), $ $ H : C\left([0,T],\mathbb{R}^m\right) \to C\left([0,T],\mathcal{L}(P,\mathbb{R}^l)\right), $ and $ \mathcal{H} : C([0,T],\mathbb{R}^m) \to C([0,T],\mathbb{R}^q). $ In this proof we denote by $(\mathbb{R}^l,\\|.\\|_u)$ the norm vector space that endows $\mathbb{R}^l$ with the $l^m$ norm $\\|v\\|_u:= \left( \sum^l_{i=1}\|v_i\|^u\right)^{\frac{1}{u}}$ for $1\leq u \leq \infty$. The normed vector space $(\mathbb{R}^{q\times l},\\|.\\|_{s,u})$ denotes the induced operator norm on matrices that map $(\mathbb{R}^l,\\|.\\|_u)$ into $(\mathbb{R}^q,\\|.\\|_s)$. Now we define an approximation on the mesh level $j$ of $\mathcal{H}$ to be $$(\mathcal{H}_j X)(t) = b(X(t)((H_j X) (t)\Pi_j \label{eq:Hj1},$$ $$(H_j X)(t) = \begin{bmatrix} h_{1,j}(a(X))(t) & 0 & \cdots & 0 \\ 0 & h_{2,j}(a(X))(t) & 0 & \vdots \\ \vdots & \cdots & \ddots & 0 \\ 0 & \cdots & 0 & h_{l,j}(a(X))(t) \end{bmatrix} \label{eq:Hj2}$$ and $$h_{i,j}(t) \circ \nu = \iint_{\Delta} \sum_{k \in \Lambda_j} 1_{\Delta_{k,j}}(s) \kappa(\xi_{j,k},t,f) \nu (s) \mathrm{d}s \label{eq:hj3}$$ for $i = 1\dots \ell$ and $\nu_i\in P_i$. To simplify the derivation or an error bound for approximation of $\mathcal{H}X(t)\circ \mu$, let $(HX)(t) \circ \mu $ be denoted by $g(t)$. We have assumed that $X \to b(X)$ and $t \to X(t)$ are continuous mappings. There fore $t \mapsto b(X(t))$ is continuous and on a compact set $[0,T]$, and $b(X(\cdot))$ $\in$ $C([0,T],\mathbb{R}^{q\times l})$. We therefore by definition have $$\begin{aligned} \\|b(X(t))\\|_{(\mathbb{R}^{q\times l},\\|\cdot\\|_{s,u})} & \leq \sup_{\tau \in [0,T]}\\|b(X(\tau))\\|_{(\mathbb{R}^{q\times l},\\|\cdot\\|_{s,u})}, \\ & = \\|b(X(\cdot))\\|_{C([0,T],(\mathbb{R}^{q\times l},\\|\cdot\\|_{s,u}))}, \\ \\|b(X(t))g(t)\\|_{(\mathbb{R}^q,\\|\cdot\\|_q)} &\leq \\|b(X(\cdot))\\|_{C([0,T],(\mathbb{R}^{q\times l},\\|\cdot\\|_{s,u}))}\\|g(t)\\|_{(\mathbb{R}^l,\\|\cdot\\|_u)}. \end{aligned}$$ with the norms explicitly denoted in the subscript. For $t\in[0,T]$, and applying these definitions, [$$\begin{aligned} \\|b(X(t))\left((HX)(t)- (H_j X)(t)\Pi_j\right)\mu\\|_{(\mathbb{R}^q,\\|\cdot\\|_s)} &\leq \\|b(X(t))\\|_{(\mathbb{R}^{q\times l},\\|\cdot\\|_{s,u})}\\|((HX)(t)-(H_j X)(t)\Pi_j) \circ \mu \\|_{(\mathbb{R}^l,\\|\cdot\\|_u)} \\ &\leq \\|b(X(\cdot))\\|_{(C([0,T],(\mathbb{R}^{q\times l},\\|\cdot\\|_{s,u}))}\\|((HX)(t)-(H_j X)(t)\Pi_j) \circ \mu \\|_{(\mathbb{R}^l,\\|\cdot\\|_u)}\end{aligned}$$ ]{} with $$\begin{aligned} \\|((&HX)(t)-(H_j X)(t)\Pi_j) \circ \mu \\|_{(\mathbb{R}^l,\\|\cdot\\|_u)} = \\ & \left \\|\begin{bsmallmatrix} h_{1,j}(a(X))(t) - h_{1,j}(a(X))(t)\Pi_j&&&&0 \\ & h_{2,j}(a(X))(t) - h_{2,j}(a(X))(t)\Pi_j &&&\\ &&\ddots &&\\ 0&&&& h_{l,j}(a(X))(t)-h_{l,j}(a(X))(t)\Pi_j \end{bsmallmatrix} \begin{bsmallmatrix} \mu_1\\ \mu_2\\ \vdots\\ \mu_l \end{bsmallmatrix}\right \\|_{(\mathbb{R}^l,\\|\cdot\\|_u)}.\end{aligned}$$ Therefore we can now write $$\begin{aligned} \left\\|((HX)(t)-(H_j X)(t)\Pi_j) \circ \mu \right\\|_{\left(\mathbb{R}^l ,\\|\cdot\\|_u\right)} \leq \\|((HX)(t)-(H_j X)(t)\Pi_j)\\|_{(\mathcal{L}(P,(\mathbb{R}^l,\\|\cdot\\|_u)))}\\|\mu\\|_{P}.\end{aligned}$$ Hence, recalling Theorem 1 we can now derive the convergence rate $$\begin{aligned} \\|((HX)(t)-(H_j X)(t)\Pi_j)\\|_{\left(\mathcal{L}\left(P,\left(\mathbb{R}^l,\\|\cdot\\|_u\right)\right)\right)} &= \sup_{\\|\mu\\|<1} \\|\left(\left(HX\right)(t)-\left(H_j X\right)(t)\Pi_j\right) \circ \mu \\|_{(\mathbb{R}^l,\\|\cdot\\|_u)} \\ &\leq \sup_{\\|\mu\\|<1} \left\| ((h_{i,j} (a(X)))(t) - (h_{i,j}(a(X)))(t)\Pi_j) \circ \mu \right\| \\ &\leq \hat{C}2^{-(\alpha+1)j}.\end{aligned}$$ Therefore we obtain the final bound $$\begin{aligned} \\|(\mathcal{H}X)(t)-(\mathcal{H}_j X)(t)\Pi_j)\\|_{(\mathbb{R}^l ,\\|\cdot\\|_u)}\lesssim 2^{-(\alpha+1)j},\end{aligned}$$ for all $t\in [0,T]$. Well-Posedness: Existence and Uniqueness {#sec:exist} ======================================== The history dependent governing equations studied in this paper are a special case of the more general class of abstract Volterra equations or functional differential equations. A general treatise on abstract Volterra equations can be found in [@corduneaunu2008], while various generalizations of theory for the existence and uniqueness of functional differential equations have been given in [@driver1962], [@rudakov1978], [@irs2002]. We have noted in Section \[sec:intro\] that the general form of the governing equations we consider in this paper have the form $$\dot{X}(t)=AX(t)+B((\mathcal{H}X)(t) \circ {\mu}+u(t)) \label{eq:first_order}$$ where the state vector $X(t) \in \mathbb {R}^{m}$, the control inputs $u(t) \in \mathbb{R}^{q}$, $A \in \mathbb{R}^{m{\times}m}=\mathbb{R}^{2n{\times}2n}$ is a Hurwitz matrix, and $B \in \mathbb{R}^{m{\times}q}$ is the control input matrix. We make the following assumptions about the history dependent operators $\mathcal{H}$: 1. $ \mathcal{H} : C([0,\infty),\mathbb{R}^m ) \mapsto C([0,\infty),P^)$ 2. $\mathcal{H}$ is causal in the sense that for all $ x,y \in C([0,\infty);\mathbb{R}^m)$, $$x(\cdot) \equiv {y} (\cdot) \; \text{on} \; [0,{\tau}] \implies (\mathcal{H}x)({t})=(\mathcal{H}y)({t})\: \quad \forall \: {t} \in [0,\tau].$$ 3. Define the closed set consisting of all continuous functions $f$ that remain within radius $r$ of the initial condition $X_0$ over the closed interval $[t,t+h]$, $$\overline{\mathcal{B}}_{[t,t+h],r} (X_0):=\begin{Bmatrix} f\in C([0,h),\mathbb{R}^m) \biggl \| f(0)=X_0 \text{ and } \\|f(s) - X_0\\|_{\mathbb{R}^m} \leq r \text{ for } s\in[t,t+h] \end{Bmatrix},$$ for a fixed $X_0\in\mathbb{R}^m$. For each $t\ge 0$, we assume that there exist $h,r,L>0$ such that $$\\| (\mathcal{H}X)(s)-(\mathcal{H}Y)(s) \\|_{P^} \leq L \\| X-Y \\|_{[t,t+h]} \quad \quad s\in[t,t+h] \label{eq:local_lip}$$ for all $X,Y \in \overline{\mathcal{B}}_{[t,t+h],r}(X_0)$. Our first result guarantees the existence and uniqueness of a local solution to Equation \[eq:1st\_order\], and also describes an important case when such local solutions can be extended to $[0,\infty)$. This theorem can be proven via the existence and uniqueness Theorem 2.3 in [@irs2002] for functional delay-differential equations. However, since we are not interested in delay differential equations in this paper, but rather on a highly structured class of integral hysteresis operators, the proof can be much simplified. \[th:exist\_general\] Suppose that the history dependent operator $\mathcal{H}$ satisfies the hypotheses (H1),(H2),(H3). Then there is a $\delta>0$ such that Equation \[eq:first\_order\] has a solution $X\in C([0,\delta), \mathbb{R}^m)$. Suppose the interval $[0,\delta)$ is extended to the maximal interval $[0,\omega)\subset [0,\delta)$ over which such a solution exists. If the solution is bounded, then $[0,\omega)=[0,\infty)$. \[th:exist\_specific\] Suppose that the history dependent operator $\mathcal{H}$ in Equation \[eq:first\_order\] is defined as in Equation \[eq:hdef1\] and \[eq:hdef2\] in terms of a globally Lipschitz, bounded continuous ridge function $\gamma: \mathbb{R}\rightarrow \mathbb{R}$ in Equation \[eq:defgamma\]. Then Equation \[eq:first\_order\] has a unique solution $X \in C([0,\infty),\mathbb{R}^m)$ for each $\mu\in P$. For completeness, we outline a simplified version the proof of Theorem \[th:exist\_general\] for our class of history and parameter dependent equations. As a point of comparison, the reader is urged to compare the proof below to the conventional proof for systems of nonlinear ordinary differential equations, such as in [@khalil]. If we integrate the equations of motion in time, we can define an operator $T:C([0,h),\mathbb{R}^m) \rightarrow C([0,h), \mathbb{R}^m)$ from $$\begin{aligned} X(t) &= X_{0} + \int_0^t AX(\tau) + B((\mathcal{H}X)(\tau) \circ \mu + u(\tau) ) \mathrm{d} \tau, \\ X(t) &= (TX)(t), \end{aligned}$$ for all $t\in[0,h]$. As introduced in hypothesis (H3), we select $h,r>0$ and define $$\overline{\mathcal{B}}_{[0,h],r}(X_0) := \Big\{ X \in C([0,h),\mathbb{R}^m) \biggl \| X(0) = X_0 , \\| X_0 - X \\|_{[0,\delta]} \leq r \Big\} .$$ such that the local Lipschitz condition in Equation \[eq:local\_lip\] holds. Now we consider restricting the equation to a subinterval $[0,\delta]\subseteq [0,h]$, and investigate conditions on $T$ that enable the application of the contraction mapping theorem. We first study what conditions on $\delta>0$ are sufficient to guarantee that $T: \overline{\mathcal{B}}_{[0,\delta],r}(X_0) \rightarrow \overline{\mathcal{B}}_{[0,\delta],r}(X_0)$. We have [$$\begin{aligned} \\| TX (t) - X_0 \\|_{\mathbb{R}^m} & \leq \int_0^t \\| AX(s) + B((\mathcal{H}X(s) \circ \mu + u(s)) \\|_{\mathbb{R}^m} \mathrm{d}s\\ & \leq \int_0^t \biggl ( \\| A \\| \\|X(s) - X_0 \\|_{\mathbb{R}^m} + \underbrace{\\|AX_0 \\|_{\mathbb{R}^m}}_{\leq\\|A\\| \\|X_0\\| = \text{$M_A$}} \\ & \qquad \qquad + \\| B \\| \underbrace{(\\| (\mathcal{H}X)(s) -(\mathcal{H}X_0)(s) \\|_{P^}}_{\text{$ \leq L \\|X-X_0\\|_{[0,\delta]} $}} \underbrace{\\| \mu \\|_P}_{\text{$M_\mu$}} + \underbrace{\\| (\mathcal{H}X_0)(s) \\|_{P^} }_{\text{$\leq M_H= \\|\mathcal{H}X_0\\|_{C([0,\delta],P^)}$ }}+ \underbrace{\\| u \\|_{C([0,h),\mathbb{R}^p)}}_{\text{$\leq M_u= \\|u\\|_{C([0,\delta],\mathbb{R}^q)}$}}) \biggr) \mathrm{d}s\\ &\leq ((\\|A\\| + \\| B \\|M_\mu L)r + M_A + \\|B\\|M_T)t\\ & \leq ((\\|A\\| + \\| B \\|M_\mu L)r + M_A + \\|B\\|M_T)\delta\end{aligned}$$ ]{} where $M_T=M_H+M_u$. Now we restrict $\delta$ so that $$\Big((\\|A\\| + \\| B \\|M_\mu L)r + M_A + \\|B\\|M_T \Big)\delta \leq r,$$ which implies $$\delta < \frac{r}{(\\|A\\| + \\| B \\|M_\mu L)r + M_A + \\|B\\|M_T)} .$$ We thereby conclude that $$\\| TX (t) - X_0 \\|_{C([0,h),\mathbb{R}^p)} \leq r \: \quad \text{for} \quad \: t \in [0,\delta],$$ and it follows that $T : \overline{\mathcal{B}}_{[0,\delta],r} \rightarrow \overline{\mathcal{B}}_{[0,\delta],r} $. Next we study conditions on $\delta$ that guarantee that $T : \overline{\mathcal{B}}_{[0,\delta],r} \rightarrow \overline{\mathcal{B}}_{[0,\delta],r} $ is a contraction. We compute directly a bound on the difference of the output as $$\begin{aligned} \\| (TX) (t) - (TY)(t) \\|_{\mathbb{R}^m} &\leq \int_0^t \\| AX(s)-AY(s) + B((\mathcal{H}X)(s) - (\mathcal{H}Y(s))\circ \mu ) \\|_{\mathbb{R}^m} \mathrm{d}s \\ &\leq (\\| A \\|+\\| B \\| M_\mu L_\mu )\\|X-Y\\|_{\mathbb{R}^m}\delta . \end{aligned}$$ If we choose $$\delta < min \biggl \{ h, \frac{r}{(\\|A\\| + \\| B \\|M_\mu L)r + M_A + \\|B\\|M_T)},\frac{1}{ \\| A \\|+\\| B \\| M_\mu L } \biggr \},$$ it is apparent that $T$ is a contraction that maps the closed set $\overline{\mathcal{B}}_{[0,\delta],r}$ into itself. There is a unique solution in $\overline{\mathcal{B}}_{[0,\delta],r}$ on $[0,\delta]$. When seek to control the error in the approximation of our history dependent equation of motion, two distinct types of errors can arise. First, since the history dependent contribution to the equations of motion often cannot be calculated in closed form, some approximations of the history dependent operators must be used. We refer to this as operator approximation error. Even if the history dependent terms are expressed exactly and without error, the resulting equations are a collection of functional differential equations. The usual collection of time stepping integration rules for ordinary differential equations are not directly applicable to such functional differential equations. In this paper, we employ the strategy first proposed in [@tavernini] for numerical time integration of functional differential equations. These techniques assume that the functional differential equations have the form $$\begin{aligned} \eta^{\prime}(t)=F(\eta,t), \quad t\in[t_b,t_c]\\ \eta(t)=\eta_a(t), \quad t\in [t_a,t_b]\end{aligned}$$ where $\eta_a \in C[t_a,t_b]$ is the initial condition and the functional $f:C[a,c]\times [a,c]\rightarrow \mathbb{R}^n$. A linear multistep method for functional differential equations constructs the recursion for the solution $\eta_h$ on a grid having time step $h$ via the formula $$\begin{aligned} a_k& \eta_h(t_{i+k-1}+r h)+a_{k-1}(r)y_k(t_{i+k-1})+\cdots \\ & +a_0(r)\eta_h(t_i)-h[b_k(r) F_h(y_h,t_{i+k})+\cdots\\ & +b_0(r) F_h(y_h,t_i)]=0 \label{eq:recursion} \end{aligned}$$ where $r \in [0,1]$, $i=0,1, \cdots, N-k$,and $h=(b-a)/N_0$. Three observations should be noted about these approximation scheme: - solution of this equation yields an extrapolation of the solution $\eta_h$ on $[t_{i+k-1},t_{i+k-1}+r h]=[t_{i+k-1},t_{i+k}]$ since $r\in[0,1]$. - The discrete equation depends on the history of the discrete approximation through the history dependent functionals $F_h(y_h,t_{i+k}$ and $F_h(y_h,t_i)$. - The term $F_h(y_h,t_{i+k})$ gives rise to implicit methods in that $y_h$ must be defined on $[t_{i+k-1},t_{i+k}]$ which is not known at time $t_i$. The last observation, in particular, means that the solution of equation \[eq:recursion\] involves in general an implicit nonlinear solver over the future history of the solution during the next time step. In this paper we generalize the strategy in [@tavernini] to employ a predictor-corrector structure. In the predictor phase, we choose the constant ${a_k}$ and ${b_k}$ so that $b_k(r)\equiv 0$. Hence the nonlinear dependence on the future solution does not appear. Subsequently, we choose a corresponding corrector in which $b_k(r)\neq 0$. In the correction step we calculate $F_h(y_h^p,t_{i+k})$ in terms of the predictor solution $y_h^p$. It is defined on the entire future interval. Numerical examples demonstrate that this approach is computationally efficient and accurate. The constants ${a_k}$ and ${b_k}$ are selected as described in section 4 of [@tavernini] based on conventional linear multistep predictor-corrector integrator schemes. Numerical Integration for History Dependent Differential equations ------------------------------------------------------------------ As discussed in the last section we use specialized integration schemes to approximate discrete solutions to Volterra Functional Differential Equations [@tavernini]. In this section we use Adam Bashforth (explicit) Predictor and Adam Moulton (implicit) Corrector to numerically solve the functional differential equation. The predictor step computes the approximate value and the corrector step refines the approximation to improve accuracy. The order of accuracy quantifies the rate of convergence for the approximation, The numerical solution is said to be $p$th order accurate if the error $E$ is proportional to $p$th power of step size $h$ i.e $E = C h^p$. The local truncation error given by $\tau_n = y_n - y_{n-1}$ is of the order $\mathcal{O}(h)^{p+1}$. The rate of convergence for the numerical solutions for two examples are presented below.Example 1 involves history dependent nonlinearity, while example 2 is a standard integro-differential equation from [@laksh] used to benchmark to validate the numerical solution against the analytical solution. #### Example 1 As a first example, we choose to modify the usual harmonic oscillator equation with an additional history dependent term, $$\begin{aligned} \begin{bmatrix} \dot{y_1}(t)\\ \dot{y_2}(t) \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -1+H & 0 \end{bmatrix} \begin{bmatrix} y_1(t)\\ y_2(t) \end{bmatrix},\end{aligned}$$ where $H$ is the history dependent term and its numerical value depends on the entire history of $y_1$. For simplicity we choose $H=\kappa(s,t,f)$ in this example. The kernel constructed using methods described in previous sections, takes the history of $y_1$ until current time step as input and gives the value of $H$ as output. We numerically integrate the above equation using predictor corrector methods. Since we do not have means to derive a closed form solution, we usually rely on the smallest step size numerical solution best approximations of the true value. We compare this with solutions having larger step size, ( h is least an order of magnitude greater than the finest step size). The order of accuracy of the numerical solution depends on the smoothness of $H$ and hence the kernel that outputs $H$. The single step error $\tau$ shown in the Figure \[fig:LocE1\] is plotted against the step size $h$ on a log-log scale ( i.e. $\log h$ vs $log \tau$). The slope in the figure corresponds to the order of accuracy and is approximately equal to $p+1$ where p is the order of the integrator. The rate of convergence for the second order predictor corrector from plot is seen to be 2.974 and for the fourth order predictor corrector the rate is 4.9. We compare these results with the rate of convergence in example 2. #### Example 2 To illustrate the utility of predictor corrector algorithm, the following integro-differential equation was numerically solved and compared to its analytical solution [@laksh]. $$\begin{aligned} u'(x) + 2u(x) + 5 \int_0^x u(t)\mathrm{d}t = \left \{ \begin{array}{lll} 1 & \text{x} \geq 0\\ 0 & \text{x} < 0 \end{array} \right.\end{aligned}$$ where $u(0)=0$. It can be verified that the closed form solution for the above problem is $$u(x) = \frac{1}{2} e^{-x}\sin{2x}.$$ The rate of convergence for the second order predictor-corrector is 3.045 and for the fourth order predictor-corrector is 5.119, which validates the expected rate of convergence $p+1$ for a given $p$th order of numerical integrator. ![Single Step Error vs Step Size for Example 1[]{data-label="fig:LocE1"}](locerrsingker.pdf){width="35.00000%"} ![Single Step Error vs Step Size for Example 2[]{data-label="fig:LocE2"}](locerrideal.pdf){width="35.00000%"} Online Identification {#sec:ident} ===================== A substantial literature has emerged that treats online estimation problems for linear or nonlinear plants governed by systems of ordinary differential equations. Approaches for these finite dimensional systems that are based on variants of Lyapunov’s direct method can be found in any of a number of good texts including, for instance, [@na2005], [@sb2012], or [@is2012]. The general strategies that have proven fruitful for such finite dimensional systems have often been extended to classes of systems whose dynamics evolve in an infinite dimensional space: distributed parameter systems. A discussion of the general considerations for identification of distributed parameter systems can be found in [@bk1989], for example, while studies that are specifically relevant to this paper include [@d1993], [@dr1994], [@dr1994pe], and [@bsdr1997]. In this section we adapt the framework introduced in [@bsdr1997] to our class of history dependent, functional differential equations. The approach in [@bsdr1997] assumes that the state equations for the distributed parameter system have first order form, and they are cast in terms of a nonlinear, parametrically dependent bilinear form that is coercive. The resulting equations that govern the error in state and in distributed parameter estimates is a nonlinear function of the state trajectory of the plant. In contrast, a similar strategy in this paper yields error equations that depend nonlinearly on the history of the state trajectory. The general online estimation problem discussed in this section assumes that we observe the value of the state $X(t)\in \mathbb{R}^m$ at each time $t\ge 0$ that depends on some unknown distributed parameter $\mu \in P$, and subsequently use the observed state to construct estimates $\hat{X}$ of the states and $\hat{\mu}$ of the distributed parameters. We construct online estimates that evolve on the state space $\mathbb{R}^m \times P$ according to the time varying, distributed parameter system equations $$\begin{aligned} \dot{\hat{X}}(t) & = A \hat{X}(t) + B\left ( \left ( \mathcal{H} X \right )(t) \circ \hat{\mu}(t) + u(t) \right ), \notag \\ \dot{\hat{\mu}}(t) & = -\left (B (\mathcal{H}X)(t) \right)^\hat{X}(t), \label{eq:infdim}\end{aligned}$$ for $t\ge 0$ where the initial conditions are $\hat{X}_0:=X_0$, $\hat{\mu}(0):=\mu_0$. In these equations, we denote the adjoint operator $L^$ for any bounded linear operator $L$. These equations can be understood as incorporating a natural choice of a parameter update law. The learning law above can be interpreted as generalization of the conventional gradient update law that features prominently in approaches for finite dimensional systems [@is2012] and that has been extended to distributed parameter systems in [@bsdr1997]. It is immediate that the error in estimation of the states $\tilde{X}:=X-\hat{X}$ and in the distributed parameters $\tilde{\mu}:=\mu-\hat{\mu}$ satisfy the homogeneous system of equations $$\begin{aligned} \begin{Bmatrix} \dot{\tilde{X}}(t) \\ \dot{\tilde{\mu}}(t) \end{Bmatrix} = \begin{bmatrix} A & B(\mathcal{H}X)(t) \\ -\left ( B(\mathcal{H}X)(t)\right)^& 0 \end{bmatrix} \begin{Bmatrix} {\tilde{X}}(t) \\ {\tilde{\mu}}(t)\end{Bmatrix}.\end{aligned}$$ Approximation of the Estimation Equations {#subsec:approx_est} ----------------------------------------- The governing system in Equations \[eq:infdim\] constitute a distributed parameter system since the functions $\hat{\mu}(t)$ evolve in the infinite dimensional space $P$. In practice these equations must be approximated by some finite dimensional system. We define $\tilde{X}_j=\hat{X}-\hat{X}_j$ and $\tilde{\mu}_j=\hat{\mu}-\hat{\mu}_j $ where $\tilde{X}_j$ and $\tilde{\mu}_j$ express approximation errors due to projection of solutions in $\mathbb{R}^m \times P $ to a finite dimensional approximation space. We construct a finite dimensional approximation of the the online estimation equations using the results of Section \[subsec:approx\_hist\] and obtain $$\begin{aligned} \dot{\hat{X}}_j(t) & = A\hat{X}_j(t) + B \left ( (\mathcal{H}_j X)(t) \Pi_j \circ \hat{\mu}_j(t) + u(t) \right ), \\ \dot{\hat{\mu}}_j(t) & = - \left ( B(\mathcal{H}_{j} X)(t) \Pi_j \right)^ X(t). \label{eq:approx_on_est}\end{aligned}$$ Suppose that the history dependent operator $\mathcal{H}$ in Equation \[eq:first\_order\] is defined as in equation \[eq:hdef1\] and \[eq:hdef2\] in terms of a globally Lipschitz, bounded continuous ridge function $\gamma: \mathbb{R}\rightarrow \mathbb{R}$ in Equation \[eq:defgamma\]. Then for any $T>0$, we have $$\begin{aligned} \\| \hat{X} - \hat{X}_j\\|_{C([0,T],\mathbb{R}^m)} & \rightarrow 0, \\ \\|\hat{\mu} - \hat{\mu}_j\\|_{C([0,T],P)} & \rightarrow 0,\end{aligned}$$ as $j\rightarrow \infty$. Define the operators $G(t):P\rightarrow \mathbb{R}^m$ and $G_j(t):P \rightarrow \mathbb{R}^m$ for each $t\geq 0$ as $$\begin{aligned} G(t) & := B (\mathcal{H}X)(t), \\ G_j(t) & := B (\mathcal{H}_jX)(t) \Pi_j.\end{aligned}$$ The time derivative of the error in approximation can be expanded as follows: [ $$\begin{aligned} \frac{1}{2} \frac{d}{dt}\left ( ( {\tilde{X}}_j, {\tilde{X}}_j )_{\mathbb{R}^m} + ({\tilde{\mu}}_j,{\tilde{\mu}}_j )_P \right ) & = ( \dot{\tilde{X}}_j, {\tilde{X}}_j )_{\mathbb{R}^m} + (\dot{\tilde{\mu}}_j,{\tilde{\mu}}_j )_P \\ &= (A\tilde{X}_j + G\hat{\mu} - G_j \hat{\mu}_j , \tilde{X}_j)_{\mathbb{R}^m} + \left ( -(G-G_j)^* X, \tilde{\mu}_j\right )_P \\ &= (A\tilde{X}_j, \tilde{X}_j)_{\mathbb{R}^m} + \left ( (G-G_j)\hat{\mu}, \tilde{X}_j \right )_{\mathbb{R}^m} + \left ( G_j(\hat{\mu}-\hat{\mu}_j), \tilde{X}_j \right )_{\mathbb{R}^m} - \left ( (G-G_j)\tilde{\mu}_j, X \right )_{\mathbb{R}^m} \\ &\leq c (\tilde{X}_j, \tilde{X}_j)_{\mathbb{R}^m } + \\|(G-G_j)\hat{\mu} \\|_{\mathbb{R}^m} \\| \tilde{X}_j \\|_{\mathbb{R}^m} +\\ & \qquad \qquad \qquad \qquad \\| G_j\\|_{\mathcal{L}(P,\mathbb{R}^m)} \\| \tilde{\mu}_j \\|_{P} \\| \tilde{X}_j \\|_{\mathbb{R}^m}+ \\ & \qquad \qquad \qquad \qquad \\| G-G_j\\|_{\mathcal{L}(P,\mathbb{R}^m)} \\|\tilde{\mu}_j \\|_{P} \\| X\\|_{\mathbb{R}^m}.\end{aligned}$$ ]{} We will next use a common inequality that can be derived from two applications of the triangle inequality. We have $$\begin{aligned} (a+b,a+b)=(a,a) + 2(a,b) + (b,b) &\geq 0, \\ (a-b,a-b)=(a,a) - 2(a,b) + (b,b) & \geq 0. \end{aligned}$$ We conclude from this pair of inequalities that $$\|(a,b)\| \leq \frac{1}{2} \left ( \\|a\\|^2 + \\|b\\|^2 \right ).$$ The specific form that we apply this theorem is written as $$\|(a,b)\|= \|(\sqrt{\epsilon} a, \frac{1}{\sqrt{\epsilon}} b)\| \leq \epsilon \frac{\\|a\\|^2}{2} + \frac{1}{\epsilon} \frac{\\|b\\|^2}{2}. \label{eq:ip_ab}$$ We apply the inequality in Equation \[eq:ip\_ab\] to each term in which $\tilde{\mu}_j$ and $\tilde{X}_j$ appear in a product. $$\begin{aligned} \frac{1}{2} \frac{d}{dt} \left ( \\| \tilde{X}_j \\|^2_{\mathbb{R}^m} + \\| \tilde{\mu}_j \\|^2_{P)} \right) & \leq c \\| \tilde{X}_j \\|_{\mathbb{R}^m}^2 + \frac{1}{2a} \\| (G-G_j)\hat{\mu} \\|^2_{\mathbb{R}^m} + \frac{a}{2} \\|\tilde{X}_j \\|^2_{\mathbb{R}^m} \\ & \text{\hspace{.2in}} + \frac{1}{2b} \\| G_j \tilde{\mu}_j\\|^2_{\mathbb{R}^m} + \frac{b}{2} \\|\tilde{X}_j \\|^2_{\mathbb{R}^m} + \frac{1}{2c} \\| \tilde{\mu}_j \\|^2_{P} \\& \text{\hspace{.2in}} + \frac{c}{2} \\| G-G_j\\|^2_{\mathcal{L}(P,\mathbb{R}^m)} \\|X\\|^2_{\mathbb{R}^m} .\end{aligned}$$ Then $$\begin{aligned} \frac{d}{dt} \left ( \\| \tilde{X}_j \\|^2_{\mathbb{R}^m} + \\| \tilde{\mu}_j \\|^2_{P} \right ) & \leq c \\| G-G_j \\|^2_{\mathcal{L}(P,\mathbb{R}^m)} \\|X\\|^2_{\mathbb{R}^m} + (2c + a + b)\\|\tilde{X}_j\\|^2_{\mathbb{R}^m} \\ & \text{\hspace{.2in}}+ \left ( \frac{1}{c} + \frac{1}{b} \\|G_j^G_j \\| \right ) \\| \tilde{\mu}_j\\|^2_{P} + \frac{1}{a} \\|G - G_j \\|^2_{\mathcal{L}(P,\mathbb{R}^m)} \\|\hat{\mu}\\|^2_{P}. \end{aligned}$$ We integrate this inequality in time from $0$ to $t$ to obtain [$$\begin{aligned} \\| \tilde{X }_j(t) \\|^2_{\mathbb{R}^m} + \\| \tilde{\mu}_j(t) \\|^2_{P} & \leq \\| \tilde{X}_j(0) \\|^2_{\mathbb{R}^m} + \\| \tilde{\mu}_j(0) \\|^2_{P} \\ & \text{\hspace{.2 in}} + \int_0^t c \\|G(s) - G_j(s) \\|^2_{\mathcal{L}(P,\mathbb{R}^m)} \\| X(s)\\|^2_{\mathbb{R}^m} ds \\ & \text{\hspace{.2in}} + \int_0^t \left \{ (2c+a+b) \\|\tilde{X}(s)\\|^2_{\mathbb{R}^m} + \left ( \frac{1}{c} + \frac{1}{b} \\| G^_j(s) G_j(s) \\| \right ) \\|\tilde{\mu}_j\\|^2_{P} \right \}ds \\ & \text{\hspace{.2 in}}+ \int_0^t \frac{1}{a} \\|G(s) - G_j(s) \\|^2_{\mathcal{L}(P,\mathbb{R}^m)} \\|\hat{\mu}\\|^2_{P} ds.\end{aligned}$$ ]{} Choose $a,b>0$ large enough so that $(2c+a+b)>0$ and set $\gamma>1$. If we define $$\begin{aligned} \gamma &:=\max \left ( 2c+a+b, \frac{1}{c}+ \frac{\eta}{b} \sup_{s\in [0,T]} \\|G^(s)G(s)\\|,1 \right ), \\ \lambda_j(t)&:= \\|(I-\Pi_j)\hat{\mu}(0)\\|_{P} + \int_0^t \\|G(s) - G_j(s)\\|^2_{\mathcal{L}(P,\mathbb{R}^m)} \left(c \\|X\\|^2_{\mathbb{R}^m}+ \frac{1}{a} \\| \hat{\mu}\\|^2_{P} \right) ds,\end{aligned}$$ then the inequality can be written as $$\begin{aligned} \\|\tilde{X}_j(t)\\|^2_{\mathbb{R}^m} + \\|\tilde{\mu}_j(t)\\|^2_{P} \leq \lambda_j(t) + \gamma \int_0^t \left ( \\|\tilde{X}_j(s)\\|^2_{\mathbb{R}^m} + \\| \tilde{\mu}_j(s) \\|^2_{P} \right ) ds.\end{aligned}$$ Gronwall’s Inequality now completes the proof of the theorem (see Appendix C). We also further investigate $\lambda_j(t)$ to derive the convergence rate for the approximate states and parameters evolving associated with level $j$ resolution. According to the convergence results obtained in Theorem 1 we have $\\|G(s)-G_j(s)\\|_{\mathcal{L}(P,\mathbb{R}^m)} = \\|B(\mathcal{H}X)(t)-B(\mathcal{H}_j X)(t)\Pi_j\\| \leq C_2 2^{-(\alpha+1)j}$. Therefore, $\\|G(s)-G_j(s)\\|^2 \leq C_2^{2} 2^{-(\alpha+1)2j}$. It then follows that $$\begin{aligned} \lambda_j(t) &= \\|(I-\Pi_j)\hat{\mu}(0)\\|_{P}+ \int_0^t 2^{-(\alpha+1)2j} \left(c \\|X\\|^2_{\mathbb{R}^m}+ \frac{1}{a} \\| \hat{\mu}\\|^2_{P} \right)\mathrm{d}s \\ &\leq \\|(I-\Pi_j)\\| \\| \hat{\mu}(0)\\|_{P}+ 2^{-(\alpha+1)2j}\left(c \\|X\\|^2_{\mathbb{R}^m}+ \frac{1}{a} \\| \hat{\mu}\\|^2_{P} \right) t.\\\end{aligned}$$ If $t$ $\simeq$ $C_3 2^{(\alpha +1)j}$, then $ \lambda_j(t) < \mathcal{O}(2^{-(\alpha+1)j}) \quad \text{for} \quad t \in [0,C_3 2^{(\alpha +1)j}]. $ Adaptive Control Synthesis {#sec:ident} ========================== In order to estimate the function $\mu$ that weighs the contribution of history dependent kernels to the equations of motion, we first map it to an n-dimensional subspace of square integrable functions using a projection operator $\Pi^n: P \mapsto P^n$. Let $$\dot{X}=AX+B((\mathcal{H}X) \circ ( \mu- \hat{\mu})+v)$$ be the governing equation of a robotic system after applying a feedback linearization control signal as mentioned in Equation \[eqn:3\] with $u=v-(\mathcal{H}X) \circ \hat{\mu}$. We substitute $\mu = \Pi^n \mu+ (I-\Pi^n) \mu$ and write $$\dot{X}=AX+B((\mathcal{H}X) \circ ( \Pi^n \mu- \hat{\mu})+v)+B((\mathcal{H}X) \circ (I- \Pi^n) \mu).$$ Finally, by replacing $d= \{(\mathcal{H}X)(I-\Pi^n) \circ \mu \}$ we obtain $$\dot{X}=AX+B((\mathcal{H}X) \circ ( \Pi^n \tilde{\mu})+v+d),$$ where $$\dot{\tilde{\mu}} = -((\mathcal{H}X) \Pi ^n )^ B^T P X.$$ Suppose the state equations have the form of Equation \[eqn:3\] and the matrix $\mathcal{P}$ is a symmetric positive definite solution of the Lyapunov equation $A^T\mathcal{P}+\mathcal{P}A=-Q$ where $Q>0$. Then by employing the update law $\dot{\tilde{\mu}} = -((\mathcal{H}X) \Pi ^n )^* B^T \mathcal{P} X$, the control signal $$\begin{aligned} v(t)= \begin{cases} -k\frac{B^T \mathcal{P} X}{\\|B^T \mathcal{P} X\\|},& \text{ if } \\|B^T \mathcal{P} X\\|\geq \epsilon\\ -\frac{k}{\epsilon}B^T \mathcal{P} X, & \text{ if } \\|B^T \mathcal{P} X\\|<\epsilon \end{cases} \label{eqn:sliding_C}\end{aligned}$$ with $k>\\|d\\|$ drives the tracking error dynamics of the closed loop system is uniformly ultimately bounded and its norm is eventually $O(\epsilon)$. We choose the Lyapunov function $$V=\frac{1}{2}X^T \mathcal{P} X + \frac{1}{2}\left( \tilde{\mu},\tilde{\mu}\right)_{P}$$ where $\mathcal{P}$ is the solution of the Lyapunov equation $ A^T \mathcal{P} + \mathcal{P} A = -Q$. The derivative of the Lyapunov function $V$ along the closed loop system trajectory is $$\begin{aligned} \dot{V} &=\frac{1}{2} (\dot{X}^T \mathcal{P} X + X^T \mathcal{P} \dot{X})+\left( \dot{\tilde{\mu}},\tilde{\mu}\right)_p\\ &= \frac{1}{2}\big(AX+B((\mathcal{H}X) \circ ( \Pi^n \tilde{\mu})+v+d)\big)^T \mathcal{P} X +X^T \mathcal{P} (AX+B((\mathcal{H}X) \circ ( \Pi^n \tilde{\mu})+v+d)\big) + \left( \dot{\tilde{\mu}},{\mu}\right)_P \\ &= \frac{1}{2}X^T (A^T \mathcal{P} + \mathcal{P} A)X+ X^T PB ( v +d) +X^T \mathcal{P} B \big((\mathcal{H}X) \circ ( \Pi^n \tilde{\mu})\big) + \left( \dot{\tilde{\mu}},{\mu}\right)_P\\ &= -\frac{1}{2}X^T Q X + X^T \mathcal{P} B (v +d) + \left( \dot{\tilde{\mu}} + ((\mathcal{H}X) \Pi ^n )^* B^T \mathcal{P} X, \tilde{\mu}\right)_P\\ &= -\frac{1}{2}X^T Q X + X^T \mathcal{P} B ( v +d).\end{aligned}$$ Therefore we have $$\begin{aligned} \dot{V} & \leq -\frac{1}{2}X^T Q X + X^T \mathcal{P} B ( v +d), \\ & \leq -\frac{1}{2}X^T Q X + \begin{cases} X^T\mathcal{P}B\left(-k\frac{B^T \mathcal{P} X}{\\|B^T \mathcal{P} X\\|} +d \right) & \text{ if } \\|B^T \mathcal{P} X\\|\geq \epsilon \\ X^T\mathcal{P}B\left( -\frac{k}{\epsilon}B^T \mathcal{P} X +d \right) & \text{ if } \\|B^T \mathcal{P} X\\|\leq \epsilon \end{cases}, \\ & \leq -\frac{1}{2}X^T Q X + \begin{cases} -\left(k-\\|d\\|\right)\\|B^T\mathcal{P}X\\|& \text{ if } \\|B^T \mathcal{P} X\\|\geq \epsilon\\ \epsilon k & \text{ if } \\|B^T \mathcal{P} X\\|\leq \epsilon \end{cases}\\ & \leq -\frac{1}{2}X^T Q X+ \epsilon k.\end{aligned}$$ By Theorem 4.18 in [@khalil] we conclude that there is a $\bar{T}>0$ and $\tau>0$ such that $\\|X(t) \\| \leq \bar{C} \epsilon $ for all $t\geq \bar{T}$. $$\label{eqn:4}\frac{d}{dt} \begin{bmatrix} X\\ {\tilde{\mu}} \end{bmatrix} = \begin{bmatrix} A&B({\mathcal{H}}X){\Pi}^{n}\\ -(B({\mathcal{H}}X){\Pi}^{n})^{}P&0 \end{bmatrix} \begin{bmatrix} X\\ \tilde{\mu} \end{bmatrix} + \begin{bmatrix} B({\mathcal{H}}X)(I-{\Pi}^{n}){\mu} \\ 0 \end{bmatrix} + \begin{bmatrix} B \\ 0 \end{bmatrix} u$$ Numerical Simulations ===================== Our principle approximation result, the proposed online identification, and adaptive control of systems with history dependent forces are verified in this section. In the first experiment, we validate the operator approximation error bound presented in Theorem 1. In the second experiment, we model a wind tunnel single wing section with a leading and trailing edge flaps and apply the proposed sliding mode adaptive controller presented in Theorem 4. We illustrate the stability of the closed loop system and convergence of the closed-loop system trajectories to the equilibrium point. \[sec:numerical\] Operator Approximation Error ---------------------------- In this section we consider a collection of numerical experiments to validate the operator approximation rates derived in Theorem 1. In order to show that Equation \[eq:error\_2\] holds, we choose a function $\mu(s)$ over $\Delta$ and then calculate $(h_j f)(t)\circ \mu_j$ for different levels of refinement. Since the computation of $(hf)(t)\circ \mu$ exactly is numerically infeasible, we choose $J \gg j$ as the finest level of refinement in our simulation. According to Theorem 1, we have $$\|(h_J f)(t)\circ \mu_J- (hf)(t)\circ \mu\| \leq C_J 2^{-(\alpha+1)J},$$ and for $j\ll J$ we see that $$\|(h_j f)(t)\circ \mu_j- (hf)(t)\circ \mu\| \leq C_j 2^{-(\alpha+1)j}.$$ Assuming $C=\max\{C_j,C_J\}$ and using the triangle inequality, we obtain $$\begin{aligned} \|(h_J f)(t)&\circ \mu_J-(h_j f)(t)\circ \mu_j\|\\ &\leq C(2^{-(\alpha+1)J}+2^{-(\alpha+1)j}) \end{aligned} \label{eq:numerical}$$ Therefore, given the weights $\mu_J$ for the finest level of refinement $J$, we can evaluate $\mu_j=\Pi_j \mu_J$ and numerically verify Equation \[eq:numerical\]. [.03]{} ![Error for different resolution simulations, $J=7$[]{data-label="fig:ErrorJ7"}](Error7_caption.pdf "fig:"){width=".5cm"} [.65]{} ![Error for different resolution simulations, $J=7$[]{data-label="fig:ErrorJ7"}](Error7_.pdf "fig:"){width="9cm"} Figure \[fig:ErrorJ7\] shows the simulation results for $J=7$ and $j=2,3,4,5$. The error term attenuates with increasing j. In order to investigate the rate of attenuation, we evaluate constant $C$ for different levels of refinements. As shown in figure \[fig:Cvsj\], $C$ is approximately constant with respect to $j$ which agrees with the result from Equation \[eq:numerical\]. [.04]{} ![$C$ for different level j refinement simulations[]{data-label="fig:Cvsj"}](Cvsj_J7_caption.pdf "fig:"){width=".65cm"} [.6]{} ![$C$ for different level j refinement simulations[]{data-label="fig:Cvsj"}](Cvsj_J7.pdf "fig:"){width="9.2cm"} Online Identification of History Dependent Aerodynamics and Adaptive Control for a Simple Wing Model {#sec:numerics2} ---------------------------------------------------------------------------------------------------- The reformatted governing equations of the system take the form of Equation \[eqn:2\] where $Q_a(t,\mu)$ is the vector of generalized history dependent aerodynamic loads. The dynamic equation of the system can be written in the form of Equation \[eq:1st\_order\], where the history dependent term $M^{-1}(q)Q_a(t,\mu)$ is rewritten in terms of a history dependent operator $(\mathcal{H}X)(t)$ acting on the distributed parameter function $\mu$. The history dependent operator includes a family of fixed history dependent kernels and the distributed parameters $\mu$ act as a weighting vector that determines the contribution of a specific history dependent kernel to the overall history dependent operator. ![Prototypical model for a wing section []{data-label="fig:wing_model"}](Wing_model2.pdf){width="12cm"} We perform an offline identification based on a set of experimental data collected from a wind tunnel experiments or CFDsimulations. These define a nominal model for the history dependent aerodynamic loads that appear in the governing equations of the system. We can exploit the model in the numerical simulations to perform an online estimation of the history dependent aerodynamics and adaptive control of a simple wing model. The details of offline identification of history dependent aerodynamics follow the steps explained in [@Dadashi2016]. The model developed in Figure \[fig:wing\_model\] is chosen to validate our proposed adaptive sliding mode controller where $w$ is the velocity of wind, $k_h$ is spring constant in plunge, $k_\theta$ is a spring constant in pitch, $\theta$ is the pitch angle, $h$ is the plunge displacement, $c_\theta$ and $c_h$ are viscous damping coefficients, $m$ and $I_\theta$ are the mass and moment of inertia and, $x_\theta$ is the non-dimensionalized distance between center of mass and the elastic axis. Finally, $L$ and $M$ are lift and moment generated by the leading and trailing edge flaps. The angles $\beta_1$ and $\beta_2$ define the rotation of the trailing edge and leading edge flaps respectively. The dynamic equations of the wing model is derived in the appendix C as $$\begin{bmatrix} m & m x_\theta \\ m x_\theta & mx_\theta^2+ I_\theta \end{bmatrix} \begin{Bmatrix} \ddot{h} \\ \ddot{\theta}\end{Bmatrix}+ \begin{bmatrix} c_h & 0 \\ 0 & c_\theta \end{bmatrix} \begin{Bmatrix} \dot{h} \\ \dot{\theta}\end{Bmatrix}+ \begin{bmatrix} k_h & 0 \\ 0 & k_\theta \end{bmatrix} \begin{Bmatrix} h \\ \theta\end{Bmatrix}= \begin{Bmatrix} L \\ 0 \end{Bmatrix} + \begin{Bmatrix} f_1(\beta_1, \beta_2) \\ f_2(\beta_1, \beta_2) \end{Bmatrix}. \label{eqn:Wing}$$ We have assumed the aerodynamic moment $M$ to be zero and the distance $x_a$ between the aerodynamic center $A$ and hinge point to be negligible to simplify the simulation. The unsteady aerodynamic lift is $ L= Q_a(t,\mu)$ where $Q_a(t,\mu)=(\mathcal{H}X)\circ \mu$ reflects the history dependent nature of aerodynamic loads. We rewrite Equation \[eqn:Wing\] to achieve the standard form presented in Equation \[eqn:2\]. The adaptive controller presented in Theorem 4 is composed of two parts. The first part compensates for the flutter generated by the history dependent aerodynamic forces through online identification of the aerodynamics. The second part employs an sliding mode controller to compensate for modeling errors. [.45]{} ![Time histories of the states and input signals for $\epsilon=0.01$, $t_h=0.001$(sec) and, $k=20$ []{data-label="fig:control_results_chattering"}](theta_chattering "fig:"){width="7.5cm"} [.45]{} ![Time histories of the states and input signals for $\epsilon=0.01$, $t_h=0.001$(sec) and, $k=20$ []{data-label="fig:control_results_chattering"}](h_chattering "fig:"){width="7.7cm"} [.45]{} ![Time histories of the states and input signals for $\epsilon=0.01$, $t_h=0.001$(sec) and, $k=20$ []{data-label="fig:control_results_chattering"}](dtheta_chattering "fig:"){width="7.5cm"} [.45]{} ![Time histories of the states and input signals for $\epsilon=0.01$, $t_h=0.001$(sec) and, $k=20$ []{data-label="fig:control_results_chattering"}](dh_chattering "fig:"){width="7.7cm"} [.45]{} ![Time histories of the states and input signals for $\epsilon=0.01$, $t_h=0.001$(sec) and, $k=20$ []{data-label="fig:control_results_chattering"}](u1_chattering "fig:"){width="7.5cm"} [.45]{} ![Time histories of the states and input signals for $\epsilon=0.01$, $t_h=0.001$(sec) and, $k=20$ []{data-label="fig:control_results_chattering"}](u2_chattering "fig:"){width="7.7cm"} It is noteworthy that numerical time integration of the evolution equations must accommodate history dependent terms. Since the dynamics of such systems are given via functional differential equations, the ordinary integration rules are not directly applicable. We exploit the predictor-corrector integration rule that has been introduced first in [@tavernini]. We also refer the interested reader to our previous paper [@dadashi2016_CDC] for details of such integration rules. Figure \[fig:control\_results\_chattering\] Shows the simulation results for the case where $\epsilon =0.01$ and $t_h=0.001$. The system response eventually enters in a $\epsilon$ neighborhood of the sliding manifold. However, as depicted in the figure a chattering behavior occurs in the control signal and system trajectories. We trace this behavior back to the integration error induced by the size of time step. When we increase $\epsilon$ or reduce the integration time step, the control signal and system trajectories become smooth. The simulation results for $\epsilon = 0.01$ and $t_h=0.0005 $ are depicted in Figure \[fig:control\_results\_smooth1\]. The system trajectories converge to a neighborhood of zero or the set $\mathcal{M}$ in Equation \[eq:M\] with time and the control signals are relatively smooth. Also, Figure \[fig:control\_results\_smooth2\] shows the case when $\epsilon = 0.1$ and $t_h=0.001 $. The convergence rate of the signals to zero is slower but the results do not show any chattering. Therefore, the proposed smooth sliding mode adaptive controller proves to be effective to identify and compensate for the unknown history dependent aerodynamic forces. [.45]{} ![Time histories of the states and input signals for $\epsilon=0.01$, $t_h=0.0005$ (sec) and, $k=20$ []{data-label="fig:control_results_smooth1"}](theta_smooth1 "fig:"){width="7.5cm"} [.45]{} ![Time histories of the states and input signals for $\epsilon=0.01$, $t_h=0.0005$ (sec) and, $k=20$ []{data-label="fig:control_results_smooth1"}](h_smooth1 "fig:"){width="7.7cm"} [.45]{} ![Time histories of the states and input signals for $\epsilon=0.01$, $t_h=0.0005$ (sec) and, $k=20$ []{data-label="fig:control_results_smooth1"}](dtheta_smooth1 "fig:"){width="7.5cm"} [.45]{} ![Time histories of the states and input signals for $\epsilon=0.01$, $t_h=0.0005$ (sec) and, $k=20$ []{data-label="fig:control_results_smooth1"}](dh_smooth1 "fig:"){width="7.7cm"} [.45]{} ![Time histories of the states and input signals for $\epsilon=0.01$, $t_h=0.0005$ (sec) and, $k=20$ []{data-label="fig:control_results_smooth1"}](u1_smooth1 "fig:"){width="7.5cm"} [.45]{} ![Time histories of the states and input signals for $\epsilon=0.01$, $t_h=0.0005$ (sec) and, $k=20$ []{data-label="fig:control_results_smooth1"}](u2_smooth1 "fig:"){width="7.7cm"} [.45]{} ![Time histories of the states and input signals for $\epsilon=0.1$, $t_h=0.001$ (sec) and, $k=20$ []{data-label="fig:control_results_smooth2"}](theta_smooth2 "fig:"){width="7.5cm"} [.45]{} ![Time histories of the states and input signals for $\epsilon=0.1$, $t_h=0.001$ (sec) and, $k=20$ []{data-label="fig:control_results_smooth2"}](h_smooth2 "fig:"){width="7.7cm"} [.45]{} ![Time histories of the states and input signals for $\epsilon=0.1$, $t_h=0.001$ (sec) and, $k=20$ []{data-label="fig:control_results_smooth2"}](dtheta_smooth2 "fig:"){width="7.5cm"} [.45]{} ![Time histories of the states and input signals for $\epsilon=0.1$, $t_h=0.001$ (sec) and, $k=20$ []{data-label="fig:control_results_smooth2"}](dh_smooth2 "fig:"){width="7.7cm"} [.45]{} ![Time histories of the states and input signals for $\epsilon=0.1$, $t_h=0.001$ (sec) and, $k=20$ []{data-label="fig:control_results_smooth2"}](u1_smooth2 "fig:"){width="7.5cm"} [.45]{} ![Time histories of the states and input signals for $\epsilon=0.1$, $t_h=0.001$ (sec) and, $k=20$ []{data-label="fig:control_results_smooth2"}](u2_smooth2 "fig:"){width="7.7cm"} The Fig.\[fig:wing\] show a simplified model of the wing. $C$ is the center of mass, $A$ is the aerodynamic center and $O$ is the point about which the wing is hinged. $K_h$ and $K_\theta$ are the linear and torsional stiffness. $h$ is the distance from origin to point $O$ in fixed reference frame $(\hat{n_1}-\hat{n_2})$.$b$ is the distance between point $O$ and center of mass $C$. whereas $a$ is the distance between $O$ and $A$. Point $O$ is the origin for body fixed reference frame $(\hat{b_1}-\hat{b_2})$. We used Euler-Lagrange technique to derive the equation of motion for the above wing model. $L(\theta,\dot{\theta})$ is the history dependent lift force acting at the aerodynamic center, and $M(\theta,\dot{\theta})$ is the history dependent aerodynamic moment about point $A$. We assume $L_\beta$ as the actuating force acting at point $D$ and $\beta$ is the angle between the wing and the actuator. The position vector of point $C$ is given as $$\overrightarrow{r}_c = h\hat{n}_1 - b\hat{b}_2$$ therefore the corresponding velocity of point $C$ is $$\dot{\overrightarrow{r}}_c = \dot{h}\hat{n}_1 + b\dot{\theta}\hat{b}_1$$ the rotation matrix for transformation between inertial frame of reference to body fixed frame of reference is given by following transformation matrix $$\begin{bmatrix} \hat{b}_1\\ \hat{b}_2 \end{bmatrix} = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} \hat{n}_1\\ \hat{n}_2 \end{bmatrix}$$ The kinetic energy is $$T = \frac{1}{2}m(\overrightarrow{r_c}.\overrightarrow{r_c} ) + \frac{1}{2}J{\dot{\theta}^2}$$ $$T=\frac{1}{2}m(\dot{h}^2+b^2 \dot{\theta}^2 +2 b \dot{h} \dot{\theta} \cos{\theta}) + \frac{1}{2}J{\dot{\theta}^2}$$ the corresponding potential energy is $$V=\frac{1}{2}K_h h^2 + \frac{1}{2}K_\theta \theta ^2 +m g (h+b \sin{\theta})$$ therefore we can write Lagrangian as $$L=T-V$$ Applying the Euler-Lagrange equations we write the equation of motion as follows: $$\begin{gathered} \begin{bmatrix} m & m b \cos{\theta} \\ m b \cos{\theta} & m b^2 + J \end{bmatrix} \begin{bmatrix} \ddot{h} \\ \ddot{\theta} \end{bmatrix} + \begin{bmatrix} 0 & -m b \dot{\theta}\sin{\theta} \\ 0 & 0 \end{bmatrix} \begin{bmatrix} \dot{h} \\ \dot{\theta} \end{bmatrix} + \begin{bmatrix} K_h & 0 \\ 0 & K_{\theta} \end{bmatrix} \begin{bmatrix} h \\ \theta \end{bmatrix}\\ = \begin{bmatrix} -m g \\ -m g b \cos{\theta} \end{bmatrix} + \begin{bmatrix} L(\theta,\dot{\theta}) \cos{\theta} \\ M(\theta,\dot{\theta}) + L(\theta,\dot{\theta})a \end{bmatrix} + \begin{bmatrix} -L_{\beta_1} \cos {(\theta + \beta_1)} \\ -L_{\beta_1} (e_1 + d_1 \cos{\beta_1}) \end{bmatrix} + \begin{bmatrix} -L_{\beta_2} \cos {(\theta + \beta_2}) \\ L_{\beta_2} (e_2 + d_2 \cos{\beta_2}) \end{bmatrix}\end{gathered}$$ The above equation is of the form $$M(q(t))\ddot{q}(t)+C(q(t),\dot{q}(t))\dot{q}(t)+K(q(t))=Q_a (t) + {u}(t)$$ where $q = [h \: \theta]^T $. We implement to proposed control strategy on the simplified model and the results are discussed in the next section. Results and Conclusion ====================== In this paper, we have derived an explicit bound for the error of approximation for certain history dependent operators that are used in construction of robotic FDE’s in [@Dadashi2016] and this paper. The numerical simulations presented validate our results. We establish uniform upper bounds on their accuracy of the approximations. The uniform $\mathcal{O}(2^{-(\alpha+1)j})$ rates of approximation for grid resolution $j$ depend on the Holder coefficient $\alpha$ that describes the smoothness of the ridge functions that define the history dependent kernels. In Section \[sec:exist\] we prove the existence and uniqueness of a local solution for the special case of functional differential equations with history dependent terms shown in Equation \[eq:first\_order\]. Since the functional differential equation of interest evolves in an infinite dimensional space, we construct finite dimensional approximations with grid resolution $j$. We further show that the solution of the finite dimensional distributed parameter system converges to the solution of the infinite dimensional FDE as the resolution is refined. Finally, we propose an adaptive control strategy to identify and compensate the unknown history dependent dynamics. Appendix A: Wavelets and Approximation Spaces over the Triangular Domain {#App:B .unnumbered} ======================================================================== We define the multiscaling functions $$\phi_{j,k}(x) =1_{\Delta_{i_1,i_2,\hdots,i_j}}(x)/\sqrt[]{m(\Delta_{i_1,i_2,\hdots,i_j})}$$ in which $$\begin{aligned} 1_{\Delta_{s}}(x)=\left \{ \begin{array}{lll} 1 & x \in \Delta_s\\ 0 & \text{otherwise} \end{array} \right.\end{aligned}$$ and $m(\Delta_{i_1,i_2,\hdots,i_j})$ is the area of a triangle in the level $j$ refinement. We have defined $ (hf)(t)\circ \mu= \iint_\Delta \kappa(s,t,f)\mu(s)ds. $ The approximation $(h_j f)(t)\circ \mu$ of this operator is given by $$\begin{aligned} (h_j f)(t)\circ \mu= \iint_\Delta \sum_{l\in \Gamma_j} 1_{\Delta_{j,l}}(s) \kappa(\xi_{j,l},t,f) \mu(s)ds,\end{aligned}$$ where $\xi_{j,l}$ is the quadrature point of number $l$ triangle of grid level $j$. We approximate $\mu(s) \approx \sum_{m\in \Gamma_j} \mu_{j,m}\phi_{j,m}(s)$. Therefore, $$\begin{aligned} (& h_j f)(t)\circ \mu_j\\ & =\iint_S \left( \sum_{l\in \Gamma_j} 1_{\Delta_{j,l}}(s) \kappa(\xi_{j,l},t,f) \sum_{m\in \Gamma_j} \mu_{j,m}\phi_{j,m}(s)\right) ds\\ & =\sum_{l\in \Gamma_j} \sum_{m\in \Gamma_j}\kappa(\xi_{j,l},t,f) \left(\iint_S 1_{\Delta_{j,l}}(s) \phi_{j,m}(s) ds\right) \mu_{j,m}\\ & =\sum_{l\in \Gamma_j} \kappa(\xi_{j,l},t,f) \sqrt{m(\Delta_{j,l})}\mu_{j,l}.\end{aligned}$$ For an orthonormal basis $\left \{ \phi_k \right \}_{k=1}^\infty$ of the separable Hilbert space $P$, we define the finite dimensional spaces for constructing approximations as $P_n:=\text{span}\left \{ \phi_k\right \}_{k=1}^n$. The approximation error $E_n$ of $P_n$ is given by $$E_n(f):= \inf_{g\in P_n} \\| f-g\\|_P.$$ The approximation space $\mathcal{A}^\alpha_2$ of order $\alpha$ is defined as the collection of functions in $P$ such that $$\mathcal{A}^\alpha_2 := \biggl \{ f\in P \biggl \| \|f\|_{\mathcal{A}^\alpha_2} := \left \{ \sum_{n=1}^\infty (n^\alpha E_n(f))^2 \frac{1}{n} \right \}^{1/2} < \infty \biggr\}.$$ For our purposes, the approximation spaces are easy to characterize: they consist of all functions $f \in P$ whose generalized Fourier coefficients decay sufficiently fast. That is, $f\in \mathcal{A}^\alpha_2$ if and only if $$\sum_{k=1}^\infty k^{2\alpha}\|(f,\phi_k)\|^2 \leq C$$ for some constant $C$. Appendix B: The Projection Operator $\Phi_{J\rightarrow j}$ {#App:B .unnumbered} =========================================================== The orthogonal projection operator $\Phi_{J\rightarrow j}: V_J\rightarrow V_j$ maps a distributed parameter $\mu_J$ to $\mu_j$ i.e. $\Phi_{J\rightarrow j}:\mu_J \mapsto \mu_j$. ![Projection Operator $\Phi_{J\rightarrow j}:V_J\rightarrow V_j$[]{data-label="fig:Proj"}](Projection.pdf){width="20.00000%"} By exploiting the orthogonality property of the operator we have $$\begin{aligned} \iint_\Delta \left( \sum_{m\in\Gamma_j} \mu_{j,m}\phi_{j,m}(s)-\sum_{l\in\Gamma_J} \mu_{J,l}\phi_{J,l}(s)\right) \phi_{j,n}(s) ds=0.\end{aligned}$$ Therefore, we can write $$\begin{aligned} \sum_{m\in\Gamma_j}& \left( \iint_\Delta \phi_{j,m}(s)\phi_{j,n}(s) ds \right)\mu_{j,m} =\sum_{l\in\Gamma_J} \left( \iint_\Delta \phi_{J,l}(s)\phi_{j,n}(s) ds \right) \mu_{J,l}.\end{aligned}$$ Since orthogonality implies $\iint_\Delta \phi_{j,m}(s)\phi_{j,n}(s) ds=\delta_{m,n},$ we conclude that $$\begin{aligned} \mu_{j,n}=\sum_{l\in\Gamma_j} \left(\iint_\Delta \phi_{j,n}(s)\phi_{J,l}(s) ds\right)\mu_{J,l}.\end{aligned}$$ From Theorem 1 we have $$\begin{aligned} \|(h_{j}f)(t)\circ \Pi_{j}\mu - (hf)(t)\circ \mu \| \leq \tilde{C}2^{-\alpha j}, \end{aligned}$$ with $$\begin{aligned} (hf)(t)\circ \mu = \iint_\Delta k(s,t,f)\mu(s)\mathrm{d}s,\\ (h_{j}f)(t)\circ\mu = \iint_\Delta \sum 1_{\Delta_j,l}(s)k(\zeta_{j,l},t,f)\mu(s)\mathrm{d}s,\end{aligned}$$ where $\mu \in P = L^{2}(\Delta) $ and we approximate $ \mu (s) \approx \sum_{l\in\Gamma_J}\mu_{J,l}\phi_{J,l}(s) \in V_J$. To implement this for the finest grid $J$, we compute $$\begin{aligned} (h_J f)(t)\circ \mu_J &= (h_j f)(t)\circ \Pi_J \mu_J,\\ &=\iint\left(\sum 1_{\Delta_J,l}(s)k(\zeta_{J,l},t,f)\sum_{m\in\Gamma_J}\mu_{J,m}\phi_{J,m}(s)\right)\mathrm{d}s,\\ &=\sum_{l\in\Gamma_J}\sum_{m\in\Gamma_J}k(\zeta_{J,l},t,f)\left(\iint 1_{\Delta_J,l}(s)\phi_{J,m}(s)\mathrm{d}s\right)\mu_{J,m},\\ &=\sum_{l\in\Gamma_J}\frac{k(\zeta_{J,l},t,f)\mu_{J,l}}{\left(\sqrt[]{m(\Delta_{J,l}})\right)},\end{aligned}$$ when $\sqrt[]{m(\Delta_{J,l})}$ is the area of the corresponding triangle $\Delta_{J,l}$ in the grid having resolution level $J$. Appendic C: Gronwall’s Inequality {#App:C .unnumbered} ================================= We employ the integral form of Gronwall’s Inequality to obtain our final convergence result. Many forms of Gronwall’s Inequality exist, and we will use a particularly simple version. See Section 3.3.4 in [@is2012]. If the piecewise continuous function $f$ satisfies the inequality $$f(t) \leq \alpha(t) + \int_0^t \beta(s) f(s) ds$$ with some piecewise continuous functions $\alpha,\beta$ where $\alpha$ is nondecreasing, then $$f(t) \leq \alpha(t) e^{\int_0^t\beta(s)ds}.$$ Appendix D: Modeling of a Prototypical Wing Section {#App:D .unnumbered} =================================================== Figure \[fig:wing\_model\] shows a simplified model of the wing. In the figure we denote the center of mass by $c.m.$, $A$ is the aerodynamic center, and $O$ is the elastic axis of the wing. The constants $K_h$ and $K_\theta$ are the linear and torsional stiffness, and $h$ is the distance from origin to point $O$ in the fixed reference frame. We denote by $x_\theta$ the distance between point $O$ and center of mass, whereas $x_a$ is the distance between $O$ and $A$. Point $O$ is the origin for the body fixed reference frame. We employ The Euler-Lagrange technique to derive the equation of motion for the depicted wing model. The function $L(\theta,\dot{\theta})$ is the history dependent lift force acting at the aerodynamic center, and $M(\theta,\dot{\theta})$ is the history dependent aerodynamic moment about point $A$. The variables $L_{\beta_1}$ and $L_{\beta_2}$ are the actuating forces acting at point $D$, and $\beta_1$, $\beta_2$ are the angles between the midchord of the wing and the trailing edge and leading edge flaps, respectively. The position vector of the mass center is given as $$\mathbf{r}_{c.m.} = h\hat{n}_1 - x_\theta\hat{b}_2,$$ and therefore the corresponding velocity of point $C$ is $$\mathbf{\dot{r}}_{c.m.} = \dot{h}\hat{n}_1 + x_\theta \dot{\theta}\hat{b}_1.$$ The rotation matrix for transformation between inertial frame of reference to body fixed frame of reference is $$\begin{bmatrix} \hat{b}_1\\ \hat{b}_2 \end{bmatrix} = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} \hat{n}_1\\ \hat{n}_2 \end{bmatrix}.$$ The kinetic energy is computed to be $$T = \frac{1}{2}m(\mathbf{r}_{c.m.}.\mathbf{r}_{c.m.} ) + \frac{1}{2} I_\theta {\dot{\theta}^2},$$ $$T=\frac{1}{2}m(\dot{h}^2+x_\theta^2 \dot{\theta}^2 +2 x_\theta \dot{h} \dot{\theta} \cos{\theta}) + \frac{1}{2}I_\theta {\dot{\theta}^2},$$ and the corresponding potential energy is $$V=\frac{1}{2}K_h h^2 + \frac{1}{2}K_\theta \theta ^2. $$ therefore we can write Lagrangian as $L=T-V$. We apply Euler-Lagrange equations to write the equation of motion as follows $$\begin{gathered} \begin{bmatrix} m & m x_\theta \cos{\theta} \\ m x_\theta \cos{\theta} & m x_\theta^2 + J \end{bmatrix} \begin{bmatrix} \ddot{h} \\ \ddot{\theta} \end{bmatrix} + \begin{bmatrix} 0 & -m x_\theta \dot{\theta}\sin{\theta} \\ 0 & 0 \end{bmatrix} \begin{bmatrix} \dot{h} \\ \dot{\theta} \end{bmatrix} + \begin{bmatrix} K_h & 0 \\ 0 & K_{\theta} \end{bmatrix} \begin{bmatrix} h \\ \theta \end{bmatrix}\\ = \begin{bmatrix} L(\theta,\dot{\theta}) \cos{\theta} \\ M(\theta,\dot{\theta}) + x_a L(\theta,\dot{\theta}) \end{bmatrix} + \begin{bmatrix} -L_{\beta_1} \cos {(\theta + \beta_1)} -L_{\beta_2} \cos {(\theta + \beta_2}) \\ -L_{\beta_1} (e_1 + d_1 \cos{\beta_1})+L_{\beta_2} (e_2 + d_2 \cos{\beta_2}). \end{bmatrix}\end{gathered}$$ The above equation is written in the form of a standard robotic equations of motion $ M(q(t))\ddot{q}(t)+C(q(t),\dot{q}(t))\dot{q}(t)+K(q(t))=Q_a (t) + \tau(t)$, where $q = [h \: \theta]^T $. 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--- abstract: 'In recent years, Deep Learning(DL) techniques have been extensively deployed for computer vision tasks, particularly visual classification problems, where new algorithms reported to achieve or even surpass the human performance. While many recent works demonstrated that DL models are vulnerable to adversarial examples. Fortunately, generating adversarial examples usually requires white-box access to the victim model, and real-world cloud-based image classification services are more complex than white-box classifier,the architecture and parameters of DL models on cloud platforms cannot be obtained by the attacker. The attacker can only access the APIs opened by cloud platforms. Thus, keeping models in the cloud can usually give a (false) sense of security. In this paper, we mainly focus on studying the security of real-world cloud-based image classification services. Specifically, (1) We propose a novel attack method, Fast Featuremap Loss PGD (FFL-PGD) attack based on Substitution model, which achieves a high bypass rate with a very limited number of queries. Instead of millions of queries in previous studies, our method finds the adversarial examples using only two queries per image; and (2) we make the first attempt to conduct an extensive empirical study of black-box attacks against real-world cloud-based classification services. Through evaluations on four popular cloud platforms including Amazon, Google, Microsoft, Clarifai, we demonstrate that FFL-PGD attack has a success rate over 90% among different classification services. (3) We discuss the possible defenses to address these security challenges in cloud-based classification services. Our defense technology is mainly divided into model training stage and image preprocessing stage.' author: - bibliography: - 'IEEEexample.bib' title: 'Transferability of Adversarial Examples to Attack Cloud-based Image Classifier Service' --- Cloud Vision API , Cloud-based Image Classification Service , Deep Learning , Adversarial Examples INTRODUCTION ============ In recent years, Deep Learning(DL) techniques have been extensively deployed for computer vision tasks, particularly visual classification problems, where new algorithms reported to achieve or even surpass the human performance. Success of DL algorithms has led to an explosion in demand. To further broaden and simplify the use of DL algorithms, cloud-based services offered by Amazon[^1], Google[^2], Microsoft[^3], Clarifai[^4], and others to offer various computer vision related services including image auto-classification, object identification and illegal image detection. Thus, users and companies can readily benefit from DL applications without having to train or host their own models. [@szegedy2013intriguing] discovered an intriguing properties of DL models in the context of image classification for the first time. They showed that despite the state-of-the-art DL models are surprisingly susceptible to adversarial attacks in the form of small perturbations to images that remain (almost) imperceptible to human vision system. These perturbations are found by optimizing the input to maximize the prediction error and the images modified by these perturbations are called as adversarial example. The profound implications of these results triggered a wide interest of researchers in adversarial attacks and their defenses for deep learning in general.The initially involved computer vision task is image classification. For that, a variety of attacking methods have been proposed, such as L-BFGS of [@szegedy2013intriguing], FGSM of [@goodfellow2014explaining], PGD of [@madry2017towards],deepfool of [@moosavi2016deepfool] ,C&W of [@Carlini2016Towards] and so on. Fortunately, generating adversarial examples usually requires white-box access to the victim model, and real-world cloud-based image detection services are more complex than white-box classification and the architecture and parameters of DL models on cloud platforms cannot be obtained by the attacker. The attacker can only access the APIs opened by cloud platforms[@goodman2019cloud; @goodman2020attacking]. Thus, keeping models in the cloud can usually give a (false) sense of security. Unfortunately, a lot of experiments have proved that attackers can successfully deceive cloud-based DL models without knowing the type, structure and parameters of the DL models[@goodman2019defcontransferability; @goodman2019hitbtransferability]. In general, in terms of applications, research of adversarial example attacks against cloud vision services can be grouped into three main categories: query-based attacks, transfer learning attacks and spatial transformation attacks. Query-based attacks are typical black-box attacks, attackers do not have the prior knowledge and get inner information of DL models through hundreds of thousands of queries to successfully generate an adversarial example [@Shokri2017Membership]. In [@ilyas2017query], thousands of queries are required for low-resolution images. For high-resolution images, it still takes tens of thousands of times. For example, they achieves a 95.5% success rate with a mean of 104342 queries to the black-box classifier. In a real attack, the cost of launching so many requests is very high.Transfer learning attacks are first examined by [@szegedy2013intriguing], which study the transferability between different models trained over the same dataset. [@Liu2016Delving] propose novel ensemble-based approaches to generate adversarial example . Their approaches enable a large portion of targeted adversarial example to transfer among multiple models for the first time.However, transfer learning attacks have strong limitations, depending on the collection of enough open source models, but for example, there are not enough open source models for pornographic and violent image recognition.Spatial transformation attacks are simple and effective.[@Hosseini2017Google] found that adding an average of 14.25% impulse noise is enough to deceive the Google’s Cloud Vision API.[@YuanStealthy] found 7 major categories of spatial transformation attacks to evade explicit content detection while still preserving their sexual appeal, even though the distortions and noise introduced are clearly observable to humans. To the best of our knowledge, no extensive empirical study has yet been conducted to black-box attacks and defences against real-world cloud-based image classification services. We summarize our main contributions as follows: - We propose a novel attack method, Fast Featuremap Loss PGD(FFL-PGD) attack based on Substitution model ,which achieves a high bypass rate with a very limited number of queries. Instead of millions of queries in previous studies, our method finds the adversarial examples using only one or twe of queries. - We make the first attempt to conduct an extensive empirical study of black-box attacks against real-world cloud-based image classification services. Through evaluations on four popular cloud platforms including Amazon, Google, Microsoft, Clarifai, we demonstrate that our FFL-PGD attack has a success rate almost 90% among different classification services. - We discuss the possible defenses to address these security challenges in cloud-based classification services. Our protection technology is mainly divided into model training stage and image preprocessing stage. THREAT MODEL AND CRITERION ========================== Threat Model ------------ In this paper, we assume that the attacker can only access the APIs opened by cloud platforms, and get inner information of DL models through limited queries to generate an adversarial example.Without any access to the training data, model, or any other prior knowledge,is a real black-box attack. Criterion and Evaluation ------------------------ The same with [@Li2019Adversarial],We choose top-1 misclassification as our criterion,which means that our attack is successful if the label with the highest probability generated by the neural networks differs from the correct label.We assume the original input is $O$,the adversarial example is $ADV$. For an RGB image $(m \times n \times 3)$, $(x,y,b)$ is a coordinate of an image for channel $b(0 \leqslant b \leqslant 2)$ at location $(x,y)$.We use Peak Signal to Noise Ratio (PSNR)[@Amer2002] to measure the quality of images. $$PSNR = 10log_{10} (MAX^2/MSE)$$ where $MAX =255$, $MSE$ is the mean square error. $$MSE = \frac{1}{mn3}\sum_{b=0}^2\sum_{i=1}^n\sum_{j=1}^m \|\|ADV(i,j,b)-O(i,j,b)\|\|^2$$ Usually, values for the PSNR are considered between 20 and 40 dB, (higher is better) [@amer2005fast].We use structural similarity (SSIM) index to measure image similarity, the details of how to compute SSIM can be found in [@wang2004image].Values for the SSIM are considered good between 0.5 and 1.0, (higher is better). BLACK-BOX ATTACK ALGORITHMS =========================== Problem Definition ------------------ A real-world cloud-based image classification service is a function $F(x) = y$ that accepts an input image $x$ and produces an output $y$. $F(.)$ assigns the label $C(x)=\arg \max_{i}F(x)_i$ to the input $x$. Original input is $O$, the adversarial example is $ADV$ and $\epsilon$ is the perturbation. Adversarial example is defined as: $$ADV=O+\epsilon$$ We make a black-box untargeted attack against real-world cloud-based classification services $F(x)$: $$C(ADV)\not=C(O)$$ We also assume that we are given a suitable loss function $L( \theta ,x,y)$,for instance the cross-entropy loss for a neural network. As usual, $\theta \in \mathbb{R}^p$ is the set of model parameters. Fast Featuremap Loss PGD based on Substitution model ---------------------------------------------------- [@Papernot2016Practical] proposed that the attacker can train a substituted model, which approaches the target model, and then generate adversarial examples on the substituted model. Their experiments showed that good transferability exists in adversarial examples.But the attack is not totally black-box. They have knowledge of the training data and test the attack with the same distributed data, and they upload the training data themselves and they know the distribution of training data [@Papernot2016Practical][@Chen2017ZOO][@Hayes2017Machine]. This leads us to propose the following strategy: 1. Substitute Model Training: the attacker queries the oracle with inputs selected by manual annotation to build a model $F{}'(x)$ approximating the oracle model $F(x)$ decision boundaries. 2. Adversarial Sample Crafting: the attacker uses substitute network $F{}'(x)$ to craft adversarial samples, which are then misclassified by oracle $F(x)$ due to the transferability.We propose Fast Featuremap Loss PGD attack to improve the success rate of transfer attack. ![Top1 vs. network. Top-1 validation accuracies for top scoring single-model architectures [@Canziani2016An].[]{data-label="VGG"}](VGG.png){width="50.00000%"} ### Substitute Model Training Algorithm We observe that a large number of machine vision tasks utilize feature networks as their backends. For examples, Faster-RCNN [@Ren2015Faster] and SSD [@Liu2016SSD] use the same VGG-16 [@simonyan2014very]. If we destroy the extracted features from the backend feature network, both of them will be influenced. We can choose one of AlexNet [@krizhevsky2012imagenet], VGG [@simonyan2014very]and ResNet [@he2016deep] which pretrained on ImageNet as our substitute model. Better top-1 accuracy means stronger feature extraction capability. As can be seen from Figure 1, ResNet-152 has relatively good top-1 accuracy, so we choose ResNet-152 as our substitute model. We fix the parameters of the feature layer and train only the full connection layer of the last layer. Our Substitute Model Training Algorithm is very simple, we train substitute model and generate adversarial example with the same images. $F{}'(x)$ ,$S$ Define architecture $F{}'(x)$ //Label the substitute training set with $F$ $S \leftarrow \left\{ (x, F(x)) : x\in S \right\}$ // Train $F{}'$ on $S$ to evaluate parameters $\theta_F{}'$ $\theta_F{}' \leftarrow \mbox{train}(F{}',S)$ $\theta_F$ ### Adversarial Sample Crafting Algorithm Previous work [@xie2017mitigating] has shown that one-step or multi-step attack algorithm such as $FGSM$ and $FGSM^k$, has better robustness in transfer attacks than $CW2$, which is based on optimization.$FGSM$ is an attack for an $\ell_{\infty}$ -bounded adversary and computes an adversarial example as: $$x+\varepsilon \operatorname{sgn}\left(\nabla_{x} L(\theta, x, y)\right)$$ A more powerful adversary is the multi-step variant $FGSM^k$, which is essentially projected gradient descent ($PGD$) on the negative loss function [@madry2017towards]: $$x^{t+1}=\Pi_{x+\mathcal{S}}\left(x^{t}+\varepsilon \operatorname{sgn}\left(\nabla_{x} L(\theta, x, y)\right)\right)$$ As usual, loss function $L( \theta ,x,y)$ is cross-entropy loss for a neural network. We propose Fast Featuremap Loss PGD attack which has a novel loss function to improve the success rate of transfer attack.The loss function $L$ is defined as: $$L = class\_loss+\betaFeatureMaps\_loss$$ Where $\varepsilon$ and $\beta$ are the relative importance of each loss function. Next, we will introduce each component of the loss function in detail. Class Loss* The core goal of generating adversarial example is to make the result of classification wrong. The first part of our loss function is class loss. Assuming that the Logits output of the classifier can be recorded as $Z(x)$, the output value corresponding to the classification label $i$ is $Z_{i}(x)$, and $t$ is the label of normal image. The greater the value of $Z_{i}(x)$, the greater the confidence that $x$ is recognized as $i$ by the classifier. $$class\_loss (x) = \max(\max \{ Z(x)_i : i \ne t\} - Z(x)_t, -\kappa).$$ $\kappa$ is a hyperparameter and $\kappa$ is a positive number, the greater the $\kappa$ is, the greater the confidence of the adversarial example is recognized as $\kappa$ by the classifier. [@Carlini2016Towards] discussed the performance of various class loss in detail in CW2 algorithm. We choose the class loss selected in CW2 algorithm. The empirical value of $\kappa$ is 200. FeatureMap Loss Only class loss and distance loss can be used to generate adversarial example, which is what the L-BFGS algorithm of [@szegedy2013intriguing] does. However, as a black-box attack, we have no knowledge of the parameters and structure of the attacked model. We can only generate adversarial example through the known substitute model of white-box attack, and then attack target model. The success rate of the attack depends entirely on the similarity between the substitute model and the attacked model. We introduce Feature Maps loss, which is the output of the last convolution layer of the substitute model, representing the highest level of semantic features of the convolution layer after feature extraction layer by layer $L_n$. We assume the original input is $O$, the adversarial example is $ADV$, and the featuremap loss is: $$FeatureMap\_loss(ADV,O)=\\|L_n(ADV) - L_n(O) \\|_2$$ [zeiler2014visualizing]{} visualizes the differences in the features extracted from each convolution layer. In cat recognition, for example, Figure \[cat\_conv\] ,the first convolution layer mainly recognizes low level features such as edges and lines. In the last convolution layer, it recognizes high level features such as eyes and nose. In machine vision tasks, convolution layer is widely used for automatic feature extraction. And a large number of models are based on the common VGG, ResNet pre-training parameters on ImageNet and finetuned the weights on the their own dataset. We assume that the feature extraction part of the main stream cloud-based image classification services are based on common open source models as VGG or others. The larger the feature Maps loss of the adversarial example and the original image, the greater the difference in the semantic level. We define the hyperparameter $\beta$. The larger the $\beta$, the better the transferability. ![image](cat_conv.png){width="80.00000%"} EXPERIMENTAL EVALUATION ======================= Datasets and Preprocessing -------------------------- 100 cat images and 100 other animal images are selected from the ImageNet val set. Because VGG19 and Resnet50 both accept input images of size $224 \times 224 \times 3$, every input image is clipped to the size of $224 \times 224 \times 3$, where 3 is the number of RGB channels. The RGB value of the image is between 0 and 255. We use these 100 images of cats as original images to generate adversarial examples and make a black-box untargeted attack against real-world cloud-based image classification services. We choose top-1 misclassification as our criterion, which means that our attack is successful if the label with the highest probability generated by the cloud-based image classification service differs from the correct label “cat”. We count the number of top-1 misclassification to calculate the escape rate. Platforms Cat Images Other Animal Images All Images ----------- ------------ --------------------- ------------ Amazon 99/100 98/100 197/200 Google 97/100 100/100 197/200 Microsoft 58/100 98/100 156/200 Clarifai 97/100 98/100 195/200 : Correct label by cloud APIs[]{data-label="tab:o"} According to Table \[tab:o\], we can learn that Amazon and Google, which label 98.5% of all images correctly, have done a better job than other cloud platforms. Fast Featuremap Loss PGD based on Substitution model ---------------------------------------------------- We choose ResNet-152 as our substitute model, fix the parameters of the feature layer and train only the full connection layer of the last layer. We launched PGD and FFL-PGD attacks against our substitute model to generate adversarial examples. PGD and FFL-PGD share the same hyperparameter of step size $\varepsilon$ ,while the hyperparameter $\beta$ of FFL-PGD set to $0.1$. The escape rates of PGD and FFL-PGD attacks are shown in Figure \[fig:ffl:a\]. From Figure \[fig:ffl:a\], we know that the cloud-based image classification services of Amazon, Google, Microsoft and Clarifai are vulnerable to PGD and FFL-PGD attacks . Step size $\epsilon$ controls the escape rate. Increasing this parameter can improve the escape rate. When $\epsilon$ is the same, FFL-PGD has a higher escape rate than PGD. It can be seen that the FeatureMap Loss which added to the loss function is beneficial to improve the escape rate, that is, to improve the robustness of transfer attacks against different cloud-based image classification services. From Figure \[fig:ffl:b\] , we know that PGD has a higher PSNR ,which is considered as better image quality .But both of them higher than 20dB when $\epsilon$ from 1 to 8, which means both of them are considered acceptable for image quality.In addition, we can find that increasing $\epsilon$ will lead to image quality degradation. From Figure \[fig:ffl:c\] , we know that FFL-PGD has a higher SSIM ,which is considered as better image similarity . FFL-PGD attack has a success rate over 90% among different cloud-based image classification services and is considered acceptable for image quality and similarity using only two queries per image. [@Liu2016Delving] adopted an ensemble-based model to improve transferability of attack and successfully attack Clarifai. We used their methods to attack the same cloud platforms and train our ensemble-based model with AlexNet, VGG-19, ResNet-50, ResNet-110 and ResNet-152. The number of iteration 10 20 50 100 200 ------------------------- ------- ------- ------- ------- ------- PSNR 26.53 27.13 33.56 37.88 42.49 SSIM 0.60 0.58 0.64 0.70 0.77 According to Table \[tab:Ensemble\], we can learn that increasing the number of iteration can increase PSNR and SSIM under ensemble-based model attack, which means better image quality and similarity . We can infer that when the number of iterations continues to increase, the perturbation $l_2$ value of adversarial examples decreases and the PSNR increases. Although the image quality can be improved, adversarial examples are over fitting the model , and the transferability decreases in the face of the pretreatment of cloud services. In [@Xie2018Mitigating], they take advantage of the weakness of iteration-based white-box attack, and use the pre-processing steps of random scaling and translation to defense the adversarial examples. DISCUSSION ========== Effect of Attacks ----------------- Our research shows that FFL-PGD attack can reduce the accuracy of mainstream image classification services in varying degrees. To make matters worse, for any image classification service, we can find a way that can be almost 90% bypassed. Defenses -------- Defense adversarial examples is a huge system engineering, involving at least two stages: model training and image preprocessing. ### Model Training [@goodfellow2014explaining] proposed adversarial training to improve the robustness of deep learning model. Retraining the model with new training data may be very helpful. Adversarial training included adversarial examples in the training stage and generated adversarial examples in every step of training and inject them into the training set. On the other hand, we can also generate adversarial samples offline, the size of adversarial samples is equal to the original data set, and then retrain the model. We have developed AdvBox[@goodman2020advbox][^5], which is convenient for developers to generate adversarial samples quickly. ### Image Preprocessing [@Dziugaite2016] evaluated the effect of JPG compression on the classification of adversarial images and their experiments demonstrate that JPG compression can reverse small adversarial perturbations. However, if the adversarial perturbations are larger, JPG compression does not reverse the adversarial perturbation. [@xie2017mitigating] proposed a randomization-based mechanism to mitigate adversarial effects and their experimental results show that adversarial examples rarely transfer between different randomization patterns, especially for iterative attacks. In addition, the proposed randomization layers are compatible to different network structures and adversarial defense methods, which can serve as a basic module for defense against adversarial examples. Although all the above efforts can only solve some problems, chatting is better than nothing. RELATED WORK ============ Previous works mainly study the security and privacy in DL models via white-box mode [@szegedy2013intriguing] [@goodfellow2014explaining] [@madry2017towards] [@moosavi2016deepfool]. In the white-box model, the attacker can obtain the adversarial examples quickly and accurately. However, it is difficult for the attacker to know the inner parameters of models in the real world, so researchers have launched some black-box attacks on DL models recently. In general, in terms of applications, research of adversarial example attacks against cloud vision services can be grouped into three main categories: query-based attacks, transfer learning attacks and spatial transformation attacks. Query-based attacks are typical black-box attacks, attackers do not have the prior knowledge and get inner information of DL models through hundreds of thousands of queries to successfully generate an adversarial example [@Shokri2017Membership].In [@ilyas2017query], thousands of queries are required for low-resolution images. For high-resolution images, it still takes tens of thousands of times. But attacking an image requires thousands of queries, which is not operable in actual attacks of real-world cloud-based image classification services. In order to reduce the number of queries,[@Papernot2016Practical] attack strategy consists in training a local model to substitute for the target DL models, using inputs synthetically generated by an adversary and labeled by the target DL models. They have knowledge of the training data and test the attack with the same distributed data, and they upload the training data themselves and they know the distribution of training data [@Papernot2016Practical][@Chen2017ZOO][@Hayes2017Machine] . Transfer learning attacks are first examined by [@szegedy2013intriguing], which study the transferability between different models trained over the same dataset. [@Liu2016Delving] propose novel ensemble-based approaches to generate adversarial example . Their approaches enable a large portion of targeted adversarial example to transfer among multiple models for the first time. Spatial transformation attacks are very interesting, [@Hosseini2017Google] evaluate the robustness of Google Cloud Vision API to input perturbation, they show that adding an average of 14.25% impulse noise is enough to deceive the API and when a noise filter is applied on input images, the API generates mostly the same outputs for restored images as for original images. [@YuanStealthy] report the first systematic study on the real-world adversarial images and their use in online illicit promotions. They categorize their techniques into 7 major categories, such as color manipulation, rotation, noising and blurring. [@Li2019Adversarial] make the first attempt to conduct an extensive empirical study of black-box attacks against real-world cloud-based image detectors such as violence, politician and pornography detection. Our FFL-PGD attack based on Substitution model can be classified as a combination of query-based attack and transfer learning attack. CONCLUSION AND FUTURE WORK ========================== In this paper, (1) We propose a novel attack method, Fast Featuremap Loss PGD (FFL-PGD) attack based on Substitution model, which achieves a high bypass rate with a very limited number of queries. Instead of millions of queries in previous studies, our method finds the adversarial examples using only two queries per image; and (2) we make the first attempt to conduct an extensive empirical study of black-box attacks against real-world cloud-based classification services. Through evaluations on four popular cloud platforms including Amazon, Google, Microsoft, Clarifai, we demonstrate that FFL-PGD attack has a success rate almost 90% among different classification services. (3) We discuss the possible defenses to address these security challenges in cloud-based classification services. Our defense technology is mainly divided into model training stage and image preprocessing stage. In the future, we aim to explore the space of adversarial examples with less perturbation in black-box and attempt to study target attack using FFL-PGD attack. On the other hand, we will focus on the defense in the cloud environment, so that AI services in the cloud environment away from cybercrime. We hope cloud service providers will not continue to forget this battlefield. [^1]: https://aws.amazon.com [^2]: https://cloud.google.com [^3]: https://azure.microsoft.com [^4]: https://www.clarifai.com/ [^5]: https://github.com/advboxes/AdvBox
--- abstract: 'The general covariance of the diffusion equation is exploited in order to explore the curvature effects appearing on brownian motion over a d-dimensional curved manifold. We use the local frame defined by the so called Riemann normal coordinates to derive a general formula for the mean-square geodesic distance (MSD) at the short-time regime. This formula is written in terms of $O(d)$ invariants that depend on the Riemann curvature tensor. We study the n-dimensional sphere case to validate these results. We also show that the diffusion for positive constant curvature is slower than the diffusion in a plane space, while the diffusion for negative constant curvature turns out to be faster. Finally the two-dimensional case is emphasized, as it is relevant for the single particle diffusion on biomembranes.' --- 2em [ Brownian motion meets Riemann curvature ]{}\ 2em [Pavel Castro Villarreal[^1]\ ]{} [ Centro de Estudios en Física y Matemáticas Básicas y Aplicadas,\ Universidad Autónoma de Chiapas, C.P. 29000, 1428 Tuxtla Gutiérrez, Chiapas, MÉXICO\ ]{} 1em Introduction ============ The brownian motion phenomenon occurs in a wide diversity of physical areas; from colloidal physics to quantum gravity and biophysics (see for instance [@Colloid],[@Duplantier1] and [@Almeida]). In the last decade, motivated by problems in biophysics (see [@Weiss; @Ivo]), an intense activity has emerged on the study of diffusion processes on curved manifolds. For instance, the transport phenomena occurring in cell membranes is an interesting and complex problem. In particular, the random motion of a single integral protein or lipid in a cell membrane is difficult to realise, mainly because of the interactions and obstacles with the remainder components of the cell. In addition to this difficulty, the thermal fluctuations produce shape undulations on the curved membrane [@Seifert; @Naji; @Seifert2]. By simplifying this problem to the study of free diffusion on a cell membrane, considered as a regular and continuos two-dimensional surface, Smolouchovski’s diffusion equation has been proposed in [@Aizenbud; @Gustaffson]. Explicit formulas for constant mean and gaussian curvatures have been presented at [@H] and [@K], respectively. Nevertheless, the analytical issues that appear on the general surface case, have motivated the incorporation of novel computer simulations [@Holyst; @Christensen] (see also [@Ivo] for a related work). From a theoretical stand point, the brownian motion can be used to probe the geometry of the manifold in the spirit of Kac’s famous question: [Can one hear the shape of a drum?]{} [@Kac], and vice versa, the geometry will cause a change in the standard randomness of the particle motion [@Einstein]. It is then imminent to analyse a quantitative contribution coming from the geometry, and to be specific, how the curvature of the manifold affects the motion. These questions have been arisen in [@K], where the local concentration of a diffused substance is obtained in terms of the local curvature. Furthermore, a detailed study of the way on how the mean-values are affected by the curvature for two-dimensional manifolds is presented in [@Faraudo], with a special emphasis on developable and isotropic surfaces. Also, the study of particular cases presented in [@Tomoyoshi] suggests that for negative Gaussian curvature, the diffusion accelerates; whereas, the diffusion decelerates for surfaces with positive Gaussian curvature. We should point out that, although with different formalism, these same questions have been posted in [@Fabrice]. In this work we explore the curvature effects on the brownian motion when the particle movement takes place on a d-dimensional riemannian manifold. These effects manifest differently for different physical observables. Here, we will use the geodesic distance as the displacement of the particle, but some other observables can also be defined [@Gustaffson; @Holyst; @Faraudo], where either intrinsic or extrinsic properties of the manifold can be probed [@Castro]. Furthermore, we will take advantage of the general covariance of the diffusion equation to use a special frame defined by the so called Riemann Normal Coordinates (RNC) [@Eisenhart]. In this frame we will compute curvature corrections for the mean-square geodesic distance. In particular, we use a technique developed in [@Denjoe], originally used to compute curvature corrections that appear in effective actions of field theory on curved spaces (see for instance [@Denjoe1; @Denjoe2]). Related work concerning the RNC can be found at [@Muller; @Hatzinikitas]. It is remarkable that, using these coordinates, the geodesic curves on the manifold look like stright lines and therefore, the square geodesic distance $s^{2}$ will have the same structure as the square distance in an euclidean space: $s^{2}=\delta_{ab}y^{a}y^{b}$ (where $y^{a}$ are the RNC). As it is shown in this paper, the mean-square geodesic distance is clearly isometric, this is because it only depends on $O(d)$ invariant combinations of the Riemann curvature tensor. This paper is organized as follows: In section 2 we summarize the geometrical concepts used to approach the brownian motion. In particular, we introduce the frame defined by the Riemann Normal Coordinates. In section 3, we present the diffusion equation on curved manifolds and we derive general remarks concerning the short time regime. In section 4, we focus on the computation of the mean-square geodesic displacement using RNC. In particular, we study the curvature effects for the manifolds with constant curvature and the two-dimensional case. The $n$-dimensional sphere is also explore to validate our results. Finally, in section 5 we sintetize the main conclusions and perspectives of this work. Geometrical preliminaries and notation ====================================== In this section we review the preliminary notions about manifolds and riemannian geometries (following [@Nakahara]) needed to describe the brownian motion. Let us call $\mathbb{M}$ a $d$-dimensional manifold with local coordinates $\varphi:U\subset \mathbb{M}\to \mathbb{R}^{d}$, where $U$ is a local neighbourhood and $\mathbb{R}^{d}$ is the $d$-dimensional euclidean space. As a consequence of the differentiablity of the map $\varphi$, the set $U$ is locally diffeomorphic to a piece of the euclidean space. The local coordinates are also denoted by $\varphi(p)=(x^{1},\cdots,x^{d})$ where $p\in\mathbb{M}$. For each point $p$ on the manifold we associate a vector space called the tangent space ${\rm T}_{p}\mathbb{M}$, whose elements are denoted by capital letters $X, Y, Z, \cdots$. We are interested in manifolds endowed with a [riemannian metric]{}. If $g_{p}:{\rm T}_{p}\mathbb{M}\times {\rm T}_{p}\mathbb{M}\to\mathbb{R}$ denotes a riemannian metric, we write $$\begin{aligned} g_{p}=g_{ab}~dx^{a}\otimes dx^{b},\end{aligned}$$ where $\left\{dx^{a}\right\}$ constitute a one-form basis of the dual tangent space ${\rm T}^{}_{p}\mathbb{M}$ and $g_{ab}$ is the metric tensor. The Einstein summation rule is adopted for the repeated indexes. The knowledge of the metric tensor components allow us to compute further geometrical quantities. The geometrical meaning of how the manifold is curved is defined in terms of the torsion tensor and the Riemann curvature tensor [@Nakahara], $$\begin{aligned} T\left(X,Y\right)&\equiv& \nabla_{X}Y-\nabla_{Y}X-\left[X,Y\right],\nonumber\\ R\left(X,Y,Z\right)&\equiv& \nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{\left[X,Y\right]}Z \label{Def} \label{curvature}\end{aligned}$$ where $\nabla$ is the affine connection. For our purposes, we will use the Levi-Civita connection, which is the only one compatible with the metric. Using the coordinate basis $\left\{e_{a}\right\}\equiv\left\{\partial_{a}\right\}$ of the tangent space ${\rm T}_{p}\mathbb{M}$, the affine connection defines the components $\Gamma^{a}_{~bc}$ by $$\begin{aligned} \nabla_{a}e_{b}\equiv\nabla_{e_{a}}e_{b}=\Gamma^{c}_{~ab}e_{c},\end{aligned}$$ where $\nabla_{a}$ stands for the covariant derivative and for the Christoffel symbols $\Gamma^{a}_{~bc}$ [^2]. The torsion is a (1,2) type tensor $T=T^{a}_{~bc}~e_{a}\otimes dx^{b}\otimes dx^{c}$ whereas the Riemann curvature is a (1,3) type one $R=R^{a}_{~bcd}~e_{a}\otimes dx^{b}\otimes dx^{c}\otimes dx^ {d}$. The components of these tensors are given in terms of the Chrystoffel symbols [^3] . Clearly, the manifold is free torsion for the Levy-Civita connection, because the Christoffel symbols are symmetric. The quantity $R_{abcd}\equiv g_{af}R^{f}_{~bcd}$ satisfies the following useful identities $R_{abcd}=-R_{bacd}=-R_{abdc}=R_{cdab}$. Using the Riemann curvature components we can defined the Ricci tensor $R_{ab}\equiv R^{c}_{~acb}$ and the scalar curvature $R_{g}=g^{ab}R_{ab}$. There is an important device introduced by Riemann, nowadays called the Riemann Normal Coordinates [@Eisenhart]. This coordinate system can be defined by mapping a point $p$ on the manifold to the origin of $\mathbb{R}^{d}$ and the following conditions $$\begin{aligned} g_{ab}(0)=\delta_{ab}, ~~~~~~~y^{a}g_{ab}(y)=y^{a}\delta_{ab}. \label{defRNC}\end{aligned}$$ As it is point out at [@Muller], the second condition is equivalent to the following gauge condition on the affine connection (see appendix A) $$\begin{aligned} y^{a}y^{b}\Gamma^{c}_{~ab}(y)=0. \label{normality}\end{aligned}$$ Furthermore, in RNC the Taylor coefficients of the metric tensor can be found in terms of the covariant derivatives of the Riemann curvature tensor [@Denjoe; @Muller; @Hatzinikitas] as $$\begin{aligned} g_{ab}\left(y\right)&=&\delta_{ab}+\frac{1}{3}R_{acdb}\left(0\right)y^{c}y^{d}+\frac{1}{6}\nabla_{e}R_{acdb}\left(0\right) y^{e}y^{c}y^{d}+\frac{2}{45}R_{acdf}\left(0\right)R^{f}_{~ghb}\left(0\right)y^{c}y^{d}y^{g}y^{h}\nonumber\\&+&\frac{1}{20}\nabla_{e}\nabla_{f}R_{acdb}\left(0\right)y^{e}y^{f}y^{c}y^{d}+\cdot\cdot\cdot\nonumber, \label{MetricRNC}\end{aligned}$$ where $y$ denotes $\varphi(q)$, the RNC, and $q$ belong to the same patch of $p$. See appendix A for a derivation of this series expansion. Using these coordinates, the geodesic curves look like straight lines passing through the point $p$. Indeed, using the gauge condition (\[normality\]) and the geodesic equation it is easy to figure that out. Therefore, the geodesic curve can be written as $y^{a}\left(s\right)=\xi^{a}s$, where $s$ is the geodesic distance and $\xi^{a}$ are constants [@Hatzinikitas]. Furthermore, as the geodesic curve is parametrized by the arc-lenght, the coeffitients $\xi^{a}$ satisfy $g_{ab}\xi^{a}\xi^{b}=1$, so the square geodesic distance is given by $$\begin{aligned} s^{2}=g_{ab}y^{a}y^{b}=\delta_{ab}y^{a}y^{b}, \label{sdispl}\end{aligned}$$ where the last equality comes from the conditions of the RNC (\[defRNC\]). This equation is remarkable, because the geodesic distance has the same form as in the euclidean geometry. Diffusion and geometry ====================== Here, we introduce the simplest model for the brownian motion of a free particle, which takes place on a d-dimensional riemannian geometry. This is a direct generalization of the brownian motion on euclidean spaces, which basically consists on replacing the euclidean laplacian by the Laplace-Beltrami operator in the diffusion equation [@Aizenbud; @Gustaffson] $$\begin{aligned} \frac{\partial P\left(x,x^{\prime},t\right)}{\partial t}&=&D\Delta_{g}P\left(x,x^{\prime},t\right),\nonumber\\ P\left(x,x^{\prime},0\right)&=&\frac{1}{\sqrt{g}}\delta^{\left(d\right)}\left(x-x^{\prime}\right). \label{difeq}\end{aligned}$$ Here, $P\left(x,x^{\prime},t\right)dv$ is the probability to find the diffusing particle in the volume element $dv=\sqrt{g}d^{d}x$ when the particle started to move at $x^{\prime}$. The probability distribution $P\left(x,x^{\prime},t\right)$ is normalized with respect to the volume $v$ of the manifold and $D$ is the diffusion coefficient. The operator $\Delta_{g}$ is the Laplace-Beltrami operator, which is defined by $$\begin{aligned} \Delta_{g}\cdot=\frac{1}{\sqrt{g}}\partial_{a}\left(\sqrt{g}g^{ab}\partial_{b}~\cdot~\right), \label{L-Bop}\end{aligned}$$ where $g=\det\left( g_{ab}\right)$. We should point out that when the manifold is not compact, we will require for the probability and all its partial derivatives to vanish at the boundary. The formal solution of the diffusion equation (\[difeq\]) on curved manifolds (see [@DeWitt; @Denjoe2]) is given in terms of the Minakshisundaram-Pleijel coeffients, which depend on both $x$ and $x^{\prime}$. This solution has already been used in order to describe the concentration of a diffused substance over curved manifolds in the limit when $x\to x^{\prime}$ [@K]. Once we have a probability distribution, we want to look at the mean-values of physical observables (for example, the mean-square displacement) in order to get information of the brownian motion. For scalar functions $\Omega$ (defined on the manifold), the expectation values are defined in the standard fashion $$\begin{aligned} \left<\Omega\left(x\right)\right>_{t}=\int_{\mathbb{M}} dv~\Omega\left(x\right)P\left(x,x^{\prime},t\right).\end{aligned}$$ Note that $\left<\Omega\left(x\right)\right>_{t}$ also depends on the initial point $x^{\prime}$. In principle, it is posible to evaluate the expectation values using the formal solution mentioned above. However, using this procedure it may be very involve because the Minakshisundaram-Pleijel coeffients, as far as we know, are not known for points $x$ different from $x^{\prime}$. Here, we use a different strategy, which will be applied only for physical observables well behaved under actions of $\Delta_{g}$. General remark on the short time asymptotics -------------------------------------------- Clearly, when $\Omega(x)$ is well behaved under any number of actions of the Laplace-Beltrami operator, its expectation value can be expanded in Taylor series in the variable $t$. The $k-{\rm th}$ derivative of $\left<\Omega\left(x\right)\right>_{t}$ at $t=0$ can be computed as follows. First, let us compute the first derivative using the diffusion equation, $$\begin{aligned} \frac{\partial\left<\Omega\left(x\right)\right>_{t}}{\partial t}&=&D\int_{\mathbb{M}} dv~\Omega\left(x\right)\Delta_{g}P\left(x,x^{\prime},t\right)\nonumber\\ &=&D\int_{\mathbb{M}} dv~\Delta_{g}\Omega\left(x\right)P\left(x,x^{\prime},t\right)+\int_{\mathbb{M}} dv ~\nabla_{a}J^{a},\end{aligned}$$ where $J^{a}$ is a boundary current given by $J^{a}=g^{ab}\partial_{b}\left(\Omega P\right)$. Using this procedure it is posible to compute the $k-{\rm th}$ Taylor coeffitient by $$\begin{aligned} \left.\frac{\partial^{k} \left<\Omega\left(x\right)\right>_{t}}{\partial t^{k}}\right \|_{t=0}=D^{k}\Delta^{k}_{g}\Omega\left(x^{\prime}\right),\end{aligned}$$ here we dropped all the boundary terms because the probability and its derivatives vanish there. The expectation value for our physical observable is then given by the formal series [@Grygorian] $$\begin{aligned} \left<\Omega\left(x\right)\right>_{t}=\left .\left[1-\frac{1}{1!}t\mathcal{H}+\frac{1}{2!}t^{2}\mathcal{H}^{2}+\cdot\cdot\cdot\right]\Omega\left(x\right)\right \|_{x=x^{\prime}} \label{computation}\end{aligned}$$ where $\mathcal{H}=-D\Delta_{g}$. This expression is very useful to access the short time regime of the brownian motion. Indeed, given a physical observable which is well behaved under the actions of the Laplace-Beltrami operator, its mean-value at the short-time regime can be calculated using (\[computation\]). In particular, for the plane case $\mathbb{R}^{d}$ we want to know the mean-square displacement $\left\|{\bf x}\right\|^{2}$ (where ${\bf x}\in \mathbb{R}^{d}$ and we have chosen ${\bf x}^{\prime}=0$). In this case, the Laplace-Beltrami reduces simply to the laplacian $\partial^{a}\partial_{a}$. Appliying formula (\[computation\]) we get the standard kinematical Einstein relation for the mean-square displacement: $\left<\left\|{\bf x}\right\|^{2}\right>_{t}=2dDt$. Observe,we can not compute the mean value for the displacement $\left\|{\bf x}\right\|$ by this method because it is not well behaved under the actions of the laplacian at ${\bf x}=0$. The mean values of $s^{2}$ and short time asymptotics ===================================================== On curved manifold there are several quantities that can be used to described the brownian motion. For instance, in [@Holyst] the particle position is given in terms of the parametrization of a manifold embedded in the ambient space $\mathbb{R}^{3}$. For this case, the displacement is given by the euclidean norm of the parametrization. However, in [@Faraudo] the brownian displacement is defined by the geodesic distance. In addition, using the Monge parametrization for a surface we can also define a projected displacement [@Gustaffson]. Here, nevertheless, we will not discern between these quantities, instead, we will stress the fact that all of them represent different manifestations of the same phenomenon. The analysis between these observables is beyond the scope of this work. In this paper, we use the geodesic distance as the definition of the displacement of the particle. As in the plane case, this quantity is rotational and traslational invariant. Furthermore, the geodesic distance is invariant under general coordinate transformations. Therefore the mean-value of $s^{2}$ will be expected to be invariant under a general coordinate transformation. In what follows we compute the expectation value of the square displacement using the Riemann normal frame centered at the point $p=\varphi^{-1}(0)$. Therefore the particle starts to move at the initial condition and the expectation value of $s^{2}$ can be computed using (\[computation\]) $$\begin{aligned} \left<s^{2}\right>_{t}=\sum^{\infty}_{k=1}\frac{G_{k}}{k!}\left(Dt\right)^{k} \label{mean-squaregeo}\end{aligned}$$ where the terms $G_{k}=\left. \Delta^{k}_{g}s^{2}\right \|_{y=0}$ ($k=1,2,3,\dots$) are purely geometric factors. This factors can be computed explicitly using the technology of the RNC. We will compute the first three factors, $G_{0}$, $G_{1}$ and $G_{3}$. Using the definition of the Laplace-Beltrami operator (\[L-Bop\]) and $s^{2}=\delta_{ab}y^{a}y^{b}$, it is not difficult to show that, for every coordinate $y$ on the manifold, $$\begin{aligned} \Delta_{g}s^{2}=2d+y\cdot\partial\left(\log g\right), \label{LBonS}\end{aligned}$$ where $g$ is the determinant of the metric and $y\cdot\partial=y^{a}\partial_{a}$. Clearly, when we evaluate at $y=0$ we get $G_{1}=2d$. The factors $G_{2}$ and $G_{3}$ can be found using the result of the appendix B, summarized as follows, if $f$ is a diferentiable function on the manifold then the Laplace-Beltrami operator acting on $f$ at $y=0$ is simply by $\left.\partial^{a}\partial_{a}f\right \|_{y=0}$. Therefore $ G_{2}=\left.\partial^{2}\left(\Delta_{g}s^{2}\right)\right\|_{y=0}$ and $G_{3}=\left.\partial^{2}\left(\Delta^{2}_{g}s^{2}\right)\right\|_{y=0}$. So, we need at least the second order Taylor expansion of $\Delta_{g}s^{2}$ and $\Delta^{2}_{g}s^{2}$. Using equation (\[LBonS\]) and the Taylor expansion of $\log g$ (see appendix A), we obtain the second order of $\Delta_{g}s^{2}$, $$\begin{aligned} \Delta_{g}s^{2}=2d-\frac{2}{3}R_{ab}\left(0\right)y^{a}y^{b}+O(y^{3}).\end{aligned}$$ Therefore, the second geometric factor is $G_{2}=-\frac{4}{3}R_{g}$, where the Ricci scalar curvature is evaluated at $y=0$. For the third gometric factor $G_{3}$, let $\Delta_{g}$ act on equation (\[LBonS\]), $$\begin{aligned} \Delta^{2}_{g}s^{2}=\frac{1}{2}\left(\partial_{a}\log g\right)g^{ab}\partial_{b}\left(y\cdot\partial\log g\right)+\left(\partial_{a}g^{ab}\right)\partial_{b}\left(y\cdot\partial\log g\right)+g^{ab}\partial_{a}\partial_{b}\left(y\cdot\partial\log g\right).\end{aligned}$$ Using the perturbative expansion of the inverse metric $g^{ab}$ and the one of the $\log g$ it is not difficult to obtain the second order of $\Delta^{2}_{g}s^{2}$ (see appendix A). Therefore, by a straightforward calculation, we get $$\begin{aligned} G_{3}=\frac{8}{15}R^{ab}R_{ab}-\frac{16}{45}R^{abcd}\left(R_{dbca}+R_{dcba}\right)-\frac{16}{5}\left(\nabla^{a}\nabla^{b}+\frac{1}{2}g^{ab}\Delta_{g}\right)R_{ab}.\end{aligned}$$ In general, for the $G_{k}$ factor we need to compute the second order pertubative expression for $\Delta^{k-1}_{g}s^{2}$. For the expectation value of $s^{2}$ at the short-time regime we have considered only the values $k=1,2,3$. Hence, this value can be written as $$\begin{aligned} \left<s^{2}\right>_{t}=2dDt-\frac{2}{3}R_{g}\left(Dt\right)^{2}&+&\frac{1}{3!}\left[\frac{8}{15}R^{ab}R_{ab}\right.-\left.\frac{16}{45}R^{abcd}\left(R_{dbca}+R_{dcba}\right)\right.\nonumber\\ &&~~~~~~~~~~~~~~~~~-\left.\frac{16}{5}\left(\nabla^{a}\nabla^{b}+\frac{1}{2}g^{ab}\Delta_{g}\right)R_{ab}\right]\left(Dt\right)^{3}+\cdot\cdot\cdot \label{mean-square}\end{aligned}$$ As we anticipate in the introduction, the mean-square geodesic distance (i.e. $\left<s^{2}\right>_{t}$) is deviated from the planar expression by terms which are $O(d)$ invariant as well as invariant under a general coordinate transformation. Clearly, for very short-times, the particle movement is not affected by the curvature of the manifold and the standard mean-square displacement is recovered [@Einstein]. This follows from the very nature of the manifold (every local neighbourhood looks like a piece of an euclidean space). However, as the particle explores away from the local boundaries, the movement is affected by the curvature of the manifold. Additionally, the curvature corrections to the planar expression are isometric. Therefore, every manifold which is isometric to an euclidean space, will have null curvature effects for this observable. In particular, this is the case for the developable surfaces [@Faraudo]. Example: the brownian motion on the $n$-sphere ---------------------------------------------- Here, the idea is to perform a cross-check calculation in order to validate equation (\[mean-square\]). In particular, we compute the mean-square geodesic distance at the short-time regime, when the brownian motion takes place on a n-dimensional sphere. The hypersphere $S^{n}$ of radius $R$ is defined by $$\begin{aligned} S^{n}=\left\{{\bf x}\in \mathbb{R}^{n+1}:{\bf x}^{2}=R^{2}\right\}.\end{aligned}$$ This manifold can be parametrized using the local coordinates $\left(\theta_{0},\theta_{1},\cdots,\theta_{n-1}\right)$, where $\theta_{0}$ takes values in $[0,\pi]$ whereas the remainder coordinates in $[0,2\pi)$. In this example, we use a parametrization concerning to the ambient space ${\mathbb{R}^{n+1}}$ by using the functions ${\bf x}=(x_{0},\cdots,x_{n})$ given by $$\begin{aligned} x_{0}&=&R\cos\theta_{0}\nonumber\\ x_{1}&=&R\sin\theta_{0}\cos\theta_{1}\nonumber\\ \vdots&&\nonumber\\ x_{n}&=&R\sin\theta_{0}\sin\theta_{1}\cdots\sin\theta_{n-1}.\end{aligned}$$ Clearly, ${\bf x}=R{\bf n}$, where ${\bf n}$ is the unit normal vector pointing outward from hypersurface of $S^{n}$. The metric tensor can be computed using $g_{ab}=\partial_{a}{\bf x}\cdot\partial_{b}{\bf x}$ and it can be written in a matrix form as $$\begin{aligned} g_{ab}=\left(\begin{array}{ccccc} 1& & & & {\Large 0}\\ &\sin^{2}\theta_{0}& & &\\ & &\sin^{2}\theta_{0}\sin^{2}\theta_{1} &&\\ & & & \ddots&\\ {\Huge 0}& && &\sin^{2}\theta_{0}\cdots\sin^{2}\theta_{n-1} \end{array} \right).\end{aligned}$$ The square root of the metric tensor determinant is $\sqrt{g}=R^{n}\sin^{n-1}\theta_{0}\sin^{n-2}\theta_{1}\cdots\sin\theta_{n-1}$. It is remarkable that using extrinsic geometry we can easily compute the Riemann curvature tensor. Indeed, using the Gauss-Codazzi equations, $R_{abcd}=K_{ac}K_{bd}-K_{ad}K_{bc}$, where $K_{ab}=\partial_{a}{\bf x}\cdot\partial_{b}{\bf n}$ are the components of the second fundamental form [@spivak], the Riemann curvature tensor is $$\begin{aligned} R_{abcd}=\frac{1}{R^{2}}\left(g_{ac}g_{bd}-g_{ad}g_{bc}\right). \label{RiemannSphere}\end{aligned}$$ The Ricci tensor and the scalar curvature are then given by $$\begin{aligned} R_{ab}=\frac{n-1}{R^{2}}g_{ab},~~~~~~~~~ R_{g}=\frac{n\left(n-1\right)}{R^{2}}, \label{RicciScalar}\end{aligned}$$ respectively. Substituting equations (\[RiemannSphere\]) and (\[RicciScalar\]) into the expressions of $G_{1}$, $G_{2}$, and $G_{3}$, obtained above, we find $$\begin{aligned} G_{1}=2n,~~~~~~G_{2}=-\frac{4}{3}\frac{n(n-1)}{R^{2}},~~~~~~ G_{3}=\frac{8}{15}\frac{n(n-1)(n-3)}{R^{4}} \label{Gs}\end{aligned}$$ Now, in order to perform an indepedient calculation we compute the mean-square geodesic distance using the equation (\[computation\]). For practical purpose, we use the geodesic curve starting at the north pole defined as the point ${\bf x}=(1,0,\cdots,0)$ of $S^{n}$. It is not difficult to show that $\theta_{0}=\frac{1}{R}s$ and $\theta_{j}=0$ (with $j=1,\cdots,n-1$) is this geodesic curve. The geodesic distance is simply given by $s=R\theta_{0}$. In this case, the Laplace-Beltrami operator on $S^{n}$ can be splitted as $$\begin{aligned} \Delta_{g}=\frac{1}{R^{2}\sin^{n-1}\theta_{0}}\partial_{0}\sin^{n-1}\theta_{0}\partial_{0}+\sum^{n-1}_{i,j=1}\frac{1}{\sqrt{g}}\partial_{i}\sqrt{g}g^{ij}\partial_{j}, \label{LBonSph}\end{aligned}$$ where $\partial_{0}\equiv\frac{\partial}{\partial\theta_{0}}$ and $\partial_{j}\equiv\frac{\partial}{\partial\theta_{j}}$. Clearly, the actions of the Laplace-Beltrami operator on $s^{2}$ will involve only the first term of (\[LBonSph\]). The first, second and third action of $\Delta_{g}$ on $s^{2}$ are given by $$\begin{aligned} \Delta_{g}s^{2}&=&2(n-1)\theta_{0}\cot\theta_{0}+2,\nonumber\\ \Delta^{2}_{g}s^{2}&=&-\frac{2(n-1)}{R^{2}}\left[n-1+(n-3)(1+\cot^{2}\theta_{0})\left(\theta_{0}\cot\theta_{0}-1\right)\right],\nonumber\\ \Delta^{3}_{g}s^{2}&=&-\frac{2(n-1)(n-3)}{R^{4}}\left\{(n-1)\cot\theta_{0}(1+\cot^{2}\theta_{0})\left[\cot\theta_{0}(3-\theta_{0}\cot\theta_{0})-\theta_{0}(1+\cot^{2}\theta_{0})\right]\right.\nonumber\\ &+&\left.4(1+\cot^{2}\theta_{0})\left[(\theta_{0}\cot\theta_{0}-1)(2\cot^{2}\theta_{0}+1)-\cot\theta_{0}(\cot\theta_{0}-\theta_{0}(1+\cot^{2}\theta_{0}))\right] \right\}.\nonumber\\\end{aligned}$$ Therefore taking the limit when $\theta_{0}\to 0$ we get (\[Gs\]). The mean-square displacement is then given by $$\begin{aligned} \left<s^{2}\right>_{t}=2nDt-\frac{2}{3}\frac{n(n-1)}{R^{2}} \left(Dt\right)^{2}+\frac{4}{45}\frac{n(n-1)(n-3)}{R^{4}}\left(Dt\right)^{3}+\cdots\end{aligned}$$ This equation is the desired result (\[mean-square\]) for the particular case of the $n$-sphere. The constant curvature spaces ----------------------------- For the constant curvature manifolds, the Riemann curvature tensor is given by [@spivak] $$\begin{aligned} R_{abcd}=\frac{R_{g}}{d(d-1)}(g_{ac}g_{bd}-g_{ad}g_{bc}),\end{aligned}$$ where $R_{g}$ is a constant that can be either positive, negative or zero. It is known [@spivak] that the only three solutions for constant curvature are the d-dimensional sphere when $R_{g}>0$, the d-dimensional hyperboloid when $R_{g}<0$ and the euclidean space for $R_{g}=0$. For these cases, the mean-square geodesic distance is given by $$\begin{aligned} \left<s^{2}\right>_{t}=2dDt-\frac{2}{3}R_{g}\left(Dt\right)^{2}+\frac{4}{45}\frac{d-3}{d(d-1)}R^{2}_{g}\left(Dt\right)^{3}+\cdots . \label{MSC}\end{aligned}$$ For theses cases, the curvature effects depends on the sign of the curvature $R_{g}$. For the hyperbolic spaces, the MSD deviates from the planar result by an increasing monotonic function (in time) whereas for the spherical space by a decreasing monotonic function. In other words, the geometry of the space affects the brownian motion in such a way that the particle‘s diffusion is accelerated for $R_{g}<0$ or decelerated for $R_{g}>0$; see figure (\[fig1\]). These same results were also observed in the two-dimensional cases in [@Tomoyoshi]. ![[The time evolution of the MSD for the spherical $R_{g}>0$, hyperbolic $R_{g}<0$ and euclidean spaces $R_{g}=0$. ]{}[]{data-label="fig1"}](BMConstCurv.pdf){width="8cm"} The two-dimensional case ------------------------ ![[A Torus divided by three regions, where $K>0$ , $K<0$ (i.e the area gridded) and $K=0$ (parallels circular curves where $\theta=\pi/2$ and $\theta=3\pi/2$). ]{}[]{data-label="Figure"}](BMonTorus2.pdf){width="8cm"} ![[The time evolution of $\left<s^{2}\right>_{t}$ versus $t$ and $\theta$, where the area gridded represents the region where $K<0$. ]{}[]{data-label="Figure2"}](BMonTorusII.pdf){width="8cm"} The two-dimensional case is the most relevant one for the diffusion on biological membranes. For this case, the Riemann curvature tensor is given by $$\begin{aligned} R_{abcd}=K\left(g_{ac}g_{bd}-g_{ad}g_{bc}\right),\end{aligned}$$ where $K=R_{g}/2$ is the Gaussian curvature of the surface [@Nakahara]. Note that $K$ is not necessarily a constant. The mean-square geodesic distance, at the short-time regime, for an arbitrary surface of Gaussian curvature $K$ is given by $$\begin{aligned} \left<s^{2}\right>_{t}=4Dt-\frac{4}{3}K\left(Dt\right)^{2}-\frac{8}{15}\left[\frac{1}{3}K^{2} +2\Delta_{g}K\right]\left(Dt\right)^{3}+\cdot\cdot\cdot . \label{MSD2}\end{aligned}$$ In what follows, we derive few consequences of last equation when the diffusion takes place on a Torus. For this case, the metric is given by $ds^{2}=r^{2}d\theta^{2}+(a+r\cos\theta)^{2}d\varphi^{2}$ (with $0<\theta, \varphi<2\pi$), where $a$ and $r$ are the two radii. The Gaussian curvature is given by $$\begin{aligned} K=\frac{\cos\theta}{r(a+r\cos\theta)}.\end{aligned}$$ At figure (\[Figure\]) three regions on the Torus are shown. These regions are defined by the conditions $K>0$, $K<0$ and $K=0$, respectively[^4]. It is shown at figure (\[Figure2\]) how the time evolution of the mean-square displacement depends on the initial position of the particle on the Torus. Clearly, the particle‘s diffusion decelerates faster at the region where $K<0$ than the way how the particle‘s diffusion proceeds at the region where $K>0$. On the contrary, at the parallels ($K=0$) the particle‘s diffusion accelerates; in fact, the MSD in the parallels is given by $$\begin{aligned} \left<s^{2}\right>_{t}=4Dt+\frac{16}{15 a^2 r^2 }(Dt)^3.\end{aligned}$$ In general, the biological membranes have a wide range of morphologies; from discoidal shapes to catenoidal ones [@Lipowsky]. Also, we can consider wavy surface like membranes with microvilli [@Aizenbud], elliptic paraboloid, hyperbolic paraboloid [@Tomoyoshi] and periodic nodal surfaces [@Holyst], where the diffusion is clearly affected by the geometry. The curvature effects on the diffusion on these surface can be quantified using equation (\[MSD2\]). Conclusions and perspectives ============================ In this paper, we have studied the brownian motion over a d-dimensional curved manifold. The geodesic distance is used as the displacement of a particle diffusing on the space. Using a Riemann normal frame we have derived a general formula for the mean-square displacement at the short-time regime. This formula reproduces the standard result for very short-times [@Einstein] and it is given by $O(d)$ isometric covariant terms depending upon the Riemann curvature tensor. In particular, we have explored the diffusion over constant curvature manifolds, where we have shown that diffusion accelerates for the hyperbolic case, whereas it decelerates for the spherical one. In addition, we have discussed the two dimensional case, as it is relevant for the diffusion on biological membranes. In particular, the behaviour of the diffusion on a Torus can be classified according to the region (defined by $K<0$, $K>0$ or $K=0$) where the particle starts to move. As we have mentioned, there are several physical observables that describe the motion of a particle diffusing on a curved space. These observables quantify the curvature effects in different ways. For instance, as it was discussed at [@Faraudo] the mean-square geodesic distance has null curvature effects for the diffusion on developable surfaces, whereas using the parametrization displacement [@Holyst] it is clear that there are curvature effects. Indeed, different physical observables have different manifestations of the same phenomenon. It is then a rather natural question to ask what is the geometrical and physical content of these physical observables for the brownian motion on curved manifolds. In a different direction, we can explore the large-time regime for the mean-values for the case of compact manifolds. In this particular case, the expectation values will have a bounded above limit as a consequence of the compactness of the manifold [@Grygorian]. Acknowledgments =============== The author would like to thank Ramon Castañeda Priego and Sendic Estrada Jiménez for many valuable discussions. The support by PROMEP/103á5/08/3291 grant is acknowledged. Appendix A. Riemann normal coordinates ====================================== The gauge condition -------------------- As it is mentioned at [@Muller] the second condition of (\[defRNC\]) is equivalent to the gauge condition (\[normality\]). In order to prove this we use the Christoffel symbols, therefore we have $$\begin{aligned} y^{a}y^{b}\Gamma^{c}_{~ab}(y)=\frac{1}{2}g^{ce}\left(2y^{a}y^{a}\partial_{a}g_{eb}-y^{a}y^{b}\partial_{e}g_{ab}\right). \label{proof}\end{aligned}$$ Now, the first derivative on the second condition of (\[defRNC\]) is $g_{ab}+y^{e}\partial_{a}g_{eb}=\delta_{ab}$, then $y^{a}y^{b}\partial_{a}g_{eb}=0$. Similarly, the second term of (\[proof\]) vanishes. An affine connection expansion ------------------------------ In order to derive (\[MetricRNC\]) we follow the same procedure presented at [@Denjoe]. Let $\omega$ be a matrix one-form connection given by $\omega=\omega_{a}dx^{a}$, where the matrix one-form component $(\omega_{a})^{b}_{c}=\Gamma^{b}_{~ ac}$ is given by the Christoffel symbols. Let us consider the Lie derivative $\mathcal{L}_{X}$ along the radial vector $X=y\cdot\partial$ acting on $\omega$, $$\begin{aligned} \mathcal{L}_{X}\omega=\left[(1+y\cdot\partial)\omega_{a}\right]dy^{a}, \label{eq1}\end{aligned}$$ where $y^{a}$ are the Riemann normal coordinates. The Lie derivative can be written as $$\begin{aligned} \mathcal{L}_{X}=di_{X}+i_{X}d, \label{identity}\end{aligned}$$ where $d$ and $i_{X}$ are the exterior and interior derivative, respectively [@Nakahara]. Observe, that the gauge condition (\[normality\]) can be written as $i_{X}\omega=0$. Therefore using the Cartan structure equation for the curvature, $R=d\omega+\omega\wedge\omega$, is easy to get $$\begin{aligned} \mathcal{L}_{X}\omega&=&i_{X}R-i_{X}\left(\omega\wedge\omega\right)\nonumber\\ &=&i_{X}R\nonumber\\ &=&y^{a}{\mathcal R}_{ab}dy^{b}. \label{eq2}\end{aligned}$$ Equating (\[eq1\]) and (\[eq2\]), we get the condition $(1+y\cdot\partial)\omega_{a}=y^{a}{\mathcal{R}}_{ab}$, where $({\mathcal R}_{ab})^{c}_{~d}=R^{c}_{~dab}$ is the Riemann curvature tensor . A Taylor expansion centered at $0$ in both sides of this condition gives $$\begin{aligned} \Gamma^{a}_{~bc}=\sum^{\infty}_{k=0}\frac{\left(y\cdot\partial\right)^{k}}{k!\left(k+2\right)}y^{d}R^{a}_{~cdb},\end{aligned}$$ where Riemann curvature tensor is evaluated at $0$. A vielbein expansion -------------------- As it is point out in [@Denjoe], it is sufficient to compute the Taylor expansion of the vielbein $\theta=\theta_{a}dx^{a}$ in order to find (\[MetricRNC\]). $\theta$ satisfy the free torsion condition $d\theta+\omega\wedge\theta=0$ and the metric can be re-written in terms of the vielbein as $g_{ab}=\theta^{i}_{~a}\theta^{j}_{~b}\delta_{ij}$. Using (\[identity\]) and the gauge condition, the Lie derivative on $\theta$ is given by $$\begin{aligned} \mathcal{L}_{X}\theta=(i_{X}\theta)\omega+di_{X}\theta. \label{1Lie}\end{aligned}$$ As $\theta$ is one-form, $i_{X}\theta$ is scalar. Therefore the second Lie derivative of last expression is $$\begin{aligned} \mathcal{L}_{X}\mathcal{L}_{X}\theta=X\left[i_{X}\theta\right]\omega+\left(i_{X}\theta\right)\mathcal{L}_{X}\omega+\mathcal{L}_{X}di_{X}\theta . \label{2Lie} \end{aligned}$$ The additional gauge condition $y^{a}g_{ab}\left(y\right)=y^{a}\delta_{ab}$ is equivalent to choose $y^{a}\theta^{i}_{a}\left(y\right)=y^{a}\delta^{i}_{a}$, then $X\left[i_{X}\theta\right]=i_{X}\theta$ and $\mathcal{L}_{X}di_{X}\theta=di_{X}\theta$. Combining equation (\[1Lie\]) with (\[2Lie\]), we get $\mathcal{L}_{X}\left(\mathcal{L}_{X}-1\right)\theta=\left(i_{X}\theta\right)\mathcal{L}_{X}\omega$. Now using equation (\[eq2\]), we find $$\begin{aligned} \mathcal{L}_{X}\left(\mathcal{L}_{X}-1\right)\theta=\left(i_{X}\theta\right)i_{X}R.\end{aligned}$$ The condition for the vielbein is then given by $ \left(y\cdot\partial+1\right)(y\cdot\partial)\theta^{i}_{a}=y^{j}y^{b}R^{i}_{~jbc}\theta^{c}_{a}$ . By Taylor expansion centered at $0$ in both sides of this condition we get $$\begin{aligned} \theta^{i}_{a}&=&\delta^{i}_{a}+\sum^{\infty}_{k=2}\frac{\left(y\cdot\partial\right)^{k-2}\left[R^{i}_{~\cdot\cdot c}\theta^{c}_{a}\right]}{k\left(k+1\right)\left(k-2\right)!},\nonumber\\ y\cdot\partial \theta^{i}_{a}&=&0\end{aligned}$$ where $R^{i}_{~\cdot\cdot c}\equiv y^{a}y^{b}R^{i}_{abc}$. This system of equations can be solved iteratively and they give the following expression for the vielbein until fourth order in $y^{a}$ $$\begin{aligned} \theta^{i}_{~a}=\delta^{i}_{a}+\frac{1}{3}R^{i}_{~\cdot\cdot a}+\frac{1}{12}y\cdot \partial R^{i}_{~\cdot\cdot a}+\frac{1}{40}\left(y\cdot \partial\right)^{2}R^{i}_{~\cdot\cdot a}+\frac{1}{120}R^{i}_{~\cdot\cdot c}R^{c}_{~\cdot\cdot a}+\cdots\end{aligned}$$ The metric (\[MetricRNC\]) is then obtained using $g_{ab}=\theta^{i}_{~a}\theta^{j}_{~b}\delta_{ij}$. Determinant of the metric ------------------------- A Taylor expansion of the determinant of the metric $g\equiv \det g_{ab}$ can be obtained using the identity $\log \det g_{ab}=tr\log g_{ab}$. Since the metric (\[MetricRNC\]) can be written as $g_{ab}=\delta_{ab}+\Lambda_{ab}\left(y\right)$, its logarithm is given by $\log(1+\Lambda)\approx \Lambda-\frac{1}{2}\Lambda^{2}$. The trace of this term is then given by $$\begin{aligned} -\log g&=&\frac{1}{3}R_{ab}\left(0\right)y^{a}y^{b}+\frac{1}{6}\nabla_{e}R_{ab}\left(0\right)y^{a}y^{b}y^{e}+\frac{1}{90}R^{a}_{cdf}\left(0\right)R^{f}_{gha}\left(0\right)y^{c}y^{d}y^{g}y^{h}\nonumber\\&+&\frac{1}{20}\nabla_{e}\nabla_{f}R_{ab}\left(0\right)y^{a}y^{b}y^{e}y^{f}+\cdot\cdot\cdot .\nonumber\\\end{aligned}$$ Geometric $G_{3}$ factor ------------------------ In order to get the geometric factor $G_{3}\equiv \Delta_{g}^{3}s^{2}$, we need at least the Taylor expansion of $\Delta^{2}_{g}s^{2}$ until second order $O(y^{2})$. By straighforward calculation this expansion is given by $$\begin{aligned} \Delta^{2}_{g}s^{2}&=&-\frac{4}{3}R_{g}-y^{a}\partial_{a}R_{g}-2y^{a}\nabla^{b}R_{ab}+ \frac{4}{3}R^{a}_{c}R_{ad}y^{c}y^{d}+\frac{4}{9}\left.R^{a}_{~cd}\right.^{b}R_{ab}y^{c}y^{d}-\frac{4}{45}\left\{-2R^{a}_{f}R^{f}_{~gha}y^{g}y^{h}\right.\nonumber\\&+&\left. R^{ag}_{~~df}R^{f}_{~gha}y^{d}y^{h}+R^{ah}_{~~df}R^{f}_{~gha}y^{d}y^{g}+R^{a}_{~cgf}R^{fg}_{~~ha}y^{c}y^{h}+R_{ac}~^{hf}R_{fgh}~^{a}y^{c}y^{g}\right\} -\frac{2}{5}\left(\nabla^{2}R_{cd}\right)y^{c}y^{d}\nonumber\\&-&\frac{2}{5}\left(\nabla_{e}\nabla_{f}R_{g}\right)y^{e}y^{f}-\frac{8}{5}\left(\nabla^{e}\nabla_{f}R_{ed}\right)y^{f}y^{d}+O(y^{3}).\end{aligned}$$ Appendix B. $\Delta_{g}$ on a scalar function ============================================= Let $\Omega:\mathbb{M}\to \mathbb{R}$ be a scalar differentiable function on the manifold. Let us take a Riemann normal system of coordinates centered at $y^{a}=0$, then the Laplace-Beltrami operator acting on $\Omega$ is given by the second derivative of $\Omega$, i.e., $$\begin{aligned} \left.\Delta_{g}\Omega\right\|_{y=0}=\left.\partial^{2}\Omega\right\|_{y=0}.\end{aligned}$$ In order to show this result we split the inverse metric and the square root of the metric determinant as $$\begin{aligned} g^{ab}&=&\delta^{ab}+\Lambda^{ab}\left(y\right)\nonumber\\ \sqrt{g}&=&1+\lambda\left(y\right),\end{aligned}$$ where $\Lambda^{ab}\left(y\right)$ and $\lambda\left(y\right)$ can be found in terms of covariant derivative of the Riemann curvature as we have seen above. This functions satisfy the following properties: $\Lambda^{ab}\left(0\right)=0$, $\partial_{c}\Lambda^{ab}\left(0\right)=0$, $\lambda\left(0\right)=0$, and $\partial_{c}\lambda\left(0\right)=0$. Thus the action of $\Delta_{g}$ on $\Omega$ can be written as $$\begin{aligned} \Delta_{g}\Omega&=&\left\{\partial^{2}\Omega+\partial_{a}\lambda\left(y\right)\partial^{a}\Omega+\lambda\left(y\right)\partial^{2}\Omega+\partial_{a}\Lambda^{ab}\left(y\right)\partial_{b}\Omega+\Lambda^{ab}\left(y\right)\partial_{a}\partial_{b}\Omega\right. \nonumber\\ &+&\left.\left.\partial_{a}\lambda\left(y\right)\Lambda^{ab}\left(y\right)\partial_{b}\Omega+\lambda\left(y\right)\partial_{a}\Lambda^{ab}\left(y\right)\partial_{b}\Omega+\lambda\left(y\right)\Lambda^{ab}\left(y\right)\partial_{a}\partial_{b}\Omega\right\}\right\|_{y=0}\nonumber\\\end{aligned}$$ Then after the substitution $y=0$, it is obtained the desired result. [99]{} Jan K.G. Dhont, An Introduction to Dynamics of Colloids, 1996 Elsevier Science. Bertrand Duplantier,[Phys. Rev. Lett.]{} [81]{} (1998). P.F.F. Almeida and W.L.C. Vaz. Handbook of Biological Physics (Elsevier Science, Amsterdam, 1995), Vol. 1, Chap. 6. 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IoP Publishing (2003). Bryce S. DeWitt, Dynamical Theory of Groups and Fields, Published by Gordon Breach (1965) (New York). Alexander GrigorÕyan, Estimates of heat kernels on Riemannian manifolds, Notes (website). Spivak M. A., A Comprehensive Introduction to Differential Geometry. Vol. Four, Third Edition, Publish or Perish, Inc. (1999). Reinhard Lipowsky, Nature, Vol. 349, p. 475 (1991). [^1]: E-mail: pcastrov@unach.mx [^2]: $\Gamma^{a}_{~bc}=\frac{1}{2}g^{ae}\left(\partial_{b}g_{ec}+\partial_{c}g_{eb}-\partial_{e}g_{bc}\right)$ [^3]: $T^{a}_{ ~bc}=\Gamma^{a}_{~bc}-\Gamma^{a}_{~cb}$, and $R^{a}_{~bcd}=\partial_{c}\Gamma^{a}_{~db}-\partial_{d}\Gamma^{a}_{~cb}+\Gamma^{e}_{~db}\Gamma^{a}_{~ce}-\Gamma^{e}_{~cb}\Gamma^{a}_{~de}$. [^4]: Clearly, the Gaussian curvature vanishes along the parallels ($\theta=\pi/2$ and $\theta=3\pi/2$). In the region given by $\pi/2<\theta<3\pi/2$, $K$ is negative whereas in the one given by $0<\theta<\pi/2$ or $\pi/2<\theta<3\pi/2$, $K$ is positive.
--- author: - \| by J. Klusoň\ Department of Theoretical Physics and Astrophysics\ Faculty of Science, Masaryk University\ Kotlářská 2, 611 37, Brno\ Czech Republic\ E-mail: title: Note About Tachyon Kink In Nontrivial Background --- Introduction {#first} ============ Study of various aspects of the tachyon dynamics on a non-BPS Dp-brane in type IIA or IIB theories has led to some understanding of the tachyon dynamics near the tachyon vacuum [^1]. The tachyon effective action (\[acg\]), describing the dynamics of the tachyon field on a non-BPS Dp-brane of type IIA and IIB theory was proposed in [@Sen:1999md; @Bergshoeff:2000dq; @Garousi:2000tr; @Kluson:2000iy]. It was argued in many papers that the tachyon effective action (\[acg\]) gives a good description of the system under condition that tachyon is large and the second and higher derivatives of the tachyon are small [^2]. A kink solution in the full tachyon effective field theory, which is supposed to describe a BPS D(p-1)-brane was also constructed in [@Sen:2003tm; @Kim:2005he; @Kim:2003in; @Kim:2003ma; @Banerjee:2004cw; @Bazeia:2004vc; @Copeland:2003df; @Brax:2003rs]. A kink solution, that by definition interpolates between the vacuua at $T=\pm \infty$ has to pass through $0$. Then we could expect that higher derivative corrections to the tachyon effective action will be needed to provide a good description of the D(p-1)-brane as a kink solution. This issue was carefully analysed in paper [@Sen:2003tm] where it was shown that the energy density of the kink in the effective field theory is localised on codimension one surface as in the case of a BPS D(p-1)-brane. It was then also shown that the worldvolume theory of the kink solution is also given by the Dirac-Born-Infeld (DBI) action on a BPS D(p-1)-brane. Thus result shows that the kink solution of the effective field theory does provide a good description of the D(p-1)-brane even without taking into account higher derivative corrections. In other words, the tachyon effective action reproduces the low energy effective action on the world-volume of the soliton without any correction terms. Since these results are very impressive it would be certainly useful to test the effective field theory description of the tachyon condensation in other, more general situations. In fact, since the DBI action describes the low energy dynamics of the BPS Dp-brane in general curved background we can ask the question whether we can construct the tachyon kink on the worldvolume of a non-BPS Dp-brane embedded in curved background and whether this kink has the interpretation as a lower dimensional D(p-1)-brane [^3]. To answer this question we begin with common presumption that the tachyon effective action for Dp-brane (\[acg\]) can be applied for the description of the tachyon dynamics in the nontrivial background [^4]as well. Then we will study the equation of motion for the tachyon and for the modes that parametrise the embedding of the unstable Dp-brane in given spacetime. We will solve these equations with the field configuration similar to the ansatz that was given in [@Sen:2003tm]. We will show that this ansatz solves the equation of motion for tachyon on condition that the mode $t$ that characterises the core of the kink (The precise meaning of this claim will be given bellow.) satisfies the equation of motion of the scalar field that describes the embedding of D(p-1)-brane in given background. This result shows that the spatial dependent tachyon condensation leads to the emergence of a D(p-1)-brane where the scalar modes that propagate on its worldvolume solve the equation of motion that arise from the DBI action for D(p-1)-brane that is moving in the same background. The structure of this paper is as follows. In the next section (\[second\]) we will analyse the equation of motion for non-BPS Dp-brane in curved background. We will find the spatial dependent tachyon solution that has interpretation as a lower dimensional D(p-1)-brane whose dynamics is governed by DBI action. In section (\[third\]) we will study some examples of the nontrivial background. The first one corresponds to the stack of $N$ NS5-branes and the second one corresponds to the background generated by the collection of $N$ coincident Dk-branes. Finally, in conclusion (\[fourth\]) we will outline our results and suggest possible extension of this work. Non-BPS Dp-brane in general background {#second} ====================================== The starting point for the analysis of the dynamics of a non-BPS Dp-brane in general background is the Dirac-Born-Infeld like tachyon effective action [@Sen:1999md; @Bergshoeff:2000dq; @Garousi:2000tr; @Kluson:2000iy] [^5] $$\begin{aligned} \label{acg} S=-\int d^{p+1}\xi e^{-\Phi}V(T)\sqrt{-\det \bA} \ , \nonumber \\ \bA_{\mu\nu}=g_{MN} \partial_\mu Y^M\partial_\nu Y^N+ F_{\mu\nu}+ \partial_\mu T\partial_\nu T \ , \mu \ , \nu=0,\dots, p \ , \nonumber \\ F_{\mu\nu}=\partial_\mu A_\nu -\partial_\nu A_\mu \ , \nonumber \\\end{aligned}$$ where $A_\mu \ , \mu,\nu=0,\dots,p$ and $ Y^{M,N} \ , M,N=0,\dots,9$ are gauge and the transverse scalar fields on the worldvolume of the non-BPS Dp-brane and $T$ is the tachyon field. Since in this paper we will restrict ourselves to the situations when the gauge fields can be consistently taken to zero we will not write $F_{\mu\nu}$ anymore. $V(T)$ is the tachyon potential that is symmetric under $T\rightarrow -T$ has maximum at $T=0$ equal to the tension of a non-BPS Dp-brane $\tau_p$ and has its minimum at $T=\pm \infty$ where it vanishes. Using the worldvolume diffeomorphism invariance we can presume that the worldvolume coordinates $\xi^{\mu}$ are equal to the spacetime coordinates $y^\mu$. Explicitly, we can write $$Y^{\mu}=\xi^{\mu} \ .$$ Then the induced metric takes the form [^6] $$\gamma_{\mu\nu}\equiv g_{MN}\partial_\mu X^M \partial_\nu X^N= g_{\mu\nu}+g_{mn} \partial_\mu Y^m\partial_\nu Y^n \ ,$$ where $Y^m \ , m,n=p+1,\dots,9$ parametrise the embedding of Dp-brane in a space transverse to its worldvolume. We should also mention that generally the metric components and dilaton are functions of $\xi^\mu$ and $Y^m$: $$g_{MN}=g_{MN}(\xi^\mu,Y^m) \ , \Phi=\Phi(\xi^\mu,Y^m) \ .$$ Now the equation of motion for $T$ and $Y^m$ that follow from (\[acg\]) take the form $$\label{eqtg} \frac{\delta V}{\delta T} e^{-\Phi}\sqrt{-\det \bA} -\partial_\mu\left[e^{-\Phi} V\sqrt{-\det \bA}\partial_\nu T \bAi^{\nu\mu}\right]=0 $$ and $$\begin{aligned} \label{eqym} \frac{\delta [e^{-\Phi}]} {\delta Y^m}V\sqrt{-\det \bA}+ \frac{e^{-\Phi}}{2}V\left[ \frac{\delta g_{\mu\nu}} {\delta Y^m} +\frac{\delta g_{np}} {\delta Y^m}\partial_\mu Y^n\partial_\nu Y^p\right] \bAi^{\nu\mu} \sqrt{-\det\bA}- \nonumber \\ -\partial_\mu\left[ e^{-\Phi}Vg_{mn} \partial_\nu Y^n \bAi^{\nu\mu} \sqrt{-\det\bA}\right]=0 \ . \nonumber \\\end{aligned}$$ Our goal is to find the solution of these equations of motions that can be interpreted as a lower dimensional D(p-1)-brane. In order to obtain such a solution we will closely follow the paper by A. Sen [@Sen:2003tm]. Let us choice one particular worldvolume coordinate, say $\xi^p\equiv x$ and consider following ansatz for the fields living on the worldvolume of Dp-brane $$\label{ans} T=f(a(x-t(\xi))) \ , Y^m=Y^m(\xi) \ ,$$ where $\xi^\alpha \ , \alpha=0,\dots,p-1$ are coordinates tangential to the kink worldvolume. As in [@Sen:2003tm] we presume that $f(u)$ satisfies following properties $$f(-u)=-f(u) \ , f'(u)>0 \ , \forall u \ , f(\pm \infty)=\pm \infty $$ but is otherwise an arbitrary function of its argument $u$. $a$ is a constant that we shall take to $\infty$ in the end. In this limit we have $T=\infty$ for $x>t(\xi)$ and $T=-\infty$ for $x<t(\xi)$. Our goal is to check that the ansatz (\[ans\]) solves the equation of motion (\[eqtg\]) and (\[eqym\]). Firstly, using (\[ans\]) the matrix $\bA_{\mu\nu}$ takes the form $$\begin{aligned} \bA_{xx}=g_{xx}+a^2f'^2 \ , \bA_{x\alpha}=g_{x\alpha}-a^2f'^2 \partial_\alpha t \ , \nonumber \\ \bA_{\beta x}=g_{\beta x}- a^2f'^2\partial_\beta t \ , \bA_{\alpha\beta}=(a^2f'^2-g_{xx}) \partial_\alpha t\partial_\beta t+ \mat_{\alpha\beta} \ , \nonumber \\ \mat_{\alpha\beta}= g_{\alpha\beta}+g_{mn} \partial_\alpha Y^m\partial_\beta Y^n+\partial_\alpha t\partial_\beta t \ . \nonumber \\\end{aligned}$$ For next purposes it will be useful to know the form of the determinant $\det\bA$. Using the following identity $$\det\bA= \det (\bA_{\alpha\beta}- \bA_{\alpha x}\frac{1}{\bA_{xx}} \bA_{x\beta}) \det \bA_{xx}$$ we get $$\label{dea} \det\bA= a^2f'^2\det(\mat_{\alpha\beta}) +O(1/a) \ .$$ As a next step we should express $\bAi$ in terms of $\mat$. After some calculations we find $$\begin{aligned} \label{ina} \bAi^{xx}=(\mat^{-1})^{\alpha\beta} \partial_\alpha t\partial_\beta t \ , \bAi^{x\beta}= \partial_\alpha t (\mat^{-1}) ^{\alpha \beta} \ , \nonumber \\ \bAi^{\alpha x}= (\mat^{-1})^{\alpha\beta}\partial_\beta t \ , \bAi^{\alpha\beta}= (\mat^{-1})^{\alpha\beta} \ \nonumber \\\end{aligned}$$ up to corrections of order $\frac{1}{a^2}$. In the following calculation we will also need an exact relation $$\bAi^{\mu x}- \bAi^{\mu\alpha}\partial_\alpha t= \frac{1}{a^2f'^2} \left(\delta^{\mu}_x- \bAi^{xx}g_{xx}\right) \ .$$ Using this expression we can now write $$\begin{aligned} \partial_\mu \left[e^{-\Phi}V \sqrt{-\det\bA} \bAi^{\mu\nu}\partial_\nu T \right]=\partial_\mu \left[ e^{-\Phi}Vaf'\frac{1}{a^2f'^2} (\delta^{\mu}_x - \bAi^{\mu x}g_{xx})\sqrt{-\det\bA}\right] \ . \nonumber \\\end{aligned}$$ Following [@Sen:2003tm] we can now argue that due to the explicit factor of $a^2f'^2$ in the denominator the leading contribution from individual terms in this expression is now of order $a$ and hence we can use the approximative results of $\det\bA$ and $\bAi$ given in (\[dea\]) and (\[ina\]) to analyse the equation of motion for tachyon $$\begin{aligned} \partial_\mu \left[ e^{-\Phi}V\sqrt{-\det\bA} af'\frac{1}{a^2f'^2} (\delta^{\mu}_x -\bAi^{\mu x}g_{xx})\right] -\nonumber \\ -e^{-\Phi}V' \sqrt{-\det\bA}= \nonumber \\ \partial_x\left[e^{-\Phi}V \sqrt{-\det \mat} (1-\mati^{\alpha\beta}g_{xx} \partial_\alpha t \partial_\beta t)\right]- \nonumber \\ -\partial_\alpha \left[e^{-\Phi} V\sqrt{-\det\mat} \mati^{\alpha\beta}g_{xx} \partial_\beta t\right] -af'e^{-\Phi}V' \sqrt{-\det\mat}= \nonumber \\ =V\left\{\partial_x \left[e^{-\Phi} \sqrt{-\det \mat} (1-\mati^{\alpha\beta}g_{xx}\partial_\alpha t \partial_\beta t )\right]- \right.\nonumber \\ -\left. \partial_\alpha \left[e^{-\Phi}\sqrt{-\det \mat } (\mati^{\alpha\beta} g_{xx}\partial_\beta t)\right] \right\}=0 \ . \nonumber \\\end{aligned}$$ This is important result that deserves deeper explanation. Firstly, from the form of the tachyon potential in the limit $a\rightarrow \infty$ we know that $V$ is equal to zero for $x-t(\xi)\neq 0$ while for $ x-t(\xi)=0$ we get $V(0)=\tau_p$. Then it is clear that the tachyon equation of motion is obeyed for $x-t(\xi)\neq 0$ while for $x=t(\xi)$ we should demand that the expression in the bracket should be equal to zero. In other words, we obtain following equation $$\begin{aligned} \label{eqtf} \frac{\delta e^{-\Phi}} {\delta x} \sqrt{-\det \mat} +\frac{e^{-\Phi}}{2} \left(\frac{\delta g_{\alpha\beta}} {\delta x}+ \frac{\delta g_{xx}} {\delta x}\partial_\alpha t \partial_\beta t+ \frac{\delta g_{mn}} {\delta x}\partial_\alpha Y^m \partial_\beta Y^n\right) \mati^{\beta\alpha} \sqrt{-\det\mat}- \nonumber \\ -\partial_\alpha \left[e^{-\Phi} \sqrt{-\det \mat} (\mati^{\alpha\beta} g_{xx}\partial_\beta t)\right] -\partial_x\left[e^{-\Phi} \sqrt{-\det\mat}\mati^{\alpha \beta}g_{xx}\right]\partial_\alpha t \partial_\beta t=0 \ . \nonumber \\\end{aligned}$$ We must stress that in (\[eqtf\]) we firstly perform the derivative with respect to $x$ and then we insert the value $x=t(\xi)$ back to the resulting equation of motion. For example, in the first term on the second line we should perform a derivative with respect to $\xi^\alpha$ with in mind that $x$ is an independent variable. After doing this we should everywhere replace $x$ with $t(\xi)$. Then the presence of the second term on the second line is crucial for an interpretation of $t(\xi)$ as an additional scalar field that parametrises the position of D(p-1)-brane in $x$ direction. Put differently, we expect that the tachyon condensation leads to an emergence of D(p-1)-brane that is localised at $x=t(\xi)$. For that reason we should compare the equation (\[eqtf\]) with the equation of motion for D(p-1)-brane embedded in the same background. As we know the dynamics of such a Dp-brane is governed by the DBI action $$\label{actDBI} S=-T_{p-1}\int d^{p}\xi e^{-\Phi} \sqrt{-\det \bA_{\alpha\beta}^{BPS}} \ ,$$ where the matrix $\bA^{BPS}_{\alpha\beta}$ takes the form $$\label{babps} \bA^{BPS}_{\alpha\beta}= g_{\alpha\beta}+ g_{xx}\partial_\alpha Y \partial_\beta Y+ g_{mn}\partial_\alpha Y^m \partial_\beta Y^n \ , m,n=p+1,\dots,9 \ ,$$ and where $T_{p-1}$ is the tension of BPS D(p-1)-brane. Recall that $T_p$ is is related to the tension of the non-BPS D(p-1)-brane $\tau_p$ as $\tau_{p-1}=\sqrt{2}T_{p-1}$. In (\[babps\]) we have chosen one particular transverse mode $Y$ in order to have a contact with the mode $t$ defined in the equation (\[ans\]). Finally, the scalar fields $Y^m$ have the same meaning as in the case of a non-BPS Dp-brane. Now the equations of motion that follow from (\[actDBI\]) take the form $$\begin{aligned} \label{Ymeq} \frac{\delta } {\delta Y^m} \left[ e^{-\Phi}\sqrt{- \det\bA^{BPS}_{\alpha\beta}}\right] -\partial_{\alpha} \left[e^{-\Phi} \sqrt{-\det \bA^{BPS}_{\alpha\beta}} \bAi_{BPS}^{\beta\alpha} g_{mn}\partial_\beta Y^n \right]=0 \ , \nonumber \\ \frac{\delta } {\delta Y} \left[ e^{-\Phi}\sqrt{- \det\bA^{BPS}_{\alpha\beta}}\right] -\partial_{\alpha} \left[e^{-\Phi} \sqrt{-\det \bA^{BPS}_{\alpha\beta}} \bAi_{BPS}^{\beta\alpha} g_{xx}\partial_\beta Y \right]=0 \ , \nonumber \\\end{aligned}$$ where the variation $\frac{\delta }{\delta Y^m} \ , \frac{\delta }{\delta Y}$ means the variation of the metric, dilaton with respect to $Y^M, \ Y$ respectively. Explicitly, the equation of motion for $Y$ can be written as $$\begin{aligned} \label{Yeq} \frac{\delta e^{-\Phi}} {\delta Y} \sqrt{-\det\bA}_{BPS}+ \nonumber \\ \frac{1}{2}e^{-\Phi} \left(\frac{\delta g_{\alpha\beta}} {\delta Y}+\frac{\delta g_{xx}} {\delta Y}\partial_\alpha Y \partial_\beta Y+ \frac{\delta g_{mn}} {\delta Y}\partial_\alpha Y^m \partial_\beta Y^n\right) \bAi^{\beta\alpha}_{BPS} \sqrt{-\det\bA}_{BPS}- \nonumber \\ -\partial_{\alpha} \left[e^{-\Phi} \sqrt{-\det \bA}_{BPS} \bAi_{BPS}^{\alpha\beta}g_{xx} \partial_\beta Y \right]=0 \ . \nonumber \\\end{aligned}$$ To see more clearly the relation with the equation (\[eqtf\]) note that the expression on the third line can be written as $$\begin{aligned} \partial_{\alpha} \left[e^{-\Phi} \sqrt{-\det \bA}_{BPS} \bAi_{BPS}^{\alpha\beta}g_{xx} \partial_\beta Y \right]=\nonumber \\ =\partial_\alpha \left[ e^{-\Phi(\xi,x)} \sqrt{-\det\bA_{BPS}(\xi,x)} \bAi^{\alpha\beta}_{BPS}(\xi,x) \partial_\beta Y\right] \nonumber \\ +\partial_x \left[e^{-\Phi(\xi,x)} \sqrt{-\det\bA_{BPS}(\xi,x)} \bAi^{\beta\alpha}_{BPS}(\xi,x) \right] \partial_\alpha Y\partial_\beta Y \ , \nonumber \\\end{aligned}$$ where on the second line the derivative with respect to $\xi^\alpha$ treats $x$ as an independent variable so that we firstly perform derivative with respect to $\xi^\alpha$ and then we replace $x$ with $Y$. We proceed in the same way with the expression on the third line where we firstly perform the variation with respect to $x$ and then we replace $x$ with $Y$. Now it is clear that this prescription is the same as the expression on the second line in (\[eqtf\]). More precisely, if we compare the equation (\[Yeq\]) with the the equation (\[eqtf\]) we see that these two expressions coincide when we identify $t$ with $Y$. In other words, the location of the tachyon kink is completely determined by field $t(\xi)$ that obeys the equation of motion of the embedding mode of D(p-1)-brane. We mean that this is very satisfactory result that shows that the Sen’s construction of the tachyon kink can be consistently performed in curved background as well. Now we will discuss the equation of motion for $Y^k$. Again, we will proceed as in [@Sen:2003tm]. We begin with the first term in (\[eqym\]) that for the ansatz (\[ans\]) takes the form $$\label{eqymp1} \frac{\delta e^{-\Phi}} {\delta Y^k} V\sqrt{-\det\bA}= af'V\frac{\delta e^{-\Phi}} {\delta Y^k}\sqrt{-\det\mat} \ .$$ In the same way we can show that the second term in (\[eqym\]) can be written as $$\begin{aligned} \label{eqymp2} e^{-\Phi}V\left[ \frac{\delta g_{\mu\nu}} {\delta Y^k} +\frac{\delta g_{mn}} {\delta Y^k} \partial_\mu Y^m\partial_\nu Y^n\right] \bAi^{\nu\mu} \sqrt{-\det\bA}= af'Ve^{-\Phi}\sqrt{-\det\mat}\times \nonumber\\ \left[\frac{\delta g_{xx}} {\delta Y^k} \mati^{\alpha\beta}\partial_\alpha t \partial_\beta t+( \frac{\delta g_{\alpha\beta}} {\delta Y^k}+ \frac{g_{mn}}{\delta Y^k} \partial_\alpha Y^m\partial_\beta Y^n) \mati^{\alpha\beta}\right] \ . \nonumber \\\end{aligned}$$ Finally, the third term in (\[eqym\]) is equal to $$\begin{aligned} \label{eqymp3} -\partial_\mu\left[ e^{-\Phi}Vg_{km}\partial_\nu Y^m \bAi^{\nu\mu} \sqrt{-\det\bA}\right]=\nonumber \\ -\partial_x\left[Ve^{-\Phi} g_{km}\partial_\alpha Y^m \bAi^{\alpha x}\sqrt{-\det\bA}\right] -\partial_\alpha \left[Ve^{-\Phi}g_{km} \partial_\beta Y^m \bAi^{\alpha\beta} \sqrt{-\det\bA}\right]= \nonumber \\ =-af'\partial_x \left[Ve^{-\Phi}g_{km} \partial_\alpha Y^m \mati^{\alpha\beta} \partial_\beta t \sqrt{-\det\mat}\right]- \nonumber\\ -af'\partial_\alpha \left[Ve^{-\Phi} g_{km}\partial_\beta Y^m \mati^{\alpha\beta} \sqrt{-\det\mat} \right]= \nonumber \\ -af'V\left(\partial_\alpha\left[e^{-\Phi} g_{km}\partial_\beta Y^m \mati^{\alpha \beta} \sqrt{-\det\mat}\right] +\partial_x\left[ e^{-\Phi}g_{km}\partial_\alpha Y^m \mati^{\alpha\beta}\partial_\beta t \sqrt{-\det\mat}\right] \right) \nonumber \\\end{aligned}$$ using the fact that $\partial_\alpha f= -af'\partial_\alpha t$. Then collecting (\[eqymp1\]), (\[eqymp2\]) and (\[eqymp3\]) together we obtain $$\begin{aligned} \label{Vym} V\left\{ \frac{\delta e^{-\Phi}} {\delta Y^k} \sqrt{-\det\mat} +\frac{1}{2}e^{-\Phi}\sqrt{-\det\mat}\times \right.\nonumber\\ \left.\times\left[\frac{\delta g_{xx}}{\delta Y^k} \partial_\alpha t \partial_\beta t+ \frac{\delta g_{\alpha\beta}} {\delta Y^k}+\frac{g_{mn}}{\delta Y^k} \partial_\alpha Y^m\partial_\beta Y^n \right] \mati^{\alpha\beta}\right. \nonumber \\ \left.-\partial_\alpha\left[e^{-\Phi} g_{km}\partial_\beta Y^m \mati^{\alpha \beta} \sqrt{-\det\mat}\right] -\partial_x\left[ e^{-\Phi}g_{km}\partial_\alpha Y^m \mati^{\alpha\beta} \sqrt{-\det\mat}\right]\partial_\beta t\right\}=0 \ . \nonumber \\\end{aligned}$$ Again it is easy to see that for $x\neq t(\xi)$ the potential vanishes for $a\rightarrow \infty$ while for $x=t(\xi)$ we have $V(0)=\tau_p$. Then in order to obey the equation of motion the expression in the bracket should be equal to zero for $x=t(\xi)$. Note also that in the second expression on the last line in (\[Vym\]) we firstly perform a derivative with respect to $x$ and then we replace $x$ with $t(\xi)$. In other words, we can rewrite the last line in (\[Vym\]) into the form $$\partial_\alpha \left[e^{-\Phi(t(\xi))} \sqrt{-\det\mat(t(\xi))} \mati^{\alpha\beta}(t(\xi)) g_{km}(t(\xi)) \partial_\beta\beta Y^m\right] \ ,$$ where we have explicitly stressed the dependence of the action on the mode $t(\xi)$ that replaces in the action the dependence on $x$. This result again supports the claim that we should identify $t(\xi)$ with an additional embedding coordinate of the D(p-1)-brane. Then by comparing the expression in the bracket in (\[Vym\]) with the equation of motion for $Y^m$ given in (\[Ymeq\]) we see that these two expressions coincide. In summary, we have shown that the tachyon kink solution on a non-BPS Dp-brane in nontrivial background can be identified as a lower dimensional D(p-1)-brane that is localised at the core of the kink. We have also shown that the dynamics of this D(p-1)-brane is governed by DBI action. Stress energy tensor -------------------- Further support for the interpretation of the tachyon kink as a lower dimensional D(p-1)-brane can be obtained from the analysis of the stress energy tensor for the non-BPS Dp-brane. In order to find its form recall that we can write the action (\[acg\]) as $$\label{dactem} S_{p}=-\int d^{10}yd^{(p+1)} \xi\delta (Y^M(\xi)-y^M)e^{-\Phi}V(T) \sqrt{-\det \bA} \ ,$$ where $$\bA_{\mu\nu} =G_{MN}\partial_\mu Y^M\partial_{\nu}Y^N+ \partial_{\mu}T \partial_{\nu}T \ ,$$ and where $\xi^\mu \ , \mu=0,\dots,p$ are worldvolume coordinates on Dp-brane. The form of action (\[dactem\]) is useful for determining the stress energy tensor $T_{MN}(y)$ of an unstable D-brane. In fact, the stress energy tensor $T_{MN}(y)$ is defined as the variation of $S_p$ with respect to $g_{MN}(y)$ $$\begin{aligned} \label{TMNg} T_{MN}(y)=-2 \frac{\delta S_{p}}{ \sqrt{-g(y)}\delta g^{MN}(y)}=\nonumber \\ =-\int d^{(p+1)}\xi\frac{\delta(Y^M(\xi) -y^M)} {\sqrt{-g(y)}}e^{-\Phi}V g_{MK}g_{NL} \partial_{\mu}Y^K\partial_{\nu} Y^L(\bA^{-1})^{\nu\mu} \sqrt{-\det \bA} \ . \nonumber \\ \end{aligned}$$ The form of the stress energy tensor for gauge fixed Dp-brane action can be obtained from (\[TMNg\]) by imposing the condition $$Y^{\mu}=\xi^{\mu} \ , \mu=0,1,\dots,p \ .$$ Then the integration over $\xi^{\mu}$ swallows up the delta function $\delta (y^{\mu}-Y^{\mu}(\xi))= \delta(y^{\mu}-\xi^{\mu})$ so that the resulting stress energy tensor takes the form $$\begin{aligned} T_{mn}=-\frac{\delta(Y^m(\xi)- y^m)}{\sqrt{-g}} e^{-\Phi}Vg_{mm}\partial_\mu Y^mg_{nn}\partial_\nu Y^n \bAi^{\nu\mu}\sqrt{-\det\bA} \ , \nonumber \\ T_{\mu\nu}= -\frac{\delta(Y^m(\xi)-y^m)} {\sqrt{-g}}e^{-\Phi}Vg_{\mu\mu} g_{\nu\nu}\bAi^{\nu\mu} \sqrt{-\det\bA} \ , \nonumber \\ T_{\mu n}= -\frac{\delta(Y^m(\xi)- y^m)} {\sqrt{-g}}e^{-\Phi}Vg_{\mu\mu} g_{nn}\partial_\nu Y^n \bAi^{\nu\mu}\sqrt{-\det\bA} \ , \nonumber \\ T_{m\nu} =-\frac{\delta(Y^m(\xi)-y^m)} {\sqrt{-g}}e^{-\Phi}V g_{mm}\partial_\mu Y^mg_{\nu\nu}\bAi^{\nu\mu} \sqrt{-\det\bA} \ , \nonumber \\\end{aligned}$$ using the fact that the metric is diagonal. If we now insert the ansatz (\[ans\]) into these expressions we get $$\begin{aligned} \label{Tans} T_{mn}=-\frac{\delta(Y^m(\xi)-x^m)} {\sqrt{-g}} Vaf'e^{-\Phi}g_{mm}\partial_\alpha Y^mg_{nn}\partial _\beta Y^n \mati^{\alpha\beta}\sqrt{-\det\mat} \ ,\nonumber \\ T_{\alpha\beta}=- \frac{\delta(Y^m(\xi)-y^m)}{\sqrt{-g}} Vf'ae^{-\Phi} g_{\alpha\alpha} g_{\beta\beta}\mati^{\alpha\beta} \sqrt{-\det\mat} \ , \nonumber \\ T_{xx}= -\frac{\delta(Y^m-y^m)}{\sqrt{-g}} Vf'ae^{-\Phi} g_{xx}\partial_\alpha t\partial_\beta t \mati^{\alpha\beta}\sqrt{-\det\mat} \ , \nonumber \\ T_{x\alpha}=T_{\alpha x}=0 \ , \nonumber \\ T_{mx}=T_{xm}=- \frac{\delta (Y^m-y^m)}{ \sqrt{-g}}Vf'ae^{-\Phi}g_{mm} \partial_\alpha Y^m g_{xx} \partial_\beta t\mati^{\alpha\beta}\sqrt{-\det\mat} \ . \nonumber \\\end{aligned}$$ From now on the notation $Y^m(\xi), t(\xi)$ means that these fields are functions of the coordinates on the worldvolume of the kink $\xi^\alpha, \alpha=0,\dots,p-1$. According to [@Sen:2003tm] the components of the stress energy tensor of the lower dimensional D(p-1)-brane arise by integrating all $T_{MN}$ given above over the direction of the tachyon condensation that in our case is $x$. Now we should be more careful since metric components generally depend on $x$. Let us introduce the following notation for the components of the stress energy tensors (\[Tans\]) $$T_{MN}=V(f(a(t(\xi)-x)))af' \tilde{T}_{MN}(x) \ ,$$where we have explicitly stressed the dependence of $\tilde{T}_{MN}$ on $x$. If we now integrate $T_{MN}$ over $x$ we get $$T_{MN}^{kink}= \int_\infty^\infty dx V(f(a(x-t(\xi))))f'a \tilde{T}_{MN}(x)= \int dm V(m)\tilde{T}_{MN}\left( \frac{f^{-1}(m)}{a}+ t(\xi)\right) \ .$$ In the limit $a\rightarrow \infty$ the term proportional to $1/a$ goes to zero and we get that the components $\tilde{T}_{MN}$ are functions of $t(\xi)$ in place of $x$. Further, we can argue, following [@Sen:2003tm] that the exponential fall off in $V(m)$ implies that in the limit $a\rightarrow \infty$ the contribution to the stress energy tensor is localised at the point where $V$ is equal to $V(0)=\tau_{p}$ which happens for $x=t(\xi)$. In other words, when we presume that the tension of BPS D(p-1)-brane is given by the integral $$T_{p-1}= \int_{-\infty}^{\infty}dm V(m)$$ we obtain the result that the components of the stress energy tensor of the kink take the form $$\begin{aligned} \label{Tansf} T^{kink}_{mn}=- \frac{T_{p-1}\delta(Y^m(\xi)-y^m) \delta(t(\xi)-x)} {\sqrt{-g}} e^{-\Phi}g_{mm}\partial_\alpha Y^mg_{nn}\partial_\beta Y^n \mati^{\alpha\beta} \sqrt{-\det\mat} \ , \nonumber \\ T^{kink}_{\alpha\beta}=- \frac{T_{p-1}\delta(Y^m(\xi)-y^m) \delta(t(\xi)-x)}{\sqrt{-g}} e^{-\Phi}g_{\alpha\alpha} g_{\beta\beta}\mati^{\alpha\beta} \sqrt{-\det\mat} \ , \nonumber \\ T^{kink}_{xx}= -\frac{T_{p-1}\delta(Y^m-y^m) \delta(t(\xi)-x)}{\sqrt{-g}} e^{-\Phi} g_{xx}\partial_\alpha t\partial_\beta t \mati^{\alpha\beta} \sqrt{-\det\mat} \ , \nonumber \\ T_{x\alpha}^{kink}=T^{kink}_{\alpha x}=0 \ , \nonumber \\ T_{mx}^{kink}= T_{xm}^{kink}=- \frac{T_{p-1}\delta (Y^m-y^m) \delta(t(\xi)-x)}{ \sqrt{-g}}e^{-\Phi}g_{mm} \partial_\alpha Y^m g_{xx} \partial_\beta t\mati^{\alpha\beta} \sqrt{-\det\mat} \ , \nonumber \\ \end{aligned}$$ where it is understood that $g_{MN}$ and $\Phi$ are functions of $\xi^{\alpha} \ , Y^m(\xi) \ , t(\xi)$. In other words, the components of the stress energy tensors (\[Tansf\]) correspond to the components of the stress energy tensor of a D(p-1)-brane localised at the points $Y^m(\xi),t(\xi)$. Examples of the tachyon condensation on a non-BPS Dp-brane in nontrivial background {#third} =================================================================================== In this section we will briefly discuss some examples of the tachyon condensation on a non-BPS Dp-brane that is embedded in nontrivial backgrounds. NS5-brane background -------------------- As the first example we will consider the background corresponding to the stack of $N$ coincident NS5-branes $$\begin{aligned} \label{NSbac} ds^2=dx_{\mu}dx^{\mu}+H_{NS} dx^mdx^m \ , \nonumber \\ e^{2\Phi}= H_{NS} \ , \nonumber \\ H_{mnp}=-\epsilon^q_{mnp} \partial_q \Phi \ , \nonumber \\\end{aligned}$$ where the harmonic function $H_{NS}$ for $N$ coincident NS5-branes is equal to $$H_{NS}(y^m)=1+\frac{2\pi N}{y^my_m} \ ,$$ where $y^m \ , m=6,\dots,9$ label directions transverse to the worldvolume of NS5-branes. The most simple case occurs when Dp-brane is stretched in the direction parallel with the worldvolume of NS5-branes [^7]. Using (\[NSbac\]) it is then easy to determine the worldvolume metric $$\begin{aligned} g_{\mu\nu}= \eta_{\mu\nu} \ , g_{m_1n_1}=\delta_{m_1n_1} \ , m_1 \ , n_1 =p+1,\dots,5 \ , \nonumber \\ g_{m_2n_2}=H_{NS}\delta_{m_2n_2} \ , m_2,n_2=6,\dots,9 \ , \nonumber \\\end{aligned}$$ where now $H_{NS}$ is function of $Y^{m_2}$ $$H_{NS}=1+\frac{2\pi N}{Y^{m_2}Y_{m_2}} \ .$$ Thanks to the manifest $SO(p)$ symmetry of the worldvolume theory all spatial coordinates $\xi^i \ , i=1,\dots,p$ are equivalent. Then we choose the direction on which the tachyon depends to be $\xi^p=x$. Now it is clear that the spatial dependent tachyon condensation studied in previous section leads to the emergence of a D(p-1)-brane that is stretched in the $x^0,\dots,x^{p-1}$ directions and and which transverse position is determined by the worldvolume fields $t(\xi),Y^{m}(\xi)$. These fields also obey the equations of motion that arise from the DBI action for a D(p-1)-brane that moves in the background of $N$ NS5-branes. Another possibility occurs when we consider a non-BPS Dp-brane stretched in some of the transverse directions to the worldvolume of NS5-branes. More precisely, let us consider an unstable Dp-brane that is stretched in $x^0,x^1,\dots,x^k$ directions and in $x^6,\dots,x^{6+p-k}$ directions. Then the metric components that appear on the worldvolume of the Dp-brane take the form $$\begin{aligned} g_{\mu_1\nu_1}= \eta_{\mu_1\nu_1} \ , \mu_1\ , \nu_1=0,\dots,k \ , \nonumber \\ g_{\mu_2\nu_2}=H_{NS} \delta_{\mu_2\nu_2} \ , \mu_2 \ , \nu_2=6,\dots, (6+p-k) \ , \nonumber \\ g_{m_1n_1}= \delta_{m_1n_1} \ , m_1 , n_1=k+1,\dots,5 \ , \nonumber \\ g_{m_2n_2}=H_{NS}\delta_{m_2 n_2} \ , m_2 \ , n_2=(7+p-k)\ , \dots, 9 \ , \nonumber \\\end{aligned}$$ where the function $H_{NS}$ has the form $$H_{NS}=1+\frac{2\pi N}{(\xi^{\mu_2} \xi_{\mu_2}+Y^{m_2}Y_{m_2})^2} \ .$$ Now there are many possibilities how to construct lower dimensional D(p-1)-brane. If we perform the spatial dependent tachyon condensation on the worldvolume of the non-BPS Dp-brane where the tachyon $t(x)$ depends on coordinate from the set $\xi^1,\dots,\xi^k$ (again, we take $x=\xi^k$) we obtain D(p-1)-brane that is localised in $x^k$ direction and that is stretched in $x^0,\dots,x^{k-1}$ and $x^{6},\dots, x^{(6+p-k)}$ directions. It is important to stress that the resulting configuration of $N$ NS5-brane and BPS D(p-1)-brane is not in general stable. Rather the dynamics of the BPS D(p-1)-brane in the background of $N$ NS5-branes is governed the equation of motions (\[Yeq\]). To find stable configuration we should perform the same analysis as in [@Tseytlin:1996hi]. Another possibility is to consider the tachyon condensation in direction from the set $\xi^6,\dots, \xi^{6+p-k}$. Let us choose $x\equiv \xi^6$. Then it is clear that the tachyon condensation leads to the emergence of D(p-1)-brane stretched in $(x^0,\dots,x^k,x^7,\dots,x^{6+p-k})$ directions and where the scalar fields on its worldvolume $t(\xi), Y^{m_1}\ , Y^{m_2}$ describing embedding of this D(p-1)-brane in nontrivial background, obey the equations of motions that arise from DBI action for BPS D(p-1)-brane. Non-BPS Dp-brane in Dk-brane background --------------------------------------- The second example that we will consider in this paper, is the spatial dependent tachyon condensation on the worldvolume of a non-BPS Dp-brane that moves in the background of $N$ BPS Dk-branes. This background is characterised by following metric and dilaton in the form $$\begin{aligned} ds^2=H_k^{-1/2}\eta_{\alpha\beta} dx^\alpha dx^\beta + H_k^{1/2}\delta_{mn}dx^mdx^n \ , \nonumber \\ \alpha,\beta=0,\dots,k \ , m,n=k+1,\dots,9 \nonumber \\ e^{-2\Phi}=H_k^{\frac{k-3}{2}} \ , \nonumber \\\end{aligned}$$ where the harmonic function $H_p$ takes the form $$H_k=1+\frac{Ng_s(2\pi)^{\frac{7-k}{2}}}{(y^my_m)^{\frac{7-k}{2}}} \ ,$$ where $y^m \ , m=k+1,\dots,9$ label the directions transverse to the worldvolume of $N$ Dk–branes. There is again many possibilities how to put in a non-BPS Dp-brane in this background. As the first possibility let us consider a non-BPS Dp-brane that is stretched in $x^0,\dots,x^p$ directions and that is localised in $Y^{m_1}, m_1=p+1,\dots,k$ directions (parallel with the worldvolume of Dk-branes). This Dp-brane is also localised in $Y^{m_2} \ , m_2=k+1,\dots,9$ directions transverse to Dk-branes worldvolume. Now the metric components on its worldvolume take the form $$g_{\mu\nu}=H_k^{-1/2} \eta_{\mu\nu} \ , g_{m_1n_1}=H_k^{-1/2}\delta_{m_1n_1} \ , g_{m_2n_2}=H_k^{1/2}\delta_{m_2n_2} \ ,$$ where $H_k$ depends on $Y^{m_2}Y_{m_2}$. It is clear that the spatial dependent tachyon condensation (Let us choose $x$ that appears in the ansatz (\[ans\]) to be equal to $\xi^p$.) leads to an emergence of a D(p-1)-brane with the worldvolume fields $Y^{m_1} \ , Y^{m_2}$ as well as with the mode $t(\xi)$ that parametrises the location of D(p-1)-brane in $x^p$ direction. Another possibility occurs when we consider Dp-brane where some of its worldvolume directions are parallel with the worldvolume of Dk-branes and other ones are stretched in the directions transverse to Dk-brane. This situation can be described by following induced metric on the worldvolume of non-BPS Dp-brane: $$\begin{aligned} g_{\mu_1\nu_1}= H_k^{-1/2}\eta_{\mu_1\nu_1} \ , \mu_1\ , \nu_1=0,\dots, l \ , \nonumber \\ g_{\mu_2\nu_2}=H_{NS}^{1/2} \delta_{\mu_2\nu_2} \ , \mu_2 \ , \nu_2=k+1,\dots, (k+1+p-l) \ , \nonumber \\ g_{m_1n_1}=H_k^{-1/2} \delta_{m_1n_1} \ , m_1 , n_1=l+1,\dots,k \ , \nonumber \\ g_{m_2n_2}=H_{NS}^{1/2} \delta_{m_2 n_2} \ , m_2\ , n_2=(k+2+p-l)\ , \dots, 9 \ , \nonumber \\\end{aligned}$$ where the function $H_k$ is equal to $$H_k=1+\frac{Ng_s(2\pi)^{\frac{7-k}{2}}} {(\xi^{\mu_2}\xi_{\mu_2}+ Y^{m_2}Y_{m_2})^{\frac{7-k}{2}}} \ .$$ If now the tachyon depends on one of the coordinates from the set $\xi^1,\dots,\xi^l$, say $x=\xi^l$, we obtain a D(p-1)-brane that is localised in $x^l$ direction and that is stretched in $x^0,\dots,x^{l-1}$ and $x^{k+1}, \dots, x^{(k+1+p-l)}$ directions. The next possibility corresponds to the tachyon condensation in the direction transverse to Dk-branes, say $\xi^{k+1}\equiv x$. Following the general recipe given in previous section it is clear that this spatial dependent tachyon condensation leads to an emergence of a D(p-1)-brane that is stretched in $(x^0,\dots,x^l,x^{k+2},\dots, x^{k+2+p-l})$ directions and its positions in the transverse space are described with the worldvolume scalar fields $Y^{m_1}(\xi),Y^{m_2}(\xi)$ and also with $t(\xi)$ that parametrises the position of D(p-1)-brane in $x^{k+1}$ direction. It is also clear that these modes obey the equations of motions that follow from the DBI action for probe D(p-1)-brane in the background of $N$ Dk-branes. Note also that the resulting configuration of $N$ Dk-branes and D(p-1)-brane is not generally stable [@Tseytlin:1996hi]. Conclusion {#fourth} ========== This paper was devoted to the study of the spatial dependent tachyon condensation on the worldvolume of a non-BPS Dp-brane that is moving in nontrivial background. We have shown that this tachyon condensation leads to an emergence of the D(p-1)-brane that is moving in the same background and where the scalar mode that determines the location of the kink on a non-BPS Dp-brane worldvolume can be interpreted as a mode that describes the transverse position of D(p-1)-brane and that obeys the equation of motion that follows from DBI action for D(p-1)-brane. We hope that this result is a nice example of the efficiently of the effective field theory description of the tachyon condensation and it also gives strong support for the form of the Dirac-Born-Infeld form of the tachyon effective action (\[acg\]). The extension of this paper is obvious. First of all we would like to study the tachyon condensation when we take into account nontrivial NS $B$ field and also nontrivial Ramond-Ramond field. It would be also interesting to study the tachyon condensation on the supersymmetric form of the non-BPS Dp-brane action. We hope to return to these problems in future. [ Acknowledgement]{} This work was supported by the Czech Ministry of Education under Contract No. MSM 0021622409. [20]{} A. Sen, “Non-BPS states and branes in string theory,” arXiv:hep-th/9904207. K. Ohmori, “A review on tachyon condensation in open string field theories,” arXiv:hep-th/0102085. W. Taylor, “Lectures on D-branes, tachyon condensation, and string field theory,” arXiv:hep-th/0301094. W. Taylor and B. Zwiebach, “D-branes, tachyons, and string field theory,” arXiv:hep-th/0311017. A. Sen, “Tachyon dynamics in open string theory,” arXiv:hep-th/0410103. A. 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[^1]: For review of the open string tachyon condensation, see [@Sen:1999mg; @Ohmori:2001am; @Taylor:2002uv; @Taylor:2003gn; @Sen:2004nf]. [^2]: For discussion of the effective field theory description of the tachyon condensation, see [@Sen:2002qa; @Fotopoulos:2003yt; @Sen:2003bc; @Kutasov:2003er; @Niarchos:2004rw]. [^3]: Some partial results considering tachyon condensation on non-BPS Dp-brane in curved background were presented in [@Kim:2005he; @Panigrahi:2004qr; @Kluson:2004yk; @Kluson:2005qx]. [^4]: The case of more general background, including NS $B$ field and Ramond-Ramond forms will be discussed in forthcoming publication. [^5]: We use the convention where the fundamental string tension has been set equal to $(2\pi)^{-1}$ (i.e. $\alpha'=1$). [^6]: We restrict ourselves in this paper to the situations when the background metric is diagonal. [^7]: In what follows we will consider the situation when we can ignore the NS two form background.
--- abstract: 'In this paper we present some algebraic properties of subgroupoids and normal subgroupoids. We define the normalizer of a wide subgroupoid ${\mathcal{H}}$ and show that, as in the case of groups, the normalizer is the greatest wide subgroupoid of the groupoid ${\mathcal{G}}$ in which ${\mathcal{H}}$ is normal. Furthermore, we give the definition of center and commutator and prove that both are normal subgroupoids, the first one of the union of all the isotropy groups of ${\mathcal{G}}$ and the second one of ${\mathcal{G}}$. Finally, we introduce the concept of inner isomorphism of $\mathcal{G}$ and show that the set of all the inner isomorphisms of ${\mathcal{G}}$ is a normal subgroupoid, which is isomorphic to the quotient groupoid of $\mathcal{G}$ by its center $\mathcal{Z}(\mathcal{G})$, which extends to groupoids a well-known result in groups.' author: - \| Jesús Ávila and Víctor Marín$^{\text{1}}$\ $^{\text{1}}$Departamento de Matemáticas y Estadística\ Universidad del Tolima\ Santa Helena, Ibagué, Colombia\ e-mail: javila@ut.edu.co, vemarinc@ut.edu.co title: 'On normal subgroupoids' --- 2010 AMS Subject Classification: Primary 20N02. Secondary 20L05.\ Key Words: Groupoid, normal subgroupoid, normalizer, center, commutator and inner isomorphisms. Introduction ============ The notion of groupoid (denominated Brandt groupoid) was first introduced in [@B], from an algebraic point of view. Next, Brandt groupoids were generalized by C. Ehresmann in [@E], where he added other structures such as topological and differentiable. Other equivalent definition of groupoid and its properties appear in [@Br], where a groupoid is defined as a small category where each morphism is invertible. In [@I Definition 1.1] the author follows the definition given by Ehresmann and presents the notion of groupoid as a particular case of an universal algebra, he defines strong homomorphism for groupoids and proves the correspondence theorem in this context. The Cayley Theorem for groupoids is also presented in [@I2 Theorem 3.1]. The definition of groupoid from an approach axiomatic, such as that of group, is presented in [@L]. In that sense, Paques and Tamusiunas give necessary and sufficient conditions to be a normal subgroupoid and they construct the quotient groupoid [@PT]. In [@AMP], the isomorphism theorems are proved and one application of these to the normal series is presented. The purpose of this paper is to introduce several concepts into groupoids, which are analogous to those defined in groups such as center, normalizer, commutator and inner isomorphism. In addition, the normality of these subgroupoids and other properties that they satisfy are studied. The paper is organized of the following manner. In Section 2 we present some preliminaries and basic results on groupoids and subgroupoids, which are used in the following sections. Next, in Section 3 we present some algebraic properties of normal subgroupoids and introduce the normalizer of a subgroupoid $\mathcal{H}$, $\mathcal{N}_{\mathcal{G}}(\mathcal{H})$. Furthermore, we show that the normalizer is the greatest wide subgroupoid of the groupoid $\mathcal{G}$ in which $\mathcal{H}$ is normal (Proposition \[normalizer\]). In Section 4 we introduce the center $\mathcal{Z}(\mathcal{G})$ and the commutator subgroupoid $\mathcal{G}'$ of $\mathcal{G}$ and prove that they are normal subgroupoids and that $\mathcal{G}/\mathcal{G}'$ is the largest abelian quotient of $\mathcal{G}$ (Propositions \[center\], \[commutator\]). Finally, in Section 5 we define inner isomorphisms of a groupoid and prove that the set of all the inner isomorphisms of $\mathcal{G}$, $\mathcal{I}(\mathcal{G})$, is a normal subgroupoid and it is isomorphic to the quotient groupoid $\mathcal{G}/\mathcal{Z}(\mathcal{G})$ (Proposition \[inner\]). Preliminaries and basic results =============================== Now, we give the definition of groupoid from purely algebraic point of view. We follow the definition presented in [@L]. [@L p. 78].\[d2\] Let $\mathcal{G}$ be a set equipped with a partial binary operation which is denoted by concatenation. If $g,h \in \mathcal{G}$ and the product $gh$ is defined, we write $\exists gh$. A element $e\in \mathcal{G}$ is called an identity if $\exists eg$ implies $eg=g$ and $\exists g'e$ implies $g'e=g'$. The set of identities of $\mathcal{G}$ is denoted $\mathcal{G}_0$. $\mathcal{G}$ is said to be a groupoid if the following axioms hold: 1. $\exists g(hl)$ if, and only if, $\exists (gh)l$, and $g(hl)=(gh)l$. 2. $\exists g(hl)$ if, and only if, $\exists gh$ and $\exists hl$. 3. For each $g\in \mathcal{G}$, there exist unique elements $d(g), r(g)\in \mathcal{G}$ such that $\exists gd(g)$ and $\exists r(g)g$ and $gd(g)=g=r(g)g$. 4. For each $g\in \mathcal{G}$, there exist a element $g^{-1}\in \mathcal{G}$ such that $d(g)=g^{-1}g$ and $r(g)=gg^{-1}$. In groupoids is important to characterize in which case exist the product of two elements. It can be proved that if $x,y\in \mathcal{G}$ then $\exists xy$ if and only if $d(g)=r(h)$ [@AMP Lemma 2.3]. The following proposition shows several important properties that are fulfilled in the groupoids. [@AMP Proposition 2.7].\[t1\] Let $\mathcal{G}$ be a groupoid. Then for each $g,h, k, l\in G$ we have 1. The element $g^{-1}$ is unique and $(g^{-1})^{-1}=g$. 2. If $\exists (gh)(kl)$, then $(gh)(kl)=g[(hk)l]$. 3. $d(gh)=d(h)$ and $r(gh)=r(g)$. 4. $\exists gh$ if, and only if, $\exists h^{-1}g^{-1}$ and, in this case, $(gh)^{-1}=h^{-1}g^{-1}$. The following results are obtained easily from the previous proposition. [@AMP Proposition 2.8].\[propiedades de d y r\] If $\mathcal{G}$ is a groupoid and $g\in G$ then $d(g)=r(g^{-1}), d(d(g))=d(g)=r(d(g))$ and $d(r(g))=r(g)=r(r(g))$. If $\mathcal{G}$ is a groupoid, then the identities of $\mathcal{G}$ are the elements $e=d(g)$ with $g\in \mathcal{G}$ [@AMP Proposition 2.10] and we set $\mathcal{G}_0=\{e=d(g)\mid g\in \mathcal{G}\}$. Now, if $e\in \mathcal{G}_0$ then by Proposition \[propiedades de d y r\] it has that $d(e)=r(e)=e$, $\exists ee$ and $ee=e$ and $e^{-1}=e$ and moreover the set ${\mathcal{G}}_e=\{g\in \mathcal{G}\mid d(g)=r(g)=e\}$ is a group with identity element $e$, which is called the isotropy group associated to $e$. These isotropy groups are very important, because they allow us to extend some concepts from groups to groupoids. For example, a groupoid $\mathcal{G}$ is called abelian if all its isotropy groups are abelian [@MA Definition 1.1]. In this way the set ${\rm Iso}({\mathcal{G}})=\bigcup_{e\in {\mathcal{G}}_0}{\mathcal{G}}_e$, which is called the isotropy subgroupoid of $\mathcal{G}$, is essential for the study of the groupoids. Now, we present the definition of subgroupoid and wide subgroupoid and prove some algebraic properties of these substructures, which are also valid in groups. [@PT p. 107]. Let $\mathcal{G}$ be a groupoid and $\mathcal{H}$ a nonempty subset of $\mathcal{G}$. $\mathcal{H}$ is said to be a subgroupoid of $\mathcal{G}$ if for all $g,h \in \mathcal{H}$ it satisfies: 1. $g^{-1}\in \mathcal{H}$; 2. $\exists gh \Rightarrow gh\in \mathcal{H}$. In this case we denote $\mathcal{H}< \mathcal{G}$. In addition, if $\mathcal{H}_0=\mathcal{G}_0$ (or equivalently $\mathcal{G}_0\subseteq \mathcal{H}$) then $\mathcal{H}$ is called a wide subgroupoid of $\mathcal{G}$. Note that if $\mathcal{G}$ is a groupoid then the sets $\{d(g)\}$ ($g\in \mathcal{G}$), $\mathcal{G}_e$ ($e\in \mathcal{G}_0$), $\mathcal{G}_0$, $Iso(\mathcal{G})$ and $\mathcal{G}$ are subgroupoids of $\mathcal{G}$. Also, it is easy to see that if $\mathcal{H}$ is a subgroupoid then the set $\mathcal{H}\cup \mathcal{G}_0$ is a wide subgroupoid of $\mathcal{G}$. Moreover, if $\mathcal{H}$ is a wide subgroupoid of $\mathcal{G}$ and $g\in \mathcal{G}$ then $g^{-1}\mathcal{H}g=\{g^{-1}hg\mid h\in \mathcal{H}\textnormal{ and } r(h)=d(h)=r(g)\}$ is a subgroupoid of $\mathcal{G}$. In fact, note that $r(g)\in \mathcal{H}$ and then $\exists g^{-1}r(g)g$ that is $d(g)=g^{-1}r(g)g\in g^{-1}\mathcal{H}g$. If $x,y\in \mathcal{G}$ then $x=g^{-1}hg$ and $y=g^{-1}tg$ with $h,t\in \mathcal{H}$ and $r(h)=d(h)=r(t)=d(t)=r(g)$. Since $d(g^{-1}hg)=d(g)=r(g^{-1})=r(g^{-1}tg)$ then $\exists (g^{-1}hg)(g^{-1}tg)$ and it has that $xy = (g^{-1}hg)(g^{-1}tg) = (g^{-1}h)(r(g))(tg) = (g^{-1}h)(r(t)t)g = g^{-1}htg$. Now since $\exists ht$ then $ht\in \mathcal{H}$ and thus $xy\in g^{-1}\mathcal{H}g$. Finally, if $x\in \mathcal{G}$ then $x=g^{-1}hg$ with $h\in \mathcal{H}$ and $r(h)=d(h)=r(g)$. Thus, $x^{-1}=g^{-1}h^{-1}g\in g^{-1}\mathcal{H}g$ because $h^{-1}\in \mathcal{H}$. Hence, the set $g^{-1}\mathcal{H}g$ is a subgroupoid of $\mathcal{G}$. Note in particular that $g^{-1}\mathcal{H}g$ is a subgroup of $\mathcal{G}_{d(g)}$ and $g^{-1}\mathcal{H}g=g^{-1}\mathcal{H}_{r(g)}g$. \[interseccion de subgrupoides\] Let $\mathcal{G}$ be a groupoid and $\left\{\mathcal{H}_i\right\}_{i\in I}$ a family of subgroupoids of $\mathcal{G}$. Then: 1. If $\bigcap _{i\in I}\mathcal{H}_i\neq \emptyset$, then $\bigcap _{i\in I}\mathcal{H}_i$ is a subgroupoid of $\mathcal{G}$. 2. If $\mathcal{H}_i$ is wide for each $i\in I$ then $\bigcap _{i\in I}\mathcal{H}_i$ is a wide subgroupoid of $\mathcal{G}$. 1\. Let $A=\bigcap _{i\in I}\mathcal{H}_i$. Since $A\neq \emptyset$, let $a,b\in A$ and suppose that $\exists ab$. Then $d(a)=r(b)$ and $a,b\in \mathcal{H}_i$ for each $i\in I$. Thus $ab\in \mathcal{H}_i$ for each $i\in I$ that is $ab\in \bigcap _{i\in I}\mathcal{H}_i=A$. Finally, if $a\in A$ then $a\in \mathcal{H}_i$ for each $i\in I$ and then $a^{-1}\in \mathcal{H}_i$ for each $i\in I$. So, $a^{-1}\in \bigcap _{i\in I}\mathcal{H}_i=A$ and the result follows. 2\. It is enough to observe that $\emptyset \neq \mathcal{G}_0\subseteq A$ and to apply item (i). \[SG\] Let $\mathcal{G}$ be a groupoid and $\emptyset \neq B\subseteq G$. Then there exists the smallest subgroupoid of $\mathcal{G}$ which contains $B$. Let $\mathfrak{F}=\{\mathcal{H}\subseteq \mathcal{G}\mid \mathcal{H}\textnormal{ is a subgrupoid of }\mathcal{G}\textnormal{ and }B\subseteq \mathcal{H}\}$. Then $\mathfrak{F}\neq \emptyset$ since $\mathcal{G}\in \mathfrak{F}$ and $\bigcap \mathfrak{F}\neq \emptyset$ because $B\subseteq \bigcap \mathfrak{F}$. Thus $\bigcap \mathfrak{F}$ is a subgroupoid of $\mathcal{G}$ by virtue of Proposition \[interseccion de subgrupoides\]. Moreover, it is clear that $\bigcap \mathfrak{F}$ is the smallest subgroupoid of $\mathcal{G}$ such that $B\subseteq \bigcap \mathfrak{F}$. The other part is clear. If $\mathcal{G}$ is a groupoid and $\emptyset \neq B\subseteq \mathcal{G}$, then the subgroupoid given in the previous proposition will be called subgroupoid generated by $B$ and it will be denoted by $\langle B\rangle$. It can be proved that the set $\langle B\rangle$ is given by $\langle B\rangle=\left\{x_1^{\alpha_1}x_2^{\alpha_2}...x_n^{\alpha_n}\mid \exists x_1^{\alpha_1}x_2^{\alpha_2}...x_n^{\alpha_n}, x_i\in B, \alpha_i\in \{1,-1\}\, \forall i, n\in \mathbb{N} \right\}$. Also, note that $\langle B\rangle _w=\langle B\rangle \cup \mathcal{G}_0$ is a wide subgroupoid and it will be called wide subgroupoid generated by $B$. If $\mathcal{G}$ is a groupoid and $\mathcal{H}$, $\mathcal{K}$ are wide subgroupoids of $\mathcal{G}$, we define the set $$\mathcal{H}\mathcal{K}:=\{hk\mid d(h)=r(k), h\in \mathcal{H}, \, k\in\mathcal{K}\}.$$ Note that $\mathcal{H}\mathcal{K}\neq \emptyset$ since for $g\in \mathcal{G}$, $d(g)\in \mathcal{H}$ and $d(g)\in \mathcal{K}$ and thus $d(g)=d(g)d(g)\in \mathcal{H}\mathcal{K}$. Hence $\mathcal{G}_0\subseteq \mathcal{H}\mathcal{K}$. \[producto HK=KH\] Let $\mathcal{G}$ be a groupoid and $\mathcal{H}$, $\mathcal{K}$ wide subgroupoids of $\mathcal{G}$. Then $\mathcal{H}\mathcal{K}$ is a wide subgroupoid of $\mathcal{G}$ if and only if $\mathcal{H}\mathcal{K}=\mathcal{K}\mathcal{H}$. Let $x\in \mathcal{H}\mathcal{K}$. By assumption, $\mathcal{H}\mathcal{K}\leq \mathcal{G}$ so $x^{-1}\in \mathcal{H}\mathcal{K}$. Then $x^{-1}=hk$ with $h\in \mathcal{H}$, $k\in \mathcal{K}$, $d(h)=r(k)$ and hence $x=(x^{-1})^{-1}=(hk)^{-1}=k^{-1}h^{-1}\in \mathcal{K}\mathcal{H}$. On the other hand, we know that $G_0\subseteq \mathcal{K}\mathcal{H}$. If $y\in \mathcal{K}\mathcal{H}$ then $y=kh$ with $k\in \mathcal{K}$, $h\in \mathcal{H}$, $d(k)=r(h)$. Then $k^{-1}\in \mathcal{K}$ and $h^{-1}\in \mathcal{H}$ and by assumption $\exists h^{-1}k^{-1}$ and $h^{-1}k^{-1}\in \mathcal{H}\mathcal{K}$. Then $y=kh=(h^{-1}k^{-1})^{-1}\in \mathcal{H}\mathcal{K}$. Conversely, suppose that $\mathcal{H}\mathcal{K}=\mathcal{K}\mathcal{H}$. First note that $\mathcal{H}\mathcal{K}\neq \emptyset$, if $x,y\in \mathcal{H}\mathcal{K}$ then $x=hk$, $h\in \mathcal{H}$, $k\in \mathcal{K}$, $d(h)=r(k)$ and $y=st$, $s\in \mathcal{H}$, $t\in \mathcal{K}$, $d(s)=r(t)$. If $\exists xy$ then $xy=(hk)(st)=h(ks)t=h(s'k')t=(hs')(k't)\in \mathcal{H}\mathcal{K}$. Finally, if $x\in \mathcal{H}\mathcal{K}$ then $x=hk$, $h\in \mathcal{H}$, $k\in \mathcal{K}$, $d(h)=r(k)$. Thus $x^{-1}=(hk)^{-1}=k^{-1}h^{-1}\in \mathcal{K}\mathcal{H}\subseteq \mathcal{H}\mathcal{K}$. That is, $\mathcal{H}\mathcal{K}\leq \mathcal{G}$ and since $G_0\subseteq \mathcal{H}\mathcal{K}$ we conclude that $\mathcal{H}\mathcal{K}$ is a wide subgroupoid of $\mathcal{G}$. Normal subgroupoids =================== In this section we present the definition of normal subgroupoid and present several properties of these classes of subgroupoids. We also introduce the normalizer of a subgroupoid $\mathcal{H}$ and prove some of its algebraic properties. In particular, we prove that the normalizer is the greatest wide subgroupoid of $\mathcal{G}$ in which $\mathcal{H}$ is normal. Let $\mathcal{G}$ be a groupoid. The subgroupoid $\mathcal{H}$ of $\mathcal{G}$ is said to be normal, denoted by $\mathcal{H}\lhd \mathcal{G}$, if $\mathcal{H}$ is wide and $g^{-1}\mathcal{H}g\subseteq \mathcal{H}$ for all $g\in \mathcal{G}$. Note that $\mathcal{G}$ is a normal subgroupoid of $\mathcal{G}$. Now, $\mathcal{G}_0$ is a wide subgroupoid of $\mathcal{G}$ and if $g\in \mathcal{G}$ then $g^{-1}\mathcal{G}_0g=\{d(g)\}\subseteq \mathcal{G}_0$. That is, $\mathcal{G}_0$ is also a normal subgroupoid of $\mathcal{G}$. Moreover, $Iso(\mathcal{G})$ is a normal subgroupoid of $\mathcal{G}$ and if $\mathcal{G}$ is an abelian groupoid and $\mathcal{H}$ is a wide subgroupoid of $\mathcal{G}$ then $Iso(\mathcal{H})$ is normal in $Iso(\mathcal{G})$. Normality also can be characterized as follows [@Br]: the subgroupoid $\mathcal{H}$ is said to be normal if $\mathcal{H}_0=\mathcal{G}_0$ and $g^{-1}\mathcal{H}_{r(g)}g=\mathcal{H}_{d(g)}$ for all $g\in \mathcal{G}$. The equivalence between the two definitions of normality can be consulted in [@PT Lemma 3.1]. The following proposition extends some well-known results from normal subgroups to normal subgroupoids. \[SIT\] Let $\mathcal{G}$ be a grupoid. Then: 1. If $\left\{\mathcal{H}_i\right\}_{i\in I}$ is a family of normal subgroupoids of $\mathcal{G}$, then $\bigcap _{i\in I}H_i$ is a normal subgroupoid of $\mathcal{G}$. 2. If $\emptyset \neq B\subseteq \mathcal{G}$, then there exist the smallest normal subgroupoid of $\mathcal{G}$ which contains $B$. 3. If $\mathcal{H}$ is a subgroupoid of $\mathcal{G}$ and $\mathcal{K}$ is a normal subgroupoid of $\mathcal{G}$ such that $d(k)=r(k)$ for all $k\in \mathcal{K}$, then $\mathcal{H}\mathcal{K}$ is a subgroupoid of $\mathcal{G}$. 4. If $\mathcal{H}$ and $\mathcal{K}$ are normal subgroupoids of $\mathcal{G}$ with $d(k)=r(k)$ for all $k\in \mathcal{K}$, then $\mathcal{H}\mathcal{K}$ is a normal subgroupoid of $\mathcal{G}$. 5. If $\mathcal{H}$ is a wide subgroupoid of $\mathcal{G}$ and $\mathcal{K}$ is a normal subgroupoid of $\mathcal{G}$, then $\mathcal{H}\cap \mathcal{K}$ is a normal subgroupoid of $\mathcal{H}$. 6. If $\mathcal{H}$ and $\mathcal{K}$ are normal subgroupoids of $\mathcal{G}$ such that $\mathcal{H}\cap \mathcal{K}=\mathcal{G}_0$, then $hk=kh$ for all $h\in \mathcal{H}$ and $k\in \mathcal{K}$ such that $r(h)=d(h)=r(k)=d(k)$. 1\. By Proposition \[interseccion de subgrupoides\] (item 2) we have that $\bigcap _{i\in I}H_i$ is a wide subgrupoid of $\mathcal{G}$. Now, let $g\in \mathcal{G}$ and $h\in \bigcap _{i\in I}H_i$ such that $r(h)=d(h)=r(g)$. Then $\exists g^{-1}hg$ and since $h\in H_i$ for each $i\in I$, we have that $g^{-1}hg\in H_i$ for each $i\in I$. That is, $g^{-1}hg\in \bigcap _{i\in I}H_i$. 2\. It is enough to take the collection of normal subgroupoids of $\mathcal{G}$ that contains $B$ and to apply previous item. 3\. First note that $\mathcal{H} \mathcal{K}\neq \emptyset$ since if $h\in \mathcal{H}$ then $h=hd(h)\in \mathcal{H}\mathcal{K}$. Let $x,y\in \mathcal{H} \mathcal{K}$ then there exist $h,s\in \mathcal{H}$ and $k,t\in \mathcal{K}$ with $d(h)=r(k)$ and $d(s)=r(t)$ such that $x=hk$ and $y=st$. If $\exists xy$ then $d(k)=r(s)$ and $xy=(hk)(st)=hkst=hr(k)kst$ and since $r(k)=d(k)=r(s)$ we have that $\exists hr(s)kst=hss^{-1}kst=(hs)(s^{-1}ks)t\in \mathcal{H}\mathcal{K}$. Finally, if $x=hk$ with $h\in \mathcal{H}$, $k\in \mathcal{K}$ and $d(h)=r(k)$ then $x^{-1}=k^{-1}h^{-1}=d(k)k^{-1}h^{-1}$ and since $d(k)=r(k)=d(h)$ we have that $\exists d(h)k^{-1}h^{-1}$ and thus $x^{-1}=d(h)k^{-1}h^{-1}=h^{-1}hk^{-1}h^{-1}=h^{-1}(hk^{-1}h^{-1})\in \mathcal{H}\mathcal{K}$ and the result follows. 4\. By previous item $\mathcal{H}\mathcal{K}$ is a subgroupoid of $\mathcal{G}$. Let $g\in \mathcal{G}$, $h\in \mathcal{H}$ and $k\in \mathcal{K}$ with $r(h)=d(k)=r(g)$ then $\exists g^{-1}hkg$ and thus $d(h)=r(k)$. Then $g^{-1}hkg=g^{-1}hr(k)kg=g^{-1}hr(g)kg=(g^{-1}hg)(g^{-1}kg)\in \mathcal{H}\mathcal{K}$. That is, $\mathcal{H}\mathcal{K}$ is a normal subgroupoid of $\mathcal{G}$. 5\. It is clear that $\mathcal{H}\cap \mathcal{K}$ is a wide subgroupoid of $\mathcal{H}$. Let $g\in \mathcal{H}$ and $h\in \mathcal{H}\cap \mathcal{K}$ with $r(h)=d(h)=r(g)$, then $\exists g^{-1}hg$ and by assumptions it follows that $g^{-1}hg\in \mathcal{H}\cap \mathcal{K}$. 6\. First note that each $d(g)$ satisfies the assumptions of proposition. If $h\in \mathcal{H}$ and $k\in \mathcal{K}$ with $r(h)=d(h)=r(k)=d(k)$, then $\exists h^{-1}k^{-1}hk$. And we obtain that $h^{-1}k^{-1}hk=h^{-1}(k^{-1}hk)\in \mathcal{H}$ and $h^{-1}k^{-1}hk=(h^{-1}k^{-1}h)k\in \mathcal{K}$. Thus, $h^{-1}k^{-1}hk\in \mathcal{G}_0$, that is, $h^{-1}k^{-1}hk=d(g)$ for some $g\in \mathcal{G}$. Then, $d(k)=d(h^{-1}k^{-1}hk)=d(g)$ and thus $r(h)=d(h)=r(k)=d(k)=d(g)$. Hence $$\begin{aligned} h^{-1}k^{-1}hk &=d(g) \\ h(h^{-1}k^{-1}hk) &=hd(h) \\ r(h)(k^{-1}hk) &= h \\ (d(k)k^{-1})hk&= h\\ k^{-1}hk & = h \\ k(k^{-1}hk) &=kh \\ r(k)(hk)& = kh \\ (r(h)h)k &=kh \\ hk&=kh. \end{aligned}$$ It is well known in groups, that given a subgroup $H$, there exists the greatest subgroup of $G$ in which $H$ is normal. Such subgroup is known as normalizer of $H$ and it satisfies some interesting properties. In our case it is natural to ask if is possible to define the normalizer of a subgroupoid. The answer to this question is affirmative as we show below. If $\mathcal{H}$ is a wide subgroupoid of $\mathcal{G}$, we define the set $$\mathcal{N}_{\mathcal{G}}({\mathcal{H}})=\{g\in \mathcal{G}\mid g^{-1}\mathcal{H}_{r(g)}g=\mathcal{H}_{d(g)}\}$$ which will be called the Normalizer of $\mathcal{H}$ in $\mathcal{G}$. It is clear that $\mathcal{N}_{\mathcal{G}}({\mathcal{H}})\neq \emptyset$ since for $r(g)\in \mathcal{G}_0$ it has $r(g)^{-1}\mathcal{H}_{r(r(g))}r(g)=r(g)\mathcal{H}_{r(g)}r(g)=\mathcal{H}_{r(g)}=\mathcal{H}_{d(r(g))}$ which implies that $r(g)\in \mathcal{N}_{\mathcal{G}}({\mathcal{H}})$ and then $\mathcal{G}_0\subseteq \mathcal{N}_{\mathcal{G}}({\mathcal{H}})$. Note that in the group case this concept coincide with the normalizer of subgroups. The next proposition extends the main properties of the normalizer in groups to the normalizer in groupoids. \[normalizer\] Let $\mathcal{G}$ be a groupoid and $\mathcal{H}$ a wide subgroupoid of $\mathcal{G}$. Then: 1. $\mathcal{N}_{\mathcal{G}}({\mathcal{H}})$ is a wide subgroupoid of $\mathcal{G}$ that contains $\mathcal{H}$. 2. $\mathcal{H}$ is a normal subgroupoid of $\mathcal{N}_{\mathcal{G}}({\mathcal{H}})$. 3. $\mathcal{N}_{\mathcal{G}}({\mathcal{H}})$ is the greatest wide subgroupoid of $\mathcal{G}$ in which $\mathcal{H}$ is normal. 4. $\mathcal{N}_{\mathcal{G}}({\mathcal{H}})=\mathcal{G}$ if and only if $\mathcal{H}\lhd \mathcal{G}$. 1\. Note that the width of $\mathcal{H}$ was proved in paragraph previous to the proposition. Now, If $h\in \mathcal{H}$ then $h^{-1}\mathcal{H}_{r(h)}h\subseteq \mathcal{H}\cap \mathcal{G}_{d(h)}=\mathcal{H}_{d(h)}$. If $m\in \mathcal{H}_{d(h)}$ then $r(m)=d(m)=d(h)$ which implies that $\exists hmh^{-1}$ and $hmh^{-1}\in \mathcal{H}_{r(h)}$. Then $\exists h^{-1}(hmh^{-1})h$ and it is clear that $m=h^{-1}(hmh^{-1})h\in h^{-1}\mathcal{H}_{r(h)}h$. Hence $\mathcal{H}_{d(h)}\subseteq h^{-1}\mathcal{H}_{r(h)}h$ and thus $h^{-1}\mathcal{H}_{r(h)}h=\mathcal{H}_{d(h)}$ that is $h\in \mathcal{N}_{\mathcal{G}}(\mathcal{H})$. Then $\mathcal{H}\subseteq \mathcal{N}_{\mathcal{G}}(\mathcal{H})$. Let $g,t\in \mathcal{N}_{\mathcal{G}}(\mathcal{H})$ and suppose that $\exists gt$. Then $g^{-1}\mathcal{H}_{r(g)}g=\mathcal{H}_{d(g)}$, $t^{-1}\mathcal{H}_{r(t)}t=\mathcal{H}_{d(t)}$, $d(g)=r(t)$ and thus $$\begin{aligned} (gt)^{-1}\mathcal{H}_{r(gt)}(gt) & =t^{-1}g^{-1}\mathcal{H}_{r(g)}gt \\ & =t^{-1}\mathcal{H}_{d(g)}t \\ & =t^{-1}\mathcal{H}_{r(t)}t \\ & =\mathcal{H}_{d(t)} \\ & =\mathcal{H}_{d(gt)}. \end{aligned}$$ If $t\in \mathcal{N}_{\mathcal{G}}(\mathcal{H})$ then $t^{-1}\mathcal{H}_{r(t)}t=\mathcal{H}_{d(t)}$ and thus $t\mathcal{H}_{d(t)}t^{-1}=\mathcal{H}_{r(t)}$. Hence $(t^{-1})^{-1}\mathcal{H}_{r(t^{-1})}t^{-1}=\mathcal{H}_{d(t^{-1})}$ and then $t^{-1}\in \mathcal{N}_{\mathcal{G}}(\mathcal{H})$. Therefore $\mathcal{N}_{\mathcal{G}}(\mathcal{H})$ is a wide subgroupoid of $\mathcal{G}$. 2\. By item 1, $\mathcal{H}\subseteq \mathcal{N}_{\mathcal{G}}(\mathcal{H})$ and since $\mathcal{H}$ is a wide subgroupoid of $\mathcal{G}$ we have that $\mathcal{H}$ is a wide subgroupoid of $\mathcal{N}_{\mathcal{G}}(\mathcal{H})$. Now, consider $m\in \mathcal{N}_{\mathcal{G}}(\mathcal{H})$, $h\in \mathcal{H}$ and suppose that $\exists m^{-1}hm$. Then $r(h)=d(h)=r(m)$, that is $h\in \mathcal{H}_{r(m)}$ and thus $m^{-1}hm\in m^{-1}\mathcal{H}_{r(m)}m=\mathcal{H}_{d(m)}\subseteq \mathcal{H}$. Hence $\mathcal{H}$ is normal in $\mathcal{N}_{\mathcal{G}}(\mathcal{H})$. 3\. Suppose that $\mathcal{T}$ is a wide subgroupoid of $\mathcal{G}$ and that $\mathcal{H}$ is normal in $\mathcal{T}$. If $t\in \mathcal{T}$ then $t^{-1}\mathcal{H}_{r(t)}t=\mathcal{H}_{d(t)}$ and thus $t\in \mathcal{N}_{\mathcal{G}}(\mathcal{H})$. Hence $\mathcal{T}\subseteq \mathcal{N}_{\mathcal{G}}(\mathcal{H})$ and the result follows. 4\. It is evident. Normal subgroups are very important in group theory, because they are necessary to construct the quotient group. In the groupoid case, given a wide subgroupoid $\mathcal{H}$ of $\mathcal{G}$, in [@PT] Paques and Tamusiunas define a relation on $\mathcal{G}$ as follows: for every $g,l \in \mathcal{G}$, $$g\equiv_{\mathcal{H}}l \Longleftrightarrow \exists l^{-1}g\,\,\,\,\text{ and} \,\,\,\,l^{-1}g\in \mathcal{H}.$$ Furthermore, they prove that this relation is a congruence, that is an equivalence relation which is compatible with products. The equivalence class of $\equiv_{\mathcal{H}}$ containing $g$ is the set $g\mathcal{H}=\{gh\mid h\in \mathcal{H} \, \land \, r(h)=d(g)\}$. This set is called left coset of $\mathcal{H}$ in $\mathcal{G}$ containing $g$. Moreover, they prove that if $\mathcal{H}$ is a normal subgroupoid of $\mathcal{G}$ and $\mathcal{G/H}$ is the set of all left cosets of $\mathcal{H}$ in $\mathcal{G}$, then $\mathcal{G/H}$ is a groupoid such that $\exists (g\mathcal{H})(l\mathcal{H})$, if and only if, $\exists gl$ and the partial binary operation is given by $(g\mathcal{H})(l\mathcal{H})=gl\mathcal{H}$ [@PT Lemma 3.12]. That groupoid is called the quotient groupoid of $\mathcal{G}$ by $\mathcal{H}$. In order to improve the understanding of this work, we finish this section by presenting the notion of groupoid (strong) homomorphism and the first isomorphism theorem. The other isomorphism theorems also remains valid in our context of groupoids. The proofs of these theorems and several examples of application can be consulted in [@AMP]. Let $\mathcal{G}$ and $\mathcal{G'}$ be groupoids. A map $\phi: \mathcal{G}\to \mathcal{G'}$ is called groupoid homomorphism if for all $x,y\in \mathcal{G}$, $\exists xy$ implies that $\exists \phi(x)\phi(y)$ and in this case $\phi(xy)=\phi(x)\phi(y)$. In addition, if $\phi$ is a groupoid homomorphism and for all $x,y\in \mathcal{G}$, $\exists \phi(x)\phi(y)$ implies that $\exists xy$ then $\phi$ is called groupoid strong homomorphism. \[teoremas de isomorfismos\] (The First Isomorphism Theorem) Let $\phi: \mathcal{G}\to \mathcal{G'}$ be a groupoid strong homomorphism. If $\phi$ is surjective then there exists an strong isomorphism $\overline{\phi}:\mathcal{G}/Ker(\phi)\to \mathcal{G'}$ such that $\phi=\overline{\phi}\circ j$, where $j$ is the canonical homomorphism of $\mathcal{G}$ onto $\mathcal{G}/Ker(\phi)$. Center and commutators ====================== In this section we introduce the center and the commutator subgroupoid and prove several properties of them, which extend well-known results in groups. Let $\mathcal{G}$ be a groupoid. We define the center of $\mathcal{G}$ as the set $\mathcal{Z}(\mathcal{G})=\{g\in Iso(\mathcal{G})\mid gh=hg \textnormal{ for all }h\in \mathcal{G}\text{ such that }d(g)=r(h)=d(h)\}$. The center of the groupoid $\mathcal{G}$ has analogous properties to those of the groups, as we show in the following proposition. \[center\] Let $\mathcal{G}$ be a groupoid and $\mathcal{Z}(\mathcal{G})$ the center of $\mathcal{G}$. Then: 1. $\mathcal{Z}(\mathcal{G})=\bigsqcup_{e\in \mathcal{G}_0}Z(\mathcal{G}_e)$. 2. $\mathcal{Z}(\mathcal{G})=Iso(\mathcal{G})$ if and only if $\mathcal{G}$ is an abelian groupoid. 3. $\mathcal{Z}(\mathcal{G})$ is a normal subgrupoid of $Iso(\mathcal{G})$. 4. If $\mathcal{H}$ is a wide subgrupoid of $\mathcal{Z}(\mathcal{G})$ then it is normal in $Iso(\mathcal{G})$. 1\. If $g\in \mathcal{Z}(\mathcal{G})$ then $g\in \mathcal{G}_e$ for some $e\in \mathcal{G}_0$ and thus $d(g)=r(g)=e$. If $h\in \mathcal{G}_e$ then $\exists gh$, $\exists hg$ and then $gh=hg$ that is $g\in Z(\mathcal{G}_e)\subseteq \bigsqcup_{e\in \mathcal{G}_0}Z(\mathcal{G}_e)$. On the other hand, if $g\in \bigsqcup_{e\in \mathcal{G}_0}Z(\mathcal{G}_e)$ then $g\in Z(\mathcal{G}_e)$ for some $e\in \mathcal{G}_0$. If $h\in \mathcal{G}$ is such that $d(g)=r(h)=d(h)$ then $e=d(h)=r(h)$ and thus $h\in \mathcal{G}_e$ and hence $gh=hg$. That is $g\in \mathcal{Z}(\mathcal{G})$. 2\. It is evident. 3\. First of all that $\mathcal{G}_0\subseteq \mathcal{Z}(\mathcal{G})$. Let $g,h\in \mathcal{Z}(\mathcal{G})$ and suppose that $\exists gh$. Then by item 1. $g\in Z(\mathcal{G}_e)$ and $h\in Z(\mathcal{G}_{e'})$ for some $e,e'\in \mathcal{G}_0$. Then $d(g)=r(g)=e$, $d(h)=r(h)=e'$ and since $d(g)=r(h)$ we have $e=e'$. Thus $h\in Z(\mathcal{G}_e)$ and hence $gh\in Z(\mathcal{G}_e)\subseteq \mathcal{Z}(\mathcal{G})$. If $g\in \mathcal{Z}(\mathcal{G})$ then $g\in Z(\mathcal{G}_e)$ for some $e\in \mathcal{G}_0$ and since $Z(\mathcal{G}_e)$ is a subgroup of $\mathcal{G}_e$ we have $g^{-1}\in Z(\mathcal{G}_e)\subseteq \mathcal{Z}(\mathcal{G})$. Thus $\mathcal{Z}(\mathcal{G})$ is an ample subgroupoid of $\mathcal{G}$. Finally, let $g\in Iso(\mathcal{G})$ and $h\in \mathcal{Z}(\mathcal{G})$ such that $r(h)=d(h)=r(g)$. Then $g\in \mathcal{G}_e$ for some $e\in \mathcal{G}_0$ and thus $r(g)=d(g)=e$ and then $r(h)=d(h)=e$. Thus $h\in Z(\mathcal{G}_e)$ and since $\exists g^{-1}hg$ we have that $g^{-1}hg=g^{-1}gh=d(g)h=eh=h\in \mathcal{Z}(\mathcal{G})$. Hence $\mathcal{Z}(\mathcal{G})$ is a normal subgroupoid of $Iso(\mathcal{G})$. 4\. Let $g\in Iso(\mathcal{G})$ and $h\in \mathcal{H}$ such that $r(h)=d(h)=r(g)$. Then $g\in \mathcal{G}_e$ for some $e\in \mathcal{G}_0$ and thus $r(g)=d(g)=e$ and then $r(h)=d(h)=e$. Thus $h\in Z(\mathcal{G}_e)$ and since $\exists g^{-1}hg$ we have that $g^{-1}hg=g^{-1}gh=d(g)h=eh=h\in \mathcal{H}$. Hence $\mathcal{H}$ is a normal subgrupoid of $Iso(\mathcal{G})$. If we wish to define the commutator subgroupoid of a groupoid $\mathcal{G}$, then we must start by defining the commutator of two elements. Thus, if $x,y\in \mathcal{G}$ then $\exists xyx^{-1}y^{-1}$ if and only if $d(x)=r(x)=d(y)=r(y)$ if and only if $x,y\in \mathcal{G}_e$ for some $e\in \mathcal{G}_0$. That is, for $x,y\in \mathcal{G}_e$ for some $e\in \mathcal{G}_0$ we define the commutator of $x,y$ as $[x,y]=x^{-1}y^{-1}xy$. Let $\mathcal{G}$ be a groupoid. The commutator subgroupoid of $\mathcal{G}$ is given by the set $\mathcal{G}'=\langle [x,y]\mid x,y\in \mathcal{G}_e, e\in \mathcal{G}_0\rangle$. Note that $[x,y]^{-1}=(x^{-1}y^{-1}xy)^{-1}=[y,x]$ and $d([x,y])=r([x,y])=e$. Moreover, $xy=yx[x,y]$ and thus $xy=yx$ if and only if $[x,y]=e$. Finally, by following Proposition \[SG\] the elements of $\mathcal{G}'$ are all the finite products of commutators in $\mathcal{G}$. That is, $$\mathcal{G}'=\{x_1x_2\cdot\cdot\cdot x_n\mid \exists x_1x_2\cdot\cdot\cdot x_n, n\geq 1 \text{ and each }x_i\text{ is a commutator}\}.$$ More generally, if $\mathcal{H},\mathcal{K}$ are wide subgroupoids of $\mathcal{G}$ then we define $[\mathcal{H},\mathcal{K}]=\langle[x,y]\mid x\in \mathcal{H}_e,y\in \mathcal{K}_e,e\in \mathcal{G}_0\rangle$ and with this notation it is clear that $\mathcal{G}'=[\mathcal{G},\mathcal{G}]$. The main properties of the commutator subgroupoid are given in the following proposition. Note that all of them are valid in groups. \[commutator\] Let $\mathcal{G}$ be a groupoid, let $x,y\in \mathcal{G}_e$, $e\in \mathcal{G}_0$, and let $\mathcal{H}$ a wide subgroupoid of $\mathcal{G}$. Then: 1. $\mathcal{G}'=\bigsqcup_{e\in \mathcal{G}_0}\mathcal{G}_e'$. 2. $\mathcal{G}'=\mathcal{G}_0$ if and only if $\mathcal{G}$ is an abelian groupoid. 3. If $\mathcal{H}\lhd \mathcal{G}$ then $[\mathcal{H},\mathcal{G}]\leq \mathcal{H}$. 4. $\mathcal{G}'$ is a normal subgroupoid of $\mathcal{G}$ and $\mathcal{G}/\mathcal{G}'$ is an abelian groupoid. 5. $\mathcal{G}/\mathcal{G}'$ is the largest abelian quotient of $\mathcal{G}$ in the sense that if $\mathcal{H}\lhd \mathcal{G}$ and $\mathcal{G}/\mathcal{H}$ is abelian, then $\mathcal{G}'\leq \mathcal{H}$. 6. If $\sigma :\mathcal{G}\to \mathcal{A}$ is any homomorphism of $\mathcal{G}$ into an abelian groupoid $\mathcal{A}$, then there exists a homomorphism $\theta :\mathcal{G}/\mathcal{G}'\to \mathcal{A}$ such that $\sigma=\theta \circ j$ where $j$ is the canonical homomorphism of $\mathcal{G}$ into $\mathcal{G}/\mathcal{G}'$. 1\. If $a\in \mathcal{G}'$, then there exist commutators $x_1,x_2,\cdot\cdot\cdot, x_n$ such that $\exists x_1x_2\cdot\cdot\cdot x_n$ and $a=x_1x_2\cdot\cdot\cdot x_n$. Then $x_1,x_2,\cdot\cdot\cdot, x_n\in \mathcal{G}_e$ for some $e\in \mathcal{G}_0$ and thus $a\in \mathcal{G}_e'$. The other inclusion is evident. 2\. Let $x,y\in \mathcal{G}_e$, $e\in \mathcal{G}_0$. Then $\exists x^{-1}y^{-1}xy$ and thus $x^{-1}y^{-1}xy\in \mathcal{G}'$. Then by assumption $x^{-1}y^{-1}xy=e$ which implies that $xy=yx$, that is, $\mathcal{G}_e$ is an abelian group. Conversely, if $\mathcal{G}$ is an Abelian groupoid then for $x,y\in \mathcal{G}_e$ it has that $[x,y]=e$ and the result follows. 3\. Since $[\mathcal{H},\mathcal{G}]$ is a subgroupoid of $\mathcal{G}$ it is enough to show that $[\mathcal{H},\mathcal{G}]\subseteq \mathcal{H}$. Then if $[x,y]\in [\mathcal{H},\mathcal{G}]$ then $x\in \mathcal{H}_e$, $y\in \mathcal{G}_e$ for some $e\in \mathcal{G}_0$ and $[x,y]=x^{-1}y^{-1}xy$. Then by assumption $y^{-1}xy\in \mathcal{H}$ and hence $[x,y]\in \mathcal{H}$ which implies that $[\mathcal{H},\mathcal{G}]\subseteq \mathcal{H}$. 4\. By item 1, $\mathcal{G}'$ is a disjoint union of groups, that is, $\mathcal{G}'$ is a groupoid. Moreover, if $e\in \mathcal{G}_0 $ then $e=[e,e]\in \mathcal{G}'$ and thus $\mathcal{G}'$ is a wide subgroupoid of $\mathcal{G}$. Let $g\in \mathcal{G}$ and $a\in \mathcal{G}'$ such that $\exists g^{-1}ag$. Then $r(a)=d(a)=r(g)$ and $a=[x_1,y_1][x_2,y_2]\cdot \cdot \cdot [x_n,y_n]$ where $x_i,y_i\in \mathcal{G}_e$ for some $e\in \mathcal{G}_0$. Then $gg^{-1}=r(g)=r(a)=d(a)=e$ and we obtain $$\begin{aligned} g^{-1}ag & =g^{-1}[x_1,y_1][x_2,y_2]\cdot \cdot \cdot [x_n,y_n]g \\ & = g^{-1}[x_1,y_1]e[x_2,y_2]e\cdot \cdot \cdot e[x_n,y_n]g \\ & = g^{-1}[x_1,y_1]gg^{-1}[x_2,y_2]g\cdot \cdot \cdot g^{-1}[x_n,y_n]g \\ & =[g^{-1}x_1g,g^{-1}y_1g][g^{-1}x_2g,g^{-1}y_2g]\cdot \cdot \cdot [g^{-1}x_ng,g^{-1}y_ng]\in \mathcal{G}'. \end{aligned}$$ Hence, $\mathcal{G}'$ is a normal subgroupoid of $\mathcal{G}$. Now, let $x\mathcal{G}',y\mathcal{G}'\in (\mathcal{G}/\mathcal{G}')_{e\mathcal{G}'}$ for some $e\mathcal{G}'\in (\mathcal{G}/\mathcal{G}')_0$. Then $\exists (x\mathcal{G}')^{-1}(y\mathcal{G}')^{-1}\linebreak (x\mathcal{G}')(y\mathcal{G}')$ and $(x\mathcal{G}')^{-1}(y\mathcal{G}')^{-1}(x\mathcal{G}')(y\mathcal{G}')=(x^{-1}\mathcal{G}')(y^{-1}\mathcal{G}') (x\mathcal{G}')(y\mathcal{G}')=\linebreak (x^{-1}y^{-1}xy)\mathcal{G}'=e\mathcal{G}'$. Which implies that $(x\mathcal{G}')(y\mathcal{G}')=(y\mathcal{G}')(x\mathcal{G}')$ that is $(\mathcal{G}/\mathcal{G}')_{e\mathcal{G}'}$ is an abelian group. 5\. Let $[x,y]\in \mathcal{G}'$, then $x,y\in \mathcal{G}_e$ for some $e\in \mathcal{G}_0$. Now, $[x,y]\mathcal{H}=(x^{-1}y^{-1}xy)\mathcal{H}=(x^{-1}\mathcal{H})(y^{-1}\mathcal{H})(x\mathcal{H})(y\mathcal{H})= (x^{-1}\mathcal{H})(x\mathcal{H})(y^{-1}\mathcal{H})(y\mathcal{H})=e\mathcal{H}$ and hence $[x,y]\mathcal{H}=e\mathcal{H}$. That is, $[x,y]\in \mathcal{H}$ and then $\mathcal{G}'\subseteq \mathcal{H}$. 6\. First note that if $[x,y]\in \mathcal{G}'$ then $x,y\in \mathcal{G}_e$ for some $e\in \mathcal{G}_0$ and thus $r(x)=d(x)=r(y)=d(y)=e$ which implies that $r(\sigma(x))=d(\sigma(x))=r(\sigma(y))=d(\sigma(y))=\sigma(e)\in \mathcal{A}_{\sigma (e)}$. Then $$\begin{aligned} \sigma ([x,y]) & =\sigma (x^{-1}y^{-1}xy) \\ & =\sigma(x^{-1})\sigma(y^{-1})\sigma(x)\sigma(y) \\ & =\sigma(x^{-1})\sigma(x)\sigma(y^{-1})\sigma(y)\\ &=\sigma(e)\in \mathcal{A}_0. \end{aligned}$$ Hence $\mathcal{G}'\subseteq Ker (\sigma)$. Know, we define $\theta:\mathcal{G}/\mathcal{G}'\to \mathcal{A}$ by $\theta (x\mathcal{G}')=\sigma (x)$ for each $x\mathcal{G}'\in \mathcal{G}/\mathcal{G}'$. If $x\mathcal{G}'=y\mathcal{G}'$ then $\exists y^{-1}x$ and $y^{-1}x\in \mathcal{G}'\subseteq Ker(\sigma)$. Then $\sigma(y^{-1}x)=\sigma (y)^{-1}\sigma (x)=e'$ for some $e'\in \mathcal{A}_0$ which implies that $\sigma (x)=\sigma (y)$. Hence $\theta$ is a well defined function. Moreover, since $\sigma$ is a homomorphism then $\theta$ is also a homomorphism and it is clear that $\sigma =\theta\circ j$. Inner isomorphisms ================== In this section we define the concept of inner isomorphism in groupoids, which coincide naturally with those of groups. We extend several results of these isomorphisms and in particular we prove that the set of all the inner isomorphisms of $\mathcal{G}$ is a normal subgroupoid and it is isomorphic to the quotient groupoid of $\mathcal{G}$ by its center. If $\mathcal{G}$ is a groupoid, we define $$\mathcal{A}(\mathcal{G})=\{f:\mathcal{G}_e\to \mathcal{G}_{e'}\mid e,e'\in\mathcal{G}_0\textnormal{ and }f\textnormal{ is an isomorphism}\}.$$ Then for $f,g\in \mathcal{A}(\mathcal{G})$ we say that $\exists f\circ g$ if and only if $D(f)=R(g)$ where $D(f)$ and $R(g)$ denotes the domain of $f$ and the range of $g$ respectively and in this case $(f\circ g)(x)=f(g(x))$ for all $x\in D(g)$. With this partial operation the set $\mathcal{A}(\mathcal{G})$ is a groupoid where for $f\in \mathcal{A}(\mathcal{G})$ it has that $d(f)=id_{D(f)}$, $r(f)=id_{R(f)}$ and $f^{-1}$ is the inverse of $f$. The elements of $\mathcal{A}(\mathcal{G})$ will be called partial isomorphisms of $\mathcal{G}$. Note that if $\mathcal{G}$ is a group then the set $\mathcal{A}(\mathcal{G})$ coincide with $Aut(\mathcal{G})$. In the group case the notion of inner automorphism is very important in several subjects. The next results show the interpretation of this concept in the groupoid case and we extend several results well-known in groups. \[isomorfismos internos\] Let $\mathcal{G}$ be a groupoid and $g\in \mathcal{G}$. Then the function $\mathcal{I}_g:\mathcal{G}_{d(g)}\to \mathcal{G}_{r(g)}$ defined as $\mathcal{I}_g(x)=gxg^{-1}$ for all $x\in \mathcal{G}_{d(g)}$ is a partial isomorphism of $\mathcal{G}$. First note that if $x\in \mathcal{G}_{d(g)}$ then $r(x)=d(x)=d(g)$ which implies that $\exists gxg^{-1}$. Moreover $r(gxg^{-1})=r(g)$ and $d(gxg^{-1})=d(g^{-1})=r(g)$ that is $gxg^{-1}\in \mathcal{G}_{r(g)}$. Then $\mathcal{I}_g$ is a well defined function. If $x,y\in \mathcal{G}_{d(g)}$ then $xy\in \mathcal{G}_{d(g)}$ and then $$\begin{aligned} \mathcal{I}_g(x) & = gxyg^{-1} \\ & =gxd(g)yg^{-1} \\ & =gxg^{-1}gyg^{-1} \\ & =\mathcal{I}_g(x)\mathcal{I}_g(y). \end{aligned}$$ So $\mathcal{I}_g$ is a homomorphism of groups, in particular it is an strong homomorphism of groupoids. Now let $x,y\in \mathcal{G}_{d(g)}$ such that $\mathcal{I}_g(x)=\mathcal{I}_g(y)$ then $g^{-1}xg=g^{-1}yg$ and by the cancellation law (valid for groupoids) we obtain $x=y$ and thus $\mathcal{I}_g$ is an injective function. If $m\in \mathcal{G}_{r(g)}$ then $r(m)=d(m)=r(g)$ which implies that $\exists g^{-1}m$ and $\exists mg$ and thus $\exists g^{-1}mg$. Note that $r(g^{-1}mg)=d(g^{-1}mg)=d(g)$ and thus $g^{-1}mg\in \mathcal{G}_{d(g)}$ and then $$\begin{aligned} \mathcal{I}_g(g^{-1}mg) & =g(g^{-1}mg)g^{-1} \\ & =gg^{-1}mgg^{-1}\\ & =r(g)mr(g) \\ & =m. \end{aligned}$$ That is $\mathcal{I}_g$ is a surjective function and we conclude that $\mathcal{I}_g$ is a partial strong isomorphism of $\mathcal{G}$. Note in particular that $\mathcal{I}_g$ is an isomorphism of groups. The isomorphisms given in Proposition \[isomorfismos internos\] will be called partial inner isomorphisms of $\mathcal{G}$ and the set of all the inner isomorphisms of $\mathcal{G}$ will be denoted by $\mathcal{I}(\mathcal{G})$. We say that the wide subgroupoid $\mathcal{H}$ of $\mathcal{G}$ is invariant by the partial inner isomorphism $\mathcal{I}_g$, $g\in \mathcal{G}$, if $\mathcal{I}_g(\mathcal{H}\cap D(\mathcal{I}_g))=\mathcal{H}\cap R(\mathcal{I}_g)$. That is, if $\mathcal{I}_g(\mathcal{H}_{r(g)})=\mathcal{I}_g(\mathcal{H}\cap \mathcal{G}_{r(g)})=\mathcal{H}\cap \mathcal{G}_{d(g)}=\mathcal{H}_{d(g)}$. \[inner\] Let $\mathcal{G}$ be a groupoid and $\mathcal{H}$ an wide subgroupoid of $\mathcal{G}$. Then 1. $\mathcal{I}(\mathcal{G})$ is a normal subgroupoid of $\mathcal{A}(\mathcal{G})$. 2. $\mathcal{I}(Iso(\mathcal{G}))=\{\mathcal{I}e\mid e\in \mathcal{G}_0\}$ if and only if $\mathcal{G}$ is a abelian groupoid. 3. The function $\Theta :\mathcal{G}\to \mathcal{I}(\mathcal{G})$ defined by $\Theta (g)=\mathcal{I}_g$ for all $g\in \mathcal{G}$ is an strong homomorphism. 4. The groupoids $\mathcal{G}/\mathcal{Z}(\mathcal{G})$ and $\mathcal{I}(\mathcal{G})$ are isomorphic. 5. $\mathcal{H}$ is normal if and only if it is invariant for all the partial inner isomorphisms of $\mathcal{G}$. 1\. If $e\in \mathcal{G}_0$ then the partial inner isomorphism $\mathcal{I}_e:\mathcal{G}_e\to \mathcal{G}_e$ is given by $\mathcal{I}_e(x)=exe^{-1}=exe=x=id_{\mathcal{G}_e}(x)$ for all $x\in \mathcal{G}_e$. That is $\mathcal{I}_e=id_{\mathcal{G}_e}$ which implies that $\mathcal{A}(\mathcal{G})_0\subseteq \mathcal{I}(\mathcal{G})$. Let $\mathcal{I}_g, \mathcal{I}_h\in \mathcal{I}(\mathcal{G})$ and suppose that $\exists \mathcal{I}_g\circ \mathcal{I}_h$. Then $D(\mathcal{I}_g)=R(\mathcal{I}_h)$ that is $\mathcal{G}_{d(g)}=\mathcal{G}_{r(h)}$ and thus $d(g)=r(h)$ which implies that $\exists gh$. Now, for $x\in \mathcal{G}_{d(h)}=\mathcal{G}_{d(gh)}$ we obtain $(\mathcal{I}_g\circ \mathcal{I}_h)(x)=\mathcal{I}_g(\mathcal{I}_h(x))=g(hxh^{-1})g^{-1}=(gh)x(gh)^{-1}=\mathcal{I}_{gh}(x)$. That is $\mathcal{I}_g\circ \mathcal{I}_h=\mathcal{I}_{gh}\in \mathcal{I}(\mathcal{G})$. Now, if $\mathcal{I}_g\in \mathcal{I}(\mathcal{G})$ then $\mathcal{I}_{g^{-1}}\in \mathcal{I}(\mathcal{G})$ and $D(\mathcal{I}_g)=\mathcal{G}_{d(g)}=\mathcal{G}_{r(g^{-1})}=R(\mathcal{I}_{g^{-1}})$ that is $\exists \mathcal{I}_g\circ \mathcal{I}_{g^{-1}}$ and $\mathcal{I}_g\circ \mathcal{I}_{g^{-1}}=\mathcal{I}_{r(g)}=id_{\mathcal{G}_{r(g)}}$. Also $D(\mathcal{I}_{g^{-1}})=\mathcal{G}_{d(g^{-1})}=\mathcal{G}_{r(g)}=R(\mathcal{I}_g)$ that is $\exists \mathcal{I}_{g^{-1}}\circ \mathcal{I}_g$ and $\mathcal{I}_{g^{-1}}\circ \mathcal{I}_g=\mathcal{I}_{d(g)}=id_{\mathcal{G}_{d(g)}}$ and hence $(\mathcal{I}_g)^{-1}=\mathcal{I}_{g^{-1}}\in \mathcal{I}(\mathcal{G})$. And thus $\mathcal{I}(\mathcal{G})$ is a wide subgroupoid of $\mathcal{A}(\mathcal{G})$. Let $\sigma\in \mathcal{A}(\mathcal{G})$, $\mathcal{I}_g\in \mathcal{I}(\mathcal{G})$ and suppose that $\exists \sigma ^{-1}\circ \mathcal{I}_g\circ \sigma$. Then $R(\mathcal{I}_g)=D(\mathcal{I}_g)=R(\sigma)$ and thus $\mathcal{G}_{r(g)}=\mathcal{G}_{d(g)}=R(\sigma)=D(\sigma ^{-1})$ which implies that $g,g^{-1}\in D(\sigma ^{-1})$. Now for $x\in D(\sigma)$ we have that $$\begin{aligned} (\sigma ^{-1}\circ \mathcal{I}_g\circ \sigma)(x) & =\sigma ^{-1}(\mathcal{I}_g(\sigma(x))) \\ & =\sigma ^{-1}(g\sigma (x)g^{-1}) \\ & =\sigma ^{-1}(g)x\sigma^{-1} (g^{-1}) \\ & =\sigma ^{-1}(g)x(\sigma^{-1} (g))^{-1} \\ & =\mathcal{I}_{\sigma^{-1}(g)}(x). \end{aligned}$$ Then $\sigma ^{-1}\circ \mathcal{I}_g\circ \sigma=\mathcal{I}_{\sigma^{-1}(g)}\in \mathcal{I}(\mathcal{G})$ and hence $\mathcal{I}(\mathcal{G})$ is a normal subgroupoid of $\mathcal{A}(\mathcal{G})$. 2\. Let $g\in \mathcal{G}_e$ for some $e\in \mathcal{G}_0$. Then the partial inner isomorphism $\mathcal{I}_g\in \mathcal{I}(Iso(\mathcal{G}))$ and thus $\mathcal{I}_g=\mathcal{I}_{e'}$ for some $e'\in \mathcal{G}_0$. Then $D(\mathcal{I}_g)=D(\mathcal{I}_{e'})$ and $R(\mathcal{I}_g)=R(\mathcal{I}_{e'})$ and since $r(g)=d(g)=e$ it has that $\mathcal{G}_e=\mathcal{G}_{d(g)}=\mathcal{G}_{r(g)}=\mathcal{G}_{e'}$. Thus $e=e'$ and then $gxg^{-1}=\mathcal{I}_g(x)=\mathcal{I}_e(x)=x$ for all $x\in \mathcal{G}_e$ that is $gx=xg$ for all $x\in \mathcal{G}_e$. Hence $\mathcal{G}_e$ is an abelian group and thus $\mathcal{G}$ is an abelian groupoid. On the other hand, let $\mathcal{I}_g\in \mathcal{I}(Iso(\mathcal{G}))$, $g\in \mathcal{G}_e$ and $e\in \mathcal{G}_0$. Then $d(g)=r(g)=e$ and then for $x\in \mathcal{G}_e$, $\mathcal{I}_g(x)=gxg^{-1}=gg^{-1}x=x=\mathcal{I}_e(x)$. That is $\mathcal{I}_g=\mathcal{I}_e$ and the result follows. 3\. If $g,h\in \mathcal{G}$ and $\exists gh$ then $d(g)=r(h)$. Then $D(\Theta (g))=D(\mathcal{I}_g)=\mathcal{G}_{d(g)}=\mathcal{G}_{r(h)}=R(\mathcal{I}_h)=R(\Theta (h))$ and thus $\exists \Theta (g)\circ \Theta (h)$. Now, if $x\in D(\mathcal{I}_{gh})=\mathcal{G}_{d(gh)}=\mathcal{G}_{d(h)}$ then $$\begin{aligned} \Theta (gh)(x) & =\mathcal{I}_{gh}(x) =(gh)x(gh)^{-1}=(gh)x(h^{-1}g^{-1}) \\ & =g(hxh^{-1})g^{-1}=\mathcal{I}_g(\mathcal{I}_h(x))=(\mathcal{I}_g\circ \mathcal{I}_h)(x) \\ &=(\Theta (g)\circ \Theta (h))(x).\end{aligned}$$ That is $\Theta (gh)=\Theta (g)\circ \Theta (h)$ and then $\Theta$ is a homomorphism of groupoids. Finally let $g,h\in \mathcal{G}$ and suppose that $\exists \Theta (g)\circ \Theta (h)$. Then $D(\Theta (g))=R(\Theta (h))$, that is $\mathcal{G}_{d(g)}=\mathcal{G}_{r(h)}$ which implies that $d(g)=r(h)$ and thus $\exists gh$. Hence $\Theta$ is an strong homomorphism. 4\. First note that the strong homomorphism $\Theta$ of item 3 is surjective. Now, we prove that $Ker(\Theta)=\mathcal{Z}(\mathcal{G})$. If $g\in Ker(\Theta)$ then $\Theta (g)=\mathcal{I}_g=\mathcal{I}_e$ for some $e\in \mathcal{G}_0$. Thus $\mathcal{G}_{d(g)}=D(\mathcal{I}_g)=D(\mathcal{I}_e)=\mathcal{G}_e$ and $\mathcal{G}_{r(g)}=R(\mathcal{I}_g)=R(\mathcal{I}_e)=\mathcal{G}_e$ which implies that $\mathcal{G}_{d(g)}=\mathcal{G}_{r(g)}=\mathcal{G}_e$ and thus $d(g)=r(g)=e$. Then $g\in \mathcal{G}_e$ and $gxg^{-1}=\mathcal{I}_g(x)=\mathcal{I}_e(x)=x$ for all $x\in \mathcal{G}_e$, that is, $gx=xg$ for all $x\in \mathcal{G}_e$. Hence $g\in Z(\mathcal{G}_e)\subseteq \mathcal{Z}(\mathcal{G})$. Finally, if $g\in \mathcal{Z}(\mathcal{G})$ then $g\in Z(\mathcal{G}_e)$ for some $e\in \mathcal{G}_0$. If $x\in \mathcal{G}_e$ then $\mathcal{I}_g(x)=gxg^{-1}=xgg^{-1}=x=\mathcal{I}_e(x)$. That is $\Theta (g)=\mathcal{I}_g=\mathcal{I}_e\in \mathcal{I}(\mathcal{G})_0$. Then by using the first isomomorphism theorem (Theorem \[teoremas de isomorfismos\]) we obtain that $\mathcal{G}/\mathcal{Z}(\mathcal{G})$ is isomorphic to $\mathcal{I}(\mathcal{G})$. 5\. If $g\in \mathcal{G}$ then since $\mathcal{H}$ is normal we have $g^{-1}\mathcal{H}_{r(g)}g=\mathcal{H}_{d(g)}$. That is $\mathcal{I}_g(\mathcal{H}_{r(g)})=\mathcal{H}_{d(g)}$ and the result follows. The converse is analogous. [54]{} J. Ávila, V. Marín and H. Pinedo, Groupoids: Substructures and homomorphisms, https://arxiv.org/abs/1905.09389. (2019). D. Bagio and A. Paques, Partial Groupoid Actions: Globalization, Morita Theory, and Galois Theory, Comm. Algebra 40 (2012) 3658-3678. H. Brandt, $\ddot{U}$ber eine Verallgemeinerung des Gruppenbegriffes, Math. Ann., 96 (1926) 360-366. R. Brown, From groups to groupoids: a brief survey, Bull. Lond. Math. Soc., 19, (1987) 113134. D. S. Dummit and R. M. Foote (2004), Abstract Algebra, Wiley and Sons Inc., New Delhi. C. Ehresmann, Oeuvres complétes. Parties I.1, I.2, Topologie algéBrique et géométrie différentielle, 1950. M. V. Lawson (1998), Inverse Semigroups: The Theory of Partial Symmetries, World Scientific, Singapore. Gh. Ivan, Strong Morphisms of Groupoids, [Balkan Journal of Geometry and Its Applications]{}, Vol. 4, No. 1, (1999) 91-102. Gh. Ivan, Algebraic constructions of Brandt Groupoids, Proceeding of the Algebra Symposium, Babes-Bolyai University Cluj, 69-90, 2002. H. Myrnouri and M. Amini, Decomposable abelian groupoids, [Int. Math. Forum.]{}, 5 (2010), 2371-2380. A. Paques, and T. Tamusiunas, The Galois correspondence theorem for groupoid actions, [J. Algebra]{}, 509, 105–123 (2018).
--- abstract: \| We investigate a family of distributions having a property of stability-under-addition, provided that the number $\nu$ of added-up random variables in the random sum is also a random variable. We call the corresponding property a $\nu$-stability and investigate the situation with the semigroup generated by the generating function of $\nu$ is commutative. Using results from the theory of iterations of analytic functions, we show that the characteristic function of such a $\nu$-stable distribution can be represented in terms of Chebyshev polynomials, and for the case of $\nu$-normal distribution, the resulting characteristic function corresponds to the hyperbolic secant distribution. We discuss some specific properties of the class and present particular examples.\ author: - 'L.B. Klebanov,A.V. Kakosyan,S.T. Rachev,G. Temnov' title: On a class of distributions stable under random summation --- Key Words: Stability, random summation, characteristic function, hyperbolic secant distribution. Introduction ============ In many applications of probability theory certain specific classes of distributions have become very useful, usually called “fat tailed” of “heavy tailed” distributions. The [Stable distributions]{} that originate from the Central Limit problem, are probably most popular among the heavy tailed distributions, however there is a wide collection of classes of distributions, all related to Stable ones in many various ways, often these relations are not at all obvious. Besides, certain generalizations of stable distributions are known, using sums of random numbers of random variables (instead of sums with deterministic number of summands), see e.g. Gnedenko [@Gn1983], Klebanov, Mania, Melamed [@KlebanovMM], for the examples of such, including the so-called [$\nu$-stable]{} distributions, introduced independently by Klebanov and Rachev [@KlRa] and Bunge [@Bung]. In the present paper, we focuse on presenting further examples of strictly $\nu$-stable random variables, that could be useful in practical applications, including applications in financial mathematics. Definition of strictly $\nu$-stable r.v.’s, properties and examples =================================================================== In the present section, we give a general insight on strictly $\nu$-stable distributions and describe some examples that have been mentioned in the literature before. Basic definitions ----------------- Let $X,X_1,X_2,\dots,X_n,\dots$ be a sequence of i.i.d. random variables, and let $\{\,\nu_p\,,\,p\in\Delta\,\}$ be a family of some discrete r.v.’s taking values in the set of natural numbers $\mathbb{N}$. Assume that this family does not depend on the sequence $\{X_j,\,j\geq1\}$, and that, for $\Delta\subset(0,1)$, $$\mathbf{E}\,\left[\,\nu_p\,\right]=\frac{1}{p}\,,\,\,\,\forall\,p\in\Delta\,.$$ We say that the r.v. $X$ has a strictly $\nu$–stable distribution, if $\forall\,\,p\in\Delta$ it holds that $$X\overset{d}{=}p^{1/\alpha}\sum_{i=1}^{\nu_p}X_j\,,$$ where $\alpha\in(0,2]$ is called the index of stability. After this general definition, a narrower class is defined for $\alpha = 1/2$. We call the r.v. $X$ a strictly $\nu$–normal r.v., if $\mathbf{E}X=0$, $\mathbf{E}X^2=\infty$, and the following holds: $$X\overset{d}{=}p^{1/2}\sum_{i=1}^{\nu_p}X_j\,,\,\,\,\,\forall\,\,p\in\Delta\,\,.$$ Closely related to the stability property is the property of infinite divisibility, so we also give the following definition. $X$ has a strictly $\nu$–infinitely divisible distribution, if for any $p\in\Delta$, there exists a r.v. $Y^{(p)}$, s.t. $$X\overset{d}{=}\sum_{j=1}^{\nu_p}Y^{(p)}_j\,,\,\,\,\,\mbox{with}\, \,Y^{(p)},Y^{(p)}_1,\dots,Y^{(p)}_n,\dots\,\,\mbox{being iid r.v.'s}\,\,.$$ A powerful tool for investigating distributions’ properties is the [generating function]{}. We shall use the generating function of the r.v. $\nu_p$ denoting it by $\mathcal{P}_p(z):=\mathbf{E}\left[z^{\nu_p}\right]$. Moreover, we denote by $\mathcal{A}$ the semigroup generated by the family $\{\,\mathcal{P}_p\,,\,\,p\in\Delta\}$, with the operation of the functions’ composition. Summary of the known results ---------------------------- With regards to the definitions above, the following results are known (see e.g. [@HeavyTailed] for proofs and details). For the family $\{\,\mathcal{P}_p\,,\,\,p\in\Delta\}$, with $\mathbf{E}\left[\nu_p\right]=\frac{1}{p}$, there exists a strictly $\nu$-normal distribution, iff the semigroup $\mathcal{A}$ is commutative. Suppose that we have a commutative semigroup $\mathcal{A}$. Then the following statements (that we refer to in the sequel as [Properties]{}) are known to be true (see [@HeavyTailed] for proofs and details): 1. The system $$\label{O3} \varphi(t)=\mathcal{P}_p(\varphi(pt)),\,\,\,\forall\,p\in\Delta\,.$$ of functional equations has a solution that satisfies the initial conditions $$\label{O4} \varphi(0)=1\,,\,\,\,\varphi^{\prime}(0)=-1\,.$$ The solution is unique. In addition, there exists a distribution function (cdf) $A(x)$ (with $A(0)=0$)such that $$\label{O5} \varphi(t)=\int\limits_0^\infty e^{-tx}dA(x).$$ 2. The characteristic function (ch.f.) of the strictly $\nu$-normal distribution has the form $$\label{O6} f(t)=\varphi(at^2)\,,\,\,\,\,a>0\,.$$ 3. A ch.f. $g(t)$ is a ch.f. of a $\nu$-infinitely divisible r.v., iff there exists a chf $h(t)$ of an infinitely divisible (in the usual sense) r.v., such that $$\label{InfDiv} f(t)=\varphi(-\ln h(t)).$$ The relation (\[InfDiv\]) allows obtaining explicit representations of ch.f. of strictly $\nu$-stable distributions. Clearly, they are obtained through applying (\[InfDiv\]) to a ch.f. $h(t)$, provided that the r.v. corresponding to $h(t)$ is strictly stable (in the usual sense). Moreover, note that the ch.f. $\varphi(ait)$, $a\in\mathbb{R}^1$, is the ch.f. of an analogue of the degenerate r.v., and that for the r.v. with such ch.f. the following analogue of the Law of Large Numbers exists. Let $X_1,X_2,\dots,X_n,\dots$ be a sequence of iid random variables with the finite absolute value of the first moment, and $\{\,\nu_p\,,\,p\in\Delta\,\}$ a family of r.v.’s taking values in $\mathbb{N}$, independent of the sequence $\{X_j,j=1,2,\dots\}$. Assume that $\mathbf{E}\left[\nu_p\right]=\frac{1}{p}$ and that the semigroup $\mathcal{A}$ is commutative. Then the series $p\sum\limits_{j=1}^{\nu_p}X_j$ is convergent is distribution, as $p\rightarrow0$, and the limit of convergence is a r.v. having the ch.f. $\varphi(ait)$. The proof of this theorem follows straightforwardly from the [Property 1]{} outlined above and from the [Transfer Theorem]{} of Gnedenko, see e.g. [@GnedKor]. In the following paragraph we discuss several particular examples of strictly $\nu$-normal and strictly $\nu$-stable distributions. Examples and the outline of the problem --------------------------------------- [Example 1.]{}\ Assume the following setup:$\nu_p=\frac{1}{p}$ with probability $1$,where $p\in\Delta=\left\{1,\frac{1}{2},\dots,\frac{1}{n},\dots\right\}$, and so $\mathcal{P}_p(z)=z^{1/p}$. Clearly, the corresponding semigroup $\mathcal{A}$ is commutative. Furthermore, $\varphi(t)=\exp\{-t\}=\int\limits_0^\infty e^{-tx}dA(x)$, where $A(x)$ is a cdf with a single unit-sized jump at $x=1$. In this setup the strictly $\nu$-normal ch.f. is the ch.f. of the normal (in the usual sense) r.v. with the zero mean. [Example 2.]{}\ Suppose, $\nu_p$ is the r.v. having a geometric distribution $$\mathbf{P}\{\nu_p=k\}=p(1-p)^{k-1}\,,\,\,\,k=1,2,\dots\,\,\,,\,p\in(0,1)\,.$$ Clearly, here $\mathbf{E}\left[\nu_p\right]=\frac{1}{p}$,and $\mathcal{P}_p(z)=\frac{pz}{1-(1-p)z}$,$p\in(0,1)$. It is quite straightforward to check that $\mathcal{A}$ is commutative. Moreover, a direct calculation gives $\varphi(t)=\frac{1}{1+t}=\int\limits_0^\infty e^{-tx}e^{-x}dx$, i.e. $A(x)$ is the cdf of the exponential distribution. So that a $\nu$-analogue of the strictly normal distribution is the Laplace distribution with the ch.f. $f(t)=\frac{1}{1+at^2}$. [Example 3.]{}\ Let $\mathcal{P}(z)$ be some generating function, with $\mathcal{P}^{\prime}(1)=\frac{1}{p_0}>1$ (so that the introduced notation is $p_0=1/\mathcal{P}^{\prime}(1)$, with the condition $p_0<1$). Consider now a family given by $\mathcal{P}^{0\,n}(z)=\mathcal{P}^{0(n-1)}\left(\mathcal{P}(z)\right)$,$n=1,2,\dots$. Related to that is another family of the r.v.’s $\nu_p$:$\mathcal{P}_p(z)=\mathcal{P}^{0\,n}(z)\,,\,\,p\in\left\{\frac{1}{p_0^n}\,, \,\,n=1,2,\dots\right\}=:\Delta$. Clearly, the semigroup $\mathcal{A}$ coincides with the family $\{\mathcal{P}_p\,,\,\,p\in\Delta\}$.The ch.f. $\varphi(t)$ is a solution of the functional equation $\varphi(t)=\mathcal{P}\left(\varphi(p_0t)\right)$. It can be noted that the content of the paper by Mallows and Shepp [@SheppMall1] is actually based on considering an example identical to the [Example 3]{} above. Probably, neither the authors of that work nor its reviewers were familiar with the works by Klebanov and Rachev [@KlRa] and Bunge [@Bung], which had dealt with exactly the same example a number of years earlier. Like mentioned in Introduction, in the present work we aim in widening the collection of examples that involve random summation with the commutative semigroup $\mathcal{A}$. For that purpose, we address the description of pairs of certain commutative generating functions $\mathcal{P}$ and $\mathcal{Q}$, i.e. the ones for which the balance equality $\mathcal{P}\circ\mathcal{Q}=\mathcal{Q}\circ\mathcal{P}$ holds, – but including only the case when [there exists no]{} such function $\mathcal{H}$ such that $\mathcal{P}=\mathcal{H}^{0k}$ and $\mathcal{Q}=\mathcal{H}^{0m}$for some $k,m\in\mathbb{N}$(which would be exactly the case of the [Example 3]{}). In a general setting, the problem of describing all such commutative pairs of generating functions appears, unfortunately, far too involved to approach. However, certain special cases are rather straightforward for consideration. In order to approach the problem, we will use certain notions typical for the theory of iterations of analytic functions, that we outline in the separate section below. Theoretic justification via iterations of analytic functions ============================================================ Let $\mathcal{P}$ be a rational function with $(\deg)\geq2$. Denote by $\mathcal{P}^{0n}$ its $n$th iteration. The functions $\mathcal{P}$ and $\mathcal{Q}$ are called [conjugates]{}, if there exists a linear-fractional function $R$, such that $\mathcal{P}\circ R=R\circ\mathcal{Q}$. A subset $E$ of the extended complex plane $\overline{\mathbb{C}}$ is called [completely invariant]{}, if its complete inverse image $\mathcal{P}^{-1}(E)$ coincides with $E$. The maximal finite completely invariant set $E(\mathcal{P})$ exists and is called the [exceptional set]{} of the function $\mathcal{P}$.It is always the case that $\mbox{card}\, E(\mathcal{P})\leq2$. Moreover, if $\mbox{card}\, E(\mathcal{P})=1$ then the function $\mathcal{P}$ is a conjugate to a polynomial, while for $\mbox{card}\,E(\mathcal{P})=2$ the function $\mathcal{P}$ is a conjugate to $\mathcal{Q}(z)=z^n\,,\,\,n\in\mathbb{Z}\backslash\{0,1\}$. Clearly, $E(\mathcal{Q})=\{0,\infty\}$. If $\mathcal{P}$ is a rational function, then it is known (see e.g. [@HeavyTailed]) that there is a finite number of open sets $F_i$,$i=1,\dots,r$,which are [left invariant]{} by the operator $\mathcal{P}$ and are such that (in the sequel, we will refer to the two points below as [Conditions]{}) 1. the union $\bigcup\limits_{i=1}^rF_i$ is [dense]{} on the plane; 2. and $\mathcal{P}$ behaves [regularly]{} and in a unique way on each of the sets $F_i$. The latter means that the termini of the sequences of iterations generated by the points of $F_j$ are either precisely the same set, which is then a finite cycle, or they are finite cycles of finite or annular shaped sets that are lying concentrically. In the first case the cycle is [attracting]{}, in the second one it is [neutral]{}. The sets $F_j$ are the [Fatou domains]{} of $\mathcal{P}$, and their union is the [Fatou set]{} $F(\mathcal{P})$ of $\mathcal{P}$. The complement of $F(\mathcal{P})$ is the [Julia set]{} $\mathcal{J}(\mathcal{P})$ of $\mathcal{P}$. Note that $\mathcal{J}(\mathcal{P})$ is either a nowhere dense set (that is, without interior points) and an uncountable set (of the same cardinality as the real numbers), or $\mathcal{J}(\mathcal{P})=\overline{\mathbb{C}}$. Like $F(\mathcal{P})$, $\mathcal{J}(\mathcal{P})$ is left invariant by $\mathcal{P}$, and on this set the iteration is [repelling]{}, meaning that $\,\,\|\,\mathcal{P}(z)-\mathcal{P}(w)\,\|\,>\,\|\,z-w\,\|\,\,$ for all elements $w$ in a neighborhood of $z$ (within $\mathcal{J}(\mathcal{P})$). This means that $\mathcal{P}(z)$ behaves chaotically on the Julia set. Although there are points in the Julia set whose sequence of iterations is finite, there is only a countable number of such points (and they make up an infinitely small part of the Julia set). The sequences generated by points outside this set behave chaotically, a phenomenon called [deterministic chaos]{}. Let $z_0$ be a repelling fixed point of the function $\mathcal{P}$, and let $\lambda=\mathcal{P}^{\prime}(z_0)$. Define $\Lambda\,:\,\,z\rightarrow\lambda z$. Then there exists a unique solution of the Poincaré equation $$F\circ\Lambda=\mathcal{P}\circ F\,,\,\,\,\,F(0)=z_0\,,\,\,\,F^{\prime}(0)=1\,,$$ that is meromorphic in $\overline{\mathbb{C}}$. Now let $$\mathcal{I}(\mathcal{P})=F^{-1}(\mathcal{J}(\mathcal{P}))\,.$$ If for two functions $\mathcal{P}$ and $\mathcal{Q}$ we have $\mathcal{P}\circ\mathcal{Q}=\mathcal{Q}\circ\mathcal{P}$, then they have the same function $F$. There are the two following possibilities: 1. $\mathcal{I}(\mathcal{P})=\mathbb{C}$,in which case $\mathcal{J}(\mathcal{P})=\overline{\mathbb{C}}$,. 2. $\mathcal{I}(\mathcal{P})$ is nowhere dense and consists of analytic cuvrves. Fatou [@Fatou], and Julia [@Julia] investigated the case. It turned out that is this case $\mathcal{P}$ and $\mathcal{Q}$ can be reduced by a conjugancy either to the form $\mathcal{P}(z)=z^m$ and $\mathcal{Q}(z)=z^n$ or to the form $\mathcal{P}(z)=T_m(z)$ and $\mathcal{Q}(z)=T_n(z)$,where $T_k$ is the Chebyshev polynomial determined by the equation $\cos(k\zeta)=T_k(\cos\zeta)$. Main results ============ A new example ------------- Let us return to the study of $\nu$-normal and $\nu$-stable random variables. Recall that we deal with the family $\{\nu_p\,,\,\,p\in\Delta\}$ taking its values in $\mathbb{N}=\{1,2,\dots\}$. As before, we work with the generating function, $\mathcal{P}_p(z)=\mathbf{E}\left[\,z^{\nu_p}\,\right]$,of $\nu_p$. The important result that we stressed says the a strictly $\nu$-normal (resp. strictly $\nu$-stable) r.v. exist iff the semigroup $\mathcal{A}$ generated by $\{\mathcal{P}_p\,,\,\,p\in\Delta\}$ is commutative. If $\mathcal{P}_p\,,\,\,p\in\Delta$, is a rational function (with $\deg\leq2$) satisfying [Condition 2]{} of the above section, then either $\mathcal{P}_p(z)$ is reduced to a form $\widetilde{\mathcal{P}}_p(z)=z^{1/p}$,$p\in\left\{\frac{1}{n}\,,\,n=1,2,\dots\right\}$, and then we deal, in fact, with the classical (deterministic) summation scheme, or $\mathcal{P}_p(z)$ is reduced to the form $\mathcal{P}_p(z)=T_{1/\sqrt{p}}(z)$,$p\in\left\{\frac{1}{n^2}\,,\,n=1,2,\dots\right\}$. Clearly, the polynomial $T_m(z)$ is not a generating function itself, however a function to which it is a conjugate, specifically the function $$\label{Cheb} \mathcal{P}_p(z)=\frac{1}{T_{1/\sqrt{p}}(z)}\,\,,\,\,\,p\in\left\{\frac{1}{n^2}\,,\, n=1,2,\dots\right\}\,\,,$$ is indeed a generating function, – the fact that we prove below. Moreover, below we consider in some details a family of r.v.’s $\left\{\nu_p\,,\,p\in\left\{\frac{1}{n}\,,\,n=1,2,\dots\right\}\,\right\}$ that have generating functions of the form (\[Cheb\]), and investigate the corresponding strictly $\nu$-normal and strictly $\nu$-stable distributions. \[Lem1\] Let $P_n(x)$ be a polynomial with $\deg P_n=k$ by the even powers of $x$, and whose zeros are all within the interval $(-1,1)$.Let $P_n(1)=1$ and polynomial’s coefficient with with $x^n$ be positive. Then for any natural number $k$, the function $$\mathcal{P}(x)=\frac{x^k}{P_n(\frac{1}{x})}$$ is a generating function. Represent $P_n(x)$ as $$P_n(x)=b_0+b_1x+\dots+b_nx^n=b_n\prod\limits_{j=1}^n\left(x-a_j\right)\,,$$ where $a_j$ ($j=1,\dots,n$)are the zeros of the polynomial $P_n$sorted in the order of ascendance. As $P_n$ is a polynomial by the even powers of $x$, then, if $a_j$ is a zero of $P_n$, then $-a_j$ is also a zero of $P_n$. Therefore, $$\begin{aligned} \label{Series} \nonumber \frac{1}{P_n(\frac{1}{x})}&=&\frac{1}{b_n\prod\limits_{j=1}^n\left(\frac{1}{x}-a_j\right)}\\ \nonumber &=&\frac{1}{b_n\prod\limits_{j=1}^{n/2}\left(\frac{1}{x}-a_j\right)\left(\frac{1}{x}+a_j\right)}\\ \nonumber &=&\frac{1}{b_n}\prod\limits_{j=1}^{n/2}\frac{1}{\left(\frac{1}{x}-a_j\right) \left(\frac{1}{x}+a_j\right)}\\ &=&\frac{1}{b_n}\prod\limits_{j=1}^{n/2}\frac{x^2}{1-a_j^2x^2}\end{aligned}$$ Obviously, $$\frac{x^2}{1-a_j^2x^2}=\sum\limits_{k=0}^{\infty}a^{2k}_jx^{2k+2}$$ is a series with positive (non-negative) coefficients, converging when $\|x\|\leq1$. From (\[Series\]), it now follows that $\mathcal{P}(x)=\frac{x^k}{P_n(\frac{1}{x})}$ is a series also convergent when $\|x\|\leq1$, having non-negative coefficients, and $\mathcal{P}(1)=1$. Hence, $\mathcal{P}(x)$ is a generating function of some random variable. Let $T_n(x)$ be a Chebyshev polynomial of degree $n$. Then $$\mathcal{P}(x)=\frac{1}{T_n(\frac{1}{x})}$$ is a generating function of some r.v. taking values in $\mathbb{N}$. When $n$ is an even number, the result follows directly from [Lemma \[Lem1\]]{} and from the properties of Chebyshev polynomials. For odd $n$, consider the representation\ $T_n(x)=xP_{n-1}(x)$, where $P_{n-1}(x)$ is a polynomial by the even degrees of $x$, satisfying the conditions of [Lemma \[Lem1\]]{}. Let us now set $\Delta:=\left\{\frac{1}{n^2}\,,\,n=1,2,\dots\right\}$.Consider the family of generating functions $$\mathcal{P}_p(z)=\frac{1}{T_{1/\sqrt{p}}(1/z)}\,,\,\,\,\,p\in\Delta\,.$$ Clearly, $\mathcal{P}_{p_1}\circ\mathcal{P}_{p_2}=\mathcal{P}_{p_2}\circ\mathcal{P}_{p_1}$ for all $p_1,p_2\in\Delta\,$, due to the well known property of Chebyshev polynomials stating that$T_n(T_m(x))=T_{n\cdot m}(x)$. In other words, semigroup generated by the family $\{\,\mathcal{P}_p\,,\,\,p\in\Delta\}$ is commutative. It follows (see e.g. [@HeavyTailed]) that there exists a solution to the system of equations $$\label{Syst} \varphi(t)=\mathcal{P}_p(\varphi(pt))\,\,,\,\,\,p\in\Delta\,,$$ satisfying initial conditions $$\label{Cond} \varphi(0)=1\,\,,\,\,\,\varphi^{\prime}(0)=-1\,,$$ and the solution is unique. Since $T_n(x)=\cos(n\cdot\arccos x)=\cosh(n\cdot\mbox{arccosh}\,x)$, the direct plugging gives that the function $$\label{PhiCosh} \varphi(t)=1/\cosh\left(\sqrt{2t}\right)$$ satisfies the system (\[Syst\]), as well as the conditions (\[Cond\]). Hence, the function $$\label{Norm} f(t)=\frac{1}{\cosh(at)}\,,\,\,\,a>0$$ is actually a ch.f. of a strictly $\nu$-normal r.v.. The ch.f. (\[Norm\]) is, in fact, well known – it is the ch.f. of the [hyperbolic secant distribution]{}. Clearly, here $a$ is the scale parameter. When $a=1$, it is the case of the [standard hyperbolic secant distribution]{}, whose pdf has the form $$p(x)=\frac{1}{2}\,\mbox{sech}\,\left(\frac{\pi x}{2}\right)\,,$$ while the cdf is $$F(x)=\frac{2}{\pi}\,\mbox{arctan}\,\left[\exp\left(\frac{\pi x}{2}\right)\right]\,.$$ Furthermore, in order to obtain the expression for the ch.f. of strictly $\nu$-stable distributions, one just needs to apply the relation (\[InfDiv\]) to the strictly stable (in the usual sense) ch.f. $h$. An interesting property ----------------------- Note that the function $\varphi$, as represented by (\[PhiCosh\]), can be viewed somewhat interesting on its own, and so we shall address its properties and consider its cdf $A(x)$(which corresponds to $\varphi(t)$ via (\[O5\])). Let $W_1(t)$ and $W_2(t)$,$t\geq0$,be two independent Wiener processes. Consider a r.v. $$\label{Xxi} \xi = \int\limits_0^1W_1^2(t)dt+\int\limits_0^1W_2^2(t)dt\,.$$ This r.v. is well studied, and it is known (see e.g. [@Talacko]) that its Laplace transform equals to $$\mathbf{E}\,\left[\,e^{-t\xi}\,\right]=\frac{1}{\cosh\left(\sqrt{2t}\right)}\,,$$ which coincides with $\varphi(t)$ as given by (\[PhiCosh\]). Hence $A(x)$ is the cdf of the r.v. $\xi$. On the other hand, as follows from Gnedenko’s Transfer Theorem, $$A(x)=\underset{p\rightarrow0} {\lim}\,\,\mathbf{P}\,\{\,p\,\nu_p<x\,\}\,\,.$$ Consequently, the following theorem is valid. \[th41\] Let $\left\{\,\nu_p<x\,,\,\,p\in\Delta\,\right\}$ be a family of r.v.’s having generating functions $$\mathcal{P}_p(z)=\frac{1}{T_{1/\sqrt{p}}(\frac{1}{z})}\,,\,\,\,\,p\in\Delta= \left\{\frac{1}{n^2}\,,\,n=1,2,\dots\right\}\,.$$ Then $$\underset{p\rightarrow0} {\lim}\,\,\mathbf{P}\,\{\,p\,\nu_p<x\,\}=\mathbf{P}\{\xi<x\}\,,$$ where the r.v. $\xi$ is the one defined via (\[Xxi\]). Theorem \[th41\] may be reformulated in the following way. [Let $$\frac{1}{T_{n}(\frac{1}{z})}= \sum_{k=0}^{\infty}p_k(n)z^k.$$ Then $$\lim_{n \to \infty} \sum_{k=0}^{[n^2x]} p_k(n) = \mathbf{P}\{\xi<x\}.$$ ]{} ![Plot of the $\sum_{k=0}^{[n^2x]} p_k(n) $ as the function of $n=2, \ldots , 50$](ChebyshevTMyAsympt2.eps) \[fig1\] On Figure 1, the plot of the $\sum_{k=0}^{[n^2x]} p_k(n)$ is given as a function of $n$ starting with $n=2$ until $n=50$. We see that the functions attains the constant level rather quickly, and therefore it is possible to use the asymptotic result for $n>25$. Let $X$ be a r.v. having the standard hyperbolic secant distribution. Then its distribution can be represented in the form of a scale mixture of a normal distribution with zero mean and standard deviation $\sqrt{\xi}$, where $\xi$ is defined via (\[Xxi\]). To prove the above, one just needs to write the ch.f. of $X$ in the form $\int\limits_0^\infty e^{-t^2x}dA(x)$,and note that $e^{-t^2x}$ is actually the ch.f. of the standard Normal r.v. $N(0,\sigma^2)$($\sigma^2=x$), while $A(x)$ is the cdf of $\xi$. Note that there is a certain analogy between the representation $A(x)$ as the cdf of the r.v. $\xi$ from (\[Xxi\]) and the corresponding result in the scheme of the random summation with geometric distribution (see e.g. [@HeavyTailed]). Specifically, considering the family $\left\{\nu_p\,,\,\,p\in(0,1)\right\}$ having the geometric distribution $\mathbf{P}\,\{\,\nu_p=k\,\}=p(1-p)^{k-1}$,$k=1,2,\dots$, the function$\varphi$turns into $$\varphi(t)=\frac{1}{1+t}=\int\limits_0^\infty e^{-tx}dA_1(x)\,,$$ where $A_1(x)$ is the cdf of the exponential distribution,i.e. $A_1(x)=1-e^{-x}$for$x>0$and $A_1=0$for$x\leq0$. It can be checked that if $\eta_1$ and $\eta_2$ are two independent standard Normal r.v.’s, then $A_1$ is a cdf of the r.v. $\xi_1=\eta^2_1+\eta^2_2$,which is, in a way, related to (\[Xxi\]). Characterizations ----------------- Let us now turn to the characterizations of the distribution of the r.v. (\[Xxi\]) and of the hyperbolic secant distribution. Let $X_1,\dots,X_n,\dots$ be a sequence of non-negative iid random variables, and $\nu_p\,,\,\,p\in\left\{\frac{1}{n^2}\,,\,n=2,\dots\right\}$, is a family of the r.v.’s having the generating function $\mathcal{P}_p(z)=\frac{1}{T_{1/\sqrt{p}}(\frac{1}{z})}$,independent of the sequence $\{X_j\,,\,\,j\geq1\}$. If, for some fixed $p\in\Delta$, $$\label{X1d} X_1\overset{d}{=}\,p\sum\limits_{j=1}^{\nu_p}X_j\,\,,$$ (where “$\overset{d}{=}$” is the equality in distribution), then $X_1$ has the distribution whose Laplace transform is $$\label{XLapl} \mathbf{E}\left[\,e^{-tX}\,\right]=\frac{1}{\cosh\left(\sqrt{at}\right)}\,,\,\,a>0\,.$$ The equality (\[X1d\]), in terms of the Laplace transform $\Psi(t)=\mathbf{E}\,e^{-tX}$,can be represented as $$\label{PhiLap} \Psi(t)=\mathcal{P}_p\left(\Psi(pt)\right)\,.$$ Clearly, the function $$\Psi_a(t)=\frac{1}{\cosh\left(\sqrt{at}\right)}$$ satisfies (\[PhiLap\]) for any $a>0$ and, moreover, is analytic in the strip $\,\|\,t\,\|\,<r$($r>0$). In the following, we use the results of the book by Kakosyan, Klebanov and Melamed [@KKM]. Example 1.3.2 of this book shows that $\{\,\Psi_a\,,\,\,a>0\,\}$ forms a strongly $\varepsilon$-positive family, where $\varepsilon$ is a set of restrictions of Laplace transforms of probability distributions given in $R_+$ on an interval $[0,T]$ $(0<T<r)$. Clearly, the operator $A\,:\,f\,\rightarrow\,\mathcal{P}_p(f(pt))$ on $\varepsilon$ is intensively monotone. The result follows from Theorem 1.1.1 of the above mentioned book (page 2). Let $X_1,\dots,X_n,\dots$ be a sequence of non-negative iid random variables, having a symmetric distribution, while $\left\{\nu_p\,,\,\,p\in\Delta\right\}$ is the same family as in the previous Theorem. If, for some fixed $p\in\Delta$, $$\label{X1d2} X_1\overset{d}{=}\,p^{1/2}\sum\limits_{j=1}^{\nu_p}X_j\,\,,$$ then $X_1$ has the hyperbolic secant distribution whose ch.f. is $$\label{XLap2} f(t)=\frac{1}{\cosh\left(at\right)}\,,\,\,a>0\,.$$ Quite analogous to the proof of the previous Theorem, with the difference that instead of Example 1.3.2, the use of the Example 1.3.1 from [@KKM] is sufficient. Other examples ============== There exist examples of the pairs of commutative functions, which are [not rational]{}. Here we refer to the two classes of such functions, the first of which was investigated by Melamed [@MLN] and the second appears at first in the present work. [Example I.]{} (See Melamed [@MLN] for detailed study) Consider the family of generating functions $$\label{Ex1} \mathcal{P}_p(z)=\frac{p^{1/m}\,z}{(1-(1-p)\,z^m)^{1/m}}\,\,,$$ where $p\in(0,1)$,and $m$ is a fixed positive integer. Obviously, in the case $m=1$, $\mathcal{P}_p(z)$ reduces to the generating function of the geometric distribution, and has already been mentioned this case above. Hence, assume that $m\geq2$. In that case, it is easy to check that $$\label{Ex1-2} \varphi(t)=\frac{1}{(1+mt)^{1/m}}\,\,,$$ and therefore the ch.f. of the strictly $\nu$-normal distribution (for the family $\left\{\nu_p\,,\,\,p\in\Delta\right\}$ having the generating function (\[Ex1\])) has the form $$\varphi(t)=\frac{1}{(1+mat^2)^{1/m}}\,\,,$$ with a parameter $a>0$. [Example II.]{} Consider the family of functions $$\label{Ex2} \mathcal{P}_p(z)=\frac{1}{\left(T_{1/\sqrt{p}}\left(\frac{1}{z^m}\right)\right)^{1/m}}\,\,,$$ where $p\in\left\{\frac{1}{n^2}\,,\,n=2,\dots\right\}$,and $m\geq1$(an integer). Using a slightly modified version of the proof of [Lemma 1]{}, it is easy to check that $\mathcal{P}_p(z)$ is a generating function of some r.v. $\nu_p\,,\,\,p\in\left\{\frac{1}{n^2}\,,\,n=2,\dots\right\}$ for any fixed whole number $m\geq1$ (surely, both $\mathcal{P}_p$and$\nu_p$ both depend on $m$, but we omit this dependence in the notation). The case $m=1$ has already been considered above. For $m\geq2$ analogous methods are applicable, and so will refer to the results only. Specifically, $$\label{Ex2-2} \varphi(t)=\frac{1}{\left(\cosh\sqrt{2mt}\right)^{1/m}}\,\,,$$ while the ch.f. of the corresponding strictly $\nu$-normal distribution has the form $$\label{Ex2-3} f(t)=\frac{1}{\left(\cosh at\right)^{1/m}}\,\,,$$ where $a>0$. Note that in the case $m=2$, we have the following expressions for the distributions whose Laplace transforms are (\[Ex1-2\]) and (\[Ex2-2\]). For $m=2$, the formula (\[Ex1-2\]) gives $$\varphi(t)=\frac{1}{\sqrt{1+2t}}\,\,.$$ This function is the Laplace transform of the distribution of the r.v. $X^2$,with $X$ being the standard Normal r.v. In a similar way, (\[Ex2-2\]) gives for $m=2$ $$\varphi(t)=\frac{1}{\sqrt{\cosh\sqrt{4t}}}\,\,.$$ This function is the Laplace transform of the distribution of the r.v. $I=\int\limits_0^1X^2(t)dt$,where $X(t)$ is the standard Wiener process. [10]{} J. Bunge (1996). Compositions semigroups and random stability. Annals of Probab., [24]{}, 1476-1489. P. Fatou (1917). Sur les substitutions rationnelles. Comptes Rendus de l’Académie des Sciences de Paris, [164]{}, 806-808 B.V. Gnedenko (1983). On some stability theorems. Lecture Notes in Math., [982]{}, 24-31, Springer, Berlin. B.V. Gnedenko and V.Yu. Korolev (1996). Random Summation: Limit Theorems and Applications. CRC Press, Boca Raton. G. Julia (1918). Mémoire sur l’iteration des fonctions rationnelles, Journal de Mathématiques Pures et Appliquées, [8]{}, 47-245. A.V. Kakosyan, L.B. Klebanov and I.A. Melamed (1984) Characterization of Distributions by the Method of Intensively Monotone Operators, Springer, Berlin-Heidelberg. L.B. Klebanov (2003). Heavy Tailed Distributions. Matfyz-press, Prague. L.B. Klebanov, G.M. Maniya and I.A. Melamed (1984). A problem of Zolotarev and analogs of infinitely divisible and stable distributions in a scheme for summing a random number of random variables. Theory Probab. Appl., [29]{}, 791–794. L.B. Klebanov and S.T. Rachev (1996). Sums of a random number of random variables and their approximations with $\nu$-accompanying infinitely divisible laws. Serdica, [22]{}, 471-498. I.A. Melamed (1989). Limit theorems in the set-up of summation of a random number of independent and identically distributed random variables. Lecture Notes in Math., [1412]{}, Springer, Berlin, 194-228. C.L. Mallows, L.A. Shepp (2005). B-stability. J. Applied Probability. J.Talacko (1956). “Perks” distributions and their role in the theory of Wiener’s stochastic variables, Trabajos de Estadistica, [7]{}, 159-–174.
--- abstract: 'The impact of coherent light propagation on the excitation and fluorescence of thorium nuclei in a crystal lattice environment is investigated theoretically. We find that in the forward direction the fluorescence signal exhibits characteristic intensity modulations dominated by an orders of magnitude faster, sped-up initial decay signal. This feature can be exploited for the optical determination of the isomeric transition energy. In order to obtain a unmistakable signature of the isomeric nuclear fluorescence, we put forward a novel scheme for the direct measurement of the transition energy via electromagnetically modified nuclear forward scattering in a three-level $\Lambda$ configuration.' author: - 'Wen-Te' - Sumanta - 'Christoph H.' - Adriana title: 'Coherence enhanced optical determination of the $^{229}$Th isomeric transition' --- While atomic clocks based on the microwave ground state hyperfine transition of $^{133}$Cs [@Confi] have revolutionized the field of time-frequency metrology, general agreement exists that the next two orders of magnitude in precision pose serious challenges [@Cam12]. To circumvent this problem, an exciting alternative is to develop clocks based on a nuclear transition. The narrow line-widths of nuclear transitions and the isolation from external perturbations promise amazing stability. The optical transition from the ground state to the isomeric, i.e., long lived, first excited state of $^{229}$Th was originally proposed by Peik and Tamm in 2003 [@Pei03] as a suitable candidate for a nuclear clock. Out of the entire known nuclear chart, the $^{229}$Th ground state doublet consisting of the ground state $^{229g}$Th and the isomeric state $^{229m}$Th offers a unique transition well within vacuum ultraviolet (VUV) with energy below 10 eV [@Bec]. This transition is spotlighted also due to its tempting potential for testing the temporal variation of the fine structure constant [@Fla06], building a nuclear laser in the optical range [@Tka11], or providing an exciting platform for nuclear quantum optics and coherent control [@Olga99; @NQO2006; @Stirap11; @xpm12] of VUV photons. A key obstacle on the way is the uncertainty about the transition energy: measurements up to date have been only indirect, suggesting an energy of $\sim 7.8\pm 0.6$ eV [@Bec]. One intensely pursued approach for a more precise measurement of the $^{229}$Th isomeric transition and the realization of a nuclear frequency standard involves nuclear spectroscopy of a macroscopic number of thorium nuclei doped in VUV-transparent crystals [@Rel10; @Kaz12]. Due to the high doping density of up to $10^{18}$ Th/cm$^3$, crystals such as LiCaAlF$_{6}$ [@Rel10] or CaF$_{2}$ [@Kaz12] offer the possibility to increase the nuclear excitation probability, an important requirement especially considering the very narrow radiative width of the transition, at present estimated at approximately 0.1 mHz [@Tka11]. Both LiCaAlF$_{6}$ and CaF$_{2}$ have large band gaps and present good transparency at the probable transition wavelength, such that the interplay with electronic shells in processes such as the electronic bridge [@eb2010], internal conversion [@newprl] or nuclear excitation via electron capture or transition [@NEECT] can be neglected. The disadvantage of the crystal lattice environment is that inhomogeneous broadening due to temperature-dependent shifts of the transition energy and in particular spin-spin relaxation compromise the Rabi [@Riehle2005], Ramsey [@Riehle2005; @Boyd2006] or hyper-Ramsey [@Yudin2010] interrogation schemes, commonly used in high-performance frequency standards. Instead, clock interrogation by fluorescence spectroscopy was proposed [@Kaz12]. Significant suppression of the inhomogeneous broadening is expected as long as all nuclei experience the same crystal lattice environment and are confined to the Lamb-Dicke regime, i.e., the recoilless transitions regime [@Dick53; @Rel10]. However, together with the broad-band nature of the excitation, these very conditions lead to coherent light propagation through the sample and enhanced transient fluorescence in the forward direction, with a speed up of the initial decay depending primarily on the sample optical thickness. These effects are well known from nuclear forward scattering of synchrotron radiation (SR) [@vanB94; @Smir96] driving Mössbauer nuclear transitions in the x-ray regime, and from the interaction of atomic systems with visible and infrared light [@crisp; @HL83; @R84], but have so far never been addressed in the context of the $^{229}$Th isomeric transition. In this work we investigate the effect of coherent light propagation on the excitation and fluorescence signal of the VUV isomeric transition in $^{229}$Th. We find that the line-width of the light scattered in the forward direction is dynamically broadened by several orders of magnitude ($\sim 10^{6}$) due to the coherent decay of the large number of $^{229}$Th nuclei in the crystal and the strong dispersion close to the resonance frequency. While the dynamical fluorescence in the forward direction would therefore not be suitable for clock interrogation, it offers an enhanced and faster signal, advantageous for both a high signal-to-noise ratio and a shorter detection time. In order to exploit the full potential of the dynamical broadening we put forward a quantum optics scheme based on quantum interference induced by two coherent fields in a three level $\Lambda$-configuration as a novel way to identify the isomeric transition energy. The proposed setup reminding of electromagnetically induced transparency (EIT) provides a clear signature for the excitation of the nuclear transition and enhanced precision in the optical determination of the transition frequency compared to a direct fluorescence experiment using only one field. The reduced transition probability of the nuclear magnetic dipole ($M1$) $^{229}$Th isomeric transition has been evaluated theoretically to $B(M1)\simeq 0.032$ Weisskopf units [@Tka98; @Tka11]. This value corresponds to the very narrow radiative transition width of approximately 0.1 mHz. Since such narrow-bandwidth sources are not available in the VUV region, broadband excitation, with either synchrotron or VUV laser light, is envisaged to be employed in order to resonantly drive the isomeric transition. As long as elastic, recoil-free scattering of the incident light occurs, the contributions of all nuclei are spatially in phase in the forward direction and interfere coherently [@vanB99]. To the forward signal contribute not only the resonant frequency components, which are absorbed and reemitted, but also the non-resonant components, which experience dispersion. Thus the time evolution of the forward scattering response does not follow a natural exponential decay as expected for fluorescence involving a single-scattering event, but exhibits pronounced intensity modulations characteristic for the coherent resonant pulse propagation [@vanB99]. This modulation is known in nuclear condensed-matter physics under the name of dynamical beat and for a single-resonance and a short $\delta(t)$-like exciting pulse has the form [@crisp; @kagan; @bible] $E(t)\propto \xi \exp^{-\tau/2} J_1(\sqrt{4\xi\tau})/\sqrt{\xi\tau}$, where $E$ is the transmitted pulse envelope, $\tau$ a dimensionless time parameter $\tau=t/t_0$ with $t_0$ denoting the natural lifetime of the nuclear isomer, $\xi$ the optical thickness and $J_1$ the Bessel function of the first kind. The optical thickness is defined as $\xi=N\sigma L /4$ [@bible], where $N$ is the number density of $^{229}$Th nuclei, $\sigma$ the resonance cross section and $L$ the sample thickness. For a dopant density of $10^{18}$ Th/cm$^3$ and a sample with $L=1$ cm, we obtain $\xi\simeq 10^6$. From the asymptotic behavior of the Bessel function of first kind we see that for early times $\tau\ll 1/(1+\xi)$ immediately after the excitation pulse (chosen as $t=\tau=0$) the response field has the form $E(t)\propto \xi \exp^{-(1+\xi)\tau/2}$, showing the speed-up of the initial decay. A parallel has been established between this faster decay and the superradiant decay in single-photon superradiance [@Scu06]. For the case of the 1 cm crystal doped with $^{229}$Th considered above, the speed-up would correspond to an enhancement of the decay rate by six orders of magnitude. This value significantly reduces the gap between the incoherent relaxation time of the excited state population (the natural radiative decay) and the short coherence time due to the crystal lattice effects, bringing inhomogeneous and dynamical broadening on the same order of magnitude. At later times, the decay becomes subradiant, i.e., with a slower rate comparable to the incoherent natural decay rate due to destructive interference between radiation emitted by nuclei located at different depths in the sample. A symmetric excitation of the sample would lead to the creation of a timed Dicke state [@Ralf2010] and a prolonged superradiant decay. So far this has been achieved for $^{57}$Fe nuclei embedded in the center of a planar waveguide in the first measurement of the collective Lamb shift in nuclei [@Ralf2010]. One should however keep in mind that the dynamical beat behavior is by no means restricted to nuclear systems and the x-ray radiation wavelength, having been observed over a large wavelength range $\lambda\simeq 1$ Å$-1$ m in a variety of systems (see [@vanB99] and references therein). In the case of $^{229}$Th doped in VUV transparent crystals both conditions for (i) recoilless, coherent excitation and decay and (ii) broadband excitation are fulfilled for scattering in the forward direction. The incident laser or SR pulse duration is much shorter than the nuclear lifetime and provides broadband excitation. Due to the crystal transparency at the nuclear transition frequency, the main limiting factor for coherent pulse propagation, namely electronic photoabsorption, is not present in this case. We are interested in the initial superradiant response which occurs on a time scale much faster than the incoherent radiative decay and can facilitate the optical determination of the isomeric transition energy. As a consequence of the hyperfine splitting affecting the $^{229}$Th nuclei in the crystal lattice, an analytical result for the scattered field intensity is not available and the numerical solution of the Maxwell-Bloch equations [@Scully2006] has to be evaluated instead. This treatment allows also to take into account the inhomogeneous broadening occurring due to spin-spin relaxation in the crystal sample. We use in the following the quadrupole structure with hyperfine level energies given by $E_{m} \simeq Q_{is(g)}(1-\gamma_{\infty})\phi_{zz}[3m^{2}-I_{is(g)}(I_{is(g)}+1)]/ [4I_{is(g)}(2I_{is(g)}-1)]$ [@Abr61], where $Q_{is(g)}$ = 1.8 $e$b (3.15 $e$b), with $e$ the electron charge, is the quadrupole moment of the isomeric (ground) nuclear state, $\gamma_{\infty} = -(100-200) $ is the antishielding factor and $(1-\gamma_{\infty})\phi_{zz} \sim -10^{18}$ V/cm$^{2}$ electric field gradient [@Tka11; @Bem88; @Fei69]. Fig. \[fig1\](a) shows the energy scheme of $^{229}$Th with the electric quadrupole splitting [@Tka11; @Kaz12] of the ground and excited $^{229}$Th nuclear states of spins $I_g$=5/2 and $I_e$=3/2, respectively. Here we adopt the recently proposed $^{229}$Th:CaF$_{2}$ crystal [@Kaz12] under the thermal equilibrium condition [@Tka11], i.e., all the nuclei are equally populating the two ground states of $m_{g}=\pm 5/2$ at sub-kelvin temperatures. Plain fluorescence spectroscopy of the isomeric nuclear transition is plagued by a small signal-to-background ratio [@Rellergert2010b]. For both determination of the exact nuclear transition frequency and for the nuclear clock interrogation schemes, this is a major issue, especially due to the weak coupling of the nuclear transition to the radiation field. According to current experimental data [@Rellergert2010b], the main source of background photons is the $\alpha$-decay of the $^{229}$Th ground state with the total counting rate on the order of MHz and $4\pi$ spatial distribution. A possible solution is offered by coherent light scattering, which ensures that a directional signal collecting in the forward direction benefits of a more favorable signal-to-background ratio. However, impurities and color centers (which unfortunately may occur during the experiment as irradiation effects) can cause residual absorption and may interfere with the $^{229}$Th optical spectroscopy even in the forward direction. This is why a clear signature of the nuclear isomeric transition $^{229g}$Th$\rightarrow$ $^{229m}$Th is desirable. In the following we put forward a quantum optics scheme designed to provide such a nuclear fluorescence signature by employing two VUV fields in a $\Lambda$-type three-level system in a setup reminding of EIT [@Scully2006]. ![\[fig1\] (a) Quadrupole level splitting for the $^{229}$Th ground and isomeric states in the $^{229}$Th:CaF$_{2}$ crystal lattice environment. (b) A left circularly polarized probe field drives the $\|\frac{5}{2}\frac{5}{2}\rangle\rightarrow\|\frac{3}{2}\frac{3}{2}\rangle$ transition. (c) The electromagnetically modified forward scattering setup. The red thick arrow denotes the strong coupling field while the blue thin arrow shows the weak probe field. The detunings of the probe and coupling fields are denoted by $\triangle_{p}$ and $\triangle_{c}$, respectively. ](fig1.eps){width="8.5cm"} A radiation pulse denoted in the following as probe driving the relevant nuclear transition shines perpendicular to the nuclear sample, as shown in Fig. \[fig1\](b). We consider a left circularly polarized weak VUV probe field driving the $m_{e}-m_{g}=-1$ magnetic dipole transition, where $m_{e}$ and $m_{g}$ denote the projections of the excited and ground state nuclear spins on the quantization axis, respectively. Additionally, as shown in Fig. \[fig1\](c), we apply a strong linearly polarized continuous wave (CW) denoted as coupling laser field driving the $m_{e}-m_{g}=0$ transition. A $\Lambda$-type scheme is formed due to the combination of initial nuclear population and the selected field polarizations. The purpose of applying the coupling field is to create a dressed state which splits the nuclear resonance into a doublet [@Shvydko1999H] via the Autler-Townes effect [@AutlerGlover], giving rise to electromagnetically modified time spectra. The resulting beating can be then used as a specific signature for the detection of the nuclear fluorescence photons. We study the coherent pulse propagation for both setups in Fig. \[fig1\](b) (assuming $\Omega_c=0$) and Fig. \[fig1\](c) by numerically determining the dynamics of the density matrix $\widehat{\rho}$ via the Maxwell-Bloch equations [@Shv99; @Scully2006; @Palffy2008; @xpm12]: $$\begin{aligned} && \partial_{t}\widehat{\rho} = \frac{1}{i\hbar}\left[ \widehat{H},\widehat{\rho}\right]+\widehat{\rho}_{s}\, , \nonumber\\ && \frac{1}{c}\partial_{t}\Omega_{p}+\partial_{y}\Omega_{p}=i\eta a_{31}\rho_{31}\, , \label{eq1}\end{aligned}$$ with the interaction Hamiltonian given by $$\widehat{H} = -\frac{\hbar}{2}\left( \begin{array}{cccc} 0 & 0 & a_{31}\Omega^{}_{p}\\ 0 & -2(\Delta_{p}-\Delta_{c}) & a_{32}\Omega^{}_{c}\\ a_{31}\Omega_{p} & a_{32}\Omega_{c} & -2\Delta_{p} \end{array} \right)\, , \nonumber$$ and the decoherence matrix $$\widehat{\rho}_{s} = -\left( \begin{array}{cccc} -a^{2}_{31}\Gamma\rho_{33} & \gamma_{21}\rho_{12} & (\gamma_{31}+\frac{\Gamma}{2})\rho_{13}\\ \gamma_{21}\rho_{21} & -a^{2}_{32}\Gamma\rho_{33} & (\gamma_{32}+\frac{\Gamma}{2})\rho_{23}\\ (\gamma_{31}+\frac{\Gamma}{2})\rho_{31} & (\gamma_{32}+\frac{\Gamma}{2})\rho_{32} & \Gamma\rho_{33} \end{array} \right). \nonumber$$ In the equations above $\rho_{jk}=A_{j}A^{}_{k}$ for $\{j,k\}\in \{1,2,3\}$ are the density matrix elements of $\widehat{\rho}$ for the nuclear wave function $\|\psi\rangle=A_{1}\|\frac{5}{2}\frac{5}{2}\rangle+A_{2}\|\frac{5}{2}\frac{3}{2}\rangle+A_{3}\|\frac{3}{2}\frac{3}{2}\rangle$ with the nuclear hyperfine levels shown in Fig. \[fig1\](a). Furthermore, $(a_{31},a_{32})=(\sqrt{2/3},-2/\sqrt{15})$ are the corresponding Clebsch-Gordan coefficients. The matrix $\widehat{\rho}_{s}$ describes the decoherence due to spin relaxation $(\gamma_{31},\gamma_{32},\gamma_{21})=2\pi\times(251,108,30)$ Hz [@Kaz12] and $\Gamma=0.1$ mHz [@Tka11] denotes the spontaneous decay. The parameter $\eta$ is defined as $\eta=\frac{\Gamma\xi}{2L}$, and we consider a target thickness of $L=1$ cm. Further notations are $\Omega_{p(c)}$ for the Rabi frequency of the probe (coupling) fields which is proportional to the electric field $\vec{E}$ of the probe (coupling) VUV radiation [@Scully2006; @Palffy2008] and $c$ the speed of light. ![\[fig2\] The time spectra of the forward scattered signal (a) without coupling field, (b) with coupling field. For all spectra the detunings $\triangle_{p}=\triangle_{c}=\triangle$ take values between $0\leq\triangle\leq 10^{10}\Gamma$. The yellow filled area below the gray dashed double-dotted line delimits the region $0\leq\triangle\leq 10^{5}\Gamma$. In (a) the values for $\triangle=10^{9}\Gamma$ and $10^{10}\Gamma$ are smaller than the displayed scale. ](fig2.eps){width="8.5cm"} Since the electric quadrupole splitting of ground states can be experimentally determined via standard nuclear magnetic resonance (NMR) techniques [@Kaz12], we may set the two detunings of the coupling and probe fields to be identical but unknown initially, i.e., $\triangle_{c}=\triangle_{p}=\triangle$. We numerically solve Eq. (\[eq1\]) with $\xi=10^{6}$ and $\Omega_{c}$ with a laser intensity of 2kW/cm$^{2}$ and scan the region $0\leq\triangle\leq 10^{10}\Gamma$. The time spectra for the setup in Fig. \[fig1\](b) are presented in Figs. \[fig2\](a). In the absence of the coupling field, the probe only interacts with a two-level nuclear system, and the corresponding time spectra are less sensitive to the detuning $\triangle$. For $\triangle=10^{9}\Gamma$ and $10^{10}\Gamma$, the probe field will not create any nuclear excitation and just passes through the crystal. The $^{229}$Th nuclei will start to coherently scatter probe photons when $\triangle\leq 10^{8}\Gamma$, and with smaller detunings the time spectra become identical for $\triangle\leq 10^{7}\Gamma$. Essentially we could approach the wanted energy of $^{229m}$Th with the precision of $10^{7}\Gamma$, i.e. about 1 kHz, by measuring the signal photons scattered in the forward direction. As a disadvantage, since the spectra do not present any specific feature to confirm the excitation of the isomer, background from other unwanted electronic processes may present an identical scattering pattern at probe laser frequencies far away from the nuclear resonance. We turn now to the two-field setup presented in Fig. \[fig1\](c), which is more detuning-sensitive and provides specific identification features in the scattering time spectra. In Fig. \[fig2\](b) we present our results for the electromagnetically modified forward scattering spectra. With the influence of $\Omega_{c}$ the spectra are much more sensitive to the detuning, and probe photons start to interact with the $^{229}$Th nuclei already at $\triangle\leq 10^{10}\Gamma$. The enhanced sensitivity of the electromagnetically modified setup in Fig. \[fig1\](c) is due to the detuning-sensitive dispersion relation which leads to a unique time spectrum for each combination of input probe field and coupling field strength and detuning. A comparison between Figs. \[fig2\](a) and \[fig2\](b) shows that the coupling field has introduced additional beatings in the spectrum which can serve as a clear signature of the nuclear excitation. Furthermore, a significant advantage of our scheme is that the shapes of the spectra are not identical until $\triangle\leq 10^{5}\Gamma$. A fit of the theoretical and experimental curves can therefore be employed to determine the nuclear transition frequency. By scanning the detunings $\triangle$, i.e., by varying the known frequencies of probe and coupling simultaneously, several $\triangle$-dependent forward scattering time spectra can be measured. A fit with the theoretical curves in Fig. \[fig2\](b) allows the determination of the detuning value and thus of the $^{229g}$Th$\rightarrow$ $^{229m}$Th transition frequency down to a precision of $10^{5}\Gamma$, i.e., 10 Hz. Once the isomeric transition frequency has been identified, the fluorescence clock interrogation scheme using the incoherent response emitted at an angle to the excitation pulse direction can be employed for the nuclear frequency standard. A brief estimate of the experimental counting rate for the signal and background photons in the forward direction is performed for the case of a $^{229}$Th:CaF$_{2}$ crystal with the size 3 mm$\times$ 3 mm $\times$ 10 mm and $^{229}$Th concentration of $10^{18}$ cm$^{-3}$. Given the 7880 yr half life of the ground state $^{229}$Th, and the assumed 0.3 background photon per $\alpha$-decay [@Rellergert2010b], the total counting rate of background photons (in $4\pi$ solid angle) is then $0.75$ MHz, which could be significantly suppressed down to $1.8$ Hz by registering only signals within $1^{\circ}\times 1^{\circ}$ in the forward direction. The background generated by the VUV pulse itself in the forward direction can be eliminated with a chopper or employing the nuclear lighthouse effect [@RalfNLE]. Thus, $20$ Hz would suffice as counting rate of the scattered probe photons in the forward direction, corresponding to a signal-background ratio larger than 10. On the other hand, the spin relaxation makes the coherences decay according to $\rho_{31}\sim e^{-\gamma_{31}t}$ since $\gamma_{31}\gg\Gamma$ [@Kaz12]. The coherent effects should therefore be observed within the time scale of ms allowing $100\sim 500$ shots per second. A CW laser source in the VUV region is presently available only within limitations. The KBe$_2$BO$_3$F$_2$ (KBBF) crystals [@kbbf] have been successful in generating narrow-band VUV radiation via harmonic generation owing to their wide transparency and large birefringence necessary for phase-matched frequency conversion processes in this frequency region [@vuv_kbbf1; @vuv_kbbf2]. In particular, a quasi-CW laser with a 10 MHz repetition rate and 20 ps pulse duration has been achieved [@vuv_kbbf2]. A CW coupling VUV laser at around 160 nm wavelength could also be generated by sum frequency mixing in metal vapors or driving a KBBF crystal with Ti:Sapphire laser [@nolting; @togashi]. The circularly polarized probe laser on the other hand may be generated via nonlinear sum-frequency mixing [@irrgang], or a harmonic of a VUV frequency comb [@jones; @ozawa] around the isomeric wavelength. In the absence of a CW coupling field for the electromagnetically modified forward scattering scheme, one can use a moderate external magnetic field ($B=1$ G) to split the single transition in Fig. \[fig1\](b) into a doublet via magnetic dipole interaction. The magnetic-field-dependent quantum beat arising offers a signature that the nuclear isomer has been excited and presents reasonable sensitivity to the laser detuning. Numerical results for the LiCAF crystal for which the calculation of the magnetic hyperfine splitting is straightforward show that the detuning sensitivity goes as far as $\Delta=10^7$, i.e., one could determine the nuclear transition frequency within 1 kHz with this method. 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--- abstract: 'We use numerical simulations to examine two-dimensional particle mixtures that strongly phase separate in equilibrium. When the system is externally driven in the presence of quenched disorder, plastic flow occurs in the form of meandering and strongly mixing channels. In some cases this can produce a fast and complete mixing of previously segregated particle species, as well as an enhancement of transverse diffusion even in the absence of thermal fluctuations. We map the mixing phase diagram as a function of external driving and quenched disorder parameters.' author: - 'A. Lib['' a]{}l$^{1,2}$, C. Reichhardt$^{3}$ and C.J. Olson Reichhardt$^{3}$' title: Enhancing Mixing and Diffusion with Plastic Flow --- There have been a growing number of experiments on collections of small particles such as colloids moving over periodic or complex energy landscapes generated by various optical methods [@Review; @Grier; @Babic; @Korda; @Bechinger; @Spalding; @Lee; @Lutz] or structured surfaces [@Ling]. Such static and dynamical substrates can produce a variety of new particle segregation mechanisms [@Grier; @Lee; @Korda; @Spalding] as well as novel types of logic devices [@Babic]. Driven particles on periodic substrates can also exhibit enhanced diffusive properties such as the recently proposed giant enhancement of the diffusion which occurs at the threshold between pinned and sliding states [@Marchesoni; @Reimann; @Jay; @Bleil; @Lacasta; @Lee]. This enhancement has been demonstrated experimentally for colloids moving over a periodic optical substrate [@Lee] and could be important for applications which require mixing and dispersing of different species of particles [@Lee]. A limiting factor for using diffusion enhancement to mix particles is that the diffusion is enhanced only in the direction of the external drive. For instance, in a two-dimensional system with a corrugated potential that is tilted in the direction of the corrugation barriers, there is no enhancement of the diffusion in the direction transverse to the corrugation barriers at the pinned to sliding threshold. It would be very valuable to identify a substrate that allows for strong enhancement of the diffusion in the direction transverse to the tilt of the substrate, or one that would facilitate the mixing of particle species that are intrinsically phase separated in equilibrium. Such a substrate could be used to perform fast mixing of species and would have applications in microfluidics, chemical synthesis, and creation of emulsions and dispersions. In this work we show that a phase separated binary assembly of interacting particles in the presence of a two-dimensional random substrate tilted by a driving field undergoes rapid mixing and has an enhancement of the diffusion transverse to the tilt direction. The motion of the particles occurs via [plastic flow]{} in the form of meandering channels which have significant excursions in the direction perpendicular to the drive, leading to mixing of the two particle species. The mixing and diffusion occur even in the absence of thermal fluctuations and arise due to the complex multi-particle interactions. We map the mixing phase diagram as a function of external drive and substrate properties and identify regimes of rapid mixing. We find that as the difference between the two particle species increases, the mixing becomes increasingly asymmetric with one species penetrating more rapidly into the other. Our work shows that plastic flow can be used as a mechanism for mixing applications, and also provides a new system for the study of collective dynamical effects. We simulate a two-dimensional system with periodic boundary conditions in the $x$ and $y$ directions containing two species of Yukawa particles labeled $A$ and $B$ with charges $q_A$ and $q_B$, respectively. The particle-particle interaction potential between particles $i$ and $j$ of charges $q_i$ and $q_j$ at positions ${\bf r}_{i}$ and ${\bf r}_{j}$ is $V(r_{ij}) = E_{0}q_{i}q_{j}\exp(-\kappa r_{ij})/r_{ij}$, where $E_{0} = Z^{* 2}/4\pi\epsilon\epsilon_0$, $\epsilon$ is the dielectric constant, $Z^{}$ is the unit of charge, $\kappa$ is the screening length, and $r_{ij}=\|{\bf r}_i-{\bf r}_j\|$. We fix $\kappa=4/a_0$ where $a_0$ is the unit of length in the simulation. The system size is $L=48a_0$. The motion of particle $i$ is determined by integration of the overdamped equation of motion $$\eta \frac{d {\bf r}_{i}}{dt} = {\bf F}^{cc}_{i} + {\bf F}^{s}_{i} + {\bf F}_{d}$$ where $\eta$ is the damping term which is set equal to unity. Here ${\bf F}^{cc}_{i}= -\sum^{N}_{i\neq j}{\bf \nabla} V(r_{ij})$ is the particle-particle interaction force, where $N$ is the total number of particles in the system. The particle density is $\rho=N/L^2$. The substrate force ${\bf F}^{s}_{i} = -\sum^{N_{p}}_{k=1}\nabla V_{p}(r_{ik})$ comes from $N_p$ parabolic trapping sites placed randomly throughout the sample. Here $V_{p}(r_{ik}) = -(F_{p}/2r_{p})(r_{ik} - r_{p})^2\Theta(r_{p}-r_{ik})$, where $F_p$ is the pinning strength, $r_p=0.2a_0$ is the pin radius, $r_{ik}=\|{\bf r}_i-{\bf r}_{k}^{(p)}\|$ is the distance between particle $i$ and a pin at position ${\bf r}_k^{(p)}$, and $\Theta$ is the Heaviside step function. The pin density is $\rho_p=N_p/L^2$. The external driving force ${\bf F}_{d}=F_d{\bf \hat{x}}$ is applied uniformly to all the particles. The units of force and time are $F_{0} = E_{0}/a_{0}$ and $\tau = \eta/E_{0}$, respectively. We neglect thermal fluctuations so that $T=0$. If the two particle species are initialized in a phase separated state, in the absence of an external drive and disorder the particles will not mix unless the temperature is raised above melting. ![ Red circles and red lines: particle positions and trajectories for species $A$; blue circles and blue lines: particle positions and trajectories for species $B$; open black circles: pinning site locations in a system with particle density $\rho=0.7$, pin density $\rho_p=0.34$, and pinning force $F_p=1.0$ at different driving forces. (a) $F_{d} = 0.0$, (b) $F_{d} = 0.1$, (c) $F_{d} = 0.4$, and (d) $F_{d} = 1.1$. ](Fig1.ps){width="3.5in"} In Fig. 1(a) we show the initial phase separated particle configuration for a 50:50 mixture of the two particle species with $q_A/q_B=3/2$ and $q_A=3$. The particles are placed in a triangular lattice of density $\rho=0.7$ which is immediately distorted by the pinning sites of density $\rho_p=0.34$ and strength $F_p=1.0$. Species $A$ occupies a larger fraction of the sample due to its larger charge $q_A$ and correspondingly larger lattice constant compared to species $B$. An external driving force $F_d$ is applied in the $x$-direction and held at a fixed value. Figure 1(b) illustrates the particle trajectories at $F_{d} = 0.1$ over a period of $10^5$ simulation steps. The trajectories form meandering riverlike structures with significant displacements in the direction transverse to the drive, producing intersecting channels that permit species $A$ to mix with species $B$. When the trajectories and particle positions are followed for a longer period of time, the amount of mixing in the system increases. The riverlike channel structures are typical of plastic flow of particles in random disorder, where a portion of the particles are temporarily trapped at pinning sites while other particles move past, so that the particles do not keep their same neighbors over time. This type of plastic flow has been observed in numerous one-component systems including vortices in type-II superconductors [@Jensen; @Dominguez; @Kolton; @Olson; @Bassler; @Higgins; @Tonomura], electron flow in metal dot arrays [@Middleton], and general fluid flow through random disorder [@Fisher; @Malk]. These works have shown that by changing the strength and size of the disorder, the amount of transverse wandering or tortuosity of the riverlike channels can be adjusted, and that these channels appear even for $T = 0$ [@Dominguez; @Kolton; @Olson; @Bassler]. In our system we measure the diffusion in the $y$-direction, $d_y = \|\langle {\bf r}_i(t)\cdot{\bf \hat{y}} - {\bf r}_i(0)\cdot{\bf \hat{y}}\rangle\|^2$, and find a long time transverse diffusive motion with $d_y(t) \propto t^{\alpha}$ and $\alpha = 1.0$, indicative of normal diffusion. Single component systems exhibiting plastic flow also show a similar transverse diffusive behavior [@Kolton]. The diffusion in our system is not induced by thermal motion but rather occurs due to the complex many-body particle interactions that give rise to the meandering riverlike channels. In Fig. 1(c) we plot the particle trajectories in the same system at $F_d=0.4$. At this drive, a larger fraction of the particles are mobile and the riverlike channels become broader. As the drive is further increased, all the particles are depinned, the meandering riverlike structures are lost, and the mixing of the particles decreases. Such a state is shown in Fig. 1(d) at $F_{d} = 1.1$. For higher values of $F_{d}>1.1$, flow similar to that shown in Fig. 1(d) appears. ![Measure of local homogeneity $H$ vs time for the system in Fig. 1 at $F_d=$ 0.05 (black), 0.1 (red), 0.25 (green), 0.4 (blue), 0.5 (brown), 0.6 (magenta), 0.7 (violet), and 1.1 (top curve). $H = 1$ for phase segregation and $H = 0.5$ for complete mixing. ](Fig2.ps){width="3.5in"} In order to quantify the mixing, for each particle we identify the closest neighboring particles by performing a Voronoi tesselation on the positions of all particles in the system. We then determine the probability $H$ that a particle is of the same species as its neighbors. If the system is thoroughly mixed, the local homogeneity $H = 0.5$, while if it is completely phase separated, $H$ is slightly less than one due to the boundary between the two species. In Fig. 2 we plot $H(t)$ for the system in Fig. 1 at different values of $F_{d}$ ranging from $F_d= 0.05$ to $F_d=1.1$. For the lower drives $F_d\le 0.1$, there are few channels and a portion of the particles remain pinned throughout the duration of the simulation so that mixing saturates near $H=0.6$ to $0.7$. For the intermediate drives $0.1 < F_d \le 0.5$ any given particle is only intermittently pinned, so at long times all the particles take part in the motion and the system fully mixes, as indicated by the saturation of $H$ to $H = 0.5$. For drives $0.5 < F_{d} < 0.9$ the system can still completely mix but the time to reach full mixing increases with $F_d$. At $F_{d} > 0.9$ where the particles are completely depinned, the mixing becomes very slow as shown by the $H(t)$ behavior for $F_{d} = 1.1$. Within the strongly mixing regime, $H(t) \propto A\exp(-t)$ at early times before complete mixing occurs. ![ Mixing phase diagram of pinning density $\rho_{p}$ vs pinning strength $F_{d}$ in the form of a height map of the local homogeneity $H$ obtained from a series of simulations with $F_{p} = 1.0$ and particle density $\rho = 0.7$. Strong mixing regions are blue and weak mixing regions are red. ](Fig3.ps){width="3.5in"} ![ (a) Black line: the average particle velocity $V$ vs $F_{d}$ for a system with $f_{p} = 1.0$, $\rho_{p}= 0.34$, and $\rho = 0.7$. Red line: the corresponding $dV/dF_d$ curve. (b) Black circles: local homogeneity $H$; red squares: net transverse displacement $d_y$ for the same system as in (a). The high mixing regime ($H < 0.6$) is correlated with enhanced transverse displacements and the peak in $dV/dF_d$. $d_y$ has been shifted down for presentation purposes. (c) Particle positions (circles) and trajectories for species $A$ (black) and species $B$ (blue) in a system with $q_A/q_B=3$ and $F_d=0.2$. The mixing is asymmetric with species $A$ moving into the region occupied by species $B$ before species $B$ moves into the area occupied by species $A$. ](Fig4.ps){width="3.3in"} In Fig. 3 we plot the mixing phase diagram of pinning density $\rho_{p}$ versus driving force $F_{d}$ as determined by the local homogeneity $H$ obtained from a series of simulations with $F_{p} = 1.0$ and $\rho = 0.7$. The value of $H$ is measured after $3\times 10^7$ simulation time steps. Blue indicates strong mixing and red indicates weak mixing. For $F_{d} > 1.0$ and all values of $\rho_{p}$, all of the particles are moving in a fashion similar to that illustrated in Fig. 1(d). Since the plastic flow is lost, mixing is very inefficient in this regime. For $F_d<0.6$ at high pinning densities $\rho_{p} > 0.7$, most of the particles are pinned, preventing a significant amount of mixing from occurring. A region of strong mixing appears at $0.6 < F_{d} < 0.9$ for all values of $\rho_{p}$. Here, the particles intermittently pin and depin, producing the large amount of plastic motion necessary to generate mixing. There is another strong region of mixing for lower pinning densities $0.2 < \rho_{p} < 0.4$ and low $F_{d} < 0.4$. In this regime there are more particles than pinning sites so that interstitial particles, which are not trapped by pinning sites but which experience a caging force from neighboring pinned particles, are present. At low drives the interstitial particles easily escape from the caging potential and move through the system; however, the pinned particles remain trapped so that the interstitial particles form meandering paths through the pinned particles. This result shows that even a moderately small amount of disorder combined with a small drive can generate mixing. As the pinning density is further decreased to $\rho_{p} < 0.15$, the amount of mixing also decreases. In Fig. 4(a) we demonstrate how the mixing phases are connected to the transport properties of the system by plotting the net particle velocity $V=\langle N^{-1}\sum_{i=1}^N {\bf v}_i \cdot {\bf \hat{x}}\rangle$ and $dV/dF_d$ versus driving force $F_d$ for a system with $\rho_{p} = 0.34$ and $F_{p} = 1.0$. Here ${\bf v}_i$ is the velocity of particle $i$. In Brownian systems, it was previously shown that an enhanced diffusion peak is correlated with a peak in the derivative of the velocity force curve [@Marchesoni; @Reimann; @Jay; @Bleil; @Lacasta]. Figure 4(a) shows that there is a peak in $dV/dF_d$ spanning $ 0.5 < F_{d} < 0.9$ which also corresponds to the region of high mixing in Fig. 3. There is also a smaller peak in $dV/dF_d$ at small drives $F_d<0.2$ produced by the easy flow of interstitial particles. For $F_{d} > 1.0$, $V$ increases linearly with $F_d$ since the entire system is sliding freely. In Fig. 4(b) we plot the local homogeneity $H$ for the same system taken from the phase diagram in Fig. 3. The maximum mixing $(H<0.6)$ falls in the same region of $F_d$ where the peak in $dV/dF_d$ occurs. Figure 4(b) also shows that the net traverse particle displacement $d_{y}$ has peaks in the strong mixing regimes. We have also examined the effect of significantly increasing $q_{A}/q_{B}$ so that the system is even more strongly phase separated. In general, we find the same mixing features described previously; however, the time required for complete mixing to occur increases with increasing $q_{A}/q_{B}$. The mixing also becomes [asymmetric]{}: the more highly charged species $A$ invades the region occupied by species $B$ before the less highly charged species $B$ spreads evenly throughout the sample. In Fig. 4(c) we illustrate the particle trajectories during the first $3 \times 10^6$ simulation time steps for a system with $q_A/q_B=3$ at $F_d=0.2$. The mixing asymmetry can be seen from the fact that the black trails corresponding to the motion of species $A$ overlap the blue trails representing the motion of species $B$, but the region originally occupied by species $A$ contains no blue trails. One issue is whether the results reported here apply more generally for other types of particle interactions. We considered only Yukawa interactions; however, the meandering channel structures which lead to the mixing are a universal feature of one-component systems undergoing plastic flow though random quenched disorder. Studies performed on systems with long-range logarithmic interactions [@Kolton] as well as short range interactions [@Malk] which show this plasticity lead us to believe that plastic flow generated by random disorder can produce enhanced mixing for a wide range of particle interactions. For our specific system of Yukawa particles, experiments on single component systems have already identified a channel-like plastic flow regime [@Ling]. In summary, we have shown that two-dimensional plastic flow induced by quenched disorder in the absence of thermal fluctuations can lead to efficient mixing and enhanced diffusion in phase separating systems. This mixing occurs due to the meandering of particles through riverlike flow structures. We map the general mixing phase diagram and find that mixing is optimized in regimes where the particles depin in an intermittent fashion. For higher external drives the mixing is strongly reduced. 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--- abstract: \| We consider the Finkelstein action describing a system of spin polarized or spinless electrons in $2+2\epsilon$ dimensions, in the presence of disorder as well as the Coulomb interactions. We extend the renormalization group analysis of our previous work and evaluate the metal-insulator transition of the electron gas to second order in an $\epsilon$ expansion. We obtain the complete scaling behavior of physical observables like the conductivity and the specific heat with varying frequency, temperature and/or electron density. We extend the results for the interacting electron gas in $2+2\epsilon$ dimensions to include the quantum critical behavior of the plateau transitions in the quantum Hall regime. Although these transitions have a very different microscopic origin and are controlled by a topological term in the action ($\theta$ term), the quantum critical behavior is in many ways the same in both cases. We show that the two independent critical exponents of the quantum Hall plateau transitions, previously denoted as $\nu$ and $p$, control not only the scaling behavior of the conductances $\sigma_{xx}$ and $\sigma_{xy}$ at finite temperatures $T$, but also the non-Fermi liquid behavior of the specific heat ($c_v \propto T^p$). To extract the numerical values of $\nu$ and $p$ it is necessary to extend the experiments on transport to include the specific heat of the electron gas. address: - \| $^{1}$ Kurchatov Institute,\ Kurchatov Square 1, 123182 Moscow, Russia - \| $^{2}$ Institute for Theoretical Physics, University of Amsterdam,\ Valckenierstraat 65, 1018XE Amsterdam, The Netherlands - \| $^{3}$L.D. Landau Institute for Theoretical Physics,\ Kosygina str. 2, 117940 Moscow, Russia author: - 'M.A. Baranov$^{1,2}$, I.S. Burmistrov$^{3}$ and A.M.M. Pruisken$^{2}$' title: 'Establishing a non-Fermi liquid theory for disordered metals near two dimensions' --- [2]{} Introduction {#Intro} ============ The integral quantum Hall regime has traditionally been viewed as a (nearly) free particle localization problem with interactions playing only a minor role. [@freepart1] Although it is well known that many features of the experimental data, taken from low mobility heterostructures, [@experiments1] can be explained as the behavior of free particles, a much sharper formulation of the problem is obtained by considering the quantum Hall plateau transitions.[@freepart] Following the experimental work by H.P Wei et al.,[@experiments] these transitions behave in all respects like a disorder driven metal-insulator transition that is characterized by two independent critical indices, i.e. a [localization length]{} exponent $\nu$ and a [phase breaking length]{} exponent $p$.[@freepart] Whereas transport measurements usually provide an experimental value of only the ratio $\kappa = p/2\nu$, it is generally not known how the values of $\nu$ and $p$ can be extracted separately. Inspite of the fact that one can not proceed without having a microscopic theory of electron-electron interaction effects, there is nevertheless a strong empirical believe in the literature [@believe] which says that the zero temperature localization length exponent $\nu $ is given precisely by the free electron value $\nu =2.3$ as obtained from numerical simulations. [@kramer] The experimental situation has not been sufficiently well understood, [@newexpt] however, to justify the bold assumption of Fermi liquid behavior. In fact, the progress that has been made over the last few years in the theory of localization and interaction effects clearly indicates that Fermi liquid principles do not exist in general. The Coulomb interaction problem lies in a different universality class of transport phenomena [@eurolett] with a previously unrecognized symmetry, called ${\cal F}$ invariance. [@bps1; @bps2; @bps3; @pbb] The theory relies in many ways on the approach as initiated by Finkelstein [@fink] and adapted to the case of the spin polarized or spinless electrons.[@eurolett] By reconciling the Finkelstein theory with the topological concept of an instanton vacuum [@inst] and the Chern Simons statistical gauge fields,[@cs] the foundations have been laid for a complete renormalization theory that unifies the quantum theory of metals with that of the abelian quantum Hall states.[@pbb] A historical problem -------------------- The unification of the integral and fractional quantum Hall regimes is based on the assumption that Finkelstein approach [@fink; @eurolett] is renormalizable and generates a strong coupling, [insulating]{} phase with a massgap. However, the traditional analyses of the Finkelstein theory have actually not provided any garantee that this is indeed so. Inspite of Finkelstein’s pioneering and deep contributions to the field, it is well known that the conventional momentum shell renormalization schemes do not facilitate any computations of the quantum theory beyond one loop order. At the same time, application of the more advanced technique of dimensional regularization has led to conceptual difficulties with such aspects like [dynamical]{} scaling.[@kb] One can therefore not rule out the possibility that there are complications, either in the idea of renormalizability, or in other aspects of the theory such as the Matsubara frequency technique. Nothing much has been clarified, however, by repeating similar kinds of analyses in a different formalism, like the Keldish technique.[@ka; @cln] What has been lacking all along is the understanding of a fundamental principle that has prevented the Finkelstein approach from becoming a fully fledged field theory for localization and interaction effects. ${\cal F}$ invariance --------------------- In our previous work [@bps1] we have shown that the Finkelstein action has an exact symmetry (${\cal F}$ invariance) that is intimidly related to the electrodynamic $U(1)$ gauge invariance of the theory. ${\cal F}$ invariance is the basic mechanism that protects the renormalization of the problem with infinitely ranged interaction potentials such as the Coulomb potential. Moreover, it has turned out that the infrared behavior of physical observables can only be extracted from ${\cal F}$ invariant quantities and correlations, and these include the linear response to external potentials. Arbitrary renormalization group schemes break the ${\cal F}$ invariance of the action and this generally complicates the attempt to obtain the temperature and/or frequency dependence of physical quantities such as the conductivity and specific heat. Quantum Hall physics is in many ways a unique laboratory for investigating and exploring the various different consequences of ${\cal F}$ invariance. For example, one of the longstanding questions in the field is whether and how the theory [dynamically]{} generates the [exact]{} quantization of the Hall conductance. Important progress has been made recently by demonstrating that the instanton vacuum, on the strong coupling side of the problem, generally displays massless excitations at the edge of the system. [@pbv] These massless edge excitations are identical to those described by the more familiar theory of chiral edge bosons. [@bps3] Our theory of massless edge excitations implies that the concept of ${\cal F}$ invariance retains its fundamental significance all the way down to the regime of strong coupling. Outline of this paper --------------------- In this paper we put the concept of ${\cal F}$ invariance at work and evaluate the renormalization behavior of the Finkelstein theory at a two loop level. As shown in our previous papers, [@bps2] the technique of dimensional regularization is a unique procedure, not only for the computation of critical indices, but also for extracting the dynamical scaling functions. In fact, the metal-insulator transition in $2+2\epsilon$ spatial dimensions is the only place in the theory where the temperature and/or frequency dependence of physical observables can be obtained explicitly. This motivates us to further investigate the problem in $2+2\epsilon $ dimensions and use it as a stage setting for the much more complex problem of the quantum Hall plateau transitions. The final results of this paper are remarkably similar to those of the more familiar classical Heisenberg ferromagnet.[@bhz] For example, unlike the free electron gas, the Coulomb interaction problem displays a conventional phase transition (metal-insulator transition) in $2+2\epsilon$ dimensions with an ordinary order parameter. The theory is therefore quite different from that of free electrons which has a different dimensionality and displays, as is well known, anomalous or multifractal density fluctuations near criticality.[@weg1] It is important to bear in mind, however, that the analogy with the Heisenberg model is rather formal and it fails on many other fronts. For example, the classification of critical operators is very different from what one is used to, in ordinary sigma models. Moreover, the Feynman diagrams of the Finkelstein theory are more complex, involving internal frequency sums which indicate that the theory effectively exists in $2+1$ space-time dimensions, rather than in two spatial dimensions alone. The complexity of ${\cal F}$ invariant systems is furthermore illustrated by the lack of such principles like Griffith analyticity that facilitates a discussion of the symmetric phase in conventional sigma models.[@bzl] In a subsequent paper we shall address the strong coupling insulating phase of the electron gas and show that the dynamics is distinctly different from that of the Goldstone (metallic) phase and controlled by different operators in the theory.[@pb01] This paper is organized as follows. After introducing the formalism (Section II) we embark on the details of the two loop contributions to the conductivity in Section III. As in our earlier work, we employ an $% {\cal F}$-invariance-breaking parameter $\alpha $ to regularize the infinite sums over frequency. This methodology actually provides numerous self consistency checks and a major part of the computation consists of finding the ways in which the various singular contributions in $\alpha $ cancel each other. The actual computation of the diagrams is described in the Appendices which contain the list of the momentum and frequency integrals that are used in the text. In tables I and II we summarize how the different singular contributions in $\alpha $ cancel each other. Table III lists the various finite contributions to the pole term in $\epsilon $. The final result for the $\beta $ function is given by Eqs. (\[z1\])–(\[pf\]). In Section IV we summarize the consequences for scaling. We extend the discussion to include the plateau transitions in the quantum Hall regime in Section IVC. We briefly address several new advancements, both from theoretical and experimental sides, that seem to have general consequences for the quantum theory of conductances. Finally, we show how the results of this paper can be used in the problem of critical exponents $p$ and $\nu$. We end this paper with a conclusion (Section V). Effective parameters ==================== Introduction {#introduction} ------------ The theory for spinless electrons involves unitary matrix field variables $% Q_{nm}^{\alpha \beta }$ where the superscripts ${\alpha \beta }$ are the [replica]{} indices, the subscripts $n$, $m$ denote the [Matsubara frequency]{} indices. The $Q$ fields obey the nonlinear constraint $Q^{2}=1$ and we are interested in the following action $$\begin{aligned} S[Q,A] &=&-\frac{\sigma _{0}}{8}\int_{x}\left( \mbox{tr}[{\vec{D}}% ,Q]^{2}+2h_{0}^{2}\mbox{tr}\Lambda Q\right) + \nonumber \\ z_{0}\pi &T& \int_{x}\left( {}\right. \sum_{\alpha n}c_{0}\mbox{tr}% I_{n}^{\alpha }Q\mbox{tr}I_{-n}^{\alpha }Q+4\mbox{tr}\eta Q-6\mbox{tr}\eta \Lambda \left. {}\right) . \label{S}\end{aligned}$$ The explanation of the symbols is as follows. The parameter $\sigma _{0}$ plays the role of [conductivity]{} of the electron gas, $z_{0}$ is the so-called [singlet interaction amplitude]{} and $T$ stands for the temperature. The parameter $c_{0}=1-\alpha $ is such that the theory interpolates between the Coulomb case ($\alpha =0$) and the free particle case ($\alpha =1$). Here, the quantity $\alpha $ breaks the ${\cal F}$ invariance of the theory and we shall eventually be interested in the limit where $\alpha $ goes to zero. For a detailed exposure to the meaning of ${\cal F}$ invariance we refer the reader to the original papers.[@bps1; @bps2] We generally need the definition of two more diagonal matrices $\Lambda$ and $\eta$, and one more off-diagonal matrix $I^{\alpha}_{n}$. They are given by $$\begin{aligned} \Lambda _{nm}^{\alpha \beta } &=&{\rm sign}(n)\delta ^{\alpha \beta }\delta _{nm}, \\ \eta _{nm}^{\alpha \beta } &=&n\delta ^{\alpha \beta }\delta _{nm}, \\ \left( I_{n}^{\alpha }\right) _{kl}^{\beta \gamma } &=&\delta ^{\alpha \beta }\delta ^{\alpha \gamma }\delta _{n,k-l}.\end{aligned}$$ Here, $\eta $, being multiplied by $2\pi T$, represents the Matsubara frequencies in matrix language. The $I_{n}^{\alpha }$ are shifted diagonals in frequency space and they generally represent the generators of the $U(1)$ gauge transformations. The term proportional to $h_{0}^{2}$ is not a part of the theory but we shall use it later on as a convenient infrared regulator of the theory. Finally, the $\vec{D}$ are covariant derivatives $$D_{a}=\nabla _{a}-i\widehat{A}_{a},$$ where $$\widehat{A}_{a}=\sum_{\alpha ,n}\left( A_{a}\right) _{n}^{\alpha }I_{n}^{\alpha },$$ and $\left( A_{a}\right) _{n}^{\alpha }$ is the Fourier transform of the homogeneous external vector potential $A_{a}^{\alpha }(\tau )$: $% A_{a}^{\alpha }(\tau )=\sum_{n}\left( A_{a}\right) _{n}^{\alpha }\exp (-i\omega _{n}\tau )$, $\omega _{n}=2\pi Tn$ is the Matsubara frequency. Linear response --------------- The ”effective” action for the external vector potential is defined according to $$\exp S_{eff}[A]=\int DQ\exp S[Q,A].$$ The quadratic part can generally be written as $$S_{eff}[A]=\int\limits_{x}\sum_{\alpha ,n>0}\sigma ^{^{\prime }}(n)n(A_{a})_{n}^{\alpha }(A_{a})_{-n}^{\alpha }.$$ The quantity $\sigma ^{^{\prime }}(n)$ is the true conductivity of the electron gas, and in terms of the $Q$ matrix fields the following Kubo like expression can be obtained $$\sigma ^{^{\prime }}(n)=\langle O_{1}\rangle +\langle O_{2}\rangle , \label{cond1}$$ where $$O_{1}=-\frac{\sigma _{0}}{4n}\mbox{tr}[I_{n}^{\alpha },Q(x)][I_{-n}^{\alpha },Q(x)]$$ and $$O_{2}=\frac{\sigma _{0}^{2}}{16nd}\int_{x-x^{\prime }}\mbox{tr}% [I_{n}^{\alpha },Q(x)]\nabla Q(x)\mbox{tr}[I_{-n}^{\alpha },Q(x^{^{\prime }})]\nabla Q(x^{^{\prime }}). \label{cond2}$$ Here the expectations are with respect to the theory without the vector potentials. The $h_0$ field --------------- Although we are interested, strictly speaking, in evaluating $\sigma ^{^{\prime }}(n)$ with varying values of external frequencies $\omega _{n}$ and temperature, the computation simplifies dramatically if we put these parameters equal to zero in the end and work with a finite value of the $% h_{0}$ field instead. This procedure has been analyzed in exhaustive detail in our previous work and, in what follows, we shall greatly benifit from the technical advantages that make the two-loop analysis of the conductivity possible. We shall return to finite frequency and temperature problem in the end of this paper (Sections IV). The infrared regularization by the $h_{0}$ field relies on the following statement $$\sigma _{0}h_{0}^{2}\langle Q({\vec{x}})\rangle =\sigma ^{^{\prime }}h^{^{\prime }2}\Lambda ,$$ which says that there is an effective mass $h^{^{\prime }}$ in the problem that is being induced by the presence of the $h_{0}$ field. It is very well known that, since the quantity $\langle Q({\vec{x}})\rangle $ is not a gauge invariant object, the definition of the $h^{^{\prime }}$ field is singular as $\alpha $ goes to zero and the theory is generally not renormalizable. However, the effective parameter $\sigma ^{^{\prime }}$ is truly defined in terms of the effective mass $h^{^{\prime }}$ rather than the bare parameter $% h_{0}$. Hence, all the non-renormalizable singularities are removed from the theory, provided we express $\sigma ^{^{\prime }}$ in terms of the $% h^{^{\prime }}$ rather than the $h_{0}$. We shall show that the ultraviolet singularities of the theory can be extracted directly from the final result for $\sigma ^{^{\prime }}(h^{^{\prime }})$. On the other hand, we can make use of our previous results [@bps2] and express the final answer in terms of frequencies and temperature, rather than the mass $h^{^{\prime }}$. Computation of conductivity in $2+2\epsilon$ dimensions {#PEC} ======================================================= Introduction {#introduction-1} ------------ To define a theory for perturbative expansions we use the following parametrization $$Q=\left( \begin{array}{cc} \sqrt{1-qq^{\dagger }} & q^{\dagger } \\ q & -\sqrt{1-q^{\dagger }q} \end{array} \right) .$$ The action can be written as an infinite series in the independent fields $% q_{n_{1}n_{2}}^{\alpha \beta }$ and $[q^{\dagger }]_{n_{4}n_{3}}^{\alpha \beta }$. We use the convention that Matsibara indices with odd subscripts: $% n_{1},n_{3},...$, run over non-negative integers, whereas those with even subscripts: $n_{2},n_{4},...$, run over negative integers. The propagators can be written in the form [@kb; @bps2] $$\begin{aligned} \langle q_{n_{1}n_{2}}^{\alpha \beta }(p)[q^{\dagger }]_{n_{4}n_{3}}^{\gamma \delta }(-p)\rangle &=&\frac{4}{\sigma _{0}}\delta ^{\alpha \delta }\delta ^{\beta \gamma }\delta _{n_{12},n_{34}}D_{p}(n_{12}) \nonumber \\ \left( {}\right. \delta _{n_{1}n_{3}} &+&\delta ^{\alpha \beta }\kappa ^{2}z_{0}c_{0}D_{p}^{c}(n_{12})\left. {}\right) ,\end{aligned}$$ where $$\lbrack D_{p}(n_{12})]^{-1}=p^{2}+h_{0}^{2}+\kappa ^{2}z_{0}n_{12},$$ $$\lbrack D_{p}^{c}(n_{12})]^{-1}=p^{2}+h_{0}^{2}+\alpha \kappa ^{2}z_{0}n_{12},$$ $$\kappa ^{2}=\frac{8\pi T}{\sigma _{0}}.$$ Here we use the notation $n_{12}=n_{1}-n_{2}$ . The expression for the DC conductivity is known to one loop order [@bps2] $$\sigma _{one}^{^{\prime }}=\sigma _{0}+\frac{4\Omega _{d}h_{0}^{2\epsilon }}{% \epsilon }\,\,,\,\,\Omega _{d}=\frac{S_{d}}{2(2\pi )^{d}}, \label{one}$$ where $S_{d}=2\pi ^{d/2}/\Gamma (d/2)$ is the surface of a $d$ dimensional unit sphere. The two-loop theory -------------------- To proceed we need the following terms obtained by expanding the action (2.1) in terms of $q$ and $q^{\dagger }$ fields: $$\begin{aligned} S_{int}^{(3)}=-\frac{a\sigma _{0}}{8}\int\limits_{x}\sum\limits_{\beta ,m>0}\left\{ {}\right. &\mbox{tr}&I_{m}^{\beta }q^{\dagger }\,\mbox{tr}% I_{-m}^{\beta }[q,q^{\dagger }]+ \nonumber \\ &\mbox{tr}&I_{-m}^{\beta }q\,\mbox{tr}I_{m}^{\beta }[q,q^{\dagger }]\left. {}\right\} ,\end{aligned}$$ $$\begin{aligned} S_{int}^{(4)}=\frac{a\sigma _{0}}{16}\int\limits_{x}\left\{ {}\right. \sum\limits_{\beta ,m>0} &\mbox{tr}&I_{-m}^{\beta }[q,q^{\dagger }]\,% \mbox{tr}I_{m}^{\beta }[q,q^{\dagger }]+ \nonumber \\ 2\sum\limits_{\beta }( &\mbox{tr}&I_{0}^{\beta }[q,q^{\dagger }])^{2}\left. {}\right\} ,\end{aligned}$$ $$\begin{aligned} S_{0}^{(4)} &=&\frac{\sigma _{0}}{32}\int\limits_{p}\delta ({\bf p}_{1}+{\bf % p}_{2}+{\bf p}_{3}+{\bf p}_{4})\sum\limits_{n_{1}n_{2}n_{3}n_{4}}^{\beta \gamma \delta \mu } \nonumber \\ &\times &q_{n_{1}n_{2}}^{\beta \gamma }(p_{1})(q^{\dagger })_{n_{2}n_{3}}^{\gamma \delta }(p_{2})q_{n_{3}n_{4}}^{\delta \mu }(p_{3})(q^{\dagger })_{n_{4}n_{1}}^{\mu \beta }(p_{1}) \nonumber \\ &\times &\left\{ {}\right. ({\bf p}_{1}+{\bf p}_{2})\cdot ({\bf p}_{3}+{\bf p} _{4})+({\bf p}_{2}+{\bf p}_{3})\cdot ({\bf p}_{1}+{\bf p}_{4}) \nonumber \\ &-&\kappa ^{2}z_{0}(n_{12}+n_{34})-2h_{0}^{2}\left. {}\right\} ,\end{aligned}$$ where we define $a=\kappa ^{2}z_{0}c_{0}$. In addition, we need the following terms obtained by expanding the expression for the conductivity, Eq. (\[cond1\]), $$\begin{aligned} O_{1}^{(2)}=-\frac{\sigma _{0}}{2} &\mbox{tr}&\left\{ {}\right. I_{n}^{\alpha }q^{\dagger }I_{-n}^{\alpha }q+I_{-n}^{\alpha }q^{\dagger }I_{n}^{\alpha }q- \nonumber \\ &2&(I_{n}^{\alpha }\Lambda I_{-n}^{\alpha }+I_{-n}^{\alpha }\Lambda I_{n}^{\alpha })[q,q^{\dagger }]\left. {}\right\} ,\end{aligned}$$ $$O_{1}^{(3)}=\frac{\sigma _{0}}{4}\mbox{tr}\left\{ {}\right. I_{n}^{\alpha }(q+q^{\dagger })I_{-n}^{\alpha }qq^{\dagger }-I_{-n}^{\alpha }(q+q^{\dagger })I_{n}^{\alpha }q^{\dagger }q\left. {}\right\} ,$$ $$\begin{aligned} O_{1}^{(4)}=\frac{\sigma _{0}}{16} &\mbox{tr}&\left\{ {}\right. (I_{n}^{\alpha }\Lambda I_{-n}^{\alpha }+I_{-n}^{\alpha }\Lambda I_{n}^{\alpha })[qq^{\dagger }q,q^{\dagger }]- \nonumber \\ &2&I_{n}^{\alpha }[q,q^{\dagger }]I_{-n}^{\alpha }[q,q^{\dagger }]\left. {}\right\} ,\end{aligned}$$ $$O_{2}^{(4)}=\frac{\sigma _{0}^{2}}{4d}\int_{x-x^{\prime }}\mbox{tr}% I_{n}^{\alpha }(q\nabla q^{\dagger }+q^{\dagger }\nabla q)\mbox{tr}% I_{-n}^{\alpha }(q\nabla q^{\dagger }+q^{\dagger }\nabla q),$$ $$\begin{aligned} O_{2}^{(5)}=\frac{\sigma _{0}^{2}}{8d}\int_{x-x^{\prime }}\left\{ {}\right. &% \mbox{tr}&I_{n}^{\alpha }(q\nabla q^{\dagger }+q^{\dagger }\nabla q)\mbox{tr}% I_{-n}^{\alpha }q(\nabla q^{\dagger })q+ \nonumber \\ &\mbox{tr}&I_{-n}^{\alpha }(q\nabla q^{\dagger }+q^{\dagger }\nabla q)% \mbox{tr}I_{n}^{\alpha }q^{\dagger }(\nabla q)q^{\dagger }\left. {}\right\} ,\end{aligned}$$ $$\begin{aligned} O_{2}^{(6)}=\frac{\sigma _{0}^{2}}{16d}\int_{x-x^{\prime }}\left\{ {}\right. &\mbox{tr}&I_{n}^{\alpha }\Lambda q^{\dagger }(\nabla q)q^{\dagger })% \mbox{tr}I_{-n}^{\alpha }\Lambda q(\nabla q^{\dagger })q)+ \nonumber \\ &\mbox{tr}&I_{n}^{\alpha }(q\nabla q^{\dagger }+q^{\dagger }\nabla q)\times \nonumber \\ &\mbox{tr}&I_{-n}^{\alpha }(qq^{\dagger }\nabla (qq^{\dagger })+q^{\dagger }q\nabla (q^{\dagger }q))+ \nonumber \\ &\mbox{tr}&I_{-n}^{\alpha }(q\nabla q^{\dagger }+q^{\dagger }\nabla q)\times \nonumber \\ &\mbox{tr}&I_{n}^{\alpha }(qq^{\dagger }\nabla (qq^{\dagger })+q^{\dagger }q\nabla (q^{\dagger }q))\left. {}\right\} .\end{aligned}$$ Next we give the complete list of two loop contributions to the conductivity as follows $$\begin{aligned} &\sigma &_{two}^{^{\prime }}(n)= \nonumber \\ &\langle &O_{1}^{(4)}+O_{1}^{(3)}S_{int}^{(3)}+O_{1}^{(2)}(S_{int}^{(4)}+S_{0}^{(4)}+% \frac{1}{2}(S_{int}^{(3)})^{2}) \nonumber \\ &+&O_{2}^{(6)}+O_{2}^{(5)}S_{int}^{(3)}+O_{2}^{(4)}(S_{int}^{(4)}+S_{0}^{(4)}+% \frac{1}{2}(S_{int}^{(3)})^{2})\rangle . \label{sig}\end{aligned}$$ The computations of the terms in Eq. (\[sig\]) are straightforward but lengthy and tedious. In what follows we present the expressions in terms of the momentum integrals, frequency sums and propagators $D$, $D^{c}$ for each term in Eq. (\[sig\]) separately, along with the final answer. In the Appendices we give the complete list of integrals and symbols that we shall make use of here. Computation of contractions --------------------------- ### $\langle O_{1}^{(4)} \rangle$ $$\begin{aligned} &&\frac{2}{\sigma _{0}}(\int\limits_{p}D_{p}(0))^{2}+\frac{2a^{2}}{\sigma _{0}}(\sum_{m>0}\int\limits_{p}DD_{q}^{c}(m))^{2} \nonumber \\ &=&\frac{\Omega _{d}^{2}h_{0}^{4\epsilon }}{\sigma _{0}\epsilon ^{2}}(2+2\ln ^{2}\alpha ) \label{start}\end{aligned}$$ with $DD_{q}^{c}(m)\equiv D_{q}(m)D_{q}^{c}(m)$. ### $\langle O_{1}^{(3)}S^{(3)}_{int}\rangle$ $$\begin{aligned} &-&\frac{8a}{\sigma _{0}}\int\limits_{p,q}\left\{ {}\right. \sum_{k>0}D_{p+q}^{c}(0)D_{q}(k)D_{p}(k) \nonumber \\ &+&a\sum_{k,m>0}D_{p}^{c}(m)DD_{q}^{c}(k)D_{p+q}(k+m)\left. {}\right\} \nonumber \\ &=&\frac{\Omega _{d}^{2}h_{0}^{4\epsilon }}{\sigma _{0}\epsilon }\left( {}\right. 4S_{0}+4A_{00}^{0}\left. {}\right) \nonumber \\ &=&\frac{\Omega _{d}^{2}h_{0}^{4\epsilon }}{\sigma _{0}\epsilon ^{2}}\left[ {}\right. -4-4\ln ^{2}\alpha +\epsilon (8+4\zeta (3))\left. {}\right] ,\end{aligned}$$ where $\zeta (z)$ is the Riemann zeta-function. ### $\langle O_{1}^{(2)}(S^{(4)}_{int}+S^{(4)}_{0}+\frac{1}{2}% (S^{(3)}_{int})^{2}))\rangle$ $$\begin{aligned} &&\frac{4a}{\sigma _{0}}\int\limits_{p,q}\left\{ {}\right. D_{p+q}^{c}(0)\sum_{k>0}D_{q}(k)D_{p}(k) \nonumber \\ &+&a\sum_{k,m>0}D_{p}^{c}(m)D_{q}^{c}(k)D_{p+q}^{2}(k+m) \nonumber \\ &+&a\sum_{k,m>0}(1+amD_{p}^{c}(m))DD_{q}^{c}(k)D_{p+q}^{2}(k+m)\left. {}\right\} \nonumber \\ &=&\frac{\Omega _{d}^{2}h_{0}^{4\epsilon }}{\sigma _{0}\epsilon }\left( {}\right. -2S_{0}-2D_{1}-2T_{01}-2A_{1,0}^{0}\left. {}\right) \nonumber \\ &=&\frac{\Omega _{d}^{2}h_{0}^{4\epsilon }}{\sigma _{0}\epsilon ^{2}}\left[ {}\right. 2+2\ln ^{2}\alpha -\epsilon (4+2\zeta (3)+2\pi ^{2}/3)\left. {}\right] .\end{aligned}$$ ### $\langle O_{2}^{(6)}\rangle$ $$\begin{aligned} &-&\frac{4}{\sigma _{0}d}\int\limits_{p,q}p^{2}\left\{ {}\right. D_{p}(0)D_{q}(0)D_{p+q}(0) \nonumber \\ &-&4a^{2}\sum_{k,m>0}D^{2}D_{p}^{c}(m)\hat{S}_{m}DD_{q}^{c}(k) \nonumber \\ &-&a^{2}\sum_{k,m>0}\left[ {}\right. D_{p}(k+m)DD_{q}^{c}(m)DD_{p+q}^{c}(k) \nonumber \\ &+&2DD_{p}^{c}(k+m)DD_{q}^{c}(m)D_{p+q}(k)\left. {}\right] \left. {}\right\} \nonumber \\ &=&\frac{\Omega _{d}^{2}h_{0}^{4\epsilon }}{\sigma _{0}\epsilon }\left( {}\right. S_{1}+4(\frac{2\ln \alpha }{\epsilon }+B_{1})+C_{01}+2C_{00}\left. {}\right) \nonumber \\ &=&\frac{\Omega _{d}^{2}h_{0}^{4\epsilon }}{\sigma _{0}\epsilon ^{2}}\left[ {}\right. 16\ln \alpha -2+\epsilon (-4\ln \alpha -\frac{\pi ^{2}}{3}+\frac{% \pi ^{2}}{2}\ln 2 \nonumber \\ &+&\frac{\pi ^{4}}{12}+\frac{11\zeta (3)}{2}+\frac{\pi ^{2}}{3}\ln ^{2}2-% \frac{1}{3}\ln ^{4}2-7\zeta (3)\ln 2 \nonumber \\ &-&8Li_{4}(\frac{1}{2}))\left. {}\right] .\end{aligned}$$ Here $D^{n}D_{q}^{c}(m)\equiv D_{q}^{n}(m)D_{q}^{c}(m)$ and $$Li_{n}(x)=\sum\limits_{k=1}^{\infty }\frac{x^{k}}{k^{n}}$$ is the polylogarithmic function ($Li_{4}(1/2)=0.517...$), and we have introduced an operator $\hat{S}_{m}$ which acts only on frequency $k$ according to the following rule $\hat{S}_{m}f(k)=f(k)+f(k+m)$. ### $\langle O_{2}^{(5)}S^{(3)}_{int}\rangle$ $$\begin{aligned} &&\frac{16a}{\sigma _{0}d}\int\limits_{p,q}{\bf p}\cdot({\bf p}-{\bf q}% )\sum_{k>0}D_{p+q}^{c}(0)D_{p}^{2}(k)D_{q}(k) \nonumber \\ &+&\frac{16a^{2}}{\sigma _{0}d}\int\limits_{p,q}p^{2}% \sum_{k,m>0}D_{p+q}^{c}(m)\left[ {}\right. D_{p}^{2}(k+m)DD_{q}^{c}(k) \nonumber \\ &+&D^{2}D_{p}^{c}(k+m)D_{q}(k)\left. {}\right] \nonumber \\ &-&\frac{16a^{2}}{\sigma _{0}d}\int\limits_{p,q}({\bf p}\cdot {\bf q})\sum_{k,m>0} \left\{ {}\right. DD_{p}^{c}(m)\hat{T}_{m}DD_{p+q}^{c}(k)D_{q}(k+m) \nonumber \\ &+&D_{p+q}^{c}(m)D^{2}D_{p}^{c}(k+m)D_{q}(k+2m)\left. {}\right\} \nonumber \\ &=&\frac{\Omega _{d}^{2}h_{0}^{4\epsilon }}{\sigma _{0}\epsilon }\left( {}\right. -4S_{00}-4A_{01}^{1}-4H_{0}-4C_{0}-4A_{0}\left. {}\right) \nonumber \\ &=&\frac{\Omega _{d}^{2}h_{0}^{4\epsilon }}{\sigma _{0}\epsilon ^{2}}\left[ {}\right. -8\ln \alpha +4+\epsilon (4\ln ^{2}\alpha +20\ln \alpha \nonumber \\ &-&12+4\zeta (3)+4\pi ^{2}/3-4A_{0}+4C_{0}^{^{\prime }})\left. {}\right] ,\end{aligned}$$ Where we have introduced yet another operator $\hat{T}_{m}$ which acts only on frequency $k$ but now according to the rule $\hat{T}_{m}f(k)=f(k)-f(k+m)$. ### $\langle O_{2}^{(4)}S^{(4)}_{0}\rangle$ $$\begin{aligned} &&\frac{8a^{2}}{\sigma _{0}d}\int\limits_{p,q}p^{2}\sum_{k,m>0} \nonumber \\ &\left\{ {}\right. &3D^{3}D_{p}^{c}(m)\hat{S}% _{m}D_{q}^{c}(k)+3D^{2}D_{p}^{c}(m)\hat{S}_{m}DD_{q}^{c}(k) \nonumber \\ &+&2akD^{2}[D_{p}^{c}]^{2}(m)\hat{T}_{m}[D_{p}(m)D_{q}^{c}(k)+DD_{q}^{c}(k)]% \left. {}\right\} \nonumber \\ &=&\frac{\Omega _{d}^{2}h_{0}^{4\epsilon }}{\sigma _{0}\epsilon }\left( {}\right. -3T_{10}^{0}-3T_{11}^{0}-\frac{12\ln \alpha }{\epsilon }% -6B_{1}-2T_{20}^{0} \nonumber \\ &+&2T_{21}^{0}-4T_{10}^{1}+4B_{2}\left. {}\right) \nonumber \\ &=&\frac{\Omega _{d}^{2}h_{0}^{4\epsilon }}{\sigma _{0}\epsilon ^{2}}\left[ {}\right. 4(\ln \alpha -1)^{2}-2 \\ &+&\epsilon (\frac{2}{\alpha }-2\ln ^{2}\alpha +\ln \alpha +44/3)\left. {}\right] .\end{aligned}$$ ### $\langle O_{2}^{(4)}(S^{(4)}_{int}+\frac{1}{2}% (S^{(3)}_{int})^{2})\rangle$ $$\begin{aligned} &&\frac{16a^{2}}{\sigma _{0}d}\int\limits_{p,q}({\bf p}\cdot {\bf q} )\sum_{k>0}kD_{p+q}(0)D_{p}^{2}(k)D_{q}^{2}(k) \nonumber \\ &-&\frac{16a}{\sigma _{0}d}\int\limits_{p,q}p^{2}\sum_{k>0}\left[ {}\right. 2akD_{p+q}^{c}(0)D_{p}^{3}(k)D_{q}(k)-D_{p}^{3}(k)D_{q}(k)\left. {}\right] \nonumber \\ &+&\frac{16a}{\sigma _{0}d}\int\limits_{p,q}({\bf pq})\sum_{k,m>0}\left\{ {}\right. 2(1+amD_{q}(k)) \nonumber \\ &\times &D_{p+q}^{c}(m)D_{p}^{2}(k+m)DD_{q}^{c}(k) \nonumber \\ &-&DD_{q}^{c}(k)DD_{p}^{c}(m)D_{p+q}(k+m)\left. {}\right\} \nonumber \\ &-&\frac{16a^{2}}{\sigma _{0}d}\int\limits_{p,q}p^{2}\sum_{k,m>0}\left\{ {}\right. (1+amD_{p+q}^{c}(m)) \nonumber \\ &\times &\left[ {}\right. (2+\hat{T}_{m}+ak\hat{T}% _{m}D_{p}^{c}(k))D^{3}D_{p}^{c}(k)D_{q}(k+m) \nonumber \\ &+&\frac{1}{2}D_{q}(k)D_{p}^{3}(k+m)(3D_{q}^{c}(k)+D_{p}^{c}(k+m))\left. {}\right] \nonumber \\ &+&\frac{3}{2}D_{q}^{c}(m)D_{p+q}^{c}(k)D_{p}^{3}(k+m) \nonumber \\ &+&(1+\hat{T}_{m}+2ak\hat{T}% _{m}D_{p}^{c}(k))D_{p+q}^{c}(m)D^{2}D_{p}^{c}(k)D_{q}(k+m) \nonumber \\ &+&ak\hat{T}_{m}D[D_{p}^{c}(m)]^{2}DD_{q}^{c}(k)D_{p+q}(k+m)\left. {}\right\} \nonumber \\ &=&\frac{\Omega _{d}^{2}h_{0}^{4\epsilon }}{\sigma _{0}\epsilon }\left( {}\right. 4S_{11}+4S_{01}+\frac{2}{\epsilon }+8A_{01}+8A_{11}-4C_{11} \nonumber \\ &+&4T_{02}+4A_{10}+2T_{12}+2A_{1}+3T_{01}+3A_{11}^{1}+\alpha T_{10}^{0} \nonumber \\ &-&T_{02}+H_{1}+3D_{2}+4C_{1}+8A_{2}+8A_{00}-4A_{3}\left. {}\right) \nonumber \\ &=&\frac{\Omega _{d}^{2}h_{0}^{4\epsilon }}{\sigma _{0}\epsilon ^{2}}\left[ {}\right. -4\ln ^{2}\alpha -4+\epsilon (-\frac{2}{\alpha }-2\ln ^{2}\alpha \nonumber \\ &-&25\ln \alpha +55/2-2\zeta (3)-\frac{8}{3}\pi ^{2}+12\ln ^{2}2 \nonumber \\ &-&44\ln 2-4C_{0}^{^{\prime }}+4A_{0}+16G-8Li_{2}(\frac{1}{2})\left. {}\right] , \label{end}\end{aligned}$$ where $G=0.915...$ denotes the Catalan constant. Results of the computations --------------------------- We proceed by presenting the final answer for all the pole terms in $% \epsilon $. By putting the external frequency equal to zero and in the limit $\alpha \rightarrow 0$ we obtain $$\sigma _{two}^{^{\prime }}(0)=\frac{\Omega _{d}^{2}h_{0}^{4\epsilon }}{% \sigma _{0}\epsilon }\left( A-8(2+\ln \alpha )\right) . \label{res}$$ Here, the $A$ stands for all the terms that are finite in $\alpha $. The complete list is as follows $$\begin{aligned} A &=&50+\frac{1}{6}-3\pi ^{2}+\frac{19}{2}\zeta (3)+16\ln ^{2}2 \nonumber \\ &-&44\ln 2+\frac{\pi ^{2}}{2}\ln 2+16G+\frac{\pi ^{4}}{12}+\frac{\pi ^{2}}{3}% \ln ^{2}2 \nonumber \\ &-&\frac{1}{3}\ln ^{4}2-7\zeta (3)\ln 2-8Li_{4}(\frac{1}{2}) \nonumber \\ &\approx &1.64.\end{aligned}$$ Before Eq. (\[res\]) is obtained, one has to deal with a host of other contributions that are more singular in $\alpha $ and/or $\epsilon $. These more singular contributions all cancel one another in the end, however. There are in total six different types of contributions that are more singular than the simple pole term $1/\epsilon $. In Tables I and II we list these terms, show where they come from and how they sum up to zero. There is one exception, namely the terms proportional to $\ln (\alpha )/\epsilon $, and their contribution is written in Eq. (\[res\]). However, these terms are absorbed in the definition of an ”effective” $h^{\prime }$ field. More specifically, from the two-loop computation of the singlet amplitude $z$ we know that the effective $h,$ field is given by [@bps2] $$h_{0}^{2}\to h^{\prime 2}=h_{0}^{2}\left( 1-\frac{2+\ln \alpha }{2}\frac{% h_{0}^{2\epsilon }t_{0}}{\epsilon }\right) .$$ Using this result, as well as Eqs. (\[one\]) and (\[res\]), we can write the total answer for the conductivity as follows $$\sigma ^{^{\prime }}=\sigma _{0}\left( {}\right. 1+\frac{h^{\prime 2\epsilon }t_{0}}{\epsilon }+A\frac{h^{\prime 4\epsilon }t_{0}^{2}}{\epsilon }\left. {}\right) . \label{sigma}$$ Here we have written $t_{0}=4\Omega _{d}/\sigma _{0}$. Eq. (\[sigma\]) no longer contains $\alpha $ and is therefore the desired result. $\beta$ and $\gamma$ functions ------------------------------ Recall that $h$ is just the effective mass in the problem and we can replace it by the effective mass that is being induced by working with finite external frequencies, or finite temperatures. However, we can use Eq. (\[sigma\]) directly for extracting the renormalization constant $Z_{1}$ for the $t$ field. Introducing the renormalized fields $t$ and $z$ as usual $$t_{0}=\mu ^{-2\epsilon }tZ_{1}(t),\;\;z_{0}=zZ_{2}(t),$$ then, following the scheme of minimal subtraction, we obtain $$\begin{aligned} Z_{1} &=&1+\frac{t}{\epsilon }+\frac{t^{2}}{\epsilon ^{2}}(1+\epsilon A) \label{z1} \\ Z_{2} &=&1-\frac{t}{2\epsilon }-\frac{t^{2}}{4\epsilon ^{2}}\left( \frac{1}{2% }+\epsilon (\frac{\pi ^{2}}{6}+2)\right) . \label{z2}\end{aligned}$$ Here, we have listed also the result for $Z_{2}$ that was obtained in Ref. 10. The $\beta $ and $\gamma $ functions are defined by $$\beta =\frac{dt}{d\ln \mu }=\frac{2\epsilon t}{1+td\ln Z_{1}/dt}, \label{genbeta}$$ $$\gamma =-\frac{d\ln z}{d\ln \mu }=\beta \frac{d\ln Z_{2}}{dt}, \label{gengamma}$$ and the final answer can be written as $$\beta =2\epsilon t-2t^{2}-4At^{3}\,\,,\,\,\gamma =-t-(\frac{\pi ^{2}}{6}% +3)t^{2}. \label{pf}$$ ------------------------------------------------------------------ --------------- ---------------------------- ------------------------ ------------------------------------ ------------------------ -------------------------------- -------------------------- Contractions Diagrams $\frac{1}{\epsilon\alpha}$ $\frac{\log^{2}\alpha% $\frac{\log \alpha}{\epsilon^{2}}$ $\frac{\log^{2}\alpha% $\frac{\log \alpha}{\epsilon}$ $\frac{1}{\epsilon^{2}}$ }{\epsilon^{2}}$ }{\epsilon}$ $\langle O_{1}^{(4)} \rangle $ = 40pt = 20pt 2 2 $\langle O_{1}^{(3)} S^{(3)}_{int} \rangle $ = 40pt = 20pt -4 -4 $\langle O_{1}^{(2)}( S^{(4)}_{int} + S^{(4)}_{0} +\frac{1}{2} ( = 40pt = 20pt 2 2 S^{(3)}_{int})^{2}) \rangle $ Total 0 0 0 0 0 0 ------------------------------------------------------------------ --------------- ---------------------------- ------------------------ ------------------------------------ ------------------------ -------------------------------- -------------------------- = 200pt = 30pt ------------------------------------------------------------------------ --------------------------------- ------------------------------- -------------------------------- ----------------------------------- ---------------------------- -------------------------------- ---------------- Contractions Diagrams $\frac{1}{\epsilon \alpha } $ $\frac{% $\frac{log\alpha}{\epsilon^{2}} $ $\frac{% $\frac{log \alpha}{\epsilon} $ $\frac{1}{% log^{2}\alpha}{\epsilon^{2}} $ log^{2}\alpha}{\epsilon} $ \epsilon^{2}}$ $\langle O^{(6)}_{2} \rangle $ = 40pt = 20pt and = 40pt = 20pt 16 -4 -2 $\langle O_{2}^{(5)} S^{(3)}_{int} \rangle^{} $ = 40pt = 20pt -8 4 20 4 $\langle O_{2}^{(4)} S^{(4)}_{0} \rangle $ = 40pt = 20pt 2 4 -8 -2 1 2 $\langle O_{2}^{(2)}( S^{(4)}_{int} +\frac{1}{2} ( S^{(3)}_{int})^{2}) = 40pt = 20pt -2 -4 -2 -25 -4 \rangle^{} $ Total 0 0 0 0 - 8 0 ------------------------------------------------------------------------ --------------------------------- ------------------------------- -------------------------------- ----------------------------------- ---------------------------- -------------------------------- ---------------- = 200pt = 30pt ------------------------------------------------------------------------ --------------- -------------------------------------------------------------------------------- Contractions Diagrams $\frac{1}{\epsilon } $ $\langle O^{(4)}_{1} \rangle $ = 40pt = 20pt $0 $ $\langle O_{1}^{(3)} S^{(3)}_{int} \rangle $ = 40pt = 20pt $8 + 4 \zeta(3) $ $\langle O_{1}^{(2)}( S^{(4)}_{int} + S^{(4)}_{0} +\frac{1}{2} ( = 40pt = 20pt $- 4 - 2\zeta(3) - 2 \pi^{2} / 3 $ S^{(3)}_{int})^{2}) \rangle $ $\langle O^{(6)}_{2} \rangle $ = 40pt = 20pt $- \frac{\pi^{2}}{3} + \frac{ \pi^{2}}{2} \log 2 + \frac{ \pi^{4}}{12} + \frac{11 \zeta(3)}{2} + \frac{\pi^{2}}{3} \log^{2} 2 - \frac{1% }{3}\log^{4} 2 $ = 40pt = 20pt $- 7 \zeta(3) \log 2 - 8 Li_{4}(\frac{1}{2}) $ $\langle O_{2}^{(5)} S^{(3)}_{int} \rangle^{} $ = 40pt = 20pt $-12 + 4 \zeta(3) + 4 \pi^{2} / 3 $ $\langle O_{2}^{(4)} S^{(4)}_{0} \rangle $ = 40pt = 20pt $44 / 3 $ $\langle O_{2}^{(2)}( S^{(4)}_{int} +\frac{1}{2} ( S^{(3)}_{int})^{2}) = 40pt = 20pt $% \rangle^{} $ 55 /2 - 2 \zeta(3) - \frac{8}{3} \pi^{2} + 12 \log^{2} 2 - 44 \log 2 +16 G - 8 Li_{2}(\frac{1}{2}) $ Total $34 + \frac{1}{6} - 3 \pi^{2} + \frac{19}{2}\zeta(3) +16 \log^{2} - 44 \log 2 + \frac{\pi^{2} }{2} \log 2 +16 G $ $+ \frac{\pi^{4}}{12} + \frac{\pi^{2}}{3} \log^{2} 2 - \frac{1}{3} \log^{4} 2 - 7 \zeta(3) \log 2 - 8 Li_{4}(\frac{1}{2}) $ ------------------------------------------------------------------------ --------------- -------------------------------------------------------------------------------- = 200pt = 30pt [2]{} Dynamical scaling {#SR} ================= Relation between $h^{^{\prime }}$ and $\omega _{s}$ --------------------------------------------------- In the Section we combine the two loop computations of this paper with those of the amplitude $z_{0}$ and establish the connection between the effective mass $h^{^{\prime }}$ and the frequency $\omega _{s}$. For this purpose, recall that the renormalization of the $z$ field was obtained from the derivative of the free energy $F$ (or rather, the grand canonical potential) with respect to $\ln T$.[@bps2] The result of the computation was as follows $$\frac{dF}{d\ln T}=2\sum_{s>0}\omega _{s}z_{0}M_{b}(t_{0},h_{s}^{2}), \label{derF}$$ where $$M_{b}(t_{0},h_{s}^{2})=1+\frac{h_{s}^{2\epsilon }t_{0}}{2\epsilon }+\frac{% h_{s}^{4\epsilon }t_{0}^{2}}{\epsilon ^{2}}\left( -\frac{1}{8}+\epsilon (% \frac{1}{4}+\frac{\pi ^{2}}{24})\right) . \label{M}$$ Here, the frequency enters through the quantity $h_{s}^{2}=\kappa ^{2}z_{0}s=% \frac{2\pi }{\Omega _{d}}\omega _{s}z_{0}t_{0}$ which has the dimension of mass squared. The frequency dependence in $\sigma ^{^{\prime }}(s)$ is restored by writing $$\sigma ^{^{\prime }}(s)=\frac{4\Omega _{d}}{t_{0}}R_{b}(t_{0},h_{s}^{2}) \label{R}$$ with $$R_{b}(t_{0},h_{s}^{2})=1+\frac{h_{s}^{2\epsilon }t_{0}}{\epsilon }+(A-1/2)% \frac{h_{s}^{4\epsilon }t_{0}^{2}}{\epsilon } \label{Rr}$$ One can easily verify that Eqs. (4.1–4.4) lead to the same expressions for $Z_{1}$ and $Z_{2}$ and, hence, the same $\beta $ and $\gamma $ functions as those of the previous Section. Eq. (4.4) is therefore the correct result. The relation between $h^{^{\prime}}$ and $\omega_s$ can now be made more explicit by writing $$h^{^{\prime}2} = h_{s}^{2} M_b (t_{0},h_{s}^{2})/ R_b (t_{0},h_{s}^{2}).$$ Here, $h^{^{\prime}}$ is the effective mass that is induced by the frequency $\omega_s$ and the result is consistent with all previous statements and explicit computations.[@bps2] The Goldstone phase {#MIT} ------------------- ### Specific heat and AC conductivity The zero of the $\beta $ function, Eq. (\[pf\]), determines a critical point $t_{c}=O(\epsilon )$ that separates the Goldstone or metallic phase ($% t<t_{c}$) from an insulating phase ($t>t_{c}$). To second order in $\epsilon $ we have $$t_{c}=\epsilon -2A\epsilon ^{2}\approx \epsilon -3.28\epsilon ^{2}$$ We see that the $\epsilon ^{2}$ contribution is rather large and the expansion can clearly not be trusted for $\epsilon =1/2$ or three spatial dimensions. This is a well-known drawback of asymptotic expansions and the two-loop theory is otherwise necessary to completely establish the scaling behavior of the electron gas in $2+2\epsilon $ spatial dimensions. To discuss this scaling behaviour, we proceed and express Eqs. (\[derF\]) and (\[R\]) in terms of the renormalized parameters $t$ and $z$. The results can be written in the following general form $$\begin{aligned} \frac{dF}{d\ln T} &=&2\sum_{s>0}\mu ^{2\epsilon }\omega _{s}zM(t,\omega _{s}z), \label{derFren} \\ \sigma ^{^{\prime }}(s) &=&\mu ^{2\epsilon }\frac{4\Omega _{d}}{t}R(t,\omega _{s}z). \label{sigmaren}\end{aligned}$$ The expressions are now finite in $\epsilon $. The AC conductivity is obtained from $\sigma ^{^{\prime }}(s)$ by replacing the imaginary frequencies $i\omega _{s}$ by real ones $\omega $. On the other hand, the specific heat of the electron gas can be expressed as [@bps2] $$c_{v}=\int_{0}^{\infty }d\omega \frac{\partial f_{BE}}{\partial T}\omega \rho _{qp}(\omega ), \label{cv}$$ where $$f_{BE}=\frac{1}{e^{\omega /T}-1}$$ and $$\rho _{qp}(\omega )=\frac{z}{\pi }(M(t,i\omega z)+M(t,-i\omega z))$$ is the density of states of bosonic quasiparticles indicating that the Coulomb system is unstable with respect to the formation of particle-hole bound states. [@es] ### Scaling results Next, from the method of characteristics we can obtain the general scaling behavior of the quantities $M$ and $R$ as usual: $$\begin{aligned} M(t,\omega _{s}z) &=&M_{0}(t)G(\omega _{s}z\xi ^{d}M_{0}(t)), \nonumber \\ R(t,\omega _{s}z) &=&R_{0}(t)H(\omega _{s}z\xi ^{d}R_{0}(t)). \label{scale}\end{aligned}$$ Here $G$ and $H$ are unspecified functions, whereas $\xi $, $R_{0}$ and $% M_{0}$ each have a clear physical significance and are identified as the correlation length, the DC conductivity and $\rho _{qp}(0)$ respectivily. They obey the following equations $$\begin{aligned} (\mu \partial _{\mu }+\beta \partial _{t})\xi (t) &=&0, \nonumber \\ (\beta \partial _{t}-2\epsilon -\beta /t)R_{0}(t) &=&0, \nonumber \\ (\beta \partial _{t}+\gamma )M_{0}(t) &=&0.\end{aligned}$$ In the metallic phase ($t<t_{c}$) the solutions can be written as follows $$R_{0}(t)=(1-t/t_{c})^{2\epsilon \nu }\,\,,\,\,M_{0}(t)=(1-t/t_{c})^{\beta _{0}},$$ $$\xi =\mu ^{-1}t^{1/2\epsilon }(1-t/t_{c})^{-\nu },$$ where the critical exponents $\nu $ and $\beta _{0}$ are obtained as $$\nu ^{-1}=\beta ^{^{\prime }}(t_{c})\,\,,\,\,\beta _{0}=-\nu \gamma (t_{c}).$$ To second order in $\epsilon $ the results are $$\begin{aligned} \nu ^{-1} &=&2\epsilon (1+2A\epsilon )\approx 2\epsilon +6.56\epsilon ^{2} \nonumber \\ \beta _{0} &=&\left( 1+(\pi ^{2}/6+3-4A)\epsilon \right) /2\approx 0.50-0.96\epsilon .\end{aligned}$$ Both the DC conductivity $R_{0}$ and the quantity $M_{0}$ vanish as one approaches the metal-insulator transition at $t_{c}$. The results are quite familiar from the Heisenberg ferromagnet where $M_{0}$ stands for the spontaneous magnetization. Unlike the free electron gas,[@inst] however, the interacting system with Coulomb interactions has a true order parameter, $M_{0}$, which is associated with a non-Fermi liquid behavior of the specific heat. ### Equations of state The explicit results of Section A can be used to completely determine the quantities $M$ and $R$ in the Goldstone phase. They take the form of an ”equation of state” [@pw] $$\frac{\omega _{s}zt}{M^{\delta }}=\left( \frac{t_{c}}{t}\right) ^{1/\epsilon }\left( 1+(2\epsilon \nu -1)(1-\frac{t}{t_{c}})-2\epsilon \nu \frac{1-t/t_{c}% }{M^{1/\beta _{0}}}\right) ^{1/\epsilon },\nonumber$$ $$\frac{\omega _{s}zt}{R^{\kappa }}=\left( \frac{t_{c}}{t}\right) ^{1/\epsilon }\left( 1-\frac{1-t/t_{c}}{R^{1/2\epsilon \nu }}\right) ^{1/\epsilon }. \label{eqs}$$ Here, the exponents $\delta $ and $\kappa $ can be obtained from the values of $\nu $ and $\beta _{0}$ following the relations $$d\nu =\beta _{0}(\delta +1)\,\,,\,\,2\epsilon \nu \kappa =\beta _{0}\delta .$$ The universal features of the ”equations of state” are the Goldstone singularities at $t=0$ and the critical singularities near $t_{c}$. As for the specific heat, we find the usual behavior $c_{v}=\gamma _{0}T$ at $t=0$ but at criticality the following algebraic behavior is found $c_{v}=\gamma _{1}T^{1+1/\delta }$. It is important to remark that the expression for the conductivity $R$ can also be used in the case of finite temperatures and we may, on simple dimensional grounds, substitute $T$ for $\omega _{s}$. The results, however, strictly hold for the Goldstone and critical phases only. The ”equations of state” cannot be analytically continued and used to obtain information on the insulating phase. As we already mentioned in the introduction, the strong coupling phase is controlled by different operators in the theory and has a distinctly different frequency and temperature dependence.[@pb01] Plateau transitions in the quantum Hall regime ---------------------------------------------- ### Introduction {#introduction-2} In this Section we briefly describe how the results of this paper are extended to include the plateau transitions in the quantum Hall regime. For this purpose recall that the theory in two spatial dimensions and strong magnetic fields is given by $$S[Q,A]\rightarrow S[Q,A]+\frac{\sigma _{xy}^{0}}{8}\int_{x}{\rm tr}\epsilon _{ij}Q[D_{i},Q][D_{j},Q].$$ The theory depends on the $\theta $ term, or $\sigma _{xy}$ term, in a non-perturbative manner and the general form of the renormalization group equations can now be written as [@eurolett] $$\begin{aligned} \frac{d\sigma _{xx}}{d\ln \mu } &=&\beta _{xx}(\sigma _{xx},\sigma _{xy}), \nonumber \\ \frac{d\sigma _{xy}}{d\ln \mu } &=&\beta _{xx}(\sigma _{xx},\sigma _{xy}), \label{RG} \\ \frac{d\ln z}{d\ln \mu } &=&\gamma (\sigma _{xx},\sigma _{xy}). \nonumber\end{aligned}$$ The interesting physics actually occurs in the strong coupling phase ($% \sigma _{xx}<1$) where the crossover takes place from the perturbative regime of quantum interference effects, as studied in this paper, to the quantum Hall regime that generally appears in the limit of much larger distances only (Fig. 1). As an important general remark we can say that the quantum Hall effect is a universal, strong coupling feature of the $\theta $ term, or [instanton vacuum]{}, and fundamental aspects of the problem have not been recognized until recently. We mention in particular the fact that the theory displays [massless excitations]{} that always exist at the edge of the system. [@pbv] This new ingredient turns out to have fundamental consequences for longstanding problems such as the [quantization of topological charge]{}, the general meaning of [instantons]{} etc. Moreover, the concept of massless chiral edge modes can be used to unravel some of the outstanding strong coupling aspects of the theory such as the [exact quantization]{} of the Hall conductance which is represented by the infrared stable fixed points at $\sigma _{xx}=0$ and $\sigma _{xy}=k$ in the scaling diagram of Fig. 1. Perhaps more surprizingly, a gapless phase seems to always exist in the theory at $\theta =\pi $ or $\sigma _{xy}^{0}$ equal to an half-integer. This fundamental aspect of the quantum Hall effect is displayed even by the $% CP^{N-1}$ theory with large values of $N$. [@pbv] These results indicate that the quantum Hall effect is a generic feature of the $\theta$ term in asymptotically free field theory and, contrary to the previous believes, the number of field components plays a secondary role only. Recall that the free electron theory, the Finkelstein approach and the $CP^{N-1}$ model with large $N$ are all topologically equivalent. They have important features in common such as asymptotic freedom and instantons. They are only different in the manner the number of field components in the theory is being handled. This does not affect the fundamentals of the quantum Hall effect, however, but only the critical singularities at $\theta = \pi$ which are different in each case. ### Scaling of conductances We next focus on the consequences of the unstable fixed points in Fig. 1, located at $\sigma _{xy}=k+\frac{1}{2}$ and $\sigma _{xx}=\sigma _{xx}^{}$ which is of order unity. These fixed points describe the critical singularities of the quantum Hall plateau transitions. [@eurolett] A finite value of $\sigma _{xx}^{}$ indicates that we are dealing with a critical metallic state which is much the same phenomenon as the metal-insulator transition that separates the Goldstone phase from the insulating phase in the theory in $% 2+2\epsilon $ dimensions. The quantum Hall regime therefore provides a unique laboratory inwhich the properties of disorder driven quantum phase transition can be explored and investgated in detail. Let us first recall the results for the conductances $\sigma_{xx}^{^{\prime}}$ and $\sigma_{xy}^{^{\prime }}$ as obtained in Ref. 8 $$\begin{aligned} \sigma _{xx}^{^{\prime }} &=&f_{xx}[(zT)^{-\kappa }(\nu _{B}-\nu _{B}^{})], \nonumber \\ \sigma _{xy}^{^{\prime }} &=&f_{xy}[(zT)^{-\kappa }(\nu _{B}-\nu _{B}^{})].\end{aligned}$$ Here, the functions $f_{xx}(X)$ and $f_{xy}(X)$ are regular (differentiable) functions for small $X$, $\nu_{B}=\sigma _{xy}^{0}\propto 1/B$ is the filling fraction of the Landau levels and $\nu _{B}^{}=k+1/2$ is the critical value, corresponding to the center of the Landau band. The exponent $\kappa =p/2\nu \approx 0.42$ has been extracted from the experimental transport data taken from low mobility heterostructures in the quantum Hall regime. [@experiments] Notice that the scaling variable $X$ can be expressed as $(h^{^{\prime }}\xi)^{-1/\nu}$ where $h^{^{\prime }}$ is the mass that is induced by finite temperatures (or frequency) $$h^{^{\prime }}=(zT)^{p/2}\;,\;\;(z\omega)^{p/2}.$$ The $\xi $ is the diverging correlation length at the center of the Landau band $$\xi \propto \|\nu _{B}-\nu _{B}^{}\|^{-\nu }=\|\sigma _{xy}-k-\frac{1}{2}\|^{-\nu}. \label{xi}$$ The critical exponent $\nu $ has the same meaning as before whereas $p$ was originally introduced as the [inelastic scattering time]{} exponent. [@freepart] Both are defined formally by the $\beta _{xy}$ and $\gamma $ functions according to $$\begin{aligned} {\nu ^{-1}} &=&\partial \beta _{xy}^{}/\partial \sigma _{xy}, \nonumber \\ p &=&1+\frac{1}{\delta }=\frac{1}{1+\gamma ^{}/2},\end{aligned}$$ where $\beta _{xy}^{}=\beta _{xy}(\sigma _{xx}^{},k+1/2)$ and $\gamma ^{}=\gamma (\sigma _{xx}^{},k+1/2)$. ### Particle-hole symmetry, duality Generally speaking, one expects the functions $f_{xx} (X)$ and $f_{xy} (X)$ to be universal scaling functions, describing the points on the renormalization group trajectory that connects the unstable fixed points with the stable ones (Fig. 1). [@freepart] There is, however, interesting physics associated with this statement of universality and the subject is an extremely important objective for experimental research. The problem with the plateau transitions is that although the [macroscopic]{} conductances $\sigma_{xx}^{\prime}$ and $\sigma_{xy}^{\prime}$ are well defined and sharply distributed at finite $T$, this is not the case for the [mesoscopic]{} conductances which are defined for finite lengthscales, of the order of the phase breaking length $1/h'$. The mesoscopic conductances are, in fact, broadly distributed and the size of the fluctuations is comparable or larger than the mean value. Since the $1/h'$ is the only length scale in the problem with Coulomb interactions and at finite $T$, it directly follows that the relation between the [mesoscopic conductance distributions]{} and the [measured]{} or [macroscopic conductance]{} must be non-trivial in general. For example, it is necessary to construct [block models]{} [@cohen] that describe the electron transport process in terms of a classical network of (mesoscopic) conductances that are randomly distributed over the different areas ([blocks]{}) in the system of size $1/h'$. The concept of [block models]{} complicates such aspects like [particle-hole symmetry]{} that is displayed by the physical observables of the electron gas. [Particle-hole symmetry]{}, just like the quantization of the Hall conductance, is a direct consequence of one of the most fundamental aspects of the instanton vacuum, namely [quantization]{} of [topological charge]{}. It can be expressed as follows $$\begin{aligned} f_{xx} (X) & = & f_{xx} (-X) \nonumber \\ f_{xy} (X) & = & 2k +1 - f_{xy} (-X)\end{aligned}$$ More generally, one can show that [particle-hole symmetry]{} is displayed by the entire distribution functions of the mesoscopic conductances, rather than by the macroscopic quantities or averaged quantities alone. It is clear that the theory of block models is particularly sensitive with regard to the many controversial issues that presently span the subject of mesoscopic fluctuations.[@altshuler] It is important to keep in mind that the quantum Hall plateau transitions take place in precisely the regime ($\sigma_{xx} < 1$) where not only the conductance fluctuation are uncontrolled, but also the infinite set of higher dimensional operators that enters in the definition of the higher order momenta of the distribution functions. Obviously, for the more difficult problems like quantum criticality in the presence of the Coulomb interactions, one can not [just assume]{} that the theory automatically takes care of itself in each and every fronts. Following Kivelson et al, [@kivel] however, one can proceed in a pragmatic fashion and employ the Chern Simons mapping of abelian quantum Hall states to show that the system has a [dual]{} symmetry. Provided one works at finite $T$ and with system sizes that are much larger than $1/h'$, the mapping of conductances is not affected by the fluctuations that occur at mesoscopic lengthscales.[@pb01] By making furthermore use of [particle-hole symmetry]{} and by identifying the functions $f_{xx} (X)$ and $f_{xy} (X)$ as the subspace of conductances that is [dual]{} under the Chern Simons mapping, one arrives at the following result $$\begin{aligned} f_{xx} (X) & = & \frac{g(X)}{1+g^2 (X)} \nonumber \\ f_{xy} (X) & = & k + \frac{1}{1+g^2 (X)} \label{ff}\end{aligned}$$ where the function $g(X) = e^{a_1 X + a_3 X^3 + ...}$ obeys the general contraint $$g(X) =g^{-1} (-X) \label{gg}$$ These results imply that the sequence of plateau transitions in the quantum Hall regime ends up at $k=0$ in a so-called [quantum Hall insulating phase]{} which means that the Hall resistance $\rho_{xy}$ remains quantized throughout the lowest Landau level. It is important to remark that the statement of duality has been carried out in a manner which is consistent with the gradient expansion that generally defines the effective action or sigma model approach. [@bps1] If, on the other hand, the effective action procedure were to fail and, say, terms of higher dimension would generally become important, [@altshuler] then the statements made by Eqs. (\[ff\]) and (\[gg\]) would clearly have no meaning and the theory of quantum transport must be largely reconsidered. With regard to the universality of the functions $f_{xx} (X)$ and $f_{xy} (X)$, the experimental situation has remained unresolved for a long time. However, recent experiments have clearly demonstrated that Eqs. (\[ff\]) and (\[gg\]) are valid, at least for the lowest Landau level. The transport data were taken from a low mobility $InGaAs/InP$ heterostructure in strong magnetic fields and at low temperatures.[@newexpt] The new results indicate that the lack of universality, that was previously found,[@experiments1; @experiments] is merely the consequence of sample inhomogeneities. This means that there is little room left for the type of complications that arose in the perturbative theory of mesoscopic fluctuations. The experiments are in favor of [duality]{} as a fundamental symmetry of the electron gas with Coulomb interactions. As shown by Eqs. (\[ff\]) and (\[gg\]), this symmetry provides fundamental support for the results of the renormalization theory. It should be mentioned that the Chern Simons mapping of conductances can be carried out for almost any type of disorder and duality by itself does therefore not provide any garantee that the system is actually in a quantum critical state. For example, it is well known that complications arise in systems with longranged potential fluctuations and the matter has been extensive addressed in Ref. 11. ### Specific heat As we have mentioned earlier, it is necessary to identify other physical observables in the problem that can in principle be measured and used to extract the value of $p$ and $\nu$ separately. The microscopic theory of the electron gas in $2+2\epsilon $ dimensions tells us that the natural quantity to consider is the specific heat, Eq. (\[cv\]). Moreover, we have shown in Ref. 8 that this quantity is unchanged under the Chern Simons mapping. By using our general knowledge on the renormalization group functions $\beta _{xx}$, $\beta _{xy}$ and $\gamma $ one can derive, in the standard manner, the scaling form of the quantity $M(\sigma _{xx},\sigma _{xy},\omega _{s}z)$ in the quantum Hall regime. This leads to the same expression as in Eq. (\[scale\]) with $M_0 (t)$ now replaced by $M_0 (\nu_B ) = \|\nu_B -\nu_B^* \|^{\beta_0}$ and $\xi$ given as in Eq. (25). At the quantum critical point ($\nu_B =\nu_B^$) we obtain the same non-Fermi liquid expression as before $$c_{v}=\gamma _{1}T^{p}.$$ In different words, the physical observable, associated with the ”inelastic scattering” exponent $p$ in quantum Hall systems, is none other than the [specific heat]{} of the electron gas. A measurement of $c_{v}$ should therefore provide the ultimate test on the consistency of the theory. This information is not present as of yet. Conclusion {#Conc} =========== In this paper we have completed the two-loop analysis of the Finkelstein theory with the singlet interaction term. We have reported the detailed computations of the conductivity which is technically the most difficult part of the analysis. We have benifitted from the regularization procedure involving the $h_0$ field, which has substantially simplified the two-loop computations. Moreover, we have obtained a general relation between the effective masses that are being induced by the $h_0$ field on the one hand, and the frequency $\omega_n$ on the other. This enables one to re-express the final answer in terms of finite frequencies and/or temperature, simply by a substitution of the $h_0$ regulating field. By combining the concept of ${\cal F}$ invariance with technique of dimensional regularization, we have extracted new physical information on the disordered electron gas with Coulomb interactions in low dimensions. In particular, we now have a non-Fermi liquid theory for the specific heat and dynamical scaling. The metal-insulator transition in $2+2\epsilon $ dimensions sets the stage for the plateau transitions in the quantum Hall regime. We have identified the specific heat $c_{v}$ as the physical observable that determines the exponent $p$, previously introduced as the exponent for “inelastic scattering.” As a final remark we can say that our knowledge of the theory is limited only by the accuracy with which one can give a numerical estimate of the critical exponents $\nu $ and $p$. Except for the fact that $p$ is bounded by $1<p<2$, [@pb01] the detailed values of $\nu $ and $p$ can only be obtained by performing the renormalization group numerically. Notice that the situation is somewhat similar for the metal-insulator transition in $2+2\epsilon $ dimensions. In that case, the limitations of the $\epsilon $expansion prevent us from having accurate exponents for the electron gas in three spatial dimensions. Acknowledgement {#Ack} =============== We are indebted to E. Brézin and A. Finkelstein for numerous conversations. One of us (I.B.) is grateful to M. Feigel’man, M. Lashkevich, D. Podolsky and P. Ostrovsky for stimulating discussions. The research has been supported in part by the Dutch Science Foundation FOM and by INTAS (Grant 99-1070). Appendix A {#AA} ========== In this Appendix we present the final results for the various integrals listed in Eqs. (\[start\])-(\[end\]). We shall follow the same methodology as used in the two-loop computation of Ref. 10 and employ the standard representation for the momentum and frequency integrals in terms of the Feynman variables $x_{1}$, $x_{2}$ and $x_{3}$. We classify the different contributions in Eqs. (\[start\])-(\[end\]) in different catagories, labeled $A$-integrals, $B$-integrals etc. In total we have seven different catagories, i.e. $A$, $B$, $C$, $D$, $H$, $S$ and $T$ respectively, which are discussed separately in Sections $A$ - $G$ of this Appendix. The last Section, $H$, contains a list of abbreviations and a list of symbols for those integrals that need not be computed explicitly because their various contributions sum up to zero in the final answer. In Appendix $B$ we present the main computational steps for a specific example, the so-called $A_{10}$-integral. We show how the integral representation of hypergeometric functions can be used to define both the $\epsilon $ expansion and the limit where $\alpha \rightarrow 0$. The A - integrals {#A} ------------------ ### Definition To set the notation, we consider the integral $$\begin{aligned} X_{\nu,\eta}^\nu & = & -\frac{2^{1+\nu} a^{2+\mu} }{\sigma_{0} d^{\nu}} \int \limits_{p q} p^{2 \nu} \sum_{k,m>0} m^{\mu} \nonumber \\ & & D_{p+q}^{c}(m) D D^{c}_{p}(k) D^{1+\mu+\eta}_{q}(k+m) . \label{pr}\end{aligned}$$ Here, the three indices $\mu$, $\nu$ and $\eta$ generally take on the values $0, 1$. We shall only need those quantities $X_{\nu,\eta}^\nu$ which have $\eta = \nu$, however. Using the Feynman trick, one can write (for the notation, see Section $H$) $$\begin{aligned} X_{\nu,\eta}^\nu & = & -\frac{2^{1+\nu} a^{2+\mu} }{\sigma_{0} d^{\nu}} \int \limits_{p q} p^{2} \int \limits_{0}^{\infty} dm \; m^{\mu} \int \limits_{0}^{\infty} dk \nonumber \\ & & \frac{\Gamma(\mu +\eta+4)}{\Gamma(\mu + \eta+1)} \int \limits_{\alpha}^{1} dz \int [] \; x_{2} x_{3}^{\mu +\eta} \nonumber \\ & & \left[ \right . h_{0}^{2} + q^{2} x_{12} + p^{2} x_{13} + 2 {\bf p} \cdot {\bf q} \; x_{1} \nonumber \\ & & + a m ( \alpha x_{1} + x_{3}) + a k (z x_{2} + x_{3} ) \left . \right ]^{-\mu - \eta - 4}\end{aligned}$$ Next, by shifting $q \rightarrow q - p x_{1} / x_{12}$, we can decouple the vector variables ${\bf p}$ and ${\bf q}$ in the denominator. The integration over $k, m, p$ and $q$ then leads to an expression that only involves the integral over $z$ and the Feynman variables $x_1$, $x_2$ and $x_3$. Write $$\begin{aligned} X_{\nu,\eta}^\nu = \frac{ \Omega_{d}^{2} h_{0}^{4 \epsilon}}{ \sigma_{0} \epsilon} A^{\nu}_{\mu,\eta}\end{aligned}$$ then $$A_{\mu \eta }^{\nu }=\int\limits_{\alpha }^{1}dz\int []\frac{x_{2}x_{3}^{1+\mu +\eta }(x_{1}+x_{2})^{\nu }(x_{i}x_{j})^{-1-\nu -\epsilon }}{(zx_{2}+x_{3})(\alpha x_{1}+x_{3})^{1+\mu }}.$$ To complete the list of $A$-integrals, we next define quantities that carry either two indices $\mu , \nu$ or only a single index $\mu$. Like $A_{\mu \eta}^{\nu}$, they all describe contractions that contain both momentum and frequency integrals. The results are all expressed in terms of integrals over $z$, $x_1$, $x_2$ and $x_3$. $$\begin{aligned} A_{\nu \mu } &=&\int\limits_{\alpha }^{1}dz(z-\alpha )^{1+\nu -\mu } \nonumber \\ &\times &\int []\frac{x_{1}^{\mu }x_{2}^{2+\nu -\mu }x_{3}^{\mu }(x_{1}+x_{3})^{1-\mu }(x_{i}x_{j})^{-2-\epsilon }}{(\alpha x_{1}+x_{3})^{1+\nu }(zx_{2}+x_{3})},\end{aligned}$$ $$A_{0}=\int\limits_{\alpha }^{1}dz(z-\alpha )\int []\frac{x_{2}^{2}x_{1}(x_{i}x_{j})^{-2-\epsilon }}{(x_{3}+zx_{2})(zx_{2}+\alpha x_{1}+2x_{3})},$$ $$\begin{aligned} A_{1} &=&\int\limits_{\alpha }^{1}dz(z-\alpha )^{2}\int []\frac{x_{2}^{3}(x_{1}+x_{3})(x_{2}+x_{3})}{(zx_{2}+x_{3})^{2}} \nonumber \\ \times ( &x_{i}&x_{j})^{-2-\epsilon }\left( {}\right. \frac{1}{(\alpha x_{1}+x_{3})^{2}}-\frac{1}{(zx_{2}+\alpha x_{1}+2x_{3})^{2}}\left. {}\right) ,\end{aligned}$$ $$\begin{aligned} A_{2} &=&\int\limits_{\alpha }^{1}dz(z-\alpha )(1-z) \nonumber \\ &\times &\int []\frac{x_{2}^{3}(x_{1}+x_{3})(x_{i}x_{j})^{-2-\epsilon }}{(zx_{2}+x_{3})(\alpha x_{1}+x_{3})(zx_{2}+\alpha x_{1}+2x_{3})}\end{aligned}$$ $$A_{3}=\int\limits_{\alpha }^{1}dz(z-\alpha )\int []\frac{x_{2}^{2}(x_{1}+x_{3})(x_{i}x_{j})^{-2-\epsilon }}{(\alpha x_{1}+zx_{2}+2x_{3})(zx_{2}+x_{3})}.$$ ### $\epsilon$ expansion The calculation of integrals is staightforward but tedious and lengthy. Here we only present the final results of those quantities that are needed. The list does not contain the final answer for the $A_0$-integral because the various contributions to $A_0$ sum up to zero in the final answer. These same holds for some other integrals that are defined in Section $H$ and that we do not specify any further. $$\begin{aligned} A_{00}^{0} &=&-\frac{\ln ^{2}\alpha }{\epsilon }+\zeta (3), \nonumber \\ A_{10}^{0} &=&-\frac{\ln ^{2}\alpha +\ln \alpha }{\epsilon }-\frac{\ln ^{2}\alpha }{2}+\frac{\pi ^{2}}{6}+\zeta (3), \nonumber \\ A_{01}^{1} &=&\frac{\ln \alpha }{\epsilon }-\frac{\ln ^{2}\alpha }{2}-2\ln \alpha -\frac{\pi ^{2}}{3}+1, \nonumber \\ A_{11}^{1} &=&\frac{\ln \alpha }{\epsilon }-\frac{\ln ^{2}\alpha }{2}-2\ln \alpha -\frac{\pi ^{2}}{3}, \nonumber \\ A_{00} &=&\frac{\ln \alpha }{\epsilon }+\frac{\ln ^{2}\alpha }{2}+2\ln \alpha +\frac{\pi ^{2}}{3}-1, \nonumber \\ A_{10} &=&-\frac{1}{\alpha }-\frac{2\ln \alpha +3}{\epsilon }-\ln ^{2}\alpha -5\ln \alpha -\frac{2\pi ^{2}}{3}+3, \nonumber \\ A_{01} &=&-\ln \alpha -\frac{\pi ^{2}}{6}+1, \nonumber \\ A_{11} &=&\frac{\ln \alpha +2}{\epsilon }+\frac{\ln ^{2}\alpha }{2}+3\ln \alpha +\frac{\pi ^{2}}{2},\end{aligned}$$ $$\begin{aligned} A_{1} &=&-\frac{2}{\alpha }+\frac{2\ln ^{2}\alpha +4\ln \alpha }{\epsilon }-3\ln ^{2}\alpha \nonumber \\ &+&8\ln 2\ln \alpha -\frac{17}{2}\ln \alpha +4K_{1}(\alpha )+8J_{3}^{^{\prime }}(\alpha ) \nonumber \\ &-&\pi ^{2}-2\zeta (3)-6\ln ^{2}2+10\ln 2-\frac{1}{2},\end{aligned}$$ $$\begin{aligned} A_{2} &=&-\frac{\ln ^{2}\alpha +2\ln \alpha }{\epsilon }-2\ln \alpha -3\ln 2\ln \alpha \nonumber \\ &-&J_{1}(\alpha )-K_{1}(\alpha )-2J_{3}^{^{\prime }}(\alpha )+A_{0}-\frac{\pi ^{2}}{6} \nonumber \\ &+&1+\zeta (3)+3\ln ^{2}2-3\ln 2-3Li_{2}(1/2),\end{aligned}$$ $$A_{3}=A_{0}-2Li_{2}(\frac{1}{2})+\frac{\pi ^{2}}{6}.$$ The B - integrals {#B} ------------------ ### Definition The $B$-integrals are similarly defined in terms of the variables $z$, $x_{1} $, $x_{2}$ and $x_{3}$. However, they describe only those contractions that contain frequency sums and no momentum integrals. $$B_{\mu }=\int\limits_{\alpha }^{1}\frac{dz}{z^{\mu }}\int []\frac{x_{1}^{\mu -1}x_{2}x_{3}^{-\mu -\epsilon }(x_{1}+x_{2})^{-\mu -\epsilon }}{(\alpha x_{2}+zx_{3}+x_{1})},$$ ### $\epsilon$ expansion $$\begin{aligned} B_{1} &=&\frac{\ln \alpha }{\epsilon }+\frac{\ln ^{2}\alpha }{2}+\ln \alpha , \nonumber \\ B_{2} &=&-\frac{1}{\alpha }+\frac{\ln ^{2}\alpha }{\epsilon }+\frac{2\ln \alpha }{\epsilon }-2\ln \alpha -2.\end{aligned}$$ The C - integrals {#C} ----------------- ### Definition The $C$-integrals contain one additional integration over $y$, besides the ones over $z$ and the Feynman variables $x_1$, $x_2$ and $x_3$. They originate from expressions involving integrations over both frequencies and momenta. We distingish between quantities with two indices $\mu $ and $\nu $ $$C_{\mu \nu }=\int\limits_{\alpha }^{1}dzdy\int []\frac{x_{1}^{\mu }x_{2}x_{3}(x_{2}+x_{3})^{1-\mu }(x_{i}x_{j})^{-2-\epsilon }}{(zx_{3}+x_{1})(yx_{2}+x_{1})^{\nu }(zx_{3}+yx_{2})^{1-\nu }}$$ and those that carry only a single index $\nu $ $$\begin{aligned} C_{\nu } &=&\int\limits_{\alpha }^{1}dz(1-z)^{\nu }\int\limits_{\alpha }^{1}dy \nonumber \\ &\times &\int []\frac{x_{2}^{2-\nu }x_{3}^{1+\nu }(x_{1}+x_{2})^{\nu }(x_{i}x_{j})^{-2-\epsilon }}{(zx_{3}+x_{1})(yx_{2}+x_{1})^{\nu }(zx_{3}+yx_{2}+2x_{1})}.\end{aligned}$$ ### $\epsilon$ expansion $$\begin{aligned} C_{00} &=&\frac{\ln \alpha }{\epsilon }-\frac{\ln ^{2}\alpha }{2}-2\ln \alpha +\frac{\pi ^{2}}{4}\ln 2-\frac{\pi ^{2}}{6}+\frac{15}{4}\zeta (3) \nonumber \\ &-&\frac{\pi ^{4}}{24}-\frac{\pi ^{2}}{6}\ln ^{2}2+\frac{1}{6}\ln ^{4}2+\frac{7}{2}\zeta (3)\ln 2+4Li_{4}(\frac{1}{2}), \nonumber \\ C_{01} &=&\frac{2\ln \alpha }{\epsilon }-\ln ^{2}\alpha -4\ln \alpha -2-\zeta (3), \nonumber \\ C_{11} &=&\zeta (3), \nonumber \\ C_{0} &=&\frac{\ln \alpha }{\epsilon }-\frac{\ln ^{2}\alpha }{2}-2\ln \alpha -1-\zeta (3)-C_{0}^{^{\prime }}, \nonumber \\ C_{1} &=&4\ln 2\ln \alpha +2J_{1}(\alpha )-C_{0}^{^{\prime }}-2-\frac{\zeta (3)}{2}, \nonumber \\ &-&4\ln 2-\frac{\pi ^{2}}{6}+4G,\end{aligned}$$ where the Catalan constant $G=0.517\ldots $ appears as the integral $$G=-\int_{0}^{1}du\frac{\ln u}{1+u^2}.$$ The D-integrals {#D} ---------------- ### Definition These are integrals over the Feynman variables only. They originate from the contractions which contain sums over both momenta and frequencies. $$D_{\nu }=\int []\frac{x_{3}^{\nu }(x_{1}+x_{2})^{\nu -1}(x_{i}x_{j})^{-\nu -\epsilon }}{(\alpha x_{1}+x_{3})(\alpha x_{2}+x_{3})}.$$ ### $\epsilon$ expansion $$\begin{aligned} D_{1} &=&-\ln ^{2}\alpha -\frac{\pi ^{2}}{6}, \nonumber \\ D_{2} &=&-2\ln \alpha .\end{aligned}$$ The H - integrals {#H} ------------------ ### Definition The $H$-integrals involve the variable $z$ and the Feynman variables. All of them originate from contractions with sums over both momenta and frequencies. $$H_{\nu }=\int\limits_{\alpha }^{1}dz(z-\alpha )^{2\nu }\int []\frac{x_{2}^{2+\nu }(x_{1}+x_{3})(x_{i}x_{j})^{-2-\epsilon }}{(\alpha x_{1}+zx_{2})(zx_{2}+x_{3})}.$$ ### $\epsilon$ expansion $$\begin{aligned} H_{0} &=&-\ln \alpha +1, \nonumber \\ H_{1} &=&-\ln \alpha .\end{aligned}$$ The S - integrals {#Ss} ------------------ ### Definition These are integrals over the Feynman variables only and they do not not contain the parameter $\alpha $. All of them originate from the expressions with sums over both momenta and frequencies. $$S_{\mu \nu }=\int []\frac{x_{1}^{\mu }x_{2}^{1+\nu -\mu }((2-\nu -\mu )x_{1}+x_{3})(x_{i}x_{j})^{-2-\epsilon }}{(x_{2}+x_{3})^{1+\nu }},$$ $$S_{\nu }=\int [](x_{1}+x_{2})^{-1+2\nu }(x_{i}x_{j})^{-1-\nu -\epsilon }.$$ ### $\epsilon$ expansion $$\begin{aligned} S_{00} &=&-\frac{1}{\epsilon }+2, \nonumber \\ S_{01} &=&-\frac{1}{3\epsilon }+\frac{8}{9}, \nonumber \\ S_{11} &=&-\frac{1}{6\epsilon }+\frac{1}{9}, \nonumber \\ S_{0} &=&-\frac{1}{\epsilon }+2, \nonumber \\ S_{1} &=&-\frac{2}{\epsilon }+2.\end{aligned}$$ The T-integrals {#T} --------------- ### Definition The integrals are over the Feynman variables only. They come from the expressions which only contain sums over frequency. $$T_{\mu \nu }^{\eta }=\frac{(1-\alpha )^{\eta }}{\alpha ^{\mu }}\int []\frac{x_{1}^{2-\eta }x_{2}^{\mu +\eta -1}x_{3}^{-1-\mu -\epsilon }(x_{1}+x_{2})^{-2-\epsilon }}{(\alpha x_{2}+\nu \alpha x_{3}+x_{1})},$$ $$T_{\mu \nu }=\int []\frac{x_{1}^{2\nu -2}(x_{1}+x_{2})^{-\nu -\epsilon }(x_{1}+x_{3}+(\alpha +\mu )x_{2})}{x_{3}^{\nu +\epsilon }(\alpha x_{2}+(1+\mu )x_{3}+x_{1})(x_{1}+x_{3}+\alpha x_{2})}.$$ ### $\epsilon$ expansion $$\begin{aligned} T_{10}^{0} &=&-\frac{1}{\alpha }+1, \nonumber \\ T_{11}^{0} &=&-\frac{1}{\alpha }+\frac{1}{\epsilon }+\ln \alpha +1, \nonumber \\ T_{20}^{0} &=&\frac{1}{6\alpha ^{2}}-\frac{1}{3\alpha }-\ln \alpha -\frac{11}{12}, \nonumber \\ T_{21}^{0} &=&\frac{1}{6\alpha ^{2}}+\frac{2}{3\alpha }+\frac{\ln \alpha +5/2}{\epsilon }+\frac{\ln ^{2}\alpha }{2}+4\ln \alpha +\frac{17}{12}, \nonumber \\ T_{10}^{1} &=&-\frac{1}{\alpha }-2\ln \alpha -2, \nonumber \\ T_{01} &=&\frac{\ln \alpha }{\epsilon }-\frac{\ln ^{2}\alpha }{2}, \nonumber \\ T_{02} &=&\frac{1}{\epsilon }, \nonumber \\ T_{12} &=&-\frac{3\ln \alpha +11/2}{\epsilon }+\frac{3\ln ^{2}\alpha }{2}+\frac{9\ln \alpha }{2} \nonumber \\ &-&4\ln 2\ln \alpha +\frac{\pi ^{2}}{6}-4Li_{2}(\frac{1}{2})-12\ln 2+\frac{27}{4}.\end{aligned}$$ List of symbols and abbreviations --------------------------------- $$\begin{aligned} \int [] &=&\int\limits_{0}^{1}dx_{1}\int\limits_{0}^{1}dx_{2}\int\limits_{0}^{1}dx_{3}\;\delta (x_{1}+x_{2}+x_{3}-1), \nonumber \\ x_{ij} &=&x_{i}+x_{j}, \nonumber \\ x_{i}x_{j} &=&x_{1}x_{2}+x_{2}x_{3}+x_{3}x_{1}.\end{aligned}$$ $$\begin{aligned} K_{1}(\alpha ) &=&\int\limits_{\alpha }^{1}dz\int []\frac{x_{2}(x_{1}(x_{2}+x_{3})+x_{3}^{2})(x_{i}x_{j})^{-2-\epsilon }}{(zx_{2}+x_{3})(\alpha x_{1}+zx_{2}+2x_{3})}, \nonumber \\ J_{3}^{^{\prime }}(\alpha ) &=&\alpha \int\limits_{\alpha }^{1}\frac{dz}{z}\int []\frac{x_{2}(x_{1}(x_{2}+x_{3})+x_{3}^{2})(x_{i}x_{j})^{-2-\epsilon }}{(\alpha x_{1}+zx_{2}+2x_{3})^{2}}, \nonumber \\ J_{1}(\alpha ) &=&\int\limits_{\alpha }^{1}dz\int []\frac{x_{1}(x_{1}+x_{3})(x_{2}+x_{3})(x_{i}x_{j})^{-2-\epsilon }}{(zx_{1}+x_{3})(zx_{1}+\alpha x_{2}+2x_{3})}, \nonumber \\ C_{0}^{^{\prime }} &=&\int\limits_{\alpha }^{1}dzdy\int []\frac{x_{1}x_{2}^{2}(x_{i}x_{j})^{-2-\epsilon }}{(x_{3}+yx_{2})(zx_{1}+yx_{2}+2x_{3})}.\end{aligned}$$ Appendix B {#AB} ========== In this appendix we present the calculation of the integral $A_{10}$ as a typical example. We start with the integral $$\begin{aligned} X_{10} &=&-\frac{32a^{3}}{\sigma _{0}d}\int\limits_{pq}p^{2}\sum_{k,m>0}m \nonumber \\ &&D_{p+q}^{c}(m)D^{3}D_{p}^{c}(k)D_{q}(k+m).\end{aligned}$$ Using the Feynman trick, one can write $$\begin{aligned} X_{10} &=&-\frac{16a^{3}}{\sigma _{0}d}\int\limits_{pq}p^{2}\int\limits_{0}^{\infty }dmm\int\limits_{0}^{\infty }dk \nonumber \\ &&\Gamma (6)\int\limits_{\alpha }^{1}dz(z-\alpha )^{2}\int []\left[ {}\right. h_{0}^{2}+q^{2}x_{13}+p^{2}x_{12}+ \nonumber \\ &&2{\bf p}\cdot {\bf q}x_{1}+am(\alpha x_{1}+x_{3})+ak(zx_{2}+x_{3}\left. {}\right] ^{-6}.\end{aligned}$$ Shifting $q\rightarrow q-px_{1}/x_{13}$, we can decouple ${\bf p}$ and ${\bf q}$ in the denominator. We are then able to perform the integration over $k,m,p$ and $q$, resulting in $$X_{10}=\frac{4\Omega _{d}^{2}h_{0}^{4\epsilon }}{\sigma _{0}\epsilon }A_{10},$$ where $$A_{10}=\int\limits_{\alpha }^{1}dz(z-\alpha )^{2}\int []\frac{x_{2}^{3}(x_{1}+x_{3})(x_{i}x_{j})^{-2-\epsilon }}{(zx_{2}+x_{3})(\alpha x_{1}+x_{3})^{2}}.$$ Next we write the integral as a sum of four terms $$\begin{aligned} A_{10} &=&\int\limits_{\alpha }^{1}\frac{dz(z-\alpha )^{2}}{z}\int []\frac{x_{2}(x_{1}+x_{3})(x_{i}x_{j})^{-1-\epsilon }}{(\alpha x_{1}+x_{3})^{2}} \nonumber \\ &\times &\left\{ {}\right. 1-x_{1}x_{3}(x_{i}x_{j})^{-1}-\frac{x_{3}(x_{1}+x_{3})(x_{i}x_{j})^{-1}}{z} \nonumber \\ &+&\frac{x_{3}^{2}(x_{1}+x_{3})(x_{i}x_{j})^{-1}}{z(zx_{2}+x_{3})}\left. {}\right\} =I_{0}-I_{1}-I_{2}+I_{3}.\end{aligned}$$ In what follows we retain the full $\epsilon $ dependence in the $I_{0}$, $I_{1}$ and $I_{2}$ and it suffices to put $\epsilon =0$ in the fourth piece $I_{3}$. Introducing a change of variables $$x_{1}=\frac{u}{s+1}\,;\,x_{2}=\frac{s}{s+1}\,;\,x_{3}=\frac{1-u}{s+1},$$ where $0<s<\infty $ and $0<u<1$, then the four different pieces can be written as follows $$\begin{aligned} I_{0} &=&(\frac{1}{2}-2\alpha )\int\limits_{0}^{1}\frac{du}{(\alpha u+1-u)^{2}}\int\limits_{0}^{\infty }ds\frac{s(s+1)^{2\epsilon }}{(s+u(1-u))^{1+\epsilon }}, \nonumber \\ I_{1} &=&\frac{1}{2}\int\limits_{0}^{1}du\frac{u(1-u)}{(\alpha u+1-u)^{2}}\int\limits_{0}^{\infty }ds\frac{s(s+1)^{2\epsilon }}{(s+u(1-u))^{2+\epsilon }}, \nonumber \\ I_{2} &=&\int\limits_{0}^{1}du\frac{(1-u)}{(\alpha u+1-u)^{2}}\int\limits_{0}^{\infty }ds\frac{s(s+1)^{2\epsilon }}{(s+u(1-u))^{2+\epsilon }}, \nonumber \\ I_{3} &=&\int\limits_{\alpha }^{1}dz\left( \frac{z-\alpha }{z}\right) ^{2}\int\limits_{0}^{1}du\frac{u(1-u)^{2}}{(\alpha u+1-u)^{2}} \nonumber \\ &\times &\int\limits_{0}^{\infty }ds\frac{(s+1-u)}{(s+u(1-u))^{2}(\alpha s+1-u)^{2}}. \label{q}\end{aligned}$$ The integrals over $s$ in Eq. (\[q\]) can now be recognized as integral representations of the hypergeometric function $_{2}F_{1}$. Write $$\begin{aligned} I_{0} &=&(\frac{1}{2}-2\alpha )\int\limits_{0}^{1}du\frac{[u(1-u)]^{1-\epsilon }}{(\alpha u+1-u)^{2}} \nonumber \\ &\times &\left[ {}\right. -\frac{1}{1+\epsilon }G_{0}(u(1-u))+\frac{1}{\epsilon }G_{1}(u(1-u))\left. {}\right] , \nonumber \\ I_{1} &=&\frac{1}{2}\int\limits_{0}^{1}du\frac{[u(1-u)]^{1-\epsilon }}{(\alpha u+1-u)^{2}} \nonumber \\ &\times &\left[ {}\right. -\frac{1}{\epsilon }G_{1}(u(1-u))-\frac{1}{1-\epsilon }G_{2}(u(1-u))\left. {}\right] , \nonumber \\ I_{2} &=&\int\limits_{0}^{1}du\frac{u^{-\epsilon }(1-u)^{1-\epsilon }}{(\alpha u+1-u)^{2}} \nonumber \\ &\times &\left[ {}\right. -\frac{1}{\epsilon }G_{1}(u(1-u))-\frac{1}{1-\epsilon }G_{2}(u(1-u))\left. {}\right] , \nonumber \\ I_{3} &=&\int\limits_{\alpha }^{1}dz\left( \frac{z-\alpha }{z}\right) ^{2}\int\limits_{0}^{1}du\frac{u}{zu+1-u} \nonumber \\ &\times &\left[ {}\right. \frac{1}{2}H_{3}(1-\alpha u)+\frac{(1-u)}{u}\frac{\Gamma (3)}{\Gamma (4)}H_{4}(1-\alpha u)\left. {}\right] ,\end{aligned}$$ then, in the limit where $\epsilon \rightarrow 0$, we can identify the functions $G_{i}$ and $H$ as follows $$\begin{aligned} G_{0}(1-z) &=&{_{2}F_{1}}(1,-2\epsilon ,-\epsilon ;z)=\frac{1+z}{1-z}, \nonumber \\ G_{1}(1-z) &=&{_{2}F_{1}}(1,-2\epsilon ,1-\epsilon ;z)=1+2\epsilon \ln (1-z), \nonumber\end{aligned}$$ $$\begin{aligned} G_{2}(1-z) &=&{_{2}F_{1}}(1,-2\epsilon ,2-\epsilon ;z)=1, \nonumber \\ H_{3}(z) &=&{_{2}F_{1}}(1,2,3;z)=-\frac{2}{z^{2}}\left( \ln (1-z)+z\right) , \nonumber \\ H_{4}(z) &=&{_{2}F_{1}}(1,2,4;z)=\frac{6}{z^{3}}\left( {}\right. (1-z)\ln (1-z) \nonumber \\ &+&z-z^{2}/2\left. {}\right) .\end{aligned}$$ Using these results we obtain $$\begin{aligned} I_{0} &=&-\frac{1}{\alpha }-\frac{\ln \alpha +2}{\epsilon }-\frac{\ln ^{2}\alpha }{2}-2\ln \alpha -\frac{\pi ^{2}}{3}, \nonumber \\ I_{1} &=&\frac{\ln \alpha +2}{\epsilon }+\frac{\ln ^{2}\alpha }{2}+2\ln \alpha +\frac{\pi ^{2}}{3}, \nonumber \\ I_{2} &=&\frac{\ln \alpha +1}{\epsilon }+\frac{\ln ^{2}\alpha }{2}+2\ln \alpha +\frac{\pi ^{2}}{3}+1, \nonumber \\ I_{3} &=&-\ln \alpha .\end{aligned}$$ The final answer is therefore $$\begin{aligned} A_{10} &=&-\frac{1}{\alpha }-\frac{2\ln \alpha +3}{\epsilon }-\ln ^{2}\alpha -5\ln \alpha \nonumber \\ &-&\frac{2\pi ^{2}}{3}+3\end{aligned}$$ For a review, see [The Quantum Hall Effect]{}, edited by R.E. Prange and S.M. Girvin (Springer Verlag, Berlin, 1990). H.P. Wei, D.C. Tsui, and A.M.M. Pruisken, Phys. Rev. B [33]{}, 1488 (1985). A.M.M. Pruisken, Phys. Rev. Lett. [61]{}, 1297 (1988). H.P. Wei, D.C. Tsui, M. 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--- address: \| Institute for Theoretical Physics, University of Vienna\ Boltzmanngasse 5, A-1090 Vienna, Austria\ $^$E-mail: walter.grimus@univie.ac.at author: - 'W. Grimus$^$' title: 'Realizations of $\mu$–$\tau$ interchange symmetry' --- Mass matrices with $\mu$–$\tau$ interchange symmetry ==================================================== At present, all data from neutrino oscillation measurements are compatible with the lepton mixing angles[@tortola] $$\label{a} \theta_{23} = 45^\circ \quad \mbox{and} \quad \theta_{13} = 0^\circ.$$ A neutrino mass matrix with these properties, in the basis in which the charged-lepton mass matrix $M_\ell$ is diagonal, exhibits a $\mu$–$\tau$ (interchange) symmetry—for early papers on this issue see Refs. . Assuming that the lepton flavours $\alpha$ are ordered in the usual way with $\alpha = e,\, \mu,\, \tau$, the $\mu$–$\tau$ symmetry for a matrix $M^{(\mathrm{S})}$ is formulated as[@ustron] $$\label{S} T M^{(\mathrm{S})} T = M^{(\mathrm{S})} \Rightarrow M^{(S)} = \left( \begin{array}{ccc} x & y & y \\ y & z & w \\ y & w & z \end{array} \right),$$ where $T$ is a permutation matrix, performing the flavour exchange $\mu \leftrightarrow \tau$. If $M^{(\mathrm{S})}$ is conceived as a Majorana neutrino mass matrix ${\mathcal{M}_\nu}$, it is easy to see that it predicts the mixing angles of Eq. (\[a\]). There are no further predictions of $M^{(\mathrm{S})}$.[@Z2model] With $\mu$–$\tau$ antisymmetry[@aizawa; @anti], one obtains $$\begin{aligned} && \nonumber T M^{(\mathrm{AS})} T = -M^{(\mathrm{AS})} \Rightarrow \\ && \label{AS} M^{(\mathrm{AS})} = \left( \begin{array}{ccc} 0 & p & -p \\ p & q & 0 \\ -p & 0 & -q \end{array} \right).\end{aligned}$$ Assuming ${\mathcal{M}_\nu}= M^{(\mathrm{AS})}$, this matrix gives[@anti] $$\label{pred-anti} \theta_{12} = \theta_{23} = 45^\circ, \; m_1 = m_2,\; m_3 = 0.$$ We gather from this result that $M^{(\mathrm{AS})}$ is not suitable as a neutrino mass matrix, however, its predictions are not excessively far from reality, if we assume $\theta_{13}$ to be sufficiently small. ${\mathcal{M}_\nu}$ versus ${\mathcal{M}_\nu}^{-1}$ =================================================== 1. \[i\] Denoting the diagonalizing matrix of ${\mathcal{M}_\nu}$ by $U$ and assuming $\det {\mathcal{M}_\nu}\neq 0$, we observe the relationship $$\begin{aligned} && \nonumber U^T {\mathcal{M}_\nu}\, U = \hat m \equiv \mbox{diag}\,( m_1, m_2, m_3 ) \; \Leftrightarrow \\ && U^\dagger \left({\mathcal{M}_\nu}\right)^{-1} U^* = \left( \hat m \right)^{-1}.\end{aligned}$$ 2. \[ii\] Next, we assume the validity of the seesaw mechanism: ${\mathcal{M}_\nu}= -M_D^T M_R^{-1} M_D$ with the neutrino Dirac-mass matrix $M_D$ and the mass matrix $M_R$ of the right-handed neutrino singlet fields $\nu_R$ whose mass Lagrangian is given by $$\label{LM} \mathcal{L}_M(\nu_R) = \frac{1}{2}\, \nu_R^T C^{-1} M_R^* \nu_R + \mbox{H.c.}$$ If $M_\ell$ is diagonal and $M_D$ has the form $M_D = \mbox{diag}\,(a,b,b)$, then $\mu$–$\tau$ symmetry (antisymmetry) of $\left({\mathcal{M}_\nu}\right)^{-1}$ is equivalent to $\mu$–$\tau$ symmetry (antisymmetry) of $M_R$. To impose $\mu$–$\tau$ symmetry on $M_R$ we simply have to require invariance of $\mathcal{L}_M(\nu_R)$ under $\nu_R \to T \nu_R$; for $\mu$–$\tau$ antisymmetry the transformation is $\nu_R \to iT \nu_R$. With the assumptions of Item (\[ii\]) it makes sense to impose conditions on the inverse mass matrix[@Z2model; @lavoura] instead of on the mass matrix itself. Obviously, $\left( {\mathcal{M}_\nu}\right)^{-1}$ can be decomposed as $$\label{decomposition} \left( {\mathcal{M}_\nu}\right)^{-1} = M^{(\mathrm{S})} + M^{(\mathrm{AS})}.$$ This rather trivial observation is the point of departure for the following discussion where we will present several models which realize the $\mu$–$\tau$ interchange symmetry within seesaw extensions of the Standard Model (SM). The framework {#framework} ============= We consider the lepton sector of the SM, enlarge the scalar sector to three Higgs doublets $\phi_j$ ($j = 1,\,2,\,3$) and add three right-handed neutrino singlets $\nu_{\alpha R}$ for the purpose of the seesaw mechanism. The left-handed lepton doublets are denoted by $D_{\alpha L}$ and the right-handed charged-lepton singlets by $\ell_{\alpha R} \equiv \alpha_R$. We impose the following symmetries: 1. \[fs\] The groups $U(1)_{L_\alpha}$ ($\alpha = e,\mu,\tau$) associated with the family lepton numbers $L_\alpha$, or, alternatively, $D_L \to \mbox{diag}\, (1,\omega,\omega^2) D_L$ and, analogously, for $\ell_R$ and $\nu_R$, with $\omega = e^{2\pi i/3}$, corresponding to the symmetry group $\mathbbm{Z}_3$. 2. \[mu-tau\] The symmetry transformation $D_L \to i^k T D_L$, $\ell_R \to i^k T \ell_R$, $\nu_R \to i^k T \nu_R$, $\phi_3 \to -\phi_3$, which either corresponds to the $\mu$–$\tau$ symmetry for $k=0$ or to the $\mu$–$\tau$ antisymmetry for $k=1$. 3. \[aux\] An auxiliary symmetry ${\mathbbm{Z}_2^{(\mathrm{aux})}}$ defined by the sign change of the fields $\nu_{\alpha R}$ ($\alpha = e,\, \mu,\, \tau$), $\phi_1$, $e_R$. It is easy to check that the most general Yukawa Lagrangian compatible with these symmetries is given by $$\begin{aligned} \lefteqn{\mathcal{L}_Y(\phi) =} \nonumber \\ && - y_1 \bar D_{eL} \nu_{eR} \tilde\phi_1 - y_2 \left( \bar D_{\mu L} \nu_{\mu R} + \bar D_{\tau L} \nu_{\tau R} \right) \tilde\phi_1 \nonumber \\ && - y_3 \bar D_{eL} e_R \phi_1 - y_4 \left( \bar D_{\mu L} \mu_R + \bar D_{\tau L} \tau_R \right) \phi_2 \nonumber \\ && \label{LY} - y_5 \left( \bar D_{\mu L} \mu_R - \bar D_{\tau L} \tau_R \right) \phi_3 + \mbox{H.c.}\end{aligned}$$ Note that the symmetries of Item (\[fs\]) enforce diagonal Yukawa couplings, Item (\[mu-tau\]) provides the $\mu$–$\tau$-symmetric strucure of the couplings, and Item (\[aux\]) makes sure that $\phi_3$ does not couple to $\nu_R$; the latter point is important for supplying the $\mu$–$\tau$-symmetric form $M_D = \mbox{diag}\,(a,b,b)$ of the neutrino Dirac-mass matrix. The $\mu$–$\tau$ (anti)symmetry is spontaneously broken by the VEV of $\phi_3$, which allows for $m_\mu \neq m_\tau$. A model based on $S_3 \times {\mathbbm{Z}_2^{(\mathrm{aux})}}$ ============================================================== The model presented in this section[@GL05] is based on the group $S_3$. We have the following representations: $D_L,\, \ell_R,\, \nu_R \in \underline{1} \oplus \underline{2}$, $\phi_{1,2} \in \underline{1}$, $\phi_3 \in\underline{1}'$. We add a complex scalar $\chi$ such that $(\chi, \chi^) \in \underline{2}$. The connexion of $S_3$ with the symmetries of the previous section is obtained via $$\underline{2}: \left\{ \begin{array}{l} (\mathit{12}) \to \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right), \\ (\mathit{123}) \to \left( \begin{array}{cc} \omega & 0 \\ 0 & \omega^2 \end{array} \right), \end{array} \right.$$ where $(\mathit{12}),\, (\mathit{123}) \in S_3$. The cyclic permutation represents the $\mathbbm{Z}_3$ symmetry of Item (\[fs\]), whereas the transposition $(\mathit{12})$ is mapped into the $\mu$–$\tau$ symmetry of Item (\[mu-tau\]). The trivial one-dimensional representation is denoted by $\underline{1}$ and $\underline{1}'$ is given by $(\mathit{12}) \to -1$, $(\mathit{123}) \to 1$. Apart from the Lagrangian (\[LY\]), the symmetries allow Yukawa couplings of the singlet scalar, described by the Lagrangian $$\begin{aligned} \lefteqn{\mathcal{L}_Y(\chi) = } \nonumber \\ && y_\chi^\ast\, \nu_{eR}^T C^{-1} \left( \nu_{\mu R} \chi^\ast + \nu_{\tau R} \chi \right) + \\ && \frac{1}{2}\,z_\chi^\ast \left( \nu_{\mu R}^T C^{-1} \nu_{\mu R} \chi + \nu_{\tau R}^T C^{-1} \nu_{\tau R} \chi^\ast \right) + {\rm H.c.}, \nonumber\end{aligned}$$ and a $\nu_R$ mass term $$\begin{aligned} \lefteqn{\mathcal{L}_\mathrm{mass} =} \\ && \frac{1}{2}\, m^\ast \nu_{eR}^T C^{-1} \nu_{eR} + {m^\prime}^\ast \nu_{\mu R}^T C^{-1} \nu_{\tau R} + {\rm H.c.} \nonumber\end{aligned}$$ We assume the VEV of $\chi$ and $m$, $m^\prime$ to be of the order of the seesaw scale. This model yields the inverse neutrino mass matrix[@GL05] $$\label{mnus3} \left( {\mathcal{M}_\nu}\right)^{-1} = \left( \begin{array}{ccc} x & y & y \\ y & \hphantom{i} u\, e^{i\psi} & w \\ y & w & u\, e^{-i\psi} \end{array} \right).$$ With the decomposition (\[decomposition\]) and Eqs. (\[S\]) and (\[AS\]), we find $z = u \cos \psi$, $q = i\,u \sin \psi$, $p = 0$. For $\psi = 0$ or $\pi$, Eq. (\[mnus3\]) yields a $\mu$–$\tau$-symmetric neutrino mass matrix. If $\psi \neq 0$, $\mu$–$\tau$ symmetry is partially broken, such that the matrix of absolute values in $\left( {\mathcal{M}_\nu}\right)^{-1}$ is still $\mu$–$\tau$-symmetric. Which case is realized, depends on the type of symmetry breaking of $S_3$: If it is broken spontaneously, then $\psi = 0$ or $\pi$; if, in addition, its $\mathbbm{Z}_3$ subgroup is broken softly via terms of dimension one and two in the scalar potential, $\sin \psi$ is non-zero. In the latter case, there are correlated deviations from Eq. (\[a\]), approximately given by[@GL05] $$\begin{aligned} \lefteqn{\cos 2\theta_{23} \simeq -2\, c_{12} s_{12}} \\ && \times \frac{\Delta m^2_\odot}{c_{12}^2 m_1^2 + s_{12}^2 m_2^2 - m_1^2 m_2^2/m_3^2} \, s_{13} \cos \delta, \nonumber \end{aligned}$$ where ${\Delta m^2_\odot}= m_2^2 - m_1^2$ is the solar mass-squared difference and $\delta$ is the CKM-type phase in $U$. For an inverted neutrino mass spectrum, $\theta_{23}$ is still maximal for all practical purposes. For a normal spectrum, possible deviations of $\theta_{23}$ from $45^\circ$ are most pronounced in the hierarchical case, namely $\cos 2\theta_{23} \sim -3\,s_{13} \cos \delta$. A class of models based on $\mu$–$\tau$ antisymmetry {#class} ==================================================== Here we discuss a class of models based on conserved lepton numbers and $\mu$–$\tau$ antisymmetry—see Sec. \[framework\], Items (\[fs\]) and (\[mu-tau\]). Since a $\mu$–$\tau$-antisymmetric $M_R$ is singular, we add complex scalar gauge singlets which carry lepton numbers. Such scalars have the general Yukawa couplings $$\label{Lchi} \mathcal{L}_Y(\chi) = \frac{1}{2} \sum_{\alpha,\beta} z_{\alpha\beta}^\, \nu_{\alpha R}^T C^{-1} \nu_{\beta R}\, \chi_{\alpha\beta} + \mbox{H.c.}$$ In Table \[scalars\] we have listed the four basic cases of scalar singlets compatible with the family symmetries. Their VEVs make $M_R$ non-singular and induce a $\mu$–$\tau$-symmetric contribution in $\left( {\mathcal{M}_\nu}\right)^{-1}$—cf. Eq. (\[decomposition\])—as shown in the last column of Table \[scalars\]. Combining $M^{(\mathrm{AS})}$ with one or two of the cases in Table \[scalars\] for the construction of $M_R$ leads to ten models—see Table \[models\]. Of these models, only five are compatible with the data, as indicated in this table. Each of the five viable models has six physical parameters in ${\mathcal{M}_\nu}$. Models (1)–(4) (four parameters in ${\mathcal{M}_\nu}$) and model (10) (five parameters in ${\mathcal{M}_\nu}$) are ruled out; properties of these models which lead to contradiction with the data are found in the last column of Table \[models\]. For the viable models, the preferred or predicted neutrino mass spectrum is indicated in that column. Let us make some comments on the models of this section. Whenever $\left( {\mathcal{M}_\nu}^{-1} \right)_{ee} = \left( {\mathcal{M}_\nu}^{-1} \right)_{\mu\tau} = 0$, then ${\Delta m^2_\odot}/{\Delta m^2_\mathrm{atm}}> 1$, where ${\Delta m^2_\mathrm{atm}}= \left\| m_3^2 - m_1^2 \right\|$ is the atmospheric mass-squared difference. This is the case for models (3), (4), (10) and the reason why they are ruled out. If $\left( {\mathcal{M}_\nu}^{-1} \right)_{ee} = 0$, then only the inverted neutrino mass spectrum is possible. Among the allowed models, this applies to (8) and (9). Finally we want to mention that model (8) is the most predictive one; e.g., a slight deviation of $\sin^2 2\theta_{23}$ from one leads to a large $s_{13}^2$, which in practice gives the lower bound $\sin^2 2\theta_{23} > 0.99$. For further details see Ref. . Conclusions =========== In this report we have combined a $\mu$–$\tau$ interchange symmetry with family symmetries which give diagonal Yukawa couplings in order to obtain a predictive neutrino mass matrix. We have considered extensions of the SM which have three Higgs doublets and right-handed neutrino singlets for the seesaw mechanism. Other important ingredients are scalar gauge singlets which induce, upon acquiring VEVs, contributions to $M_R$. With diagonal Yukawa couplings, lepton mixing stems solely from a non-diagonal $M_R$ and conditions on $M_R$ are translated into conditions on $\left( {\mathcal{M}_\nu}\right)^{-1}$. While exact $\mu$–$\tau$ symmetry in ${\mathcal{M}_\nu}$ or $\left( {\mathcal{M}_\nu}\right)^{-1}$ leads to Eq. (\[a\]), deviations from exact $\mu$–$\tau$ symmetry can lead to interesting correlations between atmospheric mixing and $\theta_{13}$. Though exact $\mu$–$\tau$ antisymmetry in ${\mathcal{M}_\nu}$ or $\left( {\mathcal{M}_\nu}\right)^{-1}$ is not viable, it is nevertheless a useful concept, in combination with the above-mentioned scalar singlets, for producing predictive models. Acknowledgments {#acknowledgments .unnumbered} =============== The author is grateful to L. Lavoura and B. Strohmaier for a critical reading of the manuscript. [9]{} See, e.g., M. Maltoni, T. Schwetz, M. Tórtola and J.W.F. Valle, ; G.L. Fogli, E. Lisi, A. Marrone and A. Palazzo, . T. Fukuyama and H. Nishiura, hep-ph/9702253, in Proc. of International Workshop on Masses and Mixings of Quarks and Leptons, ed. Y. Koide (World Scientific, Singapore, 1998); R.N. Mohapatra and S. Nussinov, ; E. Ma and M. Raidal, \[\]; C.S. Lam, ; K.R.S. Balaji, W. Grimus and T. Schwetz, ; E. Ma, ; P.F. Harrison and W.G. Scott, . W. Grimus and L. Lavoura, ; W. Grimus and L. Lavoura, . W. Grimus and L. Lavoura, . T. Kitabayashi and M. Yasuè, ; I. Aizawa, T. Kitabayashi and M. Yasuè, ; T. Kitabayashi and M. Yasuè, . W. Grimus, S. Kaneko, L. Lavoura, H. Sawanaka and M. Tanimoto, . L. Lavoura, . W. Grimus and L. Lavoura, .
--- abstract: 'In this paper we consider the Moutard transformation [@mou] which is a two-dimensional version of the well-known Darboux transformation. We give an algebraic interpretation of the Moutard transformation as a conjugation in an appropriate ring and the corresponding version of the algebro-geometric formalism for two-dimensional Schrödinger operators. An application to some problems of the spectral theory of two-dimensional Schrödinger operators and to the $(2+1)$-dimensional Novikov–Veselov equation is sketched.' address: - 'Institute of Mathematics, 630090 Novosibirsk, Russia' - 'Siberian Federal University, Svobodnyi avenue, 79, 660041, Krasnoyarsk, Russia' author: - 'Iskander A. Taimanov' - 'Sergey P. Tsarev' title: 'The Moutard transformation: an algebraic formalism via pseudodifferential operators and applications' --- [^1] Introduction {#sec-dm .unnumbered} ============ The Moutard transformation plays a fundamental role in projective–differential geometry of surfaces and was extensively investigated by Bianchi, Darboux, Demoulin, Guichard and others. In recent publications [@TT07; @TT08; @TT-DAN08] we gave an application of the Moutard transformation to the explicit construction of two-dimensional Schrödinger operators $$H = -\Delta + u= -(\partial_x^2 +\partial_y^2) + u(x,y)$$ with fast decaying smooth rational potentials such that their $L_2$-kernels contain at least two-dimensional subspaces spanned by rational eigenfunctions as well as a $(2+1)$-dimensional extension of the Moutard transformation which was able to produce explicit rational blowing-up solutions to the Novikov–Veselov (NV) equation with fast decaying smooth rational Cauchy data. Purely algebraic constructions of the aforementioned papers imply that one should look for an algebraic interpretation of the Moutard transformation as a simple transformation in the ring of partial differential operators. The one-dimensional case of Darboux transformation was studied in [@EK]. Here we give an analogue of their results; as it turned out, for the Moutard transformation one needs to recourse to a more complicated ring of rational pseudodifferential operators (Ore localization of the ring of partial differential operators). As a by-product, we obtain a more natural version of the Dubrovin-Krichever-Novikov formalism [@DKN; @K76] of algebro-geometric two-dimensional Schrödinger operators. In the last Section we briefly sketch our previous results [@TT07; @TT08; @TT-DAN08]. First we give an account of the algebraic theory of the Darboux transformations. The Darboux transformation -------------------------- Let $$H = -\frac{d^2}{dx^2} + u(x)$$ be a one-dimensional Schrödinger operator and let $\omega$ satisfy the equation $$H \omega = 0.$$ The function $\omega$ determines a factorization of $H$: $$\label{onefactor} H = A^\top A, \ \ \ A = -\frac{d}{dx} + v, \ \ \ A^\top = \frac{d}{dx}+v, \ \ \ v = \frac{\omega_x}{\omega}.$$ Indeed we have $$A^\top A = \left(\frac{d}{dx} + v\right)\left(-\frac{d}{dx} + v\right) = -\frac{d^2}{dx^2} + v^2 + v_x$$ and the equation $$v_x + v^2 = u$$ is equivalent to $H\omega = 0$. If $v$ is real-valued we have $A^\ast = A^\top. $ [The Darboux transformation]{}[^2] is the swapping of $A^\top$ and $A$: $$H = A^\top A \to \widetilde{H} = AA^\top,$$ or in terms of $u$: $$u = v^2 + v_x \to \widetilde{u} = v^2 - v_x.$$ It is easy to check by simple computations that \[prop1\] If $\varphi$ satisfies the equation $H \varphi = E \varphi$ with $E = \mathrm{const}$ then $\widetilde{\varphi} = A \varphi$ satisfies the equation $\widetilde{H} \widetilde{\varphi} = E \widetilde{\varphi}.$ [Remark.]{} In general the Darboux transformation is defined for any solution to the equation $H \omega = c\omega$ with $c=\mathrm{const}$ (see, for instance, [@EK]). In this case it reduces to the transformation of $H^\prime = H-c$ for which $H^\prime\omega = 0$. The Moutard transformation -------------------------- This transformation was invented by Th. Moutard for the hyperbolic equation of the form $\psi_{xy}-u(x,y)\psi=0$ in the context of local differential geometry of surfaces and was extensively studied and used in many areas of the local surface theory [@Dar]. Here we give an elliptic version [@TT07; @TT08; @TT-DAN08] of this transformation suitable for our purposes. Let $H$ be a two-dimensional potential Schrödinger operator and let $\omega$ be a solution to the equation $$H \omega = (- \Delta + u )\omega = 0.$$ Then [the Moutard transformation]{} of $H$ is defined as $$\widetilde{H} = -\Delta + u -2\Delta \log \omega = -\Delta - u + 2\frac{\omega_x^2+\omega_y^2}{\omega^2}.$$ \[prop3\] If $\varphi$ satisfies the equation $H \varphi = 0$, then the function $\theta$ defined from the system $$\label{eigen-moutard} (\omega \theta)_x = -\omega^2 \left(\frac{\varphi}{\omega}\right)_y, \ \ \ (\omega \theta)_y = \omega^2 \left(\frac{\varphi}{\omega}\right)_x$$ satisfies $\widetilde{H}\theta = 0$. By this definition, if $\theta$ satisfies (\[eigen-moutard\]) then $$\label{theta-family} \theta + \frac{t}{\omega}, \ \ \ t = \mathrm{const},$$ satisfies (\[eigen-moutard\]) for any constant $t$. We shall use the following notation for the Moutard transformation: $$M_\omega(u) = \widetilde{u} = u - 2\Delta \log \omega, \ \ \ M_\omega(\varphi) = \{\theta+\frac{t}{\omega}, \ t \in \C\}.$$ For one-dimensional potentials the Moutard transformation reduces to the Darboux transformation. Indeed, let $u=u(x)$ depend on $x$ only and $\omega=f(x)e^{\sqrt{c}y}$. Then $f$ satisfies the one-dimensional Schrödinger equation $$H_0 f = \left( - \frac{d^2}{dx^2} + u\right) f = cf$$ and the Moutard transformation reduces to the Darboux transformation of $H_0$ defined by $f$: $$H = H_0 - \frac{\partial^2}{\partial y^2} \ \ \longrightarrow \ \ \widetilde{H} = \widetilde{H_0} -\frac{\partial^2}{\partial y^2}.$$ If $g=g(x)$ satisfies $H_0 g = E g$, then $H \varphi = 0$ with $\varphi = e^{\sqrt{E}y}g(x)$. We derive from (\[eigen-moutard\]) that $\theta = e^{\sqrt{E}y}h(x)$ satisfies $\widetilde{H}\theta = 0$ if $h = \frac{1}{\sqrt{c}+\sqrt{E}} \left(\frac{d}{dx} - \frac{f_x}{f}\right)g$, i.e. $h$ is a multiple of the Darboux transform of $g$: $h = - \frac{1}{\sqrt{c}+\sqrt{E}}Ag$ where $H_0 - c = A^\top A$ is the factorization of $H_0-c$ defined by $f$. The inverse Darboux transformation is given by $g = \frac{1}{\sqrt{c}-\sqrt{E}}\left( \frac{d}{dx} + \frac{f_x}{f}\right)h$. [Remark.]{} We can rewrite (\[eigen-moutard\]) as $$\label{eigen-moutard2} \left(\bar{\partial} + \frac{\omega_{\bar{z}}}{\omega}\right) \theta = i \left(\bar{\partial} - \frac{\omega_{\bar{z}}}{\omega}\right) \varphi, \ \ \ \left(\partial + \frac{\omega_z}{\omega}\right) \theta = -i \left(\partial - \frac{\omega_z}{\omega}\right) \varphi$$ which implies $$\label{dress-moutard} \omega^{-1}\cdot \bar{\partial}\cdot \omega (\theta) = i \omega \cdot \bar{\partial}\cdot \omega^{-1}(\varphi), \ \ \omega^{-1}\cdot \partial\cdot \omega (\theta) = - i \omega \cdot \partial \cdot \omega^{-1}(\varphi),$$ where, as usual, $z=x+i\,y$, $\partial=\partial_z=\frac{1}{2}(\partial_x- i \partial_y)$, $\bar\partial=\partial_{\bar z}=\frac{1}{2}(\partial_x + i \partial_y)$. This representation will be used later. There is another two-dimensional generalization of the Darboux transformation called the Laplace transformation. Its relation to integrable systems was recently studied in [@NV]. The spectral curves of algebraic Schrödinger operators ====================================================== Since weakly algebraic two-dimensional Schrödinger operators are defined in terms of their spectral curves on the zero energy level we recall the definitions of the spectral curves of one- and two-dimensional Schrödinger operators. One-dimensional Schrödinger operators ------------------------------------- Let a one-dimensional Schrödinger operator $$H = -\frac{d^2}{d x^2} +u(x)$$ commute with an ordinary differential operator $L$ of order $2n+1$. Then the Burchnall–Chaundy theorem [@BC] guarantees that these two operators satisfy a polynomial equation $$Q(H,L) = 0, \ \ \ Q(\lambda,E) = \lambda^2 - P_{2n+1}(E) = 0,$$ where $P_{2n+1}(E)$ is a polynomial of degree $2n+1$. This equation defines [the spectral curve]{} $\Gamma$ of the operator $H$. Its points parameterize the joint “eigenfunctions”, i.e. solutions of the equations $$H\psi = E\psi, \ \ \ L\psi = \lambda \psi.$$ Moreover these functions are glued (after some normalization) into a function $\psi(P,x)$ which is meromorphic in $P \in \Gamma$ and after completing $\Gamma$ by adding an infinity point $E=\infty$ the eigenfunction $\psi$ gets a singularity at $\infty$: $$\psi(P,x) \approx e^{ikx}$$ where $k^{-1} = \frac{1}{\sqrt{E}}$ is a local parameter on $\Gamma$ near the infinity point $\infty$ [@N; @DMN]. Therewith it is said that the operator $H$ is [algebraic]{} (or [algebro-geometric]{}). [Example.]{} [The spectral curve of a constant potential.]{} Let $u(x)= c$ be a constant potential, then $\Gamma$ is given by $\lambda^2 = E-c$, $$\psi_{\pm} = e^{\pm i\lambda x},$$ and the covering $(\lambda,E) \to E$ ramifies at $E=c$ where $\psi_\pm(x,c)=1$. Two-dimensional Schrödinger operators ------------------------------------- Given a partial differential operator $H$, one says that operators $L_1, \dots, L_n$ generate a commutative $(\mod H)$ algebra ${\mathcal A}$ if they satisfy the following commutation relations: $$[L_i,L_j]=D_{ij}H, \ \ \ [L_i,H] = D_i H$$ where $D_{ij}, D_i, 1 \leq i,j \leq n$, are partial differential operators. This definition was introduced in [@DKN] in which they considered in detail the particular case when $H$ is the two-dimensional Schrödinger operator (probably with an electromagnetic field) in two variables and introduced the following definition: - a two-dimensional operator $$H = \partial\bar{\partial}+ v \bar{\partial} + u$$ is called [(weakly) algebraic]{} if it is included in a nontrivial commutative $(\mod H)$ algebra ${\mathcal A}$ generated by operators $L_1$ and $L_2$ in two variables, the operators $L_1$ and $L_2$ satisfy the polynomial relation $$Q(L_1,L_2) = 0 (\mod H)$$ and to a generic point of the algebraic curve $Q(\lambda_1,\lambda_2)=0$ there corresponds a $k$-dimensional space of functions $\psi$ which satisfy the equations $$H\psi =0, \ \ \ L_i \psi = \lambda_i \psi, \ i=1,2.$$ The dimension $k$ is called the rank of ${\mathcal A}$. The conjugation representation ============================== The Darboux transformation -------------------------- The Darboux transformation admits the well-known representation as a conjugation in the ring of pseudodifferential operators which implies interesting corollaries concerning algebraic operators. Let $H \omega = 0$ and $A = -\frac{d}{dx} = \frac{\omega_x}{\omega}$. Then in the ring of pseudodifferential operators in $x$ we have $$A = - \omega \cdot \frac{d}{dx} \cdot \omega^{-1}, \ \ A^\top = \omega^{-1} \cdot \frac{d}{dx} \cdot \omega, \ \ H = A^\top A,$$ where the functions $\omega$ and $\omega^{-1}$ are identified with the operator of multiplication by them. From that we conclude $$\label{darboux1} \widetilde{H} = A \cdot H \cdot A^{-1}.$$ Fixing $\omega$ let us denote by $$\widetilde{M} = A \cdot M \cdot A^{-1}$$ the conjugation of a pseudodifferential operator $M$. It is clear that if $L$ commutes with $H$, then $\widetilde{L}$ commutes with $\widetilde{H}$: $$[H,L]= 0 \ \ \ \Rightarrow \ \ \ [\widetilde{H},\widetilde{L}]=0.$$ However, given an arbitrary differential operator $M$, we have $$M \cdot \omega = M\omega + M^\prime \cdot \frac{d}{dx}$$ where $M\omega$ is the function obtained by applying $M$ to $\omega$ and $M^\prime$ is a differential operator. Hence $$\widetilde{M} = \omega \cdot \frac{d}{dx} \cdot \omega^{-1} \cdot M \cdot \omega \cdot \left(\frac{d}{dx}\right)^{-1} \cdot \omega^{-1} =$$ $$\omega \cdot \frac{d}{dx} \cdot \omega^{-1} \cdot M\omega \cdot \left(\frac{d}{dx}\right)^{-1} \cdot \omega^{-1} + \omega \cdot \frac{d}{dx} \cdot \omega^{-1} \cdot M^\prime \cdot \omega^{-1}$$ and we conclude that [$\widetilde{M}$ is a differential operator if and only if $\omega^{-1} \cdot M\omega = \lambda = \mathrm{const}$, which means $$M\omega = \lambda \omega.$$ ]{} Let $H\omega=0$ and let $L$ be a differential operator of odd order which commutes with $H$. We assume that its order is minimal with respect to this property. Let, by the Burchnall–Chaundy theorem, $$Q(H,L) = 0, \ \ \ Q(E,\lambda) = \lambda^2 - P(E).$$ Then we have (see [@EK]): 1. if $L \omega =\lambda\omega$ and $P(E)$ has no a multiple root at $E=0$, then $\widetilde{H}$ and $\widetilde{L}$ generate a commutative ring of differential operators and $$Q(\widetilde{H},\widetilde{L}) = 0;$$ 2. if $\omega^{-1} \cdot L\omega \neq \mathrm{const}$, then $\widetilde{H}$ and $L_0 = \widetilde{LH}$ generate a commutative ring of differential operators and $$Q_+(\widetilde{H},L_0) = 0, \ \ \ Q_+(E,\lambda) = \lambda^2 - E^2P(E);$$ 3. if $L \omega =\lambda\omega$ and $P(E)$ has a multiple root at $E=0$, then the action of the Darboux transformation is inverse to one described in the previous statement: $\widetilde{H}$ and some operator $L_0$ generate commutative ring of differential operators with $$Q_-(\widetilde{H},L_0) = 0, \ \ \ Q_-(E,\lambda) = \lambda^2 - E^{-2}P(E).$$ In particular, this implies that - The Darboux transformation preserves the class of algebro-geometric one-dimensional Schrödinger operators. Moreover it always preserves the normalization of the spectral curve. In [@EK] it is noted that some facts mentioned above were discovered by Burchnall, Chaundy and Drach. The first two cases are demonstrated by the following [Example.]{} [The Darboux transformation of the constant potential.]{} Let $u =c = \mathrm{const}$ and $L = \frac{d}{dx}$. 1. Let $\omega = e^{\sqrt{c-c_0}x}$. Then $v = \frac{\omega_x}{\omega} = \sqrt{c-c_0}$, $v_x=0$, and the Darboux transformation is even trivial. 2. Let $\omega = \frac{1}{2}(e^{\sqrt{c-c_0}x}+e^{-\sqrt{c-c_0}x}) = \cos (\sqrt{c-c_0}x)$. Then $$v = - \sqrt{c-c_0} \tan (\sqrt{c-c_0}x), \ \ \ v_x = \frac{c_0-c}{\cos^2 (\sqrt{c-c_0}x)},$$ $$u=c \to \widetilde{u} = c + \frac{2(c-c_0)}{\cos^2 (\sqrt{c-c_0}x)}.$$ If $c-c_0 <0$, then $\widetilde{u}$ is not periodic. An example of the third case is easily derived from the following explicitly computable examples which are interesting in themselves. [Example.]{} [Rational solitons via the Darboux transformation.]{} Let $u =0$ and $\omega = x$. Then $$v = \frac{1}{x}, \ \ \ v_x = - \frac{1}{x^2}, \ \ \ \widetilde{u} = \frac{2}{x^2}.$$ The spectral curve $\Gamma = \{\lambda^2 = E\}$ is transformed to $\widetilde{\Gamma} = \{\lambda^2 = E^3\}$ and $$\psi = \left(1 - \frac{1}{i\sqrt{E}x}\right)e^{i\sqrt{E}x}$$ (we normalize it by condition $\psi \approx e^{i\sqrt{E}x}$ as $E \to \infty$). The iterations of the Darboux transformation initially applied to the trivial potential $u=0$ give all rational solitons discovered in [@AMM]. The spectral curve of the potential $$u_n = \frac{n(n+1)}{2x^2},$$ obtained after $n$ iterations, is given by the equation $\lambda^2 = E^{2n+1}$. The spectral curves of these are singular: topologically they are spheres but at $E=0$ they have the following singularities: [any rational function $f$ on $\Gamma_n$, the spectral curve of $u_n$, which is holomorphic near $E=0$ satisfies the condition]{} $$\label{singular-1} f^\prime = f^{\prime\prime\prime} = \dots = f^{(2n-1)}=0.$$ This is easily explained by the normalization mapping $\C \to \Gamma= \{\lambda^2=E^{2n+1}\}$ which has the form $$t \to (\lambda = t^{2n+1}, E = t^2).$$ Indeed any rational function $f$ which is holomorphic at $E=0$ is a ratio of polynomials $\frac{P(\lambda,E)}{Q(\lambda,E)}$ such that $Q(0,0) \neq 0$ and any such function written in terms of $t$ satisfies the conditions (\[singular-1\]). The Ore localization {#sec-ore} -------------------- In order to give an analogous interpretation of the Moutard transformation as a conjugation we first recall some basic facts from the Ore theory of localization of noncommutative rings and introduce the rings $F(\partial_y)$ and $F(\partial_y)[\partial_x]$ which we shall use. We recall that a ring $R$ is an algebra with two operations, the addition and the multiplication, such that $R$ is a commutative group with respect to the addition and the distribute laws $$a(b+c) = ab+ ac, \ \ \ (a+b)c = ac + bc \ \ \ \mbox{for all $a,b,c \in R$}$$ hold. It is said that a ring $R$ is regular if it satisfies [the Ore conditions]{}: - for $a \neq 0, b \neq 0$ there exist $r \neq 0, s \neq 0$ such that $ar = bs$; - if $ab=ac$ or $ba=ca$ for $a \neq 0$ then $b=c$ (so it has no zero divisors). Ore showed in [@Ore] that [any regular ring can be embedded as a subring into a non-commutative field (skew field of fractions)]{} as follows. Let us consider the set $S$ of all formal fractions $$ab^{-1} = \left(\frac{a}{b}\right), \ \ \ b \neq 0.$$ Note that due to non-commutativity of $R$ one shall always keep in mind that in our construction the denominators $b^{-1}$ are always on the right of the numerators $a$! It is easy to propose an analogous construction with denominators standing on the left, but the resulting skew field will be isomorphic to the field we are to construct. We say that two such fractions are equal: $$\left(\frac{a}{b}\right) = \left(\frac{c}{d}\right)$$ if and only if $$ar = cs$$ where $r$ and $s$ satisfy the equality (see the first Ore condition) $$\label{ore-pair} br = ds.$$ It is easy to show that the equality is independent on the choice of $r$ and $s$ satisfying the last equality. Then the addition is defined by $$\left(\frac{a}{b}\right) + \left(\frac{c}{d}\right) = \left( \frac{ar+cs}{br}\right) = \left(\frac{ar+cs}{ds}\right)$$ where $r$ and $s$ satisfy (\[ore-pair\]). The multiplication is given by $$\left(\frac{a}{b}\right)\left(\frac{c}{d}\right) = \left(\frac{at}{du}\right)$$ where $$bt = au.$$ The unit element is defined as $$\left(\frac{a}{a}\right) = 1, \ \ \ a \neq 0,$$ this definition is independent on the choice of $a$ and with such a law of multiplication we have $\left(\frac{a}{b}\right)\left(\frac{b}{a}\right) = 1$. We say that $S$ (with these operations) is [the Ore localization]{} of $R$. One can routinely check all the usual properties of the addition and multiplication operations defined on $S$, which becomes a skew field. This means that we can use expressions like $A^{-1}$ for arbitrary elements of this skew field. All operations in $S$ given above are constructive, unlike the case of the ring of formal pseudodifferential operators defined as infinite series customary in solitonics. For a commutative ring $R$ without divisors of zero the Ore localization coincides with the standard localization. Let us introduce the main example with non-commutative ring $R$ which we shall use. [Example.]{} Let $F = {\bf k}(x_1,\dots,x_n)$ be a field formed by all ${\bf k}$-valued functions of the form $$\frac{P(x_1,\dots,x_n)}{Q(x_1,\dots,x_n)}$$ where $P$ and $Q$ are analytical functions of $x_1,\dots,x_n$ or even formal power series in these variables. We consider two cases which are 1\) ${\bf k} = \R$ and $x_1,\dots,x_n \in \R$; 2\) ${\bf k} = \C$, $n=2m$ and $(x_1,\dots,x_n) = (z_1,\bar{z}_1,\dots,z_m,\bar{z}_m)$, and, for simplicity and our needs, we even assume that $n=2$, $x_1 = x$, and $x_2 = y$. Let us consider $F$ as an operator algebra $R$ which acts on itself by multiplications $$f(g) = fg, \ \ \ f , g \in F,$$ take the partial derivative operators $\partial_x = \frac{\partial}{\partial x}, \partial_y = \frac{\partial}{\partial y}$ and consider the operator algebra $F[\partial_y]$ which is generated by elements of $R$ and by the operator $\partial_y$ that acts on $F$. The ring $F[\partial_y]$ is noncommutative because $$\label{comm} [\partial_y,f] = \partial_y \cdot f - f \cdot \partial_y = \frac{\partial f}{\partial y} = f_y \ \ \mbox{for $f \in F$}.$$ It is straightforward to check $F[\partial_y]$ and $F[\partial_x,\partial_y]$ satisfy the Ore conditions. Let us take the Ore localization $F(\partial_y)$ of $F[\partial_y]$. We have \[commutator\] For any $f \in F$ we have $$[\partial_y^{-1},f] = - \partial_y^{-1}\cdot \frac{\partial f}{\partial y}\cdot \partial^{-1}_y.$$ [Proof.]{} Let us take the equality (\[comm\]), multiply every its term by $\partial_y^{-1}$ from both sides and obtain $[f,\partial_y] = \partial_y^{-1} \cdot f_y \cdot \partial_y^{-1}$. Proposition is proved. Let us consider $F(\partial_y)[\partial_x]$, the ring of formal differential operators in $x$ with coefficients from $F(\partial_y)$. This ring is embedded into its Ore localization $F(\partial_x,\partial_y)$. We need to define the commutation of $\partial_x^{\pm 1}$ with elements of $F(\partial_y)[\partial_x]$. Since $$[\partial_x^{-1},f] = - \partial_x^{-1}[\partial_x,f]\partial_x^{-1},$$ it is enough to define the commutators of $f \in F(\partial_y)[\partial_x]$ with $\partial_x$. Here we do that even for the Ore localization ring $F(\partial_x,\partial_y)$. If $L \in F[\partial_x,\partial_y]$ is a differential operator in $x$ and $y$ with coefficients from $F$, then its derivative in $x$ (i.e. the operator with coefficients being the derivatives of the respective coefficients of $L$) may be defined by the commutation formula $$L_x = [\partial_x, L].$$ Let us take this formula as a definition of the derivative of any element of $P \in F(\partial_x,\partial_y)$ w.r.t. $x$. Using the Leibnitz identity $$(LM)_x = L_xM + LM_x, \ \ \ L,M \in F(\partial_x,\partial_y),$$ we derive the following proposition by straightforward computations. Given $P = M \cdot L^{-1}$ with $L, M \in F[\partial_x,\partial_y]$, we have $$P_x \stackrel{\mathrm{def}}{=} [\partial_x,P] = M(L^{-1})_x + M_x L^{-1} = -M\,L^{-1}L_xL^{-1} + M_x L^{-1}.$$ \[cor1\] If $P \in F(\partial_y)$, then $P_x = [\partial_x,P] \in F(\partial_y)$, i.e. $P_x$ contains no derivations in $x$. The Moutard transformation -------------------------- Let us consider the general Moutard transformation which is applied to an operator: $$H=\partial_r\partial_s - u(r,s)$$ where $$r=x, \ \ s = y, \ \ x,y \in \R \ \ \ \mbox{(the hyperbolic case)},$$ or $$r = z, \ \ s = \bar{z}, \ \ z \in \C \ \ \ \mbox{(the elliptic case).}$$ Via the formulas $$\label{M0} (\omega \theta)_r = - \omega^2 \left(\frac{\varphi}{\omega}\right)_r, \ \ \ (\omega\theta)_s = \omega^2\left(\frac{\varphi}{\omega}\right)_s,$$ it relates solutions $\varphi$ and $\theta$ to the equations $$H\varphi =0, \ \ \ \widetilde{H} \theta =0$$ where $$\widetilde{H} = \partial_r\partial_s - \widetilde{u}, \ \ \widetilde{u} = u - 2\partial_r\partial_s \log \omega= - u + 2 \frac{\omega_r\omega_s}{\omega^2}.$$ Let us consider the Ore localization $F(\partial_s)$ of the ring of differential operators in $s$ and the ring $F(\partial_s)[\partial_r]$ of differential operators in $r$ with coefficients from (noncommutative) ring $F(\partial_s)$. Given $\omega$, a solution to the equation $H\omega=0$, we consider the differential operators $A,B \in F[\partial_s]$:$$A=\omega^{-1}\cdot\partial_s\cdot\omega, \ \ \ B=\omega\cdot \partial_s\cdot{\omega^{-1}}$$ and their “ratio”, the operator $\Omega$ of the form $$\Omega = A^{-1}\cdot B = \Big(\omega^{-1}\cdot\partial_s\cdot\omega\Big)^{-1}\cdot\Big(\omega\cdot \partial_s\cdot{\omega^{-1}}\Big)=A^{-1} B.$$ We denote by $M$ and $\widetilde{M}$ the following formal operators of the first order from $F(\partial_s)[\partial_r]$: $$M = \partial_s^{-1}\cdot H = \partial_r - \partial_s^{-1}\cdot u, $$ M = \_s\^[-1]{} = \_r - \_s\^[-1]{}. $$ \[th1\] Given $\omega$ satisfying $H\omega = (\partial_r\partial_s + u)\omega=0$ and the Moutard transform $\widetilde{u}$ of $u$ (we assume that the transformation is generated by $\omega$), the operators $M$ and $\widetilde{M}$ are conjugated in $F(\partial_s)[\partial_r]$ by $\Omega$: $$\widetilde{M} = \Omega \cdot M \cdot \Omega^{-1}.$$ [Remark.]{} It is easy to check that $H$ and $\widetilde{H}$ are not conjugated by $\Omega$: $ \widetilde{H} \neq \Omega \cdot H \cdot \Omega^{-1}$. [Proof.]{} First we expose some auxiliary facts. \[l3\] $$\label{OM2} \Omega=A^{-1} B = 1 - \frac{2}{\omega}\cdot \partial_s^{-1} \cdot \omega_s.$$ [Proof of Lemma.]{} It is enough to show that $B = A\left(1 - \frac{2}{\omega}\cdot \partial_s^{-1} \cdot \omega_s\right)$. Let us write down the right-hand side of this equality as $$A - \omega^{-1}\cdot\partial_s\cdot\omega\frac{2}{\omega}\cdot \partial_s^{-1} \cdot \omega_s=\omega^{-1}\cdot\partial_s\cdot\omega - 2\omega^{-1}\omega_s=$$ $$= \omega^{-1}(\omega_s +\omega\cdot\partial_s) - 2\omega^{-1}\omega_s= -\omega^{-1}\omega_s + \partial_s=B.$$ This proves the lemma. \[sled4\] $$\label{OMx} (A^{-1} B)_r = \left(1 - \frac{2}{\omega}\cdot \partial_s^{-1}\cdot \omega_s\right)_r= \frac{2\omega_r}{\omega^2}\cdot \partial_s^{-1}\cdot \omega_s - \frac{2}{\omega}\cdot \partial_s^{-1}\cdot \omega_{rs}.$$ Now we are ready to prove Theorem \[th1\]. It to enough to establish the equivalent equality $$\label{sopr2} \widetilde H \cdot \Omega = \partial_s\cdot \Omega\cdot \partial_s^{-1}\cdot H,$$ i.e. $$\label{sopr3} (\partial_r\partial_s - \widetilde u) \cdot A^{-1} \cdot B = \partial_s \cdot A^{-1} \cdot B \cdot \partial_s^{-1}\cdot H.$$ By Lemma \[l3\], the left-hand side of (\[sopr3\]) is $$(\partial_r\partial_s - \widetilde u)\cdot A^{-1} \cdot B = $$ \_s-(-u+)(1 - \_s\^[-1]{}\_s)= $$$$ = + $$$$ +u(1 - \_s\^[-1]{}\_s) - (1 - \_s\^[-1]{}\_s) = $$$$ = (()\_s \_s\^[-1]{}\_s + \_s \_s\^[-1]{}\_s . . - ()\_s \_s\^[-1]{}\_[rs]{} - \_s\_s\^[-1]{}\_[rs]{}) + $$$$ + \_sA\^[-1]{} B\_r +u - \_s\^[-1]{}\_s - + \_s\^[-1]{}\_s = $$$$ = (\_s\^[-1]{}\_s - \_s\^[-1]{}\_s + + ()\_s\^[-1]{}\_[rs]{} - \_[rs]{})+ $$$$ + \_sA\^[-1]{} B\_r +u - \_s\^[-1]{}\_s - + \_s\^[-1]{}\_s = $$$$ =()\_s\^[-1]{}(u) - 2u + \_sA\^[-1]{} B\_r + u = $$$$ =()\_s\^[-1]{}(u) + \_sA\^[-1]{} B\_r - u . $$In the right-hand side of (\ref{sopr3}) we have$$ \_sA\^[-1]{} B\_s\^[-1]{}(\_r\_s - u)= \_sA\^[-1]{} B\_r - \_s(1 - \_s\^[-1]{}\_s)\_s\^[-1]{} u= $$$$ = \_sA\^[-1]{} B\_r - u + ()\_s\_s\^[-1]{} \_s\_s\^[-1]{} u + ()\_s\_s\^[-1]{}\_s\_s\^[-1]{} u= $$$$ = \_sA\^[-1]{} B\_r - u - \_s\^[-1]{}\_s\_s\^[-1]{} u + \_s\^[-1]{} u. $$ Let us apply Proposition \[commutator\] to the third term $\partial_s^{-1}\cdot \omega_s\cdot \partial_s^{-1} = \omega\cdot \partial_s^{-1} - \partial_s^{-1} \cdot \omega$ in the last formula and finally derive $$\partial_s\cdot A^{-1} B\cdot\partial_r - u - \frac{2\omega_s}{\omega^2}\cdot\left(\omega\cdot \partial_s^{-1} - \partial_s^{-1} \cdot \omega\right) \cdot u + \frac{2\omega_s}{\omega}\cdot \partial_s^{-1} \cdot u={}$$ $${}=\partial_s\cdot A^{-1} B\cdot\partial_r - u + \frac{2\omega_s}{\omega^2}\cdot \partial_s^{-1} \cdot \omega \cdot u,$$ which coincides with the left-hand side of (\[sopr3\]). Theorem is proved. Let us assume that $H$ is a (weakly) algebraic operator, i.e., there are differential operators $L_1$ and $L_2$ such that $$\label{Kri1} [L_1,L_2] = D_0 H, \ \ \ [L_1,H] = D_1 H, \ \ \ [L_2,H] = D_2 H$$ and there is a polynomial $Q$ in two variables with constant coefficients such that $$Q(L_1,L_2) = 0\ (\mod H).$$ By applying the Euclid algorithm (division with remainder in the ring of formal linear ordinary differential operators $F(\partial_s)[\partial_r]$) we obtain $R_1, R_2 \in F(\partial_s)$ such that $$\label{L2R} L_1 - Q_1\cdot M=R_1 \in F(\partial_s), \quad L_2 - Q_2\cdot M=R_2 \in F(\partial_s)$$ with $Q_i \in F(\partial_s)[\partial_r]$. \[th2\] 1. $[R_1,R_2] = 0$; 2. $Q(R_1,R_2) = 0$; 3. $[R_1,M] = [R_2,M]=0$. [Proof.]{} 1. Using (\[Kri1\]) we compute that $$[R_1,R_2]=[L_1-Q_1 H,L_2-Q_2 H]=S_1\cdot H, \ \ \ S_1 \in F(\partial_s)[\partial_r].$$ However in the left-hand side we have an element of $F(\partial_s)[\partial_r]$ which contains no derivations in $r$ which implies that $S_1=0$. 2. The equality $Q(L_1,L_2) = 0\ (\mod H)$ means that $Q(L_1,L_2) = U \cdot H$ in $F[\partial_r,\partial_s]$ which, by (\[Kri1\]), implies $$Q(R_1,R_2) = S_2 \cdot H \ \ \ \in F(\partial_s)[\partial_r].$$ However in the left-hand side we have an element of $F(\partial_s)[\partial_r]$ with no derivations in $r$ which implies $S_2 = 0$. 3. We have $$[R_1,H]=[L_1-Q_1 H, H]=S_3\cdot H,$$ $$[R_1,M]=[R_1,(\partial_s)^{-1}H]=[R_1,(\partial_s)^{-1}]H+ $$ (\_s)\^[-1]{}\[R\_1,H\]=S\_4H=S\_5M, $$ where $S_i \in F(\partial_s)[\partial_r]$ for $i=3,4,5$. By Corollary \[cor1\], this implies $$[R_1,M]=[R_1,\partial_r - (\partial_s)^{-1}\cdot u(r,s)]= $$ -(R\_1)\_r + \[R\_1,(\_s)\^[-1]{}u(r,s)\] F(\_s), $$ and again as above we see that $S_5=0$. The proof of the equality $[R_2,M]=0$ is completely the same. Theorem is proved. In view of Theorems \[th1\] and \[th2\] we guess that the Moutard transformation should preserve the class of weakly algebraic two-dimensional Schrödinger operators. Applications of the Moutard transformation ========================================== In [@TT07; @TT08; @TT-DAN08] we gave a simple examples of fast decaying rational potentials for the two-dimensional Schrödinger operator which has a degenerated $L_2$-kernel. These examples are constructed by using the Moutard transformation as follows. [Main construction.]{} Let $$H_0 = - \Delta = -\Delta + u_0$$ be an operator with a potential $u_0(x,y)$ and let $\omega_1(x,y)$ and $\omega_2(x,y)$ satisfy the equations $$H_0 \omega_1 = H_0 \omega_2 = 0.$$ We take the Moutard transformations $M_{\omega_1}$ and $M_{\omega_2}$ defined by $\omega_1$ and $\omega_2$ and obtain the operators $$H_1 = -\Delta + u_1, \ \ \ H_2 = -\Delta + u_2$$ where $u_1 = M_{\omega_1}(u_0), u_2 = M_{\omega_2}(u_0)$. By the construction, we have $$H_1 M_{\omega_1}(\omega_2) = 0, \ \ \ H_2 M_{\omega_2}(\omega_1) = 0.$$ Let us choose some function $$\theta_1 \in M_{\omega_1}(\omega_2)$$ and put $$\theta_2 = -\frac{\omega_1}{\omega_2} \theta_1 \in M_{\omega_2}(\omega_1).$$ These functions define the Moutard transformations of $H_1$ and $H_2$ and we obtain the operators $H_{12}$ and $H_{21}$ with the potentials $$u_{12} = M_{\theta_1}(u_1), \ \ \ \ u_{21} = M_{\theta_2}(u_2).$$ The following key lemma is checked by straightforward computations which we omit. 1. $u_{12}=u_{21} = u$; 2. For $\psi_1 = \frac{1}{\theta_1}$ and $\psi_2 = \frac{1}{\psi_2}$ we have $$H\psi_1 = H\psi_2 = 0$$ where $H = -\Delta+u$. We note that in this construction we have a free scalar parameter $t$ (see (\[theta-family\])) for the choice of $\theta_1 \in M_{\omega_1}(\omega_2)$. This parameter can be used in some cases to build a non-singular potential $u$ and functions $\psi_1$ and $\psi_2$. For example if we apply this construction to the situation when $u_0 = 0$ and $\omega_1$ and $\omega_2$ are real-valued harmonic polynomials $$\omega_1 = x +2(x^2-y^2)+xy, \ \ \ \omega_2 = x+y+\frac{3}{2}(x^2-y^2)+5xy,$$ then for some appropriate constant of integration in $\theta_1$ we obtain $$\label{ord2-1} u = -\frac{5120 (1 + 8 x + 2y + 17 x^2 + 17 y^2)}{(160 + 4 x^2 + 4y^2 + 16 x^3 + 4 x^2y + 16 x y^2 + 4 y^3 + 17(x^2+y^2)^2)^2} =$$ $$= -\frac{5120\|1+(4-i)z\|^2}{(160+\|z\|^2\|2+(4-i)z\|^2)^2}$$ and $$\label{ord2-2} \begin{split} \psi_1 = \frac{x + 2 x^2 + x y - 2 y^2}{160 + 4 x^2 + 4y^2 + 16 x^3 + 4 x^2 y + 16 x y^2 + 4 y^3 + 17 (x^2+y^2)^2}, \\ \psi_2 = \frac{2 x + 2y + 3 x^2 + 10 x y - 3 y^2}{160 + 4 x^2 + 4y^2 + 16 x^3 + 4 x^2 y + 16 x y^2 + 4 y^3 + 17 (x^2+y^2)^2} \end{split}$$ (here we simplify $\psi_1$ and $\psi_2$ by multiplying by some constant). The potential $u$ given by (\[ord2-1\]) is smooth, rational, and decays like $1/r^6$ for $r \to \infty$ (here $r = \sqrt{x^2+y^2}$). The functions $\psi_1$ and $\psi_2$ given by (\[ord2-2\]) are smooth, rational, decay like $1/r^2$ for $r \to \infty$ and span a two-dimensional space in the kernel of the operator $L = -\Delta +u: L_2(\R^2) \to L_2(\R^2)$. If one takes appropriate two harmonic polynomials of the third order $\omega_1$ and $\omega_2$ (cf. [@TT07; @TT08]) then one can construct the potential $u$ and the functions $\psi_1$ and $\psi_2$ which are smooth, rational, the potential decays like $1/r^8$ for $r \to \infty$, the functions $\psi_1$ and $\psi_2$ decay like $1/r^3$ for $r \to \infty$ and span a two-dimensional space in the kernel of the operator $L = -\Delta +u: L_2(\R^2) \to L_2(\R^2)$. [Remark.]{} We guess that for every $N > 0$ by applying this construction to other harmonic polynomials one can construct smooth rational potentials $u$ and the eigenfunctions $\psi_1$ and $\psi_2$ decaying faster than $\frac{1}{r^N}$. In [@TT08; @TT-DAN08] we used a time-dependent extension of the Moutard transformation constructing explicit solutions of the Novikov–Veselov equation [@NV1; @NV2] $$\label{nv} \begin{split} U_t = \partial^3 U + \bar{\partial}^3 U + 3\partial(VU) + 3\bar{\partial} (\bar{V}U) =0, \\ \bar{\partial}V = \partial U. \end{split}$$ Some of our solutions show a very special behavior: the initial data for $t=0$ are smooth decaying rational functions of $x$ and $y$; nevertheless for $t \geq t_0>0$ the solutions to the NV equation (\[nv\]) blow up (become singular). In particular the following solution $U(z,\bar{z},t)$ of the Novikov–Veselov equation can be obtained using this technology: $$U= \frac{H_1}{H_2},$$ with $$\begin{array}{rcl} H_1 &=&-12\Big(24 t x^2 + 12 t x + 24 t y^2 + 12 t y + x^5 - 3 x^4 y + 2 x^4 - 2 x^3 y^2 - 4 x^3 y \\ & & \ \ \ \ \ \ \ \ - 2 x^2 y^3 - 60 x^2 - 3 x y^4 - 4 x y^3 - 30 x + y^5 + 2 y^4 - 60 y^2 - 30 y\Big), \end{array}$$ $$H_2=(3 x^4 + 4 x^3 + 6 x^2 y^2 + 3 y^4 + 4 y^3 + 30 - 12 t)^2.$$ This solution decays as $r^{-3}$ at infinity, it is nonsingular for $0\leq t < T_\ast = \frac{29}{12}$ and have singularities for $t\geq T_\ast = \frac{29}{12}$. For $t \to T_\ast$, as we can see on Figure 1, the solution $U(x,y,t)$ oscillates with growing amplitudes in the neighborhoods of the points $P = (-1,0)$ and $Q = (0,-1)$ since the denominator $H_2$ vanishes at these points for $t= T_\ast$. The numerator $H_1$ has zeros or order 3 at the points $P$ and $Q$ so their respective neighborhoods are subdivided by smooth lines into 6 sectors with different signs of the numerator. The complicated behavior of the potential in the neighborhood of one of the singular points for $t = T_\ast$ is shown on Figure 2. 0.5mm (64.67,130.00) (-20.67,17.00)[![image](two-sing.jpg){width="60.00000%"}]{} (5.67,5.00)[(0,0)\[lc\][Figure 1: The potential $U$ for $t = \frac{29}{12}$.]{}]{} (120.67,17.00)[![image](critical.jpg){width="60.00000%"}]{} (145.67,5.00)[(0,0)\[lc\][Figure 2: The potential $U$ for $t = \frac{29}{12}$]{}]{} (133.67,-5.00)[(0,0)\[lc\][ in the neighborhood of the point $(-1,0)$.]{}]{} [MMM]{} <span style="font-variant:small-caps;">H. Airault, H.P. McKean and J. Moser</span>, Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem, Comm. Pure Appl. Math. [30]{} (1977), 95–148. <span style="font-variant:small-caps;">J.L. Burchnall and T.W. Chaundy</span>, Commutative ordinary differential operators, Proc. London Soc., Ser. 2, [21]{} (1923), 420–440. <span style="font-variant:small-caps;">M.M. Crum</span>, Associated Sturm–Liouville systems, Quart. J. Math., Ser. 2 [6]{} (1955), 121–127. <span style="font-variant:small-caps;">G. Darboux</span>, Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal. T. 2. Paris: Gautier-Villars, 1915. <span style="font-variant:small-caps;">B.A. Dubrovin, I.M. Krichever and S.P. Novikov</span>, The Schrödinger equation in a periodic field and Riemann surfaces, Soviet Math. Dokl. [17]{} (1976), 947–952. <span style="font-variant:small-caps;">B.A. Dubrovin, V.B. Matveev and S.P. Novikov</span>, Nonlinear equations of Korteweg–de Vries type, finite-zone linear operators, and Abelian varieties, Russian Math. Surveys [31]{}:1 (1976), 59–146. <span style="font-variant:small-caps;">F. Ehlers and H. Knörrer</span>, An algebro-geometric interpretation of the Bäcklund transformation for the Korteweg–de Vries equation, Comm. Math. Helv. [57]{} (1982), 1–10. <span style="font-variant:small-caps;">I.M. Krichever</span>, Methods of algebraic geometry in the theory of non-linear equations, Russian Math. Surveys [32]{}:6 (1977), 185–213. <span style="font-variant:small-caps;">Th. Moutard</span>, Sur la construction des équations de la forme $\frac{1}{z} \frac{d^2 z}{d x\, d y} = \lambda(x,y)$, qui admettent une intégrale générale explicite, J. École Polytechnique. [45]{} (1878), 1–11. <span style="font-variant:small-caps;">S.P. Novikov</span>, A periodic problem for the Korteweg-de Vries equation. I, Functional Anal. Appl. [8]{} (1974), 236–246. <span style="font-variant:small-caps;">S.P. Novikov and A.P. Veselov</span>, Finite-zone, two-dimensional Schrödinger operators. Potential operators, Soviet Math. Dokl. [30]{} (1984), 705–708. <span style="font-variant:small-caps;">S.P. Novikov and A.P. Veselov</span>, Finite-zone, two-dimensional potential Schrödinger operators. Explicit formulas and evolution equations, Soviet Math. Dokl. [30]{} (1984), 588–591. <span style="font-variant:small-caps;">S.P. Novikov and A.P. Veselov</span>, Exactly solvable two-dimensional Schrödinger operators and Laplace transformations, Translations of the Amer. Math. Soc., Ser. 2, [179]{} (1997), 109–132. <span style="font-variant:small-caps;">O. Ore</span>, Linear equations in non-commutative fields, Ann. of Math. [32]{} (1931), 463–477. <span style="font-variant:small-caps;">I.A. Taimanov and S.P. Tsarev</span>, Two-dimensional Schrödinger operators with fast decaying rational potential and multidimensional $L_2$-kernel, Russian Mathematical Surveys, [62]{}:3 (2007), 631-633. <span style="font-variant:small-caps;">I.A. Taimanov and S.P. Tsarev</span>, Two-dimensional rational solitons and their blowup via the Moutard transformation, Theoretical and Mathematical Physics, [157]{}:2 (2008), 1525–1541. <span style="font-variant:small-caps;">I.A. Taimanov and S.P. Tsarev</span>, Blowing up solutions of the Novikov–Veselov equation, Doklady Mathematics, [77]{}:3 (2008), 467–468. [^1]: This research was partially supported by Siberian Branch of RAS (the interdisciplinary integration project No. 65) (the first author) and the RFBR Grants 09-01-00762-a and 06-01-89507-NNS-a (the second author). [^2]: Sometimes it is also called the Crum transformation due to [@Crum]. Burchnall and Chaundy called it transference [@BC].
--- abstract: 'Observations revealed rich dynamics within prominences, the cool ($10^4$ K), macroscopic (sizes of order 100 Mm) ‘clouds’ in the million degree solar corona. Even quiescent prominences are continuously perturbed by hot, rising bubbles. Since prominence matter is hundredfold denser than coronal plasma, this bubbling is related to Rayleigh-Taylor instabilities. Here we report on true macroscopic simulations well into this bubbling phase, adopting a magnetohydrodynamic description from chromospheric layers up to 30 Mm height. Our virtual prominences rapidly establish fully non-linear (magneto)convective motions where hot bubbles interplay with falling pillars, with dynamical details including upwelling pillars forming within bubbles. Our simulations show impacting Rayleigh-Taylor fingers reflecting on transition region plasma, ensuring that cool, dense chromospheric material gets mixed with prominence matter up to very large heights. This offers an explanation for the return mass cycle mystery for prominence material. Synthetic views at extreme ultraviolet wavelengths show remarkable agreement with observations, with clear indications of shear-flow induced fragmentations.' author: - 'R. Keppens and C. Xia' - 'O. Porth' title: 'Solar prominences: ‘double, double …boil and bubble’' --- Rayleigh-Taylor in prominences {#intro} ============================== Prominences are among ‘the most common features of the solar atmosphere’ [@parenti14] and are a consequence of the Lorentz force levitating solar plasma against gravity. This creates density inversions in the hot solar corona favourable to Rayleigh-Taylor driven dynamics. Rayleigh-Taylor instability – the reason why water falls out of a cup turned upside down – can occur whenever a fluid or gas gets accelerated or pushed into a denser substance and is at the heart of many dynamical phenomena in astrophysical plasmas. Recent magnetohydrodynamic (MHD) modeling of Rayleigh-Taylor evolution for the Crab nebula showed a clear tendency to self-organize into larger-scale structures, with filament sizes reaching up to a quarter of the entire pulsar wind nebula radius [@porthrtcrab14]. Magnetic field-guided accretion processes onto magnetized stars are enriched by equatorially accreting Rayleigh-Taylor plasma tongues, protruding into the magnetosphere from the inner accretion disk edge [@kulkarni08]. In the solar context, Rayleigh-Taylor filamentary structure can form during flux emergence and its reconnection with pre-existing coronal fields, as demonstrated by means of high resolution MHD simulations [@isobe05]. Solar prominences also demonstrate Rayleigh-Taylor mediated dynamics, with Hinode Solar Optical Telescope observations [@berger08] revealing how quiescent prominences show bright downflowing filaments of several hundred kilometres in width, typical speeds of ${\cal O}(10) \,\mathrm{km}/\mathrm{s}$ and ten minute lifetimes. At the same time, dark inclusions mark upflows at 20 $\mathrm{km}/\mathrm{s}$, rising up to 18 Mm heights and shedding voids that in turn grow to Mm sizes. Detailed observations showed how such dark upflows originate at the top of the chromosphere and can grow to 4-6 Mm plumes and rise to 15 Mm heights [@berger10]. They can form large-scale (20-50 Mm) buoyant arches or bubbles, and these rising bubbles were found to contain 25-120 times hotter material than the prominence proper [@berger11], strengthening the argument for a magneto-thermal convection process typical for coronal cavity-prominence configurations. Using a local model for a dipped prominence bottom boundary, Rayleigh-Taylor mode development in three-dimensional (3D) MHD simulations demonstrated both upflows [@hillier12] and interchange reconnection leading to downward blob motions [@hillier12b], in general agreement with observed local details. Recent modeling efforts have included partial ionization effects in local box models of the prominence-corona transition region [@khomenko14], finding clear differences with pure MHD approaches, as neutrals experience faster descents. Recent theoretical findings quantified the potentially stabilizing role of magnetic shear in idealized incompressible settings, important in the linear stages of Rayleigh-Taylor activity [@ruderman14]. A step towards global modeling of prominence dynamics in arcade systems confirmed this role of sheared magnetic fields as well as the effects of line-tying on prominence stability [@terradas15], but lacked the resolution to follow their development into the strongly nonlinear regime. In this paper, we set forth to realize this step, for the first time including chromosphere-transition region variations in a 3D prominence setup, and simulating well into the observed magneto-thermal convective motions. Numerical setup {#setup} =============== Our simulation box extends for 30 Mm horizontally ($x$) and vertically ($y$), and has a width ($z$) of 14 Mm. Using three levels of dynamic grid refinement we achieve a resolution of $600\times 600\times 280$, i.e. grid cell sizes down to 50 km. With the open-source MPI-AMRVAC software [@amrvac12; @amrvac14], we perform 3D, ideal MHD simulations for two cases that differ most markedly in the prevailing magnetic field strength throughout the domain: a weak field case at $\approx 8$ G, and a stronger field one at $\approx 20$ G. These are representative values for quiescent prominence conditions. The initial magnetic field $\mathbf{B}=(B_x=0.1998 \,\mathrm{G},0,B_z(y))$ is purely planar and non-uniform, due to an exponential decrease of the strongest $B_z(y)$ component in a layer of 2.5 Mm thickness above the initial prominence heigth $y_\mathrm{p}=12.5 \,\mathrm{Mm}$. The analytic form for the initial $B_z(y)$ is given by $$\begin{aligned} B_z & = & B_{z0} \,\,\,\,\,\,\, \mathrm{for} \,\, y<y_\mathrm{p} \,, \nonumber \\ & = & B_{z0} \exp\left[-(y-y_\mathrm{p})/\lambda_{B}\right]\,\,\,\,\, \mathrm{for}\,\, y_\mathrm{p}\leq y \leq y_\mathrm{b} \,\, \nonumber \\ & = & B_{z0} \exp\left[-(y_\mathrm{b}-y_\mathrm{p})/\lambda_{B}\right]\,\,\,\,\, \mathrm{for}\,\, y_\mathrm{b}\leq y \,. \end{aligned}$$ The parameters are set to $\lambda_B=15$ Mm, and $y_\mathrm{b}=15$ Mm, while $B_{z0}$ differs between the weak and strong field case. This induces a local shear in magnetic field, and establishes an upward magnetic pressure force that lifts matter against solar gravity. The horizontal $B_x$ field component leads to stabilization by tension forces against purely planar $(x,y)$ Rayleigh-Taylor instabilities for all wavelengths exceeding about 33 km, slightly below our numerical resolution. Nonlinear effects quickly dominate the dynamics at lengthscales fully captured in our study, a result corroborated by high resolution purely two-and-a-half dimensional simulations. The initial magneto-hydrostatic stratification introduces a transition region height at $y_{\mathrm{tr}}=2.5 \,\mathrm{Mm}$ where the temperature smoothly connects an $8000 \,\mathrm{K}$ chromosphere to a $1.8 \,\mathrm{MK}$ corona. Figure 1 shows in top and bottom left panels the temperature and density structure for the strongest magnetic field case. The dashed profiles above $y_\mathrm{p}=12.5 \,\mathrm{Mm}$ quantify $T(y)$ and $\rho(y)$ exterior to the prominence, where the corona is isothermal but the density shows an increase due to magnetic levitation. Inside the prominence, the vertical stratification follows the solid curves shown in Figure 1: the prominence temperature is 12000 K in $y\in[y_\mathrm{p}, 15 \,\mathrm{Mm}]$, increases linearly with height to 60000 K at $25\, \mathrm{Mm}$, and there connects again to coronal temperatures above the prominence structure. This initial condition has the essential characteristics of solar filaments, as the density contrast $\rho_{\mathrm{prom}}/\rho_{\mathrm{cor}}$ below the prominence is about 127.3 in this strong field case. When we set the overall dimensions of the prominence segment at 30 Mm length and 5 Mm width, we find that the total prominence mass is $7.5 \times 10^{13} \mathrm{g}$ for the weak field case, going up to $2.9 \times 10^{14}\mathrm{g}$ for the strong field case. These masses, together with the overall dimensions, all fall within their observationally known ranges. Global evolution {#evolve} ================ The initial condition – though vertically in force balance – is out of pressure equilibrium in the $z$-direction across the prominence structure. This leads to a transient phase of successive compressions of the prominence matter (and in the coronal region above it), with shock waves traversing the periodic $z$-direction. These alter the detailed temperature-density variations throughout prominence and coronal regions upward from $y=y_\mathrm{p}$, but largely retain their essential characteristics, keeping the total prominence mass and typical corona-prominence density and temperature contrasts. More importantly for our study targeting long-term prominence internal dynamics, these transverse motions quickly become dominated by vertical and horizontal ($x$) velocity components, as demonstrated in the right panel of Figure 1, where the (scaled) kinetic energy evolution is plotted for each velocity component, for the strong field case (the weak field case behaves similarly in its energetic evolution). After about 4 minutes, vertical motions (solid line) dominate in kinetic energy, and they saturate before 10 minutes. Lateral movements ($x$-direction, dotted line) peak at about 11 minutes, while we ran our models for close to 15 minutes. The growth in vertical kinetic energy directly relates to Rayleigh-Taylor modes throughout the prominence segment, which are triggered by a superposition of 50 small-amplitude velocity perturbations that fit the periodic $x$-direction with random phases. Each individual flow perturbation represents a planar $(v_x,v_y,0)$ incompressible velocity field, and is localized about the bottom prominence position $y_\mathrm{p}$ and its midplane $z=0$. Figure 2 gives a clear overview of the resulting prominence deformation and dynamics, by collecting a number of depth-integrated views taken at 6.9 minutes. This figure is for the strong field case. Panels (a) and (c) provide views on the prominence when integrated along its length, showing its entire embedding within coronal plasma, while the nonlinear Rayleigh-Taylor development has created downward falling pillars that just reached transition region heights. Panel (a) integrates an additionally advected scalar, where green values correspond to prominence matter, dark purple indicates chromosphere plasma, and white is used for coronal material. Panel (c) relates to the instantaneous temperature structure, with white indicating cool (chromospheric and prominence) matter, and red is used for coronal values. This panel also shows a thin layer of hot matter just above the prominence structure, which locates shock-heated, initially evacuated matter found there. Animated views for the entire simulation in the format of Figure 2 are provided as online material, where the mentioned transverse compressions and their transient nature become evident. From our earlier simulations of actual prominence formation due to chromospheric evaporation, thermal instability and runaway catastrophic cooling events [@xia11; @xia12; @fang13; @keppens14], these transient shock waves mimic the rebound shock fronts found to result from siphon flow driven impacts on the prominence-corona transition region. These rebound shocks ultimately repeatedly impact on the prominence structure, as a result of successive reflections when they reach chromosphere-corona transition regions along the fieldlines. The edge-on views shown in panels (b) and (d) of Figure 2 contain the $z$-integrated density structure, clearly dominated by falling Rayleigh-Taylor pillars with widths of up to 1000 km, and bubbles of upwardly curved prominence segments with lateral dimensions between 3000-4000 km. The resulting magnetic field deformation is visualized in panel (b), where streamlines, colored by the tracer from panel (a), are given for the $z$-integrated in-plane magnetic field components. This shows how the falling pillars indeed interchange magnetic field structure, where we note that the prevailing plasma beta is typically 0.16 (for the strong field case, and 0.38 for the weak one). The main displacements, as also seen in Figure 1, rapidly turn vertical and lateral, in accord with interchange modes that try to minimize field line bending, as the dominant magnetic field component is $B_z$ at all times. The same information can be deduced from panel (d) in Figure 2, where the superposed arrows likewise quantify the ($z$-integrated in-plane) velocity structure. Clearly, regions with Rayleigh-Taylor fingers are overall downward-moving at this time, while hot coronal plasma shows the fastest upwelling flows within several of the bubbles. An interesting detail is the upwelling Rayleigh-Taylor finger seen in panels (b)-(d) at horizontal distance $x\approx 12.5\,\mathrm{Mm}$ that is seen to start at $y\approx 10 \,\mathrm{Mm}$ working its way upwards from the bottom region of a bubble. This bubble has just been closed from below, by the merging of two downwelling fingers that jointly continue their fall. A localized dense protrusion then swirls upwards, indicating that the relative acceleration (between light and dense matter) causing the Rayleigh-Taylor event now acts upwards in the bottom region of this bubble. A three-dimensional view on the prominence structure is given for about the same time in the left panel of Figure 3. This shows the temperature variation in vertical bounding planes at $x=0$ and $z=-7 \,\mathrm{Mm}$. It also shows an isosurface of the temperature marking the 30000 K isosurface, showing that all cool material is found in the downward pillars and at the lower regions of the bubbles. This isosurface also nicely traces out the location of the chromosphere-corona transition region, which has hardly been perturbed at this point in the evolution. The grey isosurface shows the rear-half of the tracer isosurface, at a value which locates the original prominence matter at all times, as well as the chromosphere-corona transition. In this view, we also see several of the upwelling Rayleigh-Taylor features, in the closed bubble discussed earlier but also near the $x\approx 30\,\mathrm{Mm}$ front end. Hence such temporary upwelling features with widths of about 500 km, should be identifiable in the early stages of prominence formation and their internal dynamics. Note that speeds associated with individual larger-scale features, such as the falling and rising filaments or bubbles, are several tens of $\mathrm{km}/\mathrm{s}$, also seen from the animated views provided. Using the tracer mentioned earlier, we quantify a prominence-material-only average vertical speed. This increases from zero up to about $-20\,\mathrm{km}/\mathrm{s}$ after 400 seconds, declining again afterwards. To quantify better actual speeds, we added to our 3D MHD simulation a set of $30\times 60\times 10$ Lagrangian particles, which originally are positioned regularly on a grid throughout the simulation box. In the snapshot shown in Figure 3, twenty-four of the 18000 particle trajectories obtained are visualized with streamlines that color from dark to white when time proceeds through the almost 15 minute interval simulated. These select initial locations all near the midplane $z=0$, half of them initially right below the $y=y_\mathrm{p}$ bottom prominence layer, while the other 12 are internal to the prominence. The right panel of Figure 3 shows the same set of particle trajectories, but viewed in $(v_y,y)$ phase space. Dotted horizontal lines mark the heights of the initial prominence edge $y_p$, as well as the transition region height at 2.5 Mm. In this view, the ones that started internal to the prominence are indicated with larger symbols, while the thinner trajectories correspond to external (coronal) matter. Figure 3 shows that downward motions at up to 60 $\mathrm{km}/\mathrm{s}$ prevail at first, but nearly all get deflected upwards after encountering the transition region. Both coronal and internal prominence matter can get accelerated up to speeds exceeding 120 $\mathrm{km}/\mathrm{s}$. They can thereby reach heights well above their starting position, as some tracks go beyond 20 Mm height. Since a typical sound speed for the corona is 200 $\mathrm{km}/\mathrm{s}$, while the internal prominence sound speed is ten times lower, the process is highly nonlinear, and a vigorous magnetoconvective flow pattern extends from the transition region up into the prominence surroundings. Prominence matter can thereby repeatedly recycle, as it traverses a large range in altitude. These Lagrangian trajectories also imply that field lines (mainly directed along $z$) indeed show significant interchange behavior. This aspect may be exaggerated in our simulation by the periodicity assumption: in reality, these field lines are part of an arcade system passing through the prominence matter, and line-tying effects play a role in determining their Rayleigh-Taylor stability properties [@terradas15]. An impression of the magnetoconvection that gets established throughout the prominence surroundings is shown in Figure 4, where the high field case is visualized at the endtime of our simulation, i.e. at 14.3 minutes. At left, we show the tracer isosurfaces that identify all prominence matter (colored by the local temperature), along with a grey isosurface that identifies the original chromosphere-corona transition region. The former isosurface shows that prominence bubbles have merged and grown into arch-shaped structures that can reach sizes up to 10 Mm in width. The latter isosurface demonstrates that the impacting Rayleigh-Taylor fingers can locally significantly perturb the transition region, and cause dense chromospheric matter to be hurled up to heights of 10 Mm or more. This provides an effective route to feed more cool, dense matter into the prominence environment, and hence plays an important role in its mass recycling. Figure 4 also shows the density structure in a vertical slicing plane at $z=0$ in the box at right. In this view, we also visualize all tracer particles found between $x=15$ and 30 Mm initially, where their color encodes the original starting height of the particle. At time zero, this color coding gives a plane-parallel green-orange-yellow-red distribution from top to bottom, while at the endtime from Figure 4, vigorous convection shows effective mixing in the entire region between 2.5 Mm and up to 23 Mm. While Figure 4 is for the high field strength case, we provide animated views for the low field strength case in the representation of Figure 4, as online material. Qualitatively similar trends occur in high and low field strength cases, although the maximal velocities attained are lower for the low field case, and the falling pillars reach the transition region a bit later in the evolution. This dynamical evolution allows one to interpret the temporal evolution of the component-wise kinetic energy through the box shown in Figure 1. The maximum correlates with impacting falling pillars on the transition region, and lateral deflections maximize when up and down welling material meet up. Synthetic observations and conclusions {#conc} ====================================== Our macroscopic simulations can be turned into extreme ultraviolet synthetic images, for direct comparison with those available from Solar Dynamics Observatory [@sdo12] (SDO) observations using the Atmospheric Imaging Assembly [@aia12] (AIA). Especially its 304 Å and 171 Å channels provide views highlighting matter at 80000 K and 800000 K, respectively. This then samples cooler prominence to transition region material. A synthetic view of both the high field (top rows) and low field case (bottom rows) in both EUV channels is given in Figure 5 at the endtime of our simulations, while animated views on the final seven minutes of evolution are provided online. One notices how cool prominence matter is found embedded in hotter material, with falling and rising structures over a fair range of lengthscales. The different wavelength channels show morphological differences between hot and cool, up and downflow streams. The simulated, late nonlinear stages, especially for the low field case, show clear substructure developing along the edges of the largest bubbles, as seen in the bottom panels of Figure 5. At this stage, strong shear flows are established all along the arcs, that now extend as 10 Mm wide structures to heights of 18 Mm. We expect similar details to develop in the later stages of the high field strength case as well, as it also shows strong shear flows. This is in direct agreement with the latest observational details, pointing to Kelvin-Helmholtz and Rayleigh-Taylor interplay at the bubble boundary [@bergerproc14]. Visualizations of also the coronal channels (193 Å and 211 Å) further reveal the complex multi-temperature structure found in the magnetoconvective dynamics. Note that, by construction, our side-on views show the prominence matter in emission, and assume that the radiation is optically thin. This, together with the pure ideal MHD nature of our simulations, thereby neglecting important effects like coronal radiative losses, is an aspect to be improved upon. Further work can strive for even higher resolutions to capture smaller-scale fine-structure development from interplaying shear flow-driven, gravitational and thermal instabilities, or modifications due to partial ionization conditions. Ultimately, ab initio simulations must be able to demonstrate the thermal instability mediated formation process of prominences [@xialetter14], and simultaneously capture Rayleigh-Taylor mode development in realistic fluxrope-embedded prominence structures. This research was supported by the Interuniversity Attraction Poles Programme (initiated by the Belgian Science Policy Office, IAP P7/08 CHARM) and by the KU Leuven GOA/2015-014. 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L. 2014, Astrophys. J., 785, 110 , J., [Soler]{}, R., [Luna]{}, M., [Oliver]{}, R., & [Ballester]{}, J. L. 2015, Astrophys. J., 799, 94 , C., [Chen]{}, P. F., & [Keppens]{}, R. 2012, Astrophys. J., 748, L26 , C., [Chen]{}, P. F., [Keppens]{}, R., & [van Marle]{}, A. J. 2011, Astrophys. J., 737, 27 , C., [Keppens]{}, R., [Antolin]{}, P., & [Porth]{}, O. 2014, Astrophys. J. Lett., 792, L38 ![The initial temperature (top left) and density (bottom left) stratification, both within (solid) and external (dotted) to the prominence. Right panel: the scaled kinetic energy evolution, plotted per velocity component: vertical (solid, $y$), lateral (dotted, $x$), and transverse (dashed, $z$). []{data-label="ffig1"}](RKfigure1.pdf){width="\textwidth"} ![Ray-traced views at about 6.9 minutes, along \[panels (a), (c)\] and across \[panels (b), (d)\] the prominence axis, showing contour views of: (a) the tracers used to identify prominence (green) and chromosphere (purple) material; (b)-(d) the density variation; (c) the temperature. Right top panel also shows fieldlines based on integrated horizontal magnetic components, while bottom right arrows quantify the in-plane velocity variation. A movie in this format (for the high field case) is provided online.[]{data-label="ffig2"}](RKfigure2.pdf){width="\textwidth"} ![Left: a 3D view on the prominence, at the same time as Figure 2, showing the temperature variation on bounding planes, as well as a (red) isocontour at 30000 K. The grey isosurface shows half of the prominence-bounding surface. Furthermore, 24 Lagrangian tracer paths are superposed, changing their color from black to white to indicate temporal variation. At right, the same 24 trajectories are displayed in a $(v_y,y)$ phase-space view. []{data-label="ffig3"}](RKfigure3.pdf){width="\textwidth"} ![After 14.3 minutes, this 3D view shows at left the temperature variation on the prominence boundary (in red to yellow), as well as (in grey) the location of the heavily perturbed chromosphere-corona transition region. The prominence is in a state of vigorous nonlinear magnetoconvection, also shown by its density variation in a cutting plane, and the tracer particles at right. The latter were originally arranged from green to red in plane-parallel fashion from top to bottom. While this figure is for the high field case, a movie in this format for the low field case is provided as online material. []{data-label="ffig4"}](RKfigure4.pdf){width="\textwidth"} ![SDO AIA views on the endstate after 14.3 minutes for both the high field case (top row) and low field case (bottom row). Left panels are at 304 Å, right panels for 171 Å. A movie comparing both cases in this format is given online, covering the last 7 minutes of evolution. []{data-label="ffig5"}](RKfigure5.pdf){width="\textwidth"}
--- abstract: 'We present $H$- and $K_\mathrm{s}$-band imaging data resolving the gap in the transitional disk around LkCa 15, revealing the surrounding nebulosity. We detect sharp elliptical contours delimiting the nebulosity on the inside as well as the outside, consistent with the shape, size, ellipticity, and orientation of starlight reflected from the far-side disk wall, whereas the near-side wall is shielded from view by the disk’s optically thick bulk. We note that forward-scattering of starlight on the near-side disk surface could provide an alternate interpretation of the nebulosity. In either case, this discovery provides confirmation of the disk geometry that has been proposed to explain the spectral energy distributions (SED) of such systems, comprising an optically thick disk with an inner truncation radius of $\sim$46AU enclosing a largely evacuated gap. Our data show an offset of the nebulosity contours along the major axis, likely corresponding to a physical pericenter offset of the disk gap. This reinforces the leading theory that dynamical clearing by at least one orbiting body is the cause of the gap. Based on evolutionary models, our high-contrast imagery imposes an upper limit of 21$M_\mathrm{Jup}$ on companions at separations outside of 01 and of 13$M_\mathrm{Jup}$ outside of 02. Thus, we find that a planetary system around LkCa 15 is the most likely explanation for the disk architecture.' author: - 'C. Thalmann, C. A. Grady, M. Goto, J. P. Wisniewski, M. Janson, T. Henning, M. Fukagawa, M. Honda, G. D. Mulders, M. Min, A. Moro-Martín, M. W. McElwain, K. W. Hodapp, J. Carson, L. Abe, W. Brandner, S. Egner, M. Feldt, T. Fukue, T. Golota, O. Guyon, J. Hashimoto, Y. Hayano, M. Hayashi, S. Hayashi, M. Ishii, R. Kandori, G. R. Knapp, T. Kudo, N. Kusakabe, M. Kuzuhara, T. Matsuo, S. Miyama, J.-I. Morino, T. Nishimura, T.-S. Pyo, E. Serabyn, H. Shibai, H. Suto, R. Suzuki, M. Takami, N. Takato, H. Terada, D. Tomono, E. L. Turner, M. Watanabe, T. Yamada, H. Takami, T. Usuda, M. Tamura' title: \| Imaging of a Transitional Disk Gap in Reflected Light:\ Indications of Planet Formation Around the Young Solar Analog LC 15 --- Introduction ============ The circumstellar disks of gas and dust around newly formed stars are believed to be the birthplaces of giant planets. In some protoplanetary disks, evidence of gaps or inner cavities has been revealed through analysis of the infrared spectral energy distribution [SED, e.g. @calvet2002] or interferometry at infrared [e.g. @ratzka2007] or millimeter wavelengths [e.g. @brown2009]. These objects have been termed “transitional disks”, since they are thought to represent a transitional state of partial disk dissipation between the protoplanetary disk stage and the debris disk stage, with a variety of mechanisms proposed to account for central cavities: Photoevaporation, disk instabilities, and dynamical clearing [@alexander2006; @chiang2007; @bryden1999]. One such system is (K5:Ve, V=11.91mag, J=9.42mag, H=8.6mag, K=8.16mag), a 0.97$M_\odot$, 3–5Myr old [@simon2000], weak-line T Tauri star with a gas-rich millimeter-bright disk [@pietu2007; @henning2010]. Recently, it was identified as a pre-transitional disk [@espaillat2007], which consists of an inner disk component near the dust sublimation radius (0.12–0.15AU), a wide gap, and an outer, disk beyond 46 AU [@espaillat2008]. Spitzer IRS spectra by @sargent2009 reveal a relatively low level of dust grain growth. With an expected angular scale of 033, the outer wall is accessible to 8-m class telescopes equipped with high-contrast imagers. Observations and data reduction =============================== ![High-contrast imaging of LkCa 15. The scale bars in all panels correspond to 1$\arcsec$ = 140AU in projected separation. The greyscale is linear. (a) HiCIAO $H$-band image after conservative LOCI ADI. The inner edge of the outer disk is clearly visible. The blacked-out area in the center represents the range of separations where the field rotation is insufficient for ADI. The negative features within 05 are areas oversubtracted by LOCI as a reaction to the nearby positive features. The brightest part of the nebulosity corresponds to $H=12.7$mag/arcsec$^2$. (b) The signal-to-noise map of the same image, calculated in concentric annuli around the star. The outer wings of the nebulosity are rendered more visible. The stretch is $[-2.5\,\sigma,2.5\,\sigma]$. (c) Interferometry image of the disk gap at 1.4mm taken from @pietu2006. The cross represents the major and minor axes of the nebulosity. (d) The HiCIAO $H$-band image after reference PSF subtraction. The circular halo out to $\sim$06 is most likely an artifact from imperfect PSF matching. (e) The NACO $K_\mathrm{s}$-band image after reference PSF subtraction, for comparison. []{data-label="f:images"}](f1.eps){width="0.95\linewidth"} Subaru/HiCIAO data ------------------ As part of the ongoing high-contrast imaging survey SEEDS [Strategic Exploration of Exoplanets and Disks with Subaru/HiCIAO; @tamura2009], we observed LkCa 15 on December 26, 2009, with the HiCIAO instrument [@hodapp2008] on the Subaru Telescope with a field of view of 20$\times$20 and a plate scale of 9.5mas/pixel. The images were taken in $H$ band (1.6m), and the image rotator was operated in pupil-tracking mode to enable angular differential imaging [ADI, @marois2006]. The data set comprises 168 frames, each containing the co-add of 3 unsaturated exposures of 1.39s, for a total integration time of 700.6s. The image resolution provided by the adaptive optics system AO188 was close to the diffraction limit, with a full width at half maximum (FWHM) of 55mas at 06 natural seeing in $H$. The data were corrected for flatfield and field distortion. In order to search for point sources such as planets, ADI was then performed on the images using the LOCI algorithm [Locally Optimized Combination of Images, @lafreniere2007]. This form of ADI is the most powerful high-contrast imaging method currently available, as evidenced by recent direct detections of substellar companions [@marois2008; @thalmann2009]. For two frames to be eligible for mutual point-spread function (PSF) subtraction at a given working radius, we require the differential field rotation arc at that radius to be at least 3FWHM = 165mas, as opposed to the 0.75FWHM typically used for the detection of point-sources. This protects extended structures of up to the given azimuthal size scale from self-subtraction. We refer to this technique as “conservative LOCI”. A circularly symmetric circumstellar structure is still eliminated under these conditions, but as our data demonstrate, a sufficiently elliptical structure can survive by intersecting the concentric annuli around the star at a steep angle. An alternate method to reveal circumstellar nebulosity is subtraction of the PSF of a reference star, which avoids disk self-subtraction at the price of less effective stellar PSF removal. We observed the star in pupil-tracking mode immediately after LkCa 15 for this purpose. A neutral density filter was used with the adaptive optics wave-front sensor to achieve the same PSF quality as with the fainter LkCa 15. The exposures were centered, flatfielded, distortion-corrected, and directly collapsed with a pixel-wise median. We subtracted the resulting reference PSF from each of the LkCa 15 frames before de-rotation and co-addition to ensure optimal matching of the pupil-stabilized stellar PSFs. Archival VLT/NACO data ---------------------- For comparison, we retrieved $K_\mathrm{s}$-band coronagraphic imaging data of LkCa 15 taken with VLT/NACO in 2007 from the ESO Science Archive Facility under the program ID 280.C-5033(A). The observations were made in field-tracking mode and thus could not be used for LOCI. However, a reference star is included in the data set, allowing PSF subtraction. The camera with 13mas pixel scale and the four-quadrant phase mask coronagraph had been used for this run, producing high contrast at a small inner working angle. Results ======= Imaging of the disk gap ----------------------- The $H$-band HiCIAO images reveal a crescent-shaped nebulosity around LkCa 15 after both conservative LOCI reduction with a frame selection criterion of 3FWHM and conventional reference PSF subtraction (Figure \[f:images\]). With a diameter of $\sim$$1\farcs2$ ($\approx 170$AU), it is elongated along the position angle $\sim$$60^\circ$, leaving localized traces in the concentric annuli around the star in which LOCI operates. The consistency of the LOCI and reference subtraction images demonstrates that the nebulosity is elliptical enough to survive conservative LOCI largely intact. Although some flux is inevitably lost in this process, the elimination of the star’s PSF in LOCI is superior to conventional reference PSF subtraction, revealing the sharp inner and outer edges of the structure clearly. The crescent is also found in the PSF-subtracted $K_\mathrm{s}$-band images from VLT/NACO, supporting the interpretation that it represents a real astronomical feature. The gap enclosed by the inner edge matches the gap predicted from the SED [@espaillat2007] as well as the localized flux deficit seen in 1.4mm interferometry [@pietu2006 cf. Figure \[f:images\]] in terms of size, ellipticity, and position angle. ![Contrast (a) and mass (b) of companions detectable at $5\,\sigma$ around LkCa 15, based on the $H$-band LOCI image after convolution with a circular aperture of 4 pixel (38mas) in diameter. The contrast values are converted into companion mass using the `COND` evolutionary models by @baraffe2003 assuming a distance of 140pc and an age of 5Myr (solid curve), 3Myr (dashed curve), and 1.5Myr (dotted curve), respectively. The vertical dotted lines mark the inner working angle of ADI in this data set. The horizontal dotted lines guide the eye in (a) and indicate the deuterium burning limit of 13.7$M_\mathrm{Jup}$ in (b). No companion candidates are detected.[]{data-label="f:planets"}](f2.eps){width="\linewidth"} The negative areas in the inner field represent oversubtraction, which is inevitable in LOCI when strong signals are present. The algorithm attempts minimize the root of mean squares of the residual image, thus the positive mean offset of a signal will be lost, given that it cannot be distinguished from the positive mean offset of the stellar PSF halo. In conservative LOCI, the frame selection criterion that allows the survival of azimuthally extended positive flux concentrations also produce extended oversubtraction areas. This effect renders the flux levels in the resulting image unreliable, but preserves sharp edges in the original flux distribution. Similarly, the location of a point source in regular LOCI is reliable, but its photometry is not, requiring the determination of a correction factor. In its brightest pixel, the crescent in the LOCI image reaches a peak intensity of $H=12.7$mag/arcsec$^2$, which is to be taken as a conservative lower limit. Reference PSF subtraction is expected to retain all of the disk flux, but the central part of the image is dominated by strong residuals from the PSF subtraction and therefore does not provide a better photometric measurement. The small extended signal $0.35\arcsec$ to the southeast of the star in the $H$-band images likely corresponds to a real physical feature, given that it is visible both after LOCI processing and after reference PSF subtraction (Figures \[f:images\]a, \[f:images\]d). Constraints on point sources ---------------------------- Pericenter offset ----------------- ![Profile of the disk wall edges derived from an 11 pixel wide strip cut from the LOCI image along the perceived major axis of the ellipses in Figure \[f:ellipses\] (position angle $-32.8^\circ$), collapsed along the strip width and then median-smoothed on the scale of 5 pixels ($\sim$1FWHM), shown as a black curve. Median smoothing conserves edges well while removing pixel-to-pixel noise. The red vertical curve in the center is the bisector of the two opposed slopes of the profile, demonstrating the asymmetry. The orange curve shows the same smoothed profile taken from the LOCI image after rescaling with a map of the squared distance $r^2$ from the star. The dashed horizontal lines guide the eye. ](f3.eps){width="\linewidth"} \[f:bisector\] Our LOCI imaging allows us to set upper limits on the point sources present in the vicinity of LkCa 15. Figure \[f:planets\] shows the companion mass we can exclude at the $5\,\sigma$ level as a function of separation, assuming the `COND` evolutionary models by @baraffe2003, a distance of 140pc, and an age of 3–5Myr @simon2000. Since LkCa 15 is part of the Taurus star-forming region, for which @watson2009 give an age of 1–2Myr, we also plot a detectable mass curve for the age of 1.5Myr. The image was convolved with a circular aperture 4 pixel in diameter, and the noise level is calculated as the standard deviation in concentric annuli. We compensate for the expected flux loss due to partial self-subtraction by implanting test point sources in the raw frames and measuring how they are affected by the LOCI algorithm. We note that the bright nebulosity dominates the noise level at all separations, thus the calculated upper limit is likely conservative, overestimating the true residual speckle noise level. We do not detect any statistically significant signals that could be considered companion candidates. present a more detailed discussion of point-source constraints around LkCa 15, including alternate evolutionary model assumptions, based on the NACO $K_\mathrm{s}$-band data. They reach the same contrast levels as we do. We find a likely positional offset of the inner and outer boundaries of the nebulosity from the star along the system semi-major axis (Figures \[f:bisector\], \[f:ellipses\]a). Furthermore, our inner and outer fitted ellipses are rotated by $-4^\circ$ and $-3^\circ$ with respect to the position angle of $150.7^\circ$ in @pietu2007, with an estimated fitting error of $2^\circ$. In order to quantify the observed offset, we cut a strip with a width of 11pixels = 010 centered on the star and oriented along the major axis of our fitted ellipses from the LOCI image, assuming a $-3.5^\circ$ offset from the 1.4mm position angle. Since the disk does not suffer from foreshortening along its apparent major axis, the distances along the strip are a direct measure of physical distance. We collapse the strip into a one-dimensional profile, plotted in Figure \[f:bisector\]. The intensity drops down to zero over a distance of 54mas ($\approx$ 8AU, 1.0FWHM) on the West side and 81mas ($\approx$ 11AU, 1.5FWHM) on the East side of the gap, indicating that the edge is sharp at or below the image resolution. The bisecting curve between the two opposed gap edges reveals a systematic offset from the star’s position by 64mas (9AU) with a standard deviation of 6mas (0.9AU). Since the star is unsaturated, its centering accuracy ($\sim$0.2 pixels = 2mas) does not contribute significantly to the uncertainty. The distance of the gap edges from the star are 345mas = 48AU on the left side and 447mas = 63AU on the right side at an accuracy of 9mas (1.3AU), consistent with the 46AU radius derived from millimeter interferometry by @pietu2006. After scaling each pixel in the strip by $r^2$ before collapsing the strip, we find that the nebulosity appears at roughly the same brightness on both sides of the gap, suggesting a $r^{-2}$ dependency of the unscaled flux levels. This is consistent with the assumption that we are looking at reflected light from material at varying distances $r$ from the star. Thus, the inner edge of the outer disk around LkCa 15 likely features a pericenter offset comparable to those observed in the disks of HD 142527 [@fukagawa2006] and Fomalhaut [@kalas2005]. Discussion ========== ![(a) Ellipse fits to the inner and outer boundaries of the scattered light nebulosity seen in the HiCIAO $H$-band LOCI image after median filtering on the spatial scale of 5 pixels $\approx$ 1FWHM and derotation by $-29.3^\circ$ [based on the position angle of $150.7^\circ$ in @pietu2007]. The inner (orange) and outer (red) ellipses are offset from the star along the major axis by 51mas and 57mas and rotated by $-4^\circ$ and $-3^\circ$, respectively. Their centers are marked by orange and red plus signs, respectively, while the star’s position is indicated by a white plus sign. The dotted lines delimit the area on which the quantitative analysis in Figure \[f:bisector\] is based (offset angle $-3.5^\circ$). (b) Sketch of the illuminated wall scenario, taken from @espaillat2008, based solely on the SED and millimeter interferometry. The inner disk is not to scale. The illuminated disk wall on the far side (light grey) is directly visible while the near-side wall blocks its bright side from view. (c) $H$-band image of the forward scattering scenario including anisotropic scattering, from the simulation presented in @mulders2010. The near-side disk surface (top) appears bright due to efficient forward scattering, whereas the far-side disk wall (bottom) is mostly shadowed by the inner disk (center, not resolved), reducing the wall image to two thin parallel arcs. []{data-label="f:ellipses"}](f4.eps){width="0.95\linewidth"} Forward scattering scenario --------------------------- One possible explanation of the observed bright nebulosity is forward-scattering, which is commonly invoked to explain the brightness asymmetries in disk surfaces seen in reflected light [e.g. @weinberger1999; @fukagawa2006]. Forward-scattering on large dust grains can be several times as efficient as backward-scattering; as a result, the near-side surface of such a disk appears brighter than the far side. This requires the outer disk surface to be illuminated by the star. The first of the two SED-compliant models in @mulders2010 includes a forward-scattering disk surface; furthermore, the far-side gap wall is shadowed by an optically thick inner disk, rendering it hard to detect in direct imaging (Figure \[f:ellipses\]c). Note that the size and shape of the dust grains in the wall are not well known. This scenario is supported by @pietu2007, whose orientation and inclination values for the LkCa 15 disk suggest that the northwest side is the near side, as well as the fact that the crescent seen after reference PSF subtraction features a bright spot along the minor axis. Illuminated wall scenario ------------------------- Another explanation for the nebulosity is that it represents the inner wall of the outer disk on the far side of the star, illuminated by the star and viewed directly through the disk gap [@espaillat2008 Figure \[f:ellipses\]b]. The illuminated surface of the near-side wall, on the other hand, is blocked from sight by the bulk of the optically thick disk. Furthermore, the wall is high enough to cast the outer surface of the disk into shadow, suppressing a forward-scattering signature. This corresponds to the second model by @mulders2010. The morphology of the nebulosity favors this scenario, given that (1) there is a sharp outer edge roughly parallel to the inner edge, (2) the nebulosity reaches across the major axis, “embracing” the star, and (3) the nebulosity is wider along the major axis than along the minor, all of which are expected for an illuminated tapered disk wall, but not for forward scattering on the disk surface. Thus, we consider the illuminated wall scenario the more likely explanation until further data become available. Gap formation mechanism ----------------------- Both scenarios presented above must invoke a disk gap to explain the sharp inner edge of the observed nebulosity. Our data therefore prove the validity of the gapped disk model predicted from the SED. Several mechanisms have been proposed to produce cavities or gaps in protoplanetary disks. Magneto-rotational instability (MRI) as described in @chiang2007 can be excluded for LkCa 15, since it acts on all disk components that can be ionized by direct X-ray illumination from the star, thus the inner dust component of LkCa 15 known from the SED could not have survived. Furthermore, the stellar mass of 0.97$M_\odot$ and stellar accretion rate of 2.4$\cdot$10$^{-9}$ $M_\odot$/yr [@espaillat2007] would require a viscosity parameter $\alpha\approx 0.0007$, one order of magnitude below the range considered in the publication. While photoevaporation can in principle evacuate gaps as large as that of LkCa 15 in simulations [@alexander2006; @gorti2009; @owen2010], those require that the inner disk drain away entirely before the gap can grow beyond 1–10AU. The presence of an inner dust disk around LkCa 15 does not fit this scenario. therefore conclude that dynamical clearing by one or more orbiting bodies is the most plausible cause of the disk gap. Our detection of a likely pericenter offset in the LkCa 15 disk gap reinforces this argument. Dynamical sculpting is commonly invoked to explain warps and eccentricities in debris disks [e.g. @roques1994; @kalas2005], and also produces off-centered gaps in simulations of protoplanetary disks [e.g. @marzari2010]. In contrast, neither MRI nor photoevaporation have been predicted to cause such effects. In the case of LkCa 15, close stellar companions are excluded by long-baseline interferometry [@pott2010], while our data conservatively exclude companions more massive than 21$M_\mathrm{Jup}$ exterior to 01 (14AU), and all but planetary-mass bodies exterior to 02 (28AU). note that a 5–10$M_\mathrm{Jup}$ body orbiting at 30AU (the equivalent of Neptune’s orbit) has a sufficiently large Hill sphere to dynamically produce a wall at 50AU. Less massive bodies orbiting closer to the wall can have similar dynamical effects. note that bodies more massive than 6$M_\mathrm{Jup}$ will suppress accretion onto a young Solar analog to levels below that still present in the LkCa 15 system. Our data indicate that any bodies in the disk of LkCa 15 at $r\ge 0\farcs2$ (28AU) from the primary in projection must have planetary masses. We therefore find dynamical clearing by one or more planets to be the most likely cause of LkCa 15’s disk gap. Given the Sun-like mass of the young star ($0.97\,M_\odot$), we might in fact be looking at a Solar System analog in the making. We thank Cornelis P. Dullemond and Dmitry Semenov for helpful discussion, and David Lafrenière for generously providing us with the source code for his LOCI algorithm. This work is partly supported by a Grant-in-Aid for Science Research in a Priority Area from MEXT and by the Mitsubishi Foundation. JPW and MWM acknowledge support from NSF Astronomy & Astrophysics Postdoctoral Fellowships AST 08-02230 and AST-0901967, respectively. JPW also acknowledges funding from a Chrétien International Research Grant. ELT gratefully acknowledges support from a Princeton University Global Collaborative Research Fund grant and the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. Part of this work was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with NASA. [Facilities:]{} , . 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--- abstract: 'AdS/CFT induced quantum dilatonic brane-world where 4d boundary is flat or de Sitter (inflationary) or Anti-de Sitter brane is considered. The classical brane tension is fixed but boundary QFT produces the effective brane tension via the account of corresponding conformal anomaly induced effective action. This results in inducing of brane-worlds in accordance with AdS/CFT set-up as warped compactification. The explicit, independent construction of quantum induced dilatonic brane-worlds in two frames: string and Einstein one is done. It is demonstrated their complete equivalency for all quantum cosmological brane-worlds under discussion, including several examples of classical brane-world black holes. This is different from quantum corrected 4d dilatonic gravity where de Sitter solution exists in Einstein but not in Jordan (string) frame. The role of quantum corrections on massive graviton perturbations around Anti-de Sitter brane is briefly discussed.' address: - 'Department of Applied Physics, National Defence Academy, Hashirimizu Yokosuka 239-8686, JAPAN' - ' Instituto de Fisica de la Universidad de Guanajuato, Lomas del Bosque 103, Apdo. Postal E-143, 37150 Leon, Gto., MEXICO ' author: - 'Shin’ichi Nojiri[^1]' - 'Octavio Obregon[^2], Sergei D. Odintsov[^3] and Vladimir I. Tkach[^4]' title: ' STRING VERSUS EINSTEIN FRAME IN AdS/CFT INDUCED QUANTUM DILATONIC BRANE-WORLD UNIVERSE' --- =5000 5[[[AdS]{}\_5]{}]{} §4[[[S]{}\_4]{}]{} Introduction ============ Brane-worlds are alternative to the standard Kaluza-Klein compactification. They naturally lead to the following nice features of mutli-dimensional theory like trapping of 4d gravity on the brane [@RS], resolution of hierarchy problem and possibly resolution of cosmological constant problem. Different aspects of brane-world cosmology (for very incomplete list of references see [@CH; @cosm]) are under very active investigation. The essential element of original brane-world models is the presence in the theory of two free parameters (bulk cosmological constant and brane tension, or brane cosmological constant). These parameters are fine-tuned (up to some extent) in order to construct the successful classical brane-world. This is most standard prescription which may be not completely satisfactory if one wishes to have the dynamical mechanism of brane tension origin. From another side, one can fix the classical action on AdS-like space from the very beginning with the help of surface terms added in accordance with AdS/CFT correspondence[@AdS]. Such terms should make the variational procedure to be well-defined and also they should eliminate the leading divergence of the action. Brane tension is not considered as free parameter anymore but it is fixed by the condition of finiteness of spacetime when brane goes to infinity. In this case, as parameters are fixed the consistent brane-world scenario is impossible, as a rule. However, other parameters may improve the situation when quantum effects are taken into account. Taking quantum CFT (including quantum gravity!) on the brane one adds its contribution (the corresponding conformal anomaly induced effective action) to the total action. As a result, it changes the brane tension, the quantum induced brane-world occurs as it has been discovered in refs.[@NOZ; @HHR]. Actually, this represents the embedding of warped compactification (brane-worlds) to AdS/CFT correspondence, hence one gets AdS/CFT induced quantum brane-worlds [@NOZ; @HHR] where 4d boundary may be flat or de Sitter or Anti-de Sitter spacetime. This is clearly the dynamical mechanism to get curved brane-world. It is easily generalized for the presence of non-trivial dilaton, i.e. AdS/CFT induced quantum dilatonic brane-worlds occur[@NOO]. In other words, brane-worlds are the consequence of the presence of quantum fields on the brane in accord with AdS/CFT set-up. Moreover, such induced dilatonic brane-worlds are even more related with AdS/CFT correspondence as 5d dilatonic gravity represents the bosonic sector of 5d gauged supergravity (special parametrization). Even more, the dynamical determination of 4d dilaton occurs. In the study of quantum induced brane-worlds, in the same way as for any other dilatonic gravity the following question appears: which frame to work with is the physical one? There are two convenient frames: string (or Jordan) one where scalar curvature explicitly couples with dilaton and Einstein frame where scalar curvature does not couple with dilaton. Basically speaking, one should expect that results obtained in these two frames are not equivalent. Indeed, in QFT the choice of different variables and (or) form of action corresponds to different parametrizations. QFT results are parametrization dependent, only S-matrix is gauge and parametrization independent. (Even the quantization procedure (for review, see [@GL]) is parametrization dependent.) As usually the consideration is one-loop ,one should expect in many cases the explicit parametrization dependence. Moreover, it is known that even for classical dilatonic gravity the (singular) solution may exist in only one parametrization. Hence, the question of frame dependence should be carefully analyzed for all solutions at hands. This is the main purpose of the present work: to compare string frame quantum induced dilatonic brane-worlds with their analogs in Einstein frame. In the next section as the simple example, 4d dilatonic (Brans-Dicke) theory with large $N$ quantum spinor corrections is considered. In the Einstein frame where spinor is dilaton coupled one the de Sitter Universe solution with decaying dilaton exists. Working with the same theory in string (Jordan) frame where spinor is getting minimal, one finds that above solution does not exist. Hence, it is shown that two frames in 4d dilatonic gravity with quantum corrections are not equivalent. In third section we consider 5d dilatonic gravity action with 4d boundary term induced by conformal anomaly of brane, dilaton coupled spinor. Explicit examples of de Sitter, flat and Anti-de Sitter dilatonic branes are constructed in Einstein frame. The dynamical mechanism to determine the dilaton on the brane is presented. In section four the same investigation is done in string frame. Brane spinor is now minimal. The same AdS/CFT induced quantum brane-worlds are proven to exist. Hence, for quantum corrected cosmological dilatonic brane-worlds one has the equivalency of string and Einstein frames. In fifth section the equivalency of string and Einstein frames is demonstrated for number of classical dilatonic brane-world black holes. In section six some remarks on massive graviton modes around dilatonic AdS4 brane are made. The role of brane quantum corrections for massive graviton modes is clarified. Brief summary and some outlook are given in final section. Jordan and Einstein frames for 4d quantum corrected dilatonic gravity ===================================================================== In the study of dilatonic gravities the interesting question appears: which frame among few possible ones is the physical one? Basically speaking, there are two convenient frames to work with: string (or Jordan) frame and Einstein frame. These two are related by conformal transformation. The best known example is provided by the standard Brans-Dicke theory (with matter). The 4-dimensional action in the Jordan frame is: S\_[BD]{}=d\^4x +S\_M, \[bdj\] where $\phi$ is the Brans-Dicke (dilaton) field with $\omega$ being the coupling constant and $S_M$ is the matter action. Performing the following conformal transformation and a redefinition of the scalar field \[phis\] \_=Gg\_ , = (G), 2+3 >0. one gets the action in the Einstein frame S=d\^4x , \[bde\] where $A=-8\sqrt{\frac{\pi G}{2\omega+3}}$. It is expected that these two actions (at least for regular solutions) should lead to equivalent results. However, the explicit consideration shows that it is not always so (for a review, see [@FGT]). That is why it was argued in ref.[@FGT] that it is Einstein frame which is physical one. Of course, such state of affairs is not satisfactory. In quantum field theory the choice of different variables corresponds to different parametrizations. It is known that generally speaking it leads to parametrization dependent results: it is only S-matrix should be the same in different parametrizations. Of course, this should be true only in complete theory where account of all loops is taken. As usually the consideration is one-loop, one should expect parametrization dependence already at one-loop. Let us consider the explicit example in Einstein frame where quantum corrections are taken into account. As matter Lagrangian we take the one associated with $N$ massless (Dirac) spinors, i.e. $L_M=\sum_{i=1}^N \bar\psi_i\gamma^\mu\nabla_\mu\psi^i$. There is no problem to add other types of matter (say scalar or vector fields). The above choice is made only for the sake of simplicity. We shall make use of the EA formalism (for an introduction, see [@BOS]). The corresponding 4d anomaly–induced EA for dilaton coupled scalars, vectors and spinors has been found in Refs. [@NO]. Hence, starting from the theory with the action (no classical background spinors) S=d\^4x , \[bde1\] we will discuss FRW type cosmologies ds\^2=-dt\^2+a(t)\^2 dl\^2, \[st4\] where $dl^2$ is the line metric element of a 3-dimensional flat space. The computation of the anomaly–induced EA for the dilaton coupled spinor field has been done in [@NO], and the result, in the non-covariant local form, reads: \[vii\] && + b’\_1(\|G -[2 3]{}\|\|R) -[1 18]{}(b + b’)\^2}, where $\sigma_1=\sigma+ A\phi /3$, the square of the Weyl tensor is given by $ F= R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} -2 R_{\mu\nu}R^{\mu\nu} + {1 \over 3}R^2 $ and Gauss-Bonnet invariant is $G=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} -4 R_{\mu\nu}R^{\mu\nu} + R^2$. For Dirac spinors $b={3N \over 60(4\pi)^2}$, $b'=-{11 N \over 360 (4\pi)^2}$. Then we find the following Einstein frame, quantum-corrected solution whose metric is expressed in Jordan frame as \[16\] ds\_J\^2&=&a\_J\^2()(-d\^2 + dl\^2 ) a\_J\^2()&=&\^[-]{}a\^2() &=& a\_0\^[-2]{} && [1 2H\_1]{} + 1 &=&[ 1 { - [3 16]{} }]{} +1 &=&-[1 8]{} . Here $a_0$ is an arbitrary constant. On the other hand, one finds the dilaton field $\phi_J$ in the Jordan frame as \[phiJJ\] =\_0\^[[1 H\_1]{} ]{} =\_0\^[2(-1)]{} , \_0= [1 a\_0 G ]{} . Let us analyze the equations of motion in the Jordan frame (for the form of transformation to string frame see section 5). The variations over $\phi$ and $\sigma$ give the following equations: \[phieqJ\] 0&=&6 (” + [’]{}\^2)\^[2]{} -[\^2 \^2]{}\^[2]{} - 2([’\^[2]{} ]{}) ,\ \[sigmaeqJ\] 0&=& [2 16]{} (6(” + [’]{}\^2) + [\^2 ]{})\^[2]{} + [6(\^[2]{})” - 12 (’\^[2]{})’ 16]{} && + 4b’”” - 4(b + b’){( ” - [’]{}\^2 )” + 2(’ (” - [’]{}\^2))’} . Here $'\equiv {d \over d\eta}$. We can check that the solution (\[16\]) and (\[phiJJ\]) does not satisfy (\[phieqJ\]). If the solution in the Jordan frame would be equivalent to that in the Einstein frame even in the quantum level, we should have $\sigma_1=\sigma_J\equiv \ln a_J$ but we have $\sigma_1=\sigma + {A\phi \over 3} =\sigma - {4 \over 3}\ln G\phi_J$ and $\sigma_J=\sigma - {1 \over 2}\ln G\phi_J$. This is an origin of the inequivalence. Thus, it is demonstrated that for the Universe model under consideration the Jordan and Einstein frames in 4d dilatonic gravity with quantum corrections are not equivalent. Different parametrizations lead to different results (parametrization choice dependence). The physical results are expecting to be the same only for S-matrix in full theory (non-perturbative regime). Inflationary dilatonic brane-world Universe in Einstein frame ============================================================= In this section we present the review of quantum induced dilatonic brane-worlds found in ref.[@NOO]. The model is discussed in Einstein frame and using euclidean notations. This scenario represents the extension to non-constant dilaton case the earlier scenario of refs.[@NOZ; @HHR] where quantum brane-worlds were realized in frames of AdS/CFT correspondence, by adding quantum CFT on the brane to effective action. We start with Euclidean signature action $S$ which is the sum of the Einstein-Hilbert action $\SEH$ including dilaton $\phi$ with potential $V(\phi)={12 \over l^2}+\Phi(\phi)$, the Gibbons-Hawking surface term $\SGH$, the surface counter term $S_1$[^5] \[Stotal\] S&=&+ + 2 S\_1 ,\ \[SEHi\] &=&[1 16G]{}d\^5 x (R\_[(5)]{} -[1 2]{}\_\^ + [12 l\^2]{}+()),\ \[GHi\] &=&[1 8G]{}d\^4 x \_n\^,\ \[S1dil\] S\_1&=& -[1 16G]{}d\^4 ( [6 l]{} + [l 4]{}()) . Here the quantities in the 5 dimensional bulk spacetime are specified by the suffices $_{(5)}$ and those in the boundary 4 dimensional spacetime are specified by $_{(4)}$. The factor $2$ in front of $S_1$ in (\[Stotal\]) is coming from that we have two bulk regions which are connected with each other by the brane. It is clear that above representation corresponds to Einstein frame. In (\[GHi\]), $n^\mu$ is the unit vector normal to the boundary. Bulk solutions -------------- In this subsection, we find some explicit solutions in the bulk space. We now assume the metric in the following form \[DP1\] ds\^2=f(y)dy\^2 + y\_[i,j=0]{}\^3g\_[ij]{}(x\^k)dx\^i dx\^j, and $\phi$ depends only on $y$: $\phi=\phi(y)$. Here $\hat g_{ij}$ is the metric of the Einstein manifold, which is defined by $r_{ij}=k\hat g_{ij}$, where $r_{ij}$ is the Ricci tensor constructed with $\hat g_{ij}$ and $k$ is a constant. Then we obtain the following equations of motion in the bulk: \[DP2\] 0&=&[3 2y\^2]{} - [2kf y]{} - [1 4]{}( [ddy]{})\^2 - ([6 l\^2]{} + [1 2]{}())f,\ \[DP3\] 0&=&[d dy]{}([y\^2 ]{}[ddy]{}) + ’()y\^2 . It is convenient to introduce the new coordinate $z$ \[c2b\] z=dy . By solving $y$ with respect to $z$, we obtain the warp factor $l^2\e^{2\hat A(z,k)}=y(z)$. Here one assumes the metric of 5 dimensional space time as follows: \[metric1\] ds\^2=dz\^2 + \^[2A(z,)]{}g\_dx\^dx\^ , g\_dx\^dx\^l\^2(d \^2 + d\^2\_3) . where $d\Omega^2_3$ corresponds to the metric of 3 dimensional unit sphere. Suppose that $A(z,\sigma)$ can be decomposed into the sum of $z$-dependent part $\hat A(z)$ and $\sigma$-dependent part and therefore $l^2\e^{2\hat A(z)} \hat g_{\mu\nu}=\e^{2 A(z,\sigma)}\tilde g_{\mu\nu}$. Then for the unit sphere ($k=3$) \[smetric\] A(z,)=A(z,k=3) - , for the flat Euclidean space ($k=0$) \[emetric\] A(z,)=A(z,k=0) + , and for the unit hyperboloid ($k=-3$) \[hmetric\] A(z,)=A(z,k=-3) - . When $\Phi(\phi)=0$, there exists the following AdS-like solution of the equations of motion [@NOtwo] \[curv2\] ds\^2&=&f(y)dy\^2 + y\_[i,j=0]{}\^[d-1]{}g\_[ij]{}(x\^k)dx\^i dx\^j f&=&[d(d-1) 4y\^2 \^2 (1 + [ c\^2 2\^2 y\^d]{} + [kd \^2 y]{})]{} &=&cdy . Here $\lambda^2={12 \over l^2}$. When $\Phi(\phi)\neq 0$, by using (\[DP2\]) and (\[DP3\]), one can delete $f$ from the equations and we obtain an equation that contains only the dilaton field $\phi$: \[DP4\] 0&=&{ [5k 2]{} - [k 4]{}y\^2 ([ddy]{})\^2 + ([3 2]{}y - [y\^3 6]{}([ddy]{})\^2 ) ([6 l\^2]{} + [1 2]{}())} [ddy]{} && + [y\^2 2]{}([2k y]{} + [6 l\^2]{} + [1 2]{}())[d\^2dy\^2]{} + ([3 4]{} - [y\^2 8]{} ([ddy]{})\^2 )’() . We now consider a solvable case where \[ts1\] [6 l\^2]{} + [1 2]{}()= - [2k y]{} . The explicit form, or $\phi$ dependence, of $\Phi(\phi)$ can be determined after solving the equations of motion as the following \[ts4\] = (m\^2 y) . Here $m^2$ is a constant of the integration. The explicit form of $\Phi(\phi)$ is: \[ts5\] ()= - [12 l\^2]{} - 4km\^2\^ . One can also find that Eq.(\[DP2\]) is trivially satisfied. Integrating (\[DP3\]), we obtain \[ts6\] f=[1 -[2ky 9]{} + [f\_0 y\^2]{}]{} . Here $f_0$ is a constant of the integration and $f_0$ should be positive in order that $f$ is positive for large $y$. There is a (curvature) singularity at $y=0$. One should also note that when $k>0$, the horizon appears at \[ts7\] y\^3 = y\_0\^3 and we find \[ts7b\] yy\_0 . Brane solutions --------------- In this subsection, we investigate if there is a solution with brane including the quantum correction from $N$ massless brane Majorana spinors coupled with the dilaton. For simplicity, only the case that the potential is constant. On the brane, one obtains the following equations by the variations over $A$ and $\phi$: \[eq2b\] 0&=&[48 l\^4 16G]{}(\_z A - [1 l]{} - [l 24]{}())\^[4A]{} ,\ \[eq2pb\] 0&=&-[l\^4 8G]{}\^[4A]{}\_z -[l\^5 32G]{}\^[4A]{}’() . With (\[ts5\]) and the solution (\[ts6\]), these equations look \[tsc1\] 0&=&[1 2R\^2]{} - [1 2l]{} + [kl 3R\^2]{},\ \[tsc2\] 0&=& + [kl kl]{} . Here we assume that the brane lies at $y=y_0$ or $z=z_0$. The radius $R$ of the brane is defined by $R=\e^{\hat A(z_0)}$. Eq.(\[tsc2\]) tells that $k\leq 0$ but by combining (\[tsc1\]) and (\[tsc2\]), we find $R^2={kl^2 \over 2}$. Then there is no consistent classical solution. We now consider the case that the matter on the brane is some QFT like QED or QCD. Of course, such a theory is classically conformally invariant one. As an explicit example in order to be able to apply large $N$-expansion we suppose that dominant contribution is due to $N$ massless Majorana spinors coupled with the dilaton, whose action is given by \[SP1\] S= \^[a]{}\_[i=1]{}\^N\|\_i \^D\_\_i . The case of minimal spinor coupling corresponds to the choice $a=0$. Note that from Brans-Dicke theory consideration one knows that for Einstein frame the non-minimal dilaton coupling with the matter is the typical case. Then the trace anomaly induced action $W$ has the following form [@NO]: \[W2\] W&=& b d\^4x F A\_1 && + b’ d\^4x{A\_1 A\_1 . && . + (G - [2 3]{}R )A\_1 }\ && -[1 12]{}{b”+ [2 3]{}(b + b’)} d\^4x \^2 .Here \[SP2\] A\_1=A+[a3]{}, and \[SP3\] b=[3N 60(4)\^2]{} ,b’=-[11 N 360 (4)\^2]{} . We also choose $b''=0$ as it may be changed by finite renormalization of classical gravitational action. In (\[W2\]), one chooses the 4 dimensional boundary metric as \[tildeg\] \_=\^[2A]{}g\_, and we specify the quantities given by $\tilde g_{\mu\nu}$ by using $\tilde{\ }$. $G$ ($\tilde G$) and $F$ ($\tilde F$) are the Gauss-Bonnet invariant and the square of the Weyl tensor, which are given as \[GF\] G&=&R\^2 -4 R\_[ij]{}R\^[ij]{} + R\_[ijkl]{}R\^[ijkl]{}, F&=&[1 3]{}R\^2 -2 R\_[ij]{}R\^[ij]{} + R\_[ijkl]{}R\^[ijkl]{} , For simplicity, we consider a constant potential ($\Phi(\phi)=0$) case. Then brane equations are \[eq2c\] 0&=&[48 l\^4 16G]{}(\_z A - [1 l]{} )\^[4A]{} +b’(4\_\^4 A\_1 - 16 \_\^2 A\_1 ) && - 4(b+b’)(\_\^4 A\_1 + 2 \_\^2 A\_1 - 6 (\_A\_1)\^2\_\^2 A\_1 ),\ \[eq2pc\] 0&=&-[l\^4 8G]{}\^[4A]{}\_z+ [4 3]{}ab’ (4\_\^4 A\_1 - 16 \_\^2 A\_1) . Then one gets \[SP4\] 0&=&[1 G l]{}{ -1 }R\^4 + 8 b’,\ \[SP5\] 0&=& - [c 8G]{} + 32 ab’ . Note that for minimal spinor coupling the second equation does not have a solution. Eq.(\[SP5\]) can be solved with respect to $c$: \[SP5b\] c=328G a b’, but the boundary value $\phi_0$ of $\phi$ becomes a free parameter. We should also note that in the classical case that $b'=0$, there is no solution for (\[SP4\]) and (\[SP5\]). From Eq.(\[SP5\]), we find $c=0$ if $b'=0$. Then if we put $c=0$ and $b'=0$ in (\[SP4\]), there is no solution. When the dilaton vanishes ($c=0$) and the brane is the unit sphere ($k=3$), the equation (\[SP4\]) reproduces the result of ref.[@HHR] for ${\cal N}=4$ $SU(N)$ super Yang-Mills theory in case of the large $N$ limit where $b'$ is replaced by $-{N^2 \over 4(4\pi )^2}$: \[slbr3\] [R\^3 l\^3]{}=[R\^4 l\^4]{} + [GN\^2 8l\^3]{} . Let us define a function $F(R, c)$ as \[FRc\] F(R,c)( -1 )R\^4 , which appears in the r.h.s. in (\[SP4\]). For the $k>0$ case, $F(R,c)$ has a minimum at $R=R_0$, where $R_0$ is defined by \[min\] 0=[8kl\^2 3R\_0\^2]{} + [k\^2 l\^4 R\_0\^4]{} - [2l\^2 c\^2 3 R\_0\^8]{} . When $k>0$, there is only one solution for $R_0$. Therefore $F(R,c)$ in the case of $k>0$ (sphere case) is a monotonically increasing function of $R$ when $R>R_0$ and a decreasing function when $R<R_0$. Since $F(R,c)$ is clearly a monotonically increasing function of $c$, we find for $k>0$ and $b'<0$ case that $R$ decreases when $c$ increases if $R>R_0$, that is, the non-trivial dilaton makes the radius smaller. We can also find that there is no solution for $R$ in (\[SP4\]) for very large $\|c\|$. We can consider the $k<0$ case. When $c=0$, there is no solution for $R$ in (\[SP4\]). We can find, however, there is a solution if $\|c\|$ is large enough: \[G8\] [\|c\| G ]{} > -8b’ . Hence, for constant bulk potential there is the possibility of quantum creation of a 4d de Sitter or a 4d hyperbolic brane living in 5d AdS bulk space. This occurs even for not exactly conformal invariant quantum brane matter. This finishes our consideration of quantum induced dilatonic brane-worlds in Einstein frame. Quantum induced dilatonic brane-worlds in string frame. ======================================================= We now transform the brane-world action in the Einstein frame (see (\[Stotal\])) into the Jordan frame. If we consider the scale transformation \[sc1\] g\_\^g\_ , with the choice \[sc3\] \^[([D2]{} -1)]{}= , () , we find that the actions (\[SEHi\]), (\[GHi\]) and (\[S1dil\]) are transformed as \[SEHiJ\] &=&[1 16G]{}d\^5 x ( R\_[(5)]{} + [43]{}\_\^ -[2]{}\_\^.&& . + ([12 l\^2]{}+())( )\^[5 3]{}),\ \[GHiJ\] &=&[1 8G]{}d\^4 x \_n\^ ,\ \[S1dilJ\] S\_1&=& -[1 16G]{}d\^4x ()\^[4 3]{}( [6 l]{} + [l 4]{}()) . Bulk solution in the string frame --------------------------------- In the bulk, the variation over $\phi$ gives the following equation of motion: \[sc7\] 0&=& R\_[(5)]{} - [43\^2]{}\_\^- [2]{}\_\^+ [5 3]{}([12 l\^2]{} + () )\^[5 3]{} \^[2 3]{} && + ’()()\^[5 3]{} - [83]{}\_([1 ]{} \^) + \_( \^) . On the other hand, the variation over the metric $g^{\mu\nu}$ gives \[sc8\] 0&=& - [1 2]{}(R\_[(5)]{} + [43]{}\_\^ - [2]{}\_\^ + ([12 l\^2]{} + ()) ()\^[5 3]{} )g\_[(5)]{} && + R\_[(5)]{} - \_\_ + g\_[(5)]{}&& + [43]{}\_\_ - [2]{}\_\_ . Thus, one gets the bulk equations of motion in string frame. Using (\[sc8\]), we have \[sc9\] 0&=&-[3 2]{}(R\_[(5)]{} + [43]{}\_\^ - [2]{}\_\^ ) && - [5 2]{}([12 l\^2]{} + ()) ()\^[5 3]{} + 4 . Substituting (\[sc9\]) into (\[sc7\]) and (\[sc8\]), one obtains \[sc10\] 0&=&\_(\^) + ’()()\^[5 3]{}\ \[sc11\] 0&=&-\_\_ - [3]{}g\_[(5)]{} + R\_[(5)]{} && + [1 3]{}([12 l\^2]{} + ()) ()\^[5 3]{} g\_[(5)]{} + [43]{}\_\_ -[2]{}\_\_ . First, let us consider $\Phi(\phi)=0$ case. In the Einstein frame, the solution is given by (\[curv2\]). The metric $g_{(5)\mu\nu}^{\rm J}$ in the Jordan frame is obtained with the help of (\[sc1\]) and (\[sc3\]), or more explicitly \[curv3\] g\_[(5)]{}\^[J]{}dx\^dx\^&=& ()\^[-[2 3]{}]{}( f(y)dy\^2 + y\_[i,j=0]{}\^[3]{}g\_[ij]{}(x\^k)dx\^i dx\^j) f&=&[l\^2 4y\^2 (1 + [ c\^2l\^2 24 y\^4]{} + [kl\^2 3 y]{})]{} &=&cdy [l 2]{} . One can check directly that the metric (\[curv3\]) satisfies Eqs.(\[sc10\]) and (\[sc11\]). Although the classical bulk solution in the Einstein frame is equivalent to the one in the Jordan frame, the physical interpretation of the spacetime is changed due to the factor of $\left(\alpha\phi \right)^{-{2 \over 3}}$. Since the transformation is conformal, the causal structure of the spacetime is not changed, especially the situation that there is a curvature singularity at $y=0$ is not changed. When $y\rightarrow \infty$, however, the spacetime is not asymptotically AdS but the metric behaves as \[sc12\] g\_[(5)]{}\^[J]{}dx\^dx\^\~ (-[cl 4]{})\^[-[2 3]{}]{}( [l\^2 4 y\^[2 3]{}]{}dy\^2 + y\^[7 3]{} \_[i,j=0]{}\^[3]{}g\_[ij]{}(x\^k)dx\^i dx\^j) . If one defines a coordinate $z$ by \[sc13\] z(-[cl 4]{})\^[-[1 3]{}]{} [3l 4]{}y\^[2 3]{} , the metric in (\[sc12\]) is rewritten by \[sc14\] g\_[(5)]{}\^[J]{}dx\^dx\^\~ dz\^2 + (-[cl 4]{})\^[[1 2]{}]{} (4z 3l)\^[7 2]{} \_[i,j=0]{}\^[3]{}g\_[ij]{}(x\^k)dx\^i dx\^j . Then the warp factor behaves as the power of $z$, instead of the exponential function in Einstein frame. One can also consider the case that the dilaton potential ${12 \over l^2} + \Phi(\phi)$ is given by (\[ts5\]). Using the relation (\[sc1\]) and (\[sc3\]) between the Einstein frame and the Jordan frame, from (\[ts4\]) and (\[ts6\]), we find the following solution: \[sc15\] g\_[(5)]{}\^[J]{}dx\^dx\^&=& ((m\^2 y))\^[-[2 3]{}]{} ([1 -[2ky 9]{} + [f\_0 y\^2]{}]{} dy\^2 + y\_[i,j=0]{}\^[3]{}g\_[ij]{}(x\^k)dx\^i dx\^j) &=& (m\^2 y) . One can again check that the above solution satisfies Eqs.(\[sc10\]) and (\[sc11\]). Then the above result is equivalent with that in the Einstein frame. Comparing the obtained metric with that in the Einstein frame in (\[ts4\]) and (\[ts6\]), there appears the factor of the logarithmic function of $y$, coming from the conformal transformation. In other words, the interpretation of lenghts in both frames is different while solutions are equivalent. Brane solutions in the string frame ----------------------------------- Having proof of explicit equivalency of bulk solutions, one can analyze the brane. From the actions in (\[SEHiJ\]), (\[GHiJ\]) and (\[S1dilJ\]), the variation over $\phi$ gives the following equation on the boundary \[Jb1\] 0&=&[l\^4\^[4A]{} 8G]{}{([83\_0]{} - \_0)\_z+ 8\_z A .&& . - [43]{} ([6 l]{} + [l 4]{}()) (\_0)\^[1 3]{} - [l 4]{}’()()\^[4 3]{} } . Here we choose the metric as in (\[metric1\]) and $\phi_0$ is the value of $\phi$ on the boundary. The variation over $A$ gives the following equation \[Jb2\] 0=[48 l\^4 16G]{}\^[4A]{}(\_0\_z A + [3]{}\_z- [1 6]{} ([6 l]{} + [l 4]{}()) (\_0)\^[4 3]{}) . The coordinate $z$ and $A$ in the warp factor are related with those in the Einstein frame, $z_E$ and $A_E$ by \[Jb3\] dz\_E=()\^[1 3]{}dz ,A\_E=A+[1 3]{}() . Then Eqs.(\[Jb1\]) and (\[Jb2\]) are rewritten as \[Jb4\] 0&=&[l\^4\^[4A\_E]{} 8G]{}{-\_[z\_E]{} + (\_0)( 8\_[z\_E]{} A\_E - [43]{} ([6 l]{} + [l 4]{}()) ..&& .. - [l 4]{}’()) }\ \[Jb5\] 0&=& [48 l\^4 16G]{}\^[4A\_E]{}{\_[z\_E]{}A\_E - [1 6]{} ([6 l]{} + [l 4]{}())} . Combining (\[Jb4\]) and (\[Jb5\]), we obtain \[Jb6\] 0=[l\^4\^[4A\_E]{} 8G]{}{-\_[z\_E]{} - [l 4]{}’() } . The obtained equations (\[Jb5\]) and (\[Jb6\]) are identical with the corresponding equations (\[eq2b\]) and (\[eq2pb\]) without the quantum correction, respectively. Choosing the metric of 5 dimensional space-time as in (\[metric1\]): \[metric1b\] ds\^2=dz\^2 + \^[2A(z,)]{}g\_dx\^dx\^ , g\_dx\^dx\^l\^2(d \^2 + d\^2\_3) , where $d\Omega^2_3$ corresponds to the metric of 3 dimensional unit sphere, we now include the quantum correction as in (\[W2\]): \[W2b\] W&=& b d\^4x F A && + b’ d\^4x{A A . && . + (G - [2 3]{}R )A }\ && -[1 12]{}{b”+ [2 3]{}(b + b’)} d\^4x \^2 .Note that as typically in Jordan frame there is no non-minimal dilaton coupling with matter we took minimal spinors, i.e. $a=0$. Then one obtains the following brane equations (instead of (\[Jb1\]) and (\[Jb2\])): \[Jb7\] 0&=&[l\^4\^[4A]{} 8G]{}{([83\_0]{} - \_0)\_z+ 8\_z A .&& . - [43]{} ([6 l]{} + [l 4]{}()) (\_0)\^[1 3]{} - [l 4]{}’()()\^[4 3]{} } && +b’(4\_\^4 A - 16 \_\^2 A ) && - 4(b+b’)(\_\^4 A + 2 \_\^2 A - 6 (\_A)\^2\_\^2 A ),\ \[Jb8\] 0&=&[48 l\^4 16G]{}\^[4A]{}(\_0\_z A + [3]{}\_z- [1 6]{} ([6 l]{} + [l 4]{}()) (\_0)\^[4 3]{}) && + [4 3]{}ab’ (4\_\^4 A - 16 \_\^2 A) . For $\Phi(\phi)=0$ case, substituting the solution in (\[curv3\]), one finds \[Jb9\] 0&=&[1 G l]{}{ -1 }(\_0)\^[4 3]{}R\^4 && + 8 b’,\ \[Jb10\] 0&=& - [c 8G]{} && + [1 G l\_0]{}{ -1 }(\_0)\^[4 3]{}R\^4 . Combining (\[Jb9\]) and (\[Jb10\]), one gets \[Jb11\] 0= - [c 8G]{} - [8b’ \_0]{} . Eq.(\[Jb11\]) has non-trivial solution and can be solved with respect to $\phi_0$: \[Jb12\] \_0=-[64G b’ c]{} . In the classical case that $b'=0$, there is no solution for (\[Jb9\]). Let us define a function $F(R, c)$ as \[FRcb\] F(R,c){ -1 }(\_0)\^[4 3]{}R\^4 , It appears in the r.h.s. in (\[Jb9\]). For $k>0$ case, $F(R,c)$ has a minimum at $R=R_0$, where $R_0$ is defined by \[minb\] 0=[8kl\^2 3(\_0)\^[2 3]{}R\_0\^2]{} + [k\^2 l\^4 (\_0)\^[4 3]{}R\_0\^4]{} - [2l\^2 c\^2 3 (\_0)\^[8 3]{}R\_0\^8]{} . When $k>0$, there is only one solution for $R_0$. Therefore $F(R,c)$ in the case of $k>0$ (sphere case) is a monotonically increasing function of $R$ when $R>R_0$ and a decreasing function when $R<R_0$. Since $F(R,c)$ is clearly a monotonically increasing function of $c$, we find for $k>0$ and $b'<0$ case that $R$ decreases when $c$ increases if $R>R_0$, that is, the non-trivial dilaton makes the radius smaller. Since one finds \[F1\] F(R\_0,c)=[kl (\_0)\^[2 3]{}R\_0\^2 4G]{}, using (\[FRcb\]) and (\[minb\]), Eq.(\[Jb9\]) has a solution if \[F2\] [kl (\_0)\^[2 3]{}R\_0\^2 4G]{}-8b’ . That puts again some bounds to the dilaton value. When $\|c\|$ is small, using (\[minb\]), one obtains \[F3\] R\_0\^4\~[2c\^2(\_0)\^[-[4 3]{}]{} 3k\^2 l\^2]{} , F(R\_0,c)\~[1 4G]{}[\|c\| ]{} . Therefore Eq.(\[F2\]) is satisfied for small $\|c\|$. On the other hand, when $c$ is large, we get \[F4\] R\_0\^6\~[c\^2(\_0)\^[-[6 3]{}]{} 4k]{} , F(R\_0,c)\~[(k \|c\| )\^[2 3]{} 4\^[4 3]{}G]{} . Therefore Eq.(\[F2\]) is not always satisfied and we have no solution for $R$ in (\[SP4\]) for very large $\|c\|$. We now consider the $k<0$ case. When $c=0$, there is no solution for $R$ in (\[Jb9\]). Let us define another function $G(R,c)$ as follows: \[G1\] G(R,c)1 + [l\^2 c\^2 24 (\_0)\^[8 3]{}R\^8]{} + [kl\^2 3(\_0)\^[2 3]{}R\^2]{} . Since $G(R,c)$ appears in the root of $F(R,c)$ in (\[FRcb\]), $G(R,c)$ must be positive. Then since \[G2\] [G(R,c) R]{}=-[l\^2 c\^2 3(\_0)\^[8 3]{}R\^9]{} - [2kl\^2 3(\_0)\^[2 3]{}R\^3]{} , $G(R,c)$ has a minimum \[G3\] 1+[kl\^2 4]{}(-[2k c\^2]{})\^[1 3]{}, when \[G4\] R\^6 = -[c\^2(\_0)\^[-[6 3]{}]{} 2k]{} . Therefore if \[G5\] c\^2 , $F(R,c)$ is real for any positive value of $R$. Since \[G6\] F(0,c)=[\|c\| G ]{}, and when $R\rightarrow \infty$ \[G7\] F(R,c)<0 , there is a solution $R$ in (\[Jb9\]) if \[G8b\] [\|c\| G ]{} > -8b’ . This is the same bound as in Einstein frame (previous section). Thus we demonstrated the complete equivalency of quantum induced inflationary (hyperbolic) dilatonic brane-worlds in Einstein and string (Jordan) frames. Note that Eq.(\[Jb9\]) is identical with the corresponding equation (\[SP4\]) in the Einstein frame if we regard $\left(\alpha\phi_0\right)^{1 \over 3}R$ as the radius $R_E$ in the Einstein frame: \[Jb13\] R=(\_0)\^[-[1 3]{}]{}R\_E . Then the solution has properties similar to those in the Einstein frame. Since $b'$ is order $N$ quantity from (\[SP3\]), Eq.(\[Jb12\]) and (\[Jb13\]) might tell that the radius $R$ in the Jordan frame is much smaller than the radius $R_E$ in the Einstein frame if $N$ is large. In case that the brane is sphere, the brane becomes de Sitter space. Since the rate of the expansion is given by ${1 \over R}$ in de Sitter space, the rate might become much larger if compare with that in the Einstein frame when $N$ is large. Thus, even having formal equivalency, the physical interpretation of results obtained in Jordan and Einstein frames may be different. Brane-world black holes in string and Einstein frames ===================================================== In analogy with Randall-Sundrum model [@RS], we now consider the following classical action of the gravity coupled with dilaton $\phi$ in the Einstein frame with Lorentzian signature: \[S\] S&=& [1 16G]{} . Here $B_{\rm hid}$ and $B_{\rm vis}$ are branes corresponding to hidden and visible sectors respectively and $U_i(\phi)$ corresponds to the vacuum energies on the branes in [@RS]. One assumes $U(\phi)$ is dilaton dependent and its form is explicitly given later on from the consistency of the equations of motion. The dilaton potential $V(\phi)$ is often given in terms of the superpotential $W(\phi)$ : \[Vi\] V=([W ]{})\^2 - [4 6 ]{} W\^2 . We assume again $\phi$ only depends on $z$ and the metric has the following form: \[Mi2\] ds\^2=dz\^2 + \^[2A(z)]{}g\_[ij]{}dx\^i dx\^j . Here $\tilde g_{ij}$ is the metric of the Einstein manifold. We also suppose the hidden and visible branes sit on $z=z_{\rm hid}$ and $z=z_{\rm vis}$, respectively. Then the equations of motion are given by \[Ei\] && ”+ 4A’’ = [V ]{} + \_[i=[hid]{},[vis]{}]{} [U\_i() ]{} (z-z\_i) ,\ \[Eii\] && 4A”+ 4(A’)\^2 + [1 2]{}(’)\^2 && = - [1 3]{}V() - [2 3]{}\_[i=[hid]{},[vis]{}]{} U\_i()(z-z\_i) ,\ \[Eiiib\] && A” + 4 (A’)\^2 = k\^[-2A]{} - [1 3]{}V() - [1 6]{}\_[i=[hid]{},[vis]{}]{} U\_i()(z-z\_i) . Here $'\equiv {d \over dz}$. Especially when $k=0$, Eqs. (\[Ei\]-\[Eiiib\]) have the following first integrals in the bulk: \[Iii\] ’= , A’ = - [1 3]{}W . Near the branes, Eqs. (\[Ei\]-\[Eiiib\]) have the following form : \[Eiv\] ” \~[U\_i()]{}(z-z\_i) , A” \~-[U\_i() 6]{}(z-z\_i) , or \[Eivb\] 2’ \~[U\_i()]{} , 2A’ \~-[U\_i() 6]{} , at $z=z_i$. Comparing (\[Eivb\]) with (\[Iii\]), we find \[Ev\] U\_[hid]{}()= 2W() ,U\_[vis]{}()=- 2W() . We should note that $k=0$ does not always mean the brane is flat. As well-known, the Einstein equations are given by, \[A1\] R\_-[1 2]{}g\_R+[1 2]{}g\_ = T\^[matter]{}\_ . Here $T^{\rm matter}_{\mu\nu}$ is the energy-momentum tensor of the matter fields. If we consider the vacuum solution where $T^{\rm matter}_{\mu\nu}=0$, Eq.(\[A1\]) can be rewritten as \[A2\] R\_=[2]{}g\_ . If we put $\Lambda=2k$, Eq.(\[A2\]) is nothing but the equation for the Einstein manifold. The Einstein manifolds are not always homogeneous manifolds like flat Minkowski, (anti-)de Sitter space \[AdS\] ds\_4\^2= -V(r)dt\^2 + V\^[-1]{}(r)dr\^2+r\^2 d\^2, V(r) = 1 - r\^2, or Nariai space \[Nsol\] ds\_4\^2=[1 ]{}( \^2 d\^2 - d\^2 - d\^2) . but they can be some black hole solutions like Schwarzschild-(anti-)de Sitter black hole \[SAdS\] ds\_4\^2= -V(r )dt\^2 + V\^[-1]{}(r )dr\^2+r\^2 d\^2, V(r) = 1- [G\_4 M r]{} - r\^2 . As a special case, one can also consider $k=0$ solution like Schwarzschild black hole, \[schw\] ds\_4\^2g\_[ij]{}dx\^i dx\^j =-(1 - [G\_4 M r]{})dt\^2 +[dr\^2 (1 - [G\_4 M r]{})]{} + r\^2 d\^2 . In (\[SAdS\]) and (\[schw\]), $M$ is the mass of the black hole on the brane and the effective gravitational constant $G_4$ on the 3-brane (here $d=4$) is given by \[schw1\] [1 G\_4]{} =[1 G]{}\_[z\_[hid]{}]{}\^[z\_[vis]{}]{} dz \^[(d-2)A]{} . In these solutions, the curvature singularity at $r=0$ has a form of line penetrating the bulk 5d universe and the horizon makes a tube surrounding the singularity. The singularity and the horizon connect the hidden and visible branes. These black holes have been discussed in ref.[@NOOO]. We now consider the Jordan frame, in order to see if singularity supports (or breaks) the equivalency on classical level. Using scale transformation given by (\[sc1\]) and (\[sc3\]) with $D=5$, the action (\[S\]) is rewritten as \[S2\] S&=&[1 16G]{}d\^5 x ( R\_[(5)]{} + [43]{}\_\^ -[2]{}\_\^.&& . - V()()\^[5 3]{}) && . - \_[i=[hid]{},[vis]{}]{} \_[B\_i]{} d\^4 x ()\^[4 3]{}U\_i()\] . Then if we choose the metric as in (\[Mi2\]) in the Jordan frame and $\phi$ only depends on $z$ again, we obtain the following equations instead of (\[Ei\]), (\[Eii\]) and (\[Eiiib\]), \[EiJ\] && (”+ 4A’’ + (’)\^2) && = [V ]{} ()\^[5 3]{} + \_[i=[hid]{},[vis]{}]{} [U\_i() ]{} ()\^[4 3]{} (z-z\_i) ,\ \[EiiJ\] && (4A”+ 4(A’)\^2) + [2]{}(’)\^2 - [43]{}(’)\^2 + [43]{}(”+ A’’) && = - [1 3]{}V()()\^[5 3]{} - [2 3]{}\_[i=[hid]{},[vis]{}]{} U\_i()()\^[4 3]{}(z-z\_i) ,\ \[EiiibJ\] && (A” + 4 (A’)\^2) + [3]{}(” + 7A’’) && = k\^[-2A]{} - [1 3]{}V()()\^[5 3]{} - [1 6]{}\_[i=[hid]{},[vis]{}]{} U\_i()()\^[4 3]{} (z-z\_i) . If one transforms the above equations to those in the Einstein frame by changing \[Tr\] A&& A - [1 3]{}() dz&& ()\^[-[1 3]{}]{}dz && ( [rcl]{} ’\_z && ()\^[1 3]{}\_z\ ”=\_z\^2 && ()\^[2 3]{}(\_z\^2 + [\_z 3]{}\_z) ) , then Eqs.(\[Ei\]), (\[Eii\]) and (\[Eiiib\]), which are the corresponding equations in the Einstein frame, are reproduced. Thus we can confirm the equivalence between the Jordan frame and the Einstein frame description of dilatonic brane-world black holes on the classical level. Their physical interpretation may be again different. Discussion ========== In summary, we discussed AdS/CFT induced quantum dilatonic brane-worlds where branes may be flat, de Sitter (inflationary) or Anti-de Sitter Universe. Actually, such objects appear in frames of AdS/CFT correspondence [@AdS] as warped compactification of relevant holographic RG flow [@NOZ; @HHR]. The role of free parameter (brane tension) is played by effective brane tension produced by conformal anomaly of QFT sitting on the brane. Hence, only brane quantum effects are considered. We compared the construction of such quantum dilatonic brane-worlds in two frames: string and Einstein one. The very nice feature of brane-worlds is discovered: in all examples under consideration the string and Einstein frames are eqiuvalent! This holds to be true also for the number of classical dilatonic brane-world black holes. This is completely different from the case of quantum corrected 4d dilatonic gravity (section 2) where de Sitter Universe with decaying dilaton exists in Einstein frame but does not exist in Jordan frame. Quantum effects may be useful in other aspects of brane-worlds. In particulary, for flat branes the bulk quantum effects (Casimir force) may be estimated [@GPT; @NOZ1; @HKP] and used for radion stabilization. Unfortunately, in usual Randall-Sundrum Universe such quantum effects are actually supporting the radion destabilization. Nevertheless, in the case of thermal Randall-Sundrum scenario [@BMNO] such quantum effects may not only stabilize the radion but also may provide the necessary mass hierarchy [@BMNO] (at least, for some temperatures). It would be extremely interesting to estimate the bulk quantum effects for dilatonic backgrounds and to understand their role (as well as frame dependence of such Casimir effect) in the creation of dilatonic brane-worlds. Another interesting line of research is related with account of quantum effects on graviton perturbations around the brane. As is demonstrated in previous section, they may modify the massive graviton modes around hyperbolic brane. Clearly, in other regimes for quantum induced dilatonic (asymptotically) AdS brane more complicated dynamics may be expected. Acknowledgements {#acknowledgements .unnumbered} ================ We thank J. Socorro for participation at the early stage of this work. SDO is grateful to L. Randall for useful discussion. The work by O.O., S.D.O. and V.I.T. has been supported in part by CONACyT grant 28454E and that by S.D.O. in part by CONACyT(CP, ref.990356). Remarks on gravitational perturbations around hyperbolic brane ============================================================== In [@KR; @KMP], the AdS$_4$ branes in AdS$_5$ were discussed and the existence of the massive normalizable mode of graviton was found. In these papers, the tensions of the branes are free parameters but in the case treated in the present paper, the tension is dynamically determined. Let us study the role of dynamically generated tension in getting of massive graviton modes. Moreover, we consider dilatonic brane-world. We now regard the brane as an object with a tension $U(\phi)$ and assume the brane can be effectively described by the folowing action: \[bten1\] S\_[brane]{}=- [1 16G]{}d\^4x U() . If one assumes the metric in the form of (\[metric1\]), then using the Einstein equation, we find \[bten2\] \_z\^2 A + 4(\_z A)\^2 =k\^[-2A]{} + [4 l\^2]{} + [() 3]{} - [U() 6]{}(z-z\_0) . Then at $z=z_0$, \[bten3\] . \_z A \|\_[z=z\_0]{}=-[U() 12]{} . For simplicity, we consider the case of the constant dilaton potential $\Phi(\phi)=0$. Comparing (\[bten3\]) with (\[eq2c\]) and (\[SP4\]), one gets \[bten4\] U() = - [12 l]{} + [96G b’ R\^4]{} . We should note that the tension becomes $R$ dependent due to the quantum correction. In case of AdS brane $k<0$, if no dilaton is included, the boundary equation (\[SP4\]) does not have any solution for $R$. When there is non-trivial dilaton and the parameter $c$ is large enough, Eq.(\[SP4\]) has a solution. If $c$ is very large \[lR\] R\^4 \~[c G]{} + 8b’ . We now consider the perturbation by assuming the metric in the following form: \[bten5\] ds\^2=\^[2A(]{}(d\^2 + (g\_ + \^[-[3 2]{}A()]{}h\_) dx\^dx\^) . By choosing the gauge conditions $h^\mu_{\ \mu}=0$ and $\nabla^\mu h_{\mu\nu}=0$, one obtains the following equation \[bten7\] (-\_\^2 + [94]{} (\_A )\^2 + [3 2]{}\_\^2 A )h\_ = m\^2 h\_ Here $m^2$ corresponds to the mass of the graviton on the brane \[bten8\] ()h\_ = m\^2 h\_ . Here $\hat\Box$ is 4- dimensional d’Alembertian constructed on $\hat g_{\mu\nu}$ and the $+$ ($-$) sign corresponds to (anti-)de Sitter brane. Since $-\e^{A}d\zeta = dz = \sqrt{f}dy$ and $\e^A={\sqrt{y} \over l}$, we find, especially for the case of the constant dilaton potential, \[bten10\] =-dy = - [l\^2 2]{} . We now consider the case that $c$ is very large, then \[bten11\] f(y)\~[6y\^2 c\^2]{} . Since $y_0=R^2$ if there is a brane at $y=y_0$, Eq.(\[lR\]) can be rewritten as \[lR2\] y\_0\^2 \~[c G]{} + 8b’ . If we choose $\zeta=0$ when $y=y_0$, Eqs.(\[bten10\]) and (\[bten11\]) give \[bten12\] \|\|=-[1 \|c\|]{}y\^[3 2]{} + \_0 , \_0y\_0\^[3 2]{} >0 . Note that the brane separates two bulk regions corresponding to $\zeta<0$ and $\zeta>0$, respectively. Since $y$ takes the value in $[0,y_0]$, $\zeta$ takes the value in $[-\zeta_0,\zeta_0]$. Since $A={1 \over 2}\ln y$, from (\[bten7\]), one gets \[bten13\] (-\_\^2 - [1 4(\|\| -\_0)\^2]{} - [1 \_0]{}() )h\_ = m\^2 h\_ The zero mode solution with $m^2$ of (\[bten13\]) is given by \[bten14\] h\_= . The general solution of (\[bten13\] with $m^2\neq 0$ is given by the Bessel functions: \[bten15\] h\_=aJ\_0(m(\_0 - \|\|)) + b N\_0(m(\_0 - \|\|)) . The coefficients $a$ and $b$ are constants of the integration and they are determined to satisfy the boundary condition \[bten16\] .[\_h\_ h\_]{} \|\_[0+]{} =-[1 2\_0]{} . Note that zero mode solution (\[bten14\]) satisfies this boundary condition (\[bten16\]). If $b\neq 0$, the solution in (\[bten15\]) diverges at $\zeta=\pm\zeta_0$ and would not be normalizable. If $b=0$, the condition (\[bten16\]) reduces to \[bten17\] J\_1(m\_0)=0 , that is \[bten18\] m\_0=0,3.8317...,7.0155..., . The non-vanishing solutions for $m^2$ give the mass of the massive graviton modes. 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--- abstract: 'We propose an active 3D mapping method for depth sensors, which allow individual control of depth-measuring rays, such as the newly emerging solid-state lidars. The method simultaneously (i) learns to reconstruct a dense 3D occupancy map from sparse depth measurements, and (ii) optimizes the reactive control of depth-measuring rays. To make the first step towards the online control optimization, we propose a fast prioritized greedy algorithm, which needs to update its cost function in only a small fraction of possible rays. The approximation ratio of the greedy algorithm is derived. An experimental evaluation on the subset of the KITTI dataset demonstrates significant improvement in the 3D map accuracy when learning-to-reconstruct from sparse measurements is coupled with the optimization of depth measuring rays.' author: - \| Karel Zimmermann, Tom[á]{}[š]{} Pet[ř]{}[í]{}[č]{}ek, Vojt[ě]{}ch [Š]{}alansk[ý]{}, and Tom[á]{}[š]{} Svoboda\ Czech Technical University in Prague, Faculty of Electrical Engineering\ [zimmerk@fel.cvut.cz]{} bibliography: - 'paper.bib' title: Learning for Active 3D Mapping --- Introduction ============ In contrast to rotating lidars, the SSL uses an optical phased array as a transmitter of depth measuring light pulses. Since the built-in electronics can independently steer pulses of light by shifting its phase as it is projected through the array, the SSL can focus its attention on the parts of the scene important for the current task. Task-driven reactive control steering hundreds of thousands of rays per second using only an on-board computer is a challenging problem, which calls for highly efficient parallelizable algorithms. As a first step towards this goal, we propose an active mapping method for SSL-like sensors, which simultaneously (i) learns to reconstruct a dense 3D voxel-map from sparse depth measurements and (ii) optimize the reactive control of depth-measuring rays, see Figure \[fig:diagram\]. The proposed method is evaluated on a subset of the KITTI dataset [@Geiger-2013-IJRR], where sparse SSL measurements are artificially synthesized from captured lidar scans, and compared to a state-of-the-art 3D reconstruction approach [@Choy-2016-ECCV]. \[fig:diagram\] ![Active 3D mapping with Solid State Lidar. Iteratively learned deep convolutional network reconstructs local dense occupancy map from sparse depth measurements. The local map is registered to a global occupancy map, which in turn serves as an input for the optimization of depth-measuring rays along the expected vehicle trajectory. The dense occupancy maps are visualized as isosurfaces. See `https://www.youtube.com/watch?v=KNex0zjeGYE`](img/diagram){width="1\linewidth"} The main contribution of this paper lies in proposing a computationally tractable approach for very high-dimensional active perception task, which couples learning of the 3D reconstruction with the optimization of depth-measuring rays. Unlike other approaches such as active object detection [@Jayaraman-2016-ECCV] or segmentation [@Mishra-2012-TPAMI], SSL-like reactive control has significantly higher dimensionality of the state-action space, which makes a direct application of unsupervised reinforcement learning [@Jayaraman-2016-ECCV] prohibitively expensive. Keeping the on-board reactive control in mind, we propose prioritized greedy optimization of depth measuring rays, which in contrast to a na[" i]{}ve greedy algorithm re-evaluates only $1/500$ rays in each iteration. We derive the approximation ratio of the proposed algorithm. Active perception has been widely studied in many robotics applications ranging from exploration and active SLAM [@MartinezCantin-2009-AR] to active object detection [@Jayaraman-2016-ECCV] and segmentation [@Mishra-2012-TPAMI]. In contrast to these applications, the active SSL-mapping has significantly higher dimensionality of the action space, which makes a direct application of known approaches such as unsupervised reinforcement learning [@Jayaraman-2016-ECCV] prohibitively expensive. Unlike the active SLAM, we are mostly interested in a local 3D map, which allows for reactive control of the vehicle, rather than the global map which would require global motion rectifications such as loop closures. In such a short horizon, the motion estimated from odometry, IMU and GPS is sufficiently accurate, therefore localization is not an issue. In contrast to the discriminative voxel reconstruction approaches, the map being reconstructed has also significantly higher dimensionality. Nevertheless we follow the main paradigm, which achieved state-of-the-art performance in the most of the active perception tasks: discriminative learning of the target perception task coupled with the active component. The 3D mapping is handled by an iteratively learned convolution neural network (CNN), as CNNs proved their superior performance in [@Choy-2016-ECCV; @Wu-2015-CVPR]. The iterative learning procedure stems from the fact that both (i) the directions in which the depth should be measured and (ii) the weights of the 3D reconstruction network are unknown. We initialize the learning procedure by selecting depth-measuring rays randomly to learn an initial 3D mapping network which estimates occupancy of each particular voxel. Then, using this network, depth-measuring rays along the expected vehicle trajectory can be planned based on the expected reconstruction (in)accuracy in each voxel. To reduce the training-planning discrepancy, the mapping network is re-learned on optimized sparse measurements and the whole process is iterated until validation error stops decreasing. Previous work ============= High performance of image-based models is demonstrated in [@Su-2015-ICCV], where a CNN pooling results from multiple rendered views outperforms commonly used 3D shape descriptors in object recognition task. Qi [@Qi-2016-CVPR] compare several volumetric and multi-view network architectures and propose an anisotropic probing kernel to close the performance gap between the two approaches. Our network architecture uses a similar design principle. Choy [@Choy-2016-ECCV] proposed a unified approach for single and multi-view 3D object reconstruction which employs a recurrent neural architecture. Despite providing competitive results in the object reconstruction domain, the architecture is not suitable for dealing with high-dimensional outputs due to its high memory requirements and would need significant modifications to train with full-resolution maps which we use. We provide a comparison of this method to ours in Sec. \[sec:choy\], in a limited setting. Model-fitting methods such as [@Shen-2012-TOG; @Sung-2015-TOG; @Rock-2015-CVPR] rely on a manually-annotated dataset of models and assume that objects can be decomposed into a predefined set of parts. Besides that these methods are suited mostly for man-made objects of rigid structure, fitting of the models and their parts to the input points is computationally very expensive; e.g., minutes per input for [@Shen-2012-TOG; @Sung-2015-TOG]. Decomposition of the scene into plane primitives as in [@Monszpart-2015-TOG] does not scale well with scene size (quadratically due to candidate pairs) and could not most likely deal with the level of sparsity we encounter. Geometrical and physical reasoning comprising stability of objects in the scene is used by Zheng [@Zheng-2013-CVPR] to improve object segmentation and 3D volumetric recovery. Their assumption of objects being aligned with coordinate axes which seems unrealistic in practice. Moreover, it is not clear how to incorporate learned shape priors for complex real-world objects which were shown to be beneficial for many tasks (e.g., in [@Nguyen-2016-CVPR]). Firman [@Firman-2016-CVPR] use a structured-output regression forest to complete unobserved geometry of tabletop-sized objects. A generative model proposed by Wu [@Wu-2015-CVPR], termed Deep Belief Network, learns joint probability distribution $p({\ensuremath{\mathrm{\mathbf{x}}}}, y)$ of complex 3D shapes ${\ensuremath{\mathrm{\mathbf{x}}}}$ across various object categories $y$. End-to-end learning of stochastic motion control policies for active object and scene categorization is proposed by Jayaraman and Grauman [@Jayaraman-2016-ECCV]. Their CNN policy successively proposes views to capture with RGB camera to minimize categorization error. The authors use a look-ahead error as an unsupervised regularizer on the classification objective. Andreopoulos [@Andreopoulos-TRO-2011] solve the problem of an active search for an object in a 3D environment. While they minimize the classification error of a single yet apriori unknown voxel containing the searched object, we minimize the expected reconstruction error of all voxels. Also, their action space is significantly smaller than ours because they consider only local viewpoint changes at the next position while the SSL planning chooses from tens of thousands of rays over a longer horizon. Overview of the active 3D mapping {#sec:overview} ================================= We assume that the vehicle follows a known path consisting of $L$ discrete positions and a depth measuring device (SSL) can capture at most $K$ rays at each position. The set of rays to be captured at position $l$ is denoted $J_l$. We denote ${\ensuremath{\mathrm{\mathbf{Y}}}}$ the global ground-truth occupancy map, $\hat{{\ensuremath{\mathrm{\mathbf{Y}}}}}$ its estimate, and ${\ensuremath{\mathrm{\mathbf{X}}}}$ the map of the sparse measurements. All these map share common global reference frame corresponding to the first position in the path. For each of these maps there are local counterparts ${\ensuremath{\mathrm{\mathbf{y}}}}_l, \hat{{\ensuremath{\mathrm{\mathbf{y}}}}}_l$, and ${\ensuremath{\mathrm{\mathbf{x}}}}_l$, respectively. Local maps corresponding to position $l$ all share a common reference frame which is aligned with the sensor and captures its local neighborhood. The global ground-truth map ${\ensuremath{\mathrm{\mathbf{Y}}}}$ is used to synthesize sensor measurements ${\ensuremath{\mathrm{\mathbf{x}}}}_l$ and to generate local ground-truth maps ${\ensuremath{\mathrm{\mathbf{y}}}}_l$ for training. The active mapping pipeline, consisting of a measure-reconstruct-plan loop, is depicted in Fig. \[fig:diagram\] and detailed in Alg. \[algo:pipeline\]. Initialize position $l \gets 0$ and select depth-measuring rays randomly. Measure depth in the directions selected for position $l$ and update global sparse measurements ${\ensuremath{\mathrm{\mathbf{X}}}}$ and dense reconstruction $\hat{{\ensuremath{\mathrm{\mathbf{Y}}}}}$ with these measurements. Obtain local measurements ${\ensuremath{\mathrm{\mathbf{x}}}}_l$ by interpolating ${\ensuremath{\mathrm{\mathbf{X}}}}$. Compute local occupancy confidence $\hat{{\ensuremath{\mathrm{\mathbf{y}}}}}_l = {\ensuremath{\mathrm{\mathbf{h}}}}_\theta({\ensuremath{\mathrm{\mathbf{x}}}}_l)$ using the mapping network ${\ensuremath{\mathrm{\mathbf{h}}}}_\theta$. Update global occupancy confidence $\hat{{\ensuremath{\mathrm{\mathbf{Y}}}}} \gets \hat{{\ensuremath{\mathrm{\mathbf{Y}}}}} + \hat{{\ensuremath{\mathrm{\mathbf{y}}}}}_l$. Plan depth-measuring rays along the expected vehicle trajectory over horizon $L$ given reconstruction $\hat{{\ensuremath{\mathrm{\mathbf{Y}}}}}$. Repeat from line 2 for next position $l \gets l + 1$. Neglecting sensor noise, the set of depth-measuring rays obtained from the planning, the measurements ${\ensuremath{\mathrm{\mathbf{x}}}}_l$, and the resulting reconstruction $\hat{{\ensuremath{\mathrm{\mathbf{Y}}}}}$ can all be seen as a deterministic function of mapping parameters $\theta$ and ${\ensuremath{\mathrm{\mathbf{Y}}}}$. If we assume that that ground-truth maps ${\ensuremath{\mathrm{\mathbf{Y}}}}$ come from a probability distribution, both learning of $\theta$ and planning of the depth-measuring rays approximately minimize common objective \_[$\mathrm{\mathbf{Y}}$]{}[$\{{\mathcal{L}}{\ensuremath{({\ensuremath{\mathrm{\mathbf{Y}}}}, \hat{{\ensuremath{\mathrm{\mathbf{Y}}}}}(\theta, {\ensuremath{\mathrm{\mathbf{Y}}}}))}}\}$]{},\[eq:common-obj\] where ${\mathcal{L}}({\ensuremath{\mathrm{\mathbf{Y}}}}, \hat{{\ensuremath{\mathrm{\mathbf{Y}}}}}) = \sum_i w_i \log(1 + \exp(-Y_i \hat{Y}_i))$ is the weighted logistic loss, $Y_i \in \{-1, 1\}$ and $\hat{Y}_i \in \mathbb{R}$ denote the elements of ${\ensuremath{\mathrm{\mathbf{Y}}}}$ and $\hat{{\ensuremath{\mathrm{\mathbf{Y}}}}}$, respectively, corresponding to voxel $i$. In learning, $w_i \ge 0$ are used to balance the two classes, empty with $Y_i=-1$ and occupied with $Y_i=1$, and to ignore the voxels with unknown occupancy. We assume independence of measurements and use, for corresponding voxels $i$, additive updates of the occupancy confidence $\hat{Y}_i \gets \hat{Y}_i + h_i({\ensuremath{\mathrm{\mathbf{x}}}}_l)$ with $h_i({\ensuremath{\mathrm{\mathbf{x}}}}_l) \approx \log(\mathrm{Pr}(Y_i = 1 \| {\ensuremath{\mathrm{\mathbf{x}}}}_l) / \mathrm{Pr}(Y_i = -1 \| {\ensuremath{\mathrm{\mathbf{x}}}}_l))$. $\mathrm{Pr}(Y_i = 1 \| {\ensuremath{\mathrm{\mathbf{x}}}}_l)$ denotes the conditional probability of voxel $i$ being occupied given measurements ${\ensuremath{\mathrm{\mathbf{x}}}}_l$ and $\sigma(\hat{Y}_i) = 1 / (1 + e^{-\hat{Y}_i})$ is its current estimate. We assume that the voxel $i$ is visible in ray $j$ which intersects sequence of voxels $R$. If all $i$th preceding voxels $R^-(i)$ are not occupied and the voxel itself or at least one of the voxels which follow $R^+(i)$ are occupied. Consequently, we estimate probability $p_{ij}$ of voxel $i$ not being visible in $j$ as $p_{ij} = 1 - \prod_{u\in R^-(i)}(1-q_{u})\left(1 - \prod_{u\in R^+(i)}(1-q_{u})\right).$ If ray $j$ does not intersect the voxel $i$, then $p_{ij} = 1$. Dense map reconstruction is tackled by the mapping network ${\ensuremath{\mathrm{\mathbf{h}}}}_\theta$. Result of the reconstruction $\hat{{\ensuremath{\mathrm{\mathbf{y}}}}}_l = {\ensuremath{\mathrm{\mathbf{h}}}}_\theta({\ensuremath{\mathrm{\mathbf{x}}}}_l)$ at position $l$ depends on the network parameters $\theta$ and sparse depth measurements ${\ensuremath{\mathrm{\mathbf{x}}}}_l$ determined by previously captured rays $J_1\dots J_l$. We define learning of $\theta$ as the minimization of the cross entropy loss $\sum_l\mathcal{H}\{\hat{{\ensuremath{\mathrm{\mathbf{y}}}}}_l, {\ensuremath{\mathrm{\mathbf{y}}}}_l\}$ between local reconstructions $\hat{{\ensuremath{\mathrm{\mathbf{y}}}}}_l$ from captured sparse measurement and the local ground truth maps ${\ensuremath{\mathrm{\mathbf{y}}}}_l$. The minimization is tackled by the Stochastic Gradient Descent detailed in Section \[sec:mapping\]. We denote one step of the SGD as follows: $$\theta^t = \texttt{SGD}(\theta^{t-1},J(\theta)) $$ where $J(\theta)$ denotes concatenation of all rays at all positions (in all training maps). The planning at position $l$ of following rays $J_{l+1}\dots J_{l+L}$ for the horizon $L$ is defined as the minimization of the expected cross-entropy loss $E_{{\ensuremath{\mathrm{\mathbf{Y}}}}\sim \hat{{\ensuremath{\mathrm{\mathbf{Y}}}}}_l}\{\mathcal{H}(\hat{{\ensuremath{\mathrm{\mathbf{Y}}}}}_{l+L}, {\ensuremath{\mathrm{\mathbf{Y}}}})\}$ subject to the limited ray budget $K$. Planning is tackled by proposed Prioritized Greedy algorithm detailed in Section \[sec:planning\]. We denote Prioritized Greedy planning of all rays $J$ at all positions (and maps) as follows: $$J^t = \texttt{PG}(\theta, J^{t-1})$$ Learning of 3D mapping network {#sec:mapping} ============================== The learning is defined as approximate minimization of Equation \[eq:common-obj\]. Since (i) the result of planning ${\ensuremath{\mathrm{\mathbf{x}}}}_l{\ensuremath{\left(\theta,{\ensuremath{\mathrm{\mathbf{Y}}}}\right)}}$ is not differentiable with respect to $\theta$ and (ii) we want to reduce variability of training data[^1], we locally approximate the criterion around a point $\theta^0$ as &\_[$\mathrm{\mathbf{Y}}$]{}{\_[l]{}([$\mathrm{\mathbf{y}}$]{}\_l,[$\mathrm{\mathbf{h}}$]{}\_([$\mathrm{\mathbf{x}}$]{}\_l(\^0,[$\mathrm{\mathbf{Y}}$]{})))} by fixing the result of planning in ${\ensuremath{\mathrm{\mathbf{x}}}}_l(\theta^0,{\ensuremath{\mathrm{\mathbf{Y}}}})$. We also introduce a canonical frame by using the local maps instead of the global ones, which helps the mapping network to capture local regularities. The learning then becomes the following iterative optimization \^t=\_\_[$\mathrm{\mathbf{Y}}$]{}[$\{\sum_l{\mathcal{L}}{\ensuremath{( {\ensuremath{\mathrm{\mathbf{y}}}}_l,{\ensuremath{\mathrm{\mathbf{h}}}}_\theta({\ensuremath{\mathrm{\mathbf{x}}}}_l(\theta^{t-1},{\ensuremath{\mathrm{\mathbf{Y}}}})) )}}\}$]{}, \[eq:opt-approx-common-obj\] Note, that in order to achieve (i) local optimality of the criterion and (ii) statistical consistency of the learning process (i.e. that the training distribution of sparse measurements ${\ensuremath{\mathrm{\mathbf{x}}}}_l$ corresponds to the one obtained by planning), one would have to find a fixed point of Equation \[eq:opt-approx-common-obj\]. Since there are no guarantees that any fixed point exists, we instead iterate the minimization until validation error is decreasing. where minimization in each iteration is tackled by Stochastic Gradient Descent. Learning is summarized in Alg. \[algo:learning\]. Initialize $t\gets 0$ and obtain dataset ${\ensuremath{D}}_0 = \{({\ensuremath{\mathrm{\mathbf{x}}}}_l, {\ensuremath{\mathrm{\mathbf{y}}}}_l)\}_l$ by running the pipeline with the rays being selected randomly, instead of using the planner. Train the mapping network on ${\ensuremath{D}}_t$ to obtain ${\ensuremath{\mathrm{\mathbf{h}}}}_{\theta^t}$ with parameters $\theta^t$. Obtain ${\ensuremath{D}}_{t+1} = \{({\ensuremath{\mathrm{\mathbf{x}}}}_l(\theta^t,{\ensuremath{\mathrm{\mathbf{Y}}}}), {\ensuremath{\mathrm{\mathbf{y}}}}_l)\}_l$ by running Alg. \[algo:pipeline\] and using ${\ensuremath{\mathrm{\mathbf{h}}}}_{\theta^t}$ for mapping. Set $t \gets t + 1$ and repeat from line 2 until validation error stops decreasing. Note, that in order to achieve (i) local optimality of the criterion and (ii) statistical consistency of the learning process (i.e., that the training distribution of sparse measurements ${\ensuremath{\mathrm{\mathbf{x}}}}_l$ corresponds to the one obtained by planning), one would have to find a fixed point of Equation \[eq:opt-approx-common-obj\]. Since there are no guarantees that any fixed point exists, we instead iterate the minimization until validation error is decreasing. Initial learning parameters $\theta^0$ are obtained by training the mapping network on dataset ${\ensuremath{D}}_0 = \{({\ensuremath{\mathrm{\mathbf{x}}}}_l, {\ensuremath{\mathrm{\mathbf{y}}}}_l)\}_l$, where sparse measurements ${\ensuremath{\mathrm{\mathbf{x}}}}_l$ are randomly generated from available depth-measuring rays. This network provides reconstructions of the global map for all positions $l$. Planning on these maps determines sparse depth measurements ${\ensuremath{\mathrm{\mathbf{x}}}}_l(\theta^0,{\ensuremath{\mathrm{\mathbf{Y}}}})$. In the following iterations $t\geq1$, the mapping network ${\ensuremath{\mathrm{\mathbf{h}}}}_{\theta^t}$ is always trained on dataset ${\ensuremath{D}}_t = \{({\ensuremath{\mathrm{\mathbf{x}}}}_l(\theta^0,{\ensuremath{\mathrm{\mathbf{Y}}}}), {\ensuremath{\mathrm{\mathbf{y}}}}_l)\}_l$. We seek a CNN which reconstructs local 3D occupancy map ${\ensuremath{\mathrm{\mathbf{y}}}}_l$ from sparse measurements ${\ensuremath{\mathrm{\mathbf{x}}}}_l$ selected by a planner minimizing the expected reconstruction error. Since sparse depth measurements are planned in the global map estimated by CNN and, on the other side, the CNN estimates the global map from the sparse depth measurements estimated by the planner, the planning and learning are mutually interconnected. ![Architecture of the mapping network. Top: An example input with sparse measurements, showing only the occupied voxels. Bottom: The corresponding reconstructed dense occupancy confidence after thresholding. Right: Schema of the network architecture, composed from the convolutional layers (denoted conv), linear rectifier units (relu), pooling layers (pool), and upsampling layers (deconv).[]{data-label="fig:net"}](img/net_l){width="0.92\linewidth"} The mapping network consists of 6 convolutional layers with $5 \times 5$ kernels followed by linear rectifier units (element-wise $\max\{x,0\}$) and, in 2 cases, by max pooling layers with $2 \times 2$ kernels and stride $2$, see Fig. \[fig:net\]. In the end, there is an fourfold upsampling layer so that the output has same size as input. The network was implemented in [MatConvNet]{} [@Vedaldi-2015-ICMM]. Planning of depth measuring rays {#sec:planning} ================================ We assume that the vehicle follows a known path consisting from $L$ discrete positions, the SSL can capture at most $K$ rays at each position. We search for the subset of rays along this path, which decreases the logistic loss the most. Since it is not clear how to quantify the impact of measuring a subset of voxels on the CNN logistic loss, we simplify the problem and assume that measuring a voxel $i$ decreases the overall logistic loss only by its current expected loss $\epsilon_i{\mathcal{L}}(Y_i, Y^_i)$. Measuring the depth in the set of rays $J$ yields for each voxel $i$, the probability ${\ensuremath{\mathrm{\mathbf{p}}}}_i(J)$ to be measured. Each voxel $i$ has probability $p_{ij}$ not* to be covered by ray $j$. Given global confidence map $Y$ we estimate the probability $q_i = 1 / \left(1 + e^{-Y_i}\right)$ of voxel $i$ being occupied. We assume that the voxel $i$ is visible in ray $j$ which intersects sequence of voxels $R$. If all $i$th preceding voxels $R^-(i)$ are not occupied and the voxel itself or at least one of the voxels which follow $R^+(i)$ are occupied. Consequently, we estimate probability $p_{ij}$ of voxel $i$ not being visible in $j$ as $p_{ij} = 1 - \prod_{u\in R^-(i)}(1-q_{u})\left(1 - \prod_{u\in R^+(i)}(1-q_{u})\right).$ If ray $j$ does not intersect the voxel $i$, then $p_{ij} = 1$. We assume that when a voxel is covered by more than one ray, their contributions are independent, therefore probability of voxel $i$ being not covered by any ray from the set of rays $J$ is equal to $p_i(J) = \prod_{j\in J} p_{ij}$. We define the planning as the search for the sequence $J=\{J_1, \dots J_L\}$ of subsets of depth-measuring rays $J_1, \dots J_L$, which minimize the expected sum of reconstruction errors $E_{I\sim p(J)}\{\sum_{i\in I} \epsilon_i\}\approx{\ensuremath{\mathrm{\mathbf{\epsilon}}}}^{\ensuremath{\mathsf{T}}}{\ensuremath{\mathrm{\mathbf{p}}}}(J) $, such that $\|J_1\|\leq K, \dots \|J_L\|\leq K$. Planning at position $l$ searches for a set of rays $J$, which approximately minimizes the expected logistic loss ${\mathcal{L}}({\ensuremath{\mathrm{\mathbf{Y}}}},{\ensuremath{\mathrm{\mathbf{h}}}}_{\theta^t}({\ensuremath{\mathrm{\mathbf{x}}}}_{l+L}))$ between ground truth map ${\ensuremath{\mathrm{\mathbf{Y}}}}$ and reconstruction obtained from sparse measurements ${\ensuremath{\mathrm{\mathbf{x}}}}_{l+L}$ at the horizon $L$. The result of planning is the set of rays $J$, which will provide measurements for a sparse set of voxels. This set of voxels is referred to as covered by $J$ and denoted as $C(J)$. While the mapping network is trained offline on the ground-truth maps, the planning have to search the subset of rays online without any explicit knowledge of the ground-truth occupancy ${\ensuremath{\mathrm{\mathbf{Y}}}}$. Since it is not clear how to directly quantify the impact of measuring a subset of voxels on the reconstruction ${\ensuremath{\mathrm{\mathbf{h}}}}_{\theta^t}({\ensuremath{\mathrm{\mathbf{x}}}}_{l+L})$, we introduce simplified reconstruction model $\hat{{\ensuremath{\mathrm{\mathbf{h}}}}}(J,\hat{{\ensuremath{\mathrm{\mathbf{Y}}}}})$, which predicts the loss based on currently available map $\hat{{\ensuremath{\mathrm{\mathbf{Y}}}}}$. This model conservatively assumes that the reconstruction in covered voxels $i\in C(J)$ is correct (i.e. ${\mathcal{L}}\big(Y_{i},\hat{h}_i(J,\hat{{\ensuremath{\mathrm{\mathbf{Y}}}}})\big) = 0$) and reconstruction of not covered voxels $i\notin C(J)$ does not change (i.e. ${\mathcal{L}}\big(Y_{i},\hat{h}_i(J,\hat{{\ensuremath{\mathrm{\mathbf{Y}}}}})\big) = {\mathcal{L}}(Y_{i},\hat{Y}_{i})$). Given this reconstruction model, the expected loss simplifies to: &-5mu\_i(Y\_[i]{},\_i(J,))= \_[iC(J)]{}(Y\_i,\_i) Since the ground-truth occupancy of voxels is apriori unknown, neither the voxel-wise loss nor the coverage are known. We model the expected loss in voxel $i$ as (Y\_i,\_i) \_[Y\_[i]{}\~((\_[i]{}))]{}(Y\_[i]{},\_[i]{}) = (((\_i))) = \_i, where $\mathcal{H}(\textrm{B}(p))$ is the entropy of the Bernoulli distribution with parameter $p$, denoting the probability of outcome $1$ from the possible outcomes $\{-1, 1\}$. The vector of concatenated losses $\epsilon_i$ is denoted ${\ensuremath{\mathrm{\mathbf{\boldsymbol\epsilon}}}}$. Planning at position $l$ approximately minimizes the expected logistic loss ${\mathcal{L}}\big({\ensuremath{\mathrm{\mathbf{Y}}}},{\ensuremath{\mathrm{\mathbf{h}}}}_\theta({\ensuremath{\mathrm{\mathbf{x}}}}_{l+L}(\theta^0,{\ensuremath{\mathrm{\mathbf{Y}}}}))\big)$ at the horizon $L$. The result of planning is the set of rays $J$, which will provide measurements for a sparse set of voxels. This set of voxels is referred to as covered by $J$ and denoted as $C(J)$. While the mapping network is trained offline on the ground-truth maps, the planning have to search the subset of rays online without any explicit knowledge of the ground-truth occupancy ${\ensuremath{\mathrm{\mathbf{Y}}}}$. Since it is not clear how to directly quantify the impact of measuring a subset of voxels on the reconstruction ${\ensuremath{\mathrm{\mathbf{h}}}}_\theta({\ensuremath{\mathrm{\mathbf{x}}}}_{l+L}(\theta^0,{\ensuremath{\mathrm{\mathbf{Y}}}}))$, we introduce simplified reconstruction model $\hat{{\ensuremath{\mathrm{\mathbf{h}}}}}(J,\hat{{\ensuremath{\mathrm{\mathbf{Y}}}}})$. This model conservatively assumes that the reconstruction in covered voxels $i\in C(J)$ is correct (i.e. ${\mathcal{L}}\big({\ensuremath{\mathrm{\mathbf{y}}}}_{l+L,i},\hat{{\ensuremath{\mathrm{\mathbf{h}}}}}_i(J,\hat{{\ensuremath{\mathrm{\mathbf{Y}}}}})\big) = 0$) and reconstruction of not covered voxels $i\notin C(J)$ does not change. Given this reconstruction model, the expected loss simplifies to: &-5mu\_i([$\mathrm{\mathbf{y}}$]{}\_[l+L,i]{},\_i(J,))= \_[iC(J)]{}([$\mathrm{\mathbf{y}}$]{}\_[l+L,i]{},\_[l+L,i]{}) Since the ground-truth occupancy of voxels is apriori unknown, neither the voxel-wise loss nor the coverage are known. We model the expected loss in voxel $i$ as -15mu\_i = \_[[$\mathrm{\mathbf{y}}$]{}\_[l+L,i]{}\~((\_[l,i]{})]{}([$\mathrm{\mathbf{y}}$]{}\_[l+L,i]{},\_[l+L,i]{})=\ =(((\_[l,i]{}))) Vector of concatenated voxel-wise losses $\epsilon_i$ is denoted as $\boldsymbol\epsilon$. The length of particular rays is also unknown, therefore coverage $C(J)$ of voxels by particular rays cannot be determined uniquely. Consequently, we introduce probability $p_{ij}$ that voxel $i$ will not be covered by ray $j\in J$. This probability is estimated from currently available map $\hat{{\ensuremath{\mathrm{\mathbf{Y}}}}}$ as the product of (i) the probability that the voxels on ray $j$ which lie between voxel $i$ and the sensor are unoccupied and (ii) the probability that at least one of the following voxels or the voxel $i$ itself are occupied. If ray $j$ does not intersect voxel $i$, then $p_{ij} = 1$. The vector of probabilities $p_{ij}$ for ray $j$ is denoted ${\ensuremath{\mathrm{\mathbf{p}}}}_j$. Assuming that rays $J$ are independent measurements, the expected loss is modeled as ${\ensuremath{\mathrm{\mathbf{\boldsymbol\epsilon}}}}^{\ensuremath{\mathsf{T}}}\prod_{j\in J} {\ensuremath{\mathrm{\mathbf{p}}}}_j$. Each ray $j$ intersects a sequence of voxels $R_j$ ordered by the distance from the sensor. We assume that voxel $i$ is visible in ray $j$ if all voxels $R_j^-(i)$ which precede voxel $i$ are empty and the voxel itself or at least one of the voxels which follow $R_j^+(i)$ are occupied. Consequently, we estimate probability $p_{ij}$ of voxel $i$ not being covered by ray $j$ as $p_{ij} = 1 - \prod_{u\in R^-(i)}(1-q_{u}(\hat{{\ensuremath{\mathrm{\mathbf{Y}}}}}))\left(1 - \prod_{u\in R^+(i)}(1-q_{u}(\hat{{\ensuremath{\mathrm{\mathbf{Y}}}}}))\right).$ &-16mu\_[[$\mathrm{\mathbf{Y}}$]{}\~\_l]{}{([$\mathrm{\mathbf{Y}}$]{},\_[l+L]{})} = \_[[$\mathrm{\mathbf{Y}}$]{}\~\_l]{}{([$\mathrm{\mathbf{Y}}$]{}, \_[l]{})}[$\mathrm{\mathbf{p}}$]{}(J,\_l) where the expected voxel-wise loss at the current position $l$ reduces to the entropy of the current map $\hat{{\ensuremath{\mathrm{\mathbf{Y}}}}}_{l}$: & \_[[$\mathrm{\mathbf{Y}}$]{}\~\_l]{}{(\_[l]{}, [$\mathrm{\mathbf{Y}}$]{})} = (1,\_[l]{})\_[l]{} + (0,\_[l]{})(1-\_[l]{})=\ &150mu=(\_[l]{}), and ${\ensuremath{\mathrm{\mathbf{p}}}}(J,\hat{{\ensuremath{\mathrm{\mathbf{Y}}}}}_l)$ is the voxel-wise probability that particular voxels will not be covered by the set of rays $J$. Note, that since the ground-truth occupancy of voxels is apriori unknown, the length of particular rays is also unknown and the coverage of voxels by particular rays cannot be determined uniquely, therefore probability ${\ensuremath{\mathrm{\mathbf{p}}}}(J,\hat{{\ensuremath{\mathrm{\mathbf{Y}}}}}_l)$ is estimated instead. Assuming that rays are independent measurements, the probability is $ {\ensuremath{\mathrm{\mathbf{p}}}}(J,\hat{{\ensuremath{\mathrm{\mathbf{Y}}}}}_l) = \prod_{j\in J} {\ensuremath{\mathrm{\mathbf{p}}}}_j, $ where ${\ensuremath{\mathrm{\mathbf{p}}}}_j$ is the probability that voxels are not covered by ray $j\in J$ and $\prod_{j\in J} {\ensuremath{\mathrm{\mathbf{p}}}}_j$ is element-wise multiplication of vectors ${\ensuremath{\mathrm{\mathbf{p}}}}_j$. Without loss of generality, we assume that planning is performed for a fixed position $l=0$, therefore index $l$ is dropped in the following text for simplicity. We assume that the current map $\hat{{\ensuremath{\mathrm{\mathbf{Y}}}}}$ determines vectorized voxel-wise probabilities ${\ensuremath{\mathrm{\mathbf{p}}}}_j$ and vectorized voxel-wise losses $\boldsymbol\epsilon = \mathcal{H}(\hat{{\ensuremath{\mathrm{\mathbf{Y}}}}})$, detailed description is available in section \[sec:appendix-coverage\_probability\]. The planning searches for the set $J= J_1\cup\dots\cup J_L$ of subsets $J_1\dots J_L$ of depth-measuring rays for the following $L$ positions, which minimize the expected loss, subject to budget constraints $\|J_1\|\leq K, \dots \|J_L\|\leq K$ -7mu J\^\* = \_J[$\mathrm{\mathbf{\boldsymbol\epsilon}}$]{}\^[$\mathsf{T}$]{}\_[jJ]{} [$\mathrm{\mathbf{p}}$]{}\_j, [$\mathrm{s.t.}\ $]{}\|J\_1\|K, …\|J\_L\|K, \[eq:planning-problem\] where $\|J_l\|$ denotes cardinality of the set $J_l$. This is a non-convex combinatorial problem[^2] which needs to be solved online repeatedly for millions of potential rays. We tried several convex approximations, however the high-dimensional optimization has been extremely time consuming and the improvement with respect to the significantly faster greedy algorithm was negligible. As a consequence of that, we have decided to use the greedy algorithm. We first introduce its simplified version (Alg. \[algo:greedy0\]) and derive its properties, the significantly faster prioritized greedy algorithm (Alg. \[algo:greedy\]) is explained later. We denote the list of available rays at position $l$ as $V_l$. At the beginning, the list of all available rays is initialized as follows $V = V_1 \cup \dots \cup V_L$. Alg. \[algo:greedy0\] successively builds the set of selected rays $J$. In each iteration the best ray $j^$ is selected, added into $J$ and removed from $V$. The position from which the ray $j^$ is chosen is denoted $l^$. If the budget $K$ of $l^$ is reached, all rays from $V_{l^}$ are removed from $V$. In order to avoid multiplication of all selected rays at each iteration, we introduce the vector ${\ensuremath{\mathrm{\mathbf{b}}}}$, which keeps voxel loss. Vector ${\ensuremath{\mathrm{\mathbf{b}}}}$ is initialized as ${\ensuremath{\mathrm{\mathbf{b}}}}={\ensuremath{\mathrm{\mathbf{\boldsymbol\epsilon}}}}$ and whenever ray $j$ is selected, voxel losses are updated as follows ${\ensuremath{\mathrm{\mathbf{b}}}} = {\ensuremath{\mathrm{\mathbf{b}}}}\odot{\ensuremath{\mathrm{\mathbf{p}}}}_j$, where $\odot$ denotes element-wise multiplication. The rest of this section is organized as follows: Section \[sec:approximation-ratio\] shows the upper bound for the approximation ratio of the greedy algorithm. Section \[sec:faster-greedy\] introduces the prioritized greedy algorithm, which in each iteration needs to re-evaluate the cost function ${\ensuremath{\mathrm{\mathbf{b}}}}^{\ensuremath{\mathsf{T}}}{\ensuremath{\mathrm{\mathbf{p}}}}_j$ only for a small fraction of rays. Approximation ratio of the greedy algorithm {#sec:approximation-ratio} ------------------------------------------- We start by couple of lemmas, which are necessary for proving the main theorem. \[lemma1\] The following upper bound holds: $$\begin{aligned} & \max_{{\ensuremath{\mathrm{\mathbf{x}}}},{\ensuremath{\mathrm{\mathbf{y}}}}} \sum_i x_iy_i \;\;\;\;\;\;\; & \leq \min\{A,B\} \\ {\ensuremath{\mathrm{s.t.}\ }}& \sum_i x_i \leq A,\; 0\leq x_i \leq 1, & \nonumber \\ & \sum_i y_i \leq B,\; 0\leq y_i \leq 1. & \nonumber \end{aligned}$$ Without loss of generality we assume that $B\leq A$. Then $\min\{A,B\}=B\geq \sum_i y_i \geq \sum_i x_i y_i$, because multiplying positive numbers $y_i$ by positive and smaller than one numbers $x_i$ can only decrease their values. \[lemma2\] For any $0\leq p \leq 1$: $$\begin{aligned} \label{lemma2-max} \max_{{\ensuremath{\mathrm{\mathbf{x}}}}} &\sum_{i=1}^k x_i& = k-1+p. \\ {\ensuremath{\mathrm{s.t.}\ }}&\prod_{i=1}^k x_i = p& \label{lemma2-eq} \\ &0\leq x_i \leq 1&\nonumber \end{aligned}$$ Let us consider two cases separately. First, for $p = 0$, a single element $x_i = 0$ is sufficient to satisfy Eq. (\[lemma2-eq\]). The criterion can then be maximized element-wise by setting $(k-1)$ elements $x_{j\neq i} = 1$. Second, for $p > 0$, $x_i > 0$ must hold for every $i$ to satisfy Eq. (\[lemma2-eq\]). Now suppose that the maximizer contains at least two elements $x_i, x_j < 1$. It follows that $$\begin{aligned} (1 - x_i) x_j &<& 1 - x_i, \nonumber \\ x_i + x_j &<& x_i x_j + 1.\label{lemma2-ineq} \end{aligned}$$ Using $x_i' = x_i x_j$ and $x_j' = 1$ instead of $x_i$ and $x_j$, respectively, gives us another feasible point, since $x_i' x_j' = (x_i x_j) 1 = x_i x_j = p / \prod_{n \neq i,j} x_n$. By adding $\sum_{n \neq i,j} x_n$ to both sides of Eq. (\[lemma2-ineq\]) it can be seen, however, that a higher value is attained at this point. This contradicts the proposition that the maximizer contains at least two elements $x_i, x_j < 1$; the maximizer must contain at most one $x_i < 1$ with the rest of elements $x_{j\neq i} = 1$. From Eq. (\[lemma2-eq\]) it follows that $x_i = p$ and Eq. (\[lemma2-max\]) holds. We define the approximation ratio of a minimization algorithm to be $\rho=\frac{f}{{{{\normalfont\textsc{opt}}}}}$, where $f$ is the cost function achieved by the algorithm and ${{{\normalfont\textsc{opt}}}}$ is the optimal value of the cost function. Given $\rho$, we know that the algorithm provides solution whose value is at most $\rho\ {{{\normalfont\textsc{opt}}}}$. In this section we derive the upper bound of the approximation ratio ${{{\normalfont\textsc{UB}}}}(\rho)$ of Algorithm \[algo:greedy0\]. Figure \[fig:approximation\_ratio\] shows values of ${{{\normalfont\textsc{UB}}}}(\rho)$ for different number of positions $L$. The greedy algorithm successively selects rays that reduce the cost function the most. To show how cost function differs from ${{{\normalfont\textsc{opt}}}}$, an upper bound on the cost function need to be derived. Let us suppose that in the beginning of an arbitrary iteration we have voxel losses given by vector ${\ensuremath{\mathrm{\mathbf{b}}}}$, the following lemma states that for arbitrary voxel $i$, there always exists a ray $j$, that reduces the cost function to $\sum_i b_i(1-\frac{1}{K}) + \frac{{{{\normalfont\textsc{opt}}}}}{K}$, where ${{{\normalfont\textsc{opt}}}}= {\ensuremath{\mathrm{\mathbf{1}}}}^{\ensuremath{\mathsf{T}}}\prod_{j=1}^K {\ensuremath{\mathrm{\mathbf{p}}}}_{j} ={\ensuremath{\mathrm{\mathbf{1}}}}^{\ensuremath{\mathsf{T}}}{\ensuremath{\mathrm{\mathbf{p}}}}^{{{{\normalfont\textsc{opt}}}}}$ is the unknown optimum value of the cost function which is achievable by $K$ rays ${\ensuremath{\mathrm{\mathbf{p}}}}_1\dots {\ensuremath{\mathrm{\mathbf{p}}}}_K$. \[lemma3\] If for some rays $\prod_{j=1}^K p_{ij}=p_i^{{{{\normalfont\textsc{opt}}}}}$ then \_[[$\mathrm{\mathbf{0}}$]{}[$\mathrm{\mathbf{b}}$]{} [$\mathrm{\mathbf{1}}$]{}]{} \_j \_[i=1]{}\^V p\_[ij]{}b\_i \_[i=1]{}\^V b\_i(1-) + Proof:* We know that there is optimal solution consisting from $K$ rays. Without loss of generality we assume that $\prod_{j=1}^K p_{ij}=p_i^{{{{\normalfont\textsc{opt}}}}}$ holds for first $K$ rays, then \_i \_[j=1]{}\^[K]{} p\_[ij]{} K-1+p\^[[[[<span style="font-variant:small-caps;">opt</span>]{}]{}]{}]{}\_i. This holds for an arbitrary positive scaling factor $b_i$, therefore \_i \_[j=1]{}\^[K]{} p\_[ij]{}b\_i (K-1+p\^[[[[<span style="font-variant:small-caps;">opt</span>]{}]{}]{}]{}\_i)b\_i. We sum up inequalities over all voxels $i$ $$\sum_{i=1}^V\sum_{j=1}^{K} p_{ij}b_i \leq \sum_{i=1}^V(K-1+p^{{{{\normalfont\textsc{opt}}}}}_i)b_i.$$ We switch sums in the left hand side of the inequality to obtain addition of $K$ terms as follows $$\sum_{i=1}^V p_{i1}b_i + \dots + \sum_{i=1}^V p_{iK}b_i \leq \sum_{i=1}^V(K-1+p^{{{{\normalfont\textsc{opt}}}}}_i)b_i$$ Hence, we know that at least one of these $K$ terms has to be smaller than or equal to $\frac{1}{K}$ of the right hand side $$\begin{aligned} \hspace{-0.4cm}\exists_j \sum_{i=1}^V p_{ij}b_i &\leq& \frac{1}{K}\sum_{i=1}^V(K-1+p^{{{{\normalfont\textsc{opt}}}}}_i)b_i= \nonumber \\ &=& \sum_{i=1}^Vb_i (1-\frac{1}{K})+\frac{1}{K}\sum_{i=1}^Vp^{{{{\normalfont\textsc{opt}}}}}_ib_i\leq \nonumber \\ &\leq & \sum_{i=1}^Vb_i (1-\frac{1}{K})+\sum_{i=1}^V\frac{p^{{{{\normalfont\textsc{opt}}}}}_i}{K}= \\ &=& \sum_{i=1}^Vb_i (1-\frac{1}{K})+\frac{{{{\normalfont\textsc{opt}}}}}{K} \nonumber\hspace{1.3cm}\square \end{aligned}$$ Especially, if there is only one position, all optimal $K$ rays ${\ensuremath{\mathrm{\mathbf{p}}}}_1\dots {\ensuremath{\mathrm{\mathbf{p}}}}_K$ are either already selected or still available. This assumption allows to derive the following upper bound on the cost function of the greedy algorithm $f^K$ after $K$ iterations for $L=1$. \[theorem1\] Upper bound $\mathit{{{{\normalfont\textsc{UB}}}}}(f^K)\geq f^K$ of the greedy algorithm after $K$ iterations is (f\^K) = E + [[[<span style="font-variant:small-caps;">opt</span>]{}]{}]{}(1-), where $E = \sum_{i=1}^V \epsilon_i$ and $e$ is Euler number. Proof: We prove the upper bound by complete induction. In the beginning no ray is selected, per-voxel loss is $b_i^0 = \epsilon_i$ and the value of the cost function $f^0 = \sum_{i=1}^V b_i^0 = E$. Using Lemma \[lemma3\], we know that there exists ray $j$ such that $\sum_{i=1}^V p_{ij}b_i^0 \leq \sum_{i=1}^V b_i^0(1-\frac{1}{k}) + \frac{{{{\normalfont\textsc{opt}}}}}{K}$, therefore we know that &f\^1 =\_[i=1]{}\^V p\_[ij]{}b\_i\^0 \_[i=1]{}\^V b\_i\^0(1-) + =\ & =E (1-) + . Greedy algorithm continues by updating the per-voxel loss $b_i^1 = b_i^0p_{ij}$. In the second iteration there are two possible cases: (i) we have either used the optimal ray in the first iteration, then the situation is better and we know there is $(K-1)$ rays which achieves optimum, or (ii) we have not selected the optimal ray in the first iteration, therefore we have still $K$ rays which achieves the optimum. Since the cost function reduction in the latter case gives the upper bound on the cost function reduction in the former one, we assume that there is still $k$ optimal rays available, therefore there exists ray $j$ such that $$\begin{aligned} \hspace{-3cm}f^2 &=&\sum_{i=1}^V p_{ij}b_i^1\leq \sum_{i=1}^V b_i^1\left(1-\frac{1}{k}\right) + \frac{{{{\normalfont\textsc{opt}}}}}{K}\leq \nonumber \\ &\leq& E \left(1-\frac{1}{K}\right)^2 + \frac{{{{\normalfont\textsc{opt}}}}}{K}\left(\left(1-\frac{1}{K}\right)+1\right). \end{aligned}$$ We assume that the following holds $$\begin{aligned} \hspace{-0.3cm} f^{t-1} &\leq& E \left(1-\frac{1}{K}\right)^{t-1} + \frac{{{{\normalfont\textsc{opt}}}}}{K}\sum_{u=0}^{t-2}\left(1-\frac{1}{K}\right)^u. \label{eq:assumption} \end{aligned}$$ and prove the inequality for $f^t$. Using the assumption (\[eq:assumption\]) and Lemma \[lemma3\], the following inequalities hold [ $$\begin{aligned} &&\hspace{-0.7cm}f^t \leq \sum_{i=1}^V b_i^{t-1} \left(1-\frac{1}{K}\right) + \frac{{{{\normalfont\textsc{opt}}}}}{K}\leq \nonumber \\ &&\hspace{-0.7cm}\leq\left[ E\left(1-\frac{1}{K}\right)^{t-1}\!\!\!\!\!\! + \frac{{{{\normalfont\textsc{opt}}}}}{K}\sum_{u=0}^{t-2}\left(1-\frac{1}{K}\right)^u\right]\left(1-\frac{1}{K}\right)+ \frac{{{{\normalfont\textsc{opt}}}}}{K} \nonumber\\ &&\hspace{-0.7cm} = E\underbrace{\left(1-\frac{1}{K}\right)^{t}}_{\alpha_t^K} + {{{\normalfont\textsc{opt}}}}\underbrace{\frac{1}{K}\sum_{u=0}^{t-1}\left(1-\frac{1}{K}\right)^u}_{\beta_t^K} \label{eq:alpha_beta} \end{aligned}$$ ]{} Since $\alpha_t^K+\beta_t^K = 1$ [^3] and $\alpha_K = \left(1-\frac{1}{K}\right)^{K} \leq \frac{1}{e}$, the upper bound for cost function of the greedy algorithm in $K$th iteration is $f^K \leq E \frac{1}{e} + {{{\normalfont\textsc{opt}}}}\left(1-\frac{1}{e}\right)$ $\square$ Theorem \[theorem1\] reveals that the approximation ratio of the greedy algorithm $\rho = \frac{f^{K}}{{{{\normalfont\textsc{opt}}}}}$ after $K$ iterations has following upper bound [$$\rho \leq \frac{{{{\normalfont\textsc{opt}}}}(\frac{E}{{{{\normalfont\textsc{opt}}}}}\frac{1}{e} + \left(1-\frac{1}{e}\right))}{{{{\normalfont\textsc{opt}}}}} \leq \frac{E}{{{{\normalfont\textsc{LB}}}}({{{\normalfont\textsc{opt}}}})e} + \left(1-\frac{1}{e}\right)$$ ]{} We can simply find ${{{\normalfont\textsc{LB}}}}({{{\normalfont\textsc{opt}}}})$ by considering for each voxel the best $K$ rays independently. So far we have assumed that the greedy algorithm chooses only $K$ rays and that all rays are available in all iterations. Since there are $L$ positions and the greedy algorithm can choose only $K$ rays at each position, some rays may be no longer available when choosing $(K+1)$th ray. In the worst case possible, the rays from the most promising position will become unavailable. Since we have not chosen optimal rays we can no longer achieve ${{{\normalfont\textsc{opt}}}}$. Nevertheless, we can still choose from rays which achieve a new optimum. We introduce $\overline{{{{\normalfont\textsc{opt}}}}}_v$ as the optimum achievable after closing $v$ positions. Obviously $\overline{{{{\normalfont\textsc{opt}}}}}_0={{{\normalfont\textsc{opt}}}}$. Let us assume that, when the first position is closed we cannot lose more than $R_1$, therefore $\overline{{{{\normalfont\textsc{opt}}}}}_1={{{\normalfont\textsc{opt}}}}+R_1$. Without any additional assumption, $R_1$ could be arbitrarily large. We discuss potential assumptions later. Similarly $\overline{{{{\normalfont\textsc{opt}}}}}_2={{{\normalfont\textsc{opt}}}}+R_1+R_2$, and $\overline{{{{\normalfont\textsc{opt}}}}}_v={{{\normalfont\textsc{opt}}}}+\sum_{l=1}^vR_l$. The following theorem states the upper bound for $f^{LK}$ as a function of $\overline{{{{\normalfont\textsc{opt}}}}}_v$. Upper bound $\mathit{{{{\normalfont\textsc{UB}}}}}(f^{LK})\geq f^{LK}$ of the greedy algorithm after $LK$ iterations is (f\^[LK]{}) = E + \_[u=0]{}\^[L-1]{}\_u, where $\gamma_u = \left(1-\sqrt[L]{\frac{1}{e}}\right)\left(\sqrt[L]{\frac{1}{e}}\right)^{L-1-u}$ We start from the result (\[eq:alpha\_beta\]) shown in the proof of Theorem \[theorem1\]. Since there is $LK$ rays achieving optimum $\overline{{{{\normalfont\textsc{opt}}}}_0} = {{{\normalfont\textsc{opt}}}}$, the cost function $f^K$ in $K$th iteration is bounded as follows -26muf\^[K]{}E\_[\_[K]{}\^[LK]{}]{} + \_[\_[K]{}\^[LK]{}]{} In the $(K+1)$th iteration, there are two possible cases: (i) rays from some position $l$ become not available and there is $K(L-1)$ rays available which can achieve a new optimum which is not higher than $\overline{{{{\normalfont\textsc{opt}}}}_1}$ or (ii) all rays are available and there is still $LK$ rays which achieve $\overline{{{{\normalfont\textsc{opt}}}}}_0={{{\normalfont\textsc{opt}}}}$. Noticing that the upper bound is increasing in $\overline{{{{\normalfont\textsc{opt}}}}}_0$ and $L$, we can cover both cases by considering there is still $LK$ rays which achieves $\overline{{{{\normalfont\textsc{opt}}}}_1}$, therefore $$\begin{aligned} &&\hspace{-1.5cm}f^{K+1}\leq (E\alpha_{K}^{LK} + \overline{{{{\normalfont\textsc{opt}}}}_0}\beta_{K}^{LK})(1-\frac{1}{LK}) + \frac{\overline{{{{\normalfont\textsc{opt}}}}_1}}{LK}= \nonumber \\ &&\hspace{-1.5cm}\;\;\;\;\;\;\;\;\;=E\alpha_{K+1}^{LK} + \overline{{{{\normalfont\textsc{opt}}}}_0}\beta_{K}^{LK}(1-\frac{1}{LK}) + \frac{\overline{{{{\normalfont\textsc{opt}}}}_1}}{LK} \end{aligned}$$ We can now continue up to the iteration $2K$ in which the upper bound is as follows $$\begin{aligned} &&\hspace{-1.5cm}f^{2K}\leq E\alpha_{2K}^{LK} + \overline{{{{\normalfont\textsc{opt}}}}_0}\beta_{K}^{LK}\alpha_{K}^{LK} + \overline{{{{\normalfont\textsc{opt}}}}_1}\beta_{K}^{LK} \end{aligned}$$ For $(2K+1)$th iteration the situation is similar as for ${(K+1)}$th iteration. In order to cover both cases, we consider that there is $LK$ rays which achieves $\overline{{{{\normalfont\textsc{opt}}}}_2}$ and continue up to the $3k$th iteration, which yields the following upper bound $$\begin{aligned} f^{3K}&\leq& E\alpha_{3K}^{LK} + \overline{{{{\normalfont\textsc{opt}}}}_0}\beta_{K}^{LK}\alpha_{2K}^{LK} + \nonumber\\ && +\overline{{{{\normalfont\textsc{opt}}}}_1}\beta_{K}^{LK}\alpha_{K}^{LK}+\overline{{{{\normalfont\textsc{opt}}}}_2}\beta_{K}^{LK} \end{aligned}$$ Finally after $LK$ iterations the upper bound is $$\begin{aligned} &&\hspace{-1.5cm}f^{LK}\leq E \alpha_{LK}^{LK} + \beta_{K}^{LK}\sum_{u=0}^{L-1}\alpha_{(L-1-u)K}^{LK}\overline{{{{\normalfont\textsc{opt}}}}_u}\leq\nonumber\\ &&\hspace{-.8cm} \leq E \frac{1}{e} + \sum_{u=0}^{L-1}\left(1-\sqrt[L]{\frac{1}{e}}\right)\left(\sqrt[L]{\frac{1}{e}}\right)^{L-1-u}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\overline{{{{\normalfont\textsc{opt}}}}_u}. \end{aligned}$$ The last inequality stems from the fact that $(\alpha_{K}^{LK})^L = \alpha_{LK}^{LK} \leq \frac{1}{e}$ and that $\alpha_{K}^{LK} + \beta_{K}^{LK} = 1$. Finally we derive the upper bound of the approximation ratio $\rho = {f^{LK}}/{{{{\normalfont\textsc{opt}}}}}$. Upper bound of the approximation ratio is + \_[u=0]{}\^[L-1]{}\_u (1+) where ${{{\normalfont\textsc{LB}}}}({{{\normalfont\textsc{opt}}}})$ is lower bound of the ${{{\normalfont\textsc{opt}}}}$. Proof: $$\begin{aligned} &&\hspace{-0.7cm}\rho = \frac{f^{LK}}{{{{\normalfont\textsc{opt}}}}} \leq \frac{{{{\normalfont\textsc{UB}}}}(f^{LK})}{{{{\normalfont\textsc{opt}}}}} = \frac{E \frac{1}{e} + \sum_{u=1}^{L}\gamma_u \overline{{{{\normalfont\textsc{opt}}}}}_u}{{{{\normalfont\textsc{opt}}}}}= \nonumber\\ &&\hspace{-0.4cm}= \frac{{{{\normalfont\textsc{opt}}}}(\frac{E}{{{{\normalfont\textsc{opt}}}}}\frac{1}{e} +\sum_{u=1}^{L}\gamma_u \frac{\overline{{{{\normalfont\textsc{opt}}}}}_u}{{{{\normalfont\textsc{opt}}}}})}{{{{\normalfont\textsc{opt}}}}}= \nonumber\\ &&\hspace{-0.4cm}= \frac{E}{{{{\normalfont\textsc{opt}}}}}\frac{1}{e} + \sum_{u=1}^{L}\gamma_u \frac{{{{\normalfont\textsc{opt}}}}+ \sum_{v=1}^uR_v}{{{{\normalfont\textsc{opt}}}}}\leq \\ &&\hspace{-0.4cm}\leq \frac{E}{{{{\normalfont\textsc{LB}}}}({{{\normalfont\textsc{opt}}}})}\frac{1}{e} + \sum_{u=0}^{L-1}\gamma_u \left(1+\frac{\sum_{v=1}^uR_v}{\small{{{{\normalfont\textsc{LB}}}}({{{\normalfont\textsc{opt}}}})}}\right)\nonumber\;\;\;\;\;\;\;\;\;\;\;\;\;\;\square \end{aligned}$$ The approximation ratio depends on the ${{{\normalfont\textsc{opt}}}}$, if ${{{\normalfont\textsc{opt}}}}=0$ then $\rho = \infty$, if ${{{\normalfont\textsc{opt}}}}=E$ then $\rho = 1$. If we make an assumption that each position covers only $\frac{1}{L}$ fraction of voxels, then $R_v\leq \frac{V}{L}$. Figure \[fig:approximation\_ratio\] shows values of ${{{\normalfont\textsc{LB}}}}(\rho)$ for different ratios of $\frac{{{{\normalfont\textsc{opt}}}}}{E}$ for this case. ![$\text{{{{\normalfont\textsc{UB}}}}}(\rho)$ as a function of $\frac{{{{\normalfont\textsc{opt}}}}}{E}$ ratios with $R_v\leq \frac{V}{L}$.[]{data-label="fig:approximation_ratio"}](img/approximation_ratio_popt){width="0.55\linewidth"} Prioritized greedy planning {#sec:faster-greedy} --------------------------- In practice we observed a significant speed up of the greedy planning (Alg. \[algo:greedy0\]) by imposing prioritized search for $\arg\min_j {\ensuremath{\mathrm{\mathbf{b}}}}^{\ensuremath{\mathsf{T}}}{\ensuremath{\mathrm{\mathbf{p}}}}_{j}$. Namely, let us denote $\Delta_j^k$ the decrease of the expected reconstruction error achieved by selecting ray $j$ in iteration $k$, $ \Delta_j^k = \sum_i (b_i^{k-1} - b_i^{k}) = \sum_i b_i^{k-1}(1 - p_{ij}), $ and prove that it is non-increasing. $\Delta_j^k \ge \Delta_j^{k+1}$. From $0 \le p_{ij'}, p_{ij}, b_i^{k-1} \le 1$ it follows that $b_i^{k-1}(1-p_{ij}) \ge b_i^{k-1}p_{ij'}(1-p_{ij})$. Summing the inequalities for all voxels $i$, we get [$$$$ ]{} for an arbitrary ray $j'$ selected in iteration $k$. Note that the relation is transitive and $\Delta_j^k \ge \Delta_j^{k+a}$ for any $a \ge 1$. and show that it is non-increasing. For $p_{ij}, p_{ij'} \in [0, 1]$ and $b_i^{k-1} \ge 0$ it follows that $b_i^{k-1}(1-p_{ij}) \ge b_i^{k-1}p_{ij'}(1-p_{ij})$. Summing the inequalities for all voxels $i$, we get [$$\Delta_j^k = \sum_i b_i^{k-1}(1 - p_{ij}) \ge \sum_i b_i^{k-1}p_{ij'}(1 - p_{ij}) = \Delta_j^{k+1}$$ ]{} for an arbitrary ray $j'$ selected in iteration $k$. Note that $\Delta_j^k \ge \Delta_j^{k+a}$ for any $a \ge 1$. Now, when we search for $j$ maximizing $\Delta_j^k$ in decreasing order of $\Delta_j^{k-a_j}$, $a_j \ge 1\ \forall j$, we can stop once $\Delta_j^k > \Delta_{j'}^{k-a_{j'}}$ for the next ray $j'$ because none of the remaining rays can be better than $j$. Moreover, we can take advantage of the fact that all the remaining rays including $j$ remained sorted when updating the priority for the next iteration. The proposed planning is detailed in Alg. \[algo:greedy\]. The number of re-evaluations of $\Delta_j$ in Alg. \[algo:greedy\] was approximately $500\times$ smaller than in Alg. \[algo:greedy0\]. Despite the sorting took about a $1/10$ of the computation time, the prioritized planning was about $30\times$ faster and took $0.3\si{s}$ on average using a single-threaded implementation. Set of rays ${\ensuremath{V}} = \{1, \ldots, N\}$ at positions ${\ensuremath{L}}$, budget $K,$ voxel costs ${\ensuremath{\mathrm{\mathbf{b}}}},$ probability vectors ${\ensuremath{\mathrm{\mathbf{p}}}}_{j}\ \forall j \in {\ensuremath{V}},$ mapping from ray to position $\lambda\colon {\ensuremath{V}} \mapsto {\ensuremath{L}}$ ${\ensuremath{J}}_{l} \gets \emptyset \; \forall {l} \in {\ensuremath{L}}$ $\Delta_j \gets \infty \; \forall {j} \in {\ensuremath{V}}$ $S \gets (1, \dots, N)$ $\Delta_{S(n)} \gets {\ensuremath{\mathrm{\mathbf{b}}}}^{\ensuremath{\mathsf{T}}}({\ensuremath{\mathrm{\mathbf{1}}}} - {\ensuremath{\mathrm{\mathbf{p}}}}_{S(n)})$ break Sort subsequence $S(1:n)$ s.t. $\Delta_{S(n')} \ge \Delta_{S(n'+1)}$ Merge sorted subsequences $S(1:n-1)$ and $S(n:\|S\|)$ $j^* \gets S(1), l^* \gets \lambda(j^)$ ${\ensuremath{J}}_{l^} \gets {\ensuremath{J}}_{l^} \cup \{j^\}$ ${\ensuremath{\mathrm{\mathbf{b}}}} \gets {\ensuremath{\mathrm{\mathbf{b}}}} \odot {\ensuremath{\mathrm{\mathbf{p}}}}_{j^}$ $S \gets S \setminus \{j : \lambda(j) = l^ \}$ $S \gets S \setminus \{j^\}$ Selected rays ${\ensuremath{J}}_l$ at every position $l \in {\ensuremath{L}}$ Experiments =========== #### Dataset All experiments were conducted on selected sequences from categories City* and Residential from the KITTI dataset [@Geiger-2013-IJRR]. We first brought the point clouds (captured by the Velodyne HDL-64E laser scanner) to a common reference frame using the localization data from the inertial navigation system (OXTS RT 3003 GPS/IMU) and created the ground-truth voxel maps from these. The voxels traced from the sensor origin towards each measured point were updated as empty except for the voxels incident with any of the end points which were updated as occupied for each incident end point. The dynamic objects were mostly removed in the process since the voxels belonging to these objects were also many times updated as empty while moving. All maps used axis-aligned voxels of edge size $\SI{0.2}{m}$. For generating the sparse measurements, we consider an SSL sensor with the field of view of $120^{\circ}$ horizontally and $90^{\circ}$ vertically discretized in $160\times120=19200$ directions. At each position, we select $K=200$ rays and ray-trace in these directions until an occupied voxel is hit or the maximum distance of $48\si{m}$ is reached. Only the rays which end up hitting an occupied voxel produce valid measurements, as is the case with the time-of-flight sensors. Local maps ${\ensuremath{\mathrm{\mathbf{x}}}}_l$ and ${\ensuremath{\mathrm{\mathbf{y}}}}_l$ contain volume of $64\si{m} \times 64\si{m} \times 6.4\si{m}$ discretized into $320\times320\times32$ voxels. Active 3D mapping ----------------- In this experiment, we used $17$ and $3$ sequences from the Residential category for training and validation, respectively, and $13$ sequences from the City category for testing. We evaluate the iterative planning-learning procedure described in Sec. \[sec:mapping\] and the corresponding mapping networks ${\ensuremath{\mathrm{\mathbf{h}}}}_0$, …, ${\ensuremath{\mathrm{\mathbf{h}}}}_t$. The initial learning rate was always set to $\alpha = 10^{-3}$, the batch size was set to $1$ and momentum to $0.99$. Training the initial network ${\ensuremath{\mathrm{\mathbf{h}}}}_0$ took $2\cdot 10^5$ iterations (updates of the weights) with learning rate successively decreased to $10^{-5}$. Training the following networks took $10^5$ iterations (approximately one day) with exponentially decreasing learning rate to $\approx 10^{-5}$. We have observed that the procedure converges already after 4 planning-learning iterations. In this experiment, we used $17$ and $3$ sequences from the Residential category for training and validation, respectively, and $13$ sequences from the City category for testing. We evaluate the iterative planning-learning procedure described in Sec. \[sec:mapping\]. For learning the mapping networks, we used learning rate $\alpha = 10^{-3} (1/8)^{\lceil i / 10 \rceil}$ based on epoch number $i$, batch size $1$, and momentum $0.99$. Networks ${\ensuremath{\mathrm{\mathbf{h}}}}_{\theta^0}, \ldots, {\ensuremath{\mathrm{\mathbf{h}}}}_{\theta^3}$ were trained for $20$ epochs. The ROC curves shown in Fig. \[fig:roc\] (left) are computed using ground-truth maps ${\ensuremath{\mathrm{\mathbf{Y}}}}$ and predicted global occupancy maps $\hat{{\ensuremath{\mathrm{\mathbf{Y}}}}}$. The performance of the ${\ensuremath{\mathrm{\mathbf{h}}}}_{\theta^3}$ network (denoted Coupled) significantly outperforms the ${\ensuremath{\mathrm{\mathbf{h}}}}_{\theta^3}$ network (Random), which shows the benefit of the proposed iterative planning-mapping procedure. Examples of reconstructed global occupancy maps are shown in Fig. \[fig:inouts\]. Note that the valid measurements covered around $3\%$ of the input voxels. ![ROC curves of occupancy prediction from active 3D mapping on test sets. Left: Random denotes the global occupancy $\hat{{\ensuremath{\mathrm{\mathbf{Y}}}}}$ obtained by using ${\ensuremath{\mathrm{\mathbf{h}}}}_{\theta^0}$ with random sparse measurements, Coupled the occupancy obtained by using ${\ensuremath{\mathrm{\mathbf{h}}}}_{\theta^3}$ with the prioritized greedy planning. The voxels which are more than $1\si{m}$ from what could possibly be measured are excluded, together with the false positives which can be attributed to discretization error (in 1-voxel distance from an occupied voxel). Right: Random denotes the local occupancy maps $\hat{{\ensuremath{\mathrm{\mathbf{y}}}}}_l$ obtained by using ${\ensuremath{\mathrm{\mathbf{h}}}}_{\theta^0}$, Coupled the maps obtained by using ${\ensuremath{\mathrm{\mathbf{h}}}}_{\theta^1}$, and Res3D-GRU-3 denotes the reconstruction obtained by the network adapted from [@Choy-2016-ECCV]. []{data-label="fig:roc"}](img/roc_rand_coupled_tol1_cut "fig:"){width="0.495\linewidth"} ![ROC curves of occupancy prediction from active 3D mapping on test sets. Left: Random denotes the global occupancy $\hat{{\ensuremath{\mathrm{\mathbf{Y}}}}}$ obtained by using ${\ensuremath{\mathrm{\mathbf{h}}}}_{\theta^0}$ with random sparse measurements, Coupled the occupancy obtained by using ${\ensuremath{\mathrm{\mathbf{h}}}}_{\theta^3}$ with the prioritized greedy planning. The voxels which are more than $1\si{m}$ from what could possibly be measured are excluded, together with the false positives which can be attributed to discretization error (in 1-voxel distance from an occupied voxel). Right: Random denotes the local occupancy maps $\hat{{\ensuremath{\mathrm{\mathbf{y}}}}}_l$ obtained by using ${\ensuremath{\mathrm{\mathbf{h}}}}_{\theta^0}$, Coupled the maps obtained by using ${\ensuremath{\mathrm{\mathbf{h}}}}_{\theta^1}$, and Res3D-GRU-3 denotes the reconstruction obtained by the network adapted from [@Choy-2016-ECCV]. []{data-label="fig:roc"}](img/roc_choy_cut "fig:"){width="0.495\linewidth"} \[fig:choy\] ![Examples of global map reconstruction. Top: Sparse measurement maps ${\ensuremath{\mathrm{\mathbf{X}}}}$. Middle: Reconstructed occupancy maps $\hat{{\ensuremath{\mathrm{\mathbf{Y}}}}}$ in form of isosurface. Bottom: Ground-truth maps ${\ensuremath{\mathrm{\mathbf{Y}}}}$. The black line denotes trajectory of the car. []{data-label="fig:inouts"}](img/annotated_comparison.pdf){width="0.99\linewidth"} Comparison to a recurrent image-based architecture {#sec:choy} -------------------------------------------------- We provide a comparison with the image-based reconstruction method of Choy [@Choy-2016-ECCV]. Namely, we modify their residual Res3D-GRU-3 network to use sparse depth maps of size $160 \times 120$ instead of RGB images. The sensor pose corresponding to the last received depth map was used for reconstruction. The number of views were fixed to $5$, with $K = 200$ randomly selected depth-measuring rays in each image. For this experiment, we used 20 sequences from the Residential category—18 for training, 1 for validation and 1 for testing. Since the Res3D-GRU-3 architecture is not suited for high-dimensional outputs due to its high memory requirements, we limit the batch size to $1$ and the size of the maps to $128 \times 128 \times 32$, which corresponds to $16 \times 16 \times 4$ recurrent units. Our mapping network was trained and tested on voxel maps instead of depth images. The corresponding ROC curves, computed from local maps ${\ensuremath{\mathrm{\mathbf{y}}}}_l$ and $\hat{{\ensuremath{\mathrm{\mathbf{y}}}}}_l$, are shown in Fig. \[fig:choy\] (right). Both ${\ensuremath{\mathrm{\mathbf{h}}}}_{\theta^0}$ and ${\ensuremath{\mathrm{\mathbf{h}}}}_{\theta^1}$ networks outperforms the Res3D-GRU-3 network. We attribute this result mostly to the fact that our method is implicitly provided the known trajectory, while the Res3D-GRU-3 network is not. Another reason may be the ray-voxel mapping which is also known implicitly in our case, compared to [@Choy-2016-ECCV]. Conclusions =========== We have proposed a computationally tractable approach for the very high-dimensional active perception task. The proposed 3D-reconstruction CNN outperforms a state-of-the-art approach by $20\%$ in recall, and it is shown that when learning is coupled with planning, recall increases by additional $8\%$ on the same false positive rate. The proposed prioritized greedy planning algorithm seems to be a promising direction with respect to on-board reactive control since it is about $30\times$ faster and requires only $1/500$ of ray evaluations compared to a na[" i]{}ve greedy solution. Acknowledgment {#acknowledgment .unnumbered} ============== The research leading to these results has received funding from the European Union under grant agreement FP7-ICT-609763 TRADR and No. 692455 Enable-S3, from the Czech Science Foundation under Project 17-08842S, and from the Grant Agency of the CTU in Prague under Project SGS16/161/OHK3/2T/13. [^1]: We introduce a canonical frame by using the local maps instead of the global ones, which helps the mapping network to capture local regularities. [^2]: In our experiments, the number of possible combinations is greater then $10^{2000}$. [^3]: $\beta_t^K = \frac{1}{K} \sum_{u=0}^{t-1}\left(1-\frac{1}{K}\right)^u = (1-a) \sum_{u=0}^{t-1} a^u = 1-a^t = 1 - \left(1-\frac{1}{K}\right)^t = 1 - \alpha_t^K$ for $a = \left(1-\frac{1}{K}\right)$.
--- abstract: 'We discuss Green’s-function solutions of the equation for a geometrically thin, axisymmetric Keplerian accretion disc with a viscosity prescription $\nu\propto R^{n}$. The mathematical problem was solved by [@LP74] for the special cases with boundary conditions of zero viscous torque and zero mass flow at the disc center. While it has been widely established that the observational appearance of astrophysical discs depend on the physical size of the central object(s), exact time-dependent solutions with boundary conditions imposed at finite radius have not been published for a general value of the power-law index $n$. We derive exact Green’s-function solutions that satisfy either a zero-torque or a zero-flux condition at a nonzero inner boundary $R_{\rm in}>0$, for an arbitrary initial surface density profile. Whereas the viscously dissipated power diverges at the disc center for the previously known solutions with $R_{\rm in}=0$, the new solutions with $R_{\rm in}>0$ have finite expressions for the disc luminosity that agree, in the limit $t\rightarrow \infty$, with standard expressions for steady-state disc luminosities. The new solutions are applicable to the evolution of the innermost regions of thin accretion discs.' author: - Takamitsu Tanaka bibliography: - 'paper.bib' title: 'Exact Time-dependent Solutions for the Thin Accretion Disc Equation: Boundary Conditions at Finite Radius' --- Introduction ============ Since its emergence in the 1970’s [@SS73; @NT73; @LP74], the theory of astrophysical accretion discs has been applied to explain the emission properties of active galactic nuclei, X-ray binaries, cataclysmic binaries, supernovae, gamma-ray bursts, and the electromagnetic signatures of mergers of supermassive black holes; to study planetary and star formation; and to model the evolution of binary and planetary systems. Because accretion discs onto compact objects can dissipate much larger fractions of baryonic rest-mass energies than nuclear reactions, they are often associated with some of the most energetic astrophysical processes in the universe. If the local gravitational potential is dominated by a central compact object or a compact binary, and if the timescale for the viscous dissipation of energy is longer than the orbital timescale, then the accretion flow near the center of the potential is expected to be nearly axisymmetric. If the gas is able to cool efficiently, then the flow will also be geometrically thin, and one only needs the radial coordinate to describe the mass distribution in the disc (any relevant vertical structure can be integrated or averaged over the disc height). The partial differential equation [@LP74 henceforth LP74] (R,t)=, \[eq:diff\] is obtained by combining the equations of mass conservation and angular momentum, and describes the surface density evolution of a thin Keplerian accretion disc due to kinematic viscosity $\nu$. In general, the viscosity $\nu$ depends on the surface density $\Sigma$ and equation is nonlinear. If, however, $\nu$ is only a function of radius, then the equation is linear and much more amenable to analytic methods. In particular, a solution that makes use of a Green’s function $G$, (R,t)=\_[R\_[in]{}]{}\^ G(R,R\^, t) (R\^, t=0) dR\^, \[eq:Green\] gives the solution $\Sigma$ for any $t>0$ given an arbitrary profile $\Sigma (R,t=0)$ and an inner boundary condition imposed at $R_{\rm in}$. A distinct advantage of the formalism is that it gives the solution $\Sigma(R,t)$ through a single ordinary integral, whereas a finite-difference algorithm would require the computation of the profile at intermediate times. Another advantage is that the initial density profile need not be differentiable. For a power-law viscosity $\nu\propto R^{n}$, [@Lust52] and derived analytic Green’s functions that satisfy a boundary condition of either zero-torque or zero-mass-flux at the coordinate origin, i.e., for the case $R_{\rm in}=0$. In reality, however, the objects at the center of astrophysical accretion discs have a finite size to which the observational appearance of the disc is sensitive: e.g., the luminosity, spectral hardness, and variability timescales of black hole discs depend strongly on the radius of the innermost stable orbit, and those of circumbinary discs depend on where the inner disc is truncated by the central tidal torques. Green’s functions with $R_{\rm in}=0$ do not capture the time-dependent behavior of accretion discs close to the central object. In fact, in solutions obtained with such Green’s functions, the integral for the total power viscously dissipated in the center of the disc diverges. Despite the astrophysical relevance of Green’s functions to the thin accretion disc equation with boundary conditions imposed at a finite radius, such solutions have not been published. [@Pringle91] derived the Green’s function with a zero-flux boundary condition at a nonzero radius in the special case $n=1$, and noted the “extreme algebraic complexity” involved in calculating a more general solution with $R_{\rm in}>0$. Time-dependent models of accretion flows have continued to employ solutions that correspond to the central objects having zero physical size [e.g., @Metzger+08; @TM10]. In order to calculate a convergent disc luminosity and spectrum, such models typically approximate analytically the effects of an inner boundary condition, e.g., by truncating the disc profile at an artificially imposed radius. In this paper we derive exact Green’s functions for equation for boundary conditions imposed at a finite radius, for any power-law viscosity $\nu\propto R^{n}$ with $n<2$. We show that mathematical difficulties can be minimized with the aid of the appropriate integral transform techniques, namely the Weber transform [@Titchm23] and the recently proved generalized Weber transform [@ZT07]. We present two specific solutions of astrophysical interest: the solution with zero torque at a radius $R_{\rm in}>0$, which is of interest for accretion discs around black holes and slowly rotating stars; and the solution with zero mass flow at $R_{\rm in}>0$, which is applicable to accretion flows that accumulate mass at the disc center due to the injection of angular momentum from the tidal torques of a binary or perhaps the strong magnetic field of the central object. This paper is organized as follows. In §2, we review the Green’s function solutions, derived by [@Lust52] and , for the thin-disc equation with boundary conditions imposed at the origin. In §3, we derive the new Green’s function solutions, which impose boundary conditions at a finite inner boundary radius. We offer our conclusions in §4. Green’s-Function Solutions with Boundary Conditions at $R=0$ ============================================================ In the special case where the viscosity is a radial power law, $\nu\propto R^{n}$, and assuming a separable ansatz of the form $\Sigma(R,t)=R^{p}\sigma(R)\exp(-\Lambda t)$, where $p$ and $\Lambda$ are real numbers and $\sigma$ is an arbitrary function of $R$, equation can be rewritten as the Bessel differential equation: R\^[2]{}+(2p+2n+)R +=0. \[eq:Bessel\] Above, $s=\nu R^{-n}$ is a constant. With the choices $p=n-1/4$ and $\Lambda=3sk^{2}$, equation has the general solution \_[k]{}(R)= R\^[-2n]{}. Above, $k$ is an arbitrary mode of the solution; $A(k)$ and $B(k)$ are the mode weights; $\ell = (4-2n)^{-1}>0$; $y(R)\equiv R^{(1-n/2)}/(1-n/2)$; and $J_{\ell}$ and $Y_{\ell}$ are the Bessel functions of the first and second kinds, respectively, and of order $\ell$. If $\ell$ is not an integer, then $Y_{\ell}$ above may be replaced without loss of generality by $J_{-\ell}$. Integrating the fundamental solution across all possible $k$-modes gives the solution: (R,t) = \_[0]{}\^R\^[-n-1/4]{} (-3s k\^[2]{}t) dk. \[eq:gensol\] The mode-weighting functions $A(k)$ and $B(k)$ are determined by the boundary conditions and the initial surface density profile $\Sigma (R,t=0)$. Our goal is to rewrite equation in the Green’s function form (equation \[eq:Green\]) and to write down an explicit symbolic expression for the Green’s function $G(R,R^{\prime},t)$. Throughout this paper, we will employ the following strategy: 1. Using the boundary condition, find an analytic relationship between the mode weights $A(k)$ and $B(k)$. 2. Identify the appropriate integral transform to express the mode weights in terms of the initial profile $\Sigma (R,t=0)$. 3. Insert the time-dependence $\exp(-3sk^{2}t)$ and integrate over all modes to find the Green’s function. 4. Derive analytic expressions for the asymptotic disc behavior at late times and small radii. Before deriving the solutions with boundary conditions at finite radius, we begin by reviewing the Green’s functions of with boundary conditions at the coordinate origin. Zero torque at $R_{\rm in}=0$ {#subsec:zerotorqueR0} ----------------------------- An inner boundary condition with zero central torque is of astrophysical interest as it can be used to describe accretion onto a black hole or a slowly rotating star, at radii much larger than the radius of innermost circular orbit or the stellar surface, respectively. The radial torque density $g$ in the disc due to viscous shear is g(R,t)=R\^[2]{}R\^[1/2]{}, where $\Omega_{\rm K}$ is the Keplerian angular velocity of the orbit. Because the functions $J_{\ell}$ and $Y_{\ell}$ have the asymptotic behaviors $J_{\ell}(ky)\propto y^{\ell}\propto R^{1/4}$ and $Y_{\ell}(ky)\propto y^{-\ell}\propto R^{-1/4}$ near the origin, at small radii the mode weight $A(k)$ will contribute to the behavior $g\propto R^{1/2}$ while $B(k)$ will contribute to $g={\rm constant}$. Therefore, for the solution to have zero viscous torque at $R=0$ the function $B(k)$ must be identically zero. We may relate the surface density distribution at $t=0$ and the weight $A(k)$ via the integral equation (R,t=0)=R\^[-n-1/4]{}\_[0]{}\^A(k)J\_(ky) dk, which may be solved with the use of the Hankel integral transform [e.g., @Ogilvie05]. A Hankel transform pair of order $\ell$ satisfies $$\begin{aligned} \phi_{\ell}(x)&=\int_{0}^{\infty}\Phi_{\ell}(k)~J_{\ell}(kx)~k~dk,\\ \Phi_{\ell}(k)&=\int_{0}^{\infty}\phi_{\ell}(x)~J_{\ell}(kx)~x~dx.\end{aligned}$$ For the problem at hand, the suitable transform pair is $$\begin{aligned} R^{n+1/4}\Sigma(R,t=0)&=\int_{0}^{\infty}\left[A(k)k^{-1}\right]J_{\ell}(ky)~k~dk,\\ A(k)k^{-1}&=\int_{0}^{\infty}\left[R^{n+1/4}\Sigma(R,t=0)\right]J_{\ell}(ky)~y~dy.\end{aligned}$$ Combining them gives us $A(k)$: A(k)=(1-)\^[-1]{}\_[0]{}\^ (y\^,0) J\_(ky\^)k R\^[5/4]{} dR\^ \[eq:sig2\] Inserting equation and $B(k)=0$ into equation , we obtain (R,t)=(1-)\^[-1]{}R\^[-n-1/4]{}\_[0]{}\^R\^[5/4]{}\_[0]{}\^ (R\^,t=0) J\_(ky\^) J\_(ky) (-3sk\^[2]{}t) kdkdR\^. To pose the solution in terms of a Green’s function $G(R,R^{\prime},t)$ (equation \[eq:Green\]), we write $$\begin{aligned} G(R,R^{\prime},t)&=\left(1-\frac{n}{2}\right)^{-1}R^{-n-1/4}R^{\prime 5/4}\int_{0}^{\infty} J_{\ell}(ky^{\prime})~J_{\ell}(ky) ~\exp\left(-3sk^{2}t\right)~ k\;dk\nonumber\\ &=(2-n)\frac{R^{-9/4}R^{\prime 5/4}}{\tau(R)} I_{\ell}\left[\frac{2\left(R^{\prime}/R\right)^{1-n/2}}{\tau(R)}\right] \exp\left[-\frac{1+\left(R^{\prime}/R\right)^{2-n}}{\tau(R)}\right]. \label{eq:G1}\end{aligned}$$ Above, $I_{\ell}$ is the modified Bessel function of the first kind, and we have substituted $\tau(R)\equiv 12(1-n/2)^{2}R^{n-2}st = 8(1-n/2)^{2}[t/t_{\nu}(R)]$, where $t_{\nu}(R)=(2/3)R^{2}/\nu(R)$ is the local viscous timescale at $R$. Although the Green’s function allows for the calculation of $\Sigma(R,t)$ for arbitrary initial surface density profiles, it is instructive to study the case where the initial surface density is a Dirac $\delta$ function, (R,t=0)=\_[0]{} (R-R\_[0]{}) R\_[0]{}, \[eq:delta\] for which the solution is (by definition) the Green’s function itself. The integral over radius in equation becomes trivial and many behaviors of the solution may be expressed analytically. Because any initial surface density profile can be described as a superposition of $\delta$-functions, studying this special case will help illuminate the general behavior of all solutions. We may evaluate the asymptotic behavior at late times and small radii by noting that for small argument $z\ltsim 0.2\sqrt{1+\ell}$, $I_{\ell}(z)\approx (z/2)^{\ell}/\Gamma(\ell+1)$. We find (R, tt\_(R))\_[0]{} ()\^[-n]{}\^[-1-]{}, \[eq:asym1\] where $t_{\nu,0}\equiv t_{\nu}(R_{0})$. Thus, for these solutions the inward radial mass flow, (R)=-2Rv\_[R]{}=6R\^[1/2]{}(R\^[1/2]{}), becomes radially constant near the origin and at late times: (tt\_(R)) \_[0]{} \^[-1-]{}. \[eq:Mdot\] Above, we have defined $\dot{M}_{0}\equiv 3\pi\nu (R_{0})\Sigma_{0}$. The power per unit area that is locally viscously dissipated from each face of the disc is $F=(9/8)\nu\Sigma \Omega^{2}$. The total power dissipated near the center of the disc diverges: L(R, tt\_(R))=\_[0]{}\^[R]{}(R\^)(R\^,t)\^[2]{}(R\^) R\^ dR\^ \_[0]{}\^[R]{}R\^[-2]{} dR\^. Astrophysical accretion flows do not extend to zero radius, and thus in practice one may truncate the disc at some plausible boundary radius, for example the radius of innermost stable circular orbit for a disc around a black hole, by approximating the effects of a finite boundary radius . Although we have used a $\delta$ function for demonstrative purposes, the quantities $R_{0}$ and $\Sigma_{0}$ that set the physical scale and normalization of the initial surface density profile, respectively, are arbitrary. The asymptotic behaviors noted above hold for any initial surface density profile: at late times, the surface density profile approaches $\Sigma\propto R^{-n}$, $\dot{M}$ becomes radially constant, and the disc luminosity $L$ formally diverges at the center. At early times and large radii, such that $t\ll \sqrt{t_{\nu}(R) ~t_{\nu}(R^{\prime})}$, we may use the fact that $I_{\ell}(z\gg1) \approx \exp(x)/\sqrt{2\pi z}$ to find G(t){-} ()\^[(3/4)(1+n)]{} . \[eq:Gearly\] In Figure \[fig:1\], we plot the solution $\Sigma(R,t)$ and the radial mass flow $\dot{M}(R,t)$, for the $\delta$-function initial condition (equation \[eq:delta\]), and for viscosity power-law index values $n=0.1$ and $n=1$. In both cases, we see the power-law behavior from equation near the origin as the solution approaches $t\sim t_{\nu,0}$. The disc spreads as the gas at inner annuli loses angular momentum to the gas at outer annuli. The gas initially accumulates near the origin, then becomes diffuse as mass is lost into the origin. Zero mass flow at $R_{\rm in}=0$ -------------------------------- If the accretion flow has a sufficiently strong central source of angular momentum, then the gas will be unable to flow in, and instead accumulate near the origin. Such solutions can be used to describe astrophysical discs around a compact binary [@Pringle91], and perhaps those around compact objects with strong central magnetic fields . For circumbinary thin discs, [@Pringle91] demonstrated that such a boundary condition characterizes quite well the effects of an explicit central torque term. In general, the mass flow has the behavior \_[0]{}\^R\^[1/2]{} (-3sk\^[2]{}t) dk. We have seen above that for solutions with $B(k)=0$ the mass flow is radially constant and finite near the origin. On the other hand, because $Y_{\ell}(ky)\propto R^{-1/4}$ near the origin, the weights $B(k)$ will all contribute no mass flow there; so for zero mass flow at $R_{\rm in}=0$, we require $A(k)=0$. We note that because the surface density will have a power-law $\Sigma \propto R^{-1/2-n}$ at the origin, for the mass contained in the disc to converge $n$ must be less than $3/2$. Thus, for physically realistic solutions with zero mass flow at the origin, $\ell$ cannot be an integer. It follows that in this case $Y_{\ell}$ in equation may be replaced by $J_{-\ell}$ without loss of generality. Then the Green’s function for this case is derived in exactly the same fashion as in the previous case, the only difference being that the order of the Hankel transforms has the opposite sign. We obtain: G(R,R\^,t)=(2-n) I\_[-]{}. As before, we evaluate the late-time behavior for the $\delta$-function initial condition (equation \[eq:delta\]) at small radii: (R, tt\_(R))\_[0]{} ()\^[-n-1/2]{}\^[-1+]{}. \[eq:asym2\] From the above expression it is clear that the boundary condition is satisfied: $\dot{M}\propto \dd(\nu\Sigma R^{1/2})\rightarrow 0$ in the limit $R\rightarrow 0$. Just as we found for the zero-torque boundary condition, the formal expression for the power dissipated at the disc center diverges for the zero-flux solution, with $L(R\le R_{0}, t\ga t_{\nu,0})\propto \int_{0}^{R_{0}}R^{-5/2}~dR$. The asymptotic behavior at early times and large radii is unaffected by the order of the function $I_{\ell}$; it is described by equation \[eq:Gearly\]. Indeed, the inner boundary condition should have no effect on the disc at large radii. Figure \[fig:2\] shows the evolution of the surface density and the radial mass flow for the boundary condition $\dot{M}(R=0)=0$. At early times, the behavior is nearly identical to the zero-torque boundary case. At late times, the zero-flux boundary condition causes the gas to accumulate instead of being lost to the origin. The central mass concentration reaches a maximum, then decreases as the disc begins to spread outward. Green’s-Function Solutions with Boundary Conditions at Finite Radii =================================================================== As we have seen above, Green’s-function solutions of thin accretion discs with $R_{\rm in}=0$ have divergent expressions for the dissipated power, and thus the innermost surface density profile must be manipulated to obtain physically realistic disc luminosities. Analytic treatment of the case with finite boundary radius was briefly discussed in and [@Pringle91], but to the author’s knowledge explicit solutions have never before been published. We show below that the Green’s functions for finite boundary radii can be derived with the aid of the appropriate integral transform techniques, and that they can be represented as ordinary integrals of analytic functions. Zero Torque at $R_{\rm in}>0$ {#ssec:new1} ----------------------------- We wish to solve the problem as in §\[subsec:zerotorqueR0\], but with $R_{\rm in}>0$, i.e. g(R\_[in]{})R\^[1/2]{}\|\_[R=R\_[in]{}]{} (R\_[in]{})R\_[in]{}\^[n+1/2]{}=0. We may relate the mode weights $A(k)$ and $B(k)$ by requiring that every mode of the solution satisfy the boundary condition, i.e.: A(k)J\_(ky\_[in]{})+B(k)Y\_(ky\_[in]{})=0, where $y_{\rm in}\equiv y(R_{\rm in})$. Substituting $C(k)=A(k)/Y_{\ell}(ky_{\rm in}) = -B(k)/J_{\ell}(ky_{\rm in})$, we obtain (R,t)=\_[0]{}\^C(k)R\^[-n-1/4]{}(-3sk\^[2]{}t) dk. The function $C(k)$ may be evaluated with the use of the Weber integral transform [@Titchm23]. A Weber transform pair satisfies $$\begin{aligned} \phi_{\ell}(x)&=\int_{0}^{\infty}\Phi_{\ell}(\kappa )\frac{J_{\ell}(\kappa x)Y_{\ell}(\kappa )-Y_{\ell}(\kappa x)J_{\ell}(\kappa )}{J_{\ell}^{2}(\kappa )+Y_{\ell}^{2}(\kappa )}~\kappa ~d\kappa ,\label{eq:Web1}\\ \Phi_{\ell}(\kappa )&=\int_{1}^{\infty}\phi_{\ell}(x)\left[J_{\ell}(\kappa x)Y_{\ell}(\kappa )-Y_{\ell}(\kappa x)J_{\ell}(\kappa )\right]~x~dx.\label{eq:Web2}\end{aligned}$$ Proceeding as before, we construct the pair $$\begin{aligned} R^{n+1/4}\Sigma(R,t=0)&=\int_{0}^{\infty}\left[C(\kappa )\kappa ^{-1}\right]\frac{J_{\ell}(\kappa x)Y_{\ell}(\kappa )-Y_{\ell}(\kappa x)J_{\ell}(\kappa )}{J_{\ell}^{2}(\kappa )+Y_{\ell}^{2}(\kappa )}~\kappa ~d\kappa , \label{eq:WebC1}\\ C(\kappa )\kappa ^{-1}&=\int_{1}^{\infty}\left[R^{n+1/4}\Sigma(R,t=0)\right]\left[J_{\ell}(\kappa x)Y_{\ell}(\kappa )-Y_{\ell}(\kappa x)J_{\ell}(\kappa )\right]~x~dx.\label{eq:WebC2}\end{aligned}$$ Above, we have substituted $x=y/y_{\rm in}\ge 1$ and $\kappa=ky_{\rm in}$. Note the lower limit of integration in equation is nonzero to account for the finite boundary radius. Combining equations and to eliminate $C(\kappa)$, and inserting the time-dependence factor $\exp(-3sk^{2}t)=\exp[-2(1-n/2)^{2}\kappa^{2}t/t_{\nu,\rm in}]$ where $t_{\rm \nu, in}=t_{\nu}(R_{\rm in})$, we obtain our new Green’s function: $$\begin{aligned} G(R,R^{\prime},t)&=\left(1-\frac{n}{2}\right)R^{-n-1/4}R^{\prime 5/4}R_{\rm in}^{n-2}\nonumber\\ &\qquad\times\int_{0}^{\infty} \frac{\left[J_{\ell}(\kappa x)Y_{\ell}(\kappa )-Y_{\ell}(\kappa x)J_{\ell}(\kappa )\right] \left[J_{\ell}(\kappa x^{\prime})Y_{\ell}(\kappa )-Y_{\ell}(\kappa x^{\prime})J_{\ell}(\kappa )\right]}{J_{\ell}^{2}(\kappa )+Y_{\ell}^{2}(\kappa )}\nonumber\\ &\qquad\times \exp\left[-2\left(1-\frac{n}{2}\right)^{2}\kappa^{2}\frac{t}{t_{\nu,\rm in}}\right]~\kappa ~d\kappa . \label{eq:G3}\end{aligned}$$ Whereas the integral over $k$ in equation has an analytic solution, to the author’s knowledge there is no analytic expression for the integral in equation . Nonetheless, equation gives an exact expression for the Green’s function. While it is somewhat more unwieldy than the solutions for $R_{\rm in}=0$, the additional computational cost of an ordinary integral is not likely to be a significant practical barrier, e.g. one could tabulate the integral in terms of the quantities $x$, $x^{\prime}$ and $t/t_{\rm \nu, in}$. The boundary condition has little effect at large radii, so in practice the behavior far from the boundary is well approximated by the $R_{\rm in}=0$ solutions. The Green’s function in equation does have a closed-form expression for the special case $n=1$ (i.e., $\ell=1/2$). As noted by [@Pringle91], in this case the Bessel functions become easier to handle analytically, with $J_{1/2}(x)=\sqrt{\pi/2}~x^{-1/2}\sin x$ and $Y_{1/2}(x)=-\sqrt{\pi/2}~x^{-1/2}\cos x$. For this value of $n$ we obtain for our Green’s function $$\begin{aligned} G(R,R^{\prime},t)&=\frac{1}{\pi R_{\rm in}}\left(\frac{R^{\prime}}{R}\right)^{5/4}\left(x~x^{\prime}\right)^{-1/2} \int_{0}^{\infty} \sin\left[\kappa(x-1)\right]\sin\left[\kappa(x^{\prime}-1)\right] ~\exp\left[-\frac{\kappa^{2}}{2}\frac{t}{t_{\nu,\rm in}}\right]~d\kappa\nonumber\\ &=\frac{R^{-3/2}R^{\prime}R_{\rm in}^{-1/2}}{2\sqrt{2\pi}}\sqrt{\frac{t_{\nu,\rm in}}{t}} \left\{ \exp\left[-\frac{\left(x-x^{\prime}\right)^{2}}{2}\frac{t}{t_{\nu,\rm in}}\right] -\exp\left[-\frac{\left(x+x^{\prime}-2\right)^{2}}{2}\frac{t}{t_{\nu,\rm in}}\right] \right\}. \label{eq:G3n1}\end{aligned}$$ Note that in the case $n=1$, $x$ and $x^{\prime}$ are simply $\sqrt{R/R_{\rm in}}$ and $\sqrt{R^{\prime}/R_{\rm in}}$, respectively. For general values of $n$, we can evaluate the behavior at late times $t\ga t_{\nu,0}>t_{\nu,\rm in}$ by noting that in this regime only the modes $\kappa^{2}\ltsim 1$ contribute to the integral in equation . For the central region $R\ltsim R_{0}$ at late times, we obtain the following analytic expression for the $\delta$-function initial condition: $$\begin{aligned} \Sigma\left(R, t\ga t_{\nu}(R)\right)\nonumber &\approx \frac{2-n}{2^{1+2\ell}\Gamma^{2}(1+\ell)}\Sigma_{0} \left(\frac{R}{R_{\rm in}}\right)^{-n}\left(\frac{R_{0}}{R_{\rm in}}\right)^{5/2} \left(1-\sqrt{\frac{R_{\rm in}}{R} }\right) \left(\sqrt{\frac{R_{0}}{R_{\rm in}} }-1\right)\nonumber\\ &\qquad\qquad\times\int_{0}^{\infty} \exp\left[-2\left(1-\frac{n}{2}\right)^{2}\kappa^{2}\frac{t}{t_{\nu,\rm in}}\right]~\kappa^{1+2\ell} ~d\kappa\nonumber\\ &=\frac{2-n}{\Gamma(1+\ell)}\Sigma_{0} \left(\frac{R}{R_{\rm in}}\right)^{-n}\left(\frac{R_{0}}{R_{\rm in}}\right)^{5/2} \left(1-\sqrt{\frac{R_{\rm in}}{R} }\right) \left(\sqrt{\frac{R_{0}}{R_{\rm in}} }-1\right) \left[8\left(1-\frac{n}{2}\right)^{2}\frac{t}{t_{\nu,\rm in}}\right]^{-1-\ell}. \label{eq:center}\end{aligned}$$ We see that the Green’s function explicitly gives the asymptotic behavior $\Sigma\propto R^{-n}(1-\sqrt{R_{\rm in}/R})$, which has been used extensively for solutions of accretion discs near zero-torque boundary surfaces (e.g., , @FKR02).[^1] This behavior near the boundary and at late times is general for any initial surface density profile; it is insensitive to the values of $\Sigma_{0}$ and $R_{0}$, and arises for any nonzero $R_{\rm in}$. This qualitative difference in the inner disc from the $R_{\rm in}=0$ case also gives a convergent value for the power dissipated in the central disc. We obtain: L(RR\_[t]{},tt\_(R))= , where $\dot{M}_{\rm ss}$ is the mass flow quantity in equation , and $R_{t}$ is the radius where $t=t_{\nu}(R)$, inside which the disc has had sufficient time to approach the asymptotic solution. In the limit $R_{0}\gg R_{\rm in}$ and $t\gg t_{\nu,0}$, $\dot{M}_{\rm ss}$ may be interpreted as the mass supply rate into the center of the disc from arbitrarily large radii. In this limit the above expression agrees precisely with the standard expression for the luminosity of a steady-state thin accretion disc. We show in Figure \[fig:3\] the exact solutions for the $\delta$-function initial condition, with the no-torque boundary condition imposed at $R_{\rm in}=R_{0}/5$. The qualitative evolution is as predicted by : at early times, far from the boundary, the disc spreads inward in very much the same manner as the solutions with $R_{\rm in}=0$, and so the $R_{\rm in}=0$ Green’s function suffices; at late times, once the gas reaches the vicinity of the boundary it exhibits the behavior $\Sigma\propto R^{-n}(1-\sqrt{R_{\rm in}/R})$ in that neighborhood. Zero Mass Flux at $R_{\rm in}>0$ --------------------------------- We now consider the boundary condition of zero mass flow at a particular radius, (R\_[in]{})(R\^[1/2]{})\_[R=R\_[in]{}]{}{y\^}\_[y=y\_[in]{}]{} =0. From the relations $\dd [x^{\ell}J_{\ell}(x)]/\dd x=x^{\ell}J_{\ell-1}(x)$ and $\dd [x^{\ell}Y_{\ell}(x)]/\dd x=x^{\ell}Y_{\ell-1}(x)$, we obtain the relationship between $A$ and $B$ corresponding to the boundary condition: = -. The solution is then (R,t)=\_[0]{}\^C()R\^[-n-1/4]{} d[@Pringle91] solved the special case $n=1$ analytically, and noted the mathematical difficulty in deriving a solution for a more general case. We find that the mode weight $C(\kappa)$ can in fact be solved for with the use of the recently proved generalized Weber transform [@ZT07], $$\begin{aligned} \phi_{\ell}(x)&=\int_{0}^{\infty}\frac{W_{\ell}(\kappa ,x; a, b)}{Q^{2}_{\ell}(\kappa ; a, b)} ~\Phi_{\ell}(\kappa )~\kappa ~d\kappa ,\\ \Phi_{\ell}(\kappa )&=\int_{1}^{\infty}W_{\ell}(\kappa ,x; a, b)~\phi_{\ell}(x)~x~dx.\end{aligned}$$ The functions $W_{\ell}(\kappa,x; a, b)$ and $Q_{\ell}^{2}(\kappa; a, b)$ are defined as follows: $$\begin{aligned} W_{\ell}(\kappa ,x; a, b)&\equiv J_{\ell}(\kappa x)\left[a Y_{\ell}(\kappa )+b\kappa ~Y_{\ell}^{\prime}(\kappa )\right] - Y_{\ell}(\kappa x)\left[a J_{\ell}(\kappa )+b\kappa ~J_{\ell}^{\prime}(\kappa )\right]\nonumber\\ &= J_{\ell}(\kappa x)\left[\left(a-\ell b\right) Y_{\ell}(\kappa )+b\kappa ~Y_{\ell-1}(\kappa )\right] - Y_{\ell}(\kappa x)\left[\left(a-\ell b\right) J_{\ell}(\kappa )+b\kappa ~J_{\ell-1}(\kappa )\right] \qquad\\ Q_{\ell}^{2}(\kappa ; a, b)&\equiv \left[a Y_{\ell}(\kappa )+b\kappa ~Y_{\ell}^{\prime}(\kappa )\right]^{2} +\left[a J_{\ell}(\kappa )+b\kappa ~J_{\ell}^{\prime}(\kappa )\right]^{2}\nonumber\\ &=\left[\left(a-\ell b\right) Y_{\ell}(\kappa )+b\kappa ~Y_{\ell-1}(\kappa )\right]^{2} +\left[\left(a-\ell b\right) J_{\ell}(\kappa )+b\kappa ~J_{\ell-1}(\kappa )\right]^{2}.\end{aligned}$$ Above, $J_{\ell}^{\prime}$ and $Y_{\ell}^{\prime}$ are the ordinary derivatives of the Bessel functions. If $a=1$ and $b=0$, the pair is identical to the ordinary Weber transform (equations \[eq:Web1\] and \[eq:Web2\]). The choice $a=\ell$ and $b=1$ corresponds to the desired boundary condition $\dot{M}(R_{\rm in},t)=0$. The Green’s function is then: $$\begin{aligned} G(R,R^{\prime},t)&=\left(1-\frac{n}{2}\right)R^{-n-1/4}R^{\prime 5/4}R_{\rm in}^{n-2}\nonumber\\ &\qquad\times\int_{0}^{\infty} \frac{\left[J_{\ell}(\kappa x)Y_{\ell-1}(\kappa )-Y_{\ell}(\kappa x)J_{\ell-1}(\kappa )\right] \left[J_{\ell}(\kappa x^{\prime})Y_{\ell-1}(\kappa )-Y_{\ell}(\kappa x^{\prime})J_{\ell-1}(\kappa )\right]}{J_{\ell-1}^{2}(\kappa )+Y_{\ell-1}^{2}(\kappa )}\nonumber\\ &\qquad\times\exp\left[-2\left(1-\frac{n}{2}\right)^{2}\kappa^{2}\frac{t}{t_{\nu,\rm in}}\right]~\kappa ~d\kappa . \label{eq:G4}\end{aligned}$$ A specific instance of the above Green’s function was derived by [@Pringle91] for the case $n=1$. We can use equation to reproduce that previous solution by noting that $J_{-1/2}(x)=-Y_{1/2}(x)=\sqrt{\pi/2}~x^{-1/2}\cos x$ and $Y_{-1/2}(x)=J_{1/2}(x)=\sqrt{\pi/2}~x^{-1/2}\sin x$. We obtain: $$\begin{aligned} G(R,R^{\prime},t)&=\frac{1}{\pi R_{\rm in}}\left(\frac{R^{\prime}}{R}\right)^{5/4}\left(x~x^{\prime}\right)^{-1/2} \int_{0}^{\infty} \cos\left[\kappa(x-1)\right]\cos\left[\kappa(x^{\prime}-1)\right] ~\exp\left[-\frac{\kappa^{2}}{2}\frac{t}{t_{\nu,\rm in}}\right]~d\kappa\nonumber\\ &=\frac{R^{-3/2}R^{\prime}R_{\rm in}^{-1/2}}{2\sqrt{2\pi}}\sqrt{\frac{t_{\nu,\rm in}}{t}} \left\{ \exp\left[-\frac{\left(x-x^{\prime}\right)^{2}}{2}\frac{t}{t_{\nu,\rm in}}\right] +\exp\left[-\frac{\left(x+x^{\prime}-2\right)^{2}}{2}\frac{t}{t_{\nu,\rm in}}\right] \right\}. \label{eq:G4n1}\end{aligned}$$ The only difference between this Green’s function and the one for $n=1$ and zero torque at $R_{\rm in}$ (equation \[eq:G3n1\]) is the sign in between the exponential functions. For general values of $n$, the analytic late-time behavior of equation turns out to be identical to that for the case $R_{\rm in}=0$ (equation \[eq:asym2\]). This can be confirmed by observing that for $\ell<1$ and small arguments $\kappa\ll 1$ and $\kappa x\ll 1$, $W_{\ell}(\kappa, x; \ell, 1)\approx \csc(\ell\pi)J_{-\ell}(\kappa x)J_{\ell-1}(\kappa)$ and $Q_{\ell}^{2}(\kappa; \ell, 1)\approx \csc^{2}(\ell\pi)J_{\ell-1}^{2}(\kappa)$, and therefore the large fraction in equation is approximately equal to $J_{-\ell}(\kappa x)J_{-\ell}(\kappa x^{\prime})$. However, because the disc does not extend to the origin for a finite boundary, the integral for the central disc luminosity converges. For the $\delta$-function initial surface density profile, we obtain: L(RR\_[t]{}, tt\_(R))\~ (-)\^, where again $\dot{M}_{\rm ss}$ is the mass supply expression defined in §\[ssec:new1\], and $R_{t}$ is the radius where $t=t_{\nu}(R)$, inside which the disc has had sufficient time to approach the asymptotic solution. The above expression for the disc luminosity is in agreement with the estimate of [@IPP99], who considered a zero-flux boundary condition in the context of a thin disc around a supermassive black hole binary. Figure \[fig:4\] shows the solution for the $\delta$-function initial condition and the zero-flux boundary condition at $R_{\rm in}=R_{0}/5$. The panels showing the mass flow clearly exhibit the desired boundary condition. Note that the case $n=1$ (panels b and d) is the case solved analytically by [@Pringle91]. The $n=1$ case, however, leads to a more rapid evolution and steeper late-time profiles than solutions with lower values for $n$; e.g., for the innermost regions of circumbinary discs around supermassive black holes, the viscosity is believed to be roughly constant with radius [@MP05; @TM10]. Conclusion ========== We have presented Green’s functions to the equation for viscous diffusion in a thin Keplerian accretion disc, in the special case of a power-law viscosity profile $\nu\propto R^{n}$, for two different types of boundary conditions, zero viscous torque or zero mass flow, imposed at a finite inner radius $R_{\rm in}>0$. They are extensions of the elegant analytic solutions derived by [@Lust52] and for the same boundary conditions applied at $R_{\rm in}=0$. While the problem of the finite-radius boundary had been mentioned previously in the literature, to the author’s knowledge these solutions have not been explicitly pursued, and are presented here for the first time. The new solutions can be used to model the time-dependent behavior of the innermost regions of accretion discs, where the finite physical size of the central objects can significantly affect the observable characteristics of the disc. Whereas the power viscously dissipated in the $R_{\rm in}=0$ solutions diverge, and require manipulation of the profile at the disc center to calculate physically plausible disc luminosities, the power for the new solutions converge to expressions that are consistent with disc luminosities inferred by other (non-Green’s function) methods. The solutions presented here complement the numerous approximate solutions and numerical treatments in the literature.[^2] The integral transforms used to derive the solutions are applicable to a wide class of boundary conditions, and may be applicable to astrophysical thin-disc systems and configurations not considered here. Because the generalized Weber transform by its nature is applicable to many second-order differential equations with intrinsic cylindrical symmetry, they may also prove to be useful in solving other mathematical equations in astrophysics and other fields. Acknowledgements {#acknowledgements .unnumbered} ================ It is a pleasure to thank Kristen Menou and Zoltán Haiman for helpful conversations and comments on the manuscript; and Jim Pringle and Jeremy Goodman for consultation regarding the literature. The author is also grateful to the Kavli Institute for Theoretical Physics, where a part of this work took place, for their hospitality. Support for this work was provided by NASA ATFP grant NNXO8AH35G (to KM and ZH), and also by the Polányi Program of the Hungarian National Office of Technology (to ZH). \[lastpage\] [^1]: The factor arises from assuming that $\Omega$ is nearly Keplerian at the radius where the torque $g\propto \dd \Omega/\dd R=0$ [@FKR02]. [^2]: For example, [@Cannizzo+90] studied the accretion of a tidally disrupted star onto a black hole via numerical solutions and analytic self-similar solutions. The problem of a thin disc with $\dot{M}=0$ at a finite radius was discussed for the non-linear case $\nu\propto \Sigma^{m}\nu^{n}$ by [@Pringle91] and [@IPP99], with both papers providing numerical solutions as well as analytic approximations.
--- abstract: 'We realise the first and second Grushin distributions as symmetry reductions of the 3-dimensional Heisenberg distribution and 4-dimensional Engel distribution respectively. Similarly, we realise the Martinet distribution as an alternative symmetry reduction of the Engel distribution. These reductions allow us to derive the integrability conditions for the Grushin and Martinet distributions and build certain complexes of differential operators. These complexes are well-behaved despite the distributions they resolve being non-regular.' address: - '-Department of Mathematics, Eastern Michigan University,Ann Arbor, MI 48197, USA' - '-Department of Mathematics and Statistics,Georgetown University, Washington, DC 20057, USADepartment of Mathematics, Fu-Jen Catholic University,Taipei, Taiwan 24205, ROC' - '-Mathematical Sciences Institute, Australian National University,ACT 0200, Australia' author: - Ovidiu Calin - 'Der-Chen Chang' - Michael Eastwood title: Integrability conditions for the Grushin and Martinet distributions --- Introduction ============ For each $k\geq 0$, the pair of linear differential operators on ${\mathbb{R}}^2$ $$X\equiv \partial/\partial x\qquad Y\equiv x^k\partial/\partial y$$ generate what is known as a [Grushin distribution]{} [@G]. For $k\geq 1$, it is not really a distribution in the classical sense because the span of $X$ and $Y$ drops rank along the $y$-axis, $\{x=0\}$. Nevertheless, the fields $X$ and $Y$ are [bracket generating]{} in the sense that taking sufficiently many Lie brackets amongst them generates all vector fields. For example, if $k=1$, then $$X\qquad Y\qquad Z\equiv[X,Y]=\partial/\partial y$$ span all vector fields at which point we notice that $$[X,Z]=0\qquad[Y,Z]=0.$$ Similarly, if $k=2$, then $$X\qquad Y\qquad Z\equiv[X,Y]=2x\partial/\partial y \qquad W\equiv[X,Z]=2\partial/\partial y$$ span all vector fields at which point all other commutators between these fields vanish. In this article, we shall only be concerned with the cases $k=1$ and $k=2$ but, in fact, the case $k=0$ is very familiar for then the integrability equation for the system $$\left.\begin{array}{crcl}Xf&=&a\\ Yf&=&b\end{array}\right\}\quad\mbox{is}\enskip Xb=Ya.$$ More precisely, if we denote by ${\mathcal{E}}$ the germs of smooth functions on ${\mathbb{R}}^2$, then the complex of differential operators $$\begin{picture}(160,40) \put(0,20){\makebox(0,0){$0$}} \put(10,20){\vector(1,0){20}} \put(40,20){\makebox(0,0){${\mathcal{E}}$}} \put(50,22){\vector(2,1){20}} \put(50,18){\vector(2,-1){20}} \put(80,34){\makebox(0,0){${\mathcal{E}}$}} \put(80,20){\makebox(0,0){$\oplus$}} \put(80,6){\makebox(0,0){${\mathcal{E}}$}} \put(90,32){\vector(2,-1){20}} \put(90,8){\vector(2,1){20}} \put(120,20){\makebox(0,0){${\mathcal{E}}$}} \put(130,20){\vector(1,0){20}} \put(160,20){\makebox(0,0){$0$}} \end{picture}$$ is locally exact except at the leftmost ${\mathcal{E}}$ where the cohomology is ${\mathbb{R}}$, the real-valued locally constant functions. This is the familiar de Rham complex with local exactness a consequence of the Poincaré Lemma. The aim of this article is to present similar integrability conditions and consequent differential complexes for the Grushin distribution in cases $k=1,2$ and also for the Martinet distribution [@M], which is a pair of differential operators on ${\mathbb{R}}^3$ as follows: $$\label{martinet} X=\partial/\partial x\qquad Y=\partial/\partial z + x^2\partial/\partial y.$$ This is not regular in the classical sense because the span of the derived vector fields $$X\qquad Y\qquad Z\equiv[X,Y]=2x\partial/\partial y$$ drops rank along the $(y,z)$-plane, $\{x=0\}$. But, again, the two fields $X$ and $Y$ are bracket generating since $$X\qquad Y\qquad Z\qquad W\equiv[X,Z]=2\partial/\partial y$$ span the full tangent space. The complexes that we shall construct are not true resolutions. In addition to having cohomology equal to ${\mathbb{R}}$ at the start, we shall allow them to have finite-dimensional cohomology in other degrees. Bearing in mind the lack of regularity, i.e. that ${\operatorname{span}}\{X,Y\}$ and other vector spaces generated by Lie bracket are allowed to jump in dimension from point to point, this is a small price to pay. We shall devote separate sections of this article to the three cases under consideration and consign to two appendices brief reviews of the Heisenberg and Engel distributions from which they will be derived. This work was initiated while the authors were participating in the International Workshop on Several Complex Variables and Complex Geometry, which was held during July 9-13, 2012 at the Institute of Mathematics, Academia Sinica, Taipei. The authors would like to thank the local organizers, especially Professor Jih-Hsin Cheng for the invitation and warm hospitality extended to them during their visit to Taiwan. The first Grushin distribution {#firstGrushin} ============================== Recall that in this case we are concerned with the three vector fields $$\label{firstgrushin} X=\partial/\partial x\qquad Y=x\partial/\partial y\qquad Z\equiv[X,Y]=\partial/\partial y$$ on ${\mathbb{R}}^2$ with coördinates $(x,y)$. We shall derive integrability conditions from the Heisenberg fields on ${\mathbb{R}}^3$ with coördinates $(x,y,t)$, namely $$X=\partial/\partial x\qquad Y=\partial/\partial t+x\partial/\partial y \qquad Z\equiv[X,Y]=\partial/\partial y.$$ The [Rumin complex]{}, discussed in Appendix \[rumin\], says that $$\label{Rumincomplex} \raisebox{-30pt}{\begin{picture}(276,65)(0,-25) \put(0,20){\makebox(0,0){$0$}} \put(10,20){\vector(1,0){20}} \put(40,20){\makebox(0,0){${\mathcal{E}}_3$}} \put(50,22){\vector(2,1){20}} \put(50,18){\vector(2,-1){20}} \put(80,34){\makebox(0,0){${\mathcal{E}}_3$}} \put(80,20){\makebox(0,0){$\oplus$}} \put(80,6){\makebox(0,0){${\mathcal{E}}_3$}} \put(90,32){\vector(4,-1){96}} \put(90,8){\vector(4,1){96}} \put(90,34){\vector(1,0){96}} \put(90,6){\vector(1,0){96}} \put(196,34){\makebox(0,0){${\mathcal{E}}_3$}} \put(196,20){\makebox(0,0){$\oplus$}} \put(196,6){\makebox(0,0){${\mathcal{E}}_3$}} \put(206,32){\vector(2,-1){20}} \put(206,8){\vector(2,1){20}} \put(236,20){\makebox(0,0){${\mathcal{E}}_3$}} \put(246,20){\vector(1,0){20}} \put(276,20){\makebox(0,0){$0$}} \put(50,-15){\makebox(0,0){\scriptsize$f\mapsto\left[\!\!\begin{array}c Xf\\ Yf\end{array}\!\!\right]$}} \put(138,-15){\makebox(0,0){\scriptsize$\left[\!\!\begin{array}c a\\ b\end{array}\!\!\right]\mapsto\left[\!\!\begin{array}c X^2b-(XY+Z)a\\ Y^2a-(YX-Z)b\end{array}\!\!\right]$}} \put(237,-15){\makebox(0,0){\scriptsize$\left[\!\!\begin{array}c c\\ d\end{array}\!\!\right]\mapsto Xd+Yc$}} \end{picture}}$$ is locally exact except at the leftmost ${\mathcal{E}}_3$ where the cohomology is ${\mathbb{R}}$, the real-valued locally constant functions. Here, we are writing ${\mathcal{E}}_3$ for the germs of smooth functions of the three variables $(x,y,t)$ and shortly we shall write ${\mathcal{E}}_2$ for the germs of smooth functions of the two variables $(x,y)$. Evidently, there is a short exact sequence $$0\to{\mathcal{E}}_2\to{\mathcal{E}}_3\xrightarrow{\,\partial/\partial t\,} {\mathcal{E}}_3\to 0.$$ Also note that the vector field $\partial/\partial t$ commutes with $X,Y,Z$. Therefore, we may consider the commutative diagram $$\begin{picture}(276,110)(0,-70) \put(0,20){\makebox(0,0){$0$}} \put(10,20){\vector(1,0){20}} \put(40,20){\makebox(0,0){${\mathcal{E}}_3$}} \put(50,22){\vector(2,1){20}} \put(50,18){\vector(2,-1){20}} \put(80,34){\makebox(0,0){${\mathcal{E}}_3$}} \put(80,20){\makebox(0,0){$\oplus$}} \put(80,6){\makebox(0,0){${\mathcal{E}}_3$}} \put(90,32){\vector(4,-1){96}} \put(90,8){\vector(4,1){96}} \put(90,34){\vector(1,0){96}} \put(90,6){\vector(1,0){96}} \put(196,34){\makebox(0,0){${\mathcal{E}}_3$}} \put(196,20){\makebox(0,0){$\oplus$}} \put(196,6){\makebox(0,0){${\mathcal{E}}_3$}} \put(206,32){\vector(2,-1){20}} \put(206,8){\vector(2,1){20}} \put(236,20){\makebox(0,0){${\mathcal{E}}_3$}} \put(246,20){\vector(1,0){20}} \put(276,20){\makebox(0,0){$0$}} \put(40,-40){\vector(0,1){50}} \put(80,-25){\vector(0,1){20}} \put(196,-25){\vector(0,1){20}} \put(236,-40){\vector(0,1){50}} \put(52,-15){\makebox(0,0){\scriptsize$\partial/\partial t$}} \put(92,-15){\makebox(0,0){\scriptsize$\partial/\partial t$}} \put(208,-15){\makebox(0,0){\scriptsize$\partial/\partial t$}} \put(248,-15){\makebox(0,0){\scriptsize$\partial/\partial t$}} \put(0,-50){\makebox(0,0){$0$}} \put(10,-50){\vector(1,0){20}} \put(40,-50){\makebox(0,0){${\mathcal{E}}_3$}} \put(50,-48){\vector(2,1){20}} \put(50,-52){\vector(2,-1){20}} \put(80,-36){\makebox(0,0){${\mathcal{E}}_3$}} \put(80,-50){\makebox(0,0){$\oplus$}} \put(80,-64){\makebox(0,0){${\mathcal{E}}_3$}} \put(90,-38){\vector(4,-1){96}} \put(90,-62){\vector(4,1){96}} \put(90,-36){\vector(1,0){96}} \put(90,-64){\vector(1,0){96}} \put(196,-36){\makebox(0,0){${\mathcal{E}}_3$}} \put(196,-50){\makebox(0,0){$\oplus$}} \put(196,-64){\makebox(0,0){${\mathcal{E}}_3$}} \put(206,-38){\vector(2,-1){20}} \put(206,-62){\vector(2,1){20}} \put(236,-50){\makebox(0,0){${\mathcal{E}}_3$}} \put(246,-50){\vector(1,0){20}} \put(276,-50){\makebox(0,0){$0$}} \end{picture}$$ to which we may apply the spectral sequences of a double complex, or indulge in diagram chasing, to conclude that, not only is there a complex of differential operators $$\begin{picture}(276,65)(0,-25) \put(0,20){\makebox(0,0){$0$}} \put(10,20){\vector(1,0){20}} \put(40,20){\makebox(0,0){${\mathcal{E}}_2$}} \put(50,22){\vector(2,1){20}} \put(50,18){\vector(2,-1){20}} \put(80,34){\makebox(0,0){${\mathcal{E}}_2$}} \put(80,20){\makebox(0,0){$\oplus$}} \put(80,6){\makebox(0,0){${\mathcal{E}}_2$}} \put(90,32){\vector(4,-1){96}} \put(90,8){\vector(4,1){96}} \put(90,34){\vector(1,0){96}} \put(90,6){\vector(1,0){96}} \put(196,34){\makebox(0,0){${\mathcal{E}}_2$}} \put(196,20){\makebox(0,0){$\oplus$}} \put(196,6){\makebox(0,0){${\mathcal{E}}_2$}} \put(206,32){\vector(2,-1){20}} \put(206,8){\vector(2,1){20}} \put(236,20){\makebox(0,0){${\mathcal{E}}_2$}} \put(246,20){\vector(1,0){20}} \put(276,20){\makebox(0,0){$0$}} \put(50,-15){\makebox(0,0){\scriptsize$f\mapsto\left[\!\!\begin{array}c Xf\\ Yf\end{array}\!\!\right]$}} \put(138,-15){\makebox(0,0){\scriptsize$\left[\!\!\begin{array}c a\\ b\end{array}\!\!\right]\mapsto\left[\!\!\begin{array}c X^2b-(XY+Z)a\\ Y^2a-(YX-Z)b\end{array}\!\!\right]$}} \put(237,-15){\makebox(0,0){\scriptsize$\left[\!\!\begin{array}c c\\ d\end{array}\!\!\right]\mapsto Xd+Yc$}} \end{picture}$$ in which $X,Y,Z$ now denote the differential operators (\[firstgrushin\]) on ${\mathbb{R}}^2$, but also that the cohomology of this complex resides in the zeroth and first degrees, where it is ${\mathbb{R}}$. In particular, we have found integrability conditions for two smooth functions $a=a(x,y)$ and $b=b(x,y)$ to be locally of the form $a=Xf$ and $b=Yf$ for some $f=f(x,y)$ as follows. Suppose $U^{\mathrm{open}}\subset{\mathbb{R}}^2$ is contractible. Then, for a pair of smooth functions $a$ and $b$ defined on $U$, $$. [rcl]{}X\^2b&=&(XY+Z)a\ Y\^2a&=&(YX-Z)b } ---------------------------------------- $\exists$ a smooth function $f$ on $U$ and a constant $C$ such that $Xf=a$ and $Yf=C+b$. ---------------------------------------- $$ The Martinet distribution {#Martinet} ========================= We shall derive integrability conditions for the Martinet fields (\[martinet\]) on ${\mathbb{R}}^3$ from the Engel complex on ${\mathbb{R}}^4$ constructed from the fields $$\label{Engelfields} \begin{array}{ll}X=\partial/\partial x&\quad Y=\partial/\partial z + x\partial/\partial t+ x^2\partial/\partial y\\ Z\equiv[X,Y]=\partial/\partial t+2x\partial/\partial y&\quad W\equiv[X,Z]=2\partial/\partial y. \end{array}$$ The Engel complex takes the form $$0\to{\mathcal{E}}_4 \begin{array}{c}\nearrow\\ \searrow\end{array}\!\! \begin{array}c{\mathcal{E}}_4\\[2pt] \oplus\\[4pt] {\mathcal{E}}_4\end{array}\!\! \begin{array}c\longrightarrow\\ \mbox{\Large\begin{picture}(0,0) \put(0,0){\makebox(0,0){$\nearrow$}} \put(0,0){\makebox(0,0){$\searrow$}} \end{picture}}\\[4pt] \longrightarrow\end{array}\!\! \begin{array}c{\mathcal{E}}_4\\[2pt] \oplus\\[4pt] {\mathcal{E}}_4\end{array}\!\! \begin{array}c\longrightarrow\\ \mbox{\Large\begin{picture}(0,0) \put(0,0){\makebox(0,0){$\nearrow$}} \put(0,0){\makebox(0,0){$\searrow$}} \end{picture}}\\[4pt] \longrightarrow\end{array}\!\! \begin{array}c{\mathcal{E}}_4\\[2pt] \oplus\\[4pt] {\mathcal{E}}_4\end{array}\!\! \begin{array}{c}\searrow\\ \nearrow\end{array} {\mathcal{E}}_4\to 0,$$ where the differential operators are $$\label{Engeloperators}\raisebox{10pt}{\makebox[200pt]{\small $\begin{array}{l}f\mapsto \left[\!\!\begin{array}{c}Xf\\ Yf\end{array}\!\!\right] \enskip \left[\!\!\begin{array}{c}a\\ b\end{array}\!\!\right]\mapsto \left[\!\!\begin{array}{c}X^3b-(X^2Y+XZ+W)a\\ Y^2a-(YX-Z)b\end{array}\!\!\right]\\[15pt] \hspace{80pt} \left[\!\!\begin{array}{c}c\\ d\end{array}\!\!\right]\mapsto \left[\!\!\begin{array}{c}X^3d+(XY+Z)c\\ Y^2c+(YX^2-ZX+W)d\end{array}\!\!\right] \enskip\left[\!\!\begin{array}{c}g\\ h\end{array}\!\!\right]\mapsto Xh-Yg. \end{array}$}}$$ Notice that $\partial/\partial t$ commutes with each of the vector fields $X,Y,Z,W$ and so we have a commutative diagram $$\begin{array}{ccccccccccccl} 0&\to&{\mathcal{E}}_4&\to&{\mathcal{E}}_4^2&\to&{\mathcal{E}}_4^2 &\to&{\mathcal{E}}_4^2&\to&{\mathcal{E}}_4&\to&0\\ &&\Big\uparrow\makebox[0pt][l]{\scriptsize$\partial/\partial t$} &&\Big\uparrow\makebox[0pt][l]{\scriptsize$\partial/\partial t$} &&\Big\uparrow\makebox[0pt][l]{\scriptsize$\partial/\partial t$} &&\Big\uparrow\makebox[0pt][l]{\scriptsize$\partial/\partial t$} &&\Big\uparrow\makebox[0pt][l]{\scriptsize$\partial/\partial t$}\\ 0&\to&{\mathcal{E}}_4&\to&{\mathcal{E}}_4^2&\to&{\mathcal{E}}_4^2 &\to&{\mathcal{E}}_4^2&\to&{\mathcal{E}}_4&\to&0, \end{array}$$ where each row is the Engel complex. Arguing as in §\[firstGrushin\], we conclude that there is a complex of differential operators on ${\mathbb{R}}^3$ $$0\to{\mathcal{E}}_3 \begin{array}{c}\nearrow\\ \searrow\end{array}\!\! \begin{array}c{\mathcal{E}}_3\\[2pt] \oplus\\[4pt] {\mathcal{E}}_3\end{array}\!\! \begin{array}c\longrightarrow\\ \mbox{\Large\begin{picture}(0,0) \put(0,0){\makebox(0,0){$\nearrow$}} \put(0,0){\makebox(0,0){$\searrow$}} \end{picture}}\\[4pt] \longrightarrow\end{array}\!\! \begin{array}c{\mathcal{E}}_3\\[2pt] \oplus\\[4pt] {\mathcal{E}}_3\end{array}\!\! \begin{array}c\longrightarrow\\ \mbox{\Large\begin{picture}(0,0) \put(0,0){\makebox(0,0){$\nearrow$}} \put(0,0){\makebox(0,0){$\searrow$}} \end{picture}}\\[4pt] \longrightarrow\end{array}\!\! \begin{array}c{\mathcal{E}}_3\\[2pt] \oplus\\[4pt] {\mathcal{E}}_3\end{array}\!\! \begin{array}{c}\searrow\\ \nearrow\end{array} {\mathcal{E}}_3\to 0,$$ in which the differential operators are exactly as in (\[Engeloperators\]) except that $X,Y,Z,W$ now stand for the vector fields $$\begin{array}{ll}X=\partial/\partial x&\quad Y=\partial/\partial z + x^2\partial/\partial y\\ Z\equiv[X,Y]=2x\partial/\partial y&\quad W\equiv[X,Z]=2\partial/\partial y \end{array}$$ on ${\mathbb{R}}^3$, the first two of which are the Martinet fields (\[martinet\]). Moreover, the local cohomology of the complex occurs only in degrees zero and one, where it is ${\mathbb{R}}$. In particular, the integrability conditions for the Martinet fields as as follows. Suppose $U^{\mathrm{open}}\subset{\mathbb{R}}^3$ is contractible. Let $X$ and $Y$ denote the Martinet fields [(\[martinet\])]{} on ${\mathbb{R}}^3$. Then, for a pair of smooth functions $a$ and $b$ defined on $U$, $$. [rcl]{}X\^3b&=&(X\^2Y+XZ+W)a\ Y\^2a&=&(YX-Z)b } ---------------------------------------- $\exists$ a smooth function $f$ on $U$ and a constant $C$ such that $Xf=a$ and $Yf=Cx+b$. ---------------------------------------- $$ Here, for convenience, we have set $Z\equiv[X,Y]$ and $W\equiv[X,Z]$. The second Grushin distribution =============================== Recall that we are concerned with the four vector fields $$\label{secondGrushinfields} X=\partial/\partial x\qquad Y=x^2\partial/\partial y\qquad Z=2x\partial/\partial y\qquad W=2\partial/\partial y$$ on ${\mathbb{R}}^2$, which we may clearly view as the four Engel fields (\[Engelfields\]) acting on smooth functions $f$ on ${\mathbb{R}}^4$ that happen to be of the form $f=f(x,y)$. Evidently, we have the exact sequence $$\label{rel_deRham}\begin{picture}(200,40) \put(0,20){\makebox(0,0){$\;0$}} \put(10,20){\vector(1,0){20}} \put(40,20){\makebox(0,0){${\mathcal{E}}_2$}} \put(50,20){\vector(1,0){20}} \put(80,20){\makebox(0,0){${\mathcal{E}}_4$}} \put(90,22){\vector(2,1){20}} \put(90,18){\vector(2,-1){20}} \put(96,34){\makebox(0,0){\scriptsize$\partial/\partial z$}} \put(96,6){\makebox(0,0){\scriptsize$\partial/\partial t$}} \put(144,34){\makebox(0,0){\scriptsize$\partial/\partial t$}} \put(144,6){\makebox(0,0){\scriptsize$-\partial/\partial z$}} \put(120,34){\makebox(0,0){${\mathcal{E}}_4$}} \put(120,20){\makebox(0,0){$\oplus$}} \put(120,6){\makebox(0,0){${\mathcal{E}}_4$}} \put(130,32){\vector(2,-1){20}} \put(130,8){\vector(2,1){20}} \put(160,20){\makebox(0,0){${\mathcal{E}}_4$}} \put(170,20){\vector(1,0){20}} \put(200,20){\makebox(0,0){$0$.}} \end{picture}$$ It is the de Rham complex in the $(z,t)$-variables. Since both $\partial/\partial z$ and $\partial/\partial t$ commute with the Engel fields (\[Engelfields\]) there is a commutative diagram $$\addtolength{\arraycolsep}{-2pt}\begin{array}{ccccccccccccl} 0&\to&{\mathcal{E}}_4&\to&{\mathcal{E}}_4^2&\to&{\mathcal{E}}_4^2 &\to&{\mathcal{E}}_4^2&\to&{\mathcal{E}}_4&\to&0\\ &&\nearrow\;\nwarrow &&\nearrow\;\nwarrow &&\nearrow\;\nwarrow &&\nearrow\;\nwarrow &&\nearrow\;\nwarrow\\ 0&\to&{\mathcal{E}}_4\oplus{\mathcal{E}}_4&\to& {\mathcal{E}}_4^2\oplus{\mathcal{E}}_4^2&\to& {\mathcal{E}}_4^2\oplus{\mathcal{E}}_4^2&\to& {\mathcal{E}}_4^2\oplus{\mathcal{E}}_4^2&\to& {\mathcal{E}}_4\oplus{\mathcal{E}}_4&\to&0\\ &&\nwarrow\;\nearrow &&\nwarrow\;\nearrow &&\nwarrow\;\nearrow &&\nwarrow\;\nearrow &&\nwarrow\;\nearrow\\ 0&\to&{\mathcal{E}}_4&\to&{\mathcal{E}}_4^2&\to&{\mathcal{E}}_4^2 &\to&{\mathcal{E}}_4^2&\to&{\mathcal{E}}_4&\to&0, \end{array}$$ in which the rows are the Engel complex with appropriate multiplicity and the columns are copies of (\[rel\_deRham\]). As in §\[firstGrushin\] and §\[Martinet\], it follows that there is a complex of differential operators on ${\mathbb{R}}^2$ $$\label{complexforGrushin2} 0\to{\mathcal{E}}_2 \begin{array}{c}\nearrow\\ \searrow\end{array}\!\! \begin{array}c{\mathcal{E}}_2\\[2pt] \oplus\\[4pt] {\mathcal{E}}_2\end{array}\!\! \begin{array}c\longrightarrow\\ \mbox{\Large\begin{picture}(0,0) \put(0,0){\makebox(0,0){$\nearrow$}} \put(0,0){\makebox(0,0){$\searrow$}} \end{picture}}\\[4pt] \longrightarrow\end{array}\!\! \begin{array}c{\mathcal{E}}_2\\[2pt] \oplus\\[4pt] {\mathcal{E}}_2\end{array}\!\! \begin{array}c\longrightarrow\\ \mbox{\Large\begin{picture}(0,0) \put(0,0){\makebox(0,0){$\nearrow$}} \put(0,0){\makebox(0,0){$\searrow$}} \end{picture}}\\[4pt] \longrightarrow\end{array}\!\! \begin{array}c{\mathcal{E}}_2\\[2pt] \oplus\\[4pt] {\mathcal{E}}_2\end{array}\!\! \begin{array}{c}\searrow\\ \nearrow\end{array} {\mathcal{E}}_2\to 0,$$ in which the differential operators are exactly as in (\[Engeloperators\]) except that $X,Y,Z,W$ now stand for the vector fields (\[secondGrushinfields\]). Moreover, this complex has local cohomology ${\mathbb{R}}$ in degree zero, ${\mathbb{R}}\oplus{\mathbb{R}}$ in degree one, and ${\mathbb{R}}$ in degree two. More explicitly, we have proved the following. Suppose $U^{\mathrm{open}}\subset{\mathbb{R}}^2$ is contractible and let $X,Y,Z,W$ denote the Grushin fields [(\[secondGrushinfields\])]{} on ${\mathbb{R}}^2$. Then, for smooth functions $a$ and $b$ defined on $U$, $$\begin{array}{rcl}X^3b&=&(X^2Y+XZ+W)a\\ Y^2a&=&(YX-Z)b\end{array}$$ if and only if there is a smooth function $f$ on $U$ and constants $C$ and $D$ such that $Xf=a$ and $Yf=Cx+D+b$. Furthermore, for smooth functions $c$ and $d$ on $U$, $$\begin{array}{rcl}X^3d+(XY+Z)c&=&0\\ Y^2c+(YX^2-ZX+W)d&=&0\end{array}$$ if and only if there are smooth functions $a$ and $b$ on $U$ and a constant $E$ such that $$\begin{array}{rcl}X^3b-(X^2Y+XZ+W)a&=&c\\ Y^2a-(YX-Z)b&=&E+d. \end{array}$$ Otherwise, the complex [(\[complexforGrushin2\])]{} is locally exact. The Rumin complex on ${\mathbb{R}}^3$ {#rumin} ===================================== There are several different viewpoints on the Rumin complex on ${\mathbb{R}}^3$ and here is not the place to go into the details concerning various subtle distinctions. In fact, the minimal structure required for the basic construction is that of a contact distribution [@R]. A more refined outcome is obtained starting with a pair of line fields that together span a contact distribution. This structure is known as [contact-Lagrangian]{} [@CSbook §4.2.3] or sometimes as [para-CR]{} [@AMT]. For our purposes, it will suffice to consider the so-called [flat model]{}, which may be defined as follows. Choose vector fields $X$ and $Y$ spanning the two line fields and let $Z\equiv[X,Y]$. The contact condition is precisely that $X,Y,Z$ be linearly independent. For the flat model we require that $X$ and $Y$ can be chosen so that $[X,Z]$ and $[Y,Z]$ both vanish (in general, there is a curvature obstruction to this being possible). Following [@Beastwood §8.1], the Rumin complex in this case can be constructed from the de Rham sequence as follows. If we denote by $\xi,\eta,\zeta$ the co-frame dual to $X,Y,Z$, then $$\label{EDS} d\zeta=\eta\wedge\xi\qquad d\xi=0\qquad d\eta=0$$ and we may contemplate the de Rham complex written with respect to this co-frame. Specifically, let us consider the diagram $$\begin{array}{ccccccccccc} &&&&0&&0\\ &&&&\uparrow&&\uparrow\\ &&&&\langle\xi,\eta\rangle&&\langle\eta\wedge\xi\rangle &&\makebox[20pt]{$\langle\eta\wedge\xi\wedge\zeta\rangle$}\\ &&&&\uparrow&&\uparrow&&\\|\\ 0&\to&\Lambda^0&\xrightarrow{\,d\,}&\Lambda^1&\xrightarrow{\,d\,} &\Lambda^2&\xrightarrow{\,d\,}&\Lambda^3&\to&0,\\ &&&&\uparrow&&\uparrow\\ &&&&\langle\zeta\rangle &&\makebox[0pt]{$\langle\xi\wedge\zeta,\zeta\wedge\eta\rangle$}\\ &&&&\uparrow&&\uparrow\\ &&&&0&&0 \end{array}$$ where $\langle\underbar\quad\rangle$ denotes the bundle spanned by the enclosed forms. Notice that the composition $$\langle\zeta\rangle\to\Lambda^1\xrightarrow{\,d\,}\Lambda^2\to \langle\eta\wedge\xi\rangle$$ is simply $$g\,\zeta\mapsto d(g\,\zeta)=dg\wedge\zeta+g\,d\zeta= dg\wedge\zeta+g\,\eta\wedge\xi\mapsto g\,\eta\wedge\xi$$ and hence defines an isomorphism between these line bundles. The Rumin complex is obtained by using this isomorphism to cancel these line bundles hence obtaining, by dint of diagram chasing, a new locally exact complex $$\label{prototypeBGG}0\to\Lambda^0 \begin{array}{c}\nearrow\\ \searrow\end{array}\!\! \begin{array}c\langle\xi\rangle\\[2pt] \oplus\\[4pt] \langle\eta\rangle\end{array}\!\! \begin{array}c\longrightarrow\\ \mbox{\Large\begin{picture}(0,0) \put(0,0){\makebox(0,0){$\nearrow$}} \put(0,0){\makebox(0,0){$\searrow$}} \end{picture}}\\[4pt] \longrightarrow\end{array}\!\! \begin{array}c\langle\xi\wedge\zeta\rangle\\[2pt] \oplus\\[4pt] \langle\zeta\wedge\eta\rangle\end{array}\!\! \begin{array}{c}\searrow\\ \nearrow\end{array} \langle\eta\wedge\xi\wedge\zeta\rangle\to 0.$$ The operators in this complex may be explicitly computed. The one-form $\omega=a\xi+b\eta$ for example, enjoys a unique lift $\tilde\omega$ annihilated by the composition $\Lambda^1\xrightarrow{\,d\,}\Lambda^2\to\langle\eta\wedge\xi\rangle$. Specifically, from (\[EDS\]) we see that $$\begin{array}{l}d(a\xi+b\eta+(Xb-Ya)\zeta)=\\[4pt] \qquad(X(Xb-Ya)-Za)\xi\wedge\zeta +(Y(Ya-Xb)+Zb)\zeta\wedge\eta \end{array}$$ and the formul[æ]{} of (\[Rumincomplex\]) emerge. In fact, as explained in [@Beastwood; @CSbook], the flat model may be identified with the homogeneous space ${\mathrm{SL}}(3,{\mathbb{R}})/B$ where $B$ is the subgroup consisting of upper triangular matrices and then the Rumin complex (\[prototypeBGG\]) is more accurately identified as a [Bernstein-Gelfand-Gelfand (BGG)]{} complex $$0\to\begin{picture}(24,5) \put(4,1.5){\line(1,0){16}} \put(4,1.5){\makebox(0,0){$\times$}} \put(20,1.5){\makebox(0,0){$\times$}} \put(4,8){\makebox(0,0){$\scriptstyle 0$}} \put(20,8){\makebox(0,0){$\scriptstyle 0$}} \end{picture}\begin{array}{c}\nearrow\\ \searrow\end{array}\!\! \begin{array}c\begin{picture}(24,5) \put(4,1.5){\line(1,0){16}} \put(4,1.5){\makebox(0,0){$\times$}} \put(20,1.5){\makebox(0,0){$\times$}} \put(4,7.5){\makebox(0,0){$\scriptstyle -2$}} \put(20,8){\makebox(0,0){$\scriptstyle 1$}} \end{picture}\\[2pt] \oplus\\[4pt] \begin{picture}(24,5) \put(4,1.5){\line(1,0){16}} \put(4,1.5){\makebox(0,0){$\times$}} \put(20,1.5){\makebox(0,0){$\times$}} \put(4,8){\makebox(0,0){$\scriptstyle 1$}} \put(20,7.5){\makebox(0,0){$\scriptstyle -2$}} \end{picture}\end{array}\!\! \begin{array}c\longrightarrow\\ \mbox{\Large\begin{picture}(0,0) \put(0,0){\makebox(0,0){$\nearrow$}} \put(0,0){\makebox(0,0){$\searrow$}} \end{picture}}\\[4pt] \longrightarrow\end{array}\!\! \begin{array}c\begin{picture}(24,5) \put(4,1.5){\line(1,0){16}} \put(4,1.5){\makebox(0,0){$\times$}} \put(20,1.5){\makebox(0,0){$\times$}} \put(4,7.5){\makebox(0,0){$\scriptstyle -3$}} \put(20,8){\makebox(0,0){$\scriptstyle 0$}} \end{picture}\\[2pt] \oplus\\[4pt] \begin{picture}(24,5) \put(4,1.5){\line(1,0){16}} \put(4,1.5){\makebox(0,0){$\times$}} \put(20,1.5){\makebox(0,0){$\times$}} \put(4,8){\makebox(0,0){$\scriptstyle 0$}} \put(20,7.5){\makebox(0,0){$\scriptstyle -3$}} \end{picture}\end{array}\!\! \begin{array}{c}\searrow\\ \nearrow\end{array} \begin{picture}(24,5) \put(4,1.5){\line(1,0){16}} \put(4,1.5){\makebox(0,0){$\times$}} \put(20,1.5){\makebox(0,0){$\times$}} \put(4,7.5){\makebox(0,0){$\scriptstyle -2$}} \put(20,7.5){\makebox(0,0){$\scriptstyle -2$}} \end{picture}\to 0.$$ The Engel complex on ${\mathbb{R}}^4$ {#engel} ===================================== This case is discussed in detail in [@BEGN §§3,7]. Here, suffice it to say that the construction is completely parallel to that just discussed and that the result is a BGG complex for the homogeneous space ${\mathrm{Sp}}(4,{\mathbb{R}})/B$ with $B$ a Borel subgroup. In the notation of [@Beastwood] we obtain $$0\to\begin{picture}(24,5) \put(5.6,0){\line(1,0){12.8}} \put(5.6,3.2){\line(1,0){12.8}} \put(4,1.5){\makebox(0,0){$\times$}} \put(20,1.5){\makebox(0,0){$\times$}} \put(12,1.5){\makebox(0,0){$\langle$}} \put(4,8){\makebox(0,0){$\scriptstyle 0$}} \put(20,8){\makebox(0,0){$\scriptstyle 0$}} \end{picture}\begin{array}{c}\nearrow\\ \searrow\end{array}\!\! \begin{array}c\begin{picture}(24,5) \put(5.6,0){\line(1,0){12.8}} \put(5.6,3.2){\line(1,0){12.8}} \put(4,1.5){\makebox(0,0){$\times$}} \put(20,1.5){\makebox(0,0){$\times$}} \put(12,1.5){\makebox(0,0){$\langle$}} \put(4,7.5){\makebox(0,0){$\scriptstyle -2$}} \put(20,8){\makebox(0,0){$\scriptstyle 1$}} \end{picture}\\[2pt] \oplus\\[4pt] \begin{picture}(24,5) \put(5.6,0){\line(1,0){12.8}} \put(5.6,3.2){\line(1,0){12.8}} \put(4,1.5){\makebox(0,0){$\times$}} \put(20,1.5){\makebox(0,0){$\times$}} \put(12,1.5){\makebox(0,0){$\langle$}} \put(4,8){\makebox(0,0){$\scriptstyle 2$}} \put(20,7.5){\makebox(0,0){$\scriptstyle -2$}} \end{picture}\end{array}\!\! \begin{array}c\longrightarrow\\ \mbox{\Large\begin{picture}(0,0) \put(0,0){\makebox(0,0){$\nearrow$}} \put(0,0){\makebox(0,0){$\searrow$}} \end{picture}}\\[4pt] \longrightarrow\end{array}\!\! \begin{array}c\begin{picture}(24,5) \put(5.6,0){\line(1,0){12.8}} \put(5.6,3.2){\line(1,0){12.8}} \put(4,1.5){\makebox(0,0){$\times$}} \put(20,1.5){\makebox(0,0){$\times$}} \put(12,1.5){\makebox(0,0){$\langle$}} \put(4,7.5){\makebox(0,0){$\scriptstyle -4$}} \put(20,8){\makebox(0,0){$\scriptstyle 1$}} \end{picture}\\[2pt] \oplus\\[4pt] \begin{picture}(24,5) \put(5.6,0){\line(1,0){12.8}} \put(5.6,3.2){\line(1,0){12.8}} \put(4,1.5){\makebox(0,0){$\times$}} \put(20,1.5){\makebox(0,0){$\times$}} \put(12,1.5){\makebox(0,0){$\langle$}} \put(4,8){\makebox(0,0){$\scriptstyle 2$}} \put(20,7.5){\makebox(0,0){$\scriptstyle -3$}} \end{picture}\end{array}\!\! \begin{array}c\longrightarrow\\ \mbox{\Large\begin{picture}(0,0) \put(0,0){\makebox(0,0){$\nearrow$}} \put(0,0){\makebox(0,0){$\searrow$}} \end{picture}}\\[4pt] \longrightarrow\end{array}\!\! \begin{array}c\begin{picture}(24,5) \put(5.6,0){\line(1,0){12.8}} \put(5.6,3.2){\line(1,0){12.8}} \put(4,1.5){\makebox(0,0){$\times$}} \put(20,1.5){\makebox(0,0){$\times$}} \put(12,1.5){\makebox(0,0){$\langle$}} \put(4,7.5){\makebox(0,0){$\scriptstyle -4$}} \put(20,8){\makebox(0,0){$\scriptstyle 0$}} \end{picture}\\[2pt] \oplus\\[4pt] \begin{picture}(24,5) \put(5.6,0){\line(1,0){12.8}} \put(5.6,3.2){\line(1,0){12.8}} \put(4,1.5){\makebox(0,0){$\times$}} \put(20,1.5){\makebox(0,0){$\times$}} \put(12,1.5){\makebox(0,0){$\langle$}} \put(4,8){\makebox(0,0){$\scriptstyle 0$}} \put(20,7.5){\makebox(0,0){$\scriptstyle -3$}} \end{picture}\end{array}\!\! \begin{array}{c}\searrow\\ \nearrow\end{array} \begin{picture}(24,5) \put(5.6,0){\line(1,0){12.8}} \put(5.6,3.2){\line(1,0){12.8}} \put(4,1.5){\makebox(0,0){$\times$}} \put(20,1.5){\makebox(0,0){$\times$}} \put(12,1.5){\makebox(0,0){$\langle$}} \put(4,7.5){\makebox(0,0){$\scriptstyle -2$}} \put(20,7.5){\makebox(0,0){$\scriptstyle -2$}} \end{picture}\to 0$$ and following the construction in [@BEGN] with a co-frame $\xi,\eta,\zeta,\omega$ such that $$d\omega=\zeta\wedge\xi\qquad d\zeta=\eta\wedge\xi\qquad d\xi=0\qquad d\eta=0$$ leads to the explicit formul[æ]{} of (\[Engeloperators\]). We remark that the Engel fields are often written as $$X=\partial/\partial x-z\partial/\partial t-t\partial/\partial y\quad Y=\partial/\partial z \quad Z=\partial/\partial t\quad W=\partial/\partial y$$ but we prefer the form (\[Engelfields\]) so as better to relate to the Martinet and Grushin distributions. [11]{} D.V. Alekseevsky, C. Medori, and A. Tomassini, [Maximally homogeneous para-CR manifolds]{}, Ann. Global Anal. Geom. [30]{} (2006), 1–27. R.J. Baston and M.G. Eastwood, [The Penrose Transform: its Interaction with Representation Theory]{}, Oxford University Press 1989. R.L. Bryant, M.G. Eastwood, A.R. Gover, and K. Neusser [Some differential complexes within and beyond parabolic geometry]{}, arXiv:1112.2142. A. Čap and J. Slovák, [Parabolic Geometries I: Background and General Theory]{}, Amer. Math. Soc. 2009. V.V. Grushin, [A certain class of hypoelliptic operators]{}, Mat. Sb. [83]{} (1970) 456–473. J. Martinet, [Sur les singularités des formes différentielles]{}, Ann. Inst. Fourier [20]{} (1970) 90–178. M. Rumin, [Un complexe de formes différentielles sur les variétés de contact]{}, Comptes Rendus Acad. Sci. Paris Math. [310*]{} (1990) 401–404.
--- abstract: \| Background : An electron localization measure was originally introduced to characterize chemical bond structures in molecules. Recently, a nucleon localization based on Hartree-Fock densities has been introduced to investigate $\alpha$-cluster structures in light nuclei. Compared to the local nucleonic densities, the nucleon localization function has been shown to be an excellent indicator of shell effects and cluster correlations. Purpose : Using the spatial nucleon localization measure, we investigate the emergence of fragments in fissioning heavy nuclei. Methods : To illustrate basic concepts of nucleon localization, we employ the self-consistent energy density functional method with a quantified energy density functional optimized for fission studies. Results : We study the particle densities and spatial nucleon localization distributions along the fission pathways of $^{264}$Fm, $^{232}$Th and $^{240}$Pu. We demonstrate that the fission fragments are formed fairly early in the evolution, well before scission. We illustrate the usefulness of the localization measure by showing how the hyperdeformed state of $^{232}$Th can be understood in terms of a quasimolecular state made of $^{132}$Sn and $^{100}$Zr fragments. Conclusions : Compared to nucleonic distributions, the nucleon localization function more effectively quantifies nucleonic clustering: its characteristic oscillating pattern, traced back to shell effects, is a clear fingerprint of cluster/fragment configurations. This is of particular interest for studies of fragment formation and fragment identification in fissioning nuclei. author: - 'C. L. Zhang (张春莉)' - 'B. Schuetrumpf' - 'W. Nazarewicz' title: Nucleon localization and fragment formation in nuclear fission --- [UTF8]{}[gbsn]{} Introduction ============ The appearance of cluster states in atomic nuclei is a ubiquitous phenomenon with many implications for both nuclear physics and astrophysics [@beck2010clusters; @beck2012clusters; @beck2014clusters; @vonOertzen2006; @DelionCluster]. While several factors are known to contribute to clustering, a comprehensive microscopic understanding of this phenomenon still remains elusive. Cluster configurations can be energetically favorable due to the large binding energy per nucleon in constituent clusters, such as $\alpha$ particles. The binding-energy argument has often been used to explain properties of $\alpha$-conjugate nuclei [@Hafstad], cluster emission [@Rose1984; @Aleksandrov1984] and fission [@BLynn], and the appearance of a gas of light clusters in low-density nuclear matter [@Ropke2006; @Girod13; @Ebran14a] and in the interior region of heavy nuclei [@BRINK1973109]. Another important factor is the coupling to decay channels; this explains [@Okolowicz12; @Okolowicz13] the very occurrence of cluster states at low excitation energies around cluster-decay thresholds [@Ikeda68]. The microscopic description of cluster states requires the use of an advanced many-body, open-system framework [@Okolowicz12; @Okolowicz13; @Cluster12] employing realistic interactions, and there has been significant progress in this area [@Yoshida13; @Epelbaum12; @Epelbaum14; @Elh15; @QMC]. For a global characterization of cluster states in nuclei, a good starting point is density functional theory [@Jones15] based on a realistic nuclear energy density functional, or its self-consistent mean-field variant with density-dependent effective interactions [@bender2003self], to which we shall refer as the energy density functional method (EDFM) in the following. Within EDFM, cluster states have a simple interpretation in terms of quasimolecular states. Since the mean-field approach is rooted in the variational principle, the binding-energy argument favors clustering in certain configurations characterized by large shell effects of constituent fragments [@Leander; @Flocard84; @Marsh86; @Freer95; @maruhn2010linear; @Ichikawa11; @ebran2012atomic; @ebran2014density]; the characteristics of cluster states can be indeed traced back to both the symmetries and geometry of the nuclear mean-field [@Hecht77; @Nazarewicz1992]. The degree of clustering in nuclei is difficult to assess quantitatively in EDFM as the single particle (s.p.) wave functions are spread throughout the nuclear volume; hence, the resulting nucleonic distributions are rather crude indicators of cluster structures as their behavior in the nuclear interior is fairly uniform. Therefore, in this study, we utilize a different measure called spatial localization, which is a more selective signature of clustering and cluster shell structure. The localization, originally introduced for the identification of localized electronic groups in atomic and molecular systems [@Becke1990; @savin1997elf; @scemama2004electron; @Kohout04; @burnus2005time; @Poater], has recently been applied to characterize clusters in light nuclei [@Reinhard2011]. In this work, we investigate the usefulness of the spatial localization as a tool to identify fission fragments in heavy fissioning nuclei. This article is organized as follows: Section \[model\] gives a brief introduction to the EDFM and the localization measure employed in this work. The results for fissioning nuclei $^{264}$Fm, $^{232}$Th, and $^{240}$Pu are presented in Sec. \[sec:heavynuclei-HFB\]. Finally, the summary and outlook are given in Sec. \[summary\]. Model ===== EDFM Implementation ------------------- In superfluid nuclear EDFM, the binding energy is expressed through the general density matrix [@ring2004nuclear; @bender2003self]. By applying the variational principle to s.p. wave functions (Kohn-Sham orbitals), the self-consistent Hartree-Fock-Bogoliubov (HFB) equations are derived. Nuclear EDFM has been successfully used to describe properties of ground states and selected collective states across the nuclear landscape [@bender2003self; @bogner2013; @erler12; @Agbemava14]. In this work, we use Skyrme energy density functionals which are expressed in terms of local nucleonic densities and currents. We employ the UNEDF1 functional optimized for fission [@kortelainen2012nuclear] in the presence of pairing treated by means of the Lipkin-Nogami approximation as in Ref. [@Sto03]. The constrained HFB equations are solved by using the symmetry-unconstrained code HFODD [@schunck2012solution]. Spatial Localization {#model:localization} -------------------- The spatial localization measure was originally introduced in atomic and molecular physics to characterize chemical bonds in electronic systems. It also turned out to be useful for visualizing cluster structures in light nuclei [@Reinhard2011]. It can be derived by considering the conditional probability of finding a nucleon within a distance $\delta$ from a given nucleon at ${\mbox{\boldmath $r$}}$ with the same spin $\sigma$ ($=\uparrow$ or $\downarrow$) and isospin $q$ ($=n$ or $p$). As discussed in [@Becke1990; @Reinhard2011], the expansion of this probability with respect to $\delta$ can be written as $$\label{eqn:probability} R_{q\sigma}({\mbox{\boldmath $r$}},\delta)\approx{1\over 3}\left(\tau_{q\sigma}-{1\over 4}\frac{\|{\mbox{\boldmath $\nabla$}}\rho_{q\sigma}\|^2}{\rho_{q\sigma}}-\frac{{\mbox{\boldmath $j$}}^2_{q\sigma}}{\rho_{q\sigma}}\right)\delta^2+\mathcal{O}(\delta^3),$$ where $\rho_{q\sigma}$, $\tau_{q\sigma}$, ${\mbox{\boldmath $j$}}_{q\sigma}$, and ${\mbox{\boldmath $\nabla$}}\rho_{q\sigma}$ are the particle density, kinetic energy density, current density, and density gradient, respectively. They can be expressed through the canonical HFB orbits $\psi_\alpha({\mbox{\boldmath $r$}}\sigma)$: $$\begin{aligned} \rho_{q\sigma}({\mbox{\boldmath $r$}})&=&\sum_{\alpha\in q}v^2_{\alpha}\|\psi_\alpha({\mbox{\boldmath $r$}}\sigma)\|^2,\\ \tau_{q\sigma}({\mbox{\boldmath $r$}})&=&\sum_{\alpha\in q}v^2_{\alpha}\|{\mbox{\boldmath $\nabla$}}\psi_\alpha({\mbox{\boldmath $r$}}\sigma)\|^2,\\ {\mbox{\boldmath $j$}}_{q\sigma}({\mbox{\boldmath $r$}})&=&\sum_{\alpha\in q}v^2_{\alpha}\mathrm{Im}[\psi^_\alpha({\mbox{\boldmath $r$}}\sigma){\mbox{\boldmath $\nabla$}}\psi_\alpha({\mbox{\boldmath $r$}}\sigma)], \\ {\mbox{\boldmath $\nabla$}}\rho_{q\sigma}({\mbox{\boldmath $r$}})&=&2\sum_{\alpha\in q}v^2_{\alpha}\mathrm{Re}[\psi^_\alpha({\mbox{\boldmath $r$}}\sigma){\mbox{\boldmath $\nabla$}}\psi_\alpha({\mbox{\boldmath $r$}}\sigma)],\end{aligned}$$ with $v^2_{\alpha}$ being the canonical occupation probability. Thus, the expression in the parentheses of Eq. (\[eqn:probability\]) can serve as a localization measure. Unfortunately, this expression is neither dimensionless nor normalized. A natural choice for normalization is the Thomas-Fermi kinetic energy density $\tau^\mathrm{TF}_{q\sigma}={3\over 5}(6\pi^2)^{2/3}\rho_{q\sigma}^{5/3}$. Considering that the spatial localization and $R_{q\sigma}$ are in an inverse relationship, a dimensionless and normalized expression for the localization measure can be written as $$\label{eqn:localization} \mathcal{C}_{q\sigma}({\mbox{\boldmath $r$}})=\left[1+\left(\frac{\tau_{q\sigma}\rho_{q\sigma}-{1\over 4}\|{\mbox{\boldmath $\nabla$}}\rho_{q\sigma}\|^2-{\mbox{\boldmath $j$}}^2_{q\sigma}}{\rho_{q\sigma}\tau^\mathrm{TF}_{q\sigma}}\right)^2\right]^{-1}.$$ We note that the combination $\tau_{q\sigma}\rho_{q\sigma}-{\mbox{\boldmath $j$}}^2_{q\sigma}$ guarantees the Galilean invariance of the localization measure [@Engel75]. In this work, time reversal symmetry is conserved; hence, ${\mbox{\boldmath $j$}}_{q\sigma}$ vanishes. A value of $\mathcal{C}$ close to one indicates that the probability of finding two nucleons with the same spin and isospin at the same spatial location is very low. Thus the nucleon’s localization is large at that point. In particular, nucleons making up the alpha particle are perfectly localized [@Reinhard2011]. Another interesting case is $\mathcal{C}=1/2$, which corresponds to a homogeneous Fermi gas as found in nuclear matter. When applied to many-electron systems, the quantity $\mathcal{C}$ is referred to as the electron localization function, or ELF. In nuclear applications, the measure of localization (\[eqn:localization\]) shall thus be called the nucleon localization function (NLF). The above definition of the NLF works well in regions with non-zero nucleonic density. When the local densities become very small in the regions outside the range of the nuclear mean field, numerical instabilities can appear. On the other hand, when the particle density is close to zero, localization is no longer a meaningful quantity. Consequently, when visualizing localizations for finite nuclei in the 2D plots shown in this paper, we multiply the NLF by a normalized particle density $\mathcal{C}({\mbox{\boldmath $r$}})\rightarrow\mathcal{C}({\mbox{\boldmath $r$}})\rho_{q\sigma}({\mbox{\boldmath $r$}})/[\mathrm{max}(\rho_{q\sigma}({\mbox{\boldmath $r$}})]$. Localization in fissioning heavy nuclei {#sec:heavynuclei-HFB} ======================================= Based on the examples shown in Ref. [@Reinhard2011], we know that the oscillating pattern of NLFs is an excellent tool for visualizing cluster structures in light nuclei. In this work, we apply this tool to monitor the development and evolution of fission fragments in $^{264}$Fm, $^{232}$Th, and $^{240}$Pu. We begin from the discussion of the symmetric fission of $^{264}$Fm, a subject of several recent DFT studies [@Staszczak09; @Sadhukhan14; @Simenel14; @Zhao15]. As shown in Ref. [@Staszczak09], at large values of the mass quadrupole moment $Q_{20}$, the static fission pathway of $^{264}$Fm is symmetric, with a neck emerging at $Q_{20}\approx 145$b, and the scission point reached at $Q_{20} \approx 265$b, above which $^{264}$Fm splits into two fragments. The appearance of the static symmetric fission pathway in $^{264}$Fm is due to shell effects in the emerging fission fragments associated with the doubly magic nucleus $^{132}$Sn [@Hulet89]. ![Nucleonic densities (in nucleons/fm$^3$) and spatial localizations for $^{264}$Fm obtained from HFB calculations with UNEDF1 for three configurations along the symmetric fission pathway corresponding to different values of the mass quadrupole moment $Q_{20}$ and decreasing neck size.[]{data-label="fig:fm264"}](plot_fm264-localization.pdf){width="\linewidth"} Figure \[fig:fm264\] shows neutron and proton densities and NLFs for $^{264}$Fm along the fission pathway. We choose three very elongated configurations corresponding to decreasing neck sizes. To study the gradual emergence of fission fragments, we performed HFB calculations for the ground state densities and NLFs of $^{132}$Sn, see Fig. \[fig:sn132\]. The nucleus $^{132}$Sn is a doubly-magic system with a characteristic shell structure. Except for a small depression at the center of proton density in Fig. \[fig:sn132\](c), the nucleonic densities are almost constant in the interior. On the other hand, the NLFs show patterns of concentric rings with enhanced localization, in which the neutron NLF exhibits one additional maximum as compared to the proton NLF; this is due to the additional closed neutron shell. As one can see, unlike in atomic systems [@Becke1990], the total number of shells cannot be directly read from the number of peaks in the NLF, because the radial distributions of wave functions belonging to different nucleonic shells vary fairly smoothly and are poorly separated in space. Nevertheless, each magic number leaves a strong and unique imprint on the spatial localization. ![Nucleonic densities (in nucleons/fm$^3$) and spatial localizations for the ground state of $^{132}$Sn.[]{data-label="fig:sn132"}](plot_sn132-localization.pdf){width="\linewidth"} ![Neutron (left) and proton (right) NLF profiles for $^{264}$Fm (blue thick line) and two $^{132}$Sn (red and green line) along the $z$ axis ($r=0$). The three panels (a-b), (c-d), and (e-f) correspond to three deformed configurations of Fig. \[fig:fm264\].[]{data-label="fig:loc-line0"}](fm264-loc-line.pdf){width="0.8\linewidth"} By comparing the results of Figs. \[fig:fm264\] and \[fig:sn132\] one can clearly see the gradual development of the $^{132}$Sn clusters within the fissioning $^{264}$Fm. It is striking to see that the ring-like pattern of NLFs develops at an early stage of fission, at which the neck is hardly formed. To illustrate this point more clearly, Fig. \[fig:loc-line0\] displays the NLFs for the elongated configurations of $^{264}$Fm shown in Fig. \[fig:fm264\] along the $z$-axis and compares them to those of $^{132}$Sn. To avoid normalization problems we present NLFs given by Eq. (\[eqn:localization\]), i.e., without applying the density form factor. It is seen that the localizations of the emerging fragments match those of $^{132}$Sn fairly well in the exterior region. Let us now discuss two examples of asymmetric fission. Figure \[fig:PES\] shows the potential energy curves of $^{232}$Th and $^{240}$Pu along the most probable fission pathway predicted, respectively, in Refs. [@McDonnell2013] and [@Sadhukhan2016]. Both curves show secondary minima associated with the superdeformed fission isomers. For $^{232}$Th, a pronounced softness is observed at large quadrupole moments $Q_{20}\approx150-200$b. In this region of collective space, a hyperdeformed third minimum is predicted by some Skyrme functionals [@McDonnell2013]. In the next step, we consider five configurations along the fission pathway to perform detailed localization analysis. ![The potential energy curves of $^{232}$Th and $^{240}$Pu calculated with UNEDF1 along the fission pathways [@McDonnell2013; @Sadhukhan2016]. The configurations further discussed in Figs. \[fig:th232\] and \[fig:pu240\] are marked by symbols. Their quadrupole and octupole moments, $Q_{20}$(b) and $Q_{30}$ (b$^{3/2}$) respectively, are indicated.[]{data-label="fig:PES"}](q20-energy.pdf){width="\linewidth"} ![Nucleonic densities (in nucleons/fm$^3$) and spatial localizations for $^{232}$Th obtained from HFB calculations with UNEDF1for five configurations along the fission pathway marked in Fig. \[fig:PES\].[]{data-label="fig:th232"}](plot_th232-localization.pdf){width="\linewidth"} Figure \[fig:th232\] shows neutron and proton densities and NLFs for $^{232}$Th along the fission pathway. The first column corresponds to the ground-state configuration where the densities do not show obvious internal structures. However, the neutron NLF shows three concentric ellipses and the proton NLF exhibits two maxima and an enhancement at the surface. The second column corresponds to the fission isomer. Here two-center distributions begin to form in both NLFs. As discussed in [@McDonnell2013], the distributions shown in the third column can be associated with a quasimolecular “third-minimum" configuration, in which one fragment bears a strong resemblance to the doubly magic nucleus $^{132}$Sn. The forth column represents the configuration close to the scission point, where two well-developed fragments are present. As seen in the last column, at larger elongations the nucleus breaks up into two fragments, one spherical and another one strongly deformed shape. ![Similar to Fig. \[fig:sn132\], but for $^{100}$Zr.[]{data-label="fig:zr100"}](plot_zr100-localization.pdf){width="\linewidth"} ![Neutron (left) and proton (right) NLF profiles for $^{232}$Th (blue thick line), $^{100}$Zr (green line), and $^{132}$Sn (red line) along the $z$ axis ($r=0$). The first, second, and third panel correspond to the configurations in the third, fourth and fifth columns of Fig. \[fig:th232\], respectively.[]{data-label="fig:loc-line"}](th232-loc-line-3.pdf){width="0.8\linewidth"} To study the evolution of fission fragments, in addition to $^{132}$Sn (Fig. \[fig:sn132\]) we study $^{100}$Zr, which is the second presumed fission product of $^{232}$Th. The calculation for $^{100}$Zr is performed at the prolate configuration with ${Q}_{20}=10$b, which corresponds to the lighter fission fragment predicted in [@McDonnell2013]. The results are shown in Fig. \[fig:zr100\]. Again, while the particle densities are almost constant in the interior, the neutron NLF shows two concentric ovals and the proton NLF exhibits two centers in the interior and one enhanced oval at the surface. The characteristic patterns seen in the NLFs of fission fragments can be spotted during the evolution of $^{232}$Th in Fig. \[fig:th232\]. To show it more clearly, Fig. \[fig:loc-line\] displays the NLFs of the three most elongated configurations of $^{232}$Th along the $z$-axis in Fig. \[fig:th232\] and compares them to those of $^{132}$Sn and $^{100}$Zr. In Figs. \[fig:loc-line\] (a) and (b), neutron and proton localizations at the center are around 0.5, which is close to the Fermi gas limit. This is expected for a fairly heavy nucleus. In the exterior, the localizations of two developing fragments match those of $^{100}$Zr and $^{132}$Sn fairly well. In panels (c) and (d), the NLFs of $^{232}$Th grow in the interior; this demonstrates that the nucleons become localized at the neck region. Finally, in panels (e) and (f), the fission fragments are separated and their NLFs are consistent with the localizations of $^{100}$Zr and $^{132}$Sn. ![Similar to Fig. \[fig:th232\] but for the configurations of $^{240}$Pu indicated in Fig. \[fig:PES\].[]{data-label="fig:pu240"}](plot_pu240-localization.pdf){width="\linewidth"} Finally, let us consider the important case of $^{240}$Pu. Recently, a microscopic modeling of mass and charge distributions in spontaneous fission of this nucleus was carried out in Ref. [@Sadhukhan2016]. To give an insight into the evolution of $^{240}$Pu along its fission pathway, in Fig. \[fig:pu240\] we illustrate the NLFs of $^{240}$Pu. The transition to the reflection-asymmetric pathway begins at $Q_{20}\approx 95$b. It is seen that two nascent fragments start developing at this configuration. At larger elongations internal parity is broken and two fragments are formed with distinct shell imprints in the corresponding NLFs. In the last column, the rings of enhanced localization are almost closed, and the fragments are nearly separated. The examples presented above show in a rather dramatic fashion that the NLFs can serve as excellent fingerprints of both the formation and evolution of cluster structures in fissioning nuclei. Conclusion {#summary} ========== In this work, we presented DFT developments pertaining to the theoretical description of fragment emergence in heavy fissioning systems. Building upon results of previous work on cluster formation in light nuclei [@Reinhard2011], we demonstrated that the nucleon spatial localization is a superb indicator of clustering in heavy nuclei; the characteristic patterns of NLFs can serve as fingerprints of the single-particle shell structure associated with cluster configurations. While the characteristic oscillating pattern of NLFs magnifies the cluster structures in light nuclei [@Reinhard2011], shell effects in nascent fragments in fissioning nuclei also leave a strong imprint on the localization. Our EDFM analysis of fission evolution of $^{264}$Fm, $^{232}$Th, and $^{240}$Pu demonstrates that the fragments are formed fairly early in the evolution, well before scission. Future applications of NLFs will include studies of clustering in medium-mass nuclei as well as the identification of fission yields. Another interesting use of NLFs will be in the description and visualization of collective rotational motion, where spin-up and spin-down localizations are different due to the breaking of time-reversal symmetry. Such a study will provide insights into the angular momentum dynamics in atomic nuclei. Useful discussions with P.-G. Reinhard, N. Schunck, and A.S. Umar, and the help of E. Olsen are gratefully acknowledged. This work was supported by the U.S. Department of Energy, Office of Science under Award Numbers DOE-DE-NA0002847 (the Stewardship Science Academic Alliances program), DE-SC0013365 (Michigan State University), and DE-SC0008511 (NUCLEI SciDAC-3 collaboration). An award of computer time was provided by the Institute for Cyber-Enabled Research at Michigan State University. We also used computational resource provided by the National Energy Research Scientific Computing Center (NERSC). 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--- abstract: 'Low-density parity-check codes, a class of capacity-approaching linear codes, are particularly recognized for their efficient decoding scheme. The decoding scheme, known as the sum-product, is an iterative algorithm consisting of passing messages between variable and check nodes of the factor graph. The sum-product algorithm is fully parallelizable, owing to the fact that all messages can be update concurrently. However, since it requires extensive number of highly interconnected wires, the fully-parallel implementation of the sum-product on chips is exceedingly challenging. Stochastic decoding algorithms, which exchange binary messages, are of great interest for mitigating this challenge and have been the focus of extensive research over the past decade. They significantly reduce the required wiring and computational complexity of the message-passing algorithm. Even though stochastic decoders have been shown extremely effective in practice, the theoretical aspect and understanding of such algorithms remains limited at large. Our main objective in this paper is to address this issue. We first propose a novel algorithm referred to as the Markov based stochastic decoding. Then, we provide concrete quantitative guarantees on its performance for tree-structured as well as general factor graphs. More specifically, we provide upper-bounds on the first and second moments of the error, illustrating that the proposed algorithm is an asymptotically consistent estimate of the sum-product algorithm. We also validate our theoretical predictions with experimental results, showing we achieve comparable performance to other practical stochastic decoders.' bibliography: - 'SD\_bibfile.bib' --- [***]{} \ Qualcomm Research Silicon Valley\ Santa Clara, CA, USA Introduction {#SecIntro} ============ Sparse graph codes, most notably low-density parity-check (LDPC), have been adopted by the latest wireless communication standards [@TTSI; @IEEE802.16e; @Ghn; @IEEEP802.3an]. They are known to approach the channel capacity [@RicUrb; @MacKay99; @Richardson01a; @Richardson01b]. What makes them even more appealing for practical purposes is their simple decoding scheme [@AjiMce00; @KasEtal01]. More specifically, LDPC codes are decoded via a message-passing algorithm called the sum-product (SP). It is an iterative algorithm consisting of passing messages between variable and check nodes in the factor graph [@AjiMce00; @KasEtal01]. The fact that all messages in the SP algorithm can be updated concurrently, makes the fully-parallel implementation—where the factor graph is directly mapped onto the chip—most efficient. However, due to complex and seemingly random connections between check and variable nodes in the factor graph, fully-parallel implementation of the SP is challenging. The wiring complexity has a big impact on the circuit area and power consumption. Also longer, more inter-connected wires can create more parasitic capacitance and limit the clock rate. Various solutions have been suggested by researchers in order to reduce the implementation complexity of the fully-parallel SP algorithm. Analog circuits have been designed for short LDPC codes [@HematiEtal06; @WinsteadEtal06]. Bit serial algorithms, where messages are transmitted serially over single wires, have been proposed [@DarabihaEtal06; @DarabihaEtal08; @BrandonEtal08; @CushonEtal10]. Splitting row-modules by partitioning check node operations has been shown to provide substantial gains in the required area and power efficiency [@MohBaa06; @MohseninEtal10]. In another prominent line of work, researchers have proposed various stochastic decoding algorithms [@GauRap03; @WinsteadEtal05; @TehraniEtal06; @TehraniEtal10; @NaderiEtal11; @KuolunEtal11; @PrimeauEtal13; @SarkisEtal13]. They are all based on stochastic representation of the SP messages. More precisely, messages are encoded via Bernoulli sequences with correct marginal probabilities. As a result, the structure of check and variable nodes are substantially simplified and the wiring complexity is significantly reduced. (Benefits of such decoders are discussed in more details in Section \[SubSecImplementation\].) Stochastic message-passing have also been used in other contexts, among which are distributed convex optimization and learning [@JuditskyEtal09; @HazanEtal07], efficient belief propagation algorithms [@NooWai13b; @NooWai13a], and efficient learning of distributions [@SarJav11]. Although experimental results have proved stochastic decoding extremely beneficial, to date mathematical understanding of such decoders are very limited and largely missing from the literature. Since the output of stochastic decoders are random by construction, it is natural to ask the following questions: how does the stochastic decoder behave on average? can it be tuned to approach the performance of SP and if so how fast? is the average performance typical or do we have a measure of concentration around average? The main contribution of this paper is answering these questions by providing theoretical analysis* for stochastic decoders. To that end, we propose a novel algorithm, referred to as Markov based stochastic decoding (MbSD), which is amenable to theoretical analysis. We provide quantitative bounds on the first and second moments of the error in terms of the underlying parameters for tree-structured (cycle free) as well as general factor graphs, showing that the performance of [MbSD ]{}converges to that of SP. The remainder of this paper is organized as follows. We begin in Section \[SecProbState\] with some background on factor graph representation of LDPC codes, the sum-product algorithm, and stochastic decoding. In Section \[SecResults\], we turn to our main results by introducing the [MbSD ]{}algorithm followed by some discussion on its hardware implementation and statements of our main theoretical results (Theorems \[ThmTree\], and \[ThmMain\]). Section \[SecProof\] is devoted to proofs, with some technical aspects deferred to appendices. Finally in Section \[SecSimulations\], we provide some experimental results, confirming our theoretical predictions. Background and Problem Setup {#SecProbState} ============================ In this section, we setup the problem and provide the necessary background. Factor Graph Representation of LDPC Codes {#SubSecFactorGraph} ----------------------------------------- A low-density parity-check code is a linear error-correcting code, satisfying a number of parity check constraints. These constraints are encoded by a sparse parity-check matrix ${\ensuremath{H}}\in \{0, 1\}^{{\ensuremath{m}}\times{\ensuremath{n}}}$. More specifically, a binary sequence ${\ensuremath{x}}\in \{0, 1\}^{{\ensuremath{n}}}$ is a valid codeword if and only if ${\ensuremath{H}}{\ensuremath{x}}{\operatorname{\stackrel{2}{\equiv}}}0$, where all operations are module two [@RicUrb]. A popular approach for modeling LDPC codes is via the notion of factor graphs [@KasEtal01]. A factor graph representing an LDPC code with the parity-check matrix ${\ensuremath{H}}$ is a bipartite graph ${\mathcal{G}}= ({\mathcal{V}}, {\mathcal{C}}, {\mathcal{E}})$, consisting of a set of variable nodes ${\mathcal{V}}{\ensuremath{:=}}\{1, 2, \ldots, {\ensuremath{n}}\}$, a set of check nodes ${\mathcal{C}}{\ensuremath{:=}}\{1, 2, \ldots, {\ensuremath{m}}\}$, and a set of edges connecting variable and check nodes ${\mathcal{E}}{\ensuremath{:=}}\{({\ensuremath{i}}, {\ensuremath{a}}) \mid {\ensuremath{i}}\in {\mathcal{V}}, {\ensuremath{a}}\in {\mathcal{C}}, \, \text{and} \, {\ensuremath{H}}({\ensuremath{a}}, {\ensuremath{i}}) = 1\}$. (In this paper, we use letters ${\ensuremath{i}}, {\ensuremath{j}}, \ldots$, and ${\ensuremath{a}}, {\ensuremath{b}}, \ldots$ to denote variable and check nodes respectively.) A typical factor graph representing an LDPC code (the Hamming code) is illustrated in Figure \[FactorGraph\]. [![Factor graph of the Hamming code. Variable nodes are represented by circles, whereas check nodes are represented by squares. []{data-label="FactorGraph"}](FactorGraph.eps "fig:"){width=".55\textwidth"}]{} The Sum-Product Algorithm {#SubSecSP} ------------------------- Suppose a transmitter sends the codeword ${\ensuremath{x}}$ to a receiver over a memory-less, noisy communication channel. Some channel models that are commonly used in practice include the additive white Gaussian noise (AWGN), the binary symmetric channel, and the binary erasure channel. Having received the impaired signal ${\ensuremath{y}}$, the receiver attempts to recover the original signal by finding either the global maximum aposteriori (MAP) estimate $\widehat{{\ensuremath{x}}} \; = \; \arg\max_{{\ensuremath{H}}{\ensuremath{x}}= 0} \, {\mathbb{P}}({\ensuremath{x}}\mid {\ensuremath{y}})$, or the bit-wise MAP estimates , for $i = 1, 2, \ldots, {\ensuremath{n}}$. Without exploiting the underlying structure of the code, or equivalently its factor graph, the MAP estimation is intractable and requires an exponential number of operations in the code length. However, this problem can be circumvented using an algorithm called the sum-product (SP), also known as the belief propagation algorithm. The SP is an iterative algorithm consisting of passing messages, in the form of probability distributions, between nodes of the factor graph [@AjiMce00; @KasEtal01]. It is known to converge to the correct bit-wise MAP estimates for cycle-free factor graphs; however, on loopy graphs, which includes almost all practical LDPC codes, such a guarantee no longer exists. Nonetheless, the SP algorithm has been shown to be extremely accurate and effective in practice [@MacKay99; @RicUrb]. We now turn to the description of the SP algorithm. For every variable node ${\ensuremath{i}}\in{\mathcal{V}}$ let ${\ensuremath{\mathcal{N}}}({\ensuremath{i}}) {\ensuremath{:=}}\{{\ensuremath{a}}\mid ({\ensuremath{i}}, {\ensuremath{a}}) \in {\mathcal{E}}\}$ denote the set of its neighboring check nodes. Similarly define ${\ensuremath{\mathcal{N}}}({\ensuremath{a}}) {\ensuremath{:=}}\{{\ensuremath{i}}\mid ({\ensuremath{i}}, {\ensuremath{a}}) \in {\mathcal{E}}\}$, the set of neighboring variable nodes for every check node ${\ensuremath{a}}\in{\mathcal{C}}$. The SP algorithm allocates two messages to every edge $({\ensuremath{i}}, {\ensuremath{a}})\in{\mathcal{E}}$, one for each direction. At each iteration ${\ensuremath{t}}= 0, 1, \ldots$, every variable node ${\ensuremath{i}}\in{\mathcal{V}}$ (check node ${\ensuremath{a}}\in{\mathcal{C}}$), calculates a message $0 < {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1} < 1$ (message $0 < {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}+1} < 1$) and transmit it to its neighboring check node ${\ensuremath{a}}\in{\ensuremath{\mathcal{N}}}({\ensuremath{i}})$ (variable node ${\ensuremath{i}}\in{\ensuremath{\mathcal{N}}}({\ensuremath{a}})$). In updating the messages, every variable node takes into account the incoming messages from its neighboring check nodes as well as the information from the channel, namely ${\ensuremath{\alpha}}_{{\ensuremath{i}}} {\ensuremath{:=}}{\mathbb{P}}({\ensuremath{x}}_{{\ensuremath{i}}} = 1 \mid {\ensuremath{y}}_{{\ensuremath{i}}})$. With this notation at hand, the description of the SP algorithm is as follows: initialize messages from variable to check nodes, ${\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{0} = {\ensuremath{\alpha}}_{{\ensuremath{i}}}$, and update messages for each edge $({\ensuremath{i}}, {\ensuremath{a}})\in{\mathcal{E}}$ and iteration ${\ensuremath{t}}= 0, 1, \ldots$ according to $$\begin{aligned} \label{EqnSPchk} {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}} \; = \; \frac{1}{2} \, - \, \frac{1}{2} \prod_{{\ensuremath{{\ensuremath{j}}\in {\ensuremath{\mathcal{N}}}({\ensuremath{a}}) \setminus \{{\ensuremath{i}}\}}}}(1 - 2\:{\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}), \end{aligned}$$ and $$\begin{aligned} \label{EqnSPvar} {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1} \; = \; \frac{{\ensuremath{\alpha}}_{{\ensuremath{i}}}\prod_{{\ensuremath{{\ensuremath{b}}\in {\ensuremath{\mathcal{N}}}({\ensuremath{i}}) \setminus \{{\ensuremath{a}}\}}}}{\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{b}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}}} {{\ensuremath{\alpha}}_{{\ensuremath{i}}}\prod_{{\ensuremath{{\ensuremath{b}}\in {\ensuremath{\mathcal{N}}}({\ensuremath{i}}) \setminus \{{\ensuremath{a}}\}}}}{\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{b}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}} \, + \, (1-{\ensuremath{\alpha}}_{{\ensuremath{i}}})\prod_{{\ensuremath{{\ensuremath{b}}\in {\ensuremath{\mathcal{N}}}({\ensuremath{i}}) \setminus \{{\ensuremath{a}}\}}}}(1-{\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{b}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}})}.\end{aligned}$$ Information flow on a factor graph is shown in Figure \[FigMessageFlow\]. Upon receiving all the incoming messages, variable node ${\ensuremath{i}}\in{\mathcal{V}}$ update its marginal probability $$\begin{aligned} {\ensuremath{\mu}}_{\ensuremath{i}}^{{\ensuremath{t}}+1} \; = \; \frac{{\ensuremath{\alpha}}_{{\ensuremath{i}}}\prod_{{\ensuremath{b}}\in{\ensuremath{\mathcal{N}}}({\ensuremath{i}})}{\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{b}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}}} {{\ensuremath{\alpha}}_{{\ensuremath{i}}}\prod_{{\ensuremath{b}}\in{\ensuremath{\mathcal{N}}}({\ensuremath{i}})}{\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{b}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}} \, + \, (1-{\ensuremath{\alpha}}_{{\ensuremath{i}}})\prod_{{\ensuremath{b}}\in{\ensuremath{\mathcal{N}}}({\ensuremath{i}})}(1-{\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{b}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}})}.\end{aligned}$$ Accordingly, the receiver estimates the ${\ensuremath{i}}$-th bit by $\widehat{{\ensuremath{x}}}_i^{{\ensuremath{t}}+1} = \mathbb{I}({\ensuremath{\mu}}_{\ensuremath{i}}^{{\ensuremath{t}}+1} > 0.5)$, where $\mathbb{I}(\cdot)$ is the indicator function. It should also be mentioned that in practice, in order to reduce the quantization error, log-likelihood ratios are mostly used as messages. Moreover, to further simplify the SP algorithm, the check node operation is approximated. The resultant is known as the Min-Sum algorithm [@FossorierEtal99]. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- [![Graphical representation of message-passing on factor graphs (a) check to variable node (b) variable to check node.[]{data-label="FigMessageFlow"}](SPchk2var.eps "fig:"){width=".42\textwidth"}]{} [![Graphical representation of message-passing on factor graphs (a) check to variable node (b) variable to check node.[]{data-label="FigMessageFlow"}](SPvar2chk.eps "fig:"){width=".42\textwidth"}]{} (a) (b) -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Stochastic Decoding of LDPC Codes {#SubSecSD} --------------------------------- Stochastic computation in the context of LDPC decoding was first introduced in 2003 by Gaudet and Rapley [@GauRap03]. Ever since, much research has been conducted in this field and numerous stochastic decoders have been proposed [@WinsteadEtal05; @TehraniEtal06; @TehraniEtal10; @NaderiEtal11; @PrimeauEtal13; @SarkisEtal13]. For instance, Tehrani et al. [@TehraniEtal06] introduced and exploited the notions of edge memory and noise dependent scaling in order to make the stochastic decoding a viable method for long, practical, LDPC codes. Estimating the probability distributions via a successive relaxation method, Leduce-Primeau et al. [@PrimeauEtal13] proposed a scheme with improved decoding gain. More recently, Sarkis et al. [@SarkisEtal13] extended the stochastic decoding to the case of non-binary LDPC codes. The underlying structure of all these methods and most relevant to our work, however, are the following: they all encode messages by Bernoulli sequences, they all consist of ‘decoding cycles’ which should not be confused with SP iterations (roughly speaking, multiple decoding cycles correspond to one SP iteration.), the check node operation is the module-two sum (i.e. the message transmitted from a check node to a variable node is equal to the module-two sum of the incoming bits.), and finally the variable node operation is the equality (i.e. the message transmitted from a variable node to a check node is equal to one if all incoming bits are one, it is equal to zero if all incoming bits are zero, and it is equal to the previous decoding cycle’s bit in case incoming messages do not agree.). The intuition behind the stochastic variable and check node operations can be obtained from the inspection of SP message updates and . More specifically, suppose ${\ensuremath{Z}}_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}$, for ${\ensuremath{{\ensuremath{j}}\in {\ensuremath{\mathcal{N}}}({\ensuremath{a}}) \setminus \{{\ensuremath{i}}\}}}$, are independent Bernoulli random variables with distributions ${\mathbb{P}}({\ensuremath{Z}}_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}} = 1) = {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}$. Then ${\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}}$, derived from equation , becomes the probability of having odd number of ones in the sequence $\{{\ensuremath{Z}}_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}\}_{{\ensuremath{{\ensuremath{j}}\in {\ensuremath{\mathcal{N}}}({\ensuremath{a}}) \setminus \{{\ensuremath{i}}\}}}}$ (see Lemma 1 in the paper [@Gallager62]). Therefore, the statistically consistent estimate of the check to variable node message is the module-two summation of the incoming bits. Similarly, to understand the stochastic variable node operation, let ${\ensuremath{Z}}_{{\ensuremath{i}}}$ and ${\ensuremath{Z}}_{{\ensuremath{{\ensuremath{b}}\to{\ensuremath{i}}}}}$, for ${\ensuremath{{\ensuremath{b}}\in {\ensuremath{\mathcal{N}}}({\ensuremath{i}}) \setminus \{{\ensuremath{a}}\}}}$, be independent Bernoulli random variables with probability distributions ${\mathbb{P}}({\ensuremath{Z}}_{{\ensuremath{i}}} = 1) = {\ensuremath{\alpha}}_{{\ensuremath{i}}}$, and ${\mathbb{P}}({\ensuremath{Z}}_{{\ensuremath{{\ensuremath{b}}\to{\ensuremath{i}}}}} = 1) = {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{b}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}}$. Then ${\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1}$, derived from equation , becomes the probability of the event $\{{\ensuremath{Z}}_{{\ensuremath{i}}} = 1, \, {\ensuremath{Z}}_{{\ensuremath{{\ensuremath{b}}\to{\ensuremath{i}}}}} = 1, \, \forall{\ensuremath{{\ensuremath{b}}\in {\ensuremath{\mathcal{N}}}({\ensuremath{i}}) \setminus \{{\ensuremath{a}}\}}}\}$, conditioned on the event $\{{\ensuremath{Z}}_{{\ensuremath{i}}} = {\ensuremath{Z}}_{{\ensuremath{{\ensuremath{b}}\to{\ensuremath{i}}}}}, \, \forall{\ensuremath{{\ensuremath{b}}\in {\ensuremath{\mathcal{N}}}({\ensuremath{i}}) \setminus \{{\ensuremath{a}}\}}}\}$, thus supporting the intuition that one must transmit the common value from variable to check nodes in case all incoming bits are equal. Algorithm and Main Results {#SecResults} ========================== In this section, we introduce the MbSD algorithm, discuss its hardware design aspect, and state some theoretical guarantees regarding its performance. The Proposed Stochastic Algorithm {#SubSecAlgorithm} --------------------------------- The MbSD algorithm consists of passing messages between variable and check nodes of the factor graph. These messages are $2{\ensuremath{k}}$-dimensional binary vectors, for a fixed ${\ensuremath{k}}$ (design parameter). However, variable and check node updates are element-wise, bit operations. Before stating the algorithm we need to define some notation. Suppose $({\ensuremath{y}}_{1}, {\ensuremath{y}}_{2}, \ldots, {\ensuremath{y}}_{{\ensuremath{n}}})$ is the received codeword with the likelihood ${\ensuremath{\alpha}}_{{\ensuremath{i}}} = {\mathbb{P}}({\ensuremath{x}}_{{\ensuremath{i}}} = 1 \| {\ensuremath{y}}_{{\ensuremath{i}}})$, for ${\ensuremath{i}}= 1, 2, \ldots, {\ensuremath{n}}$. Our algorithm, involves messages from the channel to variable nodes at every iteration ${\ensuremath{t}}=0,1,2,\ldots$. More specifically, let ${\ensuremath{Z}}_{{\ensuremath{i}}}^{{\ensuremath{t}}} \in \{0, 1\}^{2{\ensuremath{k}}}$ be the $2{\ensuremath{k}}$-dimensional binary message from the channel to the variable node ${\ensuremath{i}}$ at time ${\ensuremath{t}}$, with independent and identically distributed (i.i.d.) entries $$\begin{aligned} {\mathbb{P}}({\ensuremath{Z}}_{{\ensuremath{i}}}^{{\ensuremath{t}}}(\ell) = 1) \;=\; {\ensuremath{\alpha}}_{{\ensuremath{i}}},\quad \text{for all $\ell = 1, 2, \ldots, 2{\ensuremath{k}}$}.\end{aligned}$$ Moreover, let ${\ensuremath{Z}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}} \in \{0, 1\}^{2{\ensuremath{k}}}$ denote the $2{\ensuremath{k}}$-dimensional binary message from the variable node ${\ensuremath{i}}$ to the check node ${\ensuremath{a}}$ at time ${\ensuremath{t}}= 0, 1, \ldots$. Similarly, let ${\ensuremath{Z}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}} \in \{0, 1\}^{2{\ensuremath{k}}}$ be the message from the check node ${\ensuremath{a}}$ to the variable node ${\ensuremath{i}}$ at time ${\ensuremath{t}}$. We also need to define the element-wise, module-two summation operator ${\ensuremath{\bigoplus}}$, as well as the “equality” operator ${\ensuremath{\bigocirc}}$. Suppose $X_1, X_2, \ldots, X_d$ are arbitrary $2{\ensuremath{k}}$-dimensional binary vectors. Then, the vector $Y = {\ensuremath{\bigoplus}}_{i=1}^{d} X_i$ denotes the module-two summation of the vectors $\{X_i\}_{i=1}^{d}$, if and only if $$\begin{aligned} Y(\ell) \;{\operatorname{\stackrel{2}{\equiv}}}\; X_1(\ell) \,+\, X_2(\ell) \,+\, \ldots \,+\, X_d(\ell),\end{aligned}$$ for all $\ell=1,2,\ldots,2{\ensuremath{k}}$. Furthermore, by $Y={\ensuremath{\bigocirc}}_{i=1}^{d} X_i$ we mean $$\begin{aligned} \label{DefnEquOpt} Y(\ell) \; = \; \left\{\begin{array}{lc} 1 & \text{if $X_i(\ell) = 1$, for all $i=1,2,\ldots,d$ } \\ 0 & \text{if $X_i(\ell) = 0$, for all $i=1,2,\ldots,d$ }\\ Y(\ell-1) & \text{otherwise} \end{array}\right.,\end{aligned}$$ for all $\ell = 1, 2, \ldots, 2{\ensuremath{k}}$. Here, we assume $Y(0)$ is either zero or one, equally likely.\ Now, the precise description of the [MbSD ]{}algorithm is as follows: 1. Initialize messages from variable nodes to check nodes at time ${\ensuremath{t}}= 0$ by ${\ensuremath{Z}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{0} = {\ensuremath{Z}}_{{\ensuremath{i}}}^{0}$. 2. For iterations ${\ensuremath{t}}= 0, 1, 2, \ldots$, and every edge $({\ensuremath{i}},{\ensuremath{a}})\in{\mathcal{E}}$: - Update messages from check nodes to variable nodes $$\begin{aligned} {\ensuremath{Z}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}} \;\; = {\ensuremath{\bigoplus}}_{{\ensuremath{{\ensuremath{j}}\in {\ensuremath{\mathcal{N}}}({\ensuremath{a}}) \setminus \{{\ensuremath{i}}\}}}} {\ensuremath{Z}}_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}.\end{aligned}$$ - Update messages from variable nodes to check nodes by following these steps: - compute the auxiliary variable $$\begin{aligned} {\ensuremath{W}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1} \;\; = {\ensuremath{\bigocirc}}_{{\ensuremath{{\ensuremath{b}}\in {\ensuremath{\mathcal{N}}}({\ensuremath{i}}) \setminus \{{\ensuremath{a}}\}}}} {\ensuremath{Z}}_{{\ensuremath{{\ensuremath{b}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}} \; {\ensuremath{\bigocirc}}\; {\ensuremath{Z}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}.\end{aligned}$$ - update the message entries ${\ensuremath{Z}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1}(\ell)$, for $\ell = 1, 2, \ldots, 2{\ensuremath{k}}$, by drawing i.i.d. samples from the set $$\begin{aligned} \{{\ensuremath{W}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1}({\ensuremath{k}}+1), \, {\ensuremath{W}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1}({\ensuremath{k}}+2), \, \ldots, \, {\ensuremath{W}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1}(2{\ensuremath{k}})\}.\end{aligned}$$ - Compute the binary vector ${\ensuremath{U}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1} = {\ensuremath{\bigocirc}}_{{\ensuremath{a}}\in {\ensuremath{\mathcal{N}}}({\ensuremath{i}})} {\ensuremath{Z}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}} {\ensuremath{\bigocirc}}{\ensuremath{Z}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}$ and update the marginal estimates according to $$\begin{aligned} \label{EqnSDmarg} {\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1} \; = \; \frac{1}{{\ensuremath{k}}} \: \sum_{\ell={\ensuremath{k}}+1}^{2{\ensuremath{k}}} {\ensuremath{U}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}(\ell),\end{aligned}$$ for all ${\ensuremath{i}}=1, 2, \ldots, {\ensuremath{n}}$. Few comments, regarding the interpretation of the algorithm, are worth mentioning at this point. The check to variable node message update (step (a)) is a statistically consistent estimate of the actual check to variable BP message update. However, same can not be stated about the variable to check node update (step (b)). As will be shown in Section \[SecProof\], the equality operator ${\ensuremath{\bigocirc}}$ generates Markov chains with desirable properties, thereby, justifying the “Markov based stochastic decoding” terminology. More specifically, the sequence $\{{\ensuremath{W}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1}(\ell)\}_{\ell=1}^{2{\ensuremath{k}}}$ is a Markov chain with the actual variable to check BP message as its stationary distribution. Our objective in step (b) is to estimate this stationary distribution. From basic Markov chain theory, we know that the marginal distribution of a chain converges to its stationary distribution. Therefore, for large enough ${\ensuremath{k}}$, the empirical distribution of the set $\{{\ensuremath{W}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1}({\ensuremath{k}}+1), {\ensuremath{W}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1}({\ensuremath{k}}+2), \ldots, {\ensuremath{W}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1}(2{\ensuremath{k}})\}$ becomes an accurate enough estimate of the stationary distribution of the Markov chain $\{{\ensuremath{W}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1}(\ell)\}_{\ell=1}^{2{\ensuremath{k}}}$. Discussion on Hardware Implementation {#SubSecImplementation} ------------------------------------- The proposed decoding scheme enjoys all the benefits of traditional stochastic decoders [@GauRap03; @TehraniEtal06]. Since messages between variable and check nodes are binary, stochastic decoding requires a substantially lower wiring complexity compared to fully-parallel sum-product or min-sum implementations. Shorter wires yield smaller circuit area, and smaller parasitic capacitance which in turn lead to higher clock frequencies and less power consumption. Another advantage of stochastic decoding algorithms is the very simple structure of check and variable nodes. As a matter of fact, check nodes can be carried out with simple XOR gates, and variable nodes can be implemented using a combination of a random number generator, a JK flip flop, and AND gates [@GauRap03]. Finally, a very beneficial property of stochastic decoding is the fact that the check node operation (XOR) is associative, i.e., can be partitioned arbitrarily without introducing any additional error. Mohsenin et al. [@MohseninEtal10], illustrated that partitioning check nodes can provide significant improvements (by a factor of four) in area, throughput, and energy efficiency. It should be noted that in this paper and for mathematical convenience, the messages between check and variable nodes are represented by binary vectors. However, to implement the [MbSD ]{}algorithm, there is no need to buffer all these vectors. We only need to count the number of ones between bits ${\ensuremath{k}}+1$, and $2{\ensuremath{k}}$ in each bulk, which can be accomplished by a simple counter. In that respect, [MbSD ]{}has a great advantage compared to the algorithm proposed by Tehrani et al. [@TehraniEtal06], which requires buffering a substantial number of bits on each edge (edge memories). As will be discussed in Section \[SecSimulations\], our algorithm has a superior bit error rate performance compared to [@TehraniEtal06], while maintaining the same order of maximum number of clocks, thereby achieving comparable if not better throughput. Moreover, [MbSD ]{}is equipped with concrete theoretical guarantees, the subject to which we now turn. Main Theoretical Results {#SubSecTheory} ------------------------ Our results concern both cases of tree-structured (cycle free) as well as general factor graphs. Since factor graphs of randomly generated LDPC codes are locally tree-like [@RicUrb], understanding the behavior of every decoding algorithm (stochastic as well as deterministic) on trees is of paramount importance. To that end, we first state some quantitative guarantees regarding the performance of the proposed stochastic decoder on tree-structured factor graphs. Recalling the fact that there exists a unique path between every two variable nodes in a tree, we denote the largest path (also known as the graph diameter) by ${\ensuremath{L}}$. Moreover, we know that estimates generated by the SP algorithm on a tree converge to true marginals after ${\ensuremath{L}}$ iterations [@AjiMce00], i.e., denoting true marginals by $\{{\ensuremath{\mu}}_{{\ensuremath{i}}}^{\ast}\}_{{\ensuremath{i}}=1}^{{\ensuremath{n}}}$, we have ${\ensuremath{\mu}}_{{\ensuremath{i}}}^{{\ensuremath{t}}} = {\ensuremath{\mu}}_{{\ensuremath{i}}}^{\ast}$, for all ${\ensuremath{i}}= 1, 2, \ldots, {\ensuremath{n}}$, and ${\ensuremath{t}}\ge {\ensuremath{L}}$. \[ThmTree\] Consider the sequence of marginals $\{{\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}\}_{{\ensuremath{t}}=0}^{\infty}$, ${\ensuremath{i}}= 1, 2, \ldots, {\ensuremath{n}}$, generated by the [MbSD ]{}algorithm on a tree-structured factor graph. Then for arbitrarily small but fixed parameter ${\ensuremath{\delta}}$, and sufficiently large ${\ensuremath{k}}= {\ensuremath{k}}({\ensuremath{\delta}}, {\mathcal{G}})$ we have: (a) The expected stochastic marginals become arbitrarily close to the true marginals, i.e., $$\begin{aligned} \max_{1\le{\ensuremath{i}}\le{\ensuremath{n}}} \;\big\|{\mathbb{E}}\big[{\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{t}}}\big]\, - \, {\ensuremath{\mu}}_{{\ensuremath{i}}}^{\ast}\big\| \; \le \; {\ensuremath{\delta}},\end{aligned}$$ for all ${\ensuremath{t}}\ge {\ensuremath{L}}$. (b) Furthermore, we have $$\begin{aligned} \max_{1\le{\ensuremath{i}}\le{\ensuremath{n}}} \; \max_{{\ensuremath{t}}\ge0} \; {\operatorname{var}}\big({\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{t}}}\big) \; = \; {\ensuremath{\mathcal{O}}}\left(\frac{1}{{\ensuremath{k}}}\right).\end{aligned}$$ #### Remarks: Theorem \[ThmTree\] provides quantitative bounds on the first and second moments of the [MbSD ]{}marginal estimates. Combining parts (a), and (b), it can be easily observed that $$\begin{aligned} \label{EqnConvRate} \nonumber \max_{1\le{\ensuremath{i}}\le{\ensuremath{n}}} \;{\mathbb{E}}\big[({\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{t}}}\, - \, {\ensuremath{\mu}}_{{\ensuremath{i}}}^{\ast})^2 \big] \; &\le \; \max_{1\le{\ensuremath{i}}\le{\ensuremath{n}}} \;\big\|{\mathbb{E}}\big[{\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{t}}}\big]\, - \, {\ensuremath{\mu}}_{{\ensuremath{i}}}^{\ast}\big\|^2 \,+\, \max_{1\le{\ensuremath{i}}\le{\ensuremath{n}}} \; {\operatorname{var}}\big({\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{t}}}\big)\\ \; & \le \; {\ensuremath{\delta}}^2 \, + \, {\ensuremath{\mathcal{O}}}\left(\frac{1}{{\ensuremath{k}}}\right),\end{aligned}$$ for all ${\ensuremath{t}}\ge {\ensuremath{L}}$. Therefore, as ${\ensuremath{k}}\to\infty$ (${\ensuremath{\delta}}\to0$), the sequence of estimates ${\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{L}}}$ (ranging over ${\ensuremath{k}}$) converges to the true marginal ${\ensuremath{\mu}}_{{\ensuremath{i}}}^{\ast}$ in the $\mathcal{L}^2$ sense. The rate of convergence, and its dependence on the underlying parameters, is fully characterized in expression . It is directly a function of the accuracy, and the factor graph structure (diameter, node degrees, etc.), and indirectly (through Lipschitz constants, etc.) a function of the signal to noise ratio (SNR).\ We now turn to the statement of results for LDPC codes with general (loopy) factor graphs. Unlike tree-structured graphs, the existence and uniqueness of the SP fixed points on general graphs is not guaranteed, nor is the convergence of SP algorithm to such fixed points. Therefore, we have to make the assumption that the LDPC code of interest is well behaved. More precisely, we make the following assumptions: \[AsssConsistency\] Suppose the SP message updates are consistent, that is ${\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}} \to {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{\ast}$, and ${\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}} \to {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{\ast}$ as ${\ensuremath{t}}\to \infty$ for all directed edges $({\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}})$, and $({\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}})$. Equivalently, there exists a sequence $\{{\ensuremath{\mu}}_{{\ensuremath{i}}}^{\ast}\}_{{\ensuremath{i}}=1}^{{\ensuremath{n}}}$ such that ${\ensuremath{\mu}}_{{\ensuremath{i}}}^{{\ensuremath{t}}} \to {\ensuremath{\mu}}_{{\ensuremath{i}}}^{\ast}$, for all ${\ensuremath{i}}= 1,2, \ldots,{\ensuremath{n}}$. For an accuracy parameter ${\ensuremath{\delta}}> 0$, arbitrarily small, we define the stopping time $$\begin{aligned} \label{EqnDefnStop} {\ensuremath{T}}\; = \; {\ensuremath{T}}({\ensuremath{\delta}}) \; {\ensuremath{:=}}\; \inf \big\{ {\ensuremath{t}}\: \| \: \max_{1\le{\ensuremath{i}}\le{\ensuremath{n}}} \|{\ensuremath{\mu}}_{{\ensuremath{i}}}^{{\ensuremath{t}}} - {\ensuremath{\mu}}_{{\ensuremath{i}}}^{\ast}\| \, \le \, {\ensuremath{\delta}}\big\}.\end{aligned}$$ According to assumption \[AsssConsistency\], the stopping time ${\ensuremath{T}}$ is always finite. \[ThmMain\] Consider the marginals $\{{\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}\}_{{\ensuremath{t}}=0}^{\infty}$ generated by the [MbSD ]{}algorithm on an LDPC code that satisfies Assumption \[AsssConsistency\]. Then for arbitrarily small but fixed parameter ${\ensuremath{\delta}}$, and sufficiently large ${\ensuremath{k}}= {\ensuremath{k}}({\ensuremath{\delta}}, {\ensuremath{T}}, {\mathcal{G}})$ we have: (a) The expected stochastic marginals become arbitrarily close to the SP marginals, i.e., $$\begin{aligned} \big\|{\mathbb{E}}\big[{\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{t}}}\big]\, - \, {\ensuremath{\mu}}_{{\ensuremath{i}}}^{{\ensuremath{t}}}\big\| \; \le \; {\ensuremath{\delta}},\end{aligned}$$ for all ${\ensuremath{i}}= 1, 2, \ldots, {\ensuremath{n}}$, and ${\ensuremath{t}}= 0, 1, \ldots, {\ensuremath{T}}$. (b) Furthermore, we have $$\begin{aligned} \max_{1\le{\ensuremath{i}}\le{\ensuremath{n}}} \; \max_{0\le{\ensuremath{t}}\le{\ensuremath{T}}} \; {\operatorname{var}}\big({\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{t}}}\big) \; = \; {\ensuremath{\mathcal{O}}}\left(\frac{1}{{\ensuremath{k}}}\right).\end{aligned}$$ #### Remarks: Theorem \[ThmMain\], in contrast to Theorem \[ThmTree\], provides quantitative bounds on the error over a finite horizon specified by the stopping time . After ${\ensuremath{T}}= {\ensuremath{T}}({\ensuremath{\delta}})$ iterations, the marginal estimates become arbitrarily close to the true marginals on average; in particular, we have $$\begin{aligned} \max_{0\le{\ensuremath{i}}\le{\ensuremath{n}}} \|{\mathbb{E}}\big[{\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{T}}}\big] - {\ensuremath{\mu}}_{{\ensuremath{i}}}^{\ast}\| \; \le \; \max_{0\le{\ensuremath{i}}\le{\ensuremath{n}}} \|{\mathbb{E}}\big[{\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{T}}}\big] - {\ensuremath{\mu}}_{{\ensuremath{i}}}^{{\ensuremath{T}}}\| + \max_{0\le{\ensuremath{i}}\le{\ensuremath{n}}} \|{\ensuremath{\mu}}_{{\ensuremath{i}}}^{{\ensuremath{T}}} - {\ensuremath{\mu}}_{{\ensuremath{i}}}^{\ast}\| \; \le \; 2\:{\ensuremath{\delta}}.\end{aligned}$$ Moreover, since ${\operatorname{var}}({\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{T}}}) = {\ensuremath{\mathcal{O}}}(1/{\ensuremath{k}})$, as ${\ensuremath{k}}\to\infty$, the random variables $\{{\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{T}}}\}_{{\ensuremath{i}}=1}^{{\ensuremath{n}}}$, become more and more concentrated around their means. Specifically, a very crude bound[^1] using Chebyshev inequality [@Grimmett] yields $$\begin{aligned} {\mathbb{P}}\left(\max_{0\le{\ensuremath{i}}\le{\ensuremath{n}}} \|{\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{T}}} - {\mathbb{E}}\big[{\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{T}}}\big]\| \ge {\ensuremath{\epsilon}}\right) \; \le \; {\ensuremath{n}}\: \max_{0\le{\ensuremath{i}}\le{\ensuremath{n}}} \: {\mathbb{P}}\left(\|{\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{T}}} - {\mathbb{E}}\big[{\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{T}}}\big]\| \ge {\ensuremath{\epsilon}}\right) \; = \; {\ensuremath{\mathcal{O}}}\left(\frac{{\ensuremath{n}}}{{\ensuremath{k}}\:{\ensuremath{\epsilon}}^2}\right).\end{aligned}$$ Therefore, it is expected that the performance of the proposed stochastic decoding converges to that of SP, as ${\ensuremath{k}}\to\infty$. Proof of the Main Results {#SecProof} ========================= Conceptually, proofs of Theorems \[ThmTree\] and \[ThmMain\] are very similar. Therefore, in this section, we only prove Theorem \[ThmTree\] and highlight its important differences with Theorem \[ThmMain\] in Appendix \[AppProofThmMain\]. Poofs make use of basic probability and Markov chain theory. At a high level, the argument consists of two parts: characterizing the expected messages and controlling the error propagation in the factor graph. As it turns out, the check node operations (module two summation ${\ensuremath{\bigoplus}}$) are consistent on average, that is expected messages from check to variable nodes are the same as SP messages. In contrast, the variable node operations (equality operator ${\ensuremath{\bigocirc}}$) are asymptotically consistent (as ${\ensuremath{k}}\to\infty$). Therefore, for a finite message dimension ${\ensuremath{k}}$, variable node operations introduce error terms which become propagated throughout the factor graph. The main challenge is to characterize and control these errors. Proof of Part $(a)$ of Theorem \[ThmTree\] {#SubSecPartA} ------------------------------------------ We start by stating a lemma which plays a key role in the sequel. Recall the definition of the equality operator ${\ensuremath{\bigocirc}}$ from . \[LemMarkovChain\] Suppose ${\ensuremath{Z}}_{i} = \{{\ensuremath{Z}}_{i}(\ell)\}_{\ell=1}^{\infty}$, for $i=1, 2, \ldots, d$, are stationary, independent, and identically distributed binary sequence with . Then assuming $\prod_{i=1}^{d} {\ensuremath{\beta}}_i + \prod_{i=1}^{d} (1-{\ensuremath{\beta}}_i) > 0$, the binary sequence ${\ensuremath{U}}= {\ensuremath{\bigocirc}}_{i=1}^{d} {\ensuremath{Z}}_i$ forms a time-reversible Markov chain with the following properties: 1. The transition probabilities are $$\begin{aligned} \label{EqnTransProbZeroOne} {\mathbb{P}}\big({\ensuremath{U}}(\ell) = 1 \,\|\, {\ensuremath{U}}(\ell-1) = 0 \big)& \; = \; \prod_{i=1}^{d}{\ensuremath{\beta}}_{i}, \quad \text{and}\\ \label{EqnTransProbOneZero} {\mathbb{P}}\big({\ensuremath{U}}(\ell) = 0 \,\|\, {\ensuremath{U}}(\ell-1) = 1 \big)& \; = \; \prod_{i=1}^{d}(1-{\ensuremath{\beta}}_{i}).\end{aligned}$$ 2. The stationary distribution is equal to $$\begin{aligned} \lim_{\ell\to\infty}{\mathbb{P}}\big({\ensuremath{U}}(\ell) = 1\big) \; = \; \frac{\prod_{i=1}^{d}{\ensuremath{\beta}}_{i}}{\prod_{i=1}^{d}{\ensuremath{\beta}}_{i} + \prod_{i=1}^{d}(1-{\ensuremath{\beta}}_{i})}.\end{aligned}$$ The proof of this lemma is straight forward and is deferred to Appendix \[AppProofMarkovChain\]. Now let ${\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}} = {\mathbb{E}}[{\ensuremath{Z}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}(\ell)] = {\mathbb{P}}({\ensuremath{Z}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}(\ell) = 1)$ be the expected message from the variable node ${\ensuremath{i}}$ to the check node ${\ensuremath{a}}$. By construction and the fact that the variables $\{{\ensuremath{Z}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}(\ell)\}_{\ell=1}^{2{\ensuremath{k}}}$ are i.i.d., the expected value ${\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}$ is independent of $\ell$. Similarly define , the expected message from the check node ${\ensuremath{a}}$ to variable node ${\ensuremath{i}}$. Taking expectation on both sides of the equation , we obtain $$\begin{aligned} \label{EqnExpMargOne} {\mathbb{E}}\big[{\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}\big] \; = \; \frac{1}{{\ensuremath{k}}} \: \sum_{\ell={\ensuremath{k}}+1}^{2{\ensuremath{k}}} {\mathbb{P}}\big({\ensuremath{U}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}(\ell) \,=\, 1\big).\end{aligned}$$ Therefore, in order to upper-bound the expected marginal ${\mathbb{E}}[{\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}]$, we need to calculate the probabilities ${\mathbb{P}}({\ensuremath{U}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}(\ell) \,=\, 1)$, for $\ell={\ensuremath{k}}+1, {\ensuremath{k}}+2, \ldots, 2{\ensuremath{k}}$. From Lemma \[LemMarkovChain\], we know that the sequence $\{{\ensuremath{U}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}(\ell)\}_{\ell=1}^{2{\ensuremath{k}}}$ is a Markov chain with the following transition probabilities: $$\begin{aligned} \label{EqnDefNodeFunBeta} {\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) \; {\ensuremath{:=}}\; {\ensuremath{\alpha}}_{{\ensuremath{i}}}\prod_{{\ensuremath{a}}\in{\ensuremath{\mathcal{N}}}({\ensuremath{i}})}{\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}}, \quad \text{and} \quad {\ensuremath{g_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) \; {\ensuremath{:=}}\; (1-{\ensuremath{\alpha}}_{{\ensuremath{i}}})\prod_{{\ensuremath{a}}\in{\ensuremath{\mathcal{N}}}({\ensuremath{i}})}(1-{\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}}),\end{aligned}$$ where ${\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}(\cdot)$, and ${\ensuremath{g_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}(\cdot)$ are multivariate functions, taking values in the space $[0 ,1]^{\|{\ensuremath{\mathcal{N}}}({\ensuremath{i}})\|}$. Recalling the basic Markov chain theory, we can calculate the probability ${\mathbb{P}}({\ensuremath{U}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}(\ell) \,=\, 1)$ in terms of the stationary distribution, the iteration number, and the second eigenvalue[^2] of the transition matrix [@Grimmett]. Doing some algebra, we obtain $$\begin{aligned} \label{EqnIndivProb} \nonumber {\mathbb{P}}\big({\ensuremath{U}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}(\ell) = 1\big) \; = & \; {\mathbb{P}}\big({\ensuremath{U}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}(0) = 0\big) \: {\mathbb{P}}\big({\ensuremath{U}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}(\ell) = 1 \mid {\ensuremath{U}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}(0) = 0\big)\\ \nonumber \, & + \, {\mathbb{P}}\big({\ensuremath{U}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}(0) = 1\big)\: {\mathbb{P}}\big({\ensuremath{U}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}(\ell) = 1 \mid {\ensuremath{U}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}(0) = 1\big) \\ \nonumber \; = & \; \Big[\frac{{\ensuremath{g_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) \: {\mathbb{P}}\big({\ensuremath{U}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}(0) = 1\big)}{{\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) + {\ensuremath{g_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}})} - \frac{{\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) \: {\mathbb{P}}\big({\ensuremath{U}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}(0) = 0\big)}{{\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) + {\ensuremath{g_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}})}\Big] \: \big(1 - {\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) - {\ensuremath{g_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}})\big)^{\ell}\\ \, &+ \, \frac{{\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}})}{{\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) + {\ensuremath{g_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}})} .\end{aligned}$$ Substituting equation into , doing some algebra simplifying the expression, and exploiting the facts $$\begin{aligned} {\ensuremath{g_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) \: {\mathbb{P}}\big({\ensuremath{U}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}(0) = 1\big) \, - \, {\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) \: {\mathbb{P}}\big({\ensuremath{U}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}(0) = 0\big) \; \le \; {\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) + {\ensuremath{g_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}})\end{aligned}$$ and $$\begin{aligned} \label{EqnLessThanOne} {\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) + {\ensuremath{g_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) \; \le \; {\ensuremath{\alpha}}_{{\ensuremath{i}}} + (1 - {\ensuremath{\alpha}}_{{\ensuremath{i}}}) \; = \; 1\end{aligned}$$ yields $$\begin{aligned} {\mathbb{E}}\big[{\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}\big] \; \le \; \frac{{\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}})}{{\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) + {\ensuremath{g_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}})} \, + \, \frac{1}{{\ensuremath{k}}} \: \frac{1}{{\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) + {\ensuremath{g_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}})} \, \big(1 - {\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) - {\ensuremath{g_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}})\big)^{{\ensuremath{k}}+1}.\end{aligned}$$ On the other hand, denoting $$\begin{aligned} \label{EqnDefNodeFunAlpha} {\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\alpha}}){\ensuremath{:=}}{\ensuremath{\alpha}}_{{\ensuremath{i}}}\prod_{{\ensuremath{a}}\in{\ensuremath{\mathcal{N}}}({\ensuremath{i}})}{\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}} \quad \text{and} \quad {\ensuremath{g_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\alpha}}){\ensuremath{:=}}(1-{\ensuremath{\alpha}}_{{\ensuremath{i}}})\prod_{{\ensuremath{a}}\in{\ensuremath{\mathcal{N}}}({\ensuremath{i}})}(1-{\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}}),\end{aligned}$$ by definition we have $$\begin{aligned} {\ensuremath{\mu}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1} \; = \; \frac{{\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\alpha}})}{{\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\alpha}}) + {\ensuremath{g_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\alpha}})}.\end{aligned}$$ Since the multivariate function ${\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}(\cdot)/({\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}(\cdot) + {\ensuremath{g_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}(\cdot))$ is Lipschitz, assuming $$\begin{aligned} \min\{{\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\alpha}}) + {\ensuremath{g_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\alpha}}), {\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) + {\ensuremath{g_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}})\} \; \ge \; {\ensuremath{c^{\ast}}},\end{aligned}$$ for some positive constant ${\ensuremath{c^{\ast}}}>0$, there exists a constant ${\ensuremath{M}}= {\ensuremath{M}}({\ensuremath{c^{\ast}}})$ such that $$\begin{aligned} \label{EqnMargError} \|{\mathbb{E}}\big[{\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}\big] - {\ensuremath{\mu}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}\|\; \le \; {\ensuremath{M}}& \: \sum_{{\ensuremath{a}}\in{\ensuremath{\mathcal{N}}}({\ensuremath{i}})}\|{\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}} - {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}}\| \, + \, \frac{1}{{\ensuremath{k}}} \: \frac{(1-{\ensuremath{c^{\ast}}})^{{\ensuremath{k}}+1}}{{\ensuremath{c^{\ast}}}}.\end{aligned}$$ Subsequently, in order to upper-bound the error we need to bound the difference between expected stochastic messages and SP messages, i.e. $\|{\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}} - {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}}\|$. The following lemma, proved in Appendix \[AppProofMessBnd\], addresses this problem. \[LemMessBnd\] On a tree-structured factor graph and for sufficiently large ${\ensuremath{k}}$, there exists a fixed positive constant such that $$\begin{aligned} \label{EqnLowBoundConst} \min \big\{{\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}({\ensuremath{\alpha}})}} + {\ensuremath{g_{{\ensuremath{i}}}^{{\ensuremath{t}}}({\ensuremath{\alpha}})}}, {\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}({\ensuremath{\beta}})}} + {\ensuremath{g_{{\ensuremath{i}}}^{{\ensuremath{t}}}({\ensuremath{\beta}})}}\big\} \; \ge \; {\ensuremath{c^{\ast}}},\end{aligned}$$ for all ${\ensuremath{t}}= 0, 1, 2, \ldots$, and ${\ensuremath{i}}= 1, 2, \ldots, {\ensuremath{n}}$. Furthermore, denoting the maximum check and variable node degrees by ${\ensuremath{d_c}}$, and ${\ensuremath{d_v}}$, respectively, we have $$\begin{aligned} \label{EqnMessBnd} \max_{({\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}})} \|{\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}} - {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}}\| \;\le\; \frac{({\ensuremath{d_c}}-1) (1-{\ensuremath{c^{\ast}}})^{{\ensuremath{k}}+1}}{{\ensuremath{k}}\: {\ensuremath{c^{\ast}}}} \: \frac{[{\ensuremath{M}}({\ensuremath{d_c}}-1)({\ensuremath{d_v}}-1)]^{{\ensuremath{L}}}} {{\ensuremath{M}}({\ensuremath{d_c}}-1)({\ensuremath{d_v}}-1) - 1},\end{aligned}$$ for all ${\ensuremath{t}}= 0, 1, 2, \ldots$. Now substituting inequality into , we obtain $$\begin{aligned} \max_{0\le{\ensuremath{i}}\le{\ensuremath{n}}} \|{\mathbb{E}}\big[{\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}\big] - {\ensuremath{\mu}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}\|\; \le \; \frac{(1-{\ensuremath{c^{\ast}}})^{{\ensuremath{k}}+1}}{{\ensuremath{k}}\: {\ensuremath{c^{\ast}}}} \, \left\{{\ensuremath{M}}({\ensuremath{d_c}}-1){\ensuremath{d_v}}\:\frac{[{\ensuremath{M}}({\ensuremath{d_c}}-1)({\ensuremath{d_v}}-1)]^{{\ensuremath{L}}}}{{\ensuremath{M}}({\ensuremath{d_c}}-1)({\ensuremath{d_v}}-1)-1} \,+\, 1\right\},\end{aligned}$$ for all ${\ensuremath{t}}= 0, 1, 2, \ldots$. Therefore, setting $$\begin{aligned} \label{EqnWhatIsK} {\ensuremath{k}}\;=\; \max\left\{\frac{\log{{\ensuremath{\delta}}} - {\ensuremath{L}}\: \log({\ensuremath{M}}({\ensuremath{d_c}}-1)({\ensuremath{d_v}}-1))}{\log(1 - {\ensuremath{c^{\ast}}})} \, , \, \frac{3}{{\ensuremath{c^{\ast}}}} \right\},\end{aligned}$$ we obtain $$\begin{aligned} \max_{0\le{\ensuremath{i}}\le{\ensuremath{n}}} \|{\mathbb{E}}\big[{\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{t}}}\big] - {\ensuremath{\mu}}_{{\ensuremath{i}}}^{{\ensuremath{t}}}\|\; \le \; {\ensuremath{\delta}},\end{aligned}$$ for all ${\ensuremath{t}}=0, 1, 2, \ldots$. Proof of Part $(b)$ of Theorem \[ThmTree\] {#SubSecPartB} ------------------------------------------ To stramline the exposition, let ${\ensuremath{U}}(\ell) {\ensuremath{:=}}{\ensuremath{U}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}(\ell)$, for fixed ${\ensuremath{i}}$, and ${\ensuremath{t}}$. As previously stated, the sequence $\{{\ensuremath{U}}(\ell)\}_{\ell=1}^{\infty}$ is a Markov chain with initial state $p_0 {\ensuremath{:=}}{\mathbb{P}}({\ensuremath{U}}(0) = 0)$, $p_1 {\ensuremath{:=}}{\mathbb{P}}({\ensuremath{U}}(0) = 1)$, and transition probabilities ${\ensuremath{f}}{\ensuremath{:=}}{\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}({\ensuremath{\beta}})}}$, and ${\ensuremath{g}}{\ensuremath{:=}}{\ensuremath{g_{{\ensuremath{i}}}^{{\ensuremath{t}}}({\ensuremath{\beta}})}}$; more specifically we have $$\begin{aligned} {\ensuremath{f}}\; = \;{\mathbb{P}}\big({\ensuremath{U}}(\ell) = 1 \,\|\, {\ensuremath{U}}(\ell-1) = 0 \big), \; \quad \text{and} \quad {\ensuremath{g}}\;=\; {\mathbb{P}}\big({\ensuremath{U}}(\ell) = 0 \,\|\, {\ensuremath{U}}(\ell-1) = 1 \big).\end{aligned}$$ Since ${\mathbb{E}}[({\ensuremath{U}}(\ell)-{\mathbb{E}}[{\ensuremath{U}}(\ell)])^2] \le 1$, in order to upper-bound the variance $$\begin{aligned} {\operatorname{var}}\big({\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}\big) \; = \; \frac{1}{{\ensuremath{k}}^2} \sum_{\ell={\ensuremath{k}}+1}^{2{\ensuremath{k}}} {\mathbb{E}}\big[\big({\ensuremath{U}}(\ell) - {\mathbb{E}}[{\ensuremath{U}}(\ell)]\big)^2\big] \, + \, \frac{2}{{\ensuremath{k}}^2} \sum_{\ell' < \ell} {\mathbb{E}}\big[\big({\ensuremath{U}}(\ell) - {\mathbb{E}}[{\ensuremath{U}}(\ell)]\big) \big({\ensuremath{U}}(\ell') - {\mathbb{E}}[{\ensuremath{U}}(\ell')]\big)\big],\end{aligned}$$ we only need to upper-bound the cross-product terms. Doing so, for $\ell > \ell'$, we have $$\begin{aligned} {\mathbb{E}}\big[\big({\ensuremath{U}}(\ell) - {\mathbb{E}}[{\ensuremath{U}}(\ell)]\big) \big({\ensuremath{U}}(\ell') - {\mathbb{E}}[{\ensuremath{U}}(\ell')]\big)\big] \; = &\; {\mathbb{E}}\big[{\ensuremath{U}}(\ell)\:{\ensuremath{U}}(\ell')\big] \, - \, {\mathbb{E}}\big[{\ensuremath{U}}(\ell)\big] \: {\mathbb{E}}\big[{\ensuremath{U}}(\ell')\big] \\ \; = &\; {\mathbb{P}}\big({\ensuremath{U}}(\ell') = 1\big) \: \big[{\mathbb{P}}\big({\ensuremath{U}}(\ell) = 1 \| {\ensuremath{U}}(\ell') = 1 \big) \,-\, {\mathbb{P}}\big({\ensuremath{U}}(\ell) = 1\big)\big]\\ \; \le& \; {\mathbb{P}}\big({\ensuremath{U}}(\ell) = 1 \| {\ensuremath{U}}(\ell') = 1 \big) \,-\, {\mathbb{P}}\big({\ensuremath{U}}(\ell) = 1\big).\end{aligned}$$ Now, exploiting the Markov property and equation , we can further simplify the aforementioned inequality $$\begin{aligned} {\mathbb{E}}\big[\big({\ensuremath{U}}(\ell) - {\mathbb{E}}[{\ensuremath{U}}(\ell)]\big) & \big({\ensuremath{U}}(\ell') - {\mathbb{E}}[{\ensuremath{U}}(\ell')]\big)\big]\\ \; & \le \; \frac{{\ensuremath{g}}}{{\ensuremath{f}}+{\ensuremath{g}}} \: (1 - {\ensuremath{f}}-{\ensuremath{g}})^{(\ell-\ell')} \, + \, \frac{{\ensuremath{f}}\:p_0}{{\ensuremath{f}}+ {\ensuremath{g}}} \: (1 - {\ensuremath{f}}-{\ensuremath{g}})^{\ell}\, - \, \frac{{\ensuremath{g}}\:p_1}{{\ensuremath{f}}+ {\ensuremath{g}}} \: (1 - {\ensuremath{f}}-{\ensuremath{g}})^{\ell}\\ \; & \stackrel{\text{(i)}}{\le} \; (1 - {\ensuremath{f}}-{\ensuremath{g}})^{(\ell-\ell')},\end{aligned}$$ where inequality (i) follows from and the fact that ${\ensuremath{g}}/({\ensuremath{f}}+{\ensuremath{g}}) + p_0{\ensuremath{f}}/({\ensuremath{f}}+{\ensuremath{g}}) \le 1$. According to Lemma \[LemMessBnd\], for sufficiently large ${\ensuremath{k}}$, we have ${\ensuremath{f}}+ {\ensuremath{g}}\ge {\ensuremath{c^{\ast}}}$. Therefore, putting the pieces together doing some algebra, we obtain $$\begin{aligned} {\operatorname{var}}\big({\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}\big) \; \le& \; \frac{1}{{\ensuremath{k}}}\, + \, \frac{2}{{\ensuremath{k}}^2} \sum_{{\ensuremath{k}}+1 \le \ell' < \ell\le 2{\ensuremath{k}}} (1 - {\ensuremath{c^{\ast}}})^{(\ell-\ell')} \\ \; =& \; \frac{1}{{\ensuremath{k}}}\, + \, \frac{2}{{\ensuremath{k}}^2} \sum_{i=1}^{{\ensuremath{k}}-1} ({\ensuremath{k}}-i)(1 - {\ensuremath{c^{\ast}}})^{i}\\ \;\le&\; \frac{1}{{\ensuremath{k}}}\, + \, \frac{2}{{\ensuremath{k}}} \sum_{i=1}^{{\ensuremath{k}}-1} (1 - {\ensuremath{c^{\ast}}})^{i} \; \le \; \frac{1 + 2/{\ensuremath{c^{\ast}}}}{{\ensuremath{k}}},\end{aligned}$$ for all ${\ensuremath{i}}= 1, 2, \ldots, {\ensuremath{n}}$, and ${\ensuremath{t}}= 0, 1, 2, \ldots$. Experimental Results {#SecSimulations} ==================== ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- [![Performance of the [MbSD ]{}algorithm on a (3,6)-LDPC code with ${\ensuremath{n}}=200$ variable nodes and ${\ensuremath{m}}=100$ check nodes. (a) Bit error rate versus the energy per bit to noise power spectral density (Eb/No) for different decoders, namely, the SP, the SD (stochastic decoding without noise dependent scaling [@TehraniEtal06]), and the [MbSD ]{}for different message dimensions ${\ensuremath{k}}\in\{64, 128, 256, [![Performance of the [MbSD ]{}algorithm on a (3,6)-LDPC code with ${\ensuremath{n}}=200$ variable nodes and ${\ensuremath{m}}=100$ check nodes. (a) Bit error rate versus the energy per bit to noise power spectral density (Eb/No) for different decoders, namely, the SP, the SD (stochastic decoding without noise dependent scaling [@TehraniEtal06]), and the [MbSD ]{}for different message dimensions ${\ensuremath{k}}\in\{64, 128, 256, 512, 1024\}$. As predicted by the theory, the performance of the [MbSD ]{}converges to that of SP. Moreover, [MbSD ]{}does not suffer from error floor, in contrast to the SD algorithm. (b) Bit error rate gap, i.e., the difference between the SP and the [MbSD ]{}bit error rates, versus the message dimension. The rate of convergence is upper bounded by ${\ensuremath{\mathcal{O}}}(1/{\ensuremath{k}})$, manifested as linear curves in the log-log domain plot.[]{data-label="FigBER_200100"}](biterr.eps "fig:"){width=".47\textwidth"}]{} 512, 1024\}$. As predicted by the theory, the performance of the [MbSD ]{}converges to that of SP. Moreover, [MbSD ]{}does not suffer from error floor, in contrast to the SD algorithm. (b) Bit error rate gap, i.e., the difference between the SP and the [MbSD ]{}bit error rates, versus the message dimension. The rate of convergence is upper bounded by ${\ensuremath{\mathcal{O}}}(1/{\ensuremath{k}})$, manifested as linear curves in the log-log domain plot.[]{data-label="FigBER_200100"}](biterr_gap.eps "fig:"){width=".47\textwidth"}]{} (a) (b) ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- To confirm our theoretical predictions, we test the [MbSD ]{}algorithm on a simple LDPC code. In our experiments, we set the block-length (variable nodes) and number of parity checks (check nodes) to be ${\ensuremath{n}}= 200$ and ${\ensuremath{m}}= 100$, respectively. Using Gallager’s construction [@RicUrb], we first generate a regular (3, 6)-LDPC code, that is, all variable nodes have degree three, whereas all check nodes have degree six. Then, considering a binary pulse-amplitude modulation (BPAM) system over an AWGN channel, we run MbSD, for ${\ensuremath{t}}= 60$ iterations, on several simulated signals in order to compute the bit error rate for different values of normalized signal to noise ratio.[^3] The test is carried out for a number of message dimensions ${\ensuremath{k}}$, and results are compared with the SP algorithm and the stochastic decoding (SD) proposed by Tehrani et al. [@TehraniEtal06] (see Figure \[FigBER\_200100\] (a)).[^4] As predicted by our theorems, the performance of the [MbSD ]{}converges to that of SP as ${\ensuremath{k}}$ grows. Therefore, in contrast to the SD algorithm, [MbSD ]{}is an asymptotically consistent estimate of the SP and does not suffer from error floor. The rate of convergence, on the other hand, can be further explored in Figure \[FigBER\_200100\] (b), wherein the bit error rate gap (i.e., the difference between SP and [MbSD ]{}bit error rates) versus the message dimension is illustrated. As can be observed, the error curves in the log-log domain plot are roughly linear with slope one. This observation is consistent with equation , suggesting the upper-bound of ${\ensuremath{\mathcal{O}}}(1/{\ensuremath{k}})$ for the rate of convergence. To improve upon the seemingly slow rate of convergence, we make use of the notion of noise dependent scaling (NDS). Sensitivity to random switching activities, referred to as ‘latching’, has been observed to be a major challenge in stochastic decoders [@WinsteadEtal05; @TehraniEtal06]. To circumvent this issue, the notion of NDS, in which the received log-likelihoods are down-scaled by a factor proportional to the SNR, was proposed and shown to be extremely effective [@TehraniEtal06]. The [MbSD ]{}algorithm suffers from the latching problem too, especially for high SNR values. Intuitively, sequences generated by Markov chains (recall Lemma \[LemMarkovChain\]) are likely to be the all-one or the all-zero sequences when the SNR is sufficiently high. As a consequence, in such cases, the positive constant ${\ensuremath{c^{\ast}}}$, defined in , is more likely to be close to zero. The rate of convergence of the expectation, specified in equation , is inversely proportional to $\log(1-{\ensuremath{c^{\ast}}})$, therefore, the smaller the ${\ensuremath{c^{\ast}}}$, the slower the rate of convergence. Resolving this issue requires increasing the switching activities of Markov chain, which is accomplished by the NDS. Figure \[FigBER\_NDS\_200100\], illustrates the bit error rate versus Eb/No for the SP, the SD using NDS, and the [MbSD ]{}using NDS.[^5] As is evident, the rate of convergence of the [MbSD ]{}algorithm, and thus its performance, is significantly improved. Moreover, having the same number of decoding cycles, [MbSD ]{}outperforms the SD for high SNRs. [![Effect of the noise dependent scaling (NDS) on the performance of the [MbSD ]{}algorithm. The panel contains several plots illustrating the bit error rate versus the energy per bit to noise power spectral density (Eb/No) for different decoders, namely, the SP, the SD (stochastic decoding with noise dependent scaling [@TehraniEtal06]), and the [MbSD ]{}for different message dimensions ${\ensuremath{k}}\in\{64, 128, 256\}$. As expected, NDS has significantly improved the performance of the MbSD. Simulations were conducted on a (3,6)-LDPC code with ${\ensuremath{n}}=200$ variable nodes and ${\ensuremath{m}}=100$ check nodes.[]{data-label="FigBER_NDS_200100"}](biterr_nds.eps "fig:"){width=".63\textwidth"}]{} Conclusion {#SecConclusion} ========== In this paper, we studied the theoretical aspect of stochastic decoders, a widely studied solution for fully-parallel implementation of LDPC decoding on chips. Generally speaking, encoding messages by binary sequences, stochastic decoders simplify check and node message updates by modulo-two summation and the equality operator, respectively. As it turns out, the check node operation is statistically consistent on average, whereas, the variable node equality operation generates a Markov chain with the desired quantity as its stationary distribution. Therefore, for a finite message dimension ${\ensuremath{k}}$, the stochastic message updates introduce error terms which become propagated in the factor graph. Controlling these errors is the main challenge in the theoretical analysis of stochastic decoders. To formalize these notions, we introduced a novel stochastic algorithm, referred to as the Markov based stochastic decoding, and provided concrete theoretical guarantees on its performance. More precisely, we showed that expected marginals produced by the [MbSD ]{}become arbitrarily close to marginals generated by the SP algorithm on tree-structured as well as general factor graphs. The rate of convergence is governed by the message dimension, the graph structure, and the Lipschitz constant, formally specified in equation . Moreover, we proved that the variance of [MbSD ]{}marginals are upper-bounded by ${\ensuremath{\mathcal{O}}}(1/{\ensuremath{k}})$. These theoretical predictions were also supported by experimental results. We showed that, maintaining the same order of decoding cycles, our algorithm does not suffer from error floor; therefore, it achieves better bit error rate performance compared to other competing methods. Acknowledgements {#acknowledgements .unnumbered} ---------------- Authors would like to thank Aman Bahtia for providing the C++ code, simulating sum-product and stochastic decoding algorithms. Proof of Lemma \[LemMarkovChain\] {#AppProofMarkovChain} ================================= By definition we have $$\begin{aligned} {\ensuremath{U}}(\ell) \; = \; \left\{\begin{array}{lc} 1 & \text{if ${\ensuremath{Z}}_{{\ensuremath{i}}}(\ell) = 1$, for all $i = 1, 2, \ldots, d$}\\ 0 & \text{if ${\ensuremath{Z}}_{{\ensuremath{i}}}(\ell) = 1$, for all $i = 1, 2, \ldots, d$}\\ {\ensuremath{U}}(\ell-1) & \text{otherwise} \end{array}\right..\end{aligned}$$ Therefore, given ${\ensuremath{U}}(\ell-1)=0$ and regardless of the sequence $\{{\ensuremath{U}}(\ell-2), \ldots, {\ensuremath{U}}(0)\}$, the event $\{{\ensuremath{U}}(\ell) = 1\}$ is equivalent to $\{{\ensuremath{Z}}_{{\ensuremath{i}}}(\ell) = 1, \forall \, {\ensuremath{i}}=1, \ldots, d\}$. Therefore, we have $$\begin{aligned} {\mathbb{P}}\big({\ensuremath{U}}(\ell)=1 \, \| \, {\ensuremath{U}}(\ell-1) = 0, {\ensuremath{U}}(\ell-2), \ldots\big) \; & = \; {\mathbb{P}}\big({\ensuremath{Z}}_{{\ensuremath{i}}}(\ell) = 1, \, \forall {\ensuremath{i}}= 1, \ldots, d,\big)\\ \; & \stackrel{(\text{i})}{=} \;\prod_{{\ensuremath{a}}=1}^{d}{\mathbb{P}}\big({\ensuremath{Z}}_{{\ensuremath{i}}}(\ell) = 1\big) \;= \; \prod_{{\ensuremath{i}}=1}^{d}{\ensuremath{\beta}}_{{\ensuremath{i}}},\end{aligned}$$ where equality (i) follows from the i.i.d. nature of the sequence. The exact same argument yields the equation . Finally, the stationary distribution can be obtained from the detailed balance condition [@Grimmett] $$\begin{aligned} \lim_{\ell\to\infty}{\mathbb{P}}\big({\ensuremath{U}}_{{\ensuremath{i}}}(\ell) = 1\big) \: \prod_{{\ensuremath{i}}=1}^{d}(1-{\ensuremath{\beta}}_{{\ensuremath{i}}}) \; = \; \lim_{\ell\to\infty} {\mathbb{P}}\big({\ensuremath{U}}_{{\ensuremath{i}}}(\ell) = 0\big) \: \prod_{{\ensuremath{i}}=1}^{d}{\ensuremath{\beta}}_{{\ensuremath{i}}}.\end{aligned}$$ Proof of Lemma \[LemMessBnd\] {#AppProofMessBnd} ============================= Recall binary messages ${\ensuremath{Z}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}}$, and ${\ensuremath{Z}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}$ from steps 2(a) and 2(b) of the main algorithm. Also recall the definition of expected messages ${\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}} = {\mathbb{P}}({\ensuremath{Z}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}}(\ell) = 1)$, and ${\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}} = {\mathbb{P}}({\ensuremath{Z}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}(\ell) = 1)$. From Lemma 1 of the paper [@Gallager62], we know that $$\begin{aligned} \label{EqnSDexpMesChk2Var} {\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}} \; = \; \frac{1}{2} \, - \, \frac{1}{2} \: \prod_{{\ensuremath{{\ensuremath{j}}\in {\ensuremath{\mathcal{N}}}({\ensuremath{a}}) \setminus \{{\ensuremath{i}}\}}}}\big(1 - 2{\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}\big).\end{aligned}$$ On the other hand, by construction we have $$\begin{aligned} {\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1} \; = \; \frac{1}{{\ensuremath{k}}} \: \sum_{\ell={\ensuremath{k}}+1}^{2{\ensuremath{k}}} {\mathbb{P}}\big({\ensuremath{W}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1}(\ell) = 1\big),\end{aligned}$$ where ${\ensuremath{W}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1} \;\; = {\ensuremath{\bigocirc}}_{{\ensuremath{{\ensuremath{b}}\in {\ensuremath{\mathcal{N}}}({\ensuremath{i}}) \setminus \{{\ensuremath{a}}\}}}} {\ensuremath{Z}}_{{\ensuremath{{\ensuremath{b}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}} \; {\ensuremath{\bigocirc}}\; {\ensuremath{Z}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}$. Since, according to Lemma \[LemMarkovChain\], the sequence $\{{\ensuremath{W}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1}(\ell)\}$ forms a Markov chain with transition probabilities $$\begin{aligned} {\ensuremath{f_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) \; {\ensuremath{:=}}\; {\ensuremath{\alpha}}_{{\ensuremath{i}}}\prod_{{\ensuremath{{\ensuremath{b}}\in {\ensuremath{\mathcal{N}}}({\ensuremath{i}}) \setminus \{{\ensuremath{a}}\}}}}{\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{b}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}}, \quad \text{and} \quad {\ensuremath{g_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) \; {\ensuremath{:=}}\; (1-{\ensuremath{\alpha}}_{{\ensuremath{i}}})\prod_{{\ensuremath{{\ensuremath{b}}\in {\ensuremath{\mathcal{N}}}({\ensuremath{i}}) \setminus \{{\ensuremath{a}}\}}}}(1-{\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{b}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}}), \end{aligned}$$ basic Markov chain theory yields $$\begin{aligned} {\mathbb{P}}&\big({\ensuremath{W}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1}(\ell) = 1\big) \; = \; \frac{{\ensuremath{f_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}})}{{\ensuremath{f_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) + {\ensuremath{g_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}})} \\ \, & + \, \Big[\frac{{\ensuremath{g_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) {\mathbb{P}}\big({\ensuremath{W}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1}(0)=1\big)}{{\ensuremath{f_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) + {\ensuremath{g_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}})} - \frac{{\ensuremath{f_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) {\mathbb{P}}\big({\ensuremath{W}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1}(0)=0\big)}{{\ensuremath{f_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) + {\ensuremath{g_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}})} \Big]\: \big(1 - {\ensuremath{f_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) - {\ensuremath{g_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}})\big)^{\ell}.\end{aligned}$$ Therefore, doing some algebra, noticing the facts that $$\begin{aligned} \frac{{\ensuremath{g_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) {\mathbb{P}}\big({\ensuremath{W}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1}(0)=1\big)}{{\ensuremath{f_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) + {\ensuremath{g_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}})} - \frac{{\ensuremath{f_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) {\mathbb{P}}\big({\ensuremath{W}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1}(0)=0\big)}{{\ensuremath{f_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) + {\ensuremath{g_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}})} \; \le \; 1\end{aligned}$$ and $$\begin{aligned} {\ensuremath{f_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) + {\ensuremath{g_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) \; \le \; {\ensuremath{\alpha}}_{{\ensuremath{i}}} + (1 - {\ensuremath{\alpha}}_{{\ensuremath{i}}}) \; = \; 1,\end{aligned}$$ we have $$\begin{aligned} \label{EqnSDexpMesVar2Chk} {\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1} \; \le \; \frac{{\ensuremath{f_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}})}{{\ensuremath{f_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) + {\ensuremath{g_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}})} \, + \, \frac{1}{{\ensuremath{k}}\:({\ensuremath{f_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) + {\ensuremath{g_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}))} \: (1 - {\ensuremath{f_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) - {\ensuremath{g_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}))^{{\ensuremath{k}}+1}.\end{aligned}$$ Equations , and characterize the stochastic decoding message updates. Similarly, we have the SP update equations for all directed edges $$\begin{aligned} \label{EqnBPmesUpdate} {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}} \; = \; \frac{1}{2} \, - \, \frac{1}{2} \: \prod_{{\ensuremath{{\ensuremath{j}}\in {\ensuremath{\mathcal{N}}}({\ensuremath{a}}) \setminus \{{\ensuremath{i}}\}}}}\big(1 - 2{\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}\big), \quad \text{and} \quad {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1} \; = \; \frac{{\ensuremath{f_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\alpha}})}{{\ensuremath{f_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\alpha}}) + {\ensuremath{g_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\alpha}})},\end{aligned}$$ where we have denoted $$\begin{aligned} {\ensuremath{f_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\alpha}})\; {\ensuremath{:=}}\; {\ensuremath{\alpha}}_{{\ensuremath{i}}}\prod_{{\ensuremath{{\ensuremath{b}}\in {\ensuremath{\mathcal{N}}}({\ensuremath{i}}) \setminus \{{\ensuremath{a}}\}}}}{\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{b}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}}, \quad \text{and} \quad {\ensuremath{g_{{\ensuremath{{\ensuremath{i}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}}}({\ensuremath{\alpha}}) \; {\ensuremath{:=}}\; (1-{\ensuremath{\alpha}}_{{\ensuremath{i}}})\prod_{{\ensuremath{{\ensuremath{b}}\in {\ensuremath{\mathcal{N}}}({\ensuremath{i}}) \setminus \{{\ensuremath{a}}\}}}}(1-{\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{b}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}}).\end{aligned}$$ Since $ 0 \le {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}\le 1$, and $0 \le {\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}\le 1$, for all ${\ensuremath{t}}$, and $({\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}})$, we have[^6] $$\begin{aligned} \label{EqnChk2VarBnd} \|{\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}+1} - {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}+1}\| \;\;\le \sum_{{\ensuremath{{\ensuremath{j}}\in {\ensuremath{\mathcal{N}}}({\ensuremath{a}}) \setminus \{{\ensuremath{i}}\}}}} \|{\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1} - {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1}\|.\end{aligned}$$ We now turn to upper-bounding the term $$\begin{aligned} \|{\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1} \, - \, {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}+1}\| \; \le & \; \Big\| \frac{{\ensuremath{f_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}({\ensuremath{\beta}})}}}{{\ensuremath{f_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}({\ensuremath{\beta}})}} + {\ensuremath{g_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}({\ensuremath{\beta}})}}} \, - \, \frac{{\ensuremath{f_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}({\ensuremath{\alpha}})}}}{{\ensuremath{f_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}({\ensuremath{\alpha}})}} + {\ensuremath{g_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}({\ensuremath{\alpha}})}}} \Big\| \\ & + \frac{1}{{\ensuremath{k}}\:({\ensuremath{f_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}({\ensuremath{\beta}})}} + {\ensuremath{g_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}({\ensuremath{\beta}})}})} \: (1 - {\ensuremath{f_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}({\ensuremath{\beta}})}} - {\ensuremath{g_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}}({\ensuremath{\beta}})}})^{{\ensuremath{k}}+1}.\end{aligned}$$ Since $0 < {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{0} = {\ensuremath{\alpha}}_{{\ensuremath{j}}} < 1$. Then by inspection of the SP updates, it is easy to see that $0< {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{{\ensuremath{t}}} <1$, and $0 <{\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{j}}}}}^{{\ensuremath{t}}} <1$, for all ${\ensuremath{t}}\ge0$, and all directed edges $({\ensuremath{{\ensuremath{a}}\to{\ensuremath{j}}}})$, and $({\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}})$. Therefore, recalling the definition , and the fact that ${\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}} = {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{\ast}$, for ${\ensuremath{t}}$ larger than the graph diameter, there exists a positive constant ${\ensuremath{c^{\ast}}}< 1$ such that $$\begin{aligned} {\ensuremath{f_{{\ensuremath{j}}}^{{\ensuremath{t}}}({\ensuremath{\alpha}})}} \, + \, {\ensuremath{g_{{\ensuremath{j}}}^{{\ensuremath{t}}}({\ensuremath{\alpha}})}} \; \ge \; 2\:{\ensuremath{c^{\ast}}}, \quad \text{for all} \quad {\ensuremath{j}}= 1, 2, \ldots, {\ensuremath{n}}, \; \text{and} \;\; {\ensuremath{t}}= 0, 1, 2, \ldots.\end{aligned}$$ Now we show that for sufficiently large ${\ensuremath{k}}$, we have $$\begin{aligned} \label{EqnClaim} {\ensuremath{f_{{\ensuremath{j}}}^{{\ensuremath{t}}}({\ensuremath{\beta}})}} \, + \, {\ensuremath{g_{{\ensuremath{j}}}^{{\ensuremath{t}}}({\ensuremath{\beta}})}} \; \ge \; {\ensuremath{c^{\ast}}}, \quad \text{for all} \quad {\ensuremath{j}}= 1, 2, \ldots, {\ensuremath{n}}, \; \text{and} \;\; {\ensuremath{t}}= 0, 1, 2, \ldots.\end{aligned}$$ Suppose for a fixed ${\ensuremath{t}}$, we have $$\begin{aligned} \label{EqnCondition} \sum_{{\ensuremath{a}}\in{\ensuremath{\mathcal{N}}}({\ensuremath{j}})} \|{\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{j}}}}}^{\tau} - {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{j}}}}}^{\tau}\| \; \le \; {\ensuremath{c^{\ast}}}, \quad \text{for all} \quad {\ensuremath{j}}= 1, 2, \ldots, {\ensuremath{n}}, \; \text{and} \;\; \tau = 0, 1, \ldots, {\ensuremath{t}}.\end{aligned}$$ Since ${\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{0} = {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{0} = {\ensuremath{\alpha}}_{{\ensuremath{j}}}$, the left hand side of the above inequality is initially equal to zero; thus the condition is satisfied for ${\ensuremath{t}}=0$. Now making use of the mean-value theorem, we obtain[^7] $$\begin{aligned} \|{\ensuremath{f_{{\ensuremath{j}}}^{\tau}({\ensuremath{\beta}})}} + {\ensuremath{g_{{\ensuremath{j}}}^{\tau}({\ensuremath{\beta}})}} - {\ensuremath{f_{{\ensuremath{j}}}^{\tau}({\ensuremath{\alpha}})}} - {\ensuremath{g_{{\ensuremath{j}}}^{\tau}({\ensuremath{\alpha}})}}\| \; \le \; \sum_{{\ensuremath{a}}\in{\ensuremath{\mathcal{N}}}({\ensuremath{j}})} \|{\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{j}}}}}^{\tau} - {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{j}}}}}^{\tau}\|.\end{aligned}$$ Putting the pieces together, assuming , yields $$\begin{aligned} {\ensuremath{f_{{\ensuremath{j}}}^{\tau}({\ensuremath{\beta}})}} \, + \, {\ensuremath{g_{{\ensuremath{j}}}^{\tau}({\ensuremath{\beta}})}} \; \ge \; {\ensuremath{c^{\ast}}}\quad \text{for all} \quad {\ensuremath{j}}= 1, 2, \ldots, {\ensuremath{n}}, \; \text{and} \;\; \tau = 0, 1, \ldots, {\ensuremath{t}}.\end{aligned}$$ Since $0 \le {\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{j}}}}}^{{\ensuremath{t}}}, {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{j}}}}}^{{\ensuremath{t}}} \le 1$, we have $$\begin{aligned} {\ensuremath{f_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{\tau}({\ensuremath{\beta}})}} \, + \, {\ensuremath{g_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{\tau}({\ensuremath{\beta}})}} \; & \ge \; {\ensuremath{f_{{\ensuremath{j}}}^{\tau}({\ensuremath{\beta}})}} \, + \, {\ensuremath{g_{{\ensuremath{j}}}^{\tau}({\ensuremath{\beta}})}} \; \ge \; {\ensuremath{c^{\ast}}}, \quad \text{and}\\ {\ensuremath{f_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{\tau}({\ensuremath{\alpha}})}} \, + \, {\ensuremath{g_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{\tau}({\ensuremath{\alpha}})}} \; & \ge \; {\ensuremath{f_{{\ensuremath{j}}}^{\tau}({\ensuremath{\alpha}})}} \, + \, {\ensuremath{g_{{\ensuremath{j}}}^{\tau}({\ensuremath{\alpha}})}} \; \ge \; 2\:{\ensuremath{c^{\ast}}};\end{aligned}$$ therefore, there exist a constant ${\ensuremath{M}}= {\ensuremath{M}}({\ensuremath{c^{\ast}}})$ such that $$\begin{aligned} \label{EqnVar2ChkBnd} \|{\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{\tau+1} - {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{j}}\to{\ensuremath{a}}}}}^{\tau+1}\| \; \le \; {\ensuremath{M}}\sum_{{\ensuremath{b}}\in{\ensuremath{\mathcal{N}}}({\ensuremath{j}})\setminus\{{\ensuremath{a}}\} } \|{\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{b}}\to{\ensuremath{j}}}}}^{\tau} - {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{b}}\to{\ensuremath{j}}}}}^{\tau}\| \, + \, \frac{1}{{\ensuremath{k}}} \frac{(1-{\ensuremath{c^{\ast}}})^{{\ensuremath{k}}+1}}{{\ensuremath{c^{\ast}}}},\end{aligned}$$ for all $\tau = 0, 1, \ldots, {\ensuremath{t}}$. Substituting the inequality into , we obtain $$\begin{aligned} \label{EqnRecursion} \|{\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{\tau+1} - {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{\tau+1}\| \;\; \le \; {\ensuremath{M}}\sum_{{\ensuremath{{\ensuremath{j}}\in {\ensuremath{\mathcal{N}}}({\ensuremath{a}}) \setminus \{{\ensuremath{i}}\}}}} \sum_{{\ensuremath{b}}\in{\ensuremath{\mathcal{N}}}({\ensuremath{j}})\setminus\{{\ensuremath{a}}\} } \|{\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{b}}\to{\ensuremath{j}}}}}^{\tau} - {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{b}}\to{\ensuremath{j}}}}}^{\tau}\| \, + \, \frac{({\ensuremath{d_c}}-1) (1-{\ensuremath{c^{\ast}}})^{{\ensuremath{k}}+1}}{{\ensuremath{k}}\: {\ensuremath{c^{\ast}}}}, \end{aligned}$$ where we have denoted ${\ensuremath{d_c}}{\ensuremath{:=}}\max_{0\le {\ensuremath{a}}\le{\ensuremath{m}}} \|{\ensuremath{\mathcal{N}}}({\ensuremath{a}})\|$. Let ${\ensuremath{e}}^{{\ensuremath{t}}} {\ensuremath{:=}}\{\|{\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}} - {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}}\|\}_{({\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}})}$ be the ${\ensuremath{r}}{\ensuremath{:=}}\sum_{{\ensuremath{a}}=1}^{{\ensuremath{m}}}\|{\ensuremath{\mathcal{N}}}({\ensuremath{a}})\|$ dimensional error vector. Now define a matrix ${\ensuremath{A}}\in{\ensuremath{\mathbb{R}}}^{{\ensuremath{r}}\times{\ensuremath{r}}}$ with entries indexed by pairs of directed edges $({\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}})$; in particular, we have $$\begin{aligned} \label{EqnDefnMat} {\ensuremath{A}}({\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}, {\ensuremath{{\ensuremath{b}}\to{\ensuremath{j}}}}) \; {\ensuremath{:=}}\; \left\{\begin{array}{cc}{\ensuremath{M}}& \text{if}\;{\ensuremath{{\ensuremath{j}}\in {\ensuremath{\mathcal{N}}}({\ensuremath{a}}) \setminus \{{\ensuremath{i}}\}}}\; \text{and}\;{\ensuremath{b}}\in{\ensuremath{\mathcal{N}}}({\ensuremath{j}})\setminus\{{\ensuremath{a}}\}\\ 0 & \text{o.w.} \end{array}\right..\end{aligned}$$ Then by stacking the scalar inequalities , we obtain the vector inequality $$\begin{aligned} \label{EqnVectorRecursion} {\ensuremath{e}}^{\tau+1} \; {\ensuremath{\preceq}}\; {\ensuremath{A}}\: {\ensuremath{e}}^{\tau} \, + \, \frac{({\ensuremath{d_c}}-1) (1-{\ensuremath{c^{\ast}}})^{{\ensuremath{k}}+1}}{{\ensuremath{k}}\: {\ensuremath{c^{\ast}}}} \: {\ensuremath{\vec{1}}},\end{aligned}$$ for all $\tau = 0, 1, \ldots, {\ensuremath{t}}$. Here, ${\ensuremath{\preceq}}$ denotes the vector inequality, i.e., for ${\ensuremath{r}}$-dimensional vectors $x$, and $y$ we say $x {\ensuremath{\preceq}}y$ if and only if $x(i) \le y(i)$, for all $i=1, 2, \ldots, {\ensuremath{r}}$. Unwrapping the recursion , noticing that ${\ensuremath{e}}^{0} = {\ensuremath{\vec{0}}}$, we obtain $$\begin{aligned} \label{EqnUnwrapped} {\ensuremath{e}}^{{\ensuremath{t}}+1} \; {\ensuremath{\preceq}}\; \frac{({\ensuremath{d_c}}-1) (1-{\ensuremath{c^{\ast}}})^{{\ensuremath{k}}+1}}{{\ensuremath{k}}\: {\ensuremath{c^{\ast}}}} \, \big(I \, + \, {\ensuremath{A}}\, + \, \ldots \, + \, {\ensuremath{A}}^{{\ensuremath{t}}}\big) \: {\ensuremath{\vec{1}}}.\end{aligned}$$ The right hand side sequence of the previous inequality have, seemingly, a growing number of terms as ${\ensuremath{t}}\to \infty$. However, according to the following lemma, proved in Appendix \[AppProofNilpotent\], the graph-respecting matrix ${\ensuremath{A}}$ is nilpotent, i.e., there exists a positive integer $\ell$ such that ${\ensuremath{A}}^{\ell} = 0$. (A similar statement regarding nilpotence of the tree-structured Markov random field have been shown in Lemma 1 [@NooWai13a].) Recall the definition of the factor graph diameter ${\ensuremath{L}}$, the largest path between any pair of variable nodes. \[LemNilpotent\] The graph-respecting matrix ${\ensuremath{A}}$, defined in , is nilpotent with degree at most the diameter of the factor graph ${\ensuremath{L}}$, that is ${\ensuremath{A}}^{{\ensuremath{L}}} = 0$. Exploiting the result of Lemma \[LemNilpotent\], we can further simplify the vector inequality $$\begin{aligned} \label{EqnUnwrappedSimple} {\ensuremath{e}}^{{\ensuremath{t}}} \; {\ensuremath{\preceq}}\; \frac{({\ensuremath{d_c}}-1) (1-{\ensuremath{c^{\ast}}})^{{\ensuremath{k}}+1}}{{\ensuremath{k}}\: {\ensuremath{c^{\ast}}}} \, \big(I \, + \, {\ensuremath{A}}\, + \, \ldots \, + \, {\ensuremath{A}}^{{\ensuremath{L}}-1}\big) \: {\ensuremath{\vec{1}}},\end{aligned}$$ for all ${\ensuremath{t}}= 0, 1, 2, \ldots$. Now we take ${\ensuremath{\\|\cdot\\|_\infty}}$ on the both sides of the inequality . Recalling the definition of the matrix norm infinity[^8], ${\|\!\|\!\| \cdot \|\!\|\!\|_{{\infty}}}$, triangle inequality, and using the fact that [@Horn85], simple algebra yields $$\begin{aligned} \max_{({\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}})} \|{\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}+1} - {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}+1}\| \; & \le \; \frac{({\ensuremath{d_c}}-1) (1-{\ensuremath{c^{\ast}}})^{{\ensuremath{k}}+1}}{{\ensuremath{k}}\: {\ensuremath{c^{\ast}}}} \; \sum_{\ell=0}^{{\ensuremath{L}}-1} {\|\!\|\!\| {\ensuremath{A}}\|\!\|\!\|_{{\infty}}}^{\ell} \\ \; & \le \; \frac{({\ensuremath{d_c}}-1)(1-{\ensuremath{c^{\ast}}})^{{\ensuremath{k}}+1}}{{\ensuremath{k}}\: {\ensuremath{c^{\ast}}}} \; \sum_{\ell=0}^{{\ensuremath{L}}-1} [{\ensuremath{M}}({\ensuremath{d_c}}-1)({\ensuremath{d_v}}-1)]^{\ell}\\ \;& \le\; \frac{({\ensuremath{d_c}}-1) (1-{\ensuremath{c^{\ast}}})^{{\ensuremath{k}}+1}}{{\ensuremath{k}}\: {\ensuremath{c^{\ast}}}} \: \frac{[{\ensuremath{M}}({\ensuremath{d_c}}-1)({\ensuremath{d_v}}-1)]^{{\ensuremath{L}}}} {{\ensuremath{M}}({\ensuremath{d_c}}-1)({\ensuremath{d_v}}-1) - 1}, \end{aligned}$$ where we have denoted ${\ensuremath{d_v}}= \max_{0\le {\ensuremath{i}}\le{\ensuremath{n}}} \|{\ensuremath{\mathcal{N}}}({\ensuremath{i}})\|$. For sufficiently large ${\ensuremath{k}}$, specifically when $$\begin{aligned} \frac{({\ensuremath{d_c}}-1) (1-{\ensuremath{c^{\ast}}})^{{\ensuremath{k}}+1}}{{\ensuremath{k}}\: {\ensuremath{c^{\ast}}}} \: \frac{[{\ensuremath{M}}({\ensuremath{d_c}}-1)({\ensuremath{d_v}}-1)]^{{\ensuremath{L}}}} {{\ensuremath{M}}({\ensuremath{d_c}}-1)({\ensuremath{d_v}}-1) - 1} \;\le\; \frac{{\ensuremath{c^{\ast}}}}{{\ensuremath{d_v}}},\end{aligned}$$ we have $$\begin{aligned} \sum_{{\ensuremath{a}}\in{\ensuremath{\mathcal{N}}}({\ensuremath{j}})} \|{\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{j}}}}}^{{\ensuremath{t}}+1} - {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{j}}}}}^{{\ensuremath{t}}+1}\| \; \le \; {\ensuremath{c^{\ast}}}, \quad \text{for all} \quad {\ensuremath{j}}= 1, 2, \ldots, {\ensuremath{n}},\end{aligned}$$ and hence ${\ensuremath{f_{{\ensuremath{j}}}^{{\ensuremath{t}}+1}({\ensuremath{\beta}})}} + {\ensuremath{g_{{\ensuremath{j}}}^{{\ensuremath{t}}+1}({\ensuremath{\beta}})}}\ge {\ensuremath{c^{\ast}}}$, which proves the claim and concludes the lemma. Proof of Lemma \[LemNilpotent\] {#AppProofNilpotent} =============================== We first show, via induction, that for any positive integer $\ell$ and edges $({\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}})$, and $({\ensuremath{{\ensuremath{b}}\to{\ensuremath{j}}}})$, the entry ${\ensuremath{A}}^{\ell}({\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}, {\ensuremath{{\ensuremath{b}}\to{\ensuremath{j}}}}) \neq 0$ if and only if there exists a directed path of length[^9] $\ell$ between the edges $({\ensuremath{{\ensuremath{b}}\to{\ensuremath{j}}}})$, and $({\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}})$. More specifically, there must exists a sequence of non-overlapping, directed (check-variable) edges $\{({\ensuremath{{\ensuremath{a}}_1\to{\ensuremath{i}}_1}}), \ldots, ({\ensuremath{{\ensuremath{a}}_{\ell-1}\to{\ensuremath{i}}_{\ell-1}}})\}$ such that $$\begin{aligned} {\ensuremath{b}}\in{\ensuremath{\mathcal{N}}}({\ensuremath{j}})\setminus\{{\ensuremath{a}}_{\ell-1}\}, \; {\ensuremath{j}}\in{\ensuremath{\mathcal{N}}}({\ensuremath{a}}_{\ell-1})\setminus\{{\ensuremath{i}}_{\ell-1}\}, \; \ldots, \; {\ensuremath{a}}_1\in{\ensuremath{\mathcal{N}}}({\ensuremath{i}}_1)\setminus\{{\ensuremath{a}}\}, \; {\ensuremath{i}}_1\in{\ensuremath{\mathcal{N}}}({\ensuremath{a}})\setminus\{{\ensuremath{i}}\}.\end{aligned}$$ The base case for $\ell = 1$, is obvious from construction . Suppose the claim is correct for $\ell$; the goal is to prove it for $\ell+1$. By definition, we have $$\begin{aligned} {\ensuremath{A}}^{\ell+1}({\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}, {\ensuremath{{\ensuremath{b}}\to{\ensuremath{j}}}}) \; = \; \sum_{({\ensuremath{c\tok}})} {\ensuremath{A}}^{\ell}({\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}, {\ensuremath{c\tok}}) \: {\ensuremath{A}}({\ensuremath{c\tok}}, {\ensuremath{{\ensuremath{b}}\to{\ensuremath{j}}}}).\end{aligned}$$ Since the matrix is non-negative, ${\ensuremath{A}}^{\ell+1}({\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}, {\ensuremath{{\ensuremath{b}}\to{\ensuremath{j}}}}) \neq 0$, if and only if there exists an edge $({\ensuremath{c\tok}})$ such that ${\ensuremath{A}}^{\ell}({\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}, {\ensuremath{c\tok}}) \neq 0$, and ${\ensuremath{A}}({\ensuremath{c\tok}}, {\ensuremath{{\ensuremath{b}}\to{\ensuremath{j}}}}) \neq 0$. Therefore, according to the induction hypothesis, there exist a sequence of non-overlapping edges such that $$\begin{aligned} c\in{\ensuremath{\mathcal{N}}}(k)\setminus\{{\ensuremath{a}}_{\ell-1}\}, \; k\in{\ensuremath{\mathcal{N}}}({\ensuremath{a}}_{\ell-1})\setminus\{{\ensuremath{i}}_{\ell-1}\}, \; \ldots, \; {\ensuremath{a}}_1\in{\ensuremath{\mathcal{N}}}({\ensuremath{i}}_1)\setminus\{{\ensuremath{a}}\}, \; {\ensuremath{i}}_1\in{\ensuremath{\mathcal{N}}}({\ensuremath{a}})\setminus\{{\ensuremath{i}}\}.\end{aligned}$$ Moreover, we should have ${\ensuremath{b}}\in{\ensuremath{\mathcal{N}}}({\ensuremath{j}})\setminus\{c\}$, and ${\ensuremath{j}}\in{\ensuremath{\mathcal{N}}}(c)\setminus\{k\}$. Putting the pieces together, there must exists a directed path of length $\ell+1$, consisting of , between edges $({\ensuremath{{\ensuremath{b}}\to{\ensuremath{j}}}})$, and $({\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}})$, which yields the claim. Finally, since there is no directed path longer than ${\ensuremath{L}}$ (diameter) between any pair of edges in a tree-structured factor graph, we must have ${\ensuremath{A}}^{{\ensuremath{L}}} = 0$ that concludes the proof. Proof of Theorem \[ThmMain\] {#AppProofThmMain} ============================ As stated previously, proof of Theorem \[ThmMain\] is similar to that of Theorem \[ThmTree\]. The major difference lie in the fact that due to its cycle-free structure, tree-respecting matrix ${\ensuremath{A}}$, defined in , is nilpotent (recall the result of Lemma \[LemNilpotent\]). However, the same may not necessarily be true for general graphs. As a consequence, for a non-tree factor graph, the right hand side of the inequality has indeed a growing number of terms as ${\ensuremath{t}}\to\infty$. However, we can upper-bound the error over a finite horizon provided by the stopping time ${\ensuremath{T}}$, defined in . More precisely, unwrapping the recursion for $\tau = 0, 1, \ldots, {\ensuremath{t}}$, taking norm-infinity on both sides of the outcome, and doing some algebra yields $$\begin{aligned} \max_{({\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}})} \|{\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}+1} - {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}+1}\| \;& \le\; \frac{({\ensuremath{d_c}}-1) (1-{\ensuremath{c^{\ast}}})^{{\ensuremath{k}}+1}}{{\ensuremath{k}}\: {\ensuremath{c^{\ast}}}} \: \frac{[{\ensuremath{M}}({\ensuremath{d_c}}-1)({\ensuremath{d_v}}-1)]^{{\ensuremath{t}}+1}} {{\ensuremath{M}}({\ensuremath{d_c}}-1)({\ensuremath{d_v}}-1) - 1},\end{aligned}$$ for all ${\ensuremath{t}}= 0, 1, \ldots, {\ensuremath{T}}-1$. Therefore, for ${\ensuremath{k}}$ sufficiently large, specifically when $$\begin{aligned} \frac{({\ensuremath{d_c}}-1) (1-{\ensuremath{c^{\ast}}})^{{\ensuremath{k}}+1}}{{\ensuremath{k}}\: {\ensuremath{c^{\ast}}}} \: \frac{[{\ensuremath{M}}({\ensuremath{d_c}}-1)({\ensuremath{d_v}}-1)]^{{\ensuremath{T}}}} {{\ensuremath{M}}({\ensuremath{d_c}}-1)({\ensuremath{d_v}}-1) - 1} \;\le\; \frac{{\ensuremath{c^{\ast}}}}{{\ensuremath{d_v}}},\end{aligned}$$ we have $$\begin{aligned} \sum_{{\ensuremath{a}}\in{\ensuremath{\mathcal{N}}}({\ensuremath{j}})} \|{\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{j}}}}}^{{\ensuremath{t}}+1} - {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{j}}}}}^{{\ensuremath{t}}+1}\| \; \le \; {\ensuremath{c^{\ast}}},\end{aligned}$$ for all ${\ensuremath{j}}= 1, 2, \ldots, {\ensuremath{n}}$, and ${\ensuremath{t}}= 0, 1, \ldots, {\ensuremath{T}}-1$, thereby completing the proof of a slightly different version of the Lemma \[LemMessBnd\] over a finite horizon. \[LemMessBndGeneral\] For sufficiently large ${\ensuremath{k}}$, there exists a fixed positive constant such that $$\begin{aligned} \min \big\{{\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}({\ensuremath{\alpha}})}} + {\ensuremath{g_{{\ensuremath{i}}}^{{\ensuremath{t}}}({\ensuremath{\alpha}})}}, {\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}({\ensuremath{\beta}})}} + {\ensuremath{g_{{\ensuremath{i}}}^{{\ensuremath{t}}}({\ensuremath{\beta}})}}\big\} \; \ge \; {\ensuremath{c^{\ast}}},\end{aligned}$$ for all ${\ensuremath{t}}= 0, 1, \ldots, {\ensuremath{T}}$, and ${\ensuremath{i}}= 1, 2, \ldots, {\ensuremath{n}}$. Furthermore, we have $$\begin{aligned} \label{EqnMessBndGeneral} \max_{({\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}})} \|{\ensuremath{\beta}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}} - {\ensuremath{\alpha}}_{{\ensuremath{{\ensuremath{a}}\to{\ensuremath{i}}}}}^{{\ensuremath{t}}}\| \;\le\; \frac{({\ensuremath{d_c}}-1) (1-{\ensuremath{c^{\ast}}})^{{\ensuremath{k}}+1}}{{\ensuremath{k}}\: {\ensuremath{c^{\ast}}}} \: \frac{[{\ensuremath{M}}({\ensuremath{d_c}}-1)({\ensuremath{d_v}}-1)]^{{\ensuremath{t}}}} {{\ensuremath{M}}({\ensuremath{d_c}}-1)({\ensuremath{d_v}}-1) - 1},\end{aligned}$$ for all ${\ensuremath{t}}= 0, 1, \ldots, {\ensuremath{T}}$. Now substituting the inequality into , setting $$\begin{aligned} {\ensuremath{k}}\;=\; \max\left\{\frac{\log{{\ensuremath{\delta}}} - {\ensuremath{T}}\log({\ensuremath{M}}({\ensuremath{d_c}}-1)({\ensuremath{d_v}}-1))}{\log(1 - {\ensuremath{c^{\ast}}})} \, , \, \frac{3}{{\ensuremath{c^{\ast}}}} \right\},\end{aligned}$$ we obtain $$\begin{aligned} \max_{0\le{\ensuremath{i}}\le{\ensuremath{n}}} \|{\mathbb{E}}\big[{\ensuremath{\hat{\eta}}}_{{\ensuremath{i}}}^{{\ensuremath{t}}}\big] - {\ensuremath{\mu}}_{{\ensuremath{i}}}^{{\ensuremath{t}}}\|\; \le \; {\ensuremath{\delta}},\end{aligned}$$ for all ${\ensuremath{t}}=0,1,\ldots,{\ensuremath{T}}$, which concludes proof of part (a) of Theorem \[ThmMain\]. Proof of part (b) follows the exact same steps outlined in the Section \[SubSecPartB\] [^1]: Tightening this bound exploiting Chernoff inequality and concentration of measure [@ChuLu06], can be further explored. [^2]: It is not hard to see that the second eigenvalue of the transition matrix of the Markov chain $\{{\ensuremath{U}}_{{\ensuremath{i}}}^{{\ensuremath{t}}+1}(\ell)\}_{\ell=0}^{\infty}$ is equal to $(1 - {\ensuremath{f_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}) - {\ensuremath{g_{{\ensuremath{i}}}^{{\ensuremath{t}}}}}({\ensuremath{\beta}}))$. [^3]: A BPAM system with transmit power one over an AWGN channel with noise variance $\sigma^2$ has the energy per bit to noise power spectral density (Eb/No) of $1/(2R\sigma^2)$, where $R$ is the code rate. [^4]: In simulating the SD algorithm, we used 30000 ‘decoding cycles’ and edge memory of length 25 without noise dependent scaling. [^5]: In our simulations we set the NDS scaling parameter to be $\sigma^2$, the optimum choice as suggested in the paper [@TehraniEtal06]. [^6]: The inequality follows from the mean-value theorem and the fact that for the function , we have $\|\partial h / \partial x_i\| \le 1$, if $0 < x_i \le 1$ for all $i=1,\ldots,d$. [^7]: Let $h(x_1, \ldots, x_d) = \alpha \prod_{i=1}^{d}x_i + (1-\alpha)\prod_{i=1}^{d}(1-x_i)$ be an arbitrary function for $0\le x_i\le1$, $i=1,\ldots,d$. Then we have $\|\partial h/ \partial x_i\| \le \alpha \prod_{j\neq i}x_j + (1 - \alpha)\prod_{j\neq i}(1 - x_j) \le 1$. [^8]: Norm infinity of a matrix is the maximum absolute row sum of the matrix [@Horn85] [^9]: Here the length of the path is equal to the number of intermediate variable nodes plus one.
--- author: - Ammar Husain bibliography: - 'clusterCategoryDefect2.bib' title: Defects in Spin Chains via Cluster Categories --- We study Picard groups and $K_0$ groups for the cluster algebras that come from the cluster categories of [@HernandezLeclerc]. This is inspired by a study of boundaries and defects in the associated spin chains. We also do some discussion for their formal deformation quantizations. Introduction ============ The algebraic Bethe ansatz is built from decorating the sites with representations of $U_q \hat{\mathfrak{g}}$. Boundaries for such systems are determined by giving a representation of a coideal subalgebra [@Letzter]. That gives a certain kind of module category. We may also talk about building bimodule categories in order to construct two-sided defects in the chain [@self]. We would like to ask about invariants that help characterize (invertible) (bi-) module categories for these. However, that is complicated due to the nature of meromorphic tensor categories [@Soibelman] as well as potential failures in the monoidal nature of $K_0$. So we perform 2 simplifications. First we look at the subcategory defined in [@HernandezLeclerc] as cluster categories. The next simplification is look at the “reduced" theory. That is where there used to be the monoidal category $\mathcal{C}_l$ assigned to all the bulk regions, we instead assign $\mathcal{A} = K_0 ( \mathcal{C}_l ) \otimes \mathbb{Q}$. In this reduced theory, the codimension 1 walls are decorated by $\mathcal{A} - \mathcal{A}$ bimodules and codimension 2 phenomena have bimodule intertwiners [@DSPS]. Physical intuition says that there should be some procedure to go from a bimodule category which is being used as a defect wall in the original theory to a bimodule which is used as a defect in analogy with a 2-dimensional topological theory.[^1] The naive guess of $K_0$ as the reduction fails unless the functor defining the module structures are bi-exact. That is an issue that has been sidestepped here by only asking about classifying (invertible) (bi)modules for these cluster algebras. The use of the cluster category provides us with the tool of cluster algebras $\mathcal{A}$. These are significantly simpler algebras so the questions of their (bi)modules is significantly easier. That means we can focus on the problems of $Pic(\mathcal{A})$/$Aut(\mathcal{A})$ and $\mathcal{A}-(bi)mod$ needed for the invertible and not necessarily invertible cases respectively. Cluster Categories and Algebras =============================== Cluster Categories ------------------ For a cluster algebra $\mathcal{A}$, a monoidal categorification thereof is defined to be an abelian monoidal category such that it’s Grothendieck ring is isomorphic to $\mathcal{A}$ and the cluster monomials are classes of real simple objects. Cluster variables are classes of real prime simple objects. Let us work with a $U_q \hat{\mathfrak{g}}$ for $\mathfrak{g}$ of ADE type of rank $n$ in the Drinfeld realization. Physically this means that we are in the Jordan-Wigner perspective rather than spins. This effects important properties because the different co-product means different entanglement structure. We are interested in representations of this as a category, but it is too large. That is the purpose of the following definition. Color the vertices of the Dynkin diagram with $\xi_i = 0/1$. Now take the full subcategory of $Rep^{fd} U_q \hat{\mathfrak{g}}$ consisting of objects such that any simple composition factor and index $i \in I$, the roots of that Drinfeld polynomial are in the set ${\{ q^{-2k+\xi_i} \}}$ for $k \in \mathbb{Z}$. Call this $\mathcal{C}_\mathbb{Z}$. Define $\mathcal{C}_l$ by only allowing $k \in [0,l]$ instead.\ $K_0 (\mathcal{C}_\mathbb{Z})$ is generated by $[V_{i,q^{2k+\xi_i}}]$ and similarly for $K_0 (\mathcal{C}_l)$ but in the $\mathcal{C}_l$ case it is a polynomial ring in those $n(l+1)$ variables.\ Cluster Algebra --------------- In order to show that $\mathcal{C}_{\mathfrak{g},l}$ is a monoidal categorification, one must say what cluster algebra it is categorifying. That is given in via a quiver. Define a quiver for the Dynkin diagram for $\mathfrak{g}$ and $l$ by the following procedure. Orient the diagram via a bipartitioning where all the vertices with $\xi_i = 0$ are colored black and those with $\xi_i = 1$ are colored white. Then orient the edges as going from black to white. Call this quiver $Q_{\mathfrak{g},0}$. Now form a new quiver with vertex set $(i,k)$ for $i \in I$ and $1 \leq k \leq l+1$ with three types of arrows. - $(i,k) \to (j , k)$ whenever $i \to j$ is an arrow in $Q_{\mathfrak{g},0}$ and all $k$\ - $(j,k) \to (i,k+1)$ for every arrow $i \to j$ in $Q_{\mathfrak{g},0}$ and all $1 \leq k \leq l$\ - $(i,k+1) \to (i,k)$ for all $i \in I$ and all $1 \leq k \leq l$\ Define the quiver cluster algebra with $x_{i,l+1}$ all frozen variables. This defines a cluster algebra $\mathcal{A}_{\mathfrak{g},l} \subset \mathbb{Q} ( x_{i,m} )$, inside a field extension of $n(l+1)$ transcendental variables. There is a ring isomorphism $K_0 ( \mathcal{C}_{l}) \otimes \mathbb{Q} \simeq \mathcal{A}_{\mathfrak{g},l}$ The conjecture is true if $\mathfrak{g}=A_1$ and arbitrary $l$. In this case the quiver $Q_{A_1,l}$ is an $A_{l+1}$ with one frozen vertex at the end.\ The conjecture is also true for $l=1$ in this case it is 2 copies connected up nontrivially. (N1) at (0,0)[$W_{1,2}$]{}; (N2) at (2,0)[$W_{2,2}$]{}; (N3) at (4,0)[$W_{3,2}$]{}; (N4) at (6,0)[$W_{4,2}$]{}; (M1) at (0,2)[$V_{1,1}$]{}; (M2) at (2,2)[$V_{2,1}$]{}; (M3) at (4,2)[$V_{3,1}$]{}; (M4) at (6,2)[$V_{4,1}$]{}; (N1.east)–(N2.west); (N2.east)–(N3.west); (N3.east)–(N4.west); (M1.east)–(M2.west); (M2.east)–(M3.west); (M3.east)–(M4.west); (N1.north)–(M1.south); (N2.north)–(M2.south); (N3.north)–(M3.south); (N4.north)–(M4.south); (M1.south)–(N2.north); (M2.south)–(N3.north); (M3.south)–(N4.north); (N1) at (0,0)[$W_{1,2}$]{}; (N2) at (2,0)[$W_{2,2}$]{}; (N3) at (4,0)[$W_{3,2}$]{}; (N4) at (6,0)[$W_{4,2}$]{}; (M1) at (0,2)[$V_{1,1}$]{}; (M2) at (2,2)[$V_{2,1}$]{}; (M3) at (4,2)[$V_{3,1}$]{}; (M4) at (6,2)[$V_{4,1}$]{}; (N1.east)–(N2.west); (N2.east)–(N3.west); (N3.east)–(N4.west); (M1.east)–(M2.west); (M2.east)–(M3.west); (M3.east)–(M4.west); (N1.north)–(M1.south); (N2.north)–(M2.south); (N3.north)–(M3.south); (N4.north)–(M4.south); (M1.south)–(N2.north); (M2.south)–(N3.north); (M3.south)–(N4.north); (M2.south)–(N2.north); (N1.east)–(N2.west); Automorphisms ------------- For a finite type cluster algebra $\mathcal{A}$ and $\mathcal{A}^{ex}$ the principal (unfrozen) part. Assuming $\mathcal{A}$ is gluing free, then the specialization map allows cluster automorphisms of $\mathcal{A}$ to give cluster automorphisms of $\mathcal{A}^{ex}$. That is $Aut(\mathcal{A}) \subset Aut(\mathcal{A}^{ex})$. The cases of $Q_{\mathfrak{g},1}$ and $Q_{A_1,l}$ will have the property that their mutable parts are orientations of a simply laced Dynkin diagram so this lemma will apply for them. There is a homomorphism $Pic \mathcal{A} \to Pic \mathcal{A}^{ex}$ Specialization gives an algebra map $\mathcal{A} \to \mathcal{A}^{ex}$ The cluster automorphisms of the principal parts for the $\mathcal{A}_{\mathfrak{g},1}$ are: $\mathfrak{g}$ Aut ---------------- ---------------------------- $A_1$ $\mathbb{Z}_2$ $A_{n > 1}$ $D_{n+3}$ $D_4$ $D_{4} \times S_3$ $D_{n \geq 5}$ $ D_n \times \mathbb{Z}_2$ $E_6 $ $D_{14}$ $E_7 $ $ D_{10}$ $E_8 $ $D_{16}$ This is much more specific then all automorphisms as rings but we still get a subgroup of all automorphisms. These types of automorphisms must take clusters to clusters and commute with mutations. By the lemma \[221\], we must look at a subgroup to account for the frozen variables. Bimodule Categories =================== When looking at the reflection equation we produce coideal subalgebras whose representations produce module categories. Similarly for two-sided defects we desire to produce comodule algebras for two copies using a folding trick. In summary we seek to produce bimodule categories. In the context of 3-dimensional topological field theories, this happens without the affinization. That is bimodule categories produce codimension 1 strata in the cobordism hypothesis with singularities. They may or may not be invertible. If $\mathcal{C}$ is the original category and $\mathcal{D}$ is the bimodule category. Let us also assume the $\mathcal{C} \times \mathcal{D} \to \mathcal{D}$ and vice versa are bi-exact. Then $K_0 (\mathcal{D})$ is both a left and right $K_0 ( \mathcal{C})$ bimodule. We may extend to $\mathbb{Q}$ and potentially produce a bimodule over $K_0 ( \mathcal{C}) \otimes \mathbb{Q}$ but now $\mathbb{Q}$ linear. If $\mathcal{C}_1 \subset \mathcal{C}_2$ is a full subcategory and $\mathcal{D}$ is a bimodule category (satisfying the exactness assumptions) for $\mathcal{C}_2$ we may restrict $K_0 ( \mathcal{D}) \otimes \mathbb{Q}$ as a $K_0 (\mathcal{C}_2) \otimes \mathbb{Q} - K_0 (\mathcal{C}_2) \otimes \mathbb{Q}$ bimodule to $K_0 (\mathcal{C}_1) \otimes \mathbb{Q} - K_0 (\mathcal{C}_1) \otimes \mathbb{Q}$. Here we let $\mathcal{C}_1$ be the cluster category and $\mathcal{C}_2$ be without constraints on the Drinfeld polynomials. A major caution here is that we have not described any operation of fusing bimodule categories. Therefore any statements about the fusing of bimodules over cluster algebras (including the criterion of invertibility) does not necessarily lift to the category level. That is we are not talking about bimodule categories that describe invertible defects, but instead simply those that go to invertible objects in the simplification. This also means that the group operation on $Pic/K_0$ is only useful in the reduction. Picard Groups and $K_0$ ======================= We may define $Pic_k (A)$ as invertible bimodules for a commutative ring $A$ where we only demand that $k$ be able to commute through the bimodule. This contrasts with the module definition ordinarily used. However, we may reduce the computation to two steps. $Pic_k (A) \simeq Aut_k (A) \ltimes Pic_A (A)$ where $Pic_A (A)$ is equivalent to the usual $Pic_{com}$ with invertible modules rather than bimodules. $Aut_k (A)$ is the automorphisms as a $k$ algebra. Useful $Pic_{com}$ Groups ------------------------- Because we are considering cluster algebras, the $\mathbb{Q}$ algebras will be very special. They will be very similar to simple polynomial and Laurent polynomial rings. $$\begin{aligned} Pic_{com} (\mathbb{Q}[t_1^\pm \cdots t_m^\pm , x_1 \cdots x_n]) = 0\end{aligned}$$ In fact true for any zero dimensional commutative ring so field is overkill. $\mathbb{Z}$ would not work because it has Krull dimension 1, but finite rings $\mathbb{Z}_n$ would.\ More generally we get $$\begin{aligned} Pic_{com} (A[t_1^\pm \cdots t_m^\pm ]) \simeq Pic_{com} (A) \bigoplus \bigsqcup_{i=1}^m LPic_{com}(A) \bigoplus \bigsqcup_{k=1}^m \bigsqcup_{i=1}^{2^k {m}\choose{k}} N^k Pic_{com}(A)\\\end{aligned}$$ $LPic_{com}(A)$ for an anodal 1 dimensional domain is trivial. Anodal means that if $b \in \bar{A}$ the integral closure and $b^3 - b^2$ and $b^2 - b$ are in $A$, then $b \in A$ as well. $NPic_{com}(A) \equiv Pic_{com}(A[t])/Pic_{com}(A)$. For example, $NPic_{com}(A) =0$ if and only if $A_{red}$ is seminormal. Normal rings like $U[x_1 \cdots x_n]$ for a UFD $U$ are a fortiori seminormal. $Pic_{com} R \simeq Pic_{com} R[x_1 \cdots x_n]$ if and only if $R$ is seminormal. For example, $$\begin{aligned} Pic_{com} \mathbb{Z} [x_1 \cdots x_n] &\simeq& 0\\\end{aligned}$$ $$\begin{aligned} Pic_{com} (\mathbb{Z}[x_1, \cdots x_n , t_1^\pm \cdots t_m^\pm ]) &\simeq& ????\\\end{aligned}$$ Let $A = \mathbb{Z}[x_1 \cdots x_n]$ $Pic_{com}(A) \simeq Pic_{com}(\mathbb{Z})$ because $\mathbb{Z}$ is seminormal. $$\begin{aligned} Pic_{com} A[t_1^\pm \cdots t_m^\pm] &\simeq& \bigsqcup_{i=1}^m LPic_{com} (A)\\\end{aligned}$$ Automorphisms \[polAut\] ------------------------ Here we gather useful automorphism groups for the commutative algebras that may show up as the cluster algebras. Easy ones include $Aut \mathbb{Z}[x] = \mathbb{Z}_2 \ltimes \mathbb{Z}$. However we had at the very simplest a polynomial ring in $n (l+1)$ variables. Always $\geq 2$ variables. That means a complicated automorphism group. It is even complicated for $\mathcal{C}_{A_1 , 1}$ as described below. $J_n (R)$ is defined as those automorphisms of $R[x_1 \cdots x_n]$ of the form $$\begin{aligned} x_1 &\to& F_1 (x_1) = \alpha_1 x_1 + \beta \in R[x_1]\\ x_2 &\to& F_2 (x_1 , x_2) = \alpha_2 x_2 + f (x_1) \in R[x_1,x_2]\\ x_i &\to& F_i (x_1 \cdots x_i ) = \alpha_i x_i + f_i (x_1 \cdots x_{i-1}) \in R[x_1 \cdots x_i]\\\end{aligned}$$ $Af_n (R)$ are those transformations of the form $$\begin{aligned} \vec{x} &\to& A \vec{x} + \vec{b}\\ A &\in& GL_n (R)\\\end{aligned}$$ The group of polynomial automorphisms of $k[x,y]$ denoted $GA_2$is generated as $$\begin{aligned} GA_2 (k) &\simeq& Af_2 (k) \star_{Bf_2(k)} J_2(k)\end{aligned}$$ For an integral domain $D$, the tame subgroup of $Aut D[x,y]$ is the subgroup generated as an amalgamated product $Af_2$ and $J_2$. However this is a proper subgroup, because there exists non-tame Nagata automorphisms like the following for all $a \neq 0$ non-unit: $$\begin{aligned} X &\to& X + a(aY - X^2)\\ Y &\to& Y + 2X(aY - X^2) + a (aY - X^2)^2\\\end{aligned}$$ As such we get more complications on $\mathbb{Q}[x_1 \cdots x_n]$ by letting $D_2=\mathbb{Q}[x_1 \cdots x_{n-2}]$. Or on $\mathbb{Z}[x_1 \cdots x_n]$ by letting $D_2 = \mathbb{Z}[x_1 \cdots x_{n-2}]$ There are many interesting subgroups inside $GA_{n}(\mathbb{Q})$. In particular inside the $GL_n (\mathbb{Q})$ subgroup, we have: Finite subgroups of $GL_n \mathbb{Q}$ with maximal order are characterized. Except for $2,4,6,7,8,9,10$ they are the orthogonal groups which have $2^n n!$. - $n=2$ has a $W(G_2)$ with order $12 > 8$.\ - $n=4$ has a $W(F_4)$ with order $1152 > 384$\ - $n=6$ has a $W(E_6) \times \mathbb{Z}_2$ with order $103680 > 46080$\ - $n=7$ has a $W(E_7)$ with order $2,903,040 > 645,120$\ - $n=8$ has a $W(E_8)$ with order $696,729,600 > 10,321,920$\ - $n=9$ has a $W(E_8) \times W(A_1)$ with order $1,393,459,200 > 185,794,560$\ - $n=10$ has a $W(E_8) \times W(G_2)$ with order $8,360,755,200 > 3,715,891,200$\ When one further demands matrices over the natural numbers with natural number inverse, there are simply the permutation matrices. Let $F \; k^N \to k^N$ be a polynomial endomorphism. $$\begin{aligned} F \begin{pmatrix}c_1\\c_2\\ \cdots\\c_N\end{pmatrix} &=& \begin{pmatrix}f_1(c_1 \cdots c_N) \\ f_2 \\ \cdots \\ f_N \end{pmatrix}\end{aligned}$$ then the Jacobian is a polynomial of the $N$ variables $X_i$. In particular, if $F$ is a polynomial automorphism having a polynomial inverse $G$, then $J_F$ has a polynomial reciprocal which makes it a nonzero constant. The conjecture says that for a field of characteristic $0$, this condition on $J_F$ is enough for existence of $G$. Non-invertible (bi)modules -------------------------- To characterize not necessarily invertible (bi)modules over cluster algebras, we consider coherent sheaves on either $Spec R$ and $Spec R \times_{Spec \mathbb{Q}} Spec R$. However we are only looking at specific (bi)modules, not accounting for maps between (bi)modules. In the discretized 1+1 picture of the Bethe ansatz, (bi)module morphisms would happen at codimension 2 at points on the walls. That means we should ignore the morphisms in these categories and only look at isomorphism classes of objects or even $K_0$ (split or ordinary). Consider a polynomial ring $\mathbb{Q}[x_1 \cdots x_n]$. If we restrict our attention to finitely generated projectives, we get that they are all free by Quillen-Suslin [@Lam]. This is similarly useful for bimodules by taking $\mathbb{Q}[x_1 \cdots x_n , y_1 \cdots y_n]$. That is $K_0 (\mathbb{Q}[x_1, \cdots x_n]) = \mathbb{Z}$ $Pic$ and $K_0$ of these cluster algebras ========================================= Now let us put all the pieces together to caluclate the associated $Pic$ and $Pic_{com}$ for both $K_0 \mathcal{C}_l \otimes \mathbb{Q}$ and it’s exchangeable parts. We can get some information about the noninvertible parts via some invariants related to $K_0$. This has 2 clusters $x$ and $w/x$ and the full cluster algebra is Laurent polynomials $\mathbb{Q}[x^\pm]$. This has a $\mathbb{Z}_2$ for automorphisms over $\mathbb{Q}$ and a trivial $Pic_A (A)$. This means we have a $\mathbb{Z}_2$ invariant. This is the cluster algebra that shows up as the principal part for $Q_{A_1,1}$. $\mathcal{A}$ itself is a polynomial ring in $n(l+1)=2$ generators. It’s automorphisms and Picard group are covered by \[polAut\] and \[polPic\]. That means our coarse invariant for invertible $\mathcal{C}_{A_1,1}$ bimodule categories is valued in $Aut_{\mathbb{Q}-alg} (\mathcal{A}) \ltimes Pic_{com} (\mathcal{A}) \simeq GA_2 (\mathbb{Q})$ Pic\_[com]{} ( ) = 0 & & Aut\_[cl]{}( ) & Aut\_[-alg]{} () = GA\_2 ()\ Pic\_[com]{} ( \^[ex]{}) = 0 & & Aut\_[cl]{}( \^[ex]{} ) = \_2 & Aut\_[-alg]{} (\^[ex]{})\ This example is more interesting . It has $(x_1,x_2)$ and 4 other clusters. Altogether $\mathcal{A} \subset \mathbb{Q}[x_1^\pm , x_2^\pm] \subset \mathbb{Q}(x_1,x_2)$. It is $\mathbb{Q}[x_1 , \frac{1+x_1}{x_2} , \frac{1+x_2}{x_1}] \simeq \mathbb{Q}[u,v,w]/(uvw-u-v-1)$ for which $\text{Spec} \mathcal{A}$ which cuts a 2 dimensional hypersurface in affine 3 space. Closure in projective space as $uvw - uz^2 - vz^2 - z^3$ which is a cubic in $P^3$.\ As above it’s cluster automorphisms contain $D_5$, but it’s Picard group involves some more algebraic geometry of this cubic.\ This quiver shows up as the principal part of $\mathcal{A}_{A_1,2}$ and $\mathcal{A}_{A_2,1}$. The $\mathcal{A}$ for these are polynomial rings in $3$ and $4$ generators respectively. Our coarse invariant for bimodule categories for these is valued in $Aut_{\mathbb{Q}-alg} (\mathcal{A}) \ltimes Pic_{com} (\mathcal{A})$ where Pic\_[com]{} ( ) = 0 & & Aut\_[cl]{}( ) & Aut\_[-alg]{} () = GA\_[3/4]{} ()\ Pic\_[com]{} ( ) & & Aut\_[cl]{}( \^[ex]{} ) = D\_5 & Aut\_[-alg]{} (\^[ex]{})\ The difference for both cases is indicated with the $3/4$ and $Aut_{cl} (\mathcal{A})$ may differ between cases. .\ The next $A_3$ has $14$ clusters that altogether form $\mathbb{Q}[x_1 , x_3 , \frac{1+x_2}{x_1}=w , \frac{1+x_2+x_1 x_3}{x_2 x_3}=t]$. This can be written as $\mathbb{Q}[x_1 , x_3 , w, t]$ quotiented by a single relation $t w x_1 x_3 - t x_3 - w x_1 - x_1 x_3 = 0$. Upon projectivization of the associated affine scheme, it becomes a quartic hypersurface in $P^4$. This gives the cluster algebra $\mathcal{A}^{ex}$.\ .\ It’s cluster automorphisms contain $D_6$ and computing the Picard group requires more algebraic geometry.\ This quiver shows up as the principal part of $\mathcal{A}_{A_1,3}$ and $\mathcal{A}_{A_3,1}$. With the frozen part, polynomial rings in $4$ or $6$ variables. Our coarse invariant for bimodule categories for these is valued in $Aut_{\mathbb{Q}-alg} (\mathcal{A}) \ltimes Pic_{com} (\mathcal{A})$ where Pic\_[com]{} ( ) = 0 & & Aut\_[cl]{}( ) & Aut\_[-alg]{} () = GA\_[4/6]{} ()\ Pic\_[com]{} ( ) & & Aut\_[cl]{}( \^[ex]{} ) = D\_6 & Aut\_[-alg]{} (\^[ex]{})\ The differences are indicated as above. $$\begin{aligned} w x_1 - 1 &=& x_2\\ t x_2 x_3 &=& t (wx_1 - 1) x_3 = 1 + x_2 + x_1 x_3\\ &=& w x_1 + x_1 x_3\\ t (w x_1 - 1)x_3 &=& w x_1 + x_1 x_3\\ t w x_1 x_3 - t x_3 - w x_1 - x_1 x_3 &=& 0\\ t w x_1 x_3 - t x_3 z^2 - w x_1 z^2 - x_1 x_3 z^2 &=& 0\\\end{aligned}$$ More detailed proofs for the above examples as cluster algebras can be found in [@Lampe]. In general for $\mathcal{C}_{\mathfrak{g} ,l} \subset Rep U_q \hat{\mathfrak{g}}$ in the cases where Leclerc’s conjecture is proven, we have an invariant for invertible bimodules valued in $GA_{n(l+1)}$. There is no information about invertible modules contained in this procedure. This structure also fits in diagrams of the form which gets access to more manageable parts of $GA_{n(l+1)}$ Pic\_[com]{} ( ) = 0 & & Aut\_[cl]{}( ) & Aut\_[-alg]{} () = GA\_[n(l+1)]{} ()\ Pic\_[com]{} ( \_[f.t.]{} ) & & Aut\_[cl]{}( \^[ex]{} ) = G & Aut\_[-alg]{} (\^[ex]{})\ where $\mathcal{X}_{f.t}$ is the specified cluster variety of finite type for the appropriate Dynkin diagram and $G$ is listed in \[clustAut\]\ When asking about not necessarily invertible finitely generated projective (bi)modules. K\_0 ( ) = & K\_0 ( \^[ex]{})\ K\_0 ( \_ ) = & K\_0 ( \^[ex]{} \_ \^[ex]{}) For a “spin" system with $\mathfrak{g}=A_1$ at arbitrary $l \geq 2$, we get a $GA_{l+1} (\mathbb{Q})$ group for $Pic(\mathcal{A}_{A_1,l})$. The invertible bimodules for $\mathcal{A}^{ex}$ have $D_{l+3} = Aut_{cl} (\mathcal{A}^{ex}) \subset Aut (\mathcal{A}^{ex})$ giving $Pic(\mathcal{A}^{ex}) \simeq Aut (\mathcal{A}^{ex}) \ltimes Pic_{com} (\mathcal{A}^{ex})$. The noninvertible finitely generated projective (bi)modules have a $\mathbb{Z}$ characterization when looking at $\mathcal{A}$. For $\mathcal{A}^{ex}$ they are rationally description using the $\delta_S$. If we had a bimodule category fusion of defects, then the cyclic subgroup determined by our favorite defect would have a map to $GA_{l+1}$. In particular if it was finite, we could have a hope of landing in a finite subgroup of maximal order. When $l=3$ or $5 \leq l \leq 9$ \[Feit\] shows that they would be especially interesting. This might address questions of orders of defects in the q-fermion. For $\mathcal{A}_{A_{n},1}$ we can also be more specific because the exchangeable part also forms an $A_n$ finite quiver. That is the same exchangeable part as $\mathcal{A}_{A_1,n}$ above. $GA_{2n}$ replaces $GA_{l+1}$ and $Aut_{cl} (\mathcal{A})$ is different because of the different frozen variables. Everything else we have described is independent of the number of variables in the polynomial ring or only involves $\mathcal{A}^{ex}$. Let $\mathcal{A}^{ex}$ be the cluster algebra given as the homogenous coordinate ring of a Grassmannian. Then $Cl(\mathcal{A}^{ex})=0$. The following correspond to the finite type cluster algebras [@Scott] - $Gr(2,n+3)$ for $A_n$\ - $Gr(3,6)$ for $D_4$\ - $Gr(3,7)$ for $E_6$\ - $Gr(3,8)$ for $E_8$\ But these are unique factorization domains [@Laface]. This then implies the Weil class group $Cl(Spec \mathcal{A}^{ex})$ is trivial. We also know that these examples are locally acyclic, therefore there can be at worst canonical singularities [@BenitoMuller]. For $K_0$ we can apply the results of [@CHWW] for $\mathcal{A}^{ex}$ the homogenous coordinate ring of $X$ of dimension $d$ to give $$\begin{aligned} K_0 ( \mathcal{A}^{ex}) &=& \mathbb{Z} \bigoplus Pic (\mathcal{A}^{ex}) \bigoplus_{p=1}^{d} \bigoplus_{k=1}^{\infty} H^p (X ,\Omega_X^p (k))\\\end{aligned}$$ Then [@Snow] gives results for each of the $H^p (X ,\Omega_X^p (k))$ Semiclassical Geometry ====================== Cluster algebras often also come as Poisson algebras. This means that we should classify (invertible) (bi)modules for these deformations. This deformation may or may not come from a deformation of the $\mathcal{C}_{\mathfrak{g},l} \subset Rep \; U_q \hat{\mathfrak{g}}$. Candidates for this may be possible using elliptic quantum groups [@Felder; @ToledanoLaredo]. Let $\mathcal{A}$ be the starting commutative unital Poisson algebra over the commutative unital ring $k$. (For us $\mathbb{Q}$). Then define $\mathcal{Q}$ to be $\mathcal{A}[[\hbar]]$ [^2] with product $\star$ as well as the classical limit maps $cl$ $$\begin{aligned} \mathcal{Q} &\to& \mathcal{A}\\ \sum_{r=0}^\infty \hbar a_r &\to& a_0\\ Pic (\mathcal{Q}) &\to& Pic( \mathcal{A})\\ K_0 ( \mathcal{Q}) &\to& K_0 ( \mathcal{A})\\\end{aligned}$$ The map on $K_0$ is an isomorphism of groups. The not necessarily invertible have no changes. The kernel of $cl$ on $Pic$ is in 1-1 correspondence with outer self-equivalences of $\mathcal{Q}$. The image can be described in terms of the action of $Pic(\mathcal{A})$ on deformations through gauge transformation equivalence. This is also called a B-field transform. In the case where taking $\mathbb{R}$ points has given a real Poisson manifold $M$, one can ask the differential geometric analog[^3] and with complex valued functions here we get: Let the deformation $\mathcal{Q}$ with $\star$ be such that: - There exists a linear map from the Poisson center $\mathcal{Z}_\pi (\mathcal{A})$ to the center $Z(\mathcal{Q})$ with $f \to f+ O(\hbar)$\ - There exists a linear map $PDer(\mathcal{A}) \to Der(\mathcal{Q})$ where the vector field $X \to \mathcal{L}_X + O(\hbar)$ and for Hamiltonian vector fields, $X_H \to \frac{i}{\hbar} ad_H$ then for the classical limit map on $Pic$. $$\begin{aligned} ker \; cl &\leftrightarrow& \frac{H_\pi^1 (M ,\mathbb{C})}{2\pi i H_\pi^1 ( M , \mathbb{Z})} + \hbar H_\pi^1 (M,\mathbb{C})[[\hbar]]\\ H_\pi^1 (M , \mathbb{C}) = 0 &\implies& ker \; cl = 0\\ &\implies& Pic(\mathcal{Q}) \hookrightarrow Diff(M) \ltimes H^2 (M,\mathbb{Z})\end{aligned}$$ If in addition $\pi = \omega^{-1}$ and $H^1 ( M ) = 0$, then $Pic( \mathcal{Q}) \hookrightarrow Diff(M) \ltimes H^2 (M , \mathbb{Z})$ In the symplectic case $H^1 (M) \simeq H_\pi^1 (M)$. Conclusion ========== Taking inspiration from the algebraic Bethe ansatz with boundaries and defects we have considered the characterization of (invertible) (bi)modules for the associated cluster algebras. We have found interesting groups that might or might not lift to the categorical level of the spin chain. There are many questions this raises. Among these are producing actual bimodule categories for these $\mathcal{C}_{\mathfrak{g},l}$. Our main tool for producing interesting bimodule categories is covariantization of coquasitriangular algebras [@self; @KolbStokman]. We would also like to work in the RTT realization. The difference of coproducts means physical properties will be drastically different between these cases. Another question is if the $\otimes \mathbb{Q}$ can be avoided. This is because the algebras were actually defined over $\mathbb{Z}$. Even for polynomial rings we get complications that can be computed as \[ZpolPic\] and \[ZpolAut\]. That gives the interesting group $GA_{n(l+1)} (\mathbb{Z})$. The exchangeable part which is not a polynomial algebra will have even more complications. For example, the singularity for $A_n$ when $n \equiv 3 (\text{mod} 4)$ [@Muller]. The Picard groups for these can be tackled with \[Weibel\]. Life in a ring is harder than life in a field. An infinitesimal formal degree $2$ parameter can be used to tropicalize by defining $\xi_i = T \log x_i$. That resembles the case of equivariant K-theory of Grassmannians [@Smirnov][^4]. $K_0 ( \mathcal{C}_{A_n,l} ) \otimes_{\mathbb{Z}} \mathbb{C}$ is also a quotient of the homogenous coordinate ring of $Gr(n+1,n+l+2)$ so relations with the Amplitudehedron are possible [@Nima]. In all cases we have provided a map $\mathbb{Z} \to K_0 (\mathcal{A}^{ex})$. We would like to calculate the image of $1$ and upon rationalization get these special elements of the Chow rings for all $\mathfrak{g}$ and $l$. The same is true for the bimodule case. We also described the analogous case for formal deformations. The differential geometric $Pic$ changed in a reasonably manageable way, but we do not know about the algebraic side. [^1]: This is a mere analogy. It can not be pushed due to the failure of separable symmetric Frobenius algebra. [^2]: $\hbar$ may not be the best notation because the system was already quantum. This is yet another deformation. [^3]: See [@Weinstein] for the Morita theory for Poisson manifolds. [^4]: The homological degee provides a caution for when one can expect convergence for numerical nonzero values of a parameter.
--- author: - bibliography: - 'biblio.bib' title: \| Beyond Metadata: Code-centric and Usage-based Analysis\ of Known Vulnerabilities in Open-source Software\ --- [Beyond Metadata: Code-centric and Usage-based Analysis\ of Known Vulnerabilities in Open-source Software]{}\ \ [Serena Elisa Ponta, Henrik Plate, Antonino Sabetta]{} <span style="font-variant:small-caps;">Abstract</span> ------------------------------------------------------------------------ [Citing this paper]{} This is a pre-print of the paper that appears in the proceedings of the 34th IEEE International Conference on Software Maintenance and Evolution (ICSME) 2018. If you wish to cite this work, please refer to it as follows: @INPROCEEDINGS{ponta2018icsme, author={Serena Elisa Ponta and Henrik Plate and Antonino Sabetta}, booktitle={2018 IEEE International Conference on Software Maintenance and Evolution (ICSME)}, title={Beyond Metadata: Code-centric and Usage-based Analysis of Known Vulnerabilities in Open-source Software}, year={2018} month={Sept}, } ------------------------------------------------------------------------ Introduction ============ Open-source software (OSS) libraries are widely used in the software industry: by some estimates, as much as 80% to 90% of the software products on the market include some OSS component [@snyk-oss-state], and each of them contains, on average, 100 distinct open source components, whose code weighs as much as 35% of the overall application size [@blackduck2016]. The same study reports that for applications developed for internal use, the proportion is as high as 75%. At the same time, the number of vulnerabilities disclosed for OSS libraries has been steadily increasing since 2009 [@snyk-oss-state]. While using OSS components with known vulnerabilities is included in the OWASP Top 10 Application Security Risks since 2013 [@owasp-top-ten-2013; @owasp-top-ten-2017], still today the problem is far from being solved. On the contrary, OSS vulnerabilities have been hitting the headlines of mainstream media many times over the past few years [@heartbleed; @shellshock]. As reported by [@snyk-databreach], OSS vulnerabilities were the root cause of the majority of the data breaches that happened in 2016. In 2017, the Equifax incident [@equifax], caused by a missed update of a widely used OSS component, compromised the personal data of over 140 millions of U.S. citizens. Establishing effective OSS vulnerability management practices, supported by adequate tools, is broadly understood as a priority in the software industry, and tools helping to detect known vulnerable libraries are available nowadays, either as OSS or as commercial products (e.g., [@owasp-dc; @whitesource]). These tools differ in terms of detection capabilities, but (to the best of our knowledge) the approaches they use rely on the assumption that the metadata associated to OSS libraries (e.g., name, version), and to vulnerability descriptions (e.g., technical details, list of affected components) are always available and accurate. Unfortunately, these metadata, which are used to map each library onto a list of known vulnerabilities that affect it, are often incomplete, inconsistent, or missing altogether. Therefore, the tools that rely on them may fail to detect vulnerabilities (false negatives), or they may report as vulnerable artifacts that do not contain the code that is the actual cause of the vulnerability (false positives). Furthermore, merely detecting the inclusion of vulnerable libraries does not cater for the needs of the entire software development life-cycle. In the early phases of development, updating a library to a more recent release is relatively unproblematic, because the necessary adaptations in the application code can be performed as part of the normal development activities. On the other hand, as soon as a project gets closer to the date of release to customers, and during the entire operational lifetime, all updates need to be carefully pondered, because they can impact the release schedule, require additional effort, cause system downtime, or introduce new defects. To evaluate precisely the need and the urgency of a library update, it is necessary to answer the key question: “is the vulnerability exploitable, given the particular way the library is used within the application?”. Answering this question is extremely difficult: vulnerabilities are typically described in advisories that consist of short, high-level, textual descriptions in natural language, whereas a reliable assessment of the exploitability and the potential impact of a vulnerability demands much lower-level, detailed, technical information. [Our previous work already]{} goes beyond the simple detection of OSS vulnerabilities: the approach we proposed in [@icsme2015] analyzes the code changes introduced by security fixes, and uses dynamic analysis to assess the impact of the vulnerability for a given application. In this paper we build on [ [@icsme2015]]{}, proposing a code-centric and usage-based approach to detect, analyze, and mitigate OSS vulnerabilities: A) We generalize the vulnerability detection approach of [@icsme2015] by considering fixes independently of the vulnerable libraries; B) We use static analysis to determine whether vulnerable code is reachable and through which call paths; C) We combine static and dynamic analysis to overcome their mutual limitations; D) We define metrics which support the choice of alternative library versions that are not vulnerable, highlighting which options are more likely to minimize the update effort and the risk of incompatibility. Our approach is implemented as a tool, [Vulas]{}, which is adopted at [SAP]{}as the officially recommended solution to scan Java software. The tool has been successfully used to scan about [500]{} applications. Vulnerable code was found reachable for 131 of them and we found that in 7.9% of the cases this was only determined by the combination of static and dynamic analysis. We report on our experience and on the lessons we learned when maturing [Vulas]{}from a research prototype to an industrial-grade solution that has been used to perform over [250000]{} scans since December 2016. The remainder of the paper is organized as follows: Section \[sec:approach\] describes the technical approach, Section \[sec:metrics\] defines the update metrics, and Section \[sec:evaluation\] illustrates our approach in practice. In Section \[sec:experience\], we report on our experience, lessons learned and the challenges we identified. Section \[sec:rel-work\] discusses related literature and Section \[sec:conclusion\] concludes the paper. Technical description of the approach {#sec:approach} ===================================== [In [@icsme2015] we already presented]{} the idea of shifting the problem of establishing whether an application is exploitable because of a known vulnerability in an OSS library, to the problem of assessing whether the vulnerable code is reachable. Sections \[subsec:background\], \[subsec:id\], and \[subsec:dynamic\] generalize [@icsme2015], whereas Sections \[subsec:static\],\[subsec:combination\] extend it with unique novel contributions, that are the basis of the update metrics presented in Section \[sec:metrics\]. Background {#subsec:background} ---------- The core of [@icsme2015] lies on the assumption that a vulnerability can be detected and analyzed considering the set of program constructs (such as methods), that were modified, added, or deleted to fix that vulnerability. We define a program construct (or simply construct) as a structural element of the source code characterized by a type (e.g., ), a language (e.g., Java, Python), and a unique identifier (e.g., the fully-qualified name[^1]). [Changes to program constructs are done by means of commits to a source code repository. The set of changes that fix a vulnerability can be obtained from the analysis of the corresponding commit, the fix commit]{}[^2]. We define a construct change as the tuple $$(c, t, \mathtt{AST}_f^{(c)},\mathtt{AST}_v^{(c)})$$ where $c$ is a construct, $t$ is a change operation (i.e., addition, deletion or modification) on the construct $c$, and $\mathtt{AST}_f^{(c)}, \mathtt{AST}_v^{(c)}$ are, respectively, the abstract syntax trees of $c$ at commit $n$ and at commit $n-1$, i.e., the AST of the fixed and vulnerable construct. Notice that for deleted (added) constructs only $\mathtt{AST}_v^{(c)}$ ($\mathtt{AST}_f^{(c)}$) exists. When a fix is implemented over multiple commits, we rely on commit timestamps to compute the set of changes by comparing the source code of the first and the last commit. If the vulnerability fix includes changes in a nested construct (e.g., a method of a class), two distinct entries are included in the set of construct changes, one for the outer construct (the class), one for the nested construct (the method). The fix commits of a vulnerability are not always communicated with its disclosure. Some OSS projects (e.g., ) provide such information via security advisories; others reference issue tracking systems which in turn describe the vulnerability being solved; some other OSS projects do not explicitly refer to vulnerabilities being fixed. Thus reconciling the information based on the textual description and code changes requires considerable manual effort [(see Section \[subsubsec:db\])]{}. A broader discussion of the data integration problem can be found in [@icsme2015]. Differently from [@icsme2015], we provide a definition of construct change and consider the ASTs of the modified program constructs. This is used in Section \[subsec:id\] to establish whether libraries include the changes introduced by the fix. Vulnerability Detection {#subsec:id} ----------------------- ![Vulnerability detection](figures/venn0.png){width="\columnwidth"} \[fig:concept-app\] Figure \[fig:concept-app\] illustrates how a vulnerability $j$ is associated to an application $a$. $C_{j}$ is the set of the constructs obtained as described above by analyzing the fix commits of $j$. The set $S_i$ is the set of all constructs of the OSS library $i$ bundled in the application $a$ whereas $S_a$ is the set of all constructs belonging to the application itself. If $C_{j} \cap S_i \neq\emptyset$ and $\forall c \in C_{j} \cap S_i, \mathtt{AST}^{(c)} = \mathtt{AST}_v^{(c)}$, then we conclude that the application includes a library $i$ with code vulnerable to $j$ (referred to as vulnerable constructs), i.e., constructs that have been changed in the fix commits of $j$. If $\exists \ c \in C_{j} \cap S_i \ \| \ \mathtt{AST}^{(c)} \neq \mathtt{AST}_v^{(c)}$, then we relax the equality constraint and use both $\mathtt{AST}_v^{(c)}$ and $\mathtt{AST}_f^{(c)}$ to establish to which of the two $\mathtt{AST}^{(c)}$ is “closest”. To this end, we use tree differencing algorithms [@gumtree; @changedistiller] and a library comparison method that we devised (whose description is omitted from this paper because of space constraints). Manual inspection is still required whenever no automated conclusion can be taken, e.g., when $\exists \ c_1,c_2 \in C_{j} \cap S_i \; \| \; \mathtt{AST}^{(c_1)} = \mathtt{AST}_v^{(c_1)}$ and $\mathtt{AST}^{(c_2)} = \mathtt{AST}_f^{(c_2)}$. Note that, even when a vulnerability is fixed by adding new methods to an existing class, the intersection $C_{j} \cap S_i$ is not empty because, as explained in the previous section, it would contain the construct for the class. If the fix includes the addition of a class, we assume that existing code is modified to invoke the new construct. Differently from [@icsme2015], we define the set of constructs $C_j$ as independent of any library $i$, and a vulnerability in a library is then detected through the intersection of its constructs with $C_j$. This approach has several advantages: First, it makes it explicit that the vulnerable constructs responsible for a vulnerability $j$ can be contained in any library $i$, hence, the approach is robust against the prominent practice of repackaging the code of OSS libraries. Second, it is sufficient that a library includes a subset of the vulnerable constructs for the vulnerability to be detected. Last, it improves the accuracy compared to approaches based on metadata, which typically flag entire open-source projects as affected, even if projects release functionalities as part of different libraries. [@poi], for instance, is developed within a single source code repository but released as multiple libraries, each offering functionalities for manipulating different types of Microsoft Office documents. Moreover, [in [@icsme2015] we focused]{} on newly-disclosed open-source vulnerabilities, and thus [could]{} assume that, at the time of disclosure, every library that includes constructs changed in the fix commit is vulnerable. While the assumption holds at that moment in time, it is not valid for old vulnerabilities. In this case, one has to establish whether a given library contains the fixed code, which we support by comparing the AST of constructs in use with those of the affected and fixed construct versions. Dynamic Assessment of Vulnerable Code {#subsec:dynamic} ------------------------------------- After having determined that an application depends on a library that includes vulnerable constructs, it is important to establish whether these constructs are reachable. [In this paper, we use the term reachable to denote both the case where dynamic analysis shows that a construct is actually executed and the case where static analysis shows potential execution paths.]{} The underlying idea is that if an application executes (or may execute) vulnerable constructs, there exists a significant risk that the vulnerability can be exploited. The dynamic assessment described here is borrowed from [@icsme2015]. [0.45]{} ![Vulnerability analysis[]{data-label="fig:analysis"}](figures/venn1.png "fig:"){width="\textwidth"} [0.45]{} ![Vulnerability analysis[]{data-label="fig:analysis"}](figures/venn2.png "fig:"){width="\textwidth"} [0.45]{} ![Vulnerability analysis[]{data-label="fig:analysis"}](figures/venn3.png "fig:"){width="\textwidth"} Figure \[fig:concept-test\] illustrates the use of dynamic analysis to assess whether the vulnerable constructs are reachable by observing actual executions. $T_{ai}$ represents the set of constructs, either part of application $a$ or its bundled library $i$, that were executed at least once during some execution of the application. The collection of actual executions of constructs can be done at different times: during unit tests, integration tests, and even during live system operation (if possible). The intersection $C_{j} \cap T_{ai}$ comprises all those constructs that are both changed in the fix commits of $j$ and executed in the context of application $a$ because of its use of library $i$. Static Assessment of Vulnerable Code {#subsec:static} ------------------------------------ In addition to the analysis of actual executions (dynamic analysis), our approach uses static analysis to determine whether the vulnerable constructs are potentially executable. Specifically, we use static analysis in two different flavors. First, we use it to complement the results of the dynamic analysis, by identifying the library constructs reachable from the application. Second, (Section \[subsec:combination\]) we combine the two techniques by using the results of the dynamic analysis as input for the static analysis, thereby overcoming limitations of both techniques: static analyzers are known to struggle with dynamic code (such as, in Java, code loaded through reflection [@DBLP:conf/icse/LandmanSV17]); on the other hand, dynamic (test-based) methods suffer from inadequate coverage of the possible execution paths. Figure \[fig:concept-a2c\] illustrates how we use static analysis to complement the results of dynamic analysis. $R_{ai}$ represents the set of all constructs, either part of application $a$ or its bundled library $i$, that are found reachable starting from the application $a$ and thus can be potentially executed. Static analysis is performed by using a static analyzer (e.g., [@wala]) to compute a graph of all library constructs reachable from the application constructs. The intersection $C_j\cap R_{ai}$ comprises all constructs that are both changed in the fix commit of $j$ and can be potentially executed. Combination of Dynamic and Static Assessment {#subsec:combination} -------------------------------------------- Figure \[fig:concept-t2c\] illustrates how we combine static and dynamic analysis. In this case the set of constructs actually executed, $T_{ai}$, is used as starting point for the static analysis. The result is the set $R_{T_{ai}}$ of constructs reachable starting from the ones executed during the dynamic analysis. The intersection $C_{j}~\cap~R_{T_{ai}}$ comprises all constructs that are both changed in the fix commit of $j$ and can be potentially executed. We explain the benefits of the combinations of the two techniques through the example in Figure \[fig:example\]. In the following, we denote a library bundled within a software program with the term dependency. Let $S_a$ be a Java application having two direct dependencies $S_1$ and $S_f$ where $S_1$ has a direct dependency $S_2$ that in turn has a direct dependency $S_3$ (thereby $S_2$ and $S_3$ are transitive dependencies for the application $S_a$). $S_1$ is a library offering a set of functionalities to be used by the application (e.g., Apache Commons FileUpload [@fileupload]). Moreover the construct $\gamma$ of $S_1$ calls the construct $\delta$ of $S_2$ dynamically, e.g., by using Java reflection, which means the construct to be called is not known at compile time. $S_f$ is what we call a “framework” providing a skeleton whose functionalities are meant to call the application defining the specific operations (e.g., Apache Struts [@struts], Spring Framework [@spring]). The key difference is the so-called inversion of control as frameworks call the application instead of the other way round. ![Vulnerability analysis example](figures/generic-example.png){width="\columnwidth"} \[fig:example\] With the vulnerability detection step of Section \[subsec:id\], our approach determines that $S_a$ includes vulnerable constructs for vulnerabilities $j_1$ and $j_2$ via the dependencies $S_f$ and $S_3$, respectively. Note that even if $S_3$ only contains two out of the three constructs of $C_{j_2}$, our approach is still able to detect the vulnerability. We start the vulnerability analysis by running the static analysis of Section \[subsec:static\] that looks for all constructs potentially reachable from the constructs of $S_a$. The result is the set $R_{a1}$ including all constructs of $S_a$ and all constructs of $S_1$ reachable from $S_a$. As expected, $S_f$ is not reachable in this case as frameworks are not called by the application. Moreover, it is well known that static analysis cannot always identify dynamic calls like those performed using Java reflection. As the call from $\gamma$ to $\delta$ uses Java reflection, in this example only $S_1$ is statically reachable from the application. As shown in Figure \[fig:example\] $R_{a1}$ does not intersect with any of the vulnerable constructs. The dynamic analysis of Figure \[fig:concept-test\] produces the set $T_a$ (omitted from the figure) of constructs that are actually executed. Though no intersection with the vulnerable constructs is found, the dynamic analysis increases the set of reachable constructs ($R_{a1}\cup T_a$ in Figure \[fig:example\]). In particular it complements static analysis revealing paths that static analysis missed. First, it contains construct $\epsilon$ of framework $S_f$ that calls construct $\alpha$ of the application. Second, it follows the dynamic call from $\gamma$ to $\delta$. Combining static and dynamic analysis, as shown in Figure \[fig:concept-t2c\], we can use static analysis with the constructs in $T_a$ as starting point. The result is the set $R_{T_a}$ (omitted in the figure) of all constructs that can be potentially executed starting from those actually executed $T_a$. After running all the analyses, we obtain the overall set $R_{a1} \cup T_a \cup R_{T_a}$ (shown with solid fill in Figure \[fig:example\]) of all constructs found reachable by at least one technique. Its intersection with $C_{j1}$ and $C_{j2}$ reveals that both vulnerabilities $j_1$ and $j_2$ are reachable, since one vulnerable construct for each of them is found in the intersection and is thus reachable ($\eta \in C_{j_1}$ is reachable from $\epsilon$ and $\omega \in C_{j_2}$ is reachable from $\delta$). Vulnerability Mitigation {#sec:metrics} ======================== The analysis presented in Sections \[subsec:dynamic\] to \[subsec:combination\] provides in-depth information about the control-flow between the code of the application and its dependencies. In the following, this information is leveraged to support application developers in mitigating vulnerable dependencies. As long as non-vulnerable library versions are available, updating to one of those is the preferred solution to fix vulnerable application dependencies. And since the approach described in Section \[subsec:background\] depends on the presence of at least one fix commit, a non-vulnerable library version becomes available when the respective open-source project releases a version including this commit. However, it is well known that developers are reluctant to update dependencies because of the risk of breaking changes, the difficulties in understanding the implications of changes, and the overall migration effort [@DBLP:journals/ese/KulaGOII18; @DBLP:conf/issta/MostafaRW17]. Such risk and effort depend on the usage the application makes of the library, and on the amount of changes between the library version currently in use and the respective non-vulnerable version. As a result of the analysis described in Sections \[subsec:dynamic\] to \[subsec:combination\], the reachable share of each library is known. Whether a construct with a given identifier is also available in other versions of a library can [always]{} be determined, for instance, by comparing compiled code with tools such as Dependency Finder [@jeant]. Among all the reachable constructs, of particular importance in the scope of mitigation are those which are called directly from the application, as they provide a measure of how many times the application developer explicitly used the library. We define a touch point as a pair of constructs $(c_1,c_2)$ such that $c_1 \in S_a$ is an application construct, $c_2 \in S_i$ is a library construct, and there exists a call from $c_1$ to $c_2$. We define callee the library construct called directly from the application, i.e., $c_2$. In the example of Figure \[fig:example\] there are two touch points: $(\alpha, \beta)$ and $(\lambda, \psi)$, with $\beta$ and $\psi$ being the callees. Given a library in use $S_i$ and its candidate replacement $S_j$, we define the following update metrics. [Callee Stability ($\mathit{CS}$). ]{} Let $c_k^{(S_i)}$ with $k=1,\ldots,n$ be the callees of $S_i$, and $c_k^{(S_j)}=1$ if $c_k^{(S_i)} \in S_j$, $0$ otherwise. Then the callee stability is the number of callees of $S_i$ that exist in $S_j$ over the number of callees of $S_i$: $$\mathit{CS} = \displaystyle\sum_{k=1}^{n} c_k^{(S_j)}/\|\{ c_1^{(S_i)},\ldots,c_n^{(S_i)}\}\| = \displaystyle\sum_{k=1}^{n} c_k^{(S_j)}/ n$$ If $S_j$ contains all the callees of $S_i$, then the callee stability is 1, to indicate that the constructs of $S_i$ called by the application exist also in library $S_j$. In case $S_j$ does not contain all the callees of $S_i$, then the callee stability is smaller than $1$ and reaches $0$ when none of the callees of $S_i$ is present in $S_j$. [Development Effort ($\mathit{DE}$). ]{} Let $a_k^{(S_i)}$ with $k=1,\ldots,n$ be the calls from the application to the callees of $S_i$, and $a_k^{(S_j)}=1$ if $a_k^{(S_i)} \not\in S_j$, $0$ otherwise. The development effort for updating from library $S_i$ in use by the application to library $S_j$ is defined as the number of application calls that require a modification due to callees of $S_i$ that do not exist in $S_j$. $$\mathit{DE} = \displaystyle\sum_{k=1}^{n} a_k^{(S_j)}$$ Compared to the callee stability, the development effort keeps into account the fact that each callee can be called multiple times within an application. In Figure \[fig:example\], for instance, each callee is called only once by $\alpha$ and $\lambda$ respectively. However, assuming that $\beta$ is called by two application constructs in addition to $\alpha$, and that it is not contained in the new library $S_j$, $\mathit{CS}=1/2$ whereas the $\mathit{DE}=3$. This reflects the fact that multiple calls need to be modified as a result of a change in a single callee. [As defined, the development effort does not take into account the complexity of each modification but rather focuses on the number of modification required by the application as each one comes at the cost not only of updating the code (which could be automated to some extent) but also of testing it.]{} [Reachable Body Stability ($\mathit{RBS}$). ]{} The reachable body stability is calculated in the same way as the callee stability, but instead of callees, it considers the reachable share of a library, i.e., the set of dynamically and/or statically reachable library constructs. Given the total number of constructs of $S_i$ reachable from the application, it measures the ratio of those that are contained as-is, i.e., with identical identifier and byte code, also in $S_j$. By quantifying the share of modified reachable constructs, this metric provides the likelihood that the behavior of the library changes from $S_i$ to $S_j$. In case all reachable constructs of $S_i$ exist in $S_j$, then $\mathit{RBS}=1$ and thus there is a higher likelihood that the library change does not break the application. [Overall Body Stability ($\mathit{OBS}$). ]{} The overall body stability is calculated similarly to $\mathit{RBS}$ but now considers all the constructs of $S_i$. This metric provides the same indication as the one above but, by considering the entire library rather than only its reachable share, it is independent of the application-specific usage. The above metrics support the application developer in estimating the effort and risks of updating a library. When several non-vulnerable libraries exist that are newer than the one in use, they are all candidate replacements. By quantifying the changes to be performed on the application and the changes that the library underwent, our update metrics allow the developer to take an informed decision. Note that the callee stability and development effort metrics only apply for dependencies that are called directly from the application, whereas the reachable and overall body stability also apply for transitive dependencies and frameworks. Evaluation {#sec:evaluation} ========== The implementation of our approach for Java, has been successfully used at [SAP]{}to perform over [250000]{} scans of about [500]{} applications since December 2016. In the following, we illustrate how our approach works in a typical scan, applying it to a [SAP]{}-internal web application that we adapted, for illustrative purposes, to include vulnerable OSS. The application allows users to upload files, such as documents or compressed archives, through an HTML form, inspects the file content and displays a summary to the user. It is developed using Maven [@maven], and depends on popular open-source libraries from the Apache Software Foundation, such as (released on 3 May 2015), (6 February 2014), (6 March 2016) and (21 February 2016). Overall, the application has 12 direct and 15 transitive compile-time dependencies. The analysis is performed using an implementation of the approach described previously: Sections \[subsec:id\], \[subsec:static\], and \[subsec:combination\] are implemented as goals of a Maven plugin; the collection of traces during the dynamic analysis (Section \[subsec:dynamic\]) is performed by instrumenting all classes of both the application and all its dependencies as described in [@icsme2015]. This happens either at runtime, when classes are loaded, or by modifying the byte code of the application before deploying it in a runtime environment such as Apache Tomcat. Detection and Analysis ---------------------- To illustrate the benefits of our approach, we go through the analysis steps and highlight selected findings. To demonstrate the added value of static analysis compared to [@icsme2015], we perform it after dynamic analysis. However, our implementation also supports changing their order (or executing only a subset of the steps). [(1) Vulnerability Detection.]{} The first step is to create a bill of materials (BOM), consisting of the constructs of the application and all its dependencies, as explained in Section \[subsec:id\]. Vulnerabilities in a library are detected by intersecting the set of constructs found in (the BOM of) that library with the vulnerable constructs of all the vulnerabilities known to our knowledge base. As an example, the bottom part of Figure \[fig:screenshot-t2c\] shows a table listing the vulnerable constructs for (columns Type and Qualified Construct Name) together with the respective change operation (column Change), as well as the information that those constructs are actually present in the Java archive corresponding to (column Contained). The vulnerability detection step reveals that our sample application includes vulnerable code related to 25 different vulnerabilities, affecting nine different compile-time dependencies: seven are direct, while the remaining two ( and [^3], pulled in through ) are transitive. [(2) Dynamic Assessment (Unit tests).]{} The execution of unit tests reveals that vulnerable constructs related to three vulnerabilities are executed, e.g., the method [^4], which is part of and subject to vulnerability [@httpclient-1803]. Another example is shown in Figure \[fig:screenshot-a2c\]: method , which is part of and subject to , is invoked in the context of unit test . The fact that reflection is heavily used inside the method (as visible from a sequence of four invocations of , see figure) makes it difficult for static analysis to determine the reachability of the vulnerable method. ![Unit tests reveal the execution of vulnerable construct in []{data-label="fig:screenshot-a2c"}](figures/a2c.png){width="\columnwidth"} ![Combined analysis reveals the reachability of in []{data-label="fig:screenshot-t2c-path"}](figures/t2c-path.png){width="\columnwidth"} [(3) Dynamic Assessment (Integration tests).]{} The execution of integration tests is done using an instrumented version of the application deployed in a runtime container. They reveal the execution of vulnerable code related to eight additional vulnerabilities, all affecting , or its dependencies and . As an example, the last line of the table in Figure \[fig:screenshot-t2c\] shows a vulnerable construct of , , whose actual execution is traced (column Traced) at the reported time. This method is included in which is part of the framework and exemplifies the inversion of control (IoC) happening when frameworks invoke application code. [(4) Static Assessment.]{} The static analysis starting from application constructs reveals that the constructor , part of and subject to , is reachable from the application. Dynamic analysis was not able to trace its execution due to the limited test coverage. On the other hand, static analysis starting from the application constructs falls short in the presence of IoC. As application methods are called by the framework, there is no path on the call graph starting from application and reaching framework constructs that are involved in the IoC mechanism. [(5) Combination of Static and Dynamic Assessment.]{} The static analysis starting from constructs traced with dynamic analysis provides additional evidence regarding the relevance of (the vulnerability that was exploited in the Equifax breach [@equifax]). In addition to the execution of method during step 3, the combination of static and dynamic analysis reveals that method , included in , is reachable with two calls from the traced method , as shown in Figure \[fig:screenshot-t2c-path\]. Its reachability is indicated with the red paw icons in the table containing the construct changes for (cf. the two right-most columns of the table in Figure \[fig:screenshot-t2c\]). ![image](figures/screenshot-t2c-cutouts.png){width="\textwidth"} The [value of combining the two analysis techniques]{} becomes more evident when considering all applications scanned with our approach: vulnerable constructs are reachable, statically or dynamically, in 131 out of [496]{}applications. In particular we observed 390 pairs of applications and vulnerabilities whose constructs were reachable. In 32 cases, the reachability could only be determined through the combination of techniques, which represents a 7.9% increase of evidence that vulnerable code is potentially executable. Mitigation ---------- During the execution of dynamic analysis (steps 2 and 3) and static analysis (steps 4 and 5), touch points and reachable constructs are collected. They are the basis for the computation of the update metrics for the application at hand. Figure \[fig:screenshot-mitigation\] shows that for , one of the direct dependencies of the application, there are nine touch points between the application and the library. The application method , for instance, calls the constructor (cf. first table in the figure). This invocation was observed during dynamic analysis, and was also found by static analysis (cf. rightmost columns in the first table in the figure). The second table of the figure shows the number of constructs of by type. For example, of the 608 constructors (), 199 were found reachable by static analysis, and 117 were actually executed during tests. The table at the bottom of Figure \[fig:screenshot-mitigation\] shows the update metrics that can guide the developer in the selection of a non-vulnerable replacement for . Each table row corresponds to a release of that is not subject to any vulnerability known to our knowledge base, hence, the developer is advised to choose among the three versions: , and . The callees of all touch points exist in all of those versions, hence, the update to any of those would not result in signature incompatibilities (cf. columns 3 and 4). The $\mathit{RBS}$ metric indicates that 872 out of 876 reachable constructs of type method and constructor are also present in release (870 out of 876 in and ). The $\mathit{OBS}$ metric is also relatively high for all three non-vulnerable releases, thus, the developer would likely choose in order to update the vulnerable library. ![image](figures/screenshot-mitigation-cutouts.png){width="\textwidth"} While the update decision is relatively straightforward for , it is more difficult for , since there are non-vulnerable replacements from both the 2.3 and the 2.5 branch (cf. Figure \[fig:screenshot-mitigation-struts\]). Here, the $\mathit{RBS}$ and $\mathit{OBS}$ metrics indicate a more significant change of constructs between the current version and the latest version of the 2.5 branch ($\mathit{RBS}$=862/887 and $\mathit{OBS}$=2781/3101) than between the current version and the latest version of the 2.3 branch ($\mathit{RBS}$=885/887 and $\mathit{OBS}$=3095/3101). Hence, the developer may be more inclined to stick to the 2.3 branch, thus updating to rather than to . ![Update metrics for []{data-label="fig:screenshot-mitigation-struts"}](figures/strus-mitigation-metrics.png){width="\columnwidth"} Related work {#sec:rel-work} ============ There exist several free [@owasp-dc] and commercial tools [@whitesource; @snyk; @blackduck; @sourceclear] for detecting vulnerabilities in OSS components. [In [@icsme2015] we showed that our approach outperforms state-of-the-art tools with respect to vulnerability detection.]{} Though [@sourceclear] claims to perform static analysis to eliminate false positives, there is no public description of their approach available. OWASP Dependency Check [@owasp-dc] is used in [@cadariu2015tracking] to create a vulnerability alert service and to perform an empirical investigation about the usage of vulnerable components in proprietary software. The results showed that 54 out of 75 of the projects analyzed have at least one vulnerable library. However the results had to be manually reviewed, as the matching of vulnerabilities to libraries showed low precision. Alqahtani et al. proposed an ontology-based approach to establish a link between vulnerability databases and software repositories [@alqahtani2016tracing]. The mapping resulting from their approach yields a precision that is 5% lower than OWASP Dependency Check. All these approaches and tools differ from ours in that they focus on vulnerability detection based on metadata, and do not provide application-specific reachability assessment nor mitigation proposals. [Our previous work [@icsme2015] already goes]{} beyond the detection of a vulnerability by performing reachability analysis: [we used]{} dynamic analysis to establish whether vulnerable code is actually executed. In this work, we extend [@icsme2015] including also static analysis and providing a novel combination of static and dynamic analysis. To the best of our knowledge none of the existing works and tools combines static and dynamic analysis, nor provides application-specific mitigation proposals. The screening test approach devised by Dashevskyi et al. [@dashevskyi2018tse] represents a scalable solution to the difficult problem of determining, at the time when a new vulnerability is disclosed for a given OSS component, which other previous versions are also affected by the same vulnerability. We tackle the same problem by comparing ASTs of the constructs involved in the vulnerability across the different versions of the affected component; however, due to space constraints, the details of our method are not covered in this paper. The empirical study conducted by Kula et al. on library migrations of 4600 GitHub projects showed that 81.5% of them do not update their direct library dependencies, not even when they are affected by publicly known vulnerabilities [@kula2017ese]. In particular, that study highlights the lack of awareness about security vulnerabilities. Considering 147 Apache software projects, [@bavota2015ese] studied the evolution of dependencies and found that applications tend to update their dependencies to newer releases containing substantial changes. Tools based on reward or incentives to trigger the update of out-of-date dependencies exist (e.g., [@david-dm; @greenkeeper]), however as shown in [@pull-requests], project developers are mostly concerned about breaking changes and mechanisms are needed to provide–next to transparency–a motivation for the update and confidence measures to estimate the risk of performing the update. By automatically detecting vulnerabilities, providing evidence about the reachability of the vulnerable code, and supporting mitigation via update metrics, our work addresses the need of motivating updates and estimating effort and risk. In [@visser2012] four metrics to measure the stability of libraries through time are proposed. In particular it considers the removal of units (constructs in our context), the amount of change in existing constructs, the ratio of change in new and old constructs, and the percentage of new constructs. Similar to our work, the metrics are meant to be representative for the amount of work required to update a certain library and thus they also consider usages of library methods in other projects. However the main focus of [@visser2012] is the library, and the metrics are used to measure its stability over time given a set of projects. Though some of their metrics ingredients can also be considered in our work, the metrics we propose are about the application-specific library usage. Moreover, our metrics benefit of our in-depth analysis of the application (e.g., some usages of the libraries that can only be observed with a combination of static and dynamic analysis). Raemaekers et al. studied breaking changes in library releases over seven years and showed that they occur with the same frequency in major and minor releases [@breakingChanges]. This shows that the rules of semantic versioning, according to which breaking changes are only allowed in major releases, are not followed in practice. It also shows that top three most frequent breaking changes involve a deletion of methods, classes, fields, respectively. This result reinforces our belief that our update metrics based on measuring the removal of program constructs provides a critical information. Mileva et al. studied the usage of different library versions and provided a tool to suggest which one to use based on the choice of the majority of similar users [@Mileva:2009]. Our work differs from theirs, in that we provide quantitative measures to support the user in selecting a non-vulnerable library. Existing works on library migration [@apiWave; @semDiff2009; @Nguyen:2010] are complementary to our approach in that they support developers in evolving their code to adapt to new libraries or library versions. [@apiWave] proposes a tool that keeps track of API popularity and migration of major frameworks/libraries, amounting to 650 Github projects resulting in 320000 APIs at the time of publication. [@semDiff2009] describes tools able to recommend replacements for framework methods accessed by a client program and deleted over time. [@Nguyen:2010] presents a tool able to recommend complex adaptations learned from already migrated clients or the library itself. Conclusion {#sec:conclusion} ========== The unique contribution of this paper is the use of static analysis and its combination with dynamic analysis to support the application-specific assessment and mitigation of open-source vulnerabilities. This approach further advances the code-centric detection and dynamic analysis of vulnerable dependencies [ we originally proposed in [@icsme2015]]{}. The accuracy and application-specific nature of our method improves over state-of-the-art approaches, which commonly depend on metadata. [Vulas]{}, the implementation of our approach for Java, was chosen by [SAP]{}among several candidates as the recommended OSS vulnerability scanner. Since December 2016, it has been used for over [250000]{} scans of about [500]{} applications, which demonstrates the viability and scalability of the approach. The variety of programming languages used in today’s software systems pushes us to extend [Vulas]{}to support languages other than Java. However, fully-qualified names can be inadequate to uniquely identify the relevant program constructs in certain languages, so we are considering the use of information extracted from the construct bodies. Finally, the problem of systematically linking open-source vulnerability information to the corresponding source code changes (the fix) remains open. Maintaining a comprehensive knowledge base of rich, detailed vulnerability data is critical to all vulnerability management approaches and requires considerable effort. While this effort could be substantially reduced creating specialized tools, we strongly believe that the maintenance of this knowledge base should become an industry-wide, coordinated effort, whose outcome would benefit the whole software industry. Acknowledgements. {#acknowledgements. .unnumbered} ----------------- We are also grateful to our colleagues Michele Bezzi, Luca Compagna, Cédric Dangremont, and Brian Duffy for their insightful comments on early drafts of this work. [^1]: As an example, is the fully-qualified name of method in class and package . [^2]: In case a commit includes not only a vulnerability fix but also unrelated changes, then a post-processing of the construct changes is required. [^3]: Maven dependencies are denoted using their artifact identifier followed by, where necessary, a colon and their version. Group identifiers are omitted for brevity. [^4]: Where possible, Java package and class names are omitted for brevity.
--- abstract: \| We introduce a new variant of the geometric Steiner arborescence problem, motivated by the layout of flow maps. Flow maps show the movement of objects between places. They reduce visual clutter by bundling lines smoothly and avoiding self-intersections. To capture these properties, our angle-restricted Steiner arborescences, or flux trees, connect several targets to a source with a tree of minimal length whose arcs obey a certain restriction on the angle they form with the source. We study the properties of optimal flux trees and show that they are planar and consist of logarithmic spirals and straight lines. Flux trees have the shallow-light property. We show that computing optimal flux trees is NP-hard. Hence we consider a variant of flux trees which uses only logarithmic spirals. Spiral trees approximate flux trees within a factor depending on the angle restriction. Computing optimal spiral trees remains NP-hard, but we present an efficient 2-approximation, which can be extended to avoid “positive monotone” obstacles. author: - Kevin Buchin - Bettina Speckmann - Kevin Verbeek date: \| Dep. of Mathematics and Computer Science, TU Eindhoven, The Netherlands.\ [k.a.buchin@tue.nl]{} title: \| Angle-Restricted Steiner Arborescences\ for Flow Map Layout[^1] --- Introduction {#sec:introduction} ============ Flow maps are a method used by cartographers to visualize the movement of objects between places [@Dent1999; @Slocum2010]. One or more sources are connected to several targets by arcs whose thickness corresponds to the amount of flow between a source and a target. Good flow maps share some common properties. They reduce visual clutter by merging (bundling) lines as smoothly and frequently as possible. Furthermore, they strive to avoid crossings between lines. Flow trees, that is, single-source flows, are drawn entirely without crossings. Flow maps that depict trade often route edges along actual shipping routes. In addition, flow maps try to avoid covering important map features with flows to aid recognizability. Most flow maps are still drawn by hand and none of the existing algorithms (that use edge bundling), can guarantee to produce crossing-free flows. ![Flow maps from our companion paper [@InfoVisFlowMap] based on angle-restricted Steiner arborescences: Migration from Colorado and whisky exports from Scotland.[]{data-label="fig:flowmaps"}](COGreen "fig:"){height="4.5cm"} ![Flow maps from our companion paper [@InfoVisFlowMap] based on angle-restricted Steiner arborescences: Migration from Colorado and whisky exports from Scotland.[]{data-label="fig:flowmaps"}](whiskyCrop "fig:"){height="4.5cm"} In this paper we introduce a new variant of geometric minimal Steiner arborescences, which captures the essential structure of flow trees and serves as a “skeleton” upon which to build high-quality flow trees. Our input consists of a point $r$, the root (source), and $n$ points $t_1, \ldots, t_n$, the terminals (targets). Visually appealing flow trees merge quickly, but smoothly. A geometric minimal Steiner arborescence on our input would result in the shortest possible tree, which naturally merges quickly. A Steiner arborescence for a given root and a set of terminals is a rooted directed Steiner tree, which contains all terminals and where all edges are directed away from the root. Without additional restrictions on the edge directions (as in the rectilinear case or in the variant proposed in this paper), a geometric Steiner arborescence is simply a geometric Steiner tree with directed edges. However, Steiner arborescences have angles of $2\pi/3$ at every internal node and hence are quite far removed from the smooth appearance of hand-drawn flow maps. Our goal is hence to connect the terminals to the root with a Steiner tree of minimal length whose arcs obey a certain restriction on the angle they form with the root. [r]{}[0.35]{} ![image](Anglerestriction) Specifically, we use a restricting angle $\alpha < \pi/2$ to control the direction of the arcs of a Steiner arborescence $T$. Consider a point $p$ on an arc $e$ from a terminal to the root (see Figure \[fig:restriction\]). Let $\gamma$ be the angle between the vector from $p$ to the root $r$ and the tangent vector of $e$ at $p$. We require that $\gamma \leq \alpha$ for all points $p$ on $T$. We refer to a Steiner arborescence that obeys this angle restriction as angle-restricted Steiner arborescence, or simply flux tree. Here and in the remainder of the paper it is convenient to direct flux trees from the terminals to the root. Also, to simplify descriptions, we often identify the nodes of a flux tree $T$ with their locations in the plane. In the context of flow maps it is important that flux trees can avoid obstacles, which model important features of the underlying geographic map. Furthermore, it is undesirable that terminals become internal nodes of a flux tree. We can ensure that our trees never pass directly through terminals by placing a small triangular obstacle just behind each terminal (as seen from the root). Hence our input also includes a set of $m$ obstacles $B_1,\ldots, B_m$. We denote the total complexity (number of vertices) of all obstacles by $M$. In the presence of obstacles our goal is to find the shortest flux tree $T$ that is planar and avoids the obstacles. The edges of flux trees are by definition “thin”, but their topology and general structure are very suitable for flow trees. In a companion paper [@InfoVisFlowMap] we describe an algorithm that thickens and smoothes a given flux tree while avoiding obstacles. Figure \[fig:flowmaps\] shows two examples of the maps computed with our algorithm, further examples and a detailed discussion of our maps can be found in [@InfoVisFlowMap]. [Related work.]{} There is a multitude of related work on both the practical and the theoretical side of our problem and consequently we cannot cover it all. One of the first systems for the automated creation of flow maps was developed by Tobler in the 1980s [@FlowMapper; @WaldoTobler1987]. His system does not use edge bundling and hence the resulting maps suffer from visual clutter. In 2005 Phan [[et al.]{}]{} [@Phan2005] presented an algorithm, based on hierarchical clustering of the terminals, which creates flow trees with bundled edges. This algorithm uses an iterative ad-hoc method to route edges and is often unable to avoid crossings. A second effect of this method is that flows are often routed along counterintuitive routes. The quality of the maps can be improved by moving the terminals, which, however, is considered to be confusing for users by cartography textbooks [@Slocum2010]. Recent papers from the information visualization community explore alternative ways to visualize flows, by using multi-view displays [@Guo2009], animations over time [@Boyandin2010], or mapping techniques close to treemaps [@Wood2010]. There are many variations on the classic Steiner tree problem which employ metrics that are related to their specific target applications. Of particular relevance to this paper is the rectilinear Steiner arborescence (RSA) problem, which is defined as follows. We are given a root (usually at the origin) and a set of terminals $t_1, \ldots, t_n$ in the northeast quadrant of the plane. The goal is to find the shortest rooted rectilinear tree $T$ with all edges directed away from the root, such that $T$ contains all points $t_1, \ldots, t_n$. For any edge of $T$ from $p = (x_p, y_p)$ to $q = (x_q, y_q)$ it must hold that $x_p \leq x_q$ and $y_p \leq y_q$. If we drop the condition of rectilinearity then we arrive at the Euclidean Steiner arborescence (ESA) problem. In both cases it is NP-hard [@Shi2000; @ss-rsap-05] to compute a tree of minimum length. Rao [[et al.]{}]{} [@Rao92] give a simple $2$-approximation algorithm for minimum rectilinear Steiner arborescences. Córdova and Lee [@Cordova94] describe an efficient heuristic which works for terminals located anywhere in the plane. Ramnath [@Ramnath03] presents a more involved $2$-approximation that can also deal with rectangular obstacles. Finally, Lu and Ruan [@Lu2000] developed a PTAS for minimum rectilinear Steiner arborescences, which is, however, more of theoretical than of practical interest. Conceptually related are gradient-constrained minimum networks which are studied by Brazil [[et al.]{}]{} [@Brazil2001; @Brazil2007] motivated by the design of underground mines. Gradient-constrained minimum networks are minimum Steiner trees in three-dimensional space, in which the (absolute) gradients of all edges are no more than an upper bound $m$ (so that heavy mining trucks can still drive up the ramps modeled by the Steiner tree). Krozel [[et al.]{}]{} [@Krozel2006] study algorithms for turn-constrained routing with thick edges in the context of air traffic control. Their paths need to avoid obstacles (bad weather systems) and arrive at a single target (the airport). The union of consecutive paths bears some similarity with flow maps, although it is not necessarily crossing-free or a tree. [Results and organization.]{} In Section \[sec:props\] we derive properties of optimal (minimum length) flux trees. In particular, we show that they are planar and that the arcs of optimal flux trees consist of (segments of) logarithmic spirals and straight lines. Flux trees have the shallow-light property [@Awerbuch1990], that is, we can bound the length of an optimal flux tree in comparison with a minimum spanning tree on the same set of terminals and we can give an upper bound on the length of a path between any point in a flux tree and the root. They also naturally induce a clustering on the terminals and smoothly bundle lines. Unfortunately we can show that it is NP-hard (Section \[sec:npproof\]) to compute optimal flux trees. Hence, in Section \[sec:spiraltrees\] we introduce a variant of flux trees, so called spiral trees. The arcs of spiral trees consist only of logarithmic spiral segments. We prove that spiral trees approximate flux trees within a factor depending on the restricting angle $\alpha$. Our experiments show that $\alpha = \pi/6$ is a reasonable restricting angle, in this case the approximation factor is . In Section \[sec:npproof\] we show that computing optimal spiral trees remains NP-hard. For a special case, we give an exact algorithm in Section \[sec:emptyregions\] that runs in $O(n^3)$ time. In Section \[sec:approximation\] we develop a 2-approximation algorithm for spiral trees that works in general and runs in $O(n \log n)$ time. Finally, in Section \[sec:obstacles\] we extend our approximation algorithm (without deteriorating the approximation factor) to include “positive monotone” obstacles. On the way, we develop a new 2-approximation algorithm for rectilinear Steiner arborescences in the presence of positive monotone obstacles. Both algorithms run in $O((n+M) \log(n+M))$ time, where $M$ is the total complexity of all obstacles. Optimal flux trees {#sec:props} ================== Recall that our input consists of a root $r$, terminals $t_1, \ldots, t_n$, and a restricting angle $\alpha < \pi/2$. Without loss of generality we assume that the root lies at the origin. Recall further that an optimal flux tree is a geometric Steiner arborescence, whose arcs are directed from the terminals to the root and that satisfies the angle restriction. We show that the arcs of an optimal flux tree consist of line segments and parts of logarithmic spirals (Property \[property:optedgeshape\]), that any node except for the root has at most two incoming arcs (Property \[property:optbinary\]), and that an optimal flux tree is planar (Property \[property:optplanar\]). Finally, flux trees (and also spiral trees) have the shallow-light property (Property \[property:shallowlight\]). [r]{}[.4]{} ![image](Spirals) [Spiral regions.]{} For a point $p$ in the plane, we consider the region $\mathcal{R}_p$ of all points that are reachable from $p$ with an angle-restricted path, that is, with a path that satisfies the angle restriction. Clearly, the root $r$ is always in $\mathcal{R}_p$. The boundaries of $\mathcal{R}_p$ consist of curves that follow one of the two directions that form exactly an angle $\alpha$ with the direction towards the root. Curves with this property are known as logarithmic spirals (see Figure \[fig:Spirals\]). Logarithmic spirals are self-similar; scaling a logarithmic spiral results in another logarithmic spiral. Logarithmic spirals are also self-approaching as defined by Aichholzer [[et al.]{}]{} [@Aichholzer2001], who give upper bounds on the lengths of (generalized) self-approaching curves. As all spirals in this paper are logarithmic, we simply refer to them as spirals. For $\alpha < \pi/2$ there are two spirals through a point. The right spiral $\mathcal{S}^{+}_p$ is given by the following parametric equation in polar coordinates, where $p = (R, \phi)$: $R(t) = R e^{-t}$ and $\phi(t) = \phi + \tan(\alpha) t$. The parametric equation of the left spiral $\mathcal{S}^{-}_p$ is the same with $\alpha$ replaced by $-\alpha$. Note that a right spiral $\mathcal{S}^{+}_p$ can never cross another right spiral $\mathcal{S}^{+}_q$ (the same holds for left spirals). The spirals $\mathcal{S}^{+}_p$ and $\mathcal{S}^{-}_p$ cross infinitely often. The reachable region $\mathcal{R}_p$ is bounded by the parts of $\mathcal{S}^{+}_p$ and $\mathcal{S}^{-}_p$ with $0 \leq t \leq \pi \cot(\alpha)$. We therefore call $\mathcal{R}_p$ the spiral region of $p$. It follows directly from the definition that for all $q \in \mathcal{R}_p$ we have that $\mathcal{R}_q \subseteq \mathcal{R}_p$. \[lem:shortestpath\] The shortest angle-restricted path between a point $p$ and a point $q \in \mathcal{R}_p$ consists of a straight segment followed by a spiral segment. Either segment can have length zero. [Proof. ]{} Consider the spirals $\mathcal{S}^{+}_q$ and $\mathcal{S}^{-}_q$ through $q$, specifically the parts with $t \leq 0$ (see Figure \[fig:Lemma1\]). Any point on the opposite side of the spirals as $p$ is unable to reach $q$. Thus any shortest path from $p$ to $q$ cannot cross either of these spirals. If we see these spirals as obstacles and ignore the angle restriction for now, the shortest path $\pi$ is simply a straight segment followed by a spiral segment. Now consider any point $u$ on $\pi$. Because $\mathcal{S}^{+}_q$ and $\mathcal{S}^{+}_u$ cannot cross (same for $\mathcal{S}^{-}_q$ and $\mathcal{S}^{-}_u$), we get that $q \in \mathcal{R}_u$. Therefore $\pi$ also satisfies the angle restriction. ![image](Lemma1){height="1.2in"} ![image](OptFlowTree){height="1.2in"} ![image](Property2){height="1.2in"} \[property:optedgeshape\] An optimal flux tree consists of straight segments and spiral segments. [Proof. ]{} Consider an optimal flux tree $T$. Now replace all edges between a terminal/the root and a Steiner node by the shortest angle-restricted path between the two points. This can only shorten $T$. By Lemma \[lem:shortestpath\], the resulting flux tree consists of only straight segments and spiral segments. \[property:optbinary\] Every node in an optimal flux tree $T$, other than the root $r$, has at most two incoming edges. [Proof. ]{} Assume $T$ contains a node at $p$ with at least three incoming edges. Pick one of the incoming edges $e$ that is not leftmost or rightmost and let $q$ be the other endpoint of $e$. Let $e_L$ and $e_R$ be the leftmost and rightmost incoming edges of $p$ (see Figure \[fig:Property2\]). Now consider the straight line $\ell$ from $q$ to the root $r$. Assume without loss of generality that $\ell$ passes $p$ on the left side (the right side is symmetric with $e_R$) or $\ell$ goes through $p$. We claim that we can locally improve the length of $T$ by moving the endpoint at $p$ of $e$ along $e_L$. The angle between $e$ and $e_L$ at $p$ is at most $\alpha$. Because $\alpha < \pi/2$ and because locally moving the endpoint of $e$ along $e_L$ will not make the spiral segment of $e$ longer, this will shorten the tree $T$. Also, locally moving the endpoint of $e$ along $e_L$ cannot suddenly violate the angle restriction (assuming that $\alpha > 0$). Contradiction. \[property:optplanar\] Every optimal flux tree is planar. Assume two edges $e_1$ (from $p_1$ to $q_1$) and $e_2$ (from $p_2$ to $q_2$) cross. Let $u$ be the crossing between $e_1$ and $e_2$. Now simply remove the part of $e_1$ from $u$ to $q_1$. There is still a connection from $p_1$ to $r$ via $q_2$, so the resulting tree is still a proper flux tree. Also, removing a segment cannot violate the angle restriction and makes the tree shorter. Contradiction. The last property requires a more involved proof. We postpone the proof of this property until Section \[sec:shallowlight\]. Let $d^T(p)$ be the distance between $p$ and $r$ in a flux tree $T$ and let $d(p)$ be the Euclidean distance between $p$ and $r$. \[property:shallowlight\] The length of an optimal flux tree $T$ is at most $O((\sec(\alpha) + \csc(\alpha)) \log n)$ times the length of the minimum spanning tree on the same set of terminals. Also, for every point $p \in T$, $d^T(p) \leq \sec(\alpha) d(p)$. Spiral trees {#sec:spiraltrees} ============ In this section we introduce spiral trees and prove that they approximate flow trees. The arcs of a spiral tree consist only of spiral segments of a given $\alpha$ (see Figure \[fig:SpiralTree\]). In other words, an optimal spiral tree is the shortest flow tree that uses only spiral segments. Spiral trees satisfy the angle restriction by definition. Any particular arc of a spiral tree can consist of arbitrarily many spiral segments. That is, any arc of the spiral tree can switch between following its right spiral and following its left spiral an arbitrary number of times. The length of a spiral segment can easily be expressed in polar coordinates. Let $p = (R_1, \phi_1)$ and $q = (R_2, \phi_2)$ be two points on a spiral, then the distance $D(p, q)$ between $p$ and $q$ on the spiral is $$\label{eqn:spiraldist} D(p, q) = \sec(\alpha) \|R_1 - R_2\| \, .$$ Consider the shortest spiral path—using only spiral segments—between a point $p$ and a point $q$ reachable from $p$. The reachable region for $p$ is still its spiral region $\mathcal{R}_p$, so necessarily $q \in \mathcal{R}_p$. The length of a shortest spiral path is given by Equation \[eqn:spiraldist\]. The shortest spiral path is not unique, in particular, any sequence of spiral segments from $p$ to $q$ is shortest, as long as we move towards the root. ![$T$ and $T''$.[]{data-label="fig:Theorem1"}](SpiralTree){width=".8\textwidth"} ![$T$ and $T''$.[]{data-label="fig:Theorem1"}](Theorem1-2) \[thm:fluxapprox\] The optimal spiral tree $T'$ is a $\sec(\alpha)$-approximation of the optimal flux tree $T$. Let $\mathcal{C}_R$ be a circle of radius $R$ with the root $r$ as center. A lower bound for the length of $T$ is given by $L(T) \geq \int_0^\infty \|T \cap \mathcal{C}_R\| dR$, where $\|T \cap \mathcal{C}_R\|$ counts the number of intersections between the tree $T$ and the circle $\mathcal{C}_R$. Using Equation \[eqn:spiraldist\], the length of $T'$ is $L(T') = \sec(\alpha) \int_0^\infty \|T' \cap \mathcal{C}_R\| dR$. Now consider the spiral tree $T''$ with the same nodes as $T$, but where all arcs between the nodes are replaced by a sequence of spiral segments (see Figure \[fig:Theorem1\]). For a given circle $\mathcal{C}_R$, this operation does not change the number of intersections of the tree with $\mathcal{C}_R$, i.e. $\|T \cap \mathcal{C}_R\| = \|T'' \cap \mathcal{C}_R\|$. So we get the following: $$L(T') \leq L(T'') = \sec(\alpha) \int_0^\infty \|T'' \cap \mathcal{C}_R\| dR = \sec(\alpha) \int_0^\infty \|T \cap \mathcal{C}_R\| dR \leq \sec(\alpha) L(T)$$ Next to the fact that optimal spiral trees are a good approximation of optimal flux trees, they also maintain important properties of optimal flux trees, namely Properties \[property:optbinary\] and \[property:optplanar\]. \[lem:spiralplandeg\] An optimal spiral tree is planar and every node, other than the root, has at most two incoming edges. The root has exactly one incoming edge. First of all, only two spirals go through a single point: the left and the right spiral. So every node other than the root $r$ has at most two incoming arcs, otherwise there is a repeated spiral segment which can be removed. Furthermore, the same arguments as in the proof of Property \[property:optplanar\] yield that also the optimal spiral tree is planar. When we approximate an optimal flux tree by a spiral tree, we can further reduce the length of the tree by replacing every arc by the shortest angle-restricted path between its endpoints. This operation does not improve the approximation factor, but it improves the tree visually. Shallow-Light Property {#sec:shallowlight} ---------------------- In the following we prove the shallow-light property for optimal spiral trees. That is, we bound the length of an optimal spiral tree in comparison with a minimum spanning tree on the same set of terminals and we give an upper bound on the length of a path between any point in a spiral tree and the root. Since for flux trees the length of such paths and the total length are not larger, we can conclude that optimal flux trees also have the shallow-light property (Property \[property:shallowlight\]). The second part of the property (shallowness) is easy to see. The path of a node to the root in any spiral tree is by Equation \[eqn:spiraldist\] bounded by $\sec(\alpha)$ times the distance of the node to the root. We now show that the length of an optimal spiral tree approximates the length of a minimum spanning tree by a factor of $O((\sec(\alpha) + \csc (\alpha)) \log n )$. We build a spiral tree in the following way. First we find a short cycle through the points. We then take a matching based on this cycle and pairwise join points by spiral segments. This results in $\lceil n/2 \rceil$ components. On these we again find a matching and pairwise join them and so on. We need to ensure that the set of spiral segments used in the construction is compatible with a spiral tree. Throughout this section we will assume that the root of the tree is placed at the origin. We call a sequence of spiral segments between two points inward going if the distance from the segments to the origin have no local maximum except possibly at the two points. In particular, if a point is in the spiral region of another, a path of decreasing distance to the root from the outer point to the inner one would be inward going. Any pair of points can be joined by an inward going sequence of two spiral segments and inward going sequences are compatible with spiral trees if we use the point with smallest distance to the root as join node. We need to bound the length of such a sequence. For this we first bound the length of a spiral segment. Let $p$, $p'$ be two points on a spiral segment with polar coordinates $p = (R,\phi)$ and $p' = (R',\phi')$. Equation \[eqn:spiraldist\] gives us a bound in terms of $R, R'$. To bound the length in terms of $R,\phi,\phi'$, let us assume $R' \leq R$. The other case is analogous. We consider the parametric equations of the spiral through $p$ and $p'$ with $(R(0),\phi(0)) = (R, \phi)$. For $p'$ we obtain the equations $R' = R e^{-t}$ and $\phi' = \phi + \tan (\alpha) t$ (or $\phi' = \phi - \tan (\alpha) t$ depending on whether the points lie on a left or right spiral). Solving for $R'$ yields $R' = R e^{- (\|\phi'-\phi\|)/\tan (\alpha)}$. Inserting this into Equation \[eqn:spiraldist\] gives $$\label{eqn:spiral2} D(p,p') = \sec (\alpha) R (1-e^{- (\|\phi'-\phi\|)/\tan (\alpha)})\, .$$ Equation \[eqn:spiral2\] has several consequences. Given two points $p$, $q$ that do not lie in the spiral regions of each other. Assume we have two sequences of spiral segments, each connecting $p$ and $q$ such that no ray through the origin intersect a sequence twice, and such that parameterized by the angle $\phi$ to the origin, one sequence has a smaller or equal distance to the origin for all $\phi$. Then sweeping over the angle and summing up the contributions of Equation \[eqn:spiral2\] gives that the sequence closer to the origin has a smaller (or equal) total arc length. Thus, the shortest connection between $p$ and $q$ is obtained by simply joining $p$ and $q$ by an inward going sequence of two spiral segments. Another consequence of Equation \[eqn:spiral2\] is the following. Again $p = (R, \phi)$, $q = (R', \phi')$ are two points that do not lie in the spiral regions of each other. Further assume $R, \phi, \phi'$ are given, but for $R'$ we only know $R' \leq R$. Then the arc length of the inward going sequence of two spiral segments joining $p$ and $q$ (using the angle range between $\phi$ and $\phi'$) is maximized for $R=R'$. This follows from the same argument as above, i.e., the resulting sequence of spiral segments dominates all others in terms of distance to the origin. So far we have not linked the arc length of the spiral segments between two points with the Euclidean distance between the points. We do this by the following lemma. Two points in the plane of distance $D$ can be connected by an inward going path of logarithmic spirals of angle $\alpha$ such that the summed length of the spiral segments is bounded by $3 D \max (\sec (\alpha), \csc (\alpha))$. The path uses at most two spiral segments. Let $p_1$, $p_2$ be two points of distance $D$ with polar coordinates $p_1 = (R_1,\phi_1)$ and $p_2 = (R_2,\phi_2)$. Without loss of generality we assume that $R_1 \leq R_2$, $\phi_1 \leq \phi_2$ and $\phi_2 - \phi_1 \leq \pi$. We first handle the case that $p_1$ lies in the spiral region of $p_2$. In this case we can connect the points by an inward going path from $p_2$ to $p_1$ using two spiral segments. By Equation \[eqn:spiraldist\] the length of this path is $\sec(\alpha) (R_2-R_1) \leq \sec(\alpha) D$, which proves the claim for this case. Next we handle the case that $p_1$ does not lie in the spiral region of $p_2$. In this case we join the points using the right spiral through $p_1$ and the left spiral through $p_2$. Let $p = (R, \phi)$ be the point where the two points first join. The summed length of the spiral segments is $L = \sec (\alpha) (R_1 + R_2 - 2R)$, which we need to bound in terms of $D$ We distinguish two cases. First assume the points have a distance of at most $3D/2$ to the root. Then we obtain the connection between the points by simply connecting both to the root. Then $L \leq \sec(\alpha) (R_1+R_2) \leq 2 \sec(\alpha) 3D/2 = 3 D \sec(\alpha)$. Next assume $R_2>3D/2$. From the discussion of Equation \[eqn:spiral2\] above we know that $L$ is maximized for $R_1=R_2$. In this case we have $\phi = (\phi_2+\phi_1)/2$ and therefore $$\begin{aligned} L &= \sec (\alpha) (2 R_2 - 2 R_2 e^{- \frac{\phi_2-\phi_1}{2\tan (\alpha)}}) = \sec (\alpha) 2 R_2 (1 - e^{- \frac{\phi_2-\phi_1}{2\tan (\alpha)}})\\ &\leq \sec (\alpha) 2 R_2 \frac{\phi_2-\phi_1}{2\tan (\alpha)} = \csc (\alpha) R_2 (\phi_2-\phi_1).\end{aligned}$$ It remains to bound $R_2 (\phi_2-\phi_1)$ in terms of the Euclidean distance $D$ of the two points. Observe that for given $p_2$ (with $R_2>3D/2$) and $D$ the angle $\phi_2-\phi_1$ is maximized if the line through the origin and $p_1$ is tangent to the circle of radius $D$ around $p_2$. Thus $\phi_2-\phi_1$ is maximized if the angle formed by $p_2$, $p_1$ and the origin is $\pi/2$. In this case $\phi_2-\phi_1 = \arcsin(D/R_2)$. Thus, in general $$\phi_2-\phi_1 \leq \arcsin \left( \frac{D}{R_2} \right) = \arctan \left( \frac{D}{\sqrt{R_2^2 - D^2}} \right) \leq \frac{D}{\sqrt{R_2^2 - D^2}} \, .$$ Since $R_2>3D/2$, we have $\sqrt{R_2^2 - D^2}\geq R_2 \sqrt{1-4/9}$. Plugging this into the above bound gives $\phi_2-\phi_1 \leq D/(R_2\sqrt{5/9})$. Now inserting this into the bound on $L$ gives $$L \leq \csc (\alpha) D/\sqrt{5/9} < 3 \csc (\alpha) D.$$ Combining the cases results in the claimed bound. The length of the optimal spiral tree of a set of points is bounded by\ $3 \lceil \log_2 n \rceil \max (\sec (\alpha), \csc (\alpha ))$ times the length of the minimum spanning tree of the set of points with the origin included. In the following we construct a spiral tree for which this bound holds. Let $L$ be the length of the minimum spanning tree on the points including the origin. Let $\kappa = 3 \max (\sec (\alpha), \csc (\alpha ))$. Let $C_1$ be a cycle through the points of length at most $2L$ (e.g., obtained by ordering the points based on a depth first search in the minimum spanning tree). We replace each edge of $C_1$ by an inward going sequence of at most two spiral segments. This results in a cycle $C'_1$ of sequences of spiral segments of length at most $2\kappa L$. By taking either every even or every odd sequence we join pairs (possibly leaving the root unmatched) of nodes by spiral segments of total length at most $\kappa L$. We repeat the construction on the join nodes (and possibly an unmatched point)) using the cycle $C_2$ induced by the order given by $C'_1$. Again we form a cycle of spiral segments through the vertices of $C_2$. If we parameterize corresponding sequences in the cycles $C'_1$ and $C'_2$ by the angle $\phi$ from the origin, the sequences in cycle $C'_2$ are closer to the origin than the corresponding sequences in $C'_1$ for all angles. Thus as consequence of Equation \[eqn:spiral2\] the length of $C'_2$ is bounded by the length of $C'_1$ and therefore by $2\kappa L$. Thus as in the previous step we can join pairs of join nodes using spiral segments of total length at most $\kappa L$. Next we construct $C'_3$ from $C'_2$ in the same way and iterate the construction. After at most $\lceil \log_2 n \rceil$ all nodes have been joined. The total length is then $\lceil \log_2 n \rceil \kappa L$ as claimed. Relation with rectilinear Steiner arborescences. {#sec:transformation} ------------------------------------------------ Both rectilinear Steiner arborescences and spiral trees contain directed paths, from the root to the terminals or vice versa. Every edge of a rectilinear Steiner arborescence is restricted to point right or up, which is similar to the angle restriction of flux and spiral trees. In fact, there exists a transformation from rectilinear Steiner arborescences into spiral trees. Consider the following transformation from the coordinates $(x, y)$ of a rectilinear Steiner arborescence to the polar coordinates $(R, \phi)$ of a spiral tree. $$\begin{aligned} \label{eqn:transformation} R =& e^{x + y} \\ \nonumber \phi =& (y - x) \tan(\alpha)\end{aligned}$$ Assume we keep one of the coordinates $x$ or $y$ fixed. Using the spiral equation from Section \[sec:props\] we see that the result is a spiral. More specifically, keeping $x$ fixed results in left spirals and keeping $y$ fixed results in right spirals. So that means that the above transformation transforms horizontal and vertical lines to right and left spirals, respectively (see Figure \[fig:RSATransform\]). The transformation maps the root of the rectilinear Steiner arborescence to $(1, 0)$. Thus, to get a valid spiral tree, we still need to connect $(1, 0)$ to $r$. The transformation in Equation \[eqn:transformation\] transforms a rectilinear Steiner arborescence into a spiral tree. Unfortunately, the transformation has several shortcomings. First of all, the transformation is not a bijection, it is a surjection. That means we can invert the transformation, but only if we restrict the domain in the rectilinear space. But most importantly, the metric does not carry over the transformation. That means that it is not necessarily true that the minimum rectilinear Steiner arborescence transforms to the optimal spiral tree. Thus the relation between the concepts cannot be used directly and algorithms developed for rectilinear Steiner arborescences cannot be simply modified to compute spiral trees. However, the same basic ideas can often be used in both settings. ![A rectilinear Steiner arborescence transformed to a spiral tree.[]{data-label="fig:RSATransform"}](RSATransform){width=".6\textwidth"} Computing spiral trees {#sec:computespirals} ====================== In this section we describe algorithms to compute (approximations of) optimal spiral trees. First we show that it is NP-hard to compute optimal flux or spiral trees. Then we give an exact algorithm for computing optimal spiral trees in the special case that all spiral regions are empty, i.e. $t_i \notin \mathcal{R}_{t_j}$ for all $i \neq j$. Finally we give an approximation algorithm for computing optimal spiral trees in the general case. Computing optimal flux and spiral trees is NP-hard {#sec:npproof} -------------------------------------------------- For the hardness proofs we will choose $\alpha = \pi/4$. The reduction is from the rectilinear Steiner arborescence (RSA) problem [@ss-rsap-05] for spiral trees, and from the Euclidean Steiner arborescence (ESA) problem [@Shi2000] for flux trees. Shi and Su [@ss-rsap-05] proved by a reduction from planar 3SAT that the decision versions of the RSA problem and the ESA problem are NP-hard. Their reduction uses points on an $O(m \times m)$ grid, where $m$ bounds the size of the 3SAT instance. We can assume the grid to be an integer grid. Now if there is a satisfying assignment, the optimal RSA and ESA have an integer length $K$, while if there is no such assignment the optimal RSA and ESA have length at least $K+1$. We can therefore state the problems for which they proved NP-hardness and from which we will reduce as follows.\ [Instance: ]{} A set of integer points $P = \{ p_1,\ldots, p_N \}$ in the first quadrant of the plane with coordinates bounded by $O(N^2)$; a positive integer $K$. [Question (RSA): ]{} Is there a RSA of total length $K$ or less? Otherwise the shortest RSA has length at least $K+1$. [Question (ESA): ]{} Is there a ESA of total length $K$ or less? Otherwise the shortest RSA has length at least $K+1$. [r]{}[.25]{} ![image](NPhard-idea-scaled) The basic idea is sketched in Figure \[fig:NPhard-idea\]. Assume we are given an instance of the Euclidean Steiner arborescence problem with polynomially bounded coordinates. We translate the set of terminals by a large polynomial factor along the diagonal with slope 1 and place a new root at the origin. If the bound on the coordinates is small (the square in Figure \[fig:NPhard-idea\]) relative to the factor of the translation, then the angle formed by the line through any of the translated points and the origin with the x-axis is “more or less $\pi/4$”. A $\pi/4$-restricted flux tree thus behaves within this square “more or less” like an Euclidean Steiner arborescence. For spiral trees we use the same setup but show that the distances on the spiral tree approximate the $L_1$-norm. However, quantifying “more or less" precisely is technically rather involved and will be done in this section. To draw the connection from Steiner arborescences to flux and spiral trees we generalize the concept of RSAs and ESAs. For flux and spiral trees the angle $\alpha$ is bounded relative to the root while for RSAs and ESAs the angle that an edge can make with the $x$-axis is bounded (or with any given line through the origin). For ESAs the angle of an edge is in $[0,\pi/2]$, while for RSAs the angle is in $\{0, \pi/2 \}$. We call a Steiner arborescence with angles in $[\beta,\pi/2-\beta]$ a $(\geq \beta)$-Steiner arborescence ($(\geq \beta)$-SA) and a Steiner arborescence with angles in $\{\beta,\pi/2-\beta\}$ a $\beta$-Steiner arborescence ($\beta$-SA). We do not restrict $\beta$ to be positive but to $-\pi/4 < \beta < \pi/4$. In the following Steiner arborescences are typically not rooted at the origin. Now let $-\pi/4 < \beta < \beta' < \pi/4$. Every $(\geq \beta')$-SA is a $(\geq \beta)$-SA but the converse does not hold. However, we can transform a $(\geq \beta)$-SA to $(\geq \beta')$-SA of similar length. Our transformation first transforms the whole tree and then connects the original points to their images under the first transformation. The transformation actually changes the location of the root. For the reduction we give this is not a problem because in the reduction we will have an additional root to which both the root of the original tree and the root of the transformed tree need to connect. Let $\eta_1 = (-\cos \beta, -\sin \beta)$ and $\eta_2 = (-\sin \beta, -\cos \beta)$. Let $p_1, \ldots, p_n$ be points with $p_i = u_i \eta_1 + v_i \eta_2$, $(u_i,v_i) \in [0,B]^2$, where $B$ may depend on $n$. Let $T$ be a $(\geq \beta)$-SA on $p_1, \ldots, p_n$ with root $B \eta_1 + B \eta_2$. Let $\lambda = \cos ( \beta + \beta')/ \cos (2 \beta')$ and $\eta'_1 = \lambda (-\cos \beta', -\sin \beta')$ and $\eta'_2 = \lambda (-\sin \beta', -\cos \beta')$. We transform $T$ by the following transformation $\tau \colon \mathbb{R}^2 \rightarrow \mathbb{R}^2$: Any point $p = u \eta_1 + v \eta_2$ is mapped to $q = u \eta'_1 + v \eta'_2$. We obtain a Steiner arborescence $T'$ on $p_1, \ldots, p_n$ with root $B \eta'_1 + B \eta'_2$ by connecting $p_i$ to $\tau (p_i)$ by a line segment. Let $\lambda' = \sin (\beta' - \beta)/ \cos (\beta + \beta')$. \[lem:transtree\] $T'$ is a $(\geq \beta')$-SA and $ \|T'\| \leq \lambda \|T\| + 2 \lambda' B n. $ [Proof.]{} If we ignore the connections between the $p_i$s and $\tau(p_i)$s the resulting transformed tree by construction fulfils the angle restriction and its length is $\lambda \|T\|$. We therefore only need to show that the connections fulfil the angle restriction and that the length of any connection is bounded by $\lambda'$. We have $p_i - \tau(p_i) = u_i (\eta_1-\eta'_1) + v_i (\eta_2-\eta'_2)$. It therefore suffices to prove that $\eta_1-\eta'_1$ and $\eta_2-\eta'_2$ fulfil the angle restriction. Since $\eta_2-\eta'_2$ is $\eta_1-\eta'_1$ mirrored at the diagonal with slope 1, it actually suffices to consider $\eta_1-\eta'_1$. [r]{}[.25]{} ![image](lemmaTransformation) Consider the triangle formed by the origin, $\eta_1$ and $\eta'_1$ (see Figure \[fig:LemmaTransformation\]). We have $\|\eta_1\| = 1$ and $\|\eta'_1\|=\lambda$. By the law of sines $\lambda = \sin \gamma / \sin \gamma'$. This equation holds for $\gamma = \pi/2 + \beta + \beta'$, since then $\gamma' = \pi - (\beta'-\beta) - \gamma = \pi/2 - 2 \beta'$ and therefore $\sin \gamma / \sin \gamma' = \sin (\pi/2 + \beta + \beta') / \sin (\pi/2 - 2 \beta') = \cos (\beta + \beta')/\cos (2 \beta') = \lambda$. On the other hand with $\gamma = \pi/2 + \beta + \beta'$ we have $\eta'_1$ is indeed reachable from $\eta_1$ in a $(\geq \beta')$-SA. The length of the connection is here the length of the third side of the triangle, which is $\sin (\beta' - \beta)/\sin \gamma = \lambda'$. More generally the length of a connection is bounded by $2B$, that is $B$ for each coordinate. Since we have $n$ such connections the bound of the lemma holds. In Lemma \[lem:transtree\] we have two summands, one depending on $\|T\|$ and one on $n$. Since the terminals lie on an integer grid and since every terminal has to connect to the tree, the length of the tree is at least of order $n$. \[obs:bound-on-n\] If the terminals of a Steiner arborescence T have integer coordinates then $n \leq 2\|T\|$. \[thm:nphard-flux\] It is NP-hard to compute the optimal flux tree of a point set. [Proof.]{} Given an ESA instance with root $(0,0)$ and with the coordinates $x$ and $y$ of any point $(x,y)$ on the tree bounded by $c n^2$, we translate every terminal by $2 n^k (1,1)$ for a constant integer $k>2$ specified later. We include the translated root in the point set but not as root. Instead we take $(0,0)$ again as root The shortest ESA is simply the originally shortest translated with one additional edge from $(2 n^k,2 n^k)$ to $(0,0)$. Now, consider a point $(c' n^k + x, c' n^k + y)$ with $c'\geq 1$ and $0\leq x,y \leq c n^2$. The angle of a line through this point and the origin with the diagonal of slope 1 is bounded by $\beta_{\max} = c n^2/n^k = c/n^{k-2}$. Now, restricted to such points every $(\geq \beta_{\max})$-SA is a flux tree with $\alpha = \pi/4$, and every such flux tree a $(\geq -\beta_{\max})$-SA. Also, every $(\geq \beta_{\max})$-SA is a Euclidean Steiner arborescence, and every Euclidean Steiner arborescence a $(\geq -\beta_{\max})$-SA. Thus, if we show that $(\geq \beta_{\max})$-SAs approximate $(\geq -\beta_{\max})$-SAs well, this directly implies that flux trees approximate Euclidean Steiner arborescences well. More specifically, if we want to show that the length of the shortest flux tree approximates the shortest Euclidean Steiner arborescence up to a precision of $1$ (so that we can make the distinction between $K$ and $K+1$) then it is sufficient to prove that a $(\geq \beta_{\max})$-SA $T'$ can approximate a $(\geq -\beta_{\max})$-SA $T$ up to this precision. By Lemma \[lem:transtree\] and Observation \[obs:bound-on-n\] we get $\|T'\| \leq \lambda \|T\| + 2 \lambda' c n^2 n \leq \|T\| (\lambda + 4 c \lambda' n^2)$. Now, $\lambda = 1/ \cos (2 \beta_{\max}) < 1 / (1 - c/n^{k-2})$ and $\lambda' = \sin (2\beta_{\max}) < 2\beta_{\max} = c/n^{k-2}$. Thus, $$\lambda + 4 c \lambda' n^2 < 1 / (1 - c/n^{k-2}) + 8c^2 /n^{k-4} = 1 + o(1/n^4)$$ for $k>8$. Since $\|T\| = O(n^4)$, this allows us to approximate the length up a o(1)-term. Note that we still need to connect the root of $\|T\|$ and the root of $\|T'\|$ to $(0,0)$. The length of this connection is slightly different because the roots of the trees are different, but the difference is negligible compared to the difference of $\|T'\|$ and $\|T\|$. It remains to prove NP-hardness for spiral trees. \[thm:nphard-spiral\] It is NP-hard to compute the optimal spiral tree of a point set. [Proof.]{} We use the same construction as above for spiral trees but we start with a rectilinear Steiner tree instance instead of a Euclidean Steiner tree instance. To adapt the reduction it suffices to show that within the relevant part of the tree, that is the part in $[2n^k, 2n^k + c n^2]$, the length of a spiral segment between two points $p$, $q$ is up to a small error the same as $c''\\|p-q\\|_1$ for a suitable constant factor $c''$. The length of the spiral is $D(p,q) = \sec (\pi/4) \|\|p\|-\|q\|\| = \sqrt{2} \|\|p\|-\|q\|\|$. Assume $x_q \leq x_p$ and $y_q \leq y_p$. Let $q' = (x_p, y_q)$. We have $\| \|p\|-\|q\| \| = (\|p\|-\|q'\|)+(\|q'\|-\|q\|)$. The difference $\|p\|-\|q'\|$ measures how much the distance to the origin decreases while moving from $p$ to $q'$. Let $y_0 = y_p-y_q$ be the length of the line segment between $p$ and $q'$ and let $\sigma \colon [0,y_0] \rightarrow \mathbb{R}^2$ be this line segment parameterized uniformly. For a point $u$ let $\gamma (u)$ be the angle formed by the line through the origin and $u$ with the $x$-axis. We have $\|p\|-\|q'\|=\int_0^{y_0} \cos \gamma (\sigma (u)) \mathrm{d}u$. Now $\pi/4 - \beta_{\max} \leq \gamma (\sigma (u)) \leq \pi/4 + \beta_{\max}$ and therefore $1/\sqrt{2} - \beta_{\max} \leq \cos \gamma (\sigma (u)) \leq 1/\sqrt{2} + \beta_{\max}$. Thus $\|\|p\|-\|q'\| - y_0/\sqrt{2}\| \leq c/n^{k-2}$. By the same argument we have that $\|\|q'\|-\|q\| - x_0/\sqrt{2}\| \leq c/n^{k-2}$, where $x_0 = x_p-x_q$. Therefore, $$\begin{aligned} \|D(p,q) - \\|p-q\\|_1\| &= \|\sqrt{2} ((\|p\|-\|q'\|)+(\|q'\|-\|q\|))- \\|p-q\\|_1\| \\ &\leq \|\sqrt{2} (x_0/\sqrt{2} + y_0/\sqrt{2} + 2c/n^{k-2}) - \\|p-q\\|_1\|\\ &= 2\sqrt{2}c/n^{k-2}.\end{aligned}$$ Since the length of the RSA instance is in $O(n^4)$, the difference between measuring the length of a spiral segment versus taking the $L_1$-distance of endpoints of segments is in $o(1)$ for $k>4$. Thus, computing the optimal spiral tree is NP-hard. To prove NP-hardness it was sufficient to consider one value of $\alpha$, namely $\alpha = \pi/4$. Nonetheless, it is an interesting problem whether with results also holds for a given smaller $\alpha$. We believe that the NP-hardness proof in [@ss-rsap-05] can be adapted to $\beta$-SAs and $(\geq \beta)$-SAs for $0<\beta \leq \pi/4$. With this we could also generalize our result to smaller $\alpha$. Optimal spiral trees with empty spiral regions {#sec:emptyregions} ---------------------------------------------- Assume we are given an input instance such that $t_i \notin \mathcal{R}_{t_j}$ for all $i \neq j$. We give an exact polynomial time algorithm that computes optimal spiral trees for input instances with this property. Before we discuss the algorithm, we first give a structural result on optimal spiral trees for these special instances. Assume all terminals are ordered radially (on angle) in counterclockwise direction around $r$ and are numbered as such. This means that the first terminal $t_1$ is arbitrary and the remaining terminals $t_2, \ldots, t_n$ follow this order. First note that, for these instances, every terminal is a leaf in any spiral tree. That is because no terminal can be reached by another terminal, so no terminal can have incoming edges. More important is the following result. \[lem:DPleaforder\] If the spiral regions of all terminals are empty, then the leaf order of any planar spiral tree follows the radial order of the terminals. Assume this is not case, so that the leaf order skips leafs $t_i, \ldots, t_j$, or in other words jumps from $t_{i-1}$ to $t_{j+1}$. Pick any terminal $t_k$ with $i \leq k \leq j$. Let $\pi$ be the path in the spiral tree from $t_{i-1}$ to $t_{j+1}$. Because the leaf order jumps from $t_{i-1}$ to $t_{j+1}$, no leaf is connected to the outside (as seen from $r$) of $\pi$. However, because $t_k \notin \mathcal{R}_{t_{i-1}}$, $t_k \notin \mathcal{R}_{t_{j+1}}$ and $t_{i-1}, t_{j+1} \notin \mathcal{R}_{t_{k}}$, the path $\pi$ cuts through $\mathcal{R}_{t_{k}}$ separating $t_k$ from $r$. So the only way for $t_k$ to be connected to $r$ is to cross $\pi$. Contradiction. If the spiral regions of all terminals are empty, then the leaf order of the optimal spiral tree follows the radial order of the terminals. ![Left: The wedge $w_{ij}$ for terminals $t_i, \ldots, t_j$. Right: $p_{ij}$ is the optimal join point.[]{data-label="fig:DPWedge"}](DPWedge) Using the above lemma we can use a simple dynamic programming algorithm to compute the optimal spiral tree. We simply solve all subproblems that ask for the optimal spiral subtree for a sequence of terminals $t_i, \ldots, t_j$. We require that this subtree is contained in the unbounded wedge $w_{ij}$ from the radial line through $t_i$ to the radial line through $t_j$ (see Figure \[fig:DPWedge\] left). Define $p_{ij}$ as the intersection of $\mathcal{S}^{+}_{t_i}$ and $\mathcal{S}^{-}_{t_j}$ ($p_{ii} = t_i$). As every internal node has exactly two incoming edges (Observation \[lem:spiralplandeg\]), we split the subtree into two subtrees at every internal node. To compute the optimal spiral tree for a sequence of terminals $t_i, \ldots, t_j$, we simply compute the optimal way to split the subtree into two subtrees by trying all possibilities. We then connect both subtrees to $p_{ij}$. Note that, by Lemma \[lem:DPleaforder\], we need to check only $j - i$ ways to split this subtree. If $F(i, j)$ is the length of the optimal spiral subtree for the terminals $t_i, \ldots, t_j$ (contained in $w_{ij}$), then we can perform dynamic programming using the following recursive relation. $$F(i, j) = \begin{cases} 0, & \mbox{if } i = j, \\ \min_k (F(i, k) + F(k+1, j) + D(p_{ik}, p_{ij}) + D(p_{(k+1)j}, p_{ij})), & \mbox{otherwise.} \end{cases}$$ Note that we allow $j < i$, because we have a cyclical order. However, the value of $k$ in the above equation must be between $i$ and $j$ in the cyclical order. The distance function $D$ is defined as in Equation \[eqn:spiraldist\]. The function $F(i, j)$ describes the length of the optimal spiral subtree for the terminals $t_i, \ldots, t_j$ contained in $w_{ij}$. We prove the lemma by induction. If $i = j$, then $F(i, j) = 0$ is clearly correct. If $i \neq j$, then, by Lemma \[lem:DPleaforder\], we compute the minimum of all possible splits for the corresponding subtree. Let this split be between $t_k$ and $t_{k+1}$. By induction, $F(i, k)$ and $F(k+1, j)$ describe the lengths of the optimal subtrees. We need to show that $p_{ij}$ is the optimal point to join the subtrees. For the sake of contradiction, assume the optimal join point is $p'_{ij}$. This point must be in the intersection of $\mathcal{R}_{t_i}$, $\mathcal{R}_{t_j}$ and $w_{ij}$ (see Figure \[fig:DPWedge\] right). This means that $p'_{ij} \in \mathcal{R}_{p_{ij}}$. We can replace the edges $p_{ik} \rightarrow p'_{ij}$ and $p_{(k+1)j} \rightarrow p'_{ij}$ by the edges $p_{ik} \rightarrow p_{ij}$, $p_{(k+1)j} \rightarrow p_{ij}$, and $p_{ij} \rightarrow p'_{ij}$. Since $p'_{ij}$ must be closer to $r$ than $p_{ij}$, it follows from the definition of $D$ in Equation \[eqn:spiraldist\] that this operation shortens the tree. Contradiction. The length of the optimal spiral tree is not necessarily given by $F(1, n)$, but it can also be any of the lengths $F(i, i-1)$ for $2 \leq i \leq n$, so we need to compute the minimum of all these values. Note that there must be at least one wedge $w_{i(i-1)}$ that contains the entire optimal spiral tree, so this will give the length of the optimal spiral tree. Using additional information we can also compute the optimal spiral tree itself in this way. From the definition of $F(i, j)$, it is clear that the algorithm runs in $O(n^3)$ time. Approximation algorithm {#sec:approximation} ----------------------- As shown in Section \[sec:npproof\], computing the optimal spiral tree is NP-hard in general. In this section we describe a simple algorithm that computes a $2$-approximation of the optimal spiral tree. Note that, using Theorem \[thm:fluxapprox\], this algorithm also directly computes a $(2 \sec(\alpha))$-approximation of the optimal flux tree. For rectilinear Steiner arborescences, Rao [[et al.]{}]{} [@Rao92] describe a simple $2$-approximation algorithm. The transformation mentioned in Section \[sec:transformation\] does not preserve length, so we cannot use this algorithm for spiral trees. However, below we show how to use the same global approach—sweep over the terminals from the outside in—to compute a $2$-approximation for optimal spiral trees in $O(n \log n)$ time. The basic idea is to iteratively join two nodes, possibly using a Steiner node, until all terminals are connected in a single tree $T$, the greedy spiral tree. Initially, $T$ is a forest. We say that a node (or terminal) is active if it does not have a parent in $T$. In every step, we join the two active nodes for which the join point is farthest from $r$. The join point $p_{uv}$ of two nodes $u$ and $v$ is the farthest point $p$ from $r$ such that $p \in \mathcal{R}_u \cap \mathcal{R}_v$. This point is unique if $u$, $v$ and $r$ are not collinear. \[lem:greedyplanar\] The greedy spiral tree is planar. [r]{}[.2]{} ![image](GreedyPlanar) [Proof.]{} Assume there is a crossing in the greedy spiral tree between two spiral segments, one between $u_1$ and its parent $v_1$, and another between $u_2$ and its parent $v_2$. Note that the intersection must be farther from $r$ than both $v_1$ and $v_2$. But that means that the intersection must have been encountered while both $u_1$ and $u_2$ were in $\mathcal{W}$, so this intersection should be a node in the greedy spiral tree. Contradiction. [r]{}[.45]{} ![image](Wavefront){width=".4\textwidth"} The algorithm sweeps a circle $\mathcal{C}$, centered at $r$, inwards over all terminals. All active nodes that lie outside of $\mathcal{C}$ form the wavefront $\mathcal{W}$ (the black nodes in Figure \[fig:Wavefront\]). $\mathcal{W}$ is implemented as a balanced binary search tree, where nodes are sorted according to the radial order around $r$. We join two active nodes $u$ and $v$ as soon as $\mathcal{C}$ passes over $p_{uv}$. For any two nodes $u, v \in \mathcal{W}$ it holds that $u \notin \mathcal{R}_v$. By Lemma \[lem:greedyplanar\] the greedy spiral tree is planar, so we can apply Lemma \[lem:DPleaforder\] to the nodes in $\mathcal{W}$. Hence, when $\mathcal{C}$ passes over $p_{uv}$ and both nodes $u$ and $v$ are still active, then $u$ and $v$ must be neighbors in $\mathcal{W}$. We process the following events. Terminal. : When $\mathcal{C}$ reaches a terminal $t$, we add $t$ to $\mathcal{W}$. We need to check whether there exists a neighbor $v$ of $t$ in $\mathcal{W}$ such that $t \in \mathcal{R}_v$. If such a node $v$ exists, then we remove $v$ from $\mathcal{W}$ and connect $v$ to $t$. Finally we compute new join point events for $t$ and its neighbors in $\mathcal{W}$. Join point. : When $\mathcal{C}$ reaches a join point $p_{uv}$ (and $u$ and $v$ are still active), we connect $u$ and $v$ to $p_{uv}$. Next, we remove $u$ and $v$ from $\mathcal{W}$ and we add $p_{uv}$ to $\mathcal{W}$ as a Steiner node. Finally we compute new join point events for $p_{uv}$ and its neighbors in $\mathcal{W}$. We store the events in a priority queue $\mathcal{Q}$, ordered by decreasing distance to $r$. Initially $\mathcal{Q}$ contains all terminal events. Every join point event adds a node to $T$ and every node generates at most two join point events, so the total number of events is $O(n)$. We can handle a single event in $O(\log n)$ time, so the total running time is $O(n \log n)$. Next we prove that the greedy spiral tree is an approximation of the optimal spiral tree. \[lem:greedycircleisects\] Let $\mathcal{C}$ be any circle centered at $r$ and let $T$ and $T'$ be the optimal spiral tree and the greedy spiral tree, respectively. Then $\|\mathcal{C} \cap T'\| \leq 2 \|\mathcal{C} \cap T\|$ holds where $\|\mathcal{C} \cap T'\|$ is the number of intersection points between $\mathcal{C}$ and $T'$. It is easy to see that $\|\mathcal{C} \cap T'\| = \|\mathcal{W}\|$ when the sweeping circle is $\mathcal{C}$. Let the nodes of $\mathcal{W}$ be $u_1, \ldots, u_k$, in radial order. Any node $u_i$ is either a terminal or it is the intersection of two spirals originating from two terminals, which we call $u^L_i$ and $u^R_i$ (see Figure \[fig:GreedyProof\]). We can assume the latter is always the case, as we can set $u^L_i = u_i = u^R_i$ if $u_i$ is a terminal. Next, let the intersections of $T$ with $\mathcal{C}$ be $v_1, \ldots, v_h$, in the same radial order as $u_1, \ldots, u_k$. As $T$ has the same terminals as $T'$, every terminal $u^L_i$ and $u^R_i$ must be able to reach a point $v_j$. Let $I^L_i$ and $I^R_i$ be the reachable parts (intervals) of $\mathcal{C}$ for $u^L_i$ and $u^R_i$, respectively (that is $I^L_i = \mathcal{C} \cap \mathcal{R}_{u^L_i}$ and $I^R_i = \mathcal{C} \cap \mathcal{R}_{u^R_i}$). Since any two neighboring nodes $u_i$ and $u_{i+1}$ have not been joined by the greedy algorithm, we know that $I^L_i \cap I^R_{i+1} = \emptyset$. Now consider the collection $\mathcal{S}_j$ of intervals that contain $v_j$. We always treat $I^L_i$ and $I^R_i$ as different intervals, even if they coincide. The union of all $\mathcal{S}_j$ has cardinality $2 k$. If $\|\mathcal{S}_j\| \geq 5$, then its intervals cannot be consecutive (i.e. $I^L_i, I^R_i, I^L_{i+1}, I^R_{i+1}$, etc.), as this would mean it contains both $I^L_i$ and $I^R_{i+1}$ for some $i$. So say the intervals of $\mathcal{S}_j$ are not consecutive and $\mathcal{S}_j$ contains $I^L_{i}$ and $I^L_{i+1}$, but not $I^R_i$ (other cases are similar). $T'$ is planar, so this is possible only if $I^R_i \subset I^L_{i+1}$ (see Figure \[fig:GreedyProof2\]). But then $I^R_i$ and $I^L_{i+1}$ are both in a collection $\mathcal{S}_{j'}$ and we can remove $I^L_{i+1}$ from $S_j$, while keeping the union of all collections the same. We repeat this process to construct reduced collections $\hat{\mathcal{S}}_j$ such that the union of all collections remains the same and all intervals in a collection $\hat{\mathcal{S}}_j$ are consecutive. As a result, $\|\hat{\mathcal{S}}_j\| \leq 4$, and hence $4h \geq 2k$ or $k \leq 2h$. ![$I^R_i \subset I^L_{i+1}$.[]{data-label="fig:GreedyProof2"}](GreedyProofClip) ![$I^R_i \subset I^L_{i+1}$.[]{data-label="fig:GreedyProof2"}](GreedyProof2Clip) The greedy spiral tree is a $2$-approximation of the optimal spiral tree and can be computed in $O(n \log n)$ time. The time bound is already mentioned above. For the approximation, recall that $L(T) = \sec(\alpha) \int_0^\infty \|T \cap \mathcal{C}_R\| dR$, where $T$ is any spiral tree and $\mathcal{C}_R$ is the circle of radius $R$ centered at $r$. Using Lemma \[lem:greedycircleisects\], we can directly conclude that the greedy spiral tree is a $2$-approximation of the optimal spiral tree. The approximation factor is most likely not tight. Experiments for rectilinear Steiner arborescences show that the greedy algorithm often computes near-optimal arborescences [@Cordova94]. Approximating spiral trees in the presence of obstacles {#sec:obstacles} ======================================================= In this section we extend the approximation algorithm of Section \[sec:approximation\] to include obstacles. Given the similarities between spiral trees and rectilinear Steiner arborescences described in Section \[sec:transformation\], it makes sense to consider existing algorithms for rectilinear Steiner arborescences in the presence of obstacles. Unfortunately, the only known algorithm for this seems to have some issues. We discuss these issues in the next section. Then we give a new algorithm for computing rectilinear Steiner arborescences in the presence of obstacles. For a certain type of obstacles, this algorithm also computes a $2$-approximation of the optimal rectilinear Steiner arborescence, although this does not hold for general obstacles. Finally we extend this algorithm to compute spiral trees in the presence of obstacles, again computing a $2$-approximation for a certain type of obstacles. Ramnath’s algorithm {#sec:ramnath} ------------------- [r]{}[.15]{} ![image](MinCoverFail) Ramnath [@Ramnath03] gives a $2$-approximation algorithm for rectilinear Steiner arborescences with rectangular obstacles. He claims that the result extends to arbitrary rectilinear obstacles. But this is not the case. Consider the configuration of points and obstacles as seen on the right. The obstacles are $L$-shaped with the longer (vertical) side of the $L$ being much longer than the shorter (horizontal) one. Between each consecutive pair of obstacles their is a terminal. What Ramnath’s algorithm does is to sweep a line of slope $-1$ starting at the root. During the sweep the arborescence is constructed greedily maintaining a minimal set of points (called cover points) on the sweep line such that all remaining points can still be connected. Thus, in the beginning the algorithm has to decide whether to grow the arborescence to the right or upwards. From these two options the algorithm picks an arbitrary one, in particular it might grow to the right. But then on the arborescence will connect to each terminal by a connection corresponding to the longer side of the $L$-shape. By making the $L$-shape sufficiently long, the approximation factor for this configuration can be made worse than any constant, in particular two. Ramnath’s paper also lacks the details to establish the claimed running time for rectangular obstacles. In particular the subdivision of a critical region (that is, a region that can be exclusively reached by one of the cover points) seems to assume that there is no obstacle strictly inside the critical region. However this case might occur and it does not seem straightforward to extend the algorithm to handle this case. Furthermore, the algorithm needs to compute the point at which the critical regions of neighboring cover points meet. This point is found by tracing paths from both of the cover points. The cost of this tracing step does not seem to be handled in the analysis and it is not clear how to account for it. Rectilinear Steiner arborescences {#sec:rectisweep} --------------------------------- We are now given a root $r$ at the origin, terminals $t_1, \ldots, t_n$ in the upper-right quadrant, and also $m$ polygonal obstacles $B_1, \ldots, B_m$ with total complexity $M$. We place a bounding square around all terminals and the root and consider the “free space” between the obstacles as a polygonal domain $P$ with $m$ holes and $M+4$ vertices. We describe a greedy algorithm that computes a rectilinear Steiner arborescence $T$, the greedy arborescence, inside $P$. Our algorithm returns only a topological representation of $T$. This can easily be extended to the explicit arborescence, which, however, can have arbitrarily high complexity. As before we incrementally join nodes until we have a complete arborescence. This time we sweep a diagonal line $L$ over $P$ towards $r$ and maintain a wavefront $\mathcal{W}$ with all active nodes that $L$ has passed. If $L$ reaches a join point $p_{uv}$ of nodes $u, v \in \mathcal{W}$, we connect $u$ and $v$ to $p_{uv}$ and add the new Steiner node to $\mathcal{W}$. Our greedy arborescence is restricted to grow inside the polygonal domain $P$. If a point $p \in P$ cannot reach $r$ with a monotone path in $P$, then $p$ is not a suitable join point. To simplify matters we compute a new polygonal domain $P'$ from $P$, such that for every $p \in P'$, there is a monotone path from $p$ to $r$ in $P$. For now we simply assume that we are given $P'$ and that it has $O(M)$ vertices. [r]{}[.2]{} ![image](AccessibleS) To compute join points we keep track of the reachable region of every node $u \in \mathcal{W}$, that is, we keep track of the part of $L$ that can be reached from $u$ via a monotone path in $P'$. As soon as two nodes $u, v \in \mathcal{W}$ can reach the same point $p$ on $L$, then $p$ is the join point $p_{uv}$ and we can connect $u$ and $v$ to $p_{uv}$. To compute the path between $u$ and $v$ and $p_{uv}$, we need some additional information. Here our definitions follow Mitchell [@Mitchell92]. Given two points $p, q \in P'$ (with $x_q \leq x_p$ and $y_q \leq y_p$), let $R(p, q)$ be the rectangle with $p$ and $q$ as corners. We say that $q$ is immediately accessible from $p$ if $p$ and $q$ are in the same connected component of $R(p, q) \cap P'$ and this connected component does not contain any other vertices or nodes. The parent of a point $p \in P'$ is the rightmost vertex or node from which $p$ is immediately accessible. The topological representation of the greedy arborescence stores only the parent information. The status of the sweep line $L$ consists of three types of intervals: (i) free intervals: points that cannot be reached by any node in $\mathcal{W}$, (ii) obstacle intervals: points not in $P'$, and (iii) reachable intervals: points reachable by a node in $\mathcal{W}$. The latter type of interval is tagged with the unique node in $\mathcal{W}$ that can reach this interval. We split the reachable intervals such that every interval has a unique parent. The intervals are stored by their endpoints in a balanced binary search tree. Initially, the status of $L$ consists of one obstacle interval. We distinguish three types of events, which are processed in order using a priority queue. [Terminal event.]{} When we encounter a terminal $t_i$, there are two cases. Either the terminal is in a free interval or in a reachable interval tagged by a node $u$. In the latter case, we connect $u$ to $t_i$ (using the parent information) and replace $u$ by $t_i$ in $\mathcal{W}$. Also, we replace all intervals tagged with $u$ by free intervals and merge them where possible. In both cases, we start a new interval for $t_i$. For the endpoints of this interval, we trace the intersections between $L$ and the horizontal and vertical line through $t_i$. Note that we also split an interval, so we add three intervals in total and remove one. For every new interval (or merged interval), we add vanishing events to the event queue. [r]{}[.25]{} ![image](VertexEventS) [Vertex event.]{} When we encounter a vertex $v$, then $v$ can be in any type of interval. If $v$ is in a free interval, then we add an obstacle interval, where the endpoints of the interval trace the edges of $P'$ connected to $v$. If $v$ is in an obstacle interval, then we add a free interval, where the endpoints of the interval trace the edges of $P'$ connected to $v$. Otherwise, $v$ is in a reachable interval or at the endpoint between a reachable interval and an obstacle interval. In the first case, we need to insert an obstacle interval at $v$, as described above. In both cases we need to set the parent of $v$ and insert a new reachable interval for $v$ (with the correct tag). Also, we need to follow the edge or edges of $P'$ connected to $v$. This can create free intervals. If one of the endpoints of the reachable interval of $v$ directly moves out of $P'$, we do not need to add this endpoint, but we can use the endpoint of the obstacle interval instead. Note that we add only a constant number of intervals. For the new intervals, we add vanishing events to the event queue. [Vanishing Interval.]{} If an interval $I$ vanishes, then there are different cases depending on the types of the neighboring intervals $I_1$ and $I_2$. Note that $I$ vanishes at a point $p$ where two endpoints meet. If $I_1$ and $I_2$ are reachable intervals with different tags $u_1$ and $u_2$, then $p$ is the join point for $u_1$ and $u_2$. We join $u_1$ and $u_2$ at $p$, as described in the terminal event. Otherwise, we need to remove one of the two endpoints. An endpoint of an interval always follows an edge of $P'$ or a vertical or horizontal line through a node in $\mathcal{W}$ or a vertex of $P'$. If $I_1$ and $I_2$ are obstacle intervals or free intervals, then we can just remove both endpoints of $I$. If $I_1$ and $I_2$ are reachable intervals with the same tag, then we keep the endpoint that follows a horizontal line (this follows the definition of a parent given above). If $I_1$ and $I_2$ are of different types, then we keep the endpoint of the obstacle interval if one is present and otherwise we keep the endpoint of the reachable interval. Again, we add vanishing event points to the event queue for every interval for which an endpoint has changed. The algorithm terminates when $L$ reaches $r$, at which point we have one node left in $\mathcal{W}$. Using the parent information in the status, we connect the final node with $r$. To compute $P'$ from $P$ we simply run the sweep line algorithm in the opposite direction, tracing the “reachable region” of $r$. The points that border a reachable interval and either a free or obstacle interval trace out $P'$. \[lem:arbruntime\] The greedy arborescence can be computed in $O((n+M) \log(n+M))$ time. First we give a bound for the number of events. Clearly, the number of terminal and vertex events are bounded by $O(n+M)$. This also means that the total number of intervals is bounded by $O(n+M)$, as we add a constant number of intervals at only these events. At every vanishing interval event we remove an interval, so the total number of events is $O(n+M)$. Also note that every event can generate only a constant number of events. This also means that $P'$ has complexity $O(M)$, as we add vertices to $P'$ only at events. It is easy to see that all events can be executed in $O(\log(n+M))$ time, except when we need to change all intervals tagged by a certain node $u$ to free intervals. We can do this in $O(n_u)$ time (by simple bookkeeping), where $n_u$ is the number of intervals tagged by $u$. An interval can only once be changed to a free interval. Merging two neighboring free intervals removes one interval, so we can charge these operations to the total number of intervals. Furthermore, the topological representation of the greedy arborescence contains only the relevant vertices and nodes to compute the paths between nodes. Every vertex or node can occur only once in this representation. So the algorithm runs in $O((n+M) \log(n+M))$ time. If $P$ has only positive monotone holes, then the greedy arborescence is a $2$-approximation of the optimal rectilinear Steiner arborescence. A hole is positive monotone if its boundary contains two points $p$ and $q$ such that both paths on the boundary from $p$ to $q$ are monotone in both the $x$-direction and the $y$-direction. In the next section we prove this result for spiral trees. The same arguments can directly be applied to prove the same result for rectilinear Steiner arborescences. \[lem:MSA\] The greedy arborescence can be computed in $O((n+M) \log(n+M))$ time. If $P$ has only positive monotone holes, then the greedy arborescence is a $2$-approximation of the optimal rectilinear Steiner arborescence. Spiral trees {#sec:spirobstacles} ------------ We now describe how to adapt our algorithm to spiral trees; we concentrate mainly on the necessary changes. We again compute only a topological representation of the output and refer to the spiral tree which we compute as the greedy spiral tree. The sweep line is replaced by a sweeping circle $\mathcal{C}$. A simple balanced binary search tree is still sufficient to store the intervals, using special cases to deal with the circular topology. [r]{}[.27]{} ![image](SpiralRectS) We need to replace horizontal and vertical lines by right and left spirals. For a given node or vertex $u$, the endpoints of its interval on $\mathcal{C}$ follow the intersections of $\mathcal{S}^{+}_u$ and $\mathcal{S}^{-}_u$ with $\mathcal{C}$. Given two points $p$ and $q$ ($q \in \mathcal{R}_p$), let $SR(p,q)$ be the spiral rectangle between $p$ and $q$. The spiral rectangle between $p$ and $q$ is bounded by the two paths (these are unique) consisting of exactly two spiral segments connecting $p$ to $q$ (this is exactly a rectangle transformed by the transformation in Section \[sec:transformation\]). The point $q$ is immediately accessible from $p$ if $p$ and $q$ are in the same connected component of $SR(p, q) \cap P'$ and this connected component does not contain any other vertices or nodes. [r]{}[.27]{} ![image](SubdividePolyS) There is one subtlety. If the left or right spiral of a vertex $v$ directly moves out of $P$, we can ignore it, as before. However, at the exact moment that this is no longer the case, we do need to trace this spiral. For rectilinear Steiner arborescences, this can only happen at vertices. For spiral trees, this can also happen at most two spiral points in the middle of an edge $e$. A point $p$ on $e$ is a spiral point if the angle between the line from $p$ to $r$ and the line through $e$ is exactly $\alpha$. We hence subdivide every edge of $P$ at the spiral points. In addition we also subdivide $e$ at the closest point to $r$ on $e$ to ensure that every edge of $P$ has a single intersection with $C$. Neither the algorithm presented in Section \[sec:rectisweep\] nor its adaptation to spiral trees gives a constant factor approximation. But, if we restrict the types of obstacles, they give 2-approximations. For rectilinear Steiner arborescences we have to use positive monotone obstacles, for spiral trees spiral monotone obstacles. An obstacle is spiral monotone if its boundary contains two points $p$ and $q$ such that both paths on the boundary from $p$ to $q$ are angle-restricted. \[lem:spiralmonotone\] Let $P$ be a polygonal domain with spiral monotone holes. Then all points on a circle $\mathcal{C}$ reachable from a node $u$ lie inside a single circular interval $I_u \subseteq \mathcal{C}$ with the property that every point in $I_u \cap P$ is reachable from $u$. Consider a point $p \in I_u \cap P$. Let $\pi_1$ and $\pi_2$ be the paths from $u$ to the endpoints of $I_u$. Repeat the following until we hit either $\pi_1$ or $\pi_2$. Move along the left spiral through $p$ going outwards (from $r$). When we hit a hole, simply follow the outline of the hole until we can follow the left spiral again. Because $P$ has only spiral monotone holes, we eventually reach either $\pi_1$ or $\pi_2$. Hence $p$ must be reachable from $u$. \[theo:greedy\] The greedy spiral tree can be computed in $O((n+M) \log(n+M))$ time. If $P$ has only spiral monotone holes, then the greedy spiral tree is a $2$-approximation of the optimal spiral tree. Correctness and running time follow from Lemma \[lem:arbruntime\] and the discussion in Section \[sec:spirobstacles\]. Assume that $P$ has only spiral monotone holes and let $T$ and $T'$ be the optimal and greedy spiral tree, respectively. By Lemma \[lem:spiralmonotone\] we can represent the part of a circle $\mathcal{C}$ that is reachable by a terminal $t$ as a single interval $I_t$. We can now follow the proof of Lemma \[lem:greedycircleisects\] with these intervals to show that $\|\mathcal{C} \cap T'\| \leq 2 \|\mathcal{C} \cap T\|$. This directly implies that, if $P$ has only spiral monotone holes, the greedy spiral tree is a $2$-approximation. [21]{} . <http://www.csiss.org/clearinghouse/FlowMapper>. O. Aichholzer, F. Aurenhammer, C. 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The rectilinear [Steiner]{} arborescence problem is [NP]{}-complete. In [Proc. 11th ACM-SIAM Symposium on Discrete Algorithms]{}, pages 780–787, 2000. W. Shi and C. Su. The rectilinear [Steiner]{} arborescence problem is [NP]{}-complete. , 35(3):729–740, 2005. T. A. Slocum, R. B. McMaster, F. C. Kessler, and H. H. Howard. . Pearson, New Jersey, 3rd edition, 2010. W. Tobler. Experiments in migration mapping by computer. , 14(2):155–163, 1987. J. Wood, J. Dykes, and A. Slingsby. Visualization of Origins, Destinations and Flows with [OD]{} Maps. , 47(2):117–129, 2010. [^1]: A preliminary version of this paper will appear at the 22nd International Symposium on Algorithms and Computation (ISAAC 2011). B. Speckmann and K. Verbeek are supported by the Netherlands Organisation for Scientific Research (NWO) under project no. 639.022.707.
--- abstract: 'We will give an elementary nonstandard proof that the family of generalized blancmange functions are nowhere differentiable. The proof follows from the intuitive characterization of differentiability at a point as almost $\delta$ affine along with the transfer of the functional equations these functions satisfy. We also give elementary nonstandard proofs of the uniform density of these functions among continuous functions. Finally, we discuss work done with the Python programming language in displaying these functions.' author: - Tom McGaffey bibliography: - 'nsabooks.bib' title: 'Nonstandard techniques and nowhere differentiable functions I: A dense family of generalized blancmange functions' --- Introduction: Monsters and nonstandard characterization of differentiability ============================================================================ As far back as Bolzano, continuous nowhere differentiable functions have been objects of fascination for mathematicians. Beginning sometimes during the first third of the 19th century, mathematicians began constructing these functions (often called “monsters” in that earlier period) to understand, refine and contrast the notions of continuity and differentiability; all in a context where the very notion of function was in contention. For a perspective embedding the production of such “pathological” functions in the controversies over generality and rigor in the nineteenth century, see eg., the paper of Chorley, [@QuestionsGeneralityChorley2009]. We became interested in these while reading the interesting study of mathematical conceptualization by Katz and Tall, [@KatzTensionIntuitiveInfinitesFormalMath]. Their infinitesimal microscopic perspective and discussion of the Takagi function, appropriately dubbed blancmange function, piqued the author’s curiosity about possible infinitesimal approaches to proving nowhere differentiability of functions defined in the manner of the blancmange function. We should note that with respect to properties of this specific nowhere differentiable function, there has been a wide range of investigations; the paper of Allaart and Kawamura, [@TakagiSurvey2011], is a good summary of this research. With some thought, the author realized that, using some elementary tools from nonstandard analysis, he could give an almost trivial proof that the blancmange function is nowhere differentiable. In particular, we will use no estimates of difference quotients. Instead, a use of the transfer of the functional equations satisfied by this function along with some elementary nonstandard tools are sufficient to give this short proof. More specifically, we used the transfer of the sequence functional equations (see the first sentence in ) evaluated at an infinite index along with essentially crude order of magnitude algebraic characterizations of differentiability. The idea to analyze the functional equation at an infinite index is inspired by the author’s recent awakening (due to the gentle prodding of Mikhail Katz) to the ingenious use of such “tricks” by Euler. (The recent paper [@TenMisconceptionsPublished] is a good introduction to the important and accumulating historical works of M. Katz and his coauthors on eg., the early history of the calculus, including recent work on Euler in manuscript form.) We believe that the arguments in eg., and were influenced by the exposure to Euler’s remarkable facility with eg., infinite sums as long finite sums and orders of magnitude numerics in place of forbidden zones of ill defined products and quotients. Maybe the best place to see these displayed is his wonderful text [@EulerAnalysisInfinite], where these brilliantly orchestrated strategies occur many times. Note that Euler typically was no more than cryptically brief in his justifications of such gymnastics. For our project, we think that viewing the infinite series defining the blancmange function as a ‘long finite sum’ (and hence being able to apply the functional relation for infinitely long sums), as well as investigating the ‘end terms’ beyond this long sum for simplifying manifestations was influenced by reading Euler. Note that the text of Kanovei and Reeken, [@Nonstandardanalysisaxiomatically], gives an enlightening nonstandard rendition of Euler’s proof of his famous product formula for sine (that appears eg., in the text of Euler already cited.) The “nonstandard analysis” text of Kanovei and Reeken and that of Gordon, Kusraev and Kutateladze, [@InfinitesAnalyGordonKusraevKutateladzeBk2002] contains several gems on the history of the calculus and eg., on Euler. We then realized that we could use almost identical arguments to establish that a wide variety of “generalized blancmange functions” are nowhere differentiable. In fact, we will show that our family, $\SB$, of continuous nowhere differentiable functions is dense in the space of continuous functions on $[0,1]$ with value $0$ at $0$ and $1$, see . Of course, it is an old standard fact, see eg., Thim’s paper, that continuous nowhere differentiable functions are not only dense, but second countable. Our fact is much different (and is apparently new): it asserts the density of $\SB$, the set of functions defined via fractal type self-similarities on a set $\SS$, of continuous piecewise linear functions. In other words, this is the family of such functions concretely defined in terms of a piecewise continuous function $s$ and a positive integer $c$ via a sequence of self similar functional identities. (For the definitions of $\SS$ and $\SB$, see the constructions around and .) In summary, we believe that the import of this paper can be summarized as follows. First, we give a concrete construction of a dense family of continuous, nowhere differentiable functions with large subfamilies having quite novel behaviors. Second, the proofs of nowhere differentiability (and density) are essentially order of magnitude algebraic arguments. Our primary references on the technical history of such functions are the extensive master’s thesis of Thim, [@ContNowhereDifferenFcns] which masterfully covers the technical history of these constructions, as well as the earlier paper of van Embe Boas, [@NowhereDifferenBoasMR0274670] giving some alternative perspectives on these constructions. In perusing the history of such functions in the papers of van Embe Boas and Thim, it appears that some of the nowhere differentiable functions constructed here have not been discussed before. We have on the one hand the wide variety of structural features of our generalized blancmange functions and on the other apparently only a handful of visual descriptions of continuous nowhere differentiable functions in the literature. So with the hope of supplementing this deficit, in the last section we will discuss work we did utilizing the Python programming language. Specifically, we wrote code to display a sequences of magnifications of a tuple of approximations of an arbitrary generalized blancmange function. We will summarize the specifications of the codes as well as display two example (using much simpler code) with the intention of giving some impression of the diversity of these functions. Nonstandard preliminaries ========================= Almost affine internal functions -------------------------------- We will assume the rudiments of (Robinsonian) nonstandard analysis; eg., elementary use of transfer for functions on Euclidean spaces and an isolated use of overflow not directly related to our proof. Good elementary introductions abound, eg., see the classic introduction of Lindstr[ø]{}m, [@Lindstrom1988]. The central idea underlying this section is the following. To require that a functions $f:\bbr^m\ra\bbr^n$ be differentiable at $x\in\bbr^m$ is to require that, for each positive infinitesimal $\d$, its restriction to $\bbr^m_\d(x)$ (the $\d$-module at $x$, see below) visually looks like an affine map, at least up to magnitudes infinitely smaller than $\d$. This is the import of . So to test for differentiability of a map at a point is to check that the map has such an almost affine structure for arbitrary positive infinitesimal $\d$. Below we will develop a few basic tools around this notion of almost affineness in order to exploit our criterion for differentiability in the following sections. We need some basic notation. Let ${\raisebox{.2ex}{}}\bbr$ denote the field of nonstandard real numbers and ${}^\s\bbr$ denote the external subfield isomorphic to the real numbers. Let ${\raisebox{.2ex}{}}\bbr_{nes}$ denote the subring of those that are nearstandard, ie., those $\Fr\in{\raisebox{.2ex}{}}\bbr$ that are infinitesimally close to a real number $a\in{}^\s\bbr$, denoted $\Fv\sim a$. Therefore, these are those nonstandard numbers $\Fv$ with a standard part, denoted $\fst(\Fv)$, in $\bbr$. It’s basic that $\fst:{\raisebox{.2ex}{}}\bbr_{nes}\ra\bbr$ is a surjective ring homomorphism with kernel the ideal (in ${\raisebox{.2ex}{}}\bbr_{nes}$) of infinitesimals, $\mu(0)$, ie., those numbers $\d\sim 0$. If $\Fr$ is a positive infinitesimal, we write $\bbr_\Fr$ for the ${\raisebox{.2ex}{}}\bbr_{nes}$ submodule of ${\raisebox{.2ex}{}}\bbr$ of all numbers $\Fv$ with $\|\Fv\|<a\Fr$ for some $a\in\bbr_+$, ie., $\Fr{\raisebox{.2ex}{}}\bbr_{nes}$. Of course, then $\bbr^m_\Fr$ will be the ${\raisebox{.2ex}{}}\bbr_{nes}$ submodule of ${\raisebox{.2ex}{}}\bbr^m$ given by the $m$-fold Cartesian product of $\bbr_\Fr$. The ${\raisebox{.2ex}{}}\bbr_{nes}$-submodules $\bbr^k_\Fr<{\raisebox{.2ex}{}}\bbr^m$, for integers $k\leq m$ will be called ${\boldsymbol{\Fr}}$-subspaces of ${\boldsymbol{{\raisebox{.2ex}{}}\bbr^m}}$. If ${}^\s\bbr$ is the external subfield of standard numbers in ${\raisebox{.2ex}{}}\bbr$, then ${}^\s\bbr_\Fr$ will denote the external subring of $\bbr_\Fr$ given by $\Fr\cdot{}^\s\bbr$. We similarly define the $\bbr$-submodule ${}^\s\bbr^m_\Fr$ of $\bbr^m_\Fr$. We will call these the ${\boldsymbol{\Fr}}$-standard vectors (in ${\boldsymbol{\bbr^m_\Fr}}$). If $\Fr,\Fs\in{\raisebox{.2ex}{}}\bbr_+$, we will let $\Fs=\Fo(\Fr)$ denote the statement $\Fs/\Fr\sim 0$ and let $\bbr_{o(\Fr)}$ denotes those $\Fs$ with $\Fs=\Fo(\Fr)$ (we include $0$ here by convention). Given this, we clearly have the decomposition $\bbr_\Fr={}^\s\bbr_\Fr+\bbr_{o(\Fr)}$ with ${}^\s\bbr_\Fr\cap\bbr_{o(\Fr)}=\{0\}$. In particular, there is a surjective ring homomorphism ${\boldsymbol{\fst_\Fr}}:\bbr_\Fr\ra{}^\s\bbr_\Fr$, the ${\boldsymbol{\Fr}}$-standard part map* satisfying $\fst_\Fr$ is the identity on ${}^\s\bbr_\Fr$. Note that the kernel of the map is clearly $\bbr_{o(\Fr)}$. Clearly, also we have the ${\raisebox{.2ex}{}}\bbr_{nes}$-module version of the above, ie., a split exact sequence of ${\raisebox{.2ex}{}}\bbr_{nes}$-modules. $$\begin{aligned} \xymatrix { {}^\s\bbr^m_{\Fo(\Fr)} \ar@{^{(}->}[r] &\bbr^m_\Fr \ar@{->>}[r]^{\fst_\Fr} &{}^\s\bbr^m_\Fr } \end{aligned}$$ If $\Fu\in{\raisebox{.2ex}{}}\bbr^m$, let ${\raisebox{.2ex}{}}\bbr^m_\Fr(\Fu)=\{\Fv+\Fu:\Fv\in{\raisebox{.2ex}{}}\bbr^m_\Fr\}$. Note that ${\raisebox{.2ex}{}}\bbr^m_\Fr(\Fu)$ has the property that if $\a,\b\in{\raisebox{.2ex}{}}\bbr_{nes}$ with $\a+\b=1$ and $\Fv,\Fw\in{\raisebox{.2ex}{}}\bbr^m_\Fr(\Fu)$, then $\a\Fv+\b\Fw\in{\raisebox{.2ex}{}}\bbr^m_\Fr(\Fu)$. Hence ${\raisebox{.2ex}{}}\bbr^m_\Fr(\Fu)$ will be called an ${\boldsymbol{\Fr}}$-affine subspace of ${\raisebox{.2ex}{}}\bbr^m$. In the usual way (via the transfer of the canonical standard affine identification $\Fu+\Fv\mapsto\Fv$) one can identify the $\Fr$-almost affine subspace $\bbr^m_\Fr(\Fu)$ with the $\Fr$-almost affine subspace $\bbr^m_\Fr$. Suppose that ${\text{\scalebox{1}[.75]{\textcursive{f}}}}:({\raisebox{.2ex}{}}\bbr^m,0)\ra({\raisebox{.2ex}{}}\bbr^n,0)$ is an internal function. We say that ${\text{\scalebox{1}[.75]{\textcursive{f}}}}$ is ${\boldsymbol{\Fr}}$-almost linear* if for all $\a,\b\in{\raisebox{.2ex}{}}\bbr_{nes}$ and $\Fv,\Fw\in{\raisebox{.2ex}{}}\bbr^m_\Fr$, we have that ${\text{\scalebox{1}[.75]{\textcursive{f}}}}(\a\Fv+\b\Fw)-\a{\text{\scalebox{1}[.75]{\textcursive{f}}}}(\Fv)-\b{\text{\scalebox{1}[.75]{\textcursive{f}}}}(\Fw)=o(\Fr)$. \[rem: \linear c= alm lin\] Note that arbitrary \linear internal maps are $\Fr$-almost linear for all $\Fr$, but don’t send $\bbr^m_\Fr$ into $\bbr^n_\Fr$. If such an ${\text{\scalebox{1}[.75]{\textcursive{f}}}}$ send a ${\raisebox{.2ex}{}}\bbr_{nes}$-basis of $\bbr^m_\Fr$ into $\bbr^n_\Fr$, then we do have ${\text{\scalebox{1}[.75]{\textcursive{f}}}}(\bbr^m_\Fr)\subset\bbr^n_\Fr$. In the case that $\Fr=1$, then ${\text{\scalebox{1}[.75]{\textcursive{f}}}}$ is $1$-almost linear implies that restricted to $\bbr^m_1={\raisebox{.2ex}{}}\bbr^m_{nes}$ it’s graph is infinitesimally close to a (possibly nonstandard) affine subspace, ie., it’s standard part is an affine subspace (with possibly vertical subspaces). Since standard functions (eg., our function $B$ below) will typically not satisfy $f(v_0)=v_0$ ($v_0$ being the point in the domain where we are testing for differentiability of $f$), we will need the corresponding nearness notion for affine maps. First, note that if we are looking at an internal map ${\text{\scalebox{1}[.75]{\textcursive{f}}}}:{\raisebox{.2ex}{}}\bbr^m\ra{\raisebox{.2ex}{}}\bbr^n$ restricted to $\bbr^m_\Fr(\Fu_0)$, then the statement in the previous paragraph implies that this restriction can be considered as a map on $\bbr^m_\Fr$. If $\Fu_0\in{\raisebox{.2ex}{}}\bbr^m$, $\Fv_0\in{\raisebox{.2ex}{}}\bbr^n$ and ${\text{\scalebox{1}[.75]{\textcursive{f}}}}(\bbr^m_\Fr(\Fu_0))\subset\bbr^n_\Fr(\Fv_0)$, we say that ${\text{\scalebox{1}[.75]{\textcursive{f}}}}$ is ${\boldsymbol{\Fr}}$-almost affine at ${\boldsymbol{\Fu}}$ if ${\text{\scalebox{1}[.75]{\textcursive{f}}}}(\a\Fv+\b\Fw)-\a{\text{\scalebox{1}[.75]{\textcursive{f}}}}(\Fv)-\b{\text{\scalebox{1}[.75]{\textcursive{(}}}}\Fw)=o(\Fr)$ holds for all $\a,\b\in{\raisebox{.2ex}{}}\bbr_{nes}$ with $\a+\b=1$ and $\Fv,\Fw\in{\raisebox{.2ex}{}}\bbr^m_\Fr$. Clearly, the sum of two $\Fr$-almost affine maps (defined on $\bbr^m_\Fr(\Fu)$ for some $\Fu$) is also $\Fr$-almost affine (with a different range). There are many other elementary properties of an $\Fr$-affine category (and relations between $\Fr$-affine and $\Fs$-affine categories) that can be straightforwardly fleshed out, but we will only develop those tools needed here. Suppose that $\SA:{\raisebox{.2ex}{}}\bbr^m\ra{\raisebox{.2ex}{}}\bbr^n$ is $\Fr$-almost affine (on $\bbr^m_\Fr(\Fu_0)$) and $\Ft_0=\SA(\Fu_0)$. Considering $\SA$ as a map on $\bbr^m_\Fr$ via the above identification, we have that $\SA-\Ft_0$ is $\Fr$-almost linear. In particular, suppose that $\SA$ is $\Fr$-almost affine. Considering $\SA$ as a map on $\bbr^m_\Fr$, we have that if $\SA(0)=0$, then $\SA$ is $\Fr$-almost linear. In particular, $\Fr$-almost affine maps are just internal translates of $\Fr$-almost linear maps. Our proof of the first statement is essentially the usual proof that an affine function fixing the origin is linear. Letting $\SL=\SA-\Ft_0$, we must first verify that for $\xi,\z\in\bbr^m_\Fr$, $\SL(\xi+\z)\stackrel{\Fr}{\sim}\SL(\xi)+\SL(\z)$. Using the definition of $\Fr$-almost affine in the case of a \affine sum with three terms ie., $\a+\b+\g=1$, in the case where $\a=\b=1$ and $\g=-1$, we get $$\begin{aligned} 0=\SL(0)=\SL(\a\xi+\b\z-\g(\xi+\z))\stackrel{\Fr}{\sim}\SL(\xi)+\SL(\z)-\SL(\xi+\z). \end{aligned}$$ We must second verify that, for $\la\in{\raisebox{.2ex}{}}\bbr_{nes}$ and $\Fv\in\bbr^m_\Fr$, $\SL(\la\Fv)\stackrel{\Fr}{\sim}\la\SL(\Fv)$. In this case, we again use three term affine sums $\a\xi+\b\z+\g\s$ where $\a+\b+\g=1$. That is, we apply $\Fr$-almost affineness in the case where $\a=1,\b=-\la,\g=\la$ and $\xi=\la\Fv,\z=\Fv$ and $\s=0$ to get $$\begin{aligned} 0=\SL(\la\Fv-\la\Fv+0)\stackrel{\d}{\sim}\SL(\la\Fv)-\la\SL(\Fv). \end{aligned}$$ Clearly, the second statement in the lemma follows from the first. If ${\text{\scalebox{1}[.75]{\textcursive{f}}}}:{\raisebox{.2ex}{}}\bbr^m\ra{\raisebox{.2ex}{}}\bbr^n$ is internal and $\Fr\in{\raisebox{.2ex}{}}\bbr$ is positive, we define the ${\boldsymbol{\Fr}}$-dilation* of ${\text{\scalebox{1}[.75]{\textcursive{f}}}}$ to be the map $\Fr^{-1}\circ{\text{\scalebox{1}[.75]{\textcursive{f}}}}\circ\Fr:{\raisebox{.2ex}{}}\bbr^m\ra\bbr^n$, ie., the map $\Fv\mapsto \Fr^{-1}{\text{\scalebox{1}[.75]{\textcursive{f}}}}(\Fr\Fv)$. An ${\boldsymbol{\Fr}}$-disk* in $\bbr^m_\Fr(\Fu)$ is a \open, \convex subset $\SD\subset\bbr^m_\Fr(\Fu)$ of the form $\Fr\cdot\!\!{\raisebox{.2ex}{}}\! D+\Fu_0$ where $D\subset\bbr^m$ is convex, open and bounded. The following lemma is essentially tautalogical; nonetheless, it is included due to its importance in our argument. \[lem: dilation result\] Suppose that ${\text{\scalebox{1}[.75]{\textcursive{f}}}}:{\raisebox{.2ex}{}}\bbr^m\ra{\raisebox{.2ex}{}}\bbr^n$ is $\Fr$-almost affine on an $\Fr$-disk $\Fr\cdot\!\!{\raisebox{.2ex}{}}\! D+\Fu\subset\bbr^m_\Fr(\Fu)$. Then ${}^\Fr{\text{\scalebox{1}[.75]{\textcursive{f}}}}=\Fr^{-1}\circ{\text{\scalebox{1}[.75]{\textcursive{f}}}}\circ\Fr$ is $1$-almost affine on ${\raisebox{.2ex}{}}D+\Fr^{-1}\cdot\Fu$. By the previous lemma, without loss of generality assume that ${\text{\scalebox{1}[.75]{\textcursive{f}}}}$ is $\Fr$-almost linear. We must show that for all $\a,\b\in{\raisebox{.2ex}{}}\bbr_{nes}$ and $\xi,\z\in{\raisebox{.2ex}{}}\bbr^m_{nes}$, we have $$\begin{aligned} {}^\Fr{\text{\scalebox{1}[.75]{\textcursive{f}}}}(\a\xi+\b\z)-\a\;{}^\Fr{\text{\scalebox{1}[.75]{\textcursive{f}}}}(\xi)-\b\;{}^\Fr{\text{\scalebox{1}[.75]{\textcursive{f}}}}(\z)=o(1). \end{aligned}$$ Writing $\xi=\ov{\xi}/\Fr$ and $\z=\ov{\z}/\Fr$ for some $\ov{\xi},\ov{\z}\in\bbr^m_\Fr$, and noting that $\ov{\xi}\mapsto \ov{\xi}/\Fr$ is a bijection $\bbr^m_\Fr\ra\bbr^m_{nes}$, we see that the previous expression holds if and only if $$\begin{aligned} \Fr^{-1}\left[{\text{\scalebox{1}[.75]{\textcursive{f}}}}(\Fr(\a\;\ov{\xi}/\Fr+\b\ov{\z}/\Fr))-\a{\text{\scalebox{1}[.75]{\textcursive{f}}}}(\Fr(\ov{\xi}/\Fr))-\b{\text{\scalebox{1}[.75]{\textcursive{f}}}}(\Fr(\ov{\z}/\Fr))\right]=o(1). \end{aligned}$$ for all $\ov{\xi},\ov{\z}\in\bbr^m_\Fr$. Noting that for a vector $\Fv\in{\raisebox{.2ex}{}}\bbr^n$, we have $\Fr^{-1}\Fv=o(1)$ if and only if $\Fv=o(\Fr)$, we see that the previous expression is equivalent to $$\begin{aligned} {\text{\scalebox{1}[.75]{\textcursive{f}}}}(\a\ov{\xi}+\b\ov{\z})-\a{\text{\scalebox{1}[.75]{\textcursive{f}}}}(\ov{\xi})-\b{\text{\scalebox{1}[.75]{\textcursive{f}}}}(\ov{\z})=o(\Fr), \end{aligned}$$ for all $\a,\b\in{\raisebox{.2ex}{}}\bbr_{nes}$ and $\ov{\xi},\ov{\z}\in\bbr^m_\Fr$, as we wanted. \[rem: \aff -> alm aff\] Note that if $\SA:{\raisebox{.2ex}{}}\bbr^m\ra{\raisebox{.2ex}{}}\bbr^n$ is \affine and ${\text{\scalebox{1}[.75]{\textcursive{f}}}}$ is $\Fr$-almost affine at $\Fu$, then ${\text{\scalebox{1}[.75]{\textcursive{f}}}}+\SA$ is $\Fr$-almost affine at $\Fu$. Nonstandard criterion for differentiability ------------------------------------------- We begin with a general fact connecting differentiability given the setup in the previous part. The following facts follow essentially from basics contained in Stroyan and Luxemburg, [@StrLux76] and an analog is stated and proved in another form in the author’s work on the inverse function theorem, [@McGaffeyInverseMappingThm2012]. The following definition and proposition are stated in stronger forms than needed in this paper. The full strength will be needed in the following paper. We say that ${\text{\scalebox{1}[.75]{\textcursive{f}}}}:{\raisebox{.2ex}{}}\bbr^m\ra{\raisebox{.2ex}{}}\bbr^n$ is ${\boldsymbol{\Fr}}$-almost affine at ${\boldsymbol{\Fu_0}}$ stably for all positive infinitesimals* ${\boldsymbol{\Fr}}$ if the following holds. There is a linear $L:\bbr^m\ra\bbr^n$ such that ${\text{\scalebox{1}[.75]{\textcursive{f}}}}$ satisfies the following for all positive infinitesimals $\Fr$. The map restricted to $\bbr_\Fr(\Fu_0)$, ie., ${\text{\scalebox{1}[.75]{\textcursive{f}}}}:\bbr^m_\Fr(\Fu_0)\ra{\raisebox{.2ex}{}}\bbr^m$ is $\Fr$-almost affine at $\Fu_0$ such that the $\Fr$-standard part of the $\Fr$-almost linear part of ${\text{\scalebox{1}[.75]{\textcursive{f}}}}$ exists and is $L$. If $f:\bbr^m\ra\bbr^n$, $\Fu\in{\raisebox{.2ex}{}}\bbr^m$ and $\Fr$ is a positive infinitesimal, let ${\boldsymbol{f^\Fr_{\Fu}}}$ denote the internal map ${\raisebox{.2ex}{}}f$ restricted to $\bbr^m_\Fr(\Fu)$. If $\Fu=0$, we write $f^\Fr$ for $f^\Fr_0$. \[prop: NS criterion for differen of f\] Suppose that $f:\bbr^m\ra\bbr^n$ and $x_0\in\bbr^m$. Then the following are equivalent. 1. $f$ is differentiable at $\Fu_0$. 2. $f^\Fr_{\Fu_0}$ is $\Fr$-almost affine at $\Fu_0$ stably for all positive infinitesimals $\Fr$. Suppose that $f$ is differentiable at $x_0$ and let $L:\bbr^m\ra\bbr^n$ denote its derivative there. Let $\Fr_0$ be a positive infinitesimal. Then we clearly have that if $\Fv\in\bbr^m_\Fr$, then ${\raisebox{.2ex}{}}f(x_0+\Fv)=f(x_0)+L(\Fv)+o(\Fr)$. In particular, if $\Fw\in\bbr^m_\Fr$ also and we have nearstandard $\a,\b$ with $\a+\b=1$, then $$\begin{aligned} \label{eqn: f is alm.affine eqn} {\raisebox{.2ex}{}}f(\a\Fv+\b\Fw)=f(x_0)+L(\a\Fv+\b\Fw)+o(\Fr). \end{aligned}$$ Similarly, we have (1)* $\a\;{\raisebox{.2ex}{}}f(\Fv)=\a f(x_0)+\a\; L(\Fv)+\Fo(\Fr)$ and (2)* $\b\;{\raisebox{.2ex}{}}f(\Fv)=\b f(x_0)+\b L(\Fv)+\Fo(\Fr)$. Subtracting (1) and (2) from , the linearity of $L$ implies $$\begin{aligned} {\raisebox{.2ex}{}}f(\a\Fv+\b\Fw)-\a{\raisebox{.2ex}{}}f(\Fv)-\b\;{\raisebox{.2ex}{}}f(\Fw)=f(x_0)-(\a +\b)f(x_0)+\Fo(\Fr), \end{aligned}$$ and so $\a+\b=1$ finishes the first half of the proof. Now suppose that ${\raisebox{.2ex}{}}f$ is $\Fr$-almost affine at $\Fu_0$ stably for all $\Fr$ with (standard) linear map $L$. This just says for each positive infinitesimal $\Fr$ and $\Fv\in\bbr^m_\Fr$, we have ${\raisebox{.2ex}{}}f(x_0+\Fv)=f(x_0)+{\raisebox{.2ex}{}}L(\Fv)+\Fo(\Fr)$. That is, fixing $\Fr$, we have (3): $\Fv\in\bbr)_\Fr\Rightarrow\| \f{1}{\Fr}({\raisebox{.2ex}{}}f(x_0+\Fv)-f(x_0))-{\raisebox{.2ex}{}}L(\Fv)\|=\Fo(1)$. We need to make internal statements in order to construct a sufficiently consequential overflow. The following statements (special restrictions of the previous) will be sufficient. Let $U_\Fr$ denote the ball consisting of those $\Fv\in\bbr_\Fr$ with $\\|\Fv\\|\leq\Fr$. Then, for all $0<\Fr\sim 0$, (3) certainly implies the weaker assertion $$\begin{aligned} \Fv\in U_\Fr\;\Rightarrow\;\Big\|\f{{\raisebox{.2ex}{}}f(x_0+\Fv)-{\raisebox{.2ex}{}}f(x_0)}{\Fr}-{\raisebox{.2ex}{}}L(\Fv)\Big\|=\Fo(1). \end{aligned}$$ The argument is finished as follows. Replacing $=\Fo(1)$ by $< c$ for an arbitrary standard positive number $c$, we get an internal statement $S(\Fr,c)$ which holds for all positive infinitesimals $\Fr$ and hence for some positive standard $b$ by overflow. But we therefore have the statement: for every positive real $c$, there is positive real $b$ such that $S(b,c)$ holds, the criterion for differentiability at $x_0$. Nowhere differentiability of the blancmange function ==================================================== Preliminaries ------------- The blancmange function is defined as follows. (See Katz and Tall’s paper for a conceptual discussion and Thim’s paper for a conventional proof.) First define $s$ on the unit interval by $s(t)=t$ for $0\leq t\leq 1/2$ and $s(t)=1-t$ for $1/2<t< 1$ and extend $s$ to a function on all of $\bbr$ by defining it to have period $1$; ie., for all $j\in\bbz$ and $t\in [0,1)$ define $s(t+j)=s(t)$. By definition, $s$ is piecewise linear and continuous. Next, define it’s dyadic dilations as follows. For $k\in\bbn$ and $t\in\bbr$, let $s_k(t)=s(2^k t)/2^k$. Finally, define, for $n\in\bbn$ and $t\in\bbr$ $$\begin{aligned} B_n(t)=\sum_{j=0}^{n-1}s_k(t)\quad \text{and}\quad B(t)=\lim_{n\ra\infty}B_n(t). \end{aligned}$$ It’s clear that the above limit exists and is continuous as $\|s_k(t)\|\leq 2^{k+1}$ for all $t\in\bbr$; and so is a uniform limit of continuous functions on $[0,1]$. Letting ${\boldsymbol{B^n(t)=B(t)-B_n(t)}}=\sum_{k=n}^\infty s_k(t)$, it’s easy to verify the following critical facts. \[lem: \fcnal eqn and \affine on interval\] For each $n\in\bbn$ and $t\in\bbr$, we have the following functional equation $B(t)=B_n(t)+B(2^nt)/2^n$. For each $n\in\bbn$, the function $B_n$ is an affine function on the interval $(j2^{-n+1},(j+1)2^{-n+1})$ for all $j\in\bbz$. \[rem: overlap\] It’s easy to see that if $C:\bbr\ra\bbr$ is another function and there is $x\in\bbr$ and $k\in\bbn$ with $C$ affine on $x+(k2^{-n+1}, (k+1)2^{-n+1})$, then for some interval $I\subset(k2^{-n+1},(k+1)2^{-n+1})$ of length at least $2^{-n}$, $B_n+C$ is affine on $x+I$. Although it is not relevant for our proof, the “slope” of $B_n$ on the interval $(j2^{-n+1},(j+1)2^{-n+1})$ ia a function (of $n$) and the dyadic expansion of the integer $j$; it will have values given by an integer between $-n$ and $n$. Proof of nowhere differentiability ---------------------------------- \[thm: 1st blancmange result\] The blancmange function is a continuous function that is differentiable at no point in $\bbr$. Suppose, by way of contradiction, that $B$ is differentiable at $t_0\in\bbr$. Let $\om\in{\raisebox{.2ex}{}}\bbn_\infty$, so that $\d=1/2^\om$ is a positive infinitesimal. Now $B$ is differentiable at $t_0$ implies that ${\raisebox{.2ex}{}}B$ is $\d$-almost affine on ${\raisebox{.2ex}{}}\bbr_\d(t_0)=t_0+\sqcup_{k\in\un{\bbn}}[k\d,(k+1)\d)$. Also by the transfer of , we have that ${\raisebox{.2ex}{}}B_\om$ is \affine on $(\Fl\d/2,(\Fl+1)\d/2)$ for all $\Fl\in{\raisebox{.2ex}{}}\bbz$ (and so eg., $\d$-almost affine in each of these intervals, see ). And then says there is an interval $I\subset(0,\d)$ of length at least $\d/2$ so that ${\raisebox{.2ex}{}}B- B_\om$ is $\d$-almost affine on $t_0+I$. Hence, this and the transfer of the functional equation gives that $\d\circ{\raisebox{.2ex}{}}B\circ\d^{-1}={\raisebox{.2ex}{}}B- B_\om$ is $\d$-almost affine on $t_0+I$. But the dilation lemma, , applied in dimension $1$, then implies that ${\raisebox{.2ex}{}}B$ is $1$-almost affine on $t_0/\d+\d^{-1}I$, which is an interval of length at least $1/2$, an absurdity by and as $B$ is a continuous function that is not affine on any interval of length $1/2$. First, note that this argument cannot work if the dilation of domain and range is not a conjugation automorphism; eg., if it is not the identity operator on the linear part of affine maps. In particular, our argument fails if we consider $s_k(t)=s(a^kt)/b^k$ for $b>a$. In fact, such functions are often differentiable, eg., see the paper of Thompson and Hagler, [@ParabolicTakagi], where the authors show that, in the case $a=2,b=4$, $B$ is just part of a parabolic curve! Next, our construction shows that $B$ fails the nonstandard test for differentiability in a very big way. That is, it fails the test for $\d$-almost linearity for $\d=1/2^\om$ for all $\om\in{\raisebox{.2ex}{}}\bbn_\infty$. This is not too surprising as $B$ is standard. Generalized blancmange functions ================================ Construction of $\SB$ {#subsec: construct SB} --------------------- Here we will see that our proof, with minor alterations, works for very large families of analogously defined functions. First, instead of the continuous piecewise affine function $s$, we will now have an open subset $\SS$ of an infinite dimensional vector space of such piecewise affine continuous function, where the function $s$ of the previous section is essentially the simplest element of this set $\SS$. (As this vector space will not play a roll here, we will leave its description to a later paper.) Further, for a given $s\in\SS$, instead of the single sequence of functional equations (generating $B$) $s_k(t)=s(2^k)/2^k$ for $k=0,1,2,\cdots$, we will have a one parameter family of such sequences $s_k(t)=s(b^kt)/b^k$ for $2<b\in\bbn$ a multiple of an integer determined by $s$. Hence, we will generate a quite large family of nowhere differentiable functions, an issue we will address after our theorem. In the following, hopefully the reader should see how the previous proof is very close to our proof below for these generalized blancmange functions. First of all, let’s define an infinite general family of generating functions, ${\boldsymbol{\SS}}$ for which our generator $s$ is a single instance. As before our generator $s$ will be defined on the interval $[0,1]$ so that it can be extended to a continuous function on $\bbr$ with a period $1$. Define $s(0)=s(1)=0$ and for some $p\in\bbn$, if $0< i<p$, let $s(i)\in\bbr$ be arbitrary with $s(i_0/p)$ nonzero for some $i_0$. Given that $s$ is now defined at the points $i/p$ for $0\leq i\leq p$, extend $s$ to a function on all of $[0,1]$ by linear interpolation so that $s$ will be a continuous function on $[0,1]$ that is affine on each of the intervals $(i/p,(i+1)/p)$ for $0\leq i<p$. As $s(0)=s(1)=0$, we can extend $s$ to a continuous function on all of $\bbr$ by defining $s(j+t)=s(t)$ for $j\in\bbz\ssm\{0\}$ and $t\in[0,1)$. Let ${\boldsymbol{\SS_p}}$ consist of the set of all such $s$ for our given integer $p>1$ and let ${\boldsymbol{\SS}}$ denote the union of all $\SS_p$ as $p>1$ varies in $\bbn$. It is no problem that this is not a disjoint union. Note that the $s$ defining our blancmange function has $p=2$ and $b=2$. For $s\in\SS_p$ and $c\in\bbn$, let $b=cp$ for some $c\in\bbn$ and, for $k\in\bbn\cup\{0\}$, define $$\begin{aligned} \label{eqn: def of little s} s_k(t)=s(b^kt)/b^k. \end{aligned}$$ As before, for $n\in\bbn$, defining $B_n(t)=\sum_{j=0}^{n-1}s_j(t)$, we find that the sequence of piecewise continuous functions $B_n$ viewed on $[0,1]$ converge uniformly to a continuous function $B={\boldsymbol{B(s,c)}}$ on $[0,1]$. So applying periodicity, we get uniform convergence on all of $\bbr$. Given this, for $s\in\SS_p$, let $$\begin{aligned} \label{eqn: def of SB} {\boldsymbol{\SB(s)}}=\{B(s,c):c\in\bbn\},\;\; {\boldsymbol{\SB(p)}}=\cup\{\SB(s):s\in\SS_p\}\\ \text{and}\;\;{\boldsymbol{\SB}}=\cup\{\SB(p):p>1\;\text{is an integer}\}\qquad\quad\notag \end{aligned}$$ denote the set of all of these continuous functions defined by a given generating function $s\in\SG$ and compatible dilation factor $c\in\bbn$. Nowhere differentiability of elements of $\SB$ ---------------------------------------------- Given the above constructions, we need a pair of lemmas before we can prove nowhere differentiability of elements of $\SB$. We begin with a simple analog of . Assume that $p>2$ for our piecewise linear function $s$ defined above. For each $n\in\bbn$ and $t\in\bbr$, we have the functional equation $B(t)=B_n(t)+B(b^nt)/a^n$. For each $n\in\bbn$, the function $B_n$ is an affine function on the interval $(j/(pb^n),(j+1)/(pb^n))$ for all $j\in\bbz$. The functional equations are easy to verify as before. On the other hand, note that the vertices of the affine function $s_k(t)$ are the points $V_k=\{i/(pb^k):i\in\bbz\}$. In particular, as $V_j\subset V_k$ for $j\leq k$, then all of the functions $s_k$ for $k\leq n-1$ are affine on each of the intervals that $s_{n-1}$ is (ie., those of the form $(i/(pb^{n-1}),(i+1)/(pb^{n-1}))$ for $i\in\bbz$). Therefore, the sum $s+s_1+s_2+\cdots+s_{n-1}$ is affine on each such interval. For a replay of our earlier proof to work, we need to prove, in contrast, that the function $B(s,b)$ is not affine on any interval of positive length in $[0,1]$, (a fact that is obvious with the original blancmange function). We prove this in the next lemma by a simple combinatorial argument. First, we need some notation. For a fixed $p>1$ in $\bbn$, $s\in\SS_p$ and $B=B(s,b)$, if $n\in\bbn$, write $B=B_n+B^n$ where $B_n=\sum_{j<n}s_j$ and $B^n=\sum_{j\geq n}s_j$. \[lem: B(s,b) not affine on I\] Suppose that $s\in\SS$ with subdivision number $p\in\bbn$ and $b=cp$ for some $c\in\bbn$ and $B=B(s,b)$ is the function defined above. Then there is no open interval $I\subset[0,1]$ such that $B\|I$ is affine. Suppose, to the contrary, such an interval $I\subset[0,1]$ exists. That is, all points on the graph of $B$ on the interval $I$ are colinear. Then, there is a minimum $m\in\bbn$ and some $j$ so that $$\begin{aligned} I_{m,j}\dot=(\f{j}{pb^m},\f{j+1}{pb^m})\subset I. \end{aligned}$$ Now one can check the following facts. (1) We have $s_{m+1}(\f{j}{pb^m})=s_{m+1}(\f{j+1}{pb^m})=0$, but $s_{m+1}(\f{j_0}{pb^{m+1}})\not=0$ for some $\f{j_0}{pb^{m+1}}\in I_{m,j}$. (2) For all positive integers $k\leq m$, $s_k$ is affine on $I_{m,j}$. (3) For all integers $k\geq m+2$, $s_k(\f{j}{pb^m})=s_k(\f{j+1}{pb^m})=s_k(\f{j_0}{pb^{m+1}})=0$. Given these three facts, we can deduce the following. First, (1) and (2) clearly imply (4): the points $B_{m+2}(\f{j}{pb^m})$, $B_{m+2}(\f{j+1}{pb^m})$ and $B_{m+2}(\f{j_0}{pb^{m+1}})$ are not colinear on the graph of $B_{m+2}$. On the other hand, fact (3) implies that (5): $B^{m+2}(\f{j}{pb^m})=B^{m+2}(\f{j+1}{pb^m})=B^{m+2}(\f{j_0}{pb^{m+1}})=0$. Clearly then, as $B(t)=B_{m+2}(t)+B^{m+2}(t)$ for all $t$, facts (4) and (5) imply that the points $B(\f{j}{pb^m})$, $B(\f{j+1}{pb^m})$ and $B(\f{j_0}{pb^{m+1}})$ on the graph of $B$ are not colinear. As these points lie in the part of the graph over $I$, we have a contradiction. We can now verify our assertion. \[thm: 2nd blancmange result\] Suppose that $s\in\SG_p$ is one of our generating functions with $p>2$, and $B\in\SB(s,c)$ for a given $c\in\bbn$. Then $B$ is continuous and nowhere differentiable. We just need to prove nowhere differentiability. Suppose, to the contrary, that $B$ is differentiable at some $t_0\in (0,1)$. Let $\om\in{\raisebox{.2ex}{}}\bbn$ be an infinite integer and let $\d=1/(pb^{\om-1})$. As $\d$ is a positive infinitesimal, then our contrary hypothesis implies that ${\raisebox{.2ex}{}}B$ is $\d$-almost affine on $\bbr_\d(t_0)\supset\sqcup\{t_0+(k\d,(k+1)\d):k\in\bbz\}$; eg., on $t_0+(0,\d)$. We also have that the transfer of (or statement (2) in the previous lemma) evaluated at $\om$ implies that ${\raisebox{.2ex}{}}B_\om$ is \affine on $(\Fj\d,(\Fj+1)\d)$ for all $\Fj\in{\raisebox{.2ex}{}}\bbz$. But there is $\Fj_0\in{\raisebox{.2ex}{}}\bbz$ so that $(\Fj_0\d,(\Fj_0+1)\d)$ and $t_0+(0,\d)$ intersect in an interval $\SI$ of length at least $\d/2$. That is,$\d\circ{\raisebox{.2ex}{}}B\circ\d^{-1}={\raisebox{.2ex}{}}B-B_\om$ is $\d$-almost affine on $\SI$. Hence, the dilation lemma, , implies that ${\raisebox{.2ex}{}}B$ is $1$-almost affine on $\d^{-1}\SI$, a \interval in ${\raisebox{.2ex}{}}\bbr$ of length at least $1/2$. That is, as ${\raisebox{.2ex}{}}B$ has \period $1$, then $B=\fst({\raisebox{.2ex}{}}B)$ must be affine on an interval of length at least $1/2$. Our contradiction then follows from . Note that our proof seems capable of giving the same conclusion with a weaker hypothesis. That is, although differentiability of $B$ at $t_0$ implies that ${\raisebox{.2ex}{}}B$ is $\d$ almost affine on all of $\bbr_\d(t_o)$, we arrived at our conclusion using the $\d$ almost affineness of $B$ only on the small segment $t_0+(0,\d)$ of $\bbr_\d(t_0)$. Further, the assumption of $\d$ almost affineness on any of the segments $t_0+(k\d,(k+1)\d)$ would yield a contradiction by the same argument. This seems to imply, for example, that $B$ does not even have one sided derivatives at any $t_0\in[0,1]$. These implication will be pursued in a later paper. Density of $\SB$ ---------------- There are a fair number of constructions of continuous nowhere differentiable functions in terms of continuous piecewise affine functions. Detailed description of this work occurs in Thim’s work, [@ContNowhereDifferenFcns]. A more limited, but more graphic display of such functions can be found in Google images under the keywords “nowhere differentiable”, “Weierstrass function”, “Takagi function”, et cetera. Beyond the blancmange function, our family of functions $\SB$ includes some described in Thim’s paper, but also includes many not yet described. For example, $\SB$, includes functions generated by elements $s\in\SS$ with arbitrarily small support. Furthermore, we can also choose $s\in\SS$ that are arbitrarily Lipschitz close to eg., $\sin(\pi x)$. (For crude, but hopefully suggestive, examples of both, see the last section.) In fact, our family is sufficiently numerous to uniformly approximate any continuous functions $f:[0,1]\ra\bbr$ sending $0$ and $1$ to $0$. Let $C^0$ denote this set of continuous functions on $[0,1]$. First, we have a lemma that is a slight generalization of Lemma 4.3 in Thim. One might notice, that besides being distinctly shorter than his proof (see pages 74-75 of his text), we use no estimates, only simple order of magnitude arguments made available by nonstandard methods. Although he claims the proof is taken essentially from Oxtoby’s classic text, [@Oxtoby], we could not find the relevant text in Oxtoby. For a function $g:[0,1]\ra\bbr$, let $\\|g\\|$ denote $\sup\{\|g(t)\|:t\in[0,1]\}$, the supremum norm. We will also use this notation in the internal realm. $\SS$ is dense in $C^0$ with respect to the uniform norm. Let $f\in C^0$ and $E\subset\bbr_+$ denote the set of $e\in\bbr_+$ such that there is $s\in\SS$ with $\\|f-s\\|<e$. It suffices to prove that ${\raisebox{.2ex}{}}E$ contains infinitesimals. So we just need to show that there is ${\text{\scalebox{1}[.75]{\textcursive{s}}}}\in{\raisebox{.2ex}{}}\SS$ with $\\|{\raisebox{.2ex}{}}f-{\text{\scalebox{1}[.75]{\textcursive{s}}}}\\|\sim 0$. Choose $\pzp\in{\raisebox{.2ex}{}}\bbn_\infty$ with $[\pzp]=\{0,1,\cdots,\pzp-1,\pzp\}$. Let $\SP=\{\Fj/\pzp:\Fj\in[\pzp]\}$ and $\SI_\Fj=[\Fj/\pzp,(\Fj+1)/\pzp]$, a \compact interval. Define ${\text{\scalebox{1}[.75]{\textcursive{s}}}}(\Fj/\pzp)={\raisebox{.2ex}{}}f(\Fj/\pzp)$ for all $\Fj\in[\pzp]$ extending it to be \affine on each $\SI_\pzp$. Clearly, ${\text{\scalebox{1}[.75]{\textcursive{s}}}}\in{\raisebox{.2ex}{}}\SS$. Fixing $\Fj\in[\pzp]\ssm\{\pzp\}$, by standard continuity of $f$, ${\raisebox{.2ex}{}}f(\Ft)\sim{\raisebox{.2ex}{}}f(\Fj/\pzp)$ for all $\Ft\in\SI_\Fj$ and so by \affineness of ${\text{\scalebox{1}[.75]{\textcursive{s}}}}$ on $\SI_\Fj$, we have ${\text{\scalebox{1}[.75]{\textcursive{s}}}}(\Ft)\sim{\text{\scalebox{1}[.75]{\textcursive{s}}}}(\Fj/\pzp)$. Put together, these say that ${\raisebox{.2ex}{}}f(\Ft)\sim{\text{\scalebox{1}[.75]{\textcursive{s}}}}(\Ft)$. That is, ${\raisebox{.2ex}{}}f(\Ft)-{\text{\scalebox{1}[.75]{\textcursive{s}}}}(\Ft)\sim 0$ for all $\Ft\in\SI_\Fj$. As $\SI_\Fj$ is \compact and ${\raisebox{.2ex}{}}f-{\text{\scalebox{1}[.75]{\textcursive{s}}}}$ is \continuous, $\e_\Fj={\raisebox{.2ex}{}}\max\{\|{\raisebox{.2ex}{}}f(\Ft)-{\text{\scalebox{1}[.75]{\textcursive{s}}}}(\Ft)\|:\Ft\in\SI_\Fj\}$ exists and is infinitesimal. But $\SA=\{\e_\Fj:\Fj\in\{0,1,\cdots,\pzp-1\}\}$ is a \finite (eg., internal) set of infinitesimals and so ${\raisebox{.2ex}{}}\max\SA\in\SA$, eg., is infinitesimal. From the above lemma, we have our assertion. \[cor: SB is dense\] $\SB$ is dense in $C^0$ in the uniform topology. By the above lemma, it suffices to verify that for a fixed $s\in\SS_p$, there is $\scrb\in{\raisebox{.2ex}{}}\SB(s)$ with $\\|{\raisebox{.2ex}{}}s-\scrb\\|\sim 0$. Choose $\Fc\in{\raisebox{.2ex}{}}\bbn_\infty$, with $\scrb={\raisebox{.2ex}{}}B(s,\Fc)$. Now $\Fb=\Fc\cdot p\in{\raisebox{.2ex}{}}\bbn_\infty$, and so letting $M=\\|s\\|$, we have for all $\Ft\in{\raisebox{.2ex}{}}[0,1]$ that $$\begin{aligned} \|{\raisebox{.2ex}{}}s(\Ft)-{\raisebox{.2ex}{}}B(s,\Fc)(\Ft)\|\leq \f{M}{\Fb}\cdot{\raisebox{.2ex}{}}\sum_{\Fj=0}^{\infty}\Fb^{-\Fj}\sim 0 \end{aligned}$$ as we wanted. Perspective ----------- In order to prove the above results, we only needed the following facts. First, we needed a (fairly crude) nonstandard characterization of differentiability at a point $t_0$, ie., that for all positive infinitesimals $\Fr$, the function restricted to $\bbr_\Fr(t_0)$ is $\Fr$-almost affine. Second, we needed the fact that dilation sends almost affine maps to almost affine maps. Finally, third we needed the transfer of the set of functional equations as well as the fact that approximations were affine on sufficiently large intervals. In particular, we did not need nuanced versions of the nonstandard characterizations of differentiability. Although such a transcription is theoretically possible, from the author’s perspective, a rewriting of this proof in standard language would seem to be a nontrivial task. One must standardize our strategy: we fixed an infinite index and did some fairly detailed combinatorics on the geometric configurations existing at that index. In our second installment, we will consider functions not generated in terms of functional equations and will use an alternative nonstandard characterization of differentiability at a point. More specifically, for a function to be differentiable at a point $t_0$, not only does ${\raisebox{.2ex}{}}f$ need to be $\d$ almost affine on $\bbr_\d$ for all infinitesimal scales, $\d$; but crudely, dilation from one infinitesimal scale to another carries our almost affine maps into each other. A computational view of elements of $\SB$ ========================================= Due to the constructive nature and broad types of behavior of these functions, the author decided to investigate some computer visualization schemes with hopes of getting some insight into the natures of these (continuous) nowhere differentiable functions. Others, eg., Thompson and Hagler, [@ParabolicTakagi], have used numerical computational tools in attempting to gain insight into continuous nowhere differentiable functions; in fact, going at least as far back as the 1961 work of Salzer and Levine, [@TableWeierstrassFcn1961]. After weeks of investigations (of Tikz, Gnuplot, Sage and other open source tools), the author decided the open source Python suite ([python(x,y)](https://code.google.com/p/pythonxy/wiki/Welcome)) of abstract computational and graphing tools was best suited for this goal. The author invested two months to learn sufficient python (and matplotlib) syntax to construct a piece of code allowing at least a multiscaled impressionistic view of these functions. We have two versions of the code. After compiling, both yield a full page with six coordinate chart “snapshots”. Each of the first five snapshots is followed by another that is a magnification (around a fixed magnification point) of the graphs on the previous coordinate chart. Each coordinate chart displays the same multicolored tuple of graphs of approximations $B_{N_1},\cdots,B_{N_k}$ of a given element $B\in\SB$. Among other parameter choices, the user can choose the center of magnification, the magnification factor and the choice of the tuple of $N_1,\cdots,N_k$, although the author has constructed the coordinate legend for a tuple of length six or less. (The legend is not totally debugged. It’s off screen on some displays, but can be pulled in using the hspace toggle of the subplot configuration tool.) Of the two code choices, the first can be copied to an interactive console (we used Spyder lite) where it runs with little prompting. To run the code with different parameters, one must manually alter the code at eg., the number and values of the vertices, the magnification point etc. Alternatively, the second version is written to prompt for these parameters, eg., for the vertices of the generator $s$, where $B=B(s,1)$ (see ), the focal point, etc. After first saving the code as a python file ( by eg., copying it to the Spyder text editor which can save it properly on prompting), one can then “run” it on Spyder with the accompanying console prompting for the desired parameters. The above is an outline of our procedure; in either case, hopefully the code is sufficiently clear (to one with an elementary knowledge of python) so that the prompts can be extended by alterations of the code allowing a more refined sequence of magnifications of the tuple of approximations of the given element of $\SB$ (for a reader who has at hand more computational power than the author’s pedestrian resources). Furthermore, the author is struggling to build computationally more efficient code (eg., using python’s multiprocessing module) and welcomes the input of pythonistas. Whatever the case, a reader who might be interested in viewing the sequence of graphs of a particular element of $\SB$ is welcome to copies of the code from the author upon request at [the author’s gmail account](mailto:mcgaffeythomas@gmail.com). We have structured the above discussed code precisely to probe the manner in which a sequence of approximations “fall away” one at a time as we continue to magnify, leaving the more intense core. On the other hand, as noted above, the family $\SB$ includes numerous examples whose graphs display novel properties. Using a greatly simplified and redirected version of the code, we’ve included graphs of a pair of such examples in figure formatted for this article. We display the generator $s$ and the generalized blancmange function $B(s,c)$ with $c=1$ arising from it. The graphs are given in terms of the approximation $B_{12}=\sum_{j=0}^{11}s_j$ of our function $B$. Recall that if $p\in\bbn$ and $s\in\SS_p$ is a generator for $B=B(s,1)\in\SB$, then $s$ is defined by the $p+1$ values $v_j=s(j/p)$ for $0\leq j\leq p$. We will denote this by ${\boldsymbol{s=[v_0,\cdots,v_p]}}$ in the graphs below. The first has a curious smooth look and the second a sparse quality. Obviously, $p$ here is a small integer; by making $p$ arbitrarily large (or $c$ a large integer) we can accentuate these behaviors greatly. ![very different elements of $\SB$ \[image: two graphs\]](betterarticleplots.pdf){width="5in" height="8.5in"}
--- abstract: 'We study the onset of chaos and statistical relaxation in two isolated dynamical quantum systems of interacting spins-1/2, one of which is integrable and the other chaotic. Our approach to identifying the emergence of chaos is based on the level of delocalization of the eigenstates with respect to the energy shell, the latter being determined by the interaction strength between particles or quasi-particles. We also discuss how the onset of chaos may be anticipated by a careful analysis of the Hamiltonian matrices, even before diagonalization. We find that despite differences between the two models, their relaxation process following a quench is very similar and can be described analytically with a theory previously developed for systems with two-body random interactions. Our results imply that global features of statistical relaxation depend on the degree of spread of the eigenstates within the energy shell and may happen to both integrable and non-integrable systems.' author: - 'L. F. Santos' - 'F. Borgonovi' - 'F. M. Izrailev' title: \| Onset of chaos and relaxation in isolated systems of interacting spins-1/2:\ energy shell approach --- Introduction ============ In recent years, a great deal of attention has been paid to the issue of thermalization in isolated quantum systems caused by interparticle interactions [@ETH; @zele; @ZelevinskyRep1996; @FIC96; @Flambaum1997b; @BGIC98; @I01; @rigol; @Fitzpatrick2011; @lea01; @lea; @recent]. Apart from theoretical aspects, this interest has been triggered by remarkable experimental progresses in the studies of quantum systems with ultracold gases trapped in optical lattices (see, e.g., [@experim]). A necessary condition for the onset of thermalization is the statistical relaxation of the system to some kind of equilibrium, which is followed by further fluctuations of the observables around their average values. In classical mechanics, as discussed in Ref.[@C97], there are two mechanisms leading to the emergence of statistical behavior in dynamical (deterministic) systems. The first scenario, known since the early days of statistical mechanics, is the thermodynamic limit in which the number of particles diverges, $N \rightarrow \infty$. In this case, the statistical description is valid even in the absence of chaos. A completely integrable system, such as the Toda-lattice, can manifest perfect statistical and thermodynamical properties for a finite, although large number of particles (practically, for $ N \gg 1$ [@CCPC97]). Even though there are initial conditions which correspond to solitons, they are rare and can be safely neglected in practice. This first mechanism, termed “linear chaos" in Ref.[@C97], is at the core of the foundation of statistical mechanics. The other mechanism, which is more recent, is based on the concept of local instability of motion in phase space. The understanding is that an isolated dynamical system can behave in a statistical way even for a very small number of interacting particles, $N \geq 2$, provided the motion is strongly chaotic (see, e.g. [@C79; @LL83]). Chaoticity does not imply “true" randomness in the equations of motion, but a “pseudo-randomness" (or, [deterministic chaos]{}), which depends on the number of particles and the strength of the interparticle interaction. Ergodicity is not essential here, provided the measure of initial conditions corresponding to regular motion is very small. In this case, an apparent irreversibility of motion emerges, since any weak external perturbation gives rise to non-recurrence of the initial conditions. It should be stressed that, although the two mechanisms above are different, in both cases the time dependence of the observables can be described by an infinite number of statistically independent frequencies (see details in Ref.[@C97]). In quantum systems, the notion of trajectories and thus of their local instabilities loses its meaning. Yet, it has been argued that thermalization may still happen, even if the system is finite and isolated, provided it is chaotic. Chaos at the quantum level refers to specific properties of spectra, eigenstates, and dynamics of the system. They were initially observed in quantum systems whose classical counterparts were chaotic, but were soon found also in quantum systems without a classical limit and in quantum systems with disordered potentials. Nowadays, the term [quantum chaos]{} is used in a broad sense when referring to those properties, irrespectively of the existence of a true classical limit. After intensive investigation, the properties of [one-body]{} quantum chaos became well understood (see, e.g., Refs. [@CC95; @ReichlBook; @S99]). In contrast, the theory of [many-body]{} chaos with respect to quantum systems of interacting Fermi or Bose particles is far from being complete. In fact, even in the classical limit, a proper analysis of chaos becomes complicated due to the large number of interacting particles and, therefore, large dimensionality of the phase space. Initial studies of quantum chaos in many-body systems focused on the statistics of the energy levels. But it soon became clear that crucial information is contained in the eigenstates. Typically, the eigenstates are written in the basis corresponding to non-interacting particles. This corresponds to using a picture where the total Hamiltonian of the model is separated into a sum of two terms, $H=H_0 + V$, where $H_0$ describes the non-interacting particles (in a more general context, quasi-particles), and $V$ absorbs the interparticle interactions. In nuclear physics the latter term is referred to as “residual interaction". The separation of the Hamiltonian into two different parts is, in fact, nothing but the mean-field (mf) approach, widely used in atomic and nuclear physics. In many cases, the choice of unperturbed mf-basis in which $H_0$ is diagonal is not well-defined (not unique). However, this choice is usually well supported physically, especially when the interaction between particles can be considered small. Examples include interactions between outer shell electrons in atoms, electrons in quantum dots, and interactions between spins. The key point of many-body quantum chaos is that the eigenfunctions (EFs) in the mf-basis spread as the interaction between particles increases and may eventually have a very large number of contributing components. However, contrary to full random matrices, where the eigenstates are completely extended independently of the choice of basis, in isolated systems with finite-range interactions, the perturbation couples only part of the unperturbed basis states $\|n\rangle$. Therefore, only a fraction of the coefficients $C_n^\alpha$ composing the full Hamiltonian eigenstates $\|\alpha \rangle =\sum_{n} C^{\alpha}_{n} \|n\rangle $ can be essentially different from zero. In the energy representation, this fraction constitutes the [energy shell]{} of the system, which can be partly or fully filled by the exact eigenstates [@Casati1993; @Casati1996]. When the number of non-zero elements $C^{\alpha}_{n}$ is a small portion of the shell, the eigenstates are [localized]{}, while a large portion implies either [sparse]{} or [ergodic]{} states [@QCC]. In ergodic eigenstates, the coefficients $C^{\alpha}_{n}$ become random variables following a Gaussian distribution around the “envelope" defined by the energy shell. This latter scenario is used as a rigorous definition of [chaotic eigenstates]{} and occurs when the interaction exceeds a critical value [@Casati1993; @Casati1996; @zele; @ZelevinskyRep1996; @FIC96; @Flambaum1997b]. An example of such chaotic eigenstates was reported in Ref.[@C85], where a careful analysis of experimental data for the cerium atom revealed that excited states with fixed total angular momentum and parity $J^{\pi} = 1^{+}$ are [random superpositions]{} of a restricted number of basis states. The energy shell is associated with the limiting form of the [strength function]{} (SF) written in the energy representation [@Casati1993; @Casati1996]. This function is obtained by projecting the unperturbed states onto the basis of exact eigenstates. It is also known as the [local density of states]{} and is broadly used in nuclear and solid state physics. SF contains much information about global properties of the interactions. It has been shown, for example, that its shape changes from Breit-Wigner (Lorentzian) to Gaussian as the interparticle interaction increases [@zele; @ZelevinskyRep1996; @FI00; @Flambaum1997b; @Kota1998; @Kota2001]. When the eigenstates are chaotic and the quantum system has a well defined classical limit, the shapes of both EFs and SFs in the energy representation have classical analogs [@Casati1993; @Casati1996; @QCC]. The first matches the distribution of the projection of the phase space surface of $H$ onto $H_0$, and the second the projection of the surface of $H_0$ onto $H$. The onset of delocalization of EFs in the energy shell is then directly related to the [chaotization]{} of the system in the classical limit [@QCC] and provides a tool to reveal the transition to quantum chaos even for dynamical quantum systems without a classical limit. The emergence of chaotic eigenstates has been related to the onset of thermalization in isolated quantum many-body systems [@ETH; @zele; @ZelevinskyRep1996; @FIC96; @Flambaum1997b; @BGIC98; @I01; @rigol; @Fitzpatrick2011; @lea01; @lea]. It has been shown, for instance, that when the eigenstates become chaotic, the distribution of occupation numbers achieves standard Fermi-Dirac or Bose-Einstein forms, thus allowing for the introduction of temperature [@FIC96; @Flambaum1997b; @BGIC98; @I01]. In particular, an analytic expression connecting the increase of temperature with the interaction strength and the number of particles was obtained using a two-body random matrix model [@Flambaum1997b]. Therefore, the interparticle interaction plays the role of a heat bath for the isolated system. Another important aspect is that since the components of chaotic eigenstates can be treated as random variables, the eigenstates close in energy are statistically similar. This fact is at the heart of the so-called [Eigenstate Thermalization Hypothesis]{} (ETH) [@ETH] and has been employed to justify the agreement between the expectation values of few-body observables and the predictions from the microcanonical ensemble [@ETH; @rigol; @lea01; @lea]. The aim of the present work is to analyze the emergence of statistical properties in isolated quantum many-body systems. We consider two dynamical models of interacting spins-1/2; one is integrable for any value of the perturbation and the other undergo a transition to chaos. Our approach is based on the concept of the energy shell, in which the eigenstates undergo a transition from localized or sparse to delocalized and random. Strictly speaking, chaotic eigenstates filling completely the energy shell appear only for the nonintegrable model. However, even for the integrable system, chaotic-like eigenstates, where a large number of mf-basis contributes to the state, may be found in the limit of strong interaction. We demonstrate that the critical strength of the interaction above which the eigenstates may be considered chaotic-like corresponds to the point where the shape of SF becomes Gaussian. We show that in comparison with the chaotic model, the lack of ergodicity of EFs in the integrable system leads to larger fluctuations of the delocalization measures and for the overlaps between neighboring eigenstates. This coincides with recent results obtained for bosonic and fermionic systems [@lea01; @lea]. In the spirit of ETH, these findings were used to explain the better agreement between eigenstate expectation values of few-body observables and thermal averages for systems in the chaotic domain. Despite differences in some static properties, the relaxation process for both models after a quench is found to be very similar, as inferred from the study of the time dependence of the Shannon entropy for initial states corresponding to mf-basis states. Our numerical data agree very well with analytical predictions developed for two-body random matrices [@Flambaum2001b], when the interaction strength is strong. In this case, the entropy shows a linear growth before reaching complete relaxation. Crucial for this behavior is that the eigenstates are delocalized (although not necessarily ergodic) in the energy shell, which may occur even when the system is integrable. We also discuss how one can predict the onset of chaotic-like eigenstates by analyzing the structure of the Hamiltonian matrices without resorting to their diagonalization. Remarkably, the estimates coincide very closely with the critical values obtained from energy level statistics and the shapes of SF and EF. The paper is organized as follows. Section \[Sec:model\] describes the models studied, their symmetries, and the structure of the Hamiltonian matrices. Section \[Sec:chaos\] analyzes the fluctuations of the energy spectrum and quantifies the level of chaoticity of the system based on the level spacing distributions. Section \[Sec:EFs\] investigates the integrable-chaos transition from the perspective of the eigenstates. We study the shape of the strength functions, the spreading of the eigenstates in the energy shell, and delocalization measures. We also propose a new signature of chaos based on correlations between neighboring eigenstates. Section \[Sec:Shannon\] focuses on the time evolution of the Shannon entropy for both integrable and nonintegrable models aiming at identifying the conditions for statistical relaxation. Both numerical and analytical results are provided. Concluding remarks are presented in Sec. \[Sec:conclusion\]. System Model {#Sec:model} ============ We consider isolated one-dimensional (1D) systems of interacting spins-1/2. These prototype quantum many-body systems are employed in the studies of a variety of subjects, ranging from quantum computing [@Gershenfeld1997; @Loss1998; @Kane1998] and quantum phase transition [@Coldea2010] to the transport behavior in magnetic compounds [@Zotos1997; @Sologubenko2000; @Heidrich2004; @Zotos2005; @Hlubek2010; @Sirker2011; @Santos2011]. The recent viability to experimentally realize such models in optical lattices [@Duan2003; @Trotzky2008; @Simon2011; @ChenARXIV] have further increased the interest in them. In 1D, these systems may remain integrable even in the presence of interaction; while the crossover to chaos can be induced by different integrability breaking terms [@Hsu1993; @Avishai2002; @Santos2004; @Rabson2004; @Kudo2005; @Dukesz2009]. This particularity turns them into natural testbeds for the analysis of the integrable-chaos transition and for comparative studies between the two regimes. Two 1D spin-1/2 systems are investigated in this work. Model 1 has only nearest-neighbor (NN) couplings and is integrable for any value of the interaction strength. Model 2 includes nearest and next-nearest-neighbor (NNN) couplings, and it becomes chaotic when the strengths of the two are comparable. Both are dynamical systems, that is they are devoid of random elements. The source of chaos in such scenarios is the complexity derived from the interparticle interactions. Hamiltonian ----------- The Hamiltonians for Model 1 and Model 2 are respectively given by $$\begin{aligned} && H_1 = H_0 + \mu V_1 , \label{model1} \\ && H_0 = \sum_{i=1}^{L-1} J \left(S_i^x S_{i+1}^x + S_i^y S_{i+1}^y \right) , \nonumber \\ && V_1 = \sum_{i=1}^{L-1} J S_i^z S_{i+1}^z, \nonumber\end{aligned}$$ and $$\begin{aligned} && H_2 = H_1 + \lambda V_2 , \label{model2} \\ && V_2 = \sum_{i=1}^{L-2} J\left[ \left( S_i^x S_{i+2}^x + S_i^y S_{i+2}^y \right) + \mu S_i^z S_{i+2}^z \right] . \nonumber\end{aligned}$$ Above, $\hbar$ is set to 1, $L$ is the number of sites, and $S^{x,y,z}_i = \sigma^{x,y,z}_i/2$ are the spin operators at site $i$, $\sigma^{x,y,z}_i$ being the Pauli matrices. The coupling parameter $J$ determines the energy scale and is set to 1. The Zeeman splittings, caused by a static magnetic field in the $z$ direction, are the same for all sites and are not shown in the Hamiltonians above. We refer to a spin pointing up in the $z$ direction as an excitation. \(i) In Model 1, $H_0$ corresponds to the unperturbed part of the Hamiltonian and $\mu$ is the strength of the perturbation. The unperturbed part is known as the flip-flop term and is responsible for moving the excitations through the chain. A system described by $H_0$ is integrable and can be mapped onto a system of noninteracting spinless fermions [@Jordan1928] or hardcore bosons [@Holstein1940]. It remains integrable with the addition of the Ising interaction $V_1$, no matter how large the anisotropy parameter $\mu $ is. The total Hamiltonian $H_1$ is referred to as the XXZ Hamiltonian and can be solved with the Bethe Ansatz [@Bethe1931; @Alcaraz1987; @Karbach1997]. We assume $J$ and $\mu$ positive, thus favoring antiferromagnetic order. \(ii) The unperturbed part of Model 2 is the XXZ Hamiltonian. The parameter $\lambda$ refers to the ratio between the NNN exchange, as determined by the perturbation $V_2$, and the NN couplings, characterized by $H_1$. A sufficiently large $\lambda$ leads to the onset of chaos. With respect to symmetries, conservation of total spin in the $z$ direction, $S^z=\sum_{i=1}^{L} S_i^z$, occurs for all parameters of Hamiltonians (\[model1\]) and (\[model2\]). Our analysis is thus restricted to a particular $S^z$-subspace. In order to deal with a reasonably large sector without resorting to very large system sizes, other symmetries [@Brown2008] are avoided as follows. $\bullet$ We deal with open boundary conditions, instead of closed boundary conditions, to prevent momentum conservation. $\bullet$ We choose subspaces filled with $L/3$ up-spins to guarantee that $S^z \neq 0$. The $S^z=0$ sector, which appears when the chain size is even and has $L/2$ up-spins, shows invariance under a $\pi$-rotation around the $x$-axis. The dimension of the $S^z$-subspace that we consider is therefore $D_{L/3} = L!/[(L/3)!(L-L/3)!]$. Unless stated otherwise, all figures are obtained for $L=15$. $\bullet$ We use $\mu \neq 1$ throughout to circumvent conservation of total spin, $S^2=(\sum_{i=1}^L \vec{S}_i)^2$. Different values of $\mu $ are studied for Model 1, but for Model 2, where the main interest is in the effects of the integrability breaking term $V_2$, we fix $\mu = 0.5$. $\bullet$ Parity is not avoided. We take it into account by analyzing even and odd eigenstates separately. The dimension of each parity sector is $D_P \sim D_{L/3}/2$. Since our numerical studies require all eigenvalues and eigenvectors of the systems, exact full diagonalization is performed. However, as it will be clear along the text, much information can be obtained just from the Hamiltonian matrix itself. Structure of the Hamiltonian matrix and strength of the perturbation -------------------------------------------------------------------- An essential point for the study of the Hamiltonian matrix is the basis considered. In general, the choice of basis is made on physical grounds, depending on the question being addressed. In the case of the Fermi-Pasta-Ulam model, for example, one focuses on the equipartition of energy among normal modes [@Berman2005]. When studying spatial localization, on the other hand, the most appropriate basis is the coordinate basis, which in the case of lattice systems corresponds to the site-basis. For systems (\[model1\]) and (\[model2\]), the site-basis corresponds to arrays of spins pointing up and down in the $z$ direction. Here, our goal is to understand the effects of the residual perturbations $V_1$ and $V_2$. They add complexity to the system, without necessarily bringing it to the chaotic domain. It becomes then essential to select a basis associated with the uncoupled particles (or quasiparticles) with which we may separate regular from complex behavior. This is the role of a mf-basis, which appears in various contexts of many-body physics. The derivation of Fermi-Dirac or Bose-Einstein distributions, for instance, requires the selection of a mf-basis. The same is true when studying the structures of nuclear and atomic systems, as well as their transition to quantum chaos. Nevertheless, there is not a well defined mathematical recipe to identifying the mf-basis; this is done based on the physical properties of the system. For the total Hamiltonians $H_1$ and $H_2$, we choose the eigenstates of $H_0$ and $H_1$,respectively, as the unperturbed basis states, $\|n\rangle $. To give an idea on how to extract information from the Hamiltonian matrix, we show in Figure \[fig:structure\] the density plot of the absolute values of the matrix elements for Model 1 (left panel) and Model 2 (right panel). The matrices are written in the mf-basis, the latter being ordered from lowest to highest energy. Light colors indicate large values. Only elements associated with even states are shown, so no trivial symmetries are present. Both matrices have large diagonal elements and significant couplings even between distant basis vectors. It is only far from the diagonal that the elements fade away, as expected for realistic physical models. More zeros are found in the matrix of Model 1, which is thus more sparse than the matrix of Model 2. Both matrices are obviously symmetric with respect to the diagonal, since $H_{nm}=H_{mn}$. In addition to this, Model 1 shows an impressive regular structure which must be related to its integrability; various curves of high density suggest strong correlations between the matrix elements. For example, for the lines in the middle, such as $mid= 121, 122, \ldots 135$, we find that several elements, but not all, satisfy the relation $\|H_{mid,1+k}\|=\|H_{mid,D_P-k}\|$. We leave it for a future publication the interesting exercise of identifying the sources of such correlations. ![(Color online.) Absolute values of the matrix elements of Model 1 (left panel) and Model 2 with $\lambda =0.5$ (right panel) for $L=12$ \[therefore $D_P\sim 250$\] and $\mu=0.5$. The mf-basis is ordered in energy. Only even states are considered. Light color indicates large values.[]{data-label="fig:structure"}](Fig01_long_aMatrix.eps){width="45.00000%"} Further details about the matrices may be obtained with the help of Figs. \[fig:ham\] and \[fig:connect\]. \(i) The diagonal elements $H_{nn}$ are shown in the top panels of Fig. \[fig:ham\]. Changes are seen as the perturbation increases, especially for Model 2. This indicates that contributions to $H_{nn}$ come not only from the unperturbed part of the Hamiltonians, but also from the perturbation. Also noticeable is an asymmetry between low and high energies, which is enhanced for larger perturbation. For Model 1, larger values of $\|H_{nn}\|$ are reached for negative energies, while the opposite occurs for Model 2. This imbalance is carried to various other properties of the systems, as will be seen later. ![(Color online.) Information about the matrix elements of Model 1 (two left columns) and Model 2 (two right columns). The matrices are written in the mf-basis, which is ordered from lowest to highest energy. The perturbation strength for each column is shown in the top panels. Top panels: diagonal elements. Bottom panels: average values of the absolute values of the off-diagonal elements vs the distance $k$ from the diagonal.[]{data-label="fig:ham"}](Fig02_long_bDiagOff.eps){width="45.00000%"} \(ii) The bottom panels of Fig. \[fig:ham\] show the average values of the absolute values of the off-diagonal elements, $\langle H_{n,n+k} \rangle =[\sum_{n=1}^{D_{L/3}-k} \|H_{n,n+k}\|]/(D_{L/3}-k)$, vs the distance $k$ from the diagonal. They are significantly smaller than the diagonal elements and decrease slowly as we move away from the diagonal. Thus, even though the Hamiltonians in the site-basis have only NN and NNN couplings, long range (but finite) interactions become present in the mf-basis. ![(Color online.) Connectivity of each line $n$ for Model 1 (left) and Model 2 (right).[]{data-label="fig:connect"}](Fig03_long_cConnect.eps){width="45.00000%"} \(iii) Figure \[fig:connect\] shows the values of the connectivity $M_n$ of each line $n$, that is the number of directly coupled basis vectors in each row. We present results for $\mu,\lambda =0.5$; they do not change much for larger values of the perturbation. To compute $M_n$, we discard the off-diagonal elements $H_{nm}$ for which $\|H_{nm}\|<\eta$, where $\eta$ is the variance of the absolute value of all off-diagonal elements. This is done, because the Hamiltonian is initially written in the site-basis and then numerically transformed into the mf-basis, which causes all matrix elements to become nonzero. The connectivity for the integrable model is significantly lower than for the chaotic system. For Model 2 in the middle of the spectrum, almost all basis vectors with the same parity are coupled. On average, for the middle of the spectrum, we find $$\begin{aligned} && \mbox{Model 1:} \hspace{0.5 cm} \langle M_n \rangle \sim D_P/4, \nonumber \\ && \mbox{Model 2:} \hspace{0.5 cm} \langle M_n \rangle \sim D_P. \label{connect_1_2}\end{aligned}$$ This confirms that $H_1$ is more sparse than $H_2$, as already observed in Fig. \[fig:structure\]. Also in connection to that figure, we see here an interesting structure of separated layers for the values of the connectivity of Model 1, which must be related to its integrability. For Model 2, on the other hand, $M_n$ has a smoother behavior with $n$. From the Hamiltonian matrix we can estimate also the relative strength of the perturbation. For this, we compare for each line the average value of the coupling strength $v_n$ with the mean level spacing $d_n$ between directly coupled states. Taking into account that not all unperturbed states are directly coupled, we define $v_n = \sum_{m\neq n} \|H_{nm}\|/M_{n}$ and compute the mean level spacing from $d_n=[\varepsilon_n^{max}-\varepsilon_n^{min}]/M_{n}$, where $\varepsilon_n^{max}$ ($\varepsilon_n^{min}$) is the unperturbed energy $H_{mm}$ corresponding to the largest (smallest) $m$ where $H_{nm}\neq0$. Strong perturbation is achieved when $v_n/d_n \gtrsim 1$. ![(Color online.) Ratio of the average coupling strength $v_n$ to the mean level spacing $d_n$ between directly coupled states for each line $n$ of Model 1 (left panels) and Model 2 (right panels). Horizontal (green) line stands for $v_n/d_n=1$.[]{data-label="fig:V_df"}](Fig04_long_VdRed.eps){width="45.00000%"} Figure \[fig:V\_df\] depicts the ratio $v_n/d_n$ for Model 1 (left panels) and Model 2 (right panels). The critical values above which the perturbation becomes strong are approximately $\mu_{cr}\sim 0.5$ and $\lambda_{cr} \sim 0.5$. As we will show later, these estimates coincide with values obtained using the eigenvalues and eigenstates of the systems. Interestingly, the ratio is not flat; it increases with $n$ for Model 1 and decreases with $n$ for Model 2. This is a reflection of the asymmetry of the diagonal elements, already seen in Fig. \[fig:ham\], and it will reappear in Sec. \[IPR\_S\] when we discuss the level of delocalization of the eigenstates. Signatures of Quantum Chaos: eigenvalues {#Sec:chaos} ======================================== Different quantities exist to identify the crossover from integrability to quantum chaos. Level spacing distribution, level number variance, and rigidity [@MehtaBook; @HaakeBook; @Guhr1998; @ReichlBook], for example, are associated with the eigenvalues, the first being the most commonly used signature of chaos. In this section, we show briefly some results for the level spacing distribution after having a look at the density of states. Density of states ----------------- We denote the eigenvalues of the system by $E_{\alpha}$ and the eigenstates by $\|\alpha\rangle$. The density of states $\rho (E_{\alpha})$ for both models are seen in Fig. \[fig:rhoE\]. Since the Hilbert space is finite, $\rho (E_{\alpha})$ consists of two parts. On the left side of the spectrum, $\rho (E_{\alpha})$ increases with energy; there the microcanonical temperature is positive. The right side corresponds to negative temperatures. The point of maximum density of states has infinite temperature. Independently of the domain, the distributions are very close to Gaussians, as typical of systems with few-body interactions (two-body in our case) [@French1970; @Brody1981]. This is to be contrasted with ensembles of full random matrices, where the density of states is semicircular [@HaakeBook; @Guhr1998; @ReichlBook]. The fact that the density of states vanishes at very low and very high energies implies that ergodic states are not expected to be found in the edges of the spectrum, even if the system is chaotic. Our analyses of the shapes of the eigenstates, developed in the next section, concentrate thus on the middle of the spectrum. ![(Color online.) Density of states for Model 1 (left panels) and Model 2 (right panels); bin size = 0.1. The solid (black) line gives the best Gaussian fit: $\mu=0.1 \rightarrow \langle E \rangle = 0.034, \sigma = 1.330$; $\mu=0.4 \rightarrow \langle E \rangle = 0.131, \sigma = 1.375$; $\mu=1.5 \rightarrow \langle E \rangle = 0.363, \sigma = 1.857$; $\lambda=0.1 \rightarrow \langle E \rangle = 0.157, \sigma = 1.400$; $\lambda=0.4 \rightarrow \langle E \rangle = 0.051, \sigma = 1.494$; and $\lambda=1.5 \rightarrow \langle E \rangle = 0.037, \sigma = 1.920$[]{data-label="fig:rhoE"}](Fig_rhoNEW.eps){width="40.00000%"} Level spacing distribution -------------------------- The analysis of the level spacing distribution requires unfolding the spectrum of each symmetry sector separately. Unfolding the spectrum consists of locally rescaling the energies, so that the mean level density of the new sequence of energies is unity [@HaakeBook; @Guhr1998; @ReichlBook]. Here, we discard 20% of the energies located at the edges of the spectrum, where the fluctuations are large, and obtain the cumulative mean level density by fitting the staircase function with a polynomial of degree 15. Quantum levels of integrable systems are not prohibited from crossing and the distribution is typically Poissonian, $$P_{ P}(s) = \exp(-s),$$ where $s$ is the normalized level spacing. This is the distribution obtained for Model 1 with any value of $\mu$, as shown in the left top panel of Fig. \[fig:Ps\]. In chaotic systems, crossings are avoided and the level spacing distribution is given by the Wigner-Dyson distribution, as predicted by random matrix theory. Ensembles of random matrices with time reversal invariance, the so-called Gaussian Orthogonal Ensembles (GOEs), lead to $$P_{ WD}(s) = (\pi s/2)\exp(-\pi s^2/4).$$ This is the distribution obtained for Model 2 in the chaotic limit, as shown in the right top panel of Fig. \[fig:Ps\]. Notice, however, that our systems, contrary to GOEs, have only finite-range-two-body interactions and do not contain random elements. Practically, $P(s)$ is not capable of detecting these differences and the same is expected for other signatures of quantum chaos associated with the energy levels, such as rigidity and level number variance. For an idea of how the results for the level number variance would look like, we refer the reader to Fig.5 in [@lea01], where an equivalent system is considered. More details about the system are found in the properties associated with the eigenstates, as further discussed in the next section. ![(Color online.) Top panels: Level spacing distribution. Bottom panels: Parameter $\beta$ of the Brody distribution vs the perturbation strength. Left panels: Model 1; right panels: Model 2.[]{data-label="fig:Ps"}](Fig06_longWD.eps){width="40.00000%"} The parameter $\beta$, used to fit $P(s)$ with the Brody distribution [@Brody1981], $$P_B(s) = (\beta +1) b s^{\beta} \exp \left( -b s^{\beta +1} \right), \hspace{0.2 cm} b= \left[\Gamma \left( \frac{\beta + 2}{\beta +1} \right)\right]^{\beta +1},$$ can be used to quantify the level of chaoticity of the system reflected by the spectrum statistics. For the integrable Model 1, $\beta$ is close to 0 for any value of $\mu$ (left bottom panel of Fig. \[fig:Ps\]), while for Model 2 (right bottom panel), it changes from 0 to 1 as $\lambda$ increases [@noteIntegrable]. The crossover from integrability to chaos is fast and occurs for $\lambda_{cr} \sim 0.5$. This value coincides with the estimate derived from the Hamiltonian matrix in Fig. \[fig:V\_df\]. It is impressive that the latter procedure, which does not require the diagonalization of the Hamiltonian, can give such satisfactory result. Signatures of Quantum Chaos: eigenstates {#Sec:EFs} ======================================== In this section we explore the features of the eigenstates $$\|\alpha\rangle=\sum_{n} C^{\alpha}_{n} \|n\rangle$$ written in the mf-basis $\|n \rangle $ for both integrable and chaotic regimes. As will soon become clear, more information about the system may be found in the structures of EFs than in the eigenvalues. Standard perturbation theory applies when the perturbation is weak, $v_n/d_n \ll 1$. In this limit, the eigenstates are very similar to the mf-basis states, having a very small number of very large components $C^{\alpha}_{n}$. As the perturbation increases, $\|\alpha \rangle$ spreads in the unperturbed basis, and the number of principal components, $N_{pc}$, eventually gets very large. This transition is illustrated in Fig. \[fig:EF\_examples\]. The eigenstates are shown as a function of the unperturbed energy $\varepsilon_n$ rather than in the basis representation, following the one-to-one correspondence between each unperturbed state $\|n \rangle $ and its energy $\varepsilon_n$. ![(Color online.) Examples of eigenstates from the center of the spectrum for Model 1 (left) and Model 2 (right). They become more extended from top to bottom.[]{data-label="fig:EF_examples"}](Fig07_longEFred.eps){width="40.00000%"} Notice that even for very large perturbation, not all vectors $\|n\rangle$ contribute to the eigenstates. The restricted number of participating basis states is a consequence of the finite range of the interactions; only part of the unperturbed states is directly coupled and therefore able to integrate the eigenstates. The limited spread of EFs is clearly seen in Fig. \[fig:EF\_matrix\], where the squared amplitudes $\|C^{\alpha}_n\|^2$ are depicted. In the figure, the basis representation is used. Each horizontal line corresponds to an eigenstate of energy $E_{\alpha}$ in the unperturbed basis. Vertical lines are the unperturbed states with energy $\varepsilon_n$ projected onto the basis of exact states. Light colors represent large $\|C^{\alpha}_n\|^2$. The widths of participating states in the vertical and horizontal lines are similar; they are broader in the middle of the spectrum and spread further as the perturbation increases. As $\mu$ and $\lambda$ increase, the differences in magnitude between diagonal and off-diagonal elements become less pronounced. The asymmetry between the edges of the spectrum observed in Figs. \[fig:ham\] and \[fig:V\_df\] is seen here again, localization being more enhanced for low energies in Model 1 and for high energies in Model 2 (see bottom panels). Also noticeable is a difference in sparsity between EF and SF depending on the system. For Model 2, just contrary to what was observed for Wigner band random matrix models [@Casati1996], EFs seem to be more sparse than SFs. 0.4 cm ![(Color online.) Matrix of squared components of the eigenstates for Model 1 (left): $\mu=0.5$ (top) and $\mu=1.5$ (bottom); and Model 2 (right): $\lambda=0.5$ (top) and $\lambda=1.0$ (bottom). Only even states are shown, $L=12$. Light color indicates large values.[]{data-label="fig:EF_matrix"}](Fig08_HamOfEFs.eps "fig:"){width="40.00000%"} Strength function and energy shell {#SFandShell} ---------------------------------- In the energy representation, the strength function corresponds to the dependence of $\|C^{\alpha}_n\|^2$ on the exact energies $E_{\alpha}$ for each fixed unperturbed energy $\varepsilon_n$. It is given by the expression, $$P_n(E) = \sum_{\alpha} \|C^{\alpha}_n\|^2 \delta (E-E_{\alpha}),$$ where the sum is performed over a small energy window centered at $E$. For an initial state $\|n_0\rangle$, $P_{n_0}$ identifies the energies $E_{\alpha}$ that become available to the state when the perturbation is turned on. The width of SF is therefore associated with the lifetime of $\|n_0\rangle $. This is clearly seen by the relation between the probability $W_{n_0}(t)$ for the system to remain in the state and SF, as given by $$\begin{aligned} W_{n_0}(t) &=& \left\| \langle n_0 \| e^{-i H t}\|n_0 \rangle \right\|^2 = \left\| \sum_{\alpha} \|C^{\alpha}_{n_0}\|^2 e^{-i E_{\alpha} t} \right\|^2 \nonumber \\ && \approx \left\| \int dE P_{n_0}(E) e^{-iEt} \right\|^2, \label{W0}\end{aligned}$$ where $$P_{n_0}(E) = \overline{\|C^{\alpha}_{n_0}\|^2} \rho(E) \label{SF_rho}$$ is SF after replacing the sum over a large number of eigenstates by an integral, the bar stands for an average in a small energy window, and $\rho(E)$ is the density of exact eigenstates. An important aspect of SF is the possibility of measuring it experimentally. In nuclear physics, this is done by exciting an unperturbed state and studying its decay. In solid state physics, SF corresponds to the local density of states, since it gives the density of states for an electron on position $\|n\rangle $. SFs, just like EFs, become more spread as the perturbation increases, as illustrated in Fig. \[fig:SF\]. We show with filled curves the average shape of SF for 5 even unperturbed states in the middle of the spectrum. SF starts as a delta function. As the interparticle interactions increase, it acquires first a Breit-Wigner (Lorentzian) shape (middle panels), and eventually becomes Gaussian (bottom panels). This agrees with previous studies of quantum many-body systems [@zele; @ZelevinskyRep1996; @Flambaum1997b; @Flambaum2000]. \(i) According to those studies, the Breit-Wigner function is given by $$P_n(E) = \frac{1}{2\pi} \frac{\Gamma_n }{(\varepsilon_n + \delta_n - E)^2 + \left[ \Gamma_n /2 \right]^2}, \label{BW_SF}$$ where the width $\Gamma_n $ is given by the Fermi Golden Rule, $$\Gamma_n \approx 2\pi \overline{\|H_{nm}\|^2} \rho_m,$$ $\delta_n$ is a correction to the unperturbed energy $\varepsilon_n$ due to the residual interaction, $\overline{\|H_{nm}\|^2}$ is the mean squared value of nonzero off-diagonal elements of the Hamiltonian, and $\rho_m$ is the density of basis states $\|m\rangle$ directly coupled to the initial state $\|n\rangle$ via $H_{nm}$. \(ii) The Gaussian form is $$P_n(E)=\frac{1}{\sqrt{2 \pi \sigma_n^2}} \exp \left( \frac{-(E-\varepsilon_n)^2} {2\sigma_n^2} \right), \label{Gaussian_SF}$$ where $$\sigma_n = \sqrt{\sum_{m\neq n} \|H_{nm}\|^2}. \label{dispersion}$$ In the following we will assume that, in the center of the band where maximal chaos is realized, $\Gamma_n = \Gamma$ and $\sigma_n = \sigma$. The transition from one shape to the other is determined by the relation between $\Gamma$ and $\sigma$ [@FI00]. Equation (\[BW\_SF\]) holds when the perturbation is small, but non-perturbative, $\Gamma \ll \sigma$, while for $\Gamma \gtrsim \sigma$, SF becomes close to a Gaussian, as in Eq. (\[Gaussian\_SF\]). 0.4 cm ![(Color online.) Strength functions for Model 1 (left) and Model 2 (right) obtained by averaging over 5 even unperturbed states in the middle of the spectrum. The average is performed after shifting the center of SFs to zero. Circles give the fitting curves. Middle panels: Breit-Wigner with $\varepsilon + \delta = -0.015$, $\Gamma = 0.302$ (left) and $\varepsilon + \delta = 0.072$, $\Gamma = 0.345 $ (right). Bottom panels: Gaussian with $\langle E \rangle = -0.072$, $\sigma = 1.322$ (left) and $\langle E \rangle = -0.022$, $\sigma = 0.936$ (right). Solid curves correspond to the Gaussian form of the energy shells with $\sigma = 0.090$ for $\mu =0.1$; $\sigma = 0.359$ for $\mu =0.4$; $\sigma = 1.345$ for $\mu =1.5$; $\sigma = 0.103$ for $\lambda =0.1$; $\sigma = 0.412$ for $\lambda =0.4$; and $\sigma = 1.029$ for $\lambda =1.0$.[]{data-label="fig:SF"}](Fig09_SFlongB.eps "fig:"){width="40.00000%"} The maximal shape of SF, as given by Eq. (\[Gaussian\_SF\]), is reached when the diagonal elements of the Hamiltonian matrix become negligible. In this case, SF coincides with the energy shell. The latter corresponds to the density of states obtained from a matrix filled only with the off-diagonal elements of the perturbation [@Casati1993; @Casati1996]. It measures the maximum number of basis states coupled by the perturbation. We computed the energy shells numerically and verified that they agree very well with the Gaussian functions (\[Gaussian\_SF\]) with dispersion (\[dispersion\]). The solid lines in Fig. \[fig:SF\] represent these functions. As follows from Eq. (\[dispersion\]), $\sigma^2$ is obtained without any diagonalization. That expression is derived from the distribution of exact eigenvalues $E_{\alpha}$ for each unperturbed state $\|n\rangle$, according to [@Flambaum1997b] $$\begin{aligned} \sigma^2 &=& \langle E_{\alpha}^2 \rangle - \langle E_{\alpha} \rangle^2 = \sum_{\alpha} \|C^{\alpha}_{n}\|^2 E_{\alpha}^2 - \left( \sum_{\alpha} \|C^{\alpha}_{n}\|^2 E_{\alpha} \right)^2 \nonumber \\ &=& \sum_{m} \langle n\|H\|m \rangle \langle m\|H \|n\rangle - \varepsilon_n^2 = \sum_{m\neq n} \|H_{nm}\|^2. \nonumber \label{sigma_shell}\end{aligned}$$ As seen in Fig. \[fig:SF\], it is only at large perturbation that SF acquires a Gaussian form and approaches the energy shell. When SF becomes Gaussian with the same width of the energy shell, maximal ergodic filling of the energy shell is realized and a statistical description becomes possible. The agreement between SF and the energy shell is another way to find the critical values $\mu_{cr}$ and $\lambda_{cr}$. We fitted our numerical data with both functions (circles in Fig. \[fig:SF\]) and verified that the transition from Breit-Wigner to Gaussian happens for the same critical values, $\mu_{cr} \, , \lambda_{cr} \approx 0.5$, obtained before from $v_n/d_n$ in Fig. \[fig:V\_df\] and from the transition to a Wigner-Dyson distribution in the case of Model 2. At large perturbation we then have an excellent agreement between the Gaussian fit and the Gaussian describing the energy shell which depends only on the off-diagonal elements of the Hamiltonian matrices. As seen in the bottom panels, these two curves become practically indistinguishable. Notice that even at very large perturbation, the width of the energy shell, and thus of the maximal SF, is narrower than the width of the density of states (cf. Fig. \[fig:SF\] and Fig. \[fig:rhoE\]), especially for Model 2. This contradicts the equality between $P_n(E)$ and $\rho(E)$ found in previous works [@Flambaum2001a] and may be due to the fact that here the perturbation acts also along the diagonal (such effect is typically removed by considering a renormalized mf-Hamiltonian that takes into account the diagonal contributions of the perturbation). Emergence of chaotic eigenstates {#chaoticEF} -------------------------------- The energy shell determines the maximum fraction of unperturbed states that are accessible to EFs. Therefore, notions of localized ($N_{pc} \sim 1$) or delocalized ($N_{pc}\gg 1$) eigenstates make sense only with respect to the energy shell. When the perturbation is not very strong, large values of $N_{pc}$ may already be found, but in this case EFs are sparse and the components fluctuate significantly. It is only at strong perturbation that the eigenstates can fill the energy shell ergodically, becoming in this way chaotic states [@Casati1993; @Casati1996] and allowing for a statistical description of the system. In this limit, the coefficients $C^{\alpha}_{n}$ become random variables from a Gaussian distribution and $\|C^{\alpha}_{n}\|^2$ fluctuate around the envelope defined by the energy shell. 0.4 cm ![(Color online.) Eigenstates for Model 1 (left) and Model 2 (right) obtained by averaging over 5 even perturbed states in the middle of the spectrum. The average is performed after shifting the center of EFs to zero. They are shown with filled curves. Solid curves correspond to the Gaussian form of the energy shells.[]{data-label="fig:EF"}](Fig10_EFlongB.eps "fig:"){width="40.00000%"} The top panels of Fig. \[fig:EF\] show strongly localized states. For Model 2, the EFs are also sparse. The transition to extended states in the energy shell occurs again at the same critical parameters $\mu_{cr} \, , \lambda_{cr} \approx 0.5$, confirming the predictions based on the estimates obtained from $v_n/d_n$ and the Gaussian form of SFs. Notice, however, that EFs from Model 1 never become completely extended, not even for $ \mu =1.5$, although they do fill a large part of the energy shell. We may argue that EFs become chaotic-like, but not truly chaotic. This lack of ergodicity has its roots in the integrability of the system. For Model 2, on the other hand, EFs fill the shell ergodically when the perturbation is strong, being therefore truly chaotic. Distinctions between integrable and chaotic regimes are thus not captured by SFs, which are ergodic for both models when $\mu$ and $\lambda$ are large. Therefore, ergodicity in SFs implies extended but not necessarily chaotic eigenstates. By comparing EFs and SFs, it becomes evident that even though their structures should be related, since both are derived from $\|C^{\alpha}_{n}\|^2$, differences do exist. Delocalization measures {#IPR_S} ----------------------- Measures quantifying the level of delocalization of individual EFs reveal further differences between integrable and nonintegrable models. Overall larger fluctuations appear for the integrable case, which agrees with recent results obtained for bosonic and fermionic systems [@lea01; @lea]. Delocalization measures [@Izrailev1990; @ZelevinskyRep1996], such as the inverse participation ratio (IPR) or the Shannon (information) entropy S, determine the degree of complexity of individual states. For eigenstates in the mf-basis, they are respectively defined as $$\mbox{IPR}_{\alpha} \equiv \frac{1}{\sum_n \|C^{\alpha}_n\|^4} \label{IPR}$$ and $$\mbox{S}_{\alpha} \equiv -\sum_n \|C^{\alpha}_n\|^2 \ln \|C^{\alpha}_n\|^2. \label{entropyS}$$ These quantities measure how much spread the eigenstates are in the unperturbed basis. To quantify the level of delocalization of the mf-basis vectors with respect to the compound states, we may simply compute the analogous quantities $\mbox{IPR}_n$ and $\mbox{S}_n$, where the sum over $n$ in Eqs. (\[IPR\]) and (\[entropyS\]) are replaced by sums over $\alpha$. Complete delocalization occurs for GOEs, where the amplitudes $C^{\alpha}_n$ are independent random variables from a Gaussian distribution and the weights $\|C^{\alpha}_n\|^2$ fluctuate around $1/D$, $D$ being the dimension of the random matrix. The average over the ensemble leads to $\mbox{IPR}_{\text{GOE}}\sim D/3$ and $\mbox{S}_{\text{GOE}} \sim \ln(0.48 D)$ [@Izrailev1990; @ZelevinskyRep1996]. For the realistic systems considered here, since their eigenstates are confined to energy shells, the values of $\mbox{IPR}$ and $\mbox{S}$ cannot reach those of GOEs [@Kota_IPR]. Figure \[fig:S\_EF\] shows $\mbox{S}$ for the eigenstates of Model 1 (left panels) and Model 2 (right panels). As expected from the shape of the density of states (see Fig. \[fig:rhoE\]), strong mixing occurs in the middle of the spectrum, $\mbox{S}$ being smaller at the edges. Interestingly however, large values of $\mbox{S}$ are still found at the borders when the perturbation is very strong. For Model 1 this happens at high energies and for Model 2 at low energies; following the same asymmetry verified before (cf. Figs. \[fig:ham\] and \[fig:V\_df\]). As the perturbation increases from top to bottom panels in Fig. \[fig:S\_EF\], the values of $\mbox{S}$ increase and the fluctuations decrease for both models. However, this reduction is much more significant for Model 2. The smooth behavior of $\mbox{S}$ in the chaotic limit (bottom right panel) indicates that the structure of eigenstates close in energy becomes statistically very similar. This fact has suggested a close relationship between chaos and the viability of thermalization [@Deutsch1991; @Srednicki1994], as numerically explored in [@lea01; @lea]. Differences between integrable and chaotic regimes, as verified in the behavior of $\mbox{S}$ and in the spreading of EFs in the energy shell (see Fig. \[fig:EF\]), appear to have their origins in the results for the connectivity shown in Fig. \[fig:connect\]. The separated values of $M_n$ seen in the integrable system must lead to EFs with different levels of delocalization, even when close in energy. This causes larger fluctuations in the values of $\mbox{S}$. For Model 2, $M_n$’s are similar for nearby states leading to the smooth behavior of $\mbox{S}$ in the bottom right panel of Fig. \[fig:S\_EF\]. ![(Color online.) Shannon entropy for all eigenstates written in the mf-basis for Model 1 (left) and Model 2 (right).[]{data-label="fig:S_EF"}](Fig11_longSh.eps){width="45.00000%"} The level of delocalization of SFs for the basis states written in terms of the eigenstates also increases with the perturbation, while the fluctuations decrease, as shown in Fig. \[fig:S\_SF\]. Here however, the width of the fluctuations are very similar for both models. This reinforces our previous statement that the SF cannot capture differences between the two models, showing comparable behavior for both integrable and nonintegrable systems. ![(Color online.) Shannon entropy for the strength functions written in the basis of the eigenstates for Model 1 (left) and Model 2 (right).[]{data-label="fig:S_SF"}](Fig12_Sh_SF.eps){width="45.00000%"} Overlap between neighboring eigenstates --------------------------------------- We define a new signature of chaos referred to as the overlap between the probability distributions of neighboring eigenstates $\|\alpha \rangle$ and $\|\alpha' \rangle$, $$\Omega_{\alpha, \alpha'} \equiv \sum_n \|C^{\alpha}_n\|^2 \|C^{\alpha'}_n\|^2.$$ It corresponds to an alternative way to capture the transition to chaos by measuring how much similar the components of neighboring states are. For GOEs, since all eigenstates are simply normalized pseudo-random vectors, one has $\Omega \sim 1/D$. These states are completely delocalized and statistically very similar. For the models studied here, the results are presented in Fig. \[fig:overlap\] and described below. ![(Color online.) Overlaps of neighboring eigenstates for Model 1 (left panels) and Model 2 (right panels). Dark (Black) and light (red) points indicate eigenstates of even or odd parity. Horizontal (green) lines indicate the GOE prediction $\Omega = 1/D$.[]{data-label="fig:overlap"}](Fig13_overlapLOG.eps){width="45.00000%"} \(i) In the limit of localized eigenstates, large fluctuations are seen. Since there are few contributing components, we find neighboring states where the probabilities $\|C^{\alpha}_n\|^2$ are nonzero and approximately the same for the same basis vectors $\|n\rangle$, but we have also pairs where the effective basis vectors do not match. There are very correlated states leading to large overlaps, and there are also uncorrelated states leading to values of $\Omega$ below the threshold from GOEs, the values reached by Model 1 being significantly lower than for Model 2. \(ii) As the perturbation increases, and the number of principal components becomes large, the maximum values of $\Omega$ decrease for both regimes, especially in the middle of the spectrum where the mixing is stronger. The fluctuations in the values of the overlaps also decrease, especially for Model 2. For the latter, a smooth behavior with energy, similar to that obtained for the Shannon entropy for EFs, is achieved. \(iii) Notice that in the limit of strong perturbation, only Model 2 does not cross the GOE threshold. In the integrable model, since EFs do not fill the energy shell completely, we may still find neighboring states that are statistically very different. At the edges of the spectrum, the overlaps tend to be larger, since there are more correlations due to finite effects. Time evolution of the Shannon entropy: statistical relaxation {#Sec:Shannon} ============================================================= We now study the quench dynamics of the system by focusing on the time evolution of the Shannon entropy for initial states corresponding to unperturbed vectors selected from the middle of the spectrum. For an initial state $\|n_0\rangle $, the entropy in the mf-basis is given by $$S_{n_0}(t)=-\sum_{n=1}^{D_P} W_{n} (t) \ln \left[ W_{n} (t) \right] \label{entropy}$$ where $$W_{n} (t) = \langle n\| e^{-iH t} \|n_0 \rangle =\left\| \sum_{\alpha} C^{\alpha}_n C^{\alpha}_{n_0} e^{-iE_{\alpha}t} \right\|^2$$ is the probability for the initial state $\|n_0 \rangle$ to be found in the state $\|n\rangle$. Numerical data are shown in Fig. \[fig:dynamics\]. To reduce fluctuations, an average is performed over 5 initial even basis states excited in a narrow energy window in the middle of the spectrum. In the limit of strong interaction, the results for both the chaotic and the integrable models agree very well with analytical expressions previously found in the context of two-body-random ensembles [@Flambaum2001b]. These expressions can be derived when the shape of SF is known, being either Breit-Wigner or Gaussian. Analytical expressions ---------------------- We reproduce here the steps of the cascade model considered in Ref. [@Flambaum2001b] to obtain an analytical expression for the time dependence of the entropy. For very short times, $t\ll \Gamma/\sigma^2$, it has been shown that the probability for the system to remain in the initial state $\|n_0\rangle $ is [@FlambaumAust; @Flambaum2001a] $$W_{n_0}(t) \approx \exp(-\sigma^2 t^2). \label{short_time}$$ For very long times the probability becomes $$W_{n_0}(t) \approx \exp(-\Gamma t), \label{long_time}$$ which means that the decay rate from the initial state is determined by $$\frac{d W_{n_0}}{dt} = -\Gamma W_{n_0}. \label{dW0}$$ Given the two-body interaction, $\|n_0\rangle $ spreads first into $N_1$ states directly coupled to it. This set is referred to as the first class of states. Subsequently, states from the first class populate those directly coupled to them, the $N_2$ basis states from the second class. The process continues successively like this as in a cascade [@Alts]. The number of states in the $k$-th class is then $$N_k = M_k \ldots M_1 M_{n_0} \approx M_{n_0}^k, \label{calM}$$ where $M_k$ is the connectivity associated with the basis states of the $k$-th class. This implies that the number of states of one class is larger than the number in the previous class, which justifies neglecting the probability of return to a previous class and allows us to write, for $k>1$ $$\frac{d {\cal C}_k}{dt} = \Gamma {\cal C}_{k-1} - \Gamma {\cal C}_k, \label{dWk}$$ where ${\cal C}_k$ is the probability for the system to be in the $k$-th class and ${\cal C}_0=W_{n_0}$. The first term on the right-hand side is the flux from the previous class and the second term is the decay of the $k$-th class. The solution of Eq. (\[dWk\]) is $${\cal C}_k = \frac{(\Gamma t)^k}{k!} e^{-\Gamma t}. \label{classes}$$ Since each $k$ class contains several basis states, ${\cal C}_k \approx N_k W_{n} $. Assuming an infinite number of classes, Eq. (\[entropy\]) becomes $$\begin{aligned} S_{n_0}(t) &\approx& - \sum_{k=0}^{\infty} {\cal C}_k \ln \left( \frac{{\cal C}_k}{N_k} \right) \nonumber \\ &=& \Gamma t \ln M_{n_0} + \Gamma t - e^{-\Gamma t} \sum_{k=0}^{\infty} \frac{(\Gamma t)^k}{k!} \ln \frac{(\Gamma t)^k}{k!}. \nonumber\end{aligned}$$ The last terms on the right-hand side of this equation are smaller than the first term, so they may be neglected, leading to a simple linear time dependence of the Shannon entropy, $$S_{n_0}(t) \approx \Gamma t \ln M_{n_0}.$$ In the limit of strong perturbation, where $\Gamma \gtrsim \sigma$ and SF is described by a Gaussian, we can write the entropy as $$S_{n_0}(t) \approx \sigma_{n_0} t \ln M_{n_0}. \label{linear}$$ Note that Eq. (\[linear\]) depends only on the elements of the Hamiltonian matrix. Yet, as seen in Fig. \[fig:dynamics\], it reproduces very well the linear increase of the entropy for both models in the regime where the eigenstates become delocalized in the energy shell. ![(Color online) Shannon entropy vs time for Model 1 (left) and Model 2 (right). Circles stand for numerical data, dashed lines show the linear dependence (\[linear\]), and solid curves correspond to Eq. (\[S\_analytical\]). The horizontal (orange) solid lines represent the value of $\mbox{S}_{\text{GOE}} \sim 6.58$.[]{data-label="fig:dynamics"}](Fig14_time.eps){width="40.00000%"} To find an expression that describes the dynamics of the system at both short and long times, Eq. (\[short\_time\]) needs to be taken into account. In Ref. [@Flambaum2001b], the following expression was proposed, $$\begin{aligned} S_{n_0}(t) = &-& W_{n_0} (t) \ln W_{n_0}(t) \nonumber \\ &-& \left[ 1- W_{n_0} (t)\right] \ln \left( \frac{1- W_{n_0} (t)}{N_{pc}}\right), \label{S_analytical}\end{aligned}$$ where $N_{pc}$ is the total number of states inside the energy shell, that is the limiting value of the entropy after relaxation. In the results shown in Fig. \[fig:dynamics\], we obtained $N_{pc}$ numerically from $N_{pc}=\langle e^S \rangle$, where the average $\langle. \rangle$ is performed over a long time interval after the entropy saturates, $t \in [100,200]$. Equation (\[S\_analytical\]) is a good approximation when the total number of classes is small, $n_c \sim 1$. This is indeed the case for Models 1 and 2. The effective number of classes in the cascade model can be obtained from $$M^{n_c} = D_P,$$ which, following Eq. (\[connect\_1\_2\]), gives $n_c\sim1.2$ for Model 1 and $n_c \sim 1$ for Model 2. In the regime of strong perturbation, Eq. (\[S\_analytical\]) captures all stages of the evolution: the initial quadratic growth, as given by perturbation theory; the linear behavior; and the final saturation. For small perturbation, the agreement with Eq. (\[S\_analytical\]) is poor. Notice, however, that the perturbation here was not sufficiently small to show oscillations as in [@Smerzi] The main aspects of the statistical relaxation process are then the linear growth of S followed by its saturation to a value close to that of a GOE: $\mbox{S}_{\text{GOE}} \sim \ln(0.48 D)$. In the limit of strong interaction, this is the behavior of the chaotic system and, to a very good approximation, also the behavior of the integrable model. This suggests that chaoticity is not essential for the emergence of statistical relaxation. The fact that EFs of both models in the limit of large interaction show significant filling of the energy shell indicates that the existence of extended eigenstates is a sufficient condition for relaxation. However, to reach a final statement, further numerical and analytical studies of one- and two-body observables are necessary. Conclusion {#Sec:conclusion} ========== We studied static and dynamic properties of two systems of interacting spins 1/2. Model 1 is integrable for any value of the perturbation and Model 2 can transition to chaos. The analysis of the Hamiltonian matrices, later combined with studies of spectrum statistics and structures of eigenstates and strength functions, suggested that aspects of the intricate behavior of complex systems can be anticipated even before diagonalization. It was shown that strength functions and eigenstates delocalize as the perturbation increases, being, however, always restricted to the energy shell. In the limit of strong perturbation, strength functions of both models in the middle of the spectrum become Gaussian and coincide with the energy shell. In the case of eigenstates, the same occurs only for the chaotic model. For the integrable system, the eigenstates become much spread, but do not fill the energy shell completely. We verified that the lack of ergodicity of the eigenstates for the integrable model is reflected in larger fluctuations of delocalization measures and of the overlaps between neighboring eigenstates. The degree of overlaps between neighboring eigenstates may be considered as a new signature of chaos. The transition to chaos occurs when the values of the overlaps becomes inversely proportional to the dimension of the Hilbert space. We also studied the time evolution of the Shannon entropy for initial states corresponding to mean-field basis vectors. Knowledge of the shape of the strength functions allowed us to describe the quench dynamics with analytical expressions originally developed and tested for systems with two-body-random interactions. They agreed very well with our numerics. The linear growth of the entropy was also well described by an expression involving parameters obtained from the analysis of the Hamiltonian matrices before diagonalization. Our results indicate that the relaxation process is very similar for integrable and nonintegrable systems, provided the eigenstates are extended in the energy shell. On the other hand, we have seen that after saturation the fluctuations of the entropy in the integrable domain are slightly larger than for the chaotic system, as observed also in [@rigol; @Balachandran2010] in the context of observables. An issue that deserves further investigation concerns the fluctuations of static and dynamic properties. A careful analysis of how they reduce with the number of particles and how the results compare for both regimes is very important for further developments of the problem of thermalization in isolated systems. L.F.S. was supported by the NSF under grant DMR-1147430. F.B. was supported by Regione Lombardia and Consorzio Interuniversitario Lombardo per L’Elaborazione Automatica through a Laboratory for Interdisciplinary Advanced Simulation Initiative grant (2010) \[http://lisa.cilea.it\]. He also acknowledges support from Universitá Cattolica Grant No. D.2.2 2010. F.M.I. acknowledges support from Consejo Nacional de Ciencia y Tecnolgía Grant No. N-161665 and thanks Yeshiva University for the hospitality during his stay. [72]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , , , , *, (). , , (). , , (). L. F. Santos and M. Rigol, Phys. Rev. E [81]{}, 036206 (2010). , (, , ). , , , , **, (). , , , , , (). , , (). , , (). , , (). , , (). , , (). , , , , , , , , , , (). , , , , (). , , , , , , (). , , , , , (). , , (). , , , , , , , , , (). , , , , (). , , (). , , , , (). , , , , , , , , , , , (). , , , , , , , (). , , , , , , . , , (). , , , , (). , , (). , , , , (). , , (). , , , , (). , , (). , , (). , , (). , , , , , , (). , , (). , , , , , (). , , (). , (, , ). , (, , ). , , , , (). , , (). , , , , , , , (). , , (). , , (). , , (). , , (). , , (). , , (). B. L. Altshuler, Y. Gefen, A. Kamenev, and L. S. Levitov, Phys. Rev. Lett. [78]{}, 2803 (1997). G. P. Berman, F. Borgonovi, F.M. Izrailev, and A. Smerzi, Phys. Rev. Lett. [92]{}, 030404 (2004). , , , , **, ().
--- abstract: 'We provide a novel analysis of low rank tensor completion based on hypergraph expanders. As a proxy for rank, we minimize the max-quasinorm of the tensor, introduced by Ghadermarzy, Plan, and Yilmaz (2018), which generalizes the max-norm for matrices. Our analysis is deterministic and shows that the number of samples required to recover an order-$t$ tensor with at most $n$ entries per dimension is linear in $n$, under the assumption that the rank and order of the tensor are $O(1)$. As steps in our proof, we find an improved expander mixing lemma for a $t$-partite, $t$-uniform regular hypergraph model and prove several new properties about tensor max-quasinorm. To the best of our knowledge, this is the first deterministic analysis of tensor completion.' address: - 'Paul G. Allen School of Computer Science and Engineering and Department of Biology, University of Washington, Seattle, WA 98195' - 'Department of Mathematics, University of California, San Diego, La Jolla, CA 92093' author: - Kameron Decker Harris - Yizhe Zhu bibliography: - 'ref.bib' title: \| Deterministic tensor completion with\ hypergraph expanders --- Introduction ============ Matrix and tensor completion ---------------------------- Classical compressed sensing considers the recovery of high-dimensional structured signals from a small number of samples. These signals are typically represented by sparse vectors or low rank matrices. A natural generalization is to study recovery of higher-order tensors, i.e. a multidimensional array of real numbers with more than two indices, using similar low rank assumptions. However, much less is understood about compressed sensing of tensors. Matrix completion is the problem of reconstructing a matrix from a subset of entries, leveraging prior knowledge such as its rank. The sparsity pattern of observed entries can be thought of as the adjacency or biadjacency matrix of a graph, where each edge corresponds to an observed entry in the matrix. There are two general sampling approaches studied for matrix completion. During probabilistic sampling, the entries in the matrix are observed at random according to either Erdős-Rényi [@keshavan2010matrix; @recht2011simpler], random regular bipartite [@gamarnik2017matrix; @brito2018spectral] or more general graph models [@klopp2014noisy]. Deterministic sampling, on the other hand, studies precisely what kind of graphs are good for matrix completion and offers some advantages: One does not have to sample different entries for new matrices, and any recovery guarantees are deterministic without failure probability. It has been shown [@heiman2014deterministic; @bhojanapalli2014universal; @burnwal2019deterministic; @burnwal2019deterministic2] that expander graphs, which have pseudo-random properties, are a good way to sample deterministically for matrix completion. A deterministic theory of matrix completion based on graph limits, a different approach, appeared very recently [@chatterjee2019deterministic]. Tensor completion, in which we observe a subset of the entries in a tensor and attempt to fill in the unobserved values, is a useful problem with a number of data science applications [@liu2012tensor; @song2019tensor]. But fewer numerical and theoretical linear algebra tools exist for working with tensors than for matrices. For example, computing the spectral norm of tensor, its low rank decomposition, and eigenvectors all turn out to be NP-hard [@hillar2013most]. Let the tensor of interest $T$ be order-$t$, each dimension of size $n$, and have $\mathrm{rank}(T)=r$, i.e. $T \in \bigotimes_{i=1}^t {\mathbb R}^n$ (we introduce our notation more fully in Section \[sec:notation\]). A fundamental question in tensor completion is how many observations are required to guarantee a mean square error of $\varepsilon$ in our reconstruction, the [sample complexity]{}. In this paper, tensor rank will always be using the canonical polyadic (CP) decomposition [@kolda2009tensor]. The CP decomposition represents $T$ using $rnt$ numbers, thus providing a lower bound for the sample complexity. No existing results that we are aware of acheive this rate, except in the case of matrices. One way to reduce a tensor problem to a matrix problem is by flattening the tensor into a matrix. Yet current results using flattening have sample complexity at best $O \left(\frac{r n^{t/2}}{\varepsilon^2}\right)$ [@gandy2011tensor; @mu2014square; @montanari2018spectral], and $n$ is potentially very large. There are a number of papers that focus on the special case of order-3 tensors, see for example [@jain2014provable; @barak2016noisy]. Ghadermarzy, Plan, and Yilmaz [@ghadermarzy2018] recently studied tensor completion without reducing it to a matrix case by minimizing a max-quasinorm (satisfying all properties of the norm except a modified triangle inequality, which we call the “max-qnorm”) as a proxy for rank. This is defined as $$\\| T \\|_\mathrm{max} = \min_{T = U^{(1)} \circ \cdots \circ U^{(t)}} \prod_{i=1}^t \\| U^{(i)} \\|_{2,\infty},$$ where the factorization is a CP decomposition of $T$ (see Definition \[def:max-qnorm\] for further details). This is a generalization of the max-norm for matrices that many have shown yields good matrix completion results [@srebro2005rank; @rennie2005fast; @foygel2011; @heiman2014deterministic; @cai2016; @foucart2017]. Assuming that the observed entries are sampled from some probability distribution, it was shown that using the max-qnorm as a penalization results in $ O \left( \frac{nt}{\varepsilon^2} \right)$ sample complexity when $r = O(1)$, and even faster rates in $\varepsilon$ for the case of zero noise [@ghadermarzy2018]. In this paper, we perform a deterministic analysis of minimizing the max-qnorm that obtains a sample complexity which is also linear in $n$, albeit with weaker dependence on other parameters. We assume that the observed entries correspond to the edges in an expander hypergraph. Expanders and mixing -------------------- The expander mixing lemma for $d$-regular graphs (e.g. [@chungspectral]) states the following: Let $G$ be a $d$-regular graph with second largest eigenvalue in absolute value $\lambda=\max \{\lambda_2,\|\lambda_n\|\}<d$. For any two sets $V_1, V_2 \subseteq V(G)$, let $$e(V_1,V_2)=\|\{(x,y)\in V_1\times V_2: xy\in E(G) \}\|$$ be the number of edges between $V_1$ and $V_2$. Then we have that $$\begin{aligned} \label{eq:expandermixing} \left\| e(V_1,V_2)-\frac{d\|V_1\|\|V_2\|}{n}\right\| \leq \lambda \sqrt{\|V_1\| \|V_2\|\left(1-\frac{V_1}{n}\right) \left(1-\frac{V_2}{n}\right)}.\end{aligned}$$ Equation \[eq:expandermixing\] tells us regular graphs with small $\lambda$ have the [expansion property]{}, where the number of edges between any two sets is well-approximated by the number of edges we would expect if they were drawn at random. The quality of such an approximation is controlled by $\lambda$. It’s known from the Alon-Boppana bound that $\lambda\geq 2\sqrt{d-1}-o(1)$, and regular graphs that achieve this bound are called Ramanujan. Deterministic and random constructions of Ramanujan (or nearly-so) graphs have been extensively studied [@ben2011combinatorial; @marcus2015interlacing; @bilu2006lifts; @burnwal2019deterministic; @mohanty2019explicit]. Higher order, i.e. hypergraph, expanders have received significant attention in combinatorics and theoretical computer science [@lubotzky2017high]. There are several expander mixing lemmas in the literature based on spectral norm of tensors [@friedman1995second; @parzanchevski2017mixing; @cohen2016inverse; @lenz2015eigenvalues]. However, an obstacle to applying such results to tensor completion is that in most cases the second eigenvalues of tensors are unknown, even approximately. In [@dumitriu2019spectra], an expander mixing lemma similar to based on the second eigenvalue of the adjacency matrix of regular hypergraphs was derived. However, for our application we need an expander mixing lemma that estimates the number of hyperedges among $t$ different vertex subsets. One exception is the work of Friedman and Widgerson [@friedman1995second], who studied a $t$-uniform hypergraph model on $n$ vertices with $d n^{t-1}$ hyperedges chosen randomly. They proved that the second eigenvalue of the associated tensor $\lambda = O(\log^{t/2} n\sqrt{d})$. However, this is a dense random hypergraph model, since the number of edges grows superlinearly with $n$ for $t>2$. Thus Friedman and Widgerson’s model only applies when one has the ability to make many measurements, as opposed to the more realistic “big data” scenario constrained to $O(n)$ observations. If we sample the original tensor according to the hyperedge set of a hypergraph, we would like the number of hyperedges to be small, in order to be able to represent a small number of samples. From previous results on matrix completion, we expect the reconstruction error should be controlled by a parameter that is related to the expansion property of the hypergraph. Main result {#thm:main} ----------- In this paper, we revisit the sparse, deterministic hypergraph construction introduced in [@bilu2004codes]. We construct the hypergraph by taking a $d$-regular “base” graph and forming a hypergraph from its walks of length $t$. In this model, each node is of degree $d^{t-1}$, corresponding to $nd^{t-1}$ samples. An advantage of our expander mixing result is that the expansion property of the hypergraph is controlled by the expansion in a $d$-regular graph (Theorem \[thm:mixing\]). This is easy to compute and optimize using known constructions of $d$-regular expanders. Based on such hypergraphs, we perform a deterministic analysis of an optimization problem similar to that analyzed by Ghadermarzy et al. [@ghadermarzy2018]. Our proof is based on the techniques used to study matrix completion in [@heiman2014deterministic] (see also [@brito2018spectral]). To have a theoretical guarantee of our algorithms, we prove several useful linear algebra facts about the max-quasinorm for tensors, which may be of separate interest. Our main result is: \[thm:main\] Given a hypercubic tensor $T$ of order-$t$, sample its entries according to a $t$-partite, $t$-uniform, $d^{t-1}$-regular hypergraph $H=(V,E)$ constructed from a $d$-regular graph $G$ of size $n$ with second eigenvalue (in absolute value) $\lambda\in (0,d)$. Then solving $$\hat{T} = \arg \min_{T'} \\| T' \\|_\mathrm{max} \mbox{\quad such that \quad $T'_{e} = T_{e}$ for all $e \in E$}$$ will result in the following mean squared error bound: $$\begin{aligned} \label{eq:errorbound} \frac{1}{n^t} \\|\hat{T}-T\\|_F^2 \leq C_t \\| T \\|_\mathrm{max}^2 \left(\left(1 + \frac{\lambda}{d} \right)^{t-1} - 1 \right),\end{aligned}$$ where $C_t \leq 2^{2t-2} c_1 c_2^t$, $c_1 \leq \frac{K_G}{5} \leq 0.36$, and $c_2 \leq 2.83$. ### Sample complexity Recall that the number of edges in $H$ is $n d^{t-1}$, equal to the number of samples. Suppose we have an expander graph $G$, where $\lambda=O( \sqrt{d})$. In order to guarantee reconstruction error $\varepsilon$, Theorems \[thm:main\] and \[thm:max-rank\] say that, assuming $r, t = O(1)$, we require $\displaystyle O \left( \frac{n}{\varepsilon^{2(t-1)}}\right)$ samples. The computations are shown in Section \[sec:sample-complexity\]. ### Dependence on rank and order Tensor completion should require at least $rnt$ samples, yet no existing bounds come close to this sample complexity. In Theorem \[thm:main\], we unfortunately have that the constant $C_t$ depends exponentially on the tensor order $t$. The dependence on rank is also exponential, since the best known dependence of the max-qnorm on rank is $\\| T \\|_\mathrm{max} = O (\sqrt{r^{t(t-1)}})$ (see Theorem \[thm:max-rank\] and [@ghadermarzy2018]). However, for matrices we have that $\\| T \\|_\mathrm{max} = O(\sqrt{r})$, which is tighter than the previous bound with $t=2$. So it could be that a better understanding of the max-qnorm for tensors will lead to better dependence on the rank. Organization of the paper ------------------------- In Section \[sec:construction\], we recall the construction of the $t$-partite, $t$-uniform hypergraph introduced in [@bilu2004codes]. In Section \[sec:expander\], we prove an expander mixing lemma for such hypergraphs. In Section \[sec:property\], we prove several useful properties of max-quasinorm for tensors. In Section \[sec:completion\], we leverage these properties to analyze the above tensor completion algorithm and prove the main result. We extend our result for tensor completion with errors in the observed entries, which can model noise or adversarial corruptions. We conclude with a discussion of limitations and future directions in Section \[sec:discussion\]. Notation {#sec:notation} -------- Tensor notation is not standard throughout the literature, which can lead to some confusion. The notations we use throughout the paper come from the review by [@kolda2009tensor], and we refer the reader there for a thorough description. In brief, we use lowercase symbols $u$ for vectors, uppercase $U$ for matrices and tensors. The symbol “$\circ$” denotes the outer product of vectors, i.e. $T= u \circ v \circ w$ denotes the order-3, rank-1 tensor with entry $T_{i,j,k} = u_i v_j w_k$. We also use this symbol for the outer product of matrices as appears in the rank-$r$ decomposition of a tensor $T = U^{(1)} \circ U^{(2)} \circ U^{(3)}$, where each matrix $U^{(i)}$ has $r$ columns, so that $T_{i,j,k} = \sum_{l=1}^r U^{(1)}_{i,l} \, U^{(2)}_{j,l} \, U^{(3)}_{k,l}$, and $T = \bigcirc_{i=1}^t U^{(i)}$ is shorthand for the same order-$t$, rank-$r$ tensor. The symbols “$\otimes$” and “$$” denote Kronecker and Hadamard products, respectively, which will be defined in Section \[sec:property\]. We use $\bigotimes_{i=1}^t {\mathbb R}^{n_i}$ for the space of all order-$t$ tensors with $n_i$ entries in the $i$th dimension. We use $1_A \in {\mathbb R}^n$ as the indicator vector of a set $A \subseteq [n]$, i.e. $(1_A)_i = 1$ if $i \in A$ and 0 otherwise. For any order-$t$ tensor $T\in \bigotimes_{i=1}^t {\mathbb R}^{n_i}$ and subsets $I_i\subseteq [n_i]$, denote $T_{I_1,\dots,I_t}$ to be the subtensor restricted on the index set $I_1\times \cdots \times I_t.$ Construction of hypergraph expanders {#sec:construction} ==================================== We start with the definition of a hypergraph and some basic properties: A hypergraph* $H$ consists of a set of vertices $V$ and a set of hyperedges $E$, where each hyperedge is a nonempty set of $V$, the vertices that participate in that hyperedge. The hypergraph $H$ is $t$-uniform for an integer $t\geq 2$ if every hyperedge $e\in E$ contains exactly $t$ vertices. The degree of vertex $i$ is the number of all hyperedges containing $i$. A hypergraph is $d$-regular if all of its vertices have degree $d$. Let $G=(V(G), E(G))$ be a connected $d$-regular graph on $n$ vertices with second largest eigenvalue (in absolute value) $\lambda\in (0,d)$. We construct a $t$-partite, $t$-uniform, $d^{t-1}$-regular hypergraph $H=(V,E)$ from $G$ as follows: Let $V=V_1\cup V_2\cup \cdots \cup V_t$ be the disjoint union of $t$ vertex sets which are copies of $V(G)$, so that $\|V_1\| = \cdots = \|V_t\|=n$. To be precise, we label all the vertices by $v_{i}^j\in V_i$ for all $1\leq i\leq t$ and $1\leq j\leq n$. The hyperedges of $H$ correspond to all walks of length $t-1$ in $G$. Therefore $\{v_1^{i_1},\dots, v_t^{i_t}\}$ is a hyperedge in $H$ if and only if $(i_1,\dots, i_{t})$ is a walk of length $t-1$ in $G$. The vertices in $(i_1,\dots, i_{t})$ may be repeated, since in the hypergraph they will occur in different partitions and correspond to different copies of the vertices in the base graph. We now introduce a slight abuse of notation that should make things less cumbersome, hopefully without creating confusion. Because our hypergraph $H$ is $t$-uniform and $t$-partite, we will identify each hyperedge $e = \{v_1^{i_1},\dots, v_t^{i_t}\} \in E(H)$ with an ordered $t$-tuple $e=(i_1,\dots, i_t)$. The reader should keep in mind that the identity of the vertex in the hypergraph corresponding to an entry in the tuple depends on its position and value. For a tensor $T \in \otimes_{i=1}^r {\mathbb R}^n$, this allows us to use the edge as shorthand for the multi-index, for example $T_e = T_{i_1, \ldots, i_t}$. Given the description above, we have $\|V\|=nt$ and $\|E\|=n d^{t-1}$, since $E$ contains all possible walks of length $t-1$ in $G$. Moreover, every vertex is contained in exactly $d^{t-1}$ many hyperedges, so $H$ is regular. From our definition of the hyperedges in $H$, the order of the walk in $G$ matters. For example, two walks $i_1 \to i_2 \to i_3$ and $i_3 \to i_2 \to i_1$ correspond to different hyperedges $(i_1,i_2,i_3)$ and $(i_3,i_2,i_1)$ in $H$ when $i_1\not=i_3$. When $t=2$, $H$ is a bipartite $d$-regular graph with $2n$ vertices. See Figure \[fig:hyp3\] for an example of the construction with $t=3$. ![An example hyperedge in the $t=3$ case: We depict the base graph $G$ on the left and a single edge in the hypergraph $H$ on the right. The set of vertices $\{v_1^{i_1}, v_2^{i_2}, v_3^{i_3} \}$ forms an hyperedge $e$ in $H$ if and only if $(i_1,i_2,i_3)$ is a walk in $G$. For convenience, we denote that hyperedge with the tuple $(i_1,i_2,i_3)$. []{data-label="fig:hyp3"}](hypergraph.pdf){width="0.8\linewidth"} This construction was used by Alon et al. [@alon1995derandomized] to de-randomize graph products and by Bilu and Hoory [@bilu2004codes] to construct error correcting codes. Both groups’ results depended on expansion. Bilu and Hoory introduced a kind of hypergraph expansion property which they called $\varepsilon$-homogeneity. Here we prove a novel and tighter expansion property in this hypergraph and apply it to the tensor completion problem. Expander mixing {#sec:expander} =============== Let $G$ be a $d$-regular graph on $n$ vertices with second largest eigenvalue $0 < \lambda < d$, and let $H$ be the corresponding $t$-partite, $t$-uniform hypergraph constructed as in Section \[sec:construction\]. We get the following mixing lemma for $H$. The mixing rate is essentially controlled by the second eigenvalue of the $d$-regular graph $G$. Given a base graph $G$, form the hypergraph $H$ following the construction in Section \[sec:construction\]. Let $W_i\subseteq V_i$, $1\leq i\leq t$ be any non-empty subsets. Denote $$\alpha_i:=\frac{\|W_i\|}{n}\in (0,1],\quad 1\leq i\leq t,$$ and let $$e(W_1,\dots, W_t):=\left\|\{(v_1,\dots, v_t)\in V_1\times \cdots \times V_t: (v_1,\dots, v_t) \in E(H)\}\right\|$$ be the number of hyperedges between $W_1,\dots, W_t$. Then we have the following expansion property: \[thm:mixing\] $$\begin{aligned} \label{eq:mixing} \frac{e(W_1,\dots, W_t)}{nd^{t-1}} &\geq \prod_{i=1}^t\alpha_i \cdot \prod_{i=1}^{t-1} \left(1- \frac{\lambda}{d} \sqrt{ \frac{(1-\alpha_i)(1-\alpha_{i+1})}{\alpha_{i}\alpha_{i+1}} } \right), \\ \frac{e(W_1,\dots, W_t)}{nd^{t-1}} & \leq \prod_{i=1}^t\alpha_i \cdot \prod_{i=1}^{t-1} \left(1+ \frac{\lambda}{d} \sqrt{ \frac{(1-\alpha_i)(1-\alpha_{i+1})}{\alpha_{i}\alpha_{i+1}} } \label{eq:bilumixing} \right). \end{aligned}$$ When $t=2$, Theorem \[thm:mixing\] reduces to the expander mixing lemma for bipartite regular graphs, see [@haemers1995interlacing; @de2012large; @brito2018spectral]. A weaker bound similar to was obtained in Lemma 3 of [@bilu2004codes]. The inequality in our Theorem \[thm:mixing\] is two-sided, and the proof is simpler. An estimate similar to is obtained in [@alon1995derandomized] under more restricted assumptions that $W_1,\dots, W_t$ have the same size and $\alpha_i\geq \frac{6\lambda}{d}$. We removed these technical assumptions here. Let $(X_1,\dots, X_t)$ be a simple random walk of length $t-1$ on the $d$-regular graph $G=(V(G),E(G))$ defined as follows. Let $X_1$ be a uniformly chosen starting vertex from $V(G)$, and for $2\leq i\leq t$ let $X_i$ be a uniformly chosen neighbor of $X_{i-1}$ in the graph $G$. Then from the construction of $H$, and the definition of simple random walks on graphs, we have $$\begin{aligned} \label{eq:mix_prop} \frac{e(W_1,\dots, W_t)}{nd^{t-1}} = \mathbb P(X_1\in W_1,\dots, X_t\in W_t).\end{aligned}$$ By the Markov property of simple random walks, $$\begin{aligned} &\mathbb P(X_1\in W_1,\dots, X_t\in W_t) \notag\\ &\quad = \mathbb P(X_1\in W_1)\cdot \mathbb P (X_2\in W_2\mid X_1\in W_1)\cdots \mathbb P(X_t\in W_t \mid X_{t-1}\in W_{t-1}) \nonumber \\ &\quad = \frac{\|W_1\|}{n} \prod_{i=1}^{t-1} \frac{e(W_i,W_{i+1})}{d\|W_i\|}=\alpha_1 \prod_{i=1}^{t-1} \frac{e(W_i,W_{i+1})}{d\|W_i\|} . \label{eq:mix_mid}\end{aligned}$$ By the expander mixing lemma for $G$ from , we have that $$\left\|e(W_i,W_{i+1})-\frac{d\|W_i\|\|W_{i+1}\|}{n}\right\| \leq \lambda \sqrt{\|W_i\|\|W_{i+1}\|\left(1-\frac{\|W_i\|}{n}\right)\left(1- \frac{\|W_{i+1}\|}{n}\right)}.$$ Expanding the absolute value and dividing through by $d \|W_i\|$ gives $$\begin{aligned} \alpha_{i+1}-\frac{\lambda}{d}\sqrt{\frac{\alpha_{i+1}}{\alpha_i}(1-\alpha_i)(1-\alpha_{i+1})} &\leq \frac{e(W_i,W_{i+1})}{d\|W_i\|} \leq \alpha_{i+1}+\frac{\lambda}{d}\sqrt{\frac{\alpha_{i+1}}{\alpha_i}(1-\alpha_i)(1-\alpha_{i+1})}, \nonumber \\ \alpha_{i+1} \left( 1- \frac{\lambda}{d} \sqrt{ \frac{(1-\alpha_i)(1-\alpha_{i+1})}{\alpha_{i+1}\alpha_i} } \right) &\leq \frac{e(W_i,W_{i+1})}{d\|W_i\|} \leq \alpha_{i+1} \left( 1 + \frac{\lambda}{d} \sqrt{ \frac{(1-\alpha_i)(1-\alpha_{i+1})}{\alpha_{i+1}\alpha_i} } \right). \label{eq:mix_bound}\end{aligned}$$ Therefore, combining , , and , we obtain and . Theorem \[thm:mixing\] bounds the ratio between $\displaystyle \frac{e(W_1,\dots, W_t)}{nd^{t-1}}$ and $\displaystyle \prod_{i=1}^t \frac{\|W_i\|}{n}$. The following lemma controls the difference between these two quantities, which will be more useful in our analysis of deterministic tensor completion in Section \[sec:completion\]. Let $H$ be the $t$-partite, $t$-uniform hypergraph defined in Section \[sec:construction\]. We have for any $W_i \subseteq V_i, 1\leq i\leq n$, \[cor:mixing\] $$\begin{aligned} \left\| \frac{e(W_1, \ldots, W_t)}{nd^{t-1}} - \prod_{i=1}^t \alpha_i \right\| \leq \left( \left(1 + \frac{\lambda}{d} \right)^{t-1} - 1 \right) \min \left\{ \frac{1}{4}, \prod_{i=1}^t \max \left\{\sqrt\alpha_i, \sqrt{1-\alpha_i} \right\} \right\}.\end{aligned}$$ If any $\alpha_i = 0$, then the discrepancy is zero and the bound holds trivially. Therefore, we can assume that all $\alpha_i \in (0, 1]$. For convenience, define $ \beta_i := \sqrt{ \frac{(1-\alpha_i) (1- \alpha_{i+1})}{\alpha_i \alpha_{i+1}}}. $ Then and imply $$\begin{aligned} &\left\| \frac{e(W_1, \ldots, W_t)}{nd^{t-1}} - \prod_{i=1}^t \alpha_i \right\|\\ &\quad \leq \prod_{i=1}^t\alpha_i \cdot \max \left\{ 1- \prod_{i=1}^{t-1} \left( 1 - \frac{\lambda}{d} \beta_i \right) , \prod_{i=1}^{t-1} \left( 1 + \frac{\lambda}{d} \beta_i \right) - 1 \right\}\\ &\quad = \prod_{i=1}^t\alpha_i \cdot \left[ \prod_{i=1}^{t-1} \left( 1 + \frac{\lambda}{d} \beta_i \right) - 1 \right] \\ &\quad = \prod_{i=1}^t \alpha_i \cdot \left[ \frac{\lambda}{d} \sum_{1\leq i_1\leq t-1} \beta_{i_1} + \left( \frac{\lambda}{d} \right)^2 \sum_{1\leq i_1 < i_2\leq t-1} \beta_{i_1} \beta_{i_2} + \ldots + \left( \frac{\lambda}{d} \right)^{t-1} \beta_1 \cdots \beta_{t-1} \right] .\end{aligned}$$ After expanding the product, we have a number of terms involving the $\alpha_i$ and $\beta_i$: $$\begin{aligned} &\left(\prod_{i=1}^t \alpha_i\right) \beta_{i_1} = \alpha_1 \cdots \alpha_{i_1-1} \sqrt{\alpha_{i_1} \alpha_{i_1+1} (1 -\alpha_{i_1})(1- \alpha_{i_1+1})} \alpha_{i_1+2} \cdots \alpha_{t}, \\ &\left(\prod_{i=1}^t \alpha_i \right) \beta_{i_1} \beta_{i_2} = \alpha_1 \cdots \alpha_{i_1-1} \sqrt{\alpha_{i_1} \alpha_{i_1+1} (1 -\alpha_{i_1})(1- \alpha_{i_1+1})} \alpha_{i_1+2} \cdots \\ & \hspace{4cm} \cdot \alpha_{i_2-1} \sqrt{\alpha_{i_2} \alpha_{i_2+1} (1 -\alpha_{i_2})(1- \alpha_{i_2+1})} \alpha_{i_2+2} \cdots \alpha_{t} , \\ & \qquad \vdots \\ & \left(\prod_{i=1}^t \alpha_i \right) \beta_1 \cdots \beta_{t-1} = \sqrt{\alpha_1 (1-\alpha_1) (1-\alpha_2) \cdots (1-\alpha_{t-1}) (1-\alpha_t) \alpha_t}.\end{aligned}$$ Because $\sqrt{x (1-x) } \leq 1/2$ for $x \in [0,1]$, each of these terms is bounded above by 1/4. On the other hand, we can also bound each term by $\prod_{i=1}^t \max \left\{\sqrt \alpha_i, \sqrt{1-\alpha_i} \right\}$. Let $$C := \min \left\{ \frac{1}{4}, \prod_{i=1}^t \max \left\{\sqrt \alpha_i, \sqrt{1-\alpha_i} \right\} \right\}$$ be the minimum of these two bounds. Taking into account their multiplicity, we find that $$\begin{aligned} \left\| \frac{e(W_1, \ldots, W_t)}{nd^{t-1}} - \prod_{i=1}^t \alpha_i \right\| &\leq C \left[ \frac{\lambda}{d} \binom{t-1}{1} + \left( \frac{\lambda}{d} \right)^2 \binom{t-1}{2} + \ldots + \left( \frac{\lambda}{d} \right)^{t-1} \binom{t-1}{t-1} \right] \\ &= C \left[ \left(1 + \frac{\lambda}{d} \right)^{t-1} - 1 \right]. \end{aligned}$$ This finishes the proof. Tensor complexity {#sec:property} ================= In order to complete a partially observed matrix or tensor, some kind of prior knowledge of its structure is required. The tensor that is output by the learning algorithm will then be the least complex one that is consistent with the observations. Consistency may be defined as either exactly matching the observed entries—in the case of zero noise—or being close to them under some loss—in the case where the observations are corrupted. We now argue for the use of the tensor max-quasinorm (see Definition \[def:max-qnorm\] below) as a measure of complexity. Towards this aim, we also show a number of previously unknown properties about the max-quasinorm. For matrices, the most common measure of complexity is the rank. In the tensor setting, there are various definitions of rank (see [@kolda2009tensor]). However, in this paper we will work with the rank defined via the CP decomposition as $$\mathrm{rank}(T) = \min \left\{ r \; \Big\| \; T = \sum_{i=1}^r u^{(1)}_i \circ \cdots \circ u^{(t)}_i \right\} , \label{eq:tensor_rank}$$ where each of the vectors $u_i^{(j)} \in {\mathbb R}^n$. Note that the decomposition above is atomic and equivalent to the decomposition used to define matrix rank when $t=2$. The sum is composed of $r$ rank-1 tensors expressed as the outer products $u_i^{(1)} \circ \cdots \circ u_i^{(t)}$. Our analysis uses Kronecker and Hadamard products of tensors. These are generalizations of the usual Kronecker and Hadamard products of matrices in the obvious way. Let $T \in \bigotimes_{i=1}^t {\mathbb R}^{n_i}$ and $S \in \bigotimes_{i=1}^t {\mathbb R}^{m_i}$. We define the [Kronecker product]{} of two tensors $(T \otimes S) \in \bigotimes_{i=1}^t {\mathbb R}^{n_i m_i}$ as the tensor with entries $$(T \otimes S)_{k_1, \ldots, k_t} = T_{i_1, \ldots, i_t} S_{j_1, \ldots, j_t}$$ for $ k_1 = j_1 + m_1 (i_1 - 1) , \ldots, k_t = j_t + m_t (i_t - 1).$ Let $T \in \bigotimes_{i=1}^t {\mathbb R}^{n_i}$ and $S \in \bigotimes_{i=1}^t {\mathbb R}^{n_i}$. We define the [Hadamard product]{} of two tensors $(T S) \in \bigotimes_{i=1}^t {\mathbb R}^{n_i}$ as the tensor with indices $ (T S)_{i_1, \ldots, i_t} = T_{i_1, \ldots, i_t} S_{i_1, \ldots, i_t} $. Max-qnorm --------- The max-norm of a matrix (also called $\gamma_2$-norm) is a common relaxation of rank. It was originally proposed in the theory of Banach spaces [@tomczak-jaegermann1989], but has found applications in communication complexity [@linial2007; @lee2008; @matousek2014] and matrix completion [@srebro2005rank; @rennie2005fast; @foygel2011; @heiman2014deterministic; @cai2016; @foucart2017]. For a matrix $A$, the max-norm of $A$ is defined as $$\\| A \\|_{\max} := \min_{U,V: A = U V^{\top}} \\| U \\|_{2,\infty} \\| V \\|_{2,\infty} .$$ We can generalize its definition to tensors, following [@ghadermarzy2018], with the caveat that it then becomes a quasinorm since the triangle inequality is not satisfied. \[def:max-qnorm\] We define the [max-quasinorm]{} (or [max-qnorm]{}) of an order-$t$ tensor $T\in \bigotimes_{i=1}^t \mathbb R^{n_i}$ as $$\\| T \\|_{\max} = \min_{T = U^{(1)} \circ \cdots \circ U^{(t)}} \prod_{i=1}^t \\| U^{(i)} \\|_{2,\infty} ,$$ where $$\\| U \\|_{2,\infty} = \max_{\\|x\\|_2=1} \\| U x\\|_\infty,$$ i.e. the maximum $\ell^2$-norm of any row of $U$, and each of the $U^{(i)} \in {\mathbb R}^{n_i \times r}$ for some $r$. The following lemma provides some basic properties of the max-quasinorm for tensors. \[lem:max-qnorm\] Let $t \geq 2$, then any two order-$t$ tensors $T$ and $S$ of the same shape satisfy the following properties: 1. $ \\|T \\|_\mathrm{max} = 0$ if and only if $T = 0$. 2. $\\| c T \\|_\mathrm{max} = \|c\| \\| T \\|_\mathrm{max}$, where $c \in {\mathbb R}$. 3. $ \\| T + S \\|_\mathrm{max} \leq \left( \\| T \\|_\mathrm{max}^{2/t} + \\|S\\|_\mathrm{max}^{2/t} \right)^{t/2} \leq 2^{t/2 - 1} \left( \\|T\\|_\mathrm{max} + \\|S\\|_\mathrm{max} \right)$. Note that property (3) in Lemma \[lem:max-qnorm\] implies that $\\| \cdot \\|_\mathrm{max}$ is a so-called $p$-norm with $p=2/t$ and also a a quasinorm with constant $2^{t/2-1}$ [@dilworth1985]. Finally, in the matrix case it is a norm. As a matrix norm, many properties and equivalent definitions of the max-norm are known, and it can be computed via semidefinite programming [@linial2007; @lee2008; @matousek2014]. In the tensor case, much less is known about the max-qnorm. We now prove generalizations of some of these properties that hold for tensors. \[thm:max-qnorm-properties\] Let $T \in \bigotimes_{i=1}^t {\mathbb R}^{n_i}$ and $S \in \bigotimes_{i=1}^t {\mathbb R}^{m_i}$. The following properties hold for the max-qnorm: 1. $ \\| T_{I_1, \ldots, I_t} \\|_\mathrm{max} \leq \\| T \\|_\mathrm{max}$ for any subsets of indices $I_i \subseteq [n_i]$. 2. $\\| T \otimes S \\|_\mathrm{max} \leq \\| T \\|_\mathrm{max} \\| S \\|_\mathrm{max}$. 3. $\\| T S \\|_\mathrm{max} \leq \\| T \otimes S \\|_\mathrm{max}$, where $T,S\in \bigotimes_{i=1}^t {\mathbb R}^{n_i}$. 4. $ \\| T T \\|_\mathrm{max} \leq \\| T \\|_\mathrm{max}^2.$ We prove claims (1–4) in order. For claim (1), write $T$ using the decomposition which attains the max-qnorm, $T = \bigcirc_{i=1}^t U^{(i)}$ for some $U^{(i)} \in {\mathbb R}^{n_i \times r}, 1\leq i\leq t$. Then a single entry of $T$ can be written as $$T_{i_1, \ldots, i_t} = \sum_{i=1}^r u^{(1)}_{i_1, i} \cdots u^{(t)}_{i_t, i},$$ and we have that $$T_{I_1, \ldots, I_t} = \bigcirc_{i=1}^t U^{(i)}_{I_i, :},$$ where $U^{(i)}_{I_i,:}$ denotes the submatrix of $U^{(i)}$ with the column restricted on $I_i$. Since the norm $\\| A \\|_{2,\infty}$ is non-increasing under removing any rows of a matrix $A$, $$\\|U_{I_i,;}^{(i)}\\|_{2,\infty}\leq \\|U^{(i)}\\|_{2,\infty}.$$ Since $T_{I_1, \ldots, I_t}$ can be factored by selecting subsets of the rows of $U^{(i)}$, by the definition of max-qnorm, we have $$\begin{aligned} \\|T_{I_1,\dots,I_t}\\|_{\max}\leq \prod_{i=1}^t \\|U_{I_i,;}^{(i)}\\|_{2,\infty}\leq \prod_{i=1}^t\\|U^{(i)}\\|_{2,\infty}=\\|T\\|_{\max}.\end{aligned}$$ This proves claim (1). For claim (2), let $$T = \bigcirc_{i=1}^t T^{(i)} \quad \text{ and }\quad S = \bigcirc_{i=1}^t S^{(i)}$$ be the rank $r_1$ and $r_2$ decompositions of $T$ and $S$ that attain their max-qnorms. Then since $$\begin{aligned} (T \otimes S)_{k_1, \ldots, k_t} &= T_{i_1, \ldots, i_t} S_{j_1, \ldots, j_t} \\ &= \left( \sum_{l=1}^{r_1} T^{(1)}_{i_1, l} \cdots T^{(t)}_{i_t, l} \right) \left( \sum_{l'=1}^{r_2} S^{(1)}_{j_1, l'} \cdots S^{(t)}_{j_t, l'} \right) \\ &= \sum_{l=1}^{r_1} \sum_{l'=1}^{r_2} \left( T^{(1)}_{i_1, l} S^{(1)}_{j_1, l'} \right) \cdots \left( T^{(t)}_{i_t, l} S^{(t)}_{j_t, l'} \right) \\ &= \sum_{p=1}^{r_1 r_2} \left( T^{(1)} \otimes S^{(1)} \right)_{k_1, p} \cdots \left( T^{(t)} \otimes S^{(t)} \right)_{k_t, p}\end{aligned}$$ for $k_s = j_s + m_s (i_s - 1)$ for all $s = 1, \ldots, t$ and $p = l' + r_2 ( l - 1)$, we have that $$T \otimes S = \bigcirc_{i=1}^t (T^{(i)} \otimes S^{(i)}).$$ Note that any matrices $A \in {\mathbb R}^{m \times n}$ and $B \in {\mathbb R}^{p \times q}$, $$\begin{aligned} \label{eq:matrix2infinity} \\|A \otimes B\\|_{2,\infty} = \\|A \\|_{2,\infty} \\|B\\|_{2,\infty} .\end{aligned}$$ To see this, assume without loss of generality that the rows of $A$ and $B$ with greatest $\ell^2$-norm are the first (all combinations of rows occur in the Kronecker product). Then the first row of $A \otimes B$ will have the largest $\ell^2$-norm of all rows in that matrix; call it $x$. Therefore, $\\| A \otimes B \\|_{2,\infty}^2 = \\| x \\|_2^2$, and $$\\| x \\|_2^2 = \sum_{i=1}^n \sum_{j=1}^q (A_{1,i} B_{1,j})^2 = \sum_{i=1}^n A_{1,i}^2 \\| B_{1,:} \\|_2^2 = \\| A_{1,:} \\|_2^2 \, \\| B_{1,:} \\|_2^2 = \\|A \\|_{2,\infty}^2 \\|B\\|_{2,\infty}^2 .$$ This implies that $\\| T^{(i)} \otimes S^{(i)} \\|_{2,\infty} = \\| T^{(i)} \\|_{2,\infty} \\| S^{(i)} \\|_{2,\infty}$ for $1\leq i\leq t$, therefore $$\begin{aligned} \\|T\otimes S\\|_{\max}\leq \prod_{i=1}^t \\|T^{(i)}\otimes S^{(i)}\\|_{2,\infty}=\prod_{i=1}^t\left(\\| T^{(i)} \\|_{2,\infty} \\| S^{(i)} \\|_{2,\infty}\right)=\\|T\\|_{\max}\\|S\\|_{\max}. \end{aligned}$$ This completes the proof of claim (2). For claim (3), note that every entry in $T S$ appears in $T \otimes S$, since $$(T S)_{i_1, \ldots, i_t} = (T \otimes S)_{i_1 + n_1 (i_1 - 1), \ldots, i_t + n_t(i_t - 1)} .$$ So we have that $T S = (T \otimes S)_{I_1, \ldots, I_t}$ for some subsets of indices $I_1,\dots, I_t$, and by claim (1) the result follows. Finally, since from claim (2) and (3), $$\begin{aligned} \\|TT\\|_{\max}\leq \\|T\otimes T\\|_{\max}\leq \\|T\\|_{\max}^2,\end{aligned}$$ claim (4) follows. This completes the proof. Part (2) of Theorem \[thm:max-qnorm-properties\] is known to hold with equality for matrices [@lee2008]. However, the proof of that uses an alternative representation of max-norm as $$\\| A \\|_\mathrm{max} = \max_{B: \\| B \\| = 1} \\| A * B \\| ,$$ where $\\| \cdot \\|$ is the spectral norm. This identity was obtained via semi-definite programming. It is not obvious whether an analogous identity exists in the tensor setting. Part (3) and (4) of Theorem \[thm:max-qnorm-properties\] are mentioned in the literature a number of times without explicit proof. These inequalities are likely quite loose, since the Kronecker product tensor is much larger than the Hadamard product. A better bound may be possible which takes into account the ranks of $T$ and $S$. In the matrix case, there is a surprising relationship between max-norm and rank. For any matrix $A$, $$\label{eq:max-rank-matrix} \\| A \\|_{1, \infty} \leq \\| A \\|_\mathrm{max} \leq \sqrt{\mathrm{rank}(A)} \cdot \\| A \\|_{1,\infty},$$ which does not depend on the size of $A$. (Recall that $\\|A \\|_{1,\infty} = \max_{i,j} \| A_{i,j} \|$.) The proof is a result of John’s theorem, and is given in [@linial2007]. For the tensor generalization, the best similar bound so far is given by the following Theorem. \[thm:max-rank\] Let $T \in \bigotimes_{i=1}^t {\mathbb R}^n$ with $\mathrm{rank}(T) = r$. Then we have $$\max_{i_1, \ldots, i_t} \|T_{i_1,\ldots, i_t}\| \leq \\| T \\|_\mathrm{max} \leq \sqrt{r^{t(t-1)}} \max_{i_1, \ldots, i_t} \|T_{i_1,\ldots, i_t}\|.$$ From Theorem \[thm:max-rank\], if we take $t=2$ we get that $\\| A \\|_\mathrm{max} \leq \mathrm{rank}(A) \cdot \\|A\\|_{1,\infty}$, which is worse than . It remains an open question whether a better bound exists for all $t \geq 2$ that reduces to for $t=2$. The numerical experiments of Ghadermarzy et al.[@ghadermarzy2018], which used a bisection method to estimate the max-qnorm of tensors of known rank, suggest that an improvement is possible. They study tensors formed from random factors, finding that increasing $t$ by one leads to approximately $\sqrt{r}$ increase. This suggests the conjecture that perhaps $ \\|T \\|_\mathrm{max} \leq \sqrt{r^{t-1}} \max_{i_1, \ldots, i_t} \|T_{i_1,\ldots, i_t}\| $ is the correct bound in terms of $r$. In any case, Theorem \[thm:max-rank\] is still useful for low rank tensor completion, as it implies that an upper bound on the generalization error in terms of the max-qnorm can be translated into a bound that depends on the rank. That upper bound does not depend on $n$, which is crucial for attaining sample complexity linear in $n$. We have found an improved lower bound on $\\|T\\|_{\max}$ via tensor matricization, sometimes called tensor unfolding or tensor flattening, defined as follows. For more details, see [@kolda2009tensor]. Let $T \in \bigotimes_{i=1}^t {\mathbb R}^{n_i}$. For $1\leq i\leq t$, the mode-$i$ matricization of $T$ is a matrix denoted by $T_{[i]} \in {\mathbb R}^{n_i} \times {\mathbb R}^{\prod_{j \neq i} n_j}$ such that for any index $(j_1,\dots, j_t)$, $$\left(T_{[i]}\right)_{j_i,k}=T_{j_1,\dots, j_t},$$ with $$k=1+\sum_{s=1,s\not=i}^t (j_s-1)N_s \quad \text{and}\quad N_s=\prod_{m=1,m\not=i}^{s-1} n_m.$$ Let $T \in \bigotimes_{i=1}^t {\mathbb R}^{n_i}$. Then $$\\| T \\|_\mathrm{max} \geq \max_{1\leq i\leq t}\\| T_{[i]} \\|_\mathrm{max}\geq \max_{i_1, \ldots, i_t} \|T_{i_1,\ldots, i_t}\|.$$ Consider the rank-$r$ factorization $T = \bigcirc_{i=1}^{t} U^{(i)}$ that attains the max-qnorm. Then, $$\begin{aligned} \\|T \\|_\mathrm{max} &= \\| U^{(1)} \\|_{2,\infty} \\| U^{(2)} \\|_{2,\infty} \cdots \\| U^{(t)} \\|_{2,\infty} \nonumber \\ &= \\| U^{(1)} \\|_{2,\infty} \\| U^{(2)} \otimes \cdots \otimes U^{(t)} \\|_{2,\infty} \nonumber \\ &\geq \\| U^{(1)} \\|_{2,\infty} \\| B \\|_{2,\infty}, \label{eq:max_lb_matrix1} \end{aligned}$$ where the second equality is from . And in the last inequality, $B = (U^{(2)} \otimes \cdots \otimes U^{(t)})_{:, I}$ is a submatrix obtained for some subset of columns $I \subseteq [r^{t-1}]$. On the other hand, the flattening may be written as [@kolda2009tensor] $$T_{[1]} = \sum_{i=1}^r U^{(1)}_{:,i} \left( U^{(2)}_{:,i} \otimes \cdots \otimes U^{(t)}_{:,i} \right)^{\top} = U^{(1)} \left( U^{(2)} \odot \cdots \odot U^{(t)} \right)^\top . \label{eq:max_lb_matrix2}$$ The symbol “$\odot$” is the Khatri-Rao product of matrices, a.k.a. the “matching columnwise” Kronecker product.[^1] Take this as the submatrix $B$ in , then by we have that $T_{[1]} = U^{(1)} B^T$ is a valid factorization, which shows that $\\|T_{[1]} \\|_\mathrm{max} \leq \\| T \\|_\mathrm{max}$. Flattening over any other mode is equivalent, so the first part of the inequality holds. Since the matrix $T_{[i]}$ and the tensor $T$ contain the same values, then by $$\\|T_{[i]}\\|_{\max} \geq \\|T_{[i]}\\|_{1,\infty} = \max_{i_1, \ldots, i_t} \|T_{i_1,\ldots, i_t}\|.$$ This completes the proof. Sign tensors ------------ In order to connect expansion properties of the hypergraph $H$ to the error of our proposed tensor completion algorithm, we will work with sign tensors. A sign tensor $S$ has all entries equal to $+1$ or $-1$, i.e. $S \in \bigotimes_{i=1}^t \{ \pm 1\}^{n_i}$. The sign rank of a sign tensor $S$ is defined as $$\mathrm{rank}_\pm (S) = \inf \left\{ r \; \Big\| \; S = \sum_{i=1}^r s^{(1)}_i \circ \cdots \circ s^{(t)}_i, \; s_i^{(j)} \in \{ \pm 1 \}^{n_i} \mbox{ for $i \in [r]$ and $j \in [t]$} \right\}. \label{eq:sign_rank}$$ Using rank-1 sign tensors as our atoms, we can construct a nuclear norm [@ghadermarzy2018]: We define the [sign nuclear norm]{} for a tensor $T$ as $$\\| T \\|_\pm = \inf \left\{ \sum_{i=1}^r \|\alpha_i\| \; \Big\| \; T = \sum_{i=1}^r \alpha_i S_i \mbox{ where } \alpha_i \in {\mathbb R}, \, \mathrm{rank}_\pm (S_i) = 1 \right\} .$$ Note that the set of all rank-1 sign tensors forms a basis for $\bigotimes_{i=1}^t {\mathbb R}^n$, so this decomposition into rank-1 sign tensors is always possible; furthermore this is a norm for tensors and matrices [@ghadermarzy2018; @heiman2014deterministic]. The sign nuclear norm is called the “atomic M-norm” by [@ghadermarzy2018] and the “atomic norm” by [@heiman2014deterministic] (who only considered matrices), but in our opinion “sign nuclear norm” is more descriptive. Inequalities between different norms are a type of fundamental result in functional analysis. The next Lemma relating $\\| \cdot \\|_\pm$ and $\\| \cdot \\|_\mathrm{max}$ follows from a multilinear generalization of Grothendieck’s inequality. We use $K_G$ to denote Grothendieck’s constant over the reals. For detailed background, see [@tomczak-jaegermann1989]. \[lem:grothendieck\] The sign nuclear norm and max-qnorm satisfy $$\\| T \\|_\pm \leq c_1 c_2^t \\| T \\|_\mathrm{max},$$ where $c_1 \leq \frac{K_G}{5} \leq 0.36$ and $c_2 \leq 2.83$. Let $$\mathbb{B}_\pm(1) = \mathrm{conv} ( \{T: T\in \{\pm 1\}^{n \times \cdots \times n}, \mathrm{rank}(T) = 1 \} ) \quad \text{and} \quad \mathbb{B}_\textrm{max}(1) = \{T: \\|T \\|_\mathrm{max} \leq 1 \}$$ be the unit balls of the sign nuclear norm and max-qnorm, respectively. By [@ghadermarzy2018], Lemma 5, a consequence of the multilinear Grothendieck inequality, we have that $$\mathbb{B}_\textrm{max}(1) \subseteq c_1 c_2^t \, \mathbb{B}_\pm(1),$$ for any tensor $T$. Rescaling by $\alpha = \\|T\\|_\mathrm{max}$ gives $T = \alpha S$, where $\\| S \\|_\mathrm{max} = 1$. This implies that $S \in c_1 c_2^t \mathbb{B}_\pm(1)$, and thus that $\\|S\\|_\pm \leq c_1 c_2^t$. Using the scalability property of $\\| \cdot \\|_\mathrm{max}$ and $\\| \cdot \\|_\pm$ establishes the upper bound. Tensor completion {#sec:completion} ================= Proof of Theorem \[thm:main\] ----------------------------- Consider a hypercubic tensor $T$ of order-$t$, i.e. $T \in {\mathbb R}^{n\times \cdots \times n} $. We sample the entry $T_e$ whenever $e = (i_1,\dots, i_t)$ is a hyperedge in $H$ defined in Section \[sec:construction\]. Then the sample size is $\|E\|=nd^{t-1}$, and for fixed $d$ and $t$ it is linear in $n$. Now we are ready to prove our main result, Theorem \[thm:main\], about deterministic tensor completion. Consider a rank-1 sign tensor $S = s_1 \circ \cdots \circ s_t$ with $s_j \in \{\pm 1\}^n$. Let $J$ be the tensor of all ones and $S' = \frac{1}{2} (S + J)$, so that $S'$ is shifted to be a tensor of zeros and ones. Then $$\begin{aligned} \label{eq:defS} S'_{i_1,\dots, i_t}=\begin{cases} 1 & \text{if } (s_1)_{i_1}\cdots (s_t)_{i_t}=1,\\ 0 & \text{if } (s_1)_{i_1}\cdots (s_t)_{i_t}=-1. \end{cases} \end{aligned}$$ Define the sets $$\begin{aligned} \label{eq:defW} W_j := \{i \in [n] : (s_j)_i = 1 \}. \end{aligned}$$ Let $\mathcal{S}_t$ is the set of even $t$-strings in $\{0,1\}^t$. An even string has an even number of 1’s in it, e.g. for $t=3$ we have 000, 110, 101, 011 as even strings. The number of these strings is $ \| \mathcal{S}_t \| = 2^{t-1} $, so we can enumerate all possible even $t$-strings from $1$ to $2^{t-1}$, denoted by $w_1,\dots, w_{2^{t-1}}\in \{0,1\}^t$. Now for all $1\leq i\leq t$ and $1\leq j\leq 2^{t-1}$, we define the sets $W_{i,j}$ by $$\begin{aligned} W_{i,j}=\begin{cases} W_i & \text{if } (w_j)_i=1, \\ W_i^c & \text{if } (w_j)_i=0. \end{cases} \end{aligned}$$ By considering the sign of entries in the components of $S$, we have the following decomposition for $S'$ as a sum of rank-$1$ tensors: $$\begin{aligned} \label{eq:sumsign} S' = \sum_{j=1}^{2^{t-1}} 1_{W_{1,j}}\circ \cdots \circ 1_{W_{t,j}} . \end{aligned}$$ Note that the right hand side in can only take values in $\{0,1\}$ for each entry from our definition of $W_{i,j}$. To see holds, we prove it for two cases: 1. For any $(i_1,\dots i_t)\in [n]^t$ that satisfies $S'_{i_1,\dots,i_t}=1$, we know $(s_1)_{i_1}\cdots (s_t)_{i_t}=1$ from . On the other hand, by our definition of $W_j$ in , we have $ \sum_{j=1}^t \mathbf{1}\{i_j\in W_j\}$ is even. Therefore we can find a corresponding even string $w_j$ such that $i_j\in W_{1,j}$ for all $1\leq j\leq t$. Then the corresponding rank-$1$ tensor satisfies $$\begin{aligned} \left( 1_{W_{1,j}} \circ\cdots\circ 1_{W_t,j} \right)_{i_1,\dots,i_t}=\prod_{k=1}^t(1_{W_{k,j}})_{i_k}=1. \end{aligned}$$ Similarly, all the other rank-$1$ tensors in the sum of that do not correspond to $w_j$ will take value $0$ at entry $(i_1,\dots, i_t)$. So in this case $$\begin{aligned} S'_{i_1,\dots,i_t} = \sum_{j=1}^{2^{t-1}} \left(1_{W_{1,j}}\circ \cdots \circ 1_{W_{t,j}}\right)_{i_1,\dots,i_t}=1. \end{aligned}$$ 2. If $S'_{i_1,\dots,i_t}=0$, then $(s_1)_{i_1}\cdots (s_t)_{i_t}=0$, which implies $\sum_{j=1}^t \mathbf{1}\{i_j\in W_j\}$ is odd and there are no corresponding even strings. Therefore all rank-$1$ tensors on the right hand side of take value $0$ at $(i_1,\dots,i_t)$, and holds. Now we consider the deviation in the sample mean from the mean over all entries in the tensor: $$\begin{aligned} \left\| \frac{1}{n^t} \sum_{e \in [n]^t} S_{e} - \frac{1}{\|E\|} \sum_{e \in E} S_{e} \right\| &= \left\| \frac{1}{n^t} \sum_{e\in [n]^t} (2 S'_e - 1) - \frac{1}{\|E\|} \sum_{e \in E} (2 S'_e - 1) \right\| \\ &= 2 \left\| \frac{1}{n^t} \sum_{e\in [n]^t} S'_e - \frac{1}{\|E\|} \sum_{e \in E} S'_e \right\| \\ &= 2 \left\| \frac{\sum_{j=1}^{2^{t-1}} \prod_{i=1}^t \|W_{i,j}\| }{n^t} - \frac{\sum_{j=1}^{2^{t-1}} e(W_{1,j}, \ldots, W_{t,j}) }{nd^{t-1}} \right\|\\ &\leq 2 \sum_{j=1}^{2^{t-1}} \left\| \frac{\|W_{1,j}\| \cdots \|W_{t,j}\|}{n^t} - \frac{ e(W_{1,j}, \ldots, W_{t,j}) }{nd^{t-1}} \right\| . \end{aligned}$$ Applying Corollary to the sets $W_{1,j}\subseteq V_1,\dots, W_{t,j}\subseteq V_t$ for each $1\leq j\leq 2^{t-1}$, we get that $$\begin{aligned} \label{eq:signbound} \left\| \frac{1}{n^t} \sum_{e \in [n]^t} S_e - \frac{1}{nd^{t-1}} \sum_{e \in E} S_e \right\| &\leq \frac{1}{2} \sum_{j=1}^{2^{t-1}} \left( \left(1 + \frac{\lambda}{d} \right)^{t-1} -1 \right) = 2^{t-2} \left( \left(1 + \frac{\lambda}{d} \right)^{t-1} -1 \right). \end{aligned}$$ We now write the tensor $T = \sum_i \alpha_i S_i$ as a sum of rank-1 sign tensors $S_i$, with coefficients $\alpha_i \in \mathbb{R}$. Let $\\| \cdot \\|_{\pm}$ be the tensor sign nuclear norm, i.e. $\\|T \\|_\pm = \sum_i \|\alpha_i\|$. Then for a general tensor $T$, we can apply to each $S_i$, and by triangle inequality we get $$\left\| \frac{1}{n^t} \sum_{e \in [n]^t} T_e - \frac{1}{nd^{t-1}} \sum_{e \in E} T_e \right\| \leq 2^{t-2} \left( \left(1 + \frac{\lambda}{d} \right)^{t-1} -1 \right) \; \\| T \\|_\pm .$$ This holds for any tensor $T$. Now we apply this inequality to the tensor of squared residuals $$R:= (\hat{T} - T ) * (\hat{T} - T).$$ Since we solve for $\hat{T}$ with equality constraints, we have that $R_e = 0$ for all $e \in E$. Thus, $$\begin{aligned} \left\| \frac{1}{n^{t}} \sum_{e \in [n]^t} R_e \right\| &\leq 2^{t-2} \left( \left(1 + \frac{\lambda}{d} \right)^{t-1} -1 \right) \; \\|R\\|_\pm & \notag\\ &\leq 2^{t-2} \left( \left(1 + \frac{\lambda}{d} \right)^{t-1} -1 \right) c_1 c_2^t \; \\|R\\|_\textrm{max} & \mbox{(Lemma~\ref{lem:grothendieck})} \notag\\ &\leq 2^{t-2} \left( \left(1 + \frac{\lambda}{d} \right)^{t-1} -1 \right) c_1 c_2^t \; \\|\hat{T} - T\\|_\textrm{max}^2 & \mbox{(Theorem \ref{thm:max-qnorm-properties} (4))}\notag \\ &\leq 2^{2t-4} \left( \left(1 + \frac{\lambda}{d} \right)^{t-1} -1 \right) c_1 c_2^t \; \left(\\|\hat{T} \\|_\mathrm{max} + \\|T \\|_\mathrm{max}\right)^2 . & \mbox{(Lemma~\ref{lem:max-qnorm})}\label{eq:maxinequality} \end{aligned}$$ Since $\hat{T}$ is the output of our optimization routine and $T$ is feasible, $\\| \hat{T} \\|_\mathrm{max} \leq \\| T \\|_\mathrm{max}$. This leads to the final result with a constant $C_t = 2^{2t-2} c_1 c_2^t$. Tensor completion with erroneous observations --------------------------------------------- Now we turn to the case when our observations $Z$ of the original tensor $T$ are corrupted by errors $\nu$. We will call this noise, but it can be anything, even chosen adversarially, so long as it’s bounded. Let $Z\in \mathbb R^n\times \cdots \times \mathbb R^n$ be the tensor we observe with $Z_{e} = 0$ if $e \not \in E$ and $Z_e = T_e + \nu_e$ for $e \in E$. In this case, we study the solution to the following optimization problem: $$\begin{aligned} \min_{X}\quad & \\|X\\|_{\max}, \label{eq:noisy_problem} \\ \text{subject to} \quad & \frac{1}{\|E\|} \sum_{ e \in E} (X_e - Z_e)^2 \leq \delta^2, \notag\end{aligned}$$ for some $\delta>0$. The parameter $\delta$ is a bound on the root mean squared error of the observations. In a probabilistic setting, we may pick this parameter so that the constraint holds with sufficiently high probability. We obtain the following corollary of Theorem \[thm:main\]. Let $E$ be the hyperedge set of $H$ defined in Section \[sec:construction\]. Suppose we observe $Z_e = T_e + \nu_e$ for all $e \in E$ with bounded error satisfying $$\begin{aligned} \label{cor:noisebound} \frac{1}{\|E\|}\sum_{e \in E} \nu_e^2\leq \delta^2. \end{aligned}$$ Then solving the optimization problem will give us a solution $\hat{T}$ that satisfies $$\begin{aligned} \frac{1}{n^t}\\|\hat{T}-T \\|_F^2\leq C_t \\| T \\|_\mathrm{max}^2 \left(\left(1 + \frac{\lambda}{d} \right)^{t-1} - 1 \right) +4\delta^2,\end{aligned}$$ where $C_t$ is the same constant as in Theorem \[thm:main\]. Let $\hat{T}$ be the solution of Problem . Define the operator $\mathcal P_{E}: {\mathbb R}^{n\times \cdots \times n} \to {\mathbb R}^{n\times \cdots \times n}$ such that $(\mathcal P_{E}(T))_{e} = T_e$ if $e \in E$ and $0$ otherwise. We have $$\begin{aligned} \left\| \frac{1}{n^t}\\|\hat{T}-T \\|_F^2-\frac{1}{\|E\|}\\|\mathcal P_E(\hat{T}-T)\\|_F^2 \right\| &= \left\| \frac{1}{n^t}\sum_{e\in [n]^t} ( \hat{T}_e - T_e )^2 - \frac{1}{\|E\|} \sum_{e \in E} ( \hat T_e - T_e )^2 \right\| \nonumber \\ &\leq C_t\left( \left(1 + \frac{\lambda}{d} \right)^{t-1} -1 \right) \\|T \\|_\mathrm{max} ^2 \label{eq:noise2},\end{aligned}$$ where follows in the same way as in due to the fact that $T$ is a feasible solution to . On the other hand, from the constraints in and , by the triangle inequality, $$\begin{aligned} \label{eq:noise3} \\|\mathcal P_{E}(\hat{T}-T)\\|_{F}\leq \\|\mathcal P_{E}(\hat{T}-Z)\\|_{F}+\\|\mathcal P_{E}(Z-T)\\|_{F}\leq 2\delta \sqrt{\|E\|}.\end{aligned}$$ With and , we have $$\begin{aligned} \frac{1}{n^t}\\|\hat{T}-T \\|_F^2 &\leq \left\| \frac{1}{n^t}\\|\hat{T}-T \\|_F^2-\frac{1}{\|E\|}\\|\mathcal P_E(\hat{T}-T)\\|_F^2 \right\| +\frac{1}{\|E\|} \\|\mathcal P_{E}(\hat{T}-T)\\|_{F}^2\\ &\leq C_t\left( \left(1 + \frac{\lambda}{d} \right)^{t-1} -1 \right) \\|T \\|_\mathrm{max} ^2+4\delta^2,\end{aligned}$$ which is the final result. Sample Complexity {#sec:sample-complexity} ----------------- To compute the sample complexity in the case of no noise, we would like to guarantee, from , that $$\label{eq:guarantee} C_t \\| T \\|_\mathrm{max}^2 \left( \left(1 + \frac{\lambda}{ d} \right)^{t-1} - 1 \right) \leq \varepsilon.$$ Then implies the following chain of inequalities: $$\begin{aligned} \left(1 + \frac{\lambda}{d} \right)^{t-1} &\leq 1 + \frac{\varepsilon}{C_t \\|T \\|_\mathrm{max}^2} \\ 1 + \frac{\lambda}{d} &\leq \left( 1 + \frac{\varepsilon}{C_t \\|T \\|_\mathrm{max}^2} \right)^{\frac{1}{t-1}} \\ \frac{\lambda}{d} &\leq \frac{\varepsilon}{(t - 1) C_t \\|T \\|_\mathrm{max}^2}\left( 1 + O(\varepsilon) \right) \\ \frac{d}{\lambda} &\geq \frac{(t - 1) C_t \\|T \\|_\mathrm{max}^2}{\varepsilon} \left( 1 + O(\varepsilon) \right) .\end{aligned}$$ Assuming that $\lambda=O(\sqrt d)$, i.e. $G$ is a good expander, then this gives that $$\|E\| = n d^{t-1} = \Omega \left( n \left( \frac{(t - 1) C_t \\|T \\|_\mathrm{max}^2}{\varepsilon} \right)^{2(t-1)} \right)$$ many samples suffice. If we assume that $r, t = O(1)$, then Theorem \[thm:max-rank\] gives that $\\|T\\|_\mathrm{max} = O(1)$ and the sample complexity $\displaystyle \|E\| = O\left( \frac{n}{\varepsilon^{2(t-1)}} \right)$. Discussion {#sec:discussion} ========== We have deterministically analyzed tensor completion using the max-qnorm as a measure of complexity and hypergraph sampling. Our main results show that, by finding the tensor with smallest max-qnorm that is consistent with the observations, one obtains a good estimate of the true tensor. The error of the estimate depends on the expansion properties of the hypergraph model. A number of theoretical and practical considerations still remain. To actually use the max-qnorm for tensor completion as proposed in this paper, one has to solve either the equality-constrained or bounded optimization problems posed in Theorem \[thm:main\] or Corollary \[cor:noisebound\]. These both appear more difficult than a penalized approach combining the noise and complexity terms with a Lagrange multiplier, as was implemented by [@ghadermarzy2018]. Since these problems are equivalent, for the correct choice of Lagrange multiplier, one would hope that penalized problem will have similar guarantees. Theoretically, it would be nice to have other constructions of sparse hypergraph expanders. In particular, completely deterministic constructions could be useful for applications. The current difficulty appears to be in computing the second eigenvalue of these hypergraphs. It also appears that using a stronger expander mixing lemma will lead to better constants and rates with few changes to our proof. Acknowledgements {#acknowledgements .unnumbered} ---------------- Thank you to Ioana Dumitriu for support and discussions. We are grateful to Paul Beame for discussions of communication complexity and to Adi Shraibman for sharing details of the proof of the Hadamard product bound on the max-norm for matrices. K.D.H. was supported by a Washington Research Foundation Postdoctoral Fellowship. Y.Z. was partially supported by NSF DMS-1712630. [^1]: For $A \in {\mathbb R}^{m \times n}$ and $B \in {\mathbb R}^{l \times n}$, we define $A \odot B$ as the $ml \times n$ matrix with entries $(A \odot B)_{ij} = A_{aj} B_{bj}$ for $i = b + l (a - 1)$.
--- abstract: 'We performed a Raman scattering study of thin films of LiTi$_2$O$_4$ spinel oxide superconductor. We detected four out of five Raman active modes, with frequencies in good accordance with our first-principles calculations. Three T$_{2g}$ modes show a Fano lineshape from 5 K to 295 K, which suggests an electron-phonon coupling in LiTi$_2$O$_4$. Interestingly, the electron-phonon coupling shows an anomaly across the negative to positive magnetoresistance transition at 50 K, which may be due to the unset of other competing orders. The strength of the electron-phonon interaction estimated from the Allen’s formula and the observed lineshape parameters suggests that the three T$_{2g}$ modes contribute little to superconductivity.' author: - 'D. Chen' - 'Y.-L. Jia' - 'T.-T. Zhang' - 'Z. Fang' - 'K. Jin' - 'P. Richard' - 'H. Ding' bibliography: - 'citation.bib' title: 'Raman study of electron-phonon coupling in thin films of LiTi$_2$O$_4$ spinel oxide superconductor' --- Because of their complexity of charge, magnetic and orbital degrees of freedom, transition-metal spinels have been studied intensely for many years [@spinel]. As the only oxide superconductor with spinel structure, LiTi$_2$O$_4$ has aroused many researches since the discovery of its superconductivity [@firstSC]. This compound was initially compared to cuprates because both materials are transition-metal oxides, although the superconducting transition temperature $T_c \approx 12$ K of LiTi$_2$O$_4$ is not very high [@calculation_cuprate_lamda; @neutron_cuprate_DOS]. Unlike cuprates, both transport and spectroscopy results indicate that LiTi$_2$O$_4$ is a typical fully gapped type-II BCS superconductor [@specificheat_BCS_lamda; @criticalfield_BCS; @Andreevreflection_BCS]. Later, with the development of lithium-ion batteries, Li$_{1+x}$Ti$_{2-x}$O$_4$ ($0 \leq x \leq \frac{1}{3}$) was studied intensively as an alternative electrode material [@4512_electrochemicalLiinsertion; @defectcalcul_battery; @4512_usr_battery; @ionicliquidgating]. Interest for LiTi$_2$O$_4$ was refreshed recently with the report of anomalous magnetoresistance at 50 K [@magnetoresistance], which suggests that spin-orbital fluctuations play an important role in LiTi$_2$O$_4$. Bosonic modes were also observed in tunneling spectra, reinforcing the assumption for an important electron-boson coupling [@anisotropic_bosonicmode]. Although several works in the literature report the lattice dynamics of LiTi$_2$O$_4$, both theoretically and experimentally [@Raman; @phononcalcul1; @A1gcalculation; @phononcalcul2; @phononcalcul3; @phononcalcul4], there is significant discrepancies among the various studies. Moreover, direct phonon measurements for coupling analysis have been limited because of the poor availability of single crystals [@review]. In order to clarify the vibration modes and investigate the strength of the electron-phonon coupling, here we report a Raman scattering study of LiTi$_2$O$_4$ single-crystalline films supported by first-principles calculations. We detect four out of five Raman active modes, with frequencies in good accordance with our first-principles calculations. A Fano line shape was observed for the three T$_{2g}$ modes from 295 K to 5 K, which suggests that electron-phonon coupling is important in LiTi$_2$O$_4$. The electron-phonon coupling shows an anomaly around 50 K, where the negative to positive magnetoresistance occurs [@magnetoresistance]. The unset of other orders below 50 K, like orbital-related states, may quench the electron-phonon coupling, resulting in its fluctuation. Allen’s formula was used to estimate the strength of the electron-phonon interaction from the observed lineshape parameters. Although they have asymmetric line shape, the three T$_{2g}$ modes contribute only little to the superconductivity, with an average electron-phonon coupling constant $\overline{\lambda}$ = 0.074. The LiTi$_2$O$_4$ films used in our Raman study were grown on MgAl$_2$O$_4$ (001) substrates by pulsed laser deposition. Films with thickness $\approx$ 200 nm have been characterized to be single-crystalline and in a pure phase [@transport]. The electrical resistivity of the films presented in Fig. \[figure1\](b) shows a sharp superconducting transition at $T_c$ $\approx$ 11.3 K. To avoid Li vacancies induced by contact to water in air [@water-oxidation; @Lideficiency], the films were measured in a ST500 (Janis) cryostat with a working vacuum better than $2\times 10^{-6}$ mbar. A long-focus distance $20\times$ objective was used for back-scattering micro-Raman measurements between 5 and 295 K. Low power 488.0 nm and 514.5 nm excitations from an Ar-Kr ion laser were used as incident light. The scattering light was analyzed by a Horiba Jobin Yvon T64000 spectrometer equipped with a nitrogen-cooled CCD camera. The confocal design of this spectrometer allows us to measure the signal from both films and substrates. Substrates without films were also measured as reference. As shown in Fig. \[figure1\](a), we define $x$, $y$ and $z$ as the directions along the unit cell axes. Raman spectra have been recorded under the $(\mathbf{\hat{e}}^i\mathbf{\hat{e}}^s)=$ ($xx$), ($yy$) and ($xy$) polarization configurations. ![\[figure1\](Color online). (a) Crystal structure of LiTi$_2$O$_4$. Li atoms fill $\frac{1}{8}$ of the tetrahedral sites (in green) and Ti atoms fill $\frac{1}{2}$ of the octahedral sites (in blue). (b) In-plane resistivity of LiTi$_2$O$_4$. The inset is a close-up of the superconducting transition.](figure1){width="\columnwidth"} Space group $Fd\overline{3}m$ (point group $O_{h}$) characterizes the crystal structure of LiTi$_2$O$_4$ [@review], which is presented in Fig. \[figure1\](a). A single unit cell contains two chemical formula units, for a total of 14 atoms. A simple group symmetry analysis [@bilbal] indicates that the phonon modes at the Brillouin zone (BZ) center $\Gamma$ decompose into \[T$_{1u}$\]+\[4T$_{1u}$\]+\[A$_{1g}$+E$_{g}$+3T$_{2g}$\]+\[2A$_{2u}$+2E$_{u}$+2T$_{2u}$+T$_{1g}$\], where the first, second, third and fourth terms represent the acoustic modes, the infrared-active modes, the Raman-active modes and the silent modes, respectively. To get estimates on the phonon frequencies, we performed first-principles calculations of the phonon modes at $\Gamma$ in the framework of the density functional perturbation theory (DFPT) [@DFPT2] without considering spin-orbit coupling. We adopted the fully-relaxed lattice parameters a = b = c = 8.4 Å, and the Wyckoff positions (Li $8a$, Ti $16d$ and O $32e$) from experimental data [@anisotropic_bosonicmode]. For all calculations, we used the Vienna ab-initio simulation package (VASP) [@VASP2] with the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof for the exchange-correlation functions [@PBE]. The projector augmented wave (PAW) method [@PAW] was employed to describe the electron-ion interactions. A plane wave cut-off energy of 500 eV was used with a uniform $6\times6\times6$ Monkhorst-Pack $k$-point mesh for a $2\times2\times2$ supercell. The real-space force constants of the supercell were calculated using DFPT [@DFPT1] and the phonon frequencies were calculated from the force constants using the PHONOPY code [@PHONOPY]. The calculated optic mode frequencies, their symmetries and optical activities, as well as the main atoms involved, are given in Table \[EXP\_CAL\_comparsion\]. Compared to previous calculation results with different methods, our results are more consistent with the experimental values. Sym. Activity Exp. (This work) Exp. (Ref. [@Raman]) Cal. (This work) Atoms involved Ref. [@phononcalcul1] Ref. [@phononcalcul2] Ref. [@phononcalcul4] ---------- ---------- ------------------ ---------------------- ------------------ ---------------- ----------------------- ----------------------- ----------------------- T$_{2u}$ Silent 116.0 Ti, O 128.5 165.0 141.2 E$_{u}$ Silent 204.0 Ti, O 286.6 236.8 275.3 T$_{1u}$ I 279.5 Li, Ti, O 289.2 210.3 247.2 T$_{2u}$ Silent 360.3 Ti, O 461.3 542.5 493.6 E$_{g}$ R - 200 366.0 O 337.4 429.0 428.6 T$_{2g}$ R 342.2 339 369.8 Li, O 288.7 344.2 196.8 T$_{1u}$ I 383.0 Li, Ti, O 389.5 424.9 412.7 T$_{1g}$ Silent 398.6 O 397.9 429.0 413.0 T$_{1u}$ I 416.4 Li, Ti, O 506.8 508.6 515.3 T$_{2g}$ R 433.3 429 466.6 Li, O 516.8 542.4 343.2 E$_{u}$ Silent 469.4 Ti, O 565.4 603.3 581.0 A$_{2u}$ Silent 498.2 Ti, O 496.3 323.8 397.6 T$_{2g}$ R 495.2 494 533.5 Li, O 687.7 652.4 596.8 T$_{1u}$ I 561.5 Li, Ti, O 696.8 668.3 695.1 A$_{2u}$ Silent 593.7 Ti, O 650.5 664.6 659.2 A$_{1g}$ R 625.4 628 625.2 O 548.5 628.0 628.0 I = infrared active, R = Raman active, Silent = not optically active. The unit of the numbers are “cm$^{-1}$".\ ![\[figure2\](Color online). Raman spectra of LiTi$_2$O$_4$ film and MgAl$_2$O$_4$ substrate recorded with 488.0 nm, 514.5 nm laser excitations under the ($xy$) and ($xx$) polarization configurations at room temperature. The red dashed lines and numbers indicate modes from the LiTi$_2$O$_4$ sample, while gray dashed lines and numbers indicate modes from substrate. “LTO" and “S" represent LiTi$_2$O$_4$ and substrate, respectively. The curves are shifted relative to each other for clarity.](figure2){width="\columnwidth"} In Fig. \[figure2\], we compare the Raman spectra of a LiTi$_2$O$_4$ film and of a substrate recorded at room temperature under different laser excitations. Based on literature, we can assign many peaks to the MgAl$_2$O$_4$ substrate. For instance, the peaks at 309.4 cm$^{-1}$ and 668.3 cm$^{-1}$ are T$_{2g}$ modes, the peak at 407.3 cm$^{-1}$ is an E$_{2g}$ mode, the one at 766.2 cm$^{-1}$ is an A$_{1g}$ mode, and the 724.3 cm$^{-1}$ excitation is from cation disorder [@MgAlO4_4; @MgAlO4_5; @MgAlO4_6; @MgAlO4_7; @MgAlO4_9]. In addition to these peaks from the substrate, we observe 4 out of 5 Raman active modes predicted for LiTi$_2$O$_4$. The Raman tensors corresponding to the $O_{h}$ symmetry group are expressed in the $xyz$ coordinates as: $$\textrm{A$_{1g}=$} \left(\begin{array}{ccc} a & 0 &0\\ 0 & a &0\\ 0 & 0 &a \end{array}\right),$$ $$\left[\begin{array}{cc}\textrm{E$_{g}=$} \left(\begin{array}{ccc} b & 0 &0\\ 0 & b &0\\ 0 & 0 &-2b\\ \end{array}\right), \left(\begin{array}{ccc} -\sqrt{3}b & 0 &0\\ 0 & \sqrt{3}b &0\\ 0 & 0 &0\\ \end{array}\right) \end{array}\right],$$ $$\left[\begin{array}{ccc}\textrm{T$_{2g}=$} \left(\begin{array}{ccc} 0 & 0 &0\\ 0 & 0 &d\\ 0 & d &0\\ \end{array}\right), \left(\begin{array}{ccc} 0 & 0 &d\\ 0 & 0 &0\\ d & 0 &0\\ \end{array}\right), \left(\begin{array}{ccc} 0 & d &0\\ d & 0 &0\\ 0 & 0 &0\\ \end{array}\right) \end{array}\right].$$ Using the polarization selection rules, it is straightforward to assign the three peaks at 342.2 cm$^{-1}$, 433.3 cm$^{-1}$ and 495.2 cm$^{-1}$ to T$_{2g}$ modes, which are detected only under ($xy$) polarization configuration. As for the peak at 625.4 cm$^{-1}$ detected under ($xx$) polarization configuration, comparison with our calculation results suggests a A$_{1g}$ mode. The missing E$_{g}$ mode predicted at 366.0 cm$^{-1}$ may have too weak scattered intensity to be detected. Since its energy is close to one of the bosonic modes at around 40 meV observed in tunneling spectra [@anisotropic_bosonicmode], it is possible that strong coupling reduces its lifetime. Apart from the phonon peaks, we observe a big hump at 1200 cm$^{-1}$ under 488.0 nm laser excitation in both the substrate and the LiTi$_2$O$_4$ thin film. Under 514.5 nm laser excitation, the hump shifts to 200 cm$^{-1}$. The humps correspond to the same transition at 520 nm (2.38 eV) that we assign to luminescence from the substrate. ![\[figure3\](Color online). (a) Waterfall plot of temperature dependent Raman spectra for the three T$_{2g}$ modes of LiTi$_2$O$_4$. The vertical dashed lines indicate these three modes. The colored dashed curves are data corrected by subtracting the substrate’s data at the corresponding temperature. The colored curves are the resulting fitted spectra with Fano functions. (b) Temperature evolution of the renormalized phonon energies and linewidths. (c) Fano asymmetry parameters $q$ of the three T$_{2g}$ modes from the fits in (a). The vertical dashed line indicates 50 K. The error bars in (b) and (c) are from the system resolution and the fitting error, respectively.](figure3){width="\columnwidth"} We notice that the three T$_{2g}$ modes of LiTi$_2$O$_4$ have quite asymmetric line shapes, which implies a Fano resonance. The Fano resonance is a quantum interference between a discrete state and a continuum [@fano]. For Raman scattering, the spectrum of the phonon mode will present an asymmetric Fano line shape if there is an electronic-phonon coupling [@Ramanfano]. To further study the role of the electronic-phonon coupling in LiTi$_2$O$_4$, we analyzed the temperature dependence of the Raman spectra of the three T$_{2g}$ modes with Fano functions. As shown in Fig. \[figure3\] (a), the Raman spectra can be well fitted by the equation: $$\label{eq} I(\omega) = \sum\limits_{i}\frac{A_{i}(q_{i}\Gamma_{i}/2+\omega-\omega_{i})^2}{(\Gamma_{i}/2)^{2}+(\omega-\omega_{i})^2}$$ where $A_{i}$ is the amplitude, $\omega_{i}$ is the resonance energy (renormalized in the presence of the coupling), $\Gamma_{i}$ is the linewidth (full-width at half-maximum) and $q_{i}$ is the asymmetric parameter for the $i^{\textrm{th}}$ T$_{2g}$ mode. The factor $\|1/q\|$ is often used to estimate the electron-phonon coupling strength. The larger $\|1/q\|$, the stronger the coupling. The fitting results including the renormalized phonon energies, the linewidths and the Fano asymmetry parameters $q$ are displayed in Figs. \[figure3\](b) and \[figure3\](c). All the three T$_{2g}$ modes have higher energies and narrow linewidths upon cooling. Interestingly, there are anomalies around 50 K for the asymmetry parameters. This indicates that the electron-phonon coupling strength varies. We note that a magnetoresistivity transition from negative to positive is reported as temperature is decreased below 50 K, which suggests the presence of an orbital-related state [@magnetoresistance]. This unset of orbital order may quench the electron-phonon coupling, inducing the abnormal behavior of $\|1/q\|$. We now follow a standard method to estimate the electron-phonon coupling strength associated to a particular mode $i$ using the Allen’s formula [@Allen1; @Allen4]: $$\label{eq} \lambda_{i} = \frac{2g_{i}\gamma_{i}}{\pi N_{\epsilon_{f}}\omega_{i}^2}$$ where $\lambda_{i}$ is the dimensionless electron-phonon coupling constant, $g_{i}$ is the mode degeneracy, $N_{\epsilon_{f}}$ is the electronic density-of-states at the Fermi surface, $\omega_{i}$ is the mode energy and $\gamma_{i}$ is the linewidth. As an approximation at 5 K with $g_i$ = 3 and $N_{\epsilon_{f}}$ = 13.44 /eV unit cell [@magnetoresistance], we get $\lambda_{1}$ = 0.089 for the T$_{2g}$(1) mode with $\omega_{1}$ = 348.9 cm$^{-1}$, $\gamma_{1}$ = 9.4 cm$^{-1}$, $\lambda_{2}$ = 0.063 for the T$_{2g}$(2) mode with $\omega_{2}$ = 442.2 cm$^{-1}$, $\gamma_{2}$ = 10.8 cm$^{-1}$, and $\lambda_{3}$ = 0.071 for the T$_{2g}$(3) mode with $\omega_{3}$ = 503.7 cm$^{-1}$, $\gamma_{3}$ = 15.8 cm$^{-1}$. The electron-phonon coupling constants for the three T$_{2g}$ modes are rather small with an average $\overline{\lambda}$ = 0.074, which is apparently inconsistent with a conventional phonon-mediated pairing mechanism. However, we caution that a more accurate evaluation of electron-phonon coupling constant would necessitate complete consideration of the contribution from all phonon modes across the entire first Brillouin zone. In summary, we reported a polarized Raman scattering study of the only oxide spinel superconductor, LiTi$_2$O$_4$. Four out of five Raman active modes were detected, with frequencies in good accordance with our first-principles calculations. A Fano line shape was observed for the three T$_{2g}$ modes from 295 K to 5 K, which suggests that electron-phonon coupling is important in LiTi$_2$O$_4$. The electron-phonon coupling shows an anomaly around 50 K, where the negative to positive magnetoresistance occurs [@magnetoresistance]. The unset of other orders below 50 K, like orbital-related states, may quench the electron-phonon coupling, resulting in this anomaly. Allen’s formula was used to estimate the strength of electron-phonon interaction from the observed lineshape parameters. Although they have electron-phonon coupling, the three T$_{2g}$ modes contribute little to the superconductivity. We acknowledge G. He for useful discussions. This work was supported by grants from the National Natural Science Foundation (Nos. 11674371, 11274362, 11474338 and 11674374) and the Ministry of Science and Technology (Nos. 2015CB921301, 2016YFA0401000, 2016YFA0300300 and 2013CB921700) from China, and the Chinese Academy of Sciences (No. XDB07000000).
--- abstract: 'An attempt is made to generalise the ideas introduced by Haldane and others regarding Bosonizing the Fermi surface. The present attempt involves introduction of Bose fields that correspond to displacements of the Fermi sea rather than just the Fermi surface. This enables the study of short wavelength fluctuations of the Fermi surface and hence the dispersion of single particle excitations with high energy. The number conserving product of two Fermi fields is represented as a simple combination of these Bose fields. It is shown that most(!) commutation rules involving these number conserving products are reproduced exactly, as are the dynamical correlation functions of the free theory. Also the work of Sharp, Menikoff and Goldin has shown that the field operator may be viewed as a Unitary representation of the local current group. An explicit realisation of this unitary representation is given in terms of canonical conjugate of the density operator.' author: - 'Girish S. Setlur' title: Expressing Products of Fermi Fields in terms of Fermi Sea Displacements --- 9.2in -0.7in 6.80in -.2in Introduction ============ Recent years have seen remarkable developments in many-body theory in the form an assortment of techniques that may be loosely termed bosonization. The beginnings of these types of techiniques may be traced back to the work of Tomonaga[@Tom] and later on by Luttinger [@Lutt] and by Leib and Mattis [@Leib]. In the 70’s an attempt was made by Luther [@Luther] at generalising these ideas to higher dimensions. Closely related to this is work by Sharp et. al. [@Sharp] in current algebra. Mention must also be made about the work of Feenberg and his collaborators[@Feen] who have a theory of correlated electrons written down in the standard wavefunction approach familiar in elementary quantum mechanics. It seems that the approach adopted here has some similarites with this work although the details are not identical. More progress was made by Haldane [@Haldane] which culminated in the explicit computation of the single particle propagator by Castro-Neto and Fradkin [@Neto] and by Houghton, Marston et.al. [@Mars] and also by Kopietz et. al. [@Kop]. Rigorous work by Frohlich and Marchetti[@Froch] is also along similar lines. The attempt made here is to generalise the concepts of Haldane [@Haldane] to accomodate short wavelengths fluctuations where the concept of a linerised bare fermion energy dispersion is no longer valid. To motivate progress in this direction one must first introduce the concept of the canonical conjugate of the Fermi density distribution. This concept is likely to be important in the construction of the field operator in terms of the Fermi sea displacements, although the latter is not completed in this article. The concept of the velocity operator being the canonical conjugate of the density has been around for a long time, and this has been exploited in the study of HeII by Sunakawa et. al. [@Sun]. However, the author is not aware of a rigorous study of the meaning of this object, in particular, an explicit formula for the canonical conjugate of the density operator has to the best of my knowledge never been written down in terms of the field operators. The work by Sharp et. al. [@Sharp] comes close to what I am attempting here. Unitary Representation of the Local Current Group ================================================= The work of Sharp et. al. [@Sharp] established that the current algebra itself does not convey the underlying particle statistics rather the statistics is hidden in a choice of a unitary representation of the local current group. Here, I try to write down some formulas that provide (possibly for the first time) an explicit and exact representation of the Fermi field operator in terms of the canonical conjugate of the density operator thereby providing an explicit realisation of the claims of Sharp et. al. [@Sharp] Some Mathematical Identities ---------------------------- $$\rho({\bf{x}}\sigma) = \psi^{\dagger}({\bf{x}}\sigma)\psi({\bf{x}}\sigma)$$ is a bose-like object; namely it satisfies $$[\rho({\bf{x}}\sigma) , \rho({\bf{x^{'}}}\sigma^{'})] = 0$$ Notation: $$N_{\sigma} = \int d{\bf{x}} \mbox{ }\rho({\bf{x}}\sigma)$$ $$n_{\sigma} = \frac{N_{\sigma}}{V}$$ $$\rho_{\sigma} = \langle n_{\sigma} \rangle$$ $ \rho_{\sigma} $ is a c-number, whereas $ N_{\sigma} $ and $ n_{\sigma} $ are operators. So it natural to introduce the canonical conjugate $ \Pi({\bf{x}}\sigma) $. $$[\Pi({\bf{x}}\sigma), \rho({\bf{x^{'}}}\sigma^{'})] = i\delta^{d}({\bf{x-x^{'}}})\delta_{\sigma,\sigma^{'}}$$ $$[\Pi({\bf{x}}\sigma), \Pi({\bf{x^{'}}}\sigma^{'})] = 0$$ Propose the ansatz $$\psi({\bf{x}}\sigma) = exp(-i\Pi({\bf{x}}\sigma)) exp(i\Phi([\rho];{\bf{x}}\sigma)) (\rho({\bf{x}}\sigma))^{\frac{1}{2}} \label{DPVA}$$ The above Eq.( \[DPVA\]) shall be called the DPVA ansatz. Here again, $$\Pi({\bf{x}}\sigma) = i\ln[ (\rho_{\sigma}^{\frac{1}{2}}+\delta\psi({\bf x}\sigma)) (\rho_{\sigma} + \delta\rho({\bf{x}}\sigma))^{-\frac{1}{2}} exp(-i\Phi([\rho];{\bf{x}}\sigma)) ] \label{Rep}$$ $$\delta\psi({\bf x}\sigma) = \psi({\bf x}\sigma) - \rho_{\sigma}^{\frac{1}{2}}$$ $$\delta\rho({\bf{x}}\sigma) = \rho({\bf{x}}\sigma) - \rho_{\sigma}$$ $ \rho_{\sigma} $ is a c-number given by $ \rho_{\sigma} = \langle n_{{\sigma}} \rangle $. Where $ \Phi([\rho];{\bf{x}}\sigma) $ is some hermitian functional to be computed later. This ansatz automatically satsfies $$\psi^{\dagger}({\bf{x}}\sigma)\psi({\bf{x}}\sigma) = \rho({\bf{x}}\sigma)$$ and $$[\psi({\bf{x}}\sigma), \rho({\bf{x^{'}}}\sigma^{'})] = \delta^{d}({\bf{x - x^{'}}}) \delta_{\sigma, \sigma^{'}} \psi({\bf{x}}\sigma)$$ Write $$\rho({\bf{x}}\sigma) = n_{\sigma} + \frac{1}{V}\sum_{{\bf{q}}\neq 0} \rho_{{\bf{q}}\sigma} exp(-i{{\bf{q.x}}})$$ where $ n_{\sigma} = \frac{N_{\sigma}}{V} $. and $$\Pi({\bf{x}}\sigma) = X_{{\bf{0}}\sigma} + \sum_{{\bf{q}}\neq 0} exp(i{{\bf{q.x}}}) X_{{\bf{q}}\sigma}$$ $$[X_{{\bf{0}}\sigma}, N_{\sigma^{'}}] = i \delta_{\sigma, \sigma^{'}}$$ $$[X_{{\bf{q}}\sigma}, \rho_{{\bf{q^{'}}}\sigma^{'}}] = i \delta_{{\bf{q}},{\bf{q^{'}}}} \delta_{\sigma, \sigma^{'}}$$ Therefore $$\psi({\bf{x}}\sigma) = exp(-i \sum_{{\bf{q}}} exp(i{{\bf{q.x}}}) X_{{\bf{q}}\sigma} ) F([\{ \rho_{{\bf{k}}\sigma^{'}} \}]; {\bf{x}}\sigma)$$ where $$F([\{ \rho_{{\bf{k}}\sigma^{'}} \}]; {\bf{x}}\sigma) = exp(i\Phi([\{ \rho_{{\bf{k}}\sigma^{'}} \}]; {\bf{x}}\sigma)) (n_{\sigma} + \frac{1}{V}\sum_{{\bf{q}}\neq 0} exp(-i{{\bf{q.x}}}) \rho_{{\bf{q}}\sigma}) ^{\frac{1}{2}}$$ It is possible to write these formulas in momentum space. This will be useful later on. First we write $$\Phi([\rho]; {\bf{x}}\sigma) = \sum_{{\bf{q}}} \phi( [\rho]; {\bf{q}}\sigma ) exp(-i{\bf{q.x}})$$ Therefore $$\psi({\bf{x}} \sigma) = exp(-i \sum_{{\bf{q}}} exp(i{\bf{q.x}}) X_{{\bf{q}}\sigma} ) exp(i \sum_{{\bf{q}}} exp(-i{\bf{q.x}}) \phi( [\rho]; {\bf{q}}\sigma ) ) (n_{\sigma} + \frac{1}{V} \sum_{{\bf{q}}\neq 0} \rho_{{\bf{q}}\sigma} exp(-i{\bf{q.x}}) )^{\frac{1}{2}}$$ It is possible to write the corresponding formula in momentum space by making the identifications $$exp(i{\bf{q.x}}) \rightarrow T_{{\bf{-q}}}({\bf{k}})$$ $$exp(-i{\bf{q.x}}) \rightarrow T_{{\bf{q}}}({\bf{k}})$$ where $$T_{{\bf{q}}}({\bf{k}}) = exp({\bf{q}}.\nabla_{{\bf{k}}})$$ $$\psi({\bf{k}}\sigma) = exp(-i \sum_{{\bf{q}}} T_{{\bf{-q}}}({\bf{k}}) X_{{\bf{q}}\sigma} ) exp(i \sum_{{\bf{q}}} T_{{\bf{q}}}({\bf{k}})\phi( [\rho]; {\bf{q}}\sigma ) ) (N_{\sigma} + \sum_{{\bf{q}}\neq 0} \rho_{{\bf{q}}\sigma} T_{{\bf{q}}}({\bf{k}}) )^{\frac{1}{2}} \delta_{{\bf{k}}, 0} \label{Fermi}$$ The translation operators translate the Kronecker delta that appears in the extreme right. It may be verified that $$[\psi({\bf{k}}\sigma), \rho_{{\bf{q}}\sigma^{'}}] = \psi({\bf{k-q}}\sigma) \delta_{\sigma,\sigma^{'}}$$ $$[\psi({\bf{k}}\sigma), N_{\sigma^{'}}] = \psi({\bf{k}}\sigma) \delta_{\sigma,\sigma^{'}}$$ Here $$\psi({\bf{x}}\sigma) = \frac{1}{{V^{\frac{1}{2}}}}\sum_{{\bf{k}}} exp(i{\bf{k.x}})\psi({\bf{k}}\sigma)$$ Also the following identities are going to be important. $$\psi({\bf{k}}\sigma) = exp( -i\sum_{\bf{q}}T_{ -{\bf{q}} }({\bf{k}})X_{ {\bf{q}} \sigma } ) f_{{\bf{k}}\sigma}([\rho])$$ $$f_{ {\bf{k}} \sigma}([\rho]) = exp( i\sum_{ {\bf{q}} }T_{ {\bf{q}} }({\bf{k}}) \phi([\rho]; {\bf{q}} \sigma) ) (N_{ \sigma } + \sum_{ {\bf{q}}\neq 0 } T_{ {\bf{q}} }({\bf{k}}) \rho_{ {\bf{q}}\sigma })^{\frac{1}{2}} \delta_{ {\bf{k, 0}} }$$ Check by expansion that $$\sum_{ {\bf{k}} } f^{\dagger}_{ {\bf{k+q/2}} \sigma}([\rho]) f_{ {\bf{k-q/2}} \sigma}([\rho]) = \rho_{ {\bf{q}}\sigma }$$ Proof (??) of the Fermion Commutation Rules ------------------------------------------- We use the theory of distributions to ( try ) prove rigorously (good enough for a physicist!) the fermion anticommutation rules. Pardon the pretense of mathematical rigor. The approach is as follows. Let $ $ be the space of all smooth functions from $ S = R^{3} \times \{ \uparrow, \downarrow \} $ to $ C $. Further, on this space define the inner product (Schwartz space). $ In: L^{2}_{C}(S) \times L^{2}_{C}(S) \rightarrow C $ $$\langle f\|g \rangle = \int d{\bf{x}} \sum_{\sigma} f^{}({\bf{x}}\sigma) g({\bf{x}}\sigma)$$ Since the Fermi fields are operator-valued distributions we can construct operators $$c_{f} = \int d{\bf{x}}\sum_{\sigma}\psi({\bf{x}}\sigma)f({\bf{x}}\sigma)$$ and we then have to prove $$\{ c_{f}, c_{g} \} = 0$$ $$c_{f}^{2} = 0$$ The above two relations shall be called $ c-c $ anticommutation rules. $$\{ c_{f}, c^{\dagger}_{g} \} = \langle g\|f \rangle$$ The above relation shall be called the $ c-c^{\dagger} $ anticommutation rule. for all $ f, g $ that belong to $ L^{2}_{C}(S) $. The claim is, that these relations are satisfied provided $ \Phi $ obeys the recursion below. $$\Phi([\{\rho({\bf{y_{1}}}\sigma_{1}) - \delta({\bf{y_{1}}}-{\bf{x}}^{'})\delta_{\sigma_{1},\sigma^{'}} \} ] ;{\bf{x}}\sigma)$$ $$+ \Phi([\rho];{\bf{x^{'}}}\sigma^{'}) - \Phi([\rho];{\bf{x}}\sigma)$$ $$-\Phi([\{\rho({\bf{y_{1}}}\sigma_{1}) - \delta({\bf{y_{1}}}-{\bf{x}})\delta_{\sigma_{1},\sigma} \} ] ;{\bf{x^{'}}}\sigma^{'}) = m\pi \label{recur}$$ where m is an odd integer. This recursion is to be satisfied for all $ ({\bf{x}}\sigma) \neq ({\bf{x^{'}}}\sigma^{'}) $. At precisely $ ({\bf{x}}\sigma) = ({\bf{x^{'}}}\sigma^{'}) $ the recursion obviously breaks down. But this is not a serious drawback as we shall soon find out. The proof involves working with a basis. Here the integer $ m $ has to be even for bosons and odd for fermions. The imposition of the commutation rules by themselves do not provide us with a formula for the hermitian functional $ \Phi $. Any redefinition of $ \Pi({\bf{x}}\sigma) $ consistent with $ [\Pi, \rho] = i\delta(...) $ may be absorbed by a suitable redefinition of $ \Phi $ (akin to gauge transformations pointed out by AHC Neto, private communication). Solution to $ \Phi([\rho];{\bf{q}}\sigma) $ by making contact with the free theory. A random choice of $ \Phi([\rho];{\bf{x}}\sigma) $ that satisfies the recursion is (A. J. Leggett : private communication) the Leggett ansatz (for fermions) (Also F.D.M. Haldane [@Haldane]). $$\Phi([\rho];{\bf{x}}\sigma) = \int \mbox{ }d{\bf{x^{'}}} \sum_{{\sigma^{'}}} \pi\theta(t({\bf{x}}\sigma) - t({\bf{x^{'}}}\sigma^{'})) \rho({\bf{x^{'}}}\sigma^{'})$$ redefinitions of $ \Phi $ that leave the statistics invariant are the ’generalised guage transformations’. Disclaimer:Please pardon my pretensions at rigor in the following paragraph. It is easy to get carried away. The relation that relates two fermi fields by a generalised guage transformation is an equivalence relation. Therefore , analogous to the claim that a state of a system is a ray in Hilbert space, one is tempted to call the equivalence class of fermi fields under these transformations the fermi distribution. Therefore a fermi distribution is not just one fermi field but a whole bunch of equivalent ones. Therefore a state containing one fermion at a space point is obtained by acting the creation fermi distribution on the vacuum. Thus one may construct the Fock space by repeatedly acting these fermi distributions on the vacuum. Therefore there are two ways by which one can choose $ \Phi $. One is a random choice which satisfies the recursion. In which case $ \Pi $ is determined by Eq.(\[Rep\]). The other choice is $ \Pi = i\frac{\delta}{\delta \rho} $. In which case, $ \Phi $ can no longer be chosen arbitrarily. It has to be determined by making contact with the free theory. Here $ \theta(x) $ is the Heaviside step function. The unpleasentness caused by the fact that the recursion for fermions does not hold when $ ({\bf{x}}\sigma) = ({\bf{x^{'}}}\sigma^{'}) $ is probably remediable by multiplying the fermi fields which are operator-valued distributions by arbitrary smooth functions and proving properties about these latter objects. More importantly, $ t $ has to invertible in order for the ansatz to satisfy the recursion. That such a bijective mapping exists is guaranteed by the theory of cardinals. However, this mapping is not continuous let alone differentiable and therefore of little practical value. (A continuous mapping would imply a homeomorphism between $ R $ and $ R^{3} $ eg.) The claim is that the state of the fermi system is prescribed by prescribing an amplitude for finding the system in a given configuration of densities. Let $$W_{FS}([\rho]) = \Theta_{H}([\rho])exp(-U_{FS}[\rho]) exp(i\theta_{FS}([\rho]))$$ be the wavefunctional of the noninteracting fermi sea. The $ U_{FS}[\rho] $ is uniquely determined by prescribing all moments of the density operator. The functional $ \theta_{FS}[\rho] $ cannot be so determined. Moreover, since the amplitude for finding the system with negative densities is zero, we must also have a prefactor, $$\Theta_{H}([\rho]) = \Pi_{{\bf{y_{1}\sigma_{1}}}} \theta_{H}(\rho({\bf{y_{1}\sigma_{1}}}) )$$ Where $ \theta_{H}(x) $ is the Heaviside unit step function. Making Contact With the Free Theory ----------------------------------- Consider the operator $$n({\bf{y}}\sigma) = \int d{\bf{x}}\mbox{ } \psi^{\dagger}({\bf{x+y/2}}\sigma)\psi({\bf{x-y/2}}\sigma) = \sum_{{\bf{k}}}\psi^{\dagger}({\bf{k}}\sigma)\psi({\bf{k}}\sigma) exp(-i{\bf{k.y}})$$ $$n({\bf{y}}\sigma)W_{FS}([\rho]) = n_{0}({\bf{y}}\sigma)W_{FS}([\rho]) \label{wavef}$$ $$n_{0}({\bf{y}}\sigma) = \sum_{{\bf{k}}}\theta(k_{F}-k) exp(-i{\bf{k.y}})$$ First assume $ {\bf{y}} \neq {\bf{0}} $. In eq. ( \[wavef\]) there are two undetermined functionals. One is $ \Phi([\rho];{\bf{x}}\sigma) $ and the other is $ \theta_{FS}([\rho]) $. We have remarked earlier that redefinitions of $ \Phi([\rho];{\bf{x}}\sigma) $ are similar to ’gauge transformations’. These functional gauge transformations leave the local density of particles invariant but alter the statistics. There is a subgroup of these functional gauge transformations that also leaves the statistics invariant. A realisation of this subgroup is achieved by for example: $$\Phi([\rho];{\bf{x}}\sigma) \rightarrow \Phi([\rho];{\bf{x}}\sigma)+\theta([\rho])$$ This may be exploited to our advantage while solving eq.( \[wavef\]). In other words $ \theta_{FS}([\rho]) $ can be absorbed into $ \Phi([\rho];{\bf{x}}\sigma) $ without altering the statistics. This leaves us with an equation for $ \Phi([\rho];{\bf{x}}\sigma) $ in terms of the properties of the free theory exclusively. Exploiting the recursion relation for $ \Phi([\rho];{\bf{x}}\sigma) $ we arrive at the result (Here $ y \neq 0 $.) $$\int d{\bf{x}} \mbox{ } (\rho({\bf{(x+y/2)}}\sigma))^{\frac{1}{2}} (\rho({\bf{(x-y/2)}}\sigma))^{\frac{1}{2}} exp[i\Phi([\rho];{\bf{(x-y/2)}}\sigma) - i\Phi([\rho];{\bf{(x+y/2)}}\sigma)]$$ $$exp(U_{FS}[\{\rho({\bf{y^{'}}}\sigma^{'}) -\delta({\bf{y^{'} - (x-y/2)}})\delta_{\sigma^{'}, \sigma} \}] -U_{FS}[\{\rho({\bf{y^{'}}}\sigma^{'}) -\delta({\bf{y^{'} - (x+y/2)}})\delta_{\sigma^{'}, \sigma} \}])$$ $$\frac{\Theta([\{\rho({\bf{y^{'}}}\sigma^{'}) -\delta({\bf{y^{'} - (x+y/2)}})\delta_{\sigma^{'}, \sigma} \}]) } {\Theta([\{\rho({\bf{y^{'}}}\sigma^{'}) -\delta({\bf{y^{'} - (x-y/2)}})\delta_{\sigma^{'}, \sigma} \}])} = - n_{0}({\bf{y}}\sigma) \label{eqnforphi}$$ The terms involving the Heaviside are indeterminate unless we expand around $$\rho({\bf{y}}\sigma) = \delta^{d}({\bf{0}}) + \delta \rho({\bf{y}}\sigma)$$ and, $$\langle \mbox{ } \rho({\bf{y}}\sigma) \mbox{ } \rangle = \delta^{d}({\bf{0}}) = \rho_{\sigma}$$ Point Splitting or no Point Splitting ? --------------------------------------- The attempts made here are partly based on the work of Ligouri and Mintchev on Generalised statistics[@Lig] and work of Goldin et. al.[@Sharp] and the series by Reed and Simon[@Reed] Here I shall attempt to provide a framework within which questions such as the existence of the canonical conjugate of the fermi-density distribution may be addressed. For reasons of clarity we shall not insist on utmost generality. The philosophy being that the quest for utmost genarality should not cloud the underlying basic principles. We start off with some preliminaries. Let $ {\mathcal{H}} $ be an infinite dimensional separable Hilbert Space. We know from textbooks that such a space possesses a countable orthonormal basis $ {\mathcal{B}} = \{ w_{i}; i \in {\mathcal{Z}} \} $ . Here, $ {\mathcal{Z}} $ is the set of all integers. Thus$ {\mathcal{H}} $ = Set of all linear combinations of vectors chosen from $ {\mathcal{B}} $. We construct the tensor product of two such spaces $${\mathcal{H}}^{{\small{\bigotimes}} 2} = {\mathcal{H}} {\bigotimes} {\mathcal{H}}$$ This is defined to be the dual space of the space of all bilinear forms on the direct sum. In plain English this means something like this. Let $ f \in {\mathcal{H}} $ and $ g \in {\mathcal{H}} $ define the object $ f {\small{\bigotimes}} g $ to be that object which acts as shown below. Let $ < v, w > $ be an element of the Cartesian product $ {\mathcal{H}} \times {\mathcal{H}} $. $$f {\small{\bigotimes}} g < v, w > = (f, v) (g, w)$$ Here, $ (f, v) $ stands for the inner product of $ f $ and $ v $. Define also the inner product of two $ f {\small{\bigotimes}} g $ and $ f^{'} {\small{\bigotimes}} g^{'} $ $$(f {\small{\bigotimes}} g, f^{'} {\small{\bigotimes}} g^{'}) = (f, f^{'} ) (g, g^{'})$$ Construct the space of all finite linear combinations of objects such as $ f {\small{\bigotimes}} g $ with different choices for $ f $ and $ g $. Lump them all into a set. You get a vector space. It is still not the vector space $ {\mathcal{H}}^{{\small{\bigotimes 2}}} $. Because the space of all finite linear combinations of objects such as $ f {\small{\bigotimes}} g $ is not complete. Not every Cauchy sequence converges. Complete the space by appending the limit points of all Cauchy sequences from the space of all finite linear combinations of vectors of the type $ f {\small{\bigotimes}} g $. This complete space is the Hilbert space $ {\mathcal{H}}^{{\small{\bigotimes 2}}} $. Similarly one can construct $ {\mathcal{H}}^{{\small{\bigotimes}} n} $ for n = 0, 1, 2, 3, ... Where we have set $ {\mathcal{H}}^{0} = {\mathcal{C}} $ the set of complex numbers. Define the Fock Space over $ {\mathcal{H}} $ as $${\mathcal{F}}({\mathcal{H}}) = {\bigoplus}_{n=0}^{\infty} {\mathcal{H}}^{{\small{\bigotimes}} n}$$ Physically, each element of it is an ordered collection of wavefunctions with different number of particles $$(\varphi_{0}, \varphi_{1}(x_{1}), \varphi_{2}(x_{1}, x_{2}), ...,\varphi_{n}, ... )$$ is a typical element of $ {\mathcal{F}}({\mathcal{H}}) $. This is the Hilbert Space which we shall be working with. Let $ {\mathcal{D}}^{n} $ be the space of all decomposable vectors. $${\mathcal{D}}^{n} = \{ f_{1} {\small{\bigotimes}}...{\small{\bigotimes}} f_{n} ; f_{i} \in {\mathcal{H}} \}$$ For each $ f \in {\mathcal{H}} $ define $$b(f): {\mathcal{D}}^{n} \rightarrow {\mathcal{D}}^{n-1}, n \geq 1$$ $$b^{}(f): {\mathcal{D}}^{n} \rightarrow {\mathcal{D}}^{n+1}, n \geq 0$$ defined by $$b(f) \mbox{ } f_{1} {\small{\bigotimes}} ... {\small{\bigotimes}} f_{n} = \sqrt{n} (f, f_{1}) f_{2} {\small{\bigotimes}} ...{\small{\bigotimes}} f_{n}$$ $$b^{}(f) \mbox{ } f_{1} {\small{\bigotimes}} ...{\small{\bigotimes}} f_{n} = \sqrt{n+1} f \bigotimes f_{1} {\small{\bigotimes}} f_{2} {\small{\bigotimes}} ... {\small{\bigotimes}} f_{n}$$ We also define $ b(f) {\mathcal{H}}^{0} = 0 $. By linearity we can extend the definitions to the space of all finite linear combinations of elements of $ {\mathcal{D}}^{n} $ namely $ {\mathcal{L}} ({\mathcal{D}}^{n}) $. For any $ \varphi \in {\mathcal{L}}({\mathcal{D}}^{n}) $ and $ \psi \in {\mathcal{L}}({\mathcal{D}}^{n+1}) $ $$\parallel b(f) \varphi \parallel \leq \sqrt{n} \parallel f \parallel \parallel \varphi \parallel$$ $$\parallel b^{}(f) \varphi \parallel \leq \sqrt{n+1} \parallel f \parallel \parallel \varphi \parallel$$ $$(\psi, b^{}(f)\varphi) = ( b(f)\psi, \varphi)$$ So long as $ \parallel f \parallel < \infty $, $ b(f) $ and $ b^{}(f) $ are bounded operators. An operator $ {\mathcal{O}} $ is said to be bounded if $$sup_{\parallel \varphi \parallel = 1} \mbox{ } \parallel {\mathcal{O}} {\mathcal{\varphi}} \parallel \mbox{ } < \mbox{ } \infty$$ $ {\mathcal{O}} $ is unbounded otherwise. The norm of a bounded operator is defined as $$\parallel {\mathcal{O}} \parallel = sup_{\parallel \varphi \parallel = 1} \mbox{ } \parallel {\mathcal{O}} {\mathcal{\varphi}} \parallel \mbox{ }$$ In order to describe fermions, it is necessary to construct orthogonal projectors on $ {\mathcal{F}}({\mathcal{H}}) $. In what follows $ c(f) $ will denote a fermi annhilation operator. $ c^{}(f) $ will denote a fermi creation operator. Physically, and naively speaking, these are the fermi operators in “momentum space” $ c_{\bf{k}} $ and $ c^{}_{\bf{k}} $. First define $ P_{-} $ to be the projection operator that projects out only the antisymmetric parts of many body wavefunctions. For example, $$P_{-} f_{1} {\small{\bigotimes}} f_{2} = \frac{1}{2} (f_{1} {\small{\bigotimes}} f_{2} - f_{2} {\small{\bigotimes}} f_{1})$$ We now have $$c(f) = P_{-}b(f)P_{-}$$ $$c^{}(f) = P_{-}b^{}(f)P_{-}$$ Let us take a more complicated example. Let us find out how $ c^{}(f) c(g) $ acts on a vector $ v = f_{1} {\small{\bigotimes}} f_{2} $. $$c^{}(f) c(g) = P_{-}b^{}(f)P_{-}P_{-}b(g)P_{-}$$ $$c^{}(f) c(g) = P_{-}b^{}(f)P_{-}b(g)P_{-}$$ $$c^{}(f) c(g) v = P_{-}b^{}(f)P_{-}b(g)P_{-} v$$ $$P_{-} v = \frac{1}{2 !}( f_{1} {\small{\bigotimes}} f_{2} - f_{2} {\small{\bigotimes}} f_{1} )$$ $$b(g)P_{-} v = \frac{1}{2 !}{\sqrt{2}}( (g, f_{1}) f_{2} - (g, f_{2}) f_{1} )$$ $$P_{-}b(g)P_{-} v = \frac{1}{2 !}{\sqrt{2}}( (g, f_{1}) f_{2} - (g, f_{2}) f_{1} )$$ $$b^{}(f)P_{-}b(g)P_{-} v = \frac{1}{2 !}{\sqrt{2}}^{2} ( (g, f_{1}) f {\small{\bigotimes}} f_{2} - (g, f_{2}) f {\small{\bigotimes}} f_{1} )$$ $$c^{}(f) \mbox{ } c(g) v = (\frac{1}{2 !})^{2}{\sqrt{2}}^{2} ( (g, f_{1})[ f {\small{\bigotimes}}f_{2} - f_{2} {\small{\bigotimes}} f] - (g, f_{2}) [ f {\small{\bigotimes}} f_{1} - f_{1} {\small{\bigotimes}} f ] )$$ Having had a feel for how the fermi operators behave, we are now equipped to pose some more pertinent questions. Choose a basis $${\mathcal{B}} = \{ w_{i}; i \in {\mathcal{Z}} \}$$ Definition of the Fermi Density Distribution -------------------------------------------- Here we would like to capture the notion of the fermi density operator. Physicists call it $ \rho(x) = \psi^{}(x) \psi(x) $. Multiplication of two fermi fields at the same point is a tricky business and we would like to make more sense out of it. For this we have to set our single particle Hilbert Space: $${\mathcal{H}} = L_{p}({\mathcal{R}}^{3}) {\bigotimes} {\mathcal{W}}$$ Here, $ L_{p}({\mathcal{R}}^{3}) $ is the space of all periodic functions with period $ L $ in each space direction. That is if $ u \in L_{p}({\mathcal{R}}^{3}) $ then $$u(x_{1} + L, x_{2}, x_{3}) = u(x_{1}, x_{2}, x_{3})$$ $$u(x_{1}, x_{2} + L, x_{3}) = u(x_{1}, x_{2}, x_{3})$$ $$u(x_{1}, x_{2}, x_{3} + L) = u(x_{1}, x_{2}, x_{3})$$ $ {\mathcal{W}} $ is the spin space spanned by two vectors. An orthonormal basis for $ {\mathcal{W}} $ $$\{ \xi_{\uparrow}, \xi_{\downarrow} \}$$ A typical element of $ {\mathcal{H}} $ is given by $ f({\bf{x}}) \bigotimes \xi_{\downarrow} $. A basis for $ {\mathcal{H}} $ is given by $${\mathcal{B}} = \{ \sqrt{ \frac{1}{L^{3}} } exp(i{\bf{q_{n}.x}}) \bigotimes \xi_{s} ; {\bf{n}} = (n_{1}, n_{2}, n_{3}) \in {\mathcal{Z}}^{3}, s \in \{ \uparrow, \downarrow \} ;$$ $${ \bf{q_{n}} } = (\frac{2 \pi n_{1}}{L}, \frac{2 \pi n_{2}}{L}, \frac{2 \pi n_{3}}{L}) \}$$ We move on to the definition of the fermi-density distribution. The Hilbert Space $ {\mathcal{H}}^{\bigotimes n} $ is the space of all n-particle wavefunctions with no symmetry restrictions. From this we may construct orthogonal subspaces $${\mathcal{H}}_{+}^{\bigotimes n} = P_{+} {\mathcal{H}}^{\bigotimes n}$$ $${\mathcal{H}}_{-}^{\bigotimes n} = P_{-} {\mathcal{H}}^{\bigotimes n}$$ Tensors from $ {\mathcal{H}}_{+}^{\bigotimes n} $ are orthogonal to tensors from $ {\mathcal{H}}_{-}^{\bigotimes n} $. The only exceptions are when $ n = 0 $ or $ n = 1 $. $${\mathcal{H}}_{+}^{\bigotimes 0} = {\mathcal{H}}_{+}^{\bigotimes 0} = {\mathcal{C}}$$ $${\mathcal{H}}_{+}^{\bigotimes 1} = {\mathcal{H}}_{+}^{\bigotimes 1} = {\mathcal{H}}$$ The space $ {\mathcal{H}}_{+}^{\bigotimes n} $ is the space of bosonic-wavefunctions and the space $ {\mathcal{H}}_{-}^{\bigotimes n} $ is the space of fermionic wavefunctions. The definition of the fermi density distribution proceeds as follows. Let $ v $ be written as $$v = \sum_{\sigma \in \{ \uparrow, \downarrow \} }a(\sigma)\xi_{\sigma}$$ The Fermi density distibution is an operator on the Fock Space, given a vector $ f \bigotimes v \in \mathcal{H} $ in the single particle Hilbert Space, and a tensor $ \varphi $ in the n-particle subspace of of $ \mathcal{F}(\mathcal{H}) $, there exists a corresponding operator $ \rho(f \bigotimes v) $ that acts as follows: $$[\rho(f \bigotimes v) \varphi ]_{n} ({\bf{x_{1}}}\sigma_{1}, {\bf{x_{2}}}\sigma_{2}, .... , {\bf{x_{n}}}\sigma_{n}) = 0$$ if $ \varphi \in {\mathcal{H}}_{+}^{\bigotimes n} $ and $$[\rho(f \bigotimes v) \varphi]_{n} ({\bf{x_{1}}}\sigma_{1}, {\bf{x_{2}}}\sigma_{2}, .... , {\bf{x_{n}}}\sigma_{n}) = \sum_{i=1}^{n} f({\bf{x_{i}}}) a(\sigma_{i}) \varphi_{n}({\bf{x_{1}}}\sigma_{1}, {\bf{x_{2}}}\sigma_{2}, .... , {\bf{x_{n}}}\sigma_{n})$$ when $ \varphi \in {\mathcal{H}}_{-}^{\bigotimes n} $. The physical meaning of this abstract operator will become clear in the next subsection. Definition of the Canonical Conjugate of the Fermi Density ---------------------------------------------------------- We introduce some notation. Let $ g = \sqrt{ \frac{1}{L^{3}} }exp(i{ \bf{k_{m}.x} } ) \bigotimes \xi_{r} $ $$\psi({ { \bf{k_{m}} } r}) = c(g)$$ $$\rho({ { \bf{k_{m}} } r}) = \rho(g)$$ This $ \rho({ { \bf{k_{m}} } r}) $ is nothing but the density operator in momentum space, familiar to Physicists $$\rho({{\bf{k_{m}}}r}) = \sum_{ {\bf{q_{n}}} }c^{\dagger}_{ {\bf{q_{n}+k_{m}}}r }c_{ {\bf{q_{n}}}r }$$ We want to define the canonical conjugate of the density operator as an operator that maps the Fock space(or a subset thereof) on to itself. If $ \varphi \in {\mathcal{H}}^{0} = {\mathcal{C}} $ then $$X_{ { \bf{q_{m}} } } \varphi = 0$$ Let $ \varphi \in {\mathcal{H}}_{+}^{n} $, $ n = 2, 3, ... $ then $$X_{ { \bf{q_{m}} } } \varphi = 0$$ The important cases are when $ \varphi \in {\mathcal{H}}_{-}^{n} $, $ n = 2, 3, ... $ or if $ \varphi \in {\mathcal{H}} $. In such a case, we set $ n = N_{s}^{0} \neq 0 $ for $ s \in \{ \downarrow, \uparrow \} $. Let us introduce some more notation. $ N_{s} = \rho_{ { \bf{q = 0} } s } $ is the number operator to be distinguished from the c-number $ N_{s}^{0} $. The eigenvalue of $ N_{s} $ is $ N_{s}^{0} $ when it acts on a state such as $ \varphi \in {\mathcal{H}}_{-}^{n} $. Some more notation. $$\delta \mbox{ } \psi({ { \bf{k_{m}} } s }) = \psi({ { \bf{k_{m}} } s }) - { \sqrt{N_{s}^{0}} } \delta_{ { \bf{k_{m}, 0} } }$$ and $$\delta \mbox{ } \rho({ { \bf{k_{m}} } s }) = \rho({ { \bf{k_{m}} } s }) - N_{s}^{0} \delta_{ { \bf{k_{m}, 0} } }$$ The Canonical Conjugate of the density distribution in real space denoted by $ \Pi_{s}({\bf{x}}) $ is defined as follows. $$\Pi_{s}(x_{1}+L, x_{2}, x_{3}) = \Pi_{s}(x_{1}, x_{2}, x_{3})$$ $$\Pi_{s}(x_{1}, x_{2}+L, x_{3}) = \Pi_{s}(x_{1}, x_{2}, x_{3})$$ $$\Pi_{s}(x_{1}, x_{2}, x_{3}+L) = \Pi_{s}(x_{1}, x_{2}, x_{3})$$ The definition is as follows. $$\Pi_{s}({\bf{x}}) = \sum_{ {\bf{q_{m}}}} exp(i{\bf{q_{m}.x}})X_{ {\bf{q_{m}}}s }$$ $$X_{ {\bf{q_{m}}}s } = i \mbox{ } {\large{ln}} [ ( 1 + \frac{1}{ { \sqrt{N_{s}^{0}} } } \sum_{ { \bf{k_{n}} } } \delta \psi({ \bf{k_{n}} } s) T_{- {\bf{k_{n}}} }({\bf{q_{m}}}))( 1 + \sum_{ { \bf{k_{n}} } }$$ $$\frac{1}{ N_{s}^{0} } \delta \rho({ \bf{k_{n}} } s) T_{{\bf{k_{n}}} }({\bf{q_{m}}}) )^{-\frac{1}{2}} exp(-i\sum_{ { \bf{k_{n}} } }\phi([\rho]; {\bf{k_{n}}} s) T_{{\bf{k_{n}}} }({\bf{q_{m}}}) ) ] \delta_{ {\bf{q_{m}}}, 0} \label{result}$$ $$\Phi([\rho]; {\bf{x}}s) = \sum_{ {\bf{k_{n}}} } \phi([\rho];{\bf{k_{n}}} s) exp(-i{\bf{k_{n}.x}})$$ $$T_{{\bf{k_{n}}} }({\bf{q_{m}}}) = exp({\bf{k_{n}.\nabla_{q_{m}}}})$$ The translation operator translates the $ {\bf{q_{m}}} $ in the Kronecker delta that appears in the extreme right by $ {\bf{k_{n}}} $ and $ \Phi([\rho]; {\bf{x}}s) $ satisfies a recursion explained in detail in the previous manuscript. The logarithm is to be interpreted as an expansion around the leading term which is either $ N_{s}^{0} $ or $ { \sqrt{N_{s}^{0}} } $. The question of existence of $ X_{ {\bf{q_{m}}}s } $ now reduces to demonstrating that this operator (possibly unbounded) maps its domain of definition(densely defined in Fock space) on to the Fock space. Defining the limit of the series expansion is likely to be the major bottleneck in demonstrating the existence of $ X_{ {\bf{q_{m}}}s } $. That this is the canonical conjugate of the density operator is not at all obvious from the above definition. A rigorous proof of that is also likely to be difficult. Considering that we arrived at this formula by first postulating the existence of $ \Pi_{s}({\bf{x}}) $, it is probably safe to just say “ it is clear that ” $ \Pi_{s}({\bf{x}}) $, is in fact the canonical conjugate of $ \rho $. The way in which the above formula can be deduced may be motivated as follows: $$\psi({\bf{x}}\sigma) = \frac{ 1 }{ V^{\frac{1}{2}} } \sum_{ {\bf{k}} }exp(i{\bf{k.x}})\psi({\bf{k}}\sigma)$$ $$= exp(-i \sum_{{\bf{q}}} exp(i{\bf{q.x}}) X_{{\bf{q}}\sigma} ) exp(i \sum_{{\bf{q}}} exp(-i{\bf{q.x}}) \phi( [\rho]; {\bf{q}}\sigma ) )$$ $$(n_{\sigma} + \frac{1}{V} \sum_{{\bf{q}}\neq 0} \rho_{{\bf{q}}\sigma} exp(-i{\bf{q.x}}) )^{\frac{1}{2}} \label{FF}$$ In the above Eq.( \[FF\]) ONLY on the right side make the replacements, $$exp(i{\bf{q.x}}) \rightarrow T_{{\bf{-q}}}({\bf{k}})$$ $$exp(-i{\bf{q.x}}) \rightarrow T_{{\bf{q}}}({\bf{k}})$$ where $$T_{{\bf{q}}}({\bf{k}}) = exp({\bf{q}}.\nabla_{{\bf{k}}})$$ and append a $ \delta_{ {\bf{k, 0}} } $ on the extreme right. Also on the LEFT side of Eqn.( \[FF\]) make the replacement $ \psi({\bf{x}}\sigma) $ by $ \psi({\bf{k}}\sigma) $. And you get a formula for $ \psi({\bf{k}}\sigma) $. In order to get a formula for $ X_{ {\bf{q_{m}} } s } $ we have to invert the relation and obtain, $$\Pi({\bf{x}}s) = i \mbox{ } ln[ ( \sqrt{N_{s}^{0}} + \sum_{ {\bf{k_{n}}}} exp(i{\bf{k_{n}.x}})\delta\psi({\bf{k_{n}}}s) )$$ $$exp(-i \sum_{{\bf{k_{n}}}} exp(-i{\bf{k_{n}.x}}) \phi( [\rho]; {\bf{k_{n}}}s ) )(N_{s}^{0} + \sum_{ {\bf{k_{n}}} } \delta\rho_{{\bf{k_{n}}}s} exp(-i{\bf{k_{n}.x}}) )^{-\frac{1}{2}} ] \label{canon}$$ Make the replacements on the right side of Eqn.( \[canon\]) $$exp(i{\bf{k_{n}.x}}) \rightarrow T_{{\bf{-k_{n}}}}({\bf{q_{m}}})$$ $$exp(-i{\bf{k_{n}.x}}) \rightarrow T_{{\bf{k_{n}}}}({\bf{q_{m}}})$$ where and append a $ \delta_{ {\bf{q_{m}, 0}} } $ on the extreme right. Also on the LEFT side of Eqn.( \[canon\]) replace $ \Pi({\bf{x}}s) $ by $ X_{ {\bf{q_{m}}}s } $. This results in formula given in Eqn. ( \[result\]). Expression in terms of Fermi Sea Displacemnts ============================================= The claim is that the following exact relation holds, $$c^{\dagger}_{ {\bf{k+q/2}} }c_{ {\bf{k-q/2}} } = n_{F}({\bf{k}})\frac{N}{\langle N \rangle}\delta_{ {\bf{q, 0}} } + (\sqrt{ \frac{N}{\langle N \rangle} }) [\Lambda_{ {\bf{k}} }( {\bf{q}} ) \theta( -{\bf{k.q}} ) a_{ {\bf{k}} }({\bf{-q}}) + \Lambda^{}_{ {\bf{k}} }( {\bf{-q}} ) \theta( {\bf{k.q}} ) a^{\dagger}_{ {\bf{k}} }({\bf{q}}) ]$$ $$+ \sum_{ { \bf{k_{1}, k_{2}} } }\sum_{ { \bf{q_{1}, q_{2}} } } \Gamma^{{ \bf{q_{1}, q_{2}} } } _{ { \bf{k_{1}, k_{2}} } } ({ \bf{k, q} })a^{\dagger}_{ {\bf{k}}_{1} }( {\bf{q}}_{1} ) a_{ {\bf{k}}_{2} }( {\bf{q}}_{2} ) \label{SEA}$$ where, $$\{c_{ {\bf{k}} }, c_{ {\bf{k}}^{'} } \} = 0$$ and $$\{c_{ {\bf{k}} }, c^{\dagger}_{ {\bf{k}}^{'} } \} = \delta_{ {\bf{k}}, {\bf{k}}^{'} }$$ and, $$[ a_{ {\bf{k}} }({\bf{q}}), a_{ {\bf{k}}^{'} }({\bf{q}}^{'}) ] = 0$$ $$[ a_{ {\bf{k}} }({\bf{q}}), a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) ] = \delta_{ {\bf{k}}, {\bf{k}}^{'} }\delta_{ {\bf{q}}, {\bf{q}}^{'} }$$ and, $$\Lambda_{ {\bf{k}} }({\bf{q}}) = \sqrt{ n_{F}({\bf{k+q/2}})(1 - n_{F}({\bf{k-q/2}})) }$$ and $$N = \sum_{ {\bf{k}} }c^{\dagger}_{ {\bf{k}} }c_{ {\bf{k}} }$$ $$\langle N \rangle = \sum_{ {\bf{k}} }n_{F}({\bf{k}})$$ and $ n_{F}({\bf{k}}) = \theta(k_{f} - \|{\bf{k}}\|) $ The $ \theta(-{\bf{k.q}}) $ is a reminder that $ \Lambda_{ {\bf{k}} }({\bf{q}}) $ is nonzero only when $ {\bf{k.q}} \leq 0 $. Also, $$[N, a_{ {\bf{k}} }({\bf{q}}) ] = 0$$ $$[c_{ {\bf{p}} }, a_{ {\bf{k}} }({\bf{q}}) ] \mbox{ } = something \mbox{ } complicated$$ and the kinetic energy operator is, $$K = \sum_{ {\bf{k}} }\epsilon_{ {\bf{k}} } c^{\dagger}_{ {\bf{k}} }c_{ {\bf{k}} }$$ in terms of the Bose fields, it is postulated to be, $$K = E_{0} + \sum_{ {\bf{k}}, {\bf{q}} } \omega_{ {\bf{k}} }( {\bf{q}} ) a^{\dagger}_{ {\bf{k}} }( {\bf{q}} ) a_{ {\bf{k}} }( {\bf{q}} ) \label{Kin}$$ $$\omega_{ {\bf{k}} }( {\bf{q}} ) = \Lambda_{{\bf{k}}}(-{\bf{q}}) \frac{ {\bf{k.q}} }{m}$$ and $ E_{0} = \sum_{ {\bf{k}} }\epsilon_{ {\bf{k}} }n_{F}({\bf{k}}) $. Also, the filled Fermi sea is identified with the Bose vacuum. $ \|FS> = \|0> $. The fact that this is true will become clearer later. At this point it is sufficient to note that $ \omega_{ {\bf{k}} }( {\bf{q}} ) \geq 0 $ for all $ {\bf{k}} $ and $ {\bf{q}} $. We have to make sure that for the right choice of $ \Gamma $, $$\langle \rho_{ {\bf{q_{1}}} }(t_{1})\rho_{ {\bf{q_{2}}} }(t_{2}) \rho_{ {\bf{-q_{1}-q_{2}}} }(t_{3}) \rangle \mbox{ }\neq \mbox{ } 0$$ The terms linear in the Fermi sea displacements in Eq.( \[SEA\]) are chosen in order to ensure that the correct dynamical four-point functions are recovered. Now for the quadratic terms. The best way to derive formulae for them is to compute the exact dynamical six point function and set it equal to the expression got from the free theory. $$\langle c^{\dagger}_{ {\bf{k+q/2}} }(t)c_{ {\bf{k-q/2}} }(t) c^{\dagger}_{ {\bf{k^{'}+q^{'}/2}} }(t^{'})c_{ {\bf{k^{'}-q^{'}/2}} }(t^{'}) c^{\dagger}_{ {\bf{k^{''}+q^{''}/2}} }(t^{''}) c_{ {\bf{k^{''}-q^{''}/2}} }(t^{''}) \rangle = I({\bf{k,q}},t;{\bf{k^{'},q^{'}}},t^{'};{\bf{k^{''},q^{''}}},t^{''})$$ From the free theory we have $$I({\bf{k,q}},t;{\bf{k^{'},q^{'}}},t^{'};{\bf{k^{''},q^{''}}},t^{''}) = exp(i\frac{ {\bf{k.q}} }{m}t)exp(i\frac{ {\bf{k^{'}.q^{'}}} }{m}t^{'}) exp(i\frac{ {\bf{k^{''}.q^{''}}} }{m}t^{''})$$ $$[\mbox{ } ( 1 - n_{F}({\bf{k-q/2}}) ) ( 1 - n_{F}({\bf{k^{'}-q^{'}/2}}) ) n_{F}({\bf{k+q/2}}) \delta_{ {\bf{k+q/2}}, {\bf{k^{''}-q^{''}/2}} } \delta_{ {\bf{k-q/2}}, {\bf{k^{'}+q^{'}/2}} } \delta_{ {\bf{k^{'}-q^{'}/2}}, {\bf{k^{''}+q^{''}/2}} }$$ $$- \delta_{ {\bf{k-q/2}}, {\bf{k^{''}+q^{''}/2}} } \delta_{ {\bf{k+q/2}}, {\bf{k^{'}-q^{'}/2}} } \delta_{ {\bf{k^{'}+q^{'}/2}}, {\bf{k^{''}-q^{''}/2}} } (1 - n_{F}({\bf{k-q/2}}) )n_{F}({\bf{k^{'}+q^{'}/2}}) n_{F}({\bf{k+q/2}}) \mbox{ }] \label{Free}$$ Also the six-point function may be evaluated in terms of the Bose fields(summation over superfluous indices is implied), $$I = \langle \Lambda_{ {\bf{k}} }({\bf{q}})a_{ {\bf{k}} }(-{\bf{q}}) exp(-i\omega_{ {\bf{k}} }(-{\bf{q}})t) \Gamma_{ {\bf{k_{1}, k_{2}}} }^{ {\bf{q_{1}, q_{2}}} } ({\bf{k}}^{'}, {\bf{q}}^{'})a^{\dagger}_{ {\bf{k_{1}}} }({\bf{q_{1}}}) a_{ {\bf{k_{2}}} }({\bf{q_{2}}})$$ $$exp(i\omega_{ {\bf{k_{1}}} }({\bf{q_{1}}})t^{'}) exp(-i\omega_{ {\bf{k_{2}}} }({\bf{q_{2}}})t^{'}) \Lambda_{ {\bf{k^{''}}} }(-{\bf{q^{''}}}) a^{\dagger}_{ {\bf{k^{''}}} }({\bf{q}}^{''}) exp(i\omega_{ {\bf{k^{''}}} }( {\bf{q}}^{''})t^{''}) \rangle \label{Bose}$$ The reduced form of the six-point function may be written as, $$I = \Lambda_{ {\bf{k}} }({\bf{q}}) exp(-i\omega_{ {\bf{k}} }(-{\bf{q}})t) \Gamma_{ {\bf{k}}_{1}, {\bf{k}}_{2} }^{ {\bf{q}}_{1}, {\bf{q}}_{2} } ({\bf{k}}^{'}, {\bf{q}}^{'}) exp(i\omega_{ {\bf{k}}_{1} }({\bf{q}}_{1})t^{'}) exp(-i\omega_{ {\bf{k}}_{2} }({\bf{q}}_{2})t^{'}) \Lambda_{ {\bf{k}}^{''} }(-{\bf{q}}^{''}) exp(i\omega_{ {\bf{k}}^{''} }({\bf{q}}^{''})t^{''})$$ $$\langle a_{ {\bf{k}} }(-{\bf{q}}) a^{\dagger}_{ {\bf{k}}_{1} }({\bf{q}}_{1}) a_{ {\bf{k}}_{2} }({\bf{q}}_{2}) a^{\dagger}_{ {\bf{k}}^{''} }({\bf{q}}^{''}) \rangle$$ This may be simplified to $$I = exp(-i\mbox{ }\omega_{ {\bf{k}} }(-{\bf{q}})\mbox{ }t) exp(i\mbox{ }\omega_{ {\bf{k}} }(-{\bf{q}})\mbox{ }t^{'}) exp(-i\mbox{ }\omega_{ {\bf{k}}^{''} }({\bf{q}}^{''})\mbox{ }t^{'}) exp(i\mbox{ }\omega_{ {\bf{k}}^{''} }({\bf{q}}^{''})\mbox{ }t^{''}) \Lambda_{ {\bf{k}} }( {\bf{q}} )$$ $$\Lambda_{ {\bf{k}}^{''} }( -{\bf{q}}^{''} ) \Gamma_{ {\bf{k}}, {\bf{k}}^{''} }^{ -{\bf{q}}, {\bf{q}}^{''} } ( {\bf{k}}^{'}, {\bf{q}}^{'} )$$ Further, since the commutation rules, $$[ c^{\dagger}_{ {\bf{k+q/2}} }c_{ {\bf{k-q/2}} }, c^{\dagger}_{ {\bf{k^{'}+q^{'}/2}} }c_{ {\bf{k^{'}-q^{'}/2}} } ] = c^{\dagger}_{ {\bf{k+q/2}} }c_{ {\bf{k^{'}-q^{'}/2}} } \delta_{ {\bf{k-q/2}}, {\bf{k^{'}+q^{'}/2}} } - c^{\dagger}_{ {\bf{k^{'}+q^{'}/2}} }c_{ {\bf{k-q/2}} } \delta_{ {\bf{k+q/2}}, {\bf{k^{'}-q^{'}/2}} }$$ This means, that the coefficents also have to satisfy, $$-\Gamma^{-q^{'}\mbox{ }q_{2}}_{k^{'}\mbox{ }k_{2}}(k,q) \Lambda_{k^{'}}(q^{'}) + \Gamma^{-q\mbox{ }q_{2}}_{k \mbox{ }k_{2}}(k^{'}q^{'}) \Lambda_{k}(q)$$ $$= \Lambda_{k^{'}+q/2}(q+q^{'})\delta_{k-q/2, k^{'}+q^{'}/2} \delta_{k_{2}, k^{'}+q/2}\delta_{q_{2}, -q-q^{'} } - \Lambda_{k^{'}-q/2}(q+q^{'})\delta_{k+q/2, k^{'}-q^{'}/2} \delta_{k_{2}, k^{'}-q/2}\delta_{q_{2}, -q-q^{'} } \label{Comm}$$ First notice that $ \Lambda_{ {\bf{k}} }({\bf{q}}) $ is either zero or one. Notice that Eq.( \[Free\]), Eq.( \[Bose\]) and Eq.( \[Comm\]) are such that they suggest to us, $$\Gamma_{ {\bf{k}}_{1}, {\bf{k}}_{2} }^{ {\bf{q}}_{1}, {\bf{q}}_{2} } ({\bf{k}}, {\bf{q}}) = 0\mbox{ } unless\mbox{ } \Lambda_{ {\bf{k}}_{1} }(-{\bf{q}}_{1}) = 1 \mbox{ } and \mbox{ } \Lambda_{ {\bf{k}}_{2} }(-{\bf{q}}_{2}) = 1$$ $$\Gamma_{ {\bf{k}}_{1}, {\bf{k}}_{2} }^{ {\bf{q}}_{1}, {\bf{q}}_{2} } ({\bf{k}}, {\bf{q}}) = \delta_{ {\bf{k}}_{1} - {\bf{q}}_{1}/2 , {\bf{k}}_{2} - {\bf{q}}_{2}/2} \delta_{ {\bf{k}}_{1} + {\bf{q}}_{1}/2 , {\bf{k}}+ {\bf{q}}/2 } \delta_{ {\bf{k}} - {\bf{q}}/2 , {\bf{k}}_{2} + {\bf{q}}_{2}/2 }$$ $$- \delta_{ {\bf{k}}_{1} + {\bf{q}}_{1}/2 , {\bf{k}}_{2} + {\bf{q}}_{2}/2} \delta_{ {\bf{k}}_{1} - {\bf{q}}_{1}/2 , {\bf{k}} - {\bf{q}}/2 } \delta_{ {\bf{k}} + {\bf{q}}/2 , {\bf{k}}_{2} - {\bf{q}}_{2}/2 } \mbox{ } for \mbox{ } \Lambda_{ {\bf{k_{1}}} }(-{\bf{q_{1}}}) \Lambda_{ {\bf{k_{2}}} }(-{\bf{q_{2}}}) = 1$$ and $$\Gamma_{ {\bf{k}}_{1}, {\bf{k}}_{2} }^{ {\bf{q}}_{1}, {\bf{q}}_{2} } ({\bf{k}}, {\bf{q}}) = 0 \mbox{ } for \mbox{ } \Lambda_{ {\bf{k_{1}}} }(-{\bf{q_{1}}}) \Lambda_{ {\bf{k_{2}}} }(-{\bf{q_{2}}}) = 0$$ From this one may verify that the formula for the kinetic energy operator written down previously, Eq.( \[Kin\]) is correct. Lastly, we must make sure that the following additional commutation rule holds. $$\Gamma_{ {\bf{k_{1}}}, {\bf{k_{3}}} }^{ {\bf{q_{1}}}, {\bf{q_{3}}} } ({\bf{k}}, {\bf{q}})\mbox{ }\Gamma_{ {\bf{k_{3}}}, {\bf{k_{2}}} } ^{ {\bf{q_{3}}}, {\bf{q_{2}}} } ({\bf{k}}^{'}, {\bf{q}}^{'}) - \Gamma_{ {\bf{k_{3}}}, {\bf{k_{2}}} }^{ {\bf{q_{3}}}, {\bf{q_{2}}} } ({\bf{k}}, {\bf{q}})\mbox{ }\Gamma_{ {\bf{k_{1}}}, {\bf{k_{3}}} } ^{ {\bf{q_{1}}}, {\bf{q_{3}}} } ({\bf{k}}^{'}, {\bf{q}}^{'})$$ $$= \Gamma^{ {\bf{q_{1}}}, {\bf{q_{2}}} }_{ {\bf{k_{1}}}, {\bf{k_{2}}} } ({\bf{k}}^{'} + {\bf{q}}/2, {\bf{q+q^{'}}} ) \delta_{ {\bf{k-q/2}}, {\bf{k^{'}+q^{'}/2}} } - \Gamma^{ {\bf{q_{1}}}, {\bf{q_{2}}} }_{ {\bf{k_{1}}}, {\bf{k_{2}}} } ({\bf{k}} + {\bf{q}}^{'}/2, {\bf{q+q^{'}}} ) \delta_{ {\bf{k^{'}-q^{'}/2}}, {\bf{k+q/2}} }$$ Verification of the above rule completes the confirmation that the formula for $ c^{\dagger}_{ {\bf{k+q/2}} }c_{ {\bf{k-q/2}} } $ in terms of the bose fields is exact. The algebra here goes through without a hitch expect in this respect, namely we have to assume that the $ \Lambda_{ {\bf{k}} }(-{\bf{q}}) = 1 $. To summarise let us write down the following: \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ [[THEOREM:]{}]{} $$c^{\dagger}_{ {\bf{k+q/2}} }c_{ {\bf{k-q/2}} } = n_{F}({\bf{k}})\frac{N}{\langle N \rangle }\delta_{ {\bf{q}}, {\bf{0}} } + (\sqrt{ \frac{N}{\langle N \rangle } }) [\Lambda_{ {\bf{k}} }({\bf{q}})a_{ {\bf{k}} }(-{\bf{q}}) + \Lambda_{ {\bf{k}} }(-{\bf{q}})a^{\dagger}_{ {\bf{k}} }({\bf{q}}) ]$$ $$+ \sum_{ {\bf{k}}_{1}, {\bf{k}}_{2} } \sum_{ {\bf{q}}_{1}, {\bf{q}}_{2} } \Gamma_{ {\bf{k}}_{1}, {\bf{k}}_{2} }^{ {\bf{q}}_{1}, {\bf{q}}_{2} } ({\bf{k}}, {\bf{q}})a^{\dagger}_{ {\bf{k}}_{1} }({\bf{q}}_{1}) a_{ {\bf{k}}_{2} }({\bf{q}}_{2})$$ together with $$\Gamma_{ {\bf{k}}_{1}, {\bf{k}}_{2} }^{ {\bf{q}}_{1}, {\bf{q}}_{2} } ({\bf{k}}, {\bf{q}}) = \Lambda_{ {\bf{k_{1}}} }(-{\bf{q_{1}}}) \Lambda_{ {\bf{k_{2}}} }(-{\bf{q_{2}}}) [ \delta_{ {\bf{k}}_{1} - {\bf{q}}_{1}/2 , {\bf{k}}_{2} - {\bf{q}}_{2}/2} \delta_{ {\bf{k}}_{1} + {\bf{q}}_{1}/2 , {\bf{k}}+ {\bf{q}}/2 } \delta_{ {\bf{k}} - {\bf{q}}/2 , {\bf{k}}_{2} + {\bf{q}}_{2}/2 }$$ $$- \delta_{ {\bf{k}}_{1} + {\bf{q}}_{1}/2 , {\bf{k}}_{2} + {\bf{q}}_{2}/2} \delta_{ {\bf{k}}_{1} - {\bf{q}}_{1}/2 , {\bf{k}} - {\bf{q}}/2 } \delta_{ {\bf{k}} + {\bf{q}}/2 , {\bf{k}}_{2} - {\bf{q}}_{2}/2 } ]$$ is an exact transformation of products of Fermi fields into Bose fields. Namely, all dynamical moments of $ c^{\dagger}_{ {\bf{k+q/2}} }c_{ {\bf{k-q/2}} } $ are recovered exactly, provided we identify the filled Fermi sea with the Bose vacuum. $$a_{ {\bf{k}} }({\bf{q}}) \|FS \rangle = 0$$ and the above is also an operator identity, namely they satisfy proper mutual commutation rules. \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ This may also be written in a simplified form as, $$c^{\dagger}_{ {\bf{k+q/2}} }c_{ {\bf{k-q/2}} } = n_{F}({\bf{k}})\frac{N}{\langle N \rangle }\delta_{ {\bf{q}}, {\bf{0}} } + (\sqrt{ \frac{N}{\langle N \rangle } }) [\Lambda_{ {\bf{k}} }({\bf{q}})a_{ {\bf{k}} }(-{\bf{q}}) + \Lambda_{ {\bf{k}} }(-{\bf{q}})a^{\dagger}_{ {\bf{k}} }({\bf{q}}) ]$$ $$+ \sum_{ {\bf{q}}_{1} }\Lambda_{ {\bf{k+q/2 - q_{1}/2}} }(-{\bf{q_{1}}}) \Lambda_{ {\bf{k - q_{1}/2}} }({\bf{q-q_{1}}}) a^{\dagger}_{ {\bf{k}} + {\bf{q}}/2 - {\bf{q}}_{1}/2 }({\bf{q}}_{1}) a_{ {\bf{k}} - {\bf{q}}_{1}/2 }(-{\bf{q}} + {\bf{q}}_{1})$$ $$- \sum_{ {\bf{q}}_{1} }\Lambda_{ {\bf{k-q/2 + q_{1}/2}} }(-{\bf{q_{1}}}) \Lambda_{ {\bf{k + q_{1}/2}} }({\bf{q-q_{1}}}) a^{\dagger}_{ {\bf{k}} - {\bf{q}}/2 + {\bf{q}}_{1}/2 }({\bf{q}}_{1}) a_{ {\bf{k}} + {\bf{q}}_{1}/2 }(-{\bf{q}} + {\bf{q}}_{1})$$ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\* The $ \frac{N}{\langle N \rangle } $ is needed to ensure that, $ \sum_{ {\bf{k}} }c^{\dagger}_{ {\bf{k}} }c_{ {\bf{k}} } = N $ (and not $ \sum_{ {\bf{k}} }c^{\dagger}_{ {\bf{k}} }c_{ {\bf{k}} } = \langle N \rangle $) The $ (\sqrt{ \frac{N}{\langle N \rangle } }) $ is needed to ensure that the commutation amongst the $ c^{\dagger}_{ {\bf{k+q/2}} }c_{ {\bf{k-q/2}} } $ come out right. However, there are other finer points that need to be discussed. For example, do the thermodynamic expectation values also come out right ? That is, assuming that the temperature is finite, can we compute the dynamical density correlation function and show that it agrees with the free theory ? The finite temp. case is most unusual. It seems that for dealing with fermions at finite temperature, we have to assume that the Bose like-excitations are always at zero temperature with zero chemical potential, only the coefficients in the expansion acquire finite temp. values. All this is very strange and on shaky ground. Thus, I shall relegate the finite temp. case to the last section where the skeptical reader may have a field day demolishing my ideas. The Fermi Field Operator ------------------------ Also how about expressing the Fermi fields themselves in terms of the Fermi sea displacements ? For this one has to first construct a canonical conjugate of the density and then use it in the DPVA ansatz Eq.( \[DPVA\]) [@Setlur] to compute the phase functional $ \Phi $. I have tried to construct a canonical conjugate of $ \rho $ ( any canonical conjugate suffices ) but it has not been successful. Just for future reference, it is useful to write down a formula for the total current operator. $${\bf{J}}_{tot} = \sum_{ {\bf{k}} }(\frac{ {\bf{k}} }{m}) c^{\dagger}_{ {\bf{k}} } c_{ {\bf{k}} } = \sum_{ {\bf{k}}_{1}, {\bf{k}}_{2} } \sum_{ {\bf{q}}_{1}, {\bf{q}}_{2} }{\bf{\Gamma}}_{ {\bf{k}}_{1}, {\bf{k}}_{2} } ^{ {\bf{q}}_{1}, {\bf{q}}_{2} }a^{\dagger}_{ {\bf{k}}_{1} } ({\bf{q}}_{1})a_{ {\bf{k}}_{2} }({\bf{q}}_{2})$$ $${\bf{\Gamma}}_{ {\bf{k}}_{1}, {\bf{k}}_{2} }^{ {\bf{q}}_{1}, {\bf{q}}_{2} } = \Lambda_{ {\bf{k}}_{1} }(-{\bf{q}}_{1}) \Lambda_{ {\bf{k}}_{2} }(-{\bf{q}}_{2}) (\frac{ {\bf{q}}_{1} }{m})\delta_{ {\bf{k}}_{1}, {\bf{k}}_{2} } \delta_{ {\bf{q}}_{1}, {\bf{q}}_{2} }$$ Therefore, $${\bf{J}}_{tot} = \sum_{ {\bf{k}} }(\frac{ {\bf{k}} }{m}) c^{\dagger}_{ {\bf{k}} } c_{ {\bf{k}} } = \sum_{ {\bf{k}}_{1}, {\bf{q}}_{1} } \Lambda_{ {\bf{k}}_{1} }(-{\bf{q}}_{1}) (\frac{ {\bf{q}}_{1} }{m})a^{\dagger}_{ {\bf{k}}_{1} } ({\bf{q}}_{1})a_{ {\bf{k}}_{1} }({\bf{q}}_{1})$$ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Thus an alternative but less systematic approach seems to be in order. For this choose, $$\psi({\bf{x}}) = exp( -i\mbox{ }U_{0}({\bf{x}}) ) exp( -i\mbox{ }U_{1}({\bf{x}}) ) exp( -i\mbox{ }U_{2}({\bf{x}}) ) ( \rho({\bf{x}}) )^{\frac{1}{2}}$$ here $ U_{0}({\bf{x}}) $ is a c-number and, $$U_{1}({\bf{x}}) = \sum_{ {\bf{k}}, {\bf{q}} }[a_{ {\bf{k}} }({\bf{q}}) f({\bf{k}}, {\bf{q}}; {\bf{x}}) + a^{\dagger}_{ {\bf{k}} }({\bf{q}}) f^{}({\bf{k}}, {\bf{q}}; {\bf{x}})]$$ $$U_{2}({\bf{x}}) = \sum_{ {\bf{k}}_{1}, {\bf{q}}_{1} } \sum_{ {\bf{k}}_{2}, {\bf{q}}_{2} } U_{2}({\bf{k}}_{1}, {\bf{q}}_{1}; {\bf{k}}_{2}, {\bf{q}}_{2}; {\bf{x}}) a^{\dagger}_{ {\bf{k}}_{1} }({\bf{q}}_{1}) a_{ {\bf{k}}_{2} }({\bf{q}}_{2})$$ Now let us evolve this using the kinetic energy operator, $$\psi({\bf{x}},t) = e^{i \mbox{ }t\mbox{ }K}\psi({\bf{x}})e^{-i \mbox{ }t\mbox{ }K} = exp(-i\mbox{ }U_{0}({\bf{x}})) exp(-i\mbox{ }U_{1}({\bf{x}},t)) exp(-i\mbox{ }U_{2}({\bf{x}},t))( \rho({\bf{x}},t) )^{\frac{1}{2}}$$ here, $$U_{1}({\bf{x}},t) = \sum_{ {\bf{k}}, {\bf{q}} }[a_{ {\bf{k}} }({\bf{q}}) e^{-i\mbox{ }t\mbox{ }\omega_{ {\bf{k}} }({\bf{q}})} f({\bf{k}}, {\bf{q}}; {\bf{x}}) + a^{\dagger}_{ {\bf{k}} }({\bf{q}}) e^{i\mbox{ }t\mbox{ }\omega_{ {\bf{k}} }({\bf{q}})} f^{}({\bf{k}}, {\bf{q}}; {\bf{x}})]$$ $$U_{2}({\bf{x}},t) = \sum_{ {\bf{k}}_{1}, {\bf{q}}_{1} } \sum_{ {\bf{k}}_{2}, {\bf{q}}_{2} } U_{2}({\bf{k}}_{1}, {\bf{q}}_{1}; {\bf{k}}_{2}, {\bf{q}}_{2}; {\bf{x}}) a^{\dagger}_{ {\bf{k}}_{1} }({\bf{q}}_{1}) a_{ {\bf{k}}_{2} }({\bf{q}}_{2}) e^{i\mbox{ }t\mbox{ } (\omega_{ {\bf{k}}_{1} }({\bf{q}}_{1})- \omega_{ {\bf{k}}_{2} }({\bf{q}}_{2}))}$$ The basic philosophy it seems, involves observing that the commutation rules satisfied by number conserving objects such as $ c^{\dagger}_{ {\bf{k+q/2}} }c_{ {\bf{k-q/2}} } $ amongst themselves is independent of the underlying statistics. Thus one is lead to postulate convenient forms for such objects in terms of other Bose fields. But not all approaches are likely to be correct. Take for example, the straightforward substitution, $$c^{\dagger}_{ {\bf{k+q/2}} }c_{ {\bf{k-q/2}} } = b^{\dagger}_{ {\bf{k+q/2}} }b_{ {\bf{k-q/2}} }$$ where $ c_{ {\bf{k}} } $ are fermions and $ b_{ {\bf{k}} } $ are bosons. This no doubt makes the commutation rules come out right but is quite obviously wrong because all the correlation functions are wrongly represented. Luttinger and Fermi Liquids =========================== Consider an interaction of the type, $$H_{I} = \sum_{ {\bf{q}} \neq 0 }\frac{ v_{ {\bf{q}} } }{2 \mbox{ }V} \sum_{ {\bf{k}}, {\bf{k}}^{'} } [\Lambda_{ {\bf{k}} }({\bf{q}})a_{ {\bf{k}} }(-{\bf{q}}) + \Lambda_{ {\bf{k}} }(-{\bf{q}})a^{\dagger}_{ {\bf{k}} }({\bf{q}})] [\Lambda_{ {\bf{k}}^{'} }(-{\bf{q}})a_{ {\bf{k}}^{'} }({\bf{q}}) + \Lambda_{ {\bf{k}}^{'} }({\bf{q}})a^{\dagger}_{ {\bf{k}}^{'} }(-{\bf{q}})]$$ $$i \mbox{ }\frac{\partial}{ \partial t }a^{t}_{ {\bf{k}} }({\bf{q}}) = \omega_{ {\bf{k}} }({\bf{q}})a^{t}_{ {\bf{k}} }({\bf{q}}) + (\frac{v_{ {\bf{q}} } }{V}) \Lambda_{ {\bf{k}} }(-{\bf{q}})\sum_{ {\bf{k}}^{'} }[ \Lambda_{ {\bf{k}}^{'} }(-{\bf{q}})a^{t}_{ {\bf{k}}^{'} }({\bf{q}}) + \Lambda_{ {\bf{k}}^{'} }({\bf{q}})a^{\dagger t}_{ {\bf{k}}^{'} }(-{\bf{q}}) ]$$ The equations of motion for the Bose propagators read as, $$(i\frac{\partial}{\partial t} - \omega_{ {\bf{k}} }({\bf{q}})) \frac{ -i\langle T a^{t}_{ {\bf{k}} }({\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle} = \delta_{ {\bf{k}}, {\bf{k}}^{'} } \delta_{ {\bf{q}}, {\bf{q}}^{'} }\delta(t)$$ $$+ (\frac{ v_{ {\bf{q}} } }{V}) \Lambda_{ {\bf{k}} }(-{\bf{q}}) \sum_{ {\bf{k}}^{''} }[\Lambda_{ {\bf{k}}^{''} }(-{\bf{q}}) \frac{ -i\langle T a^{t}_{ {\bf{k}}^{''} }({\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle} + \Lambda_{ {\bf{k}}^{''} }({\bf{q}}) \frac{ -i\langle T a^{\dagger t}_{ {\bf{k}}^{''} }(-{\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle}]$$ $$(i\frac{\partial}{\partial t} + \omega_{ {\bf{k}} }(-{\bf{q}})) \frac{ -i\langle T a^{\dagger t}_{ {\bf{k}} }(-{\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle}$$ $$= -(\frac{ v_{ {\bf{q}} } }{V}) \Lambda_{ {\bf{k}} }({\bf{q}}) \sum_{ {\bf{k}}^{''} }[\Lambda_{ {\bf{k}}^{''} }({\bf{q}}) \frac{ -i\langle T a^{\dagger t}_{ {\bf{k}}^{''} }(-{\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle} + \Lambda_{ {\bf{k}}^{''} }(-{\bf{q}}) \frac{ -i\langle T a^{t}_{ {\bf{k}}^{''} }({\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle} ]$$ The boundary conditions on these propagators may be written down as, $$\frac{ -i\langle T a^{\dagger t}_{ {\bf{k}} }(-{\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle} = \frac{ -i\langle T a^{\dagger (t - i\beta)}_{ {\bf{k}} }(-{\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle}$$ $$\frac{ -i\langle T a^{t}_{ {\bf{k}} }({\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle} = \frac{ -i\langle T a^{(t - i\beta)}_{ {\bf{k}} }({\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle}$$ $$\delta(t) = (\frac{1}{-i \mbox{ }\beta})\sum_{n} exp(\omega_{n}t)$$ $$\theta(t) = (\frac{1}{-i \mbox{ }\beta})\sum_{n} \frac{ exp(\omega_{n}t) }{\omega_{n}}$$ The boundary conditions imply that we may write, $$\frac{ -i\langle T a^{t}_{ {\bf{k}} }({\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle} = \sum_{n}\mbox{ }exp(\omega_{n}t)\mbox{ } \frac{ -i\langle T a^{n}_{ {\bf{k}} }({\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle}$$ $$\frac{ -i\langle T a^{\dagger t}_{ {\bf{k}} }(-{\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle} = \sum_{n}\mbox{ }exp(\omega_{n}t)\mbox{ } \frac{ -i\langle T a^{\dagger n}_{ {\bf{k}} }(-{\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle}$$ and, $ \omega_{n} = (2\mbox{ }\pi \mbox{ }n)/\beta $. Thus, $$(i\omega_{n} - \omega_{ {\bf{k}} }({\bf{q}})) \frac{ -i\langle T a^{n}_{ {\bf{k}} }({\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle} = \frac{ \delta_{ {\bf{k}}, {\bf{k}}^{'} } \delta_{ {\bf{q}}, {\bf{q}}^{'} } }{-i \mbox{ }\beta}$$ $$+ (\frac{ v_{ {\bf{q}} } }{V}) \Lambda_{ {\bf{k}} }(-{\bf{q}}) \sum_{ {\bf{k}}^{''} }[\Lambda_{ {\bf{k}}^{''} }(-{\bf{q}}) \frac{ -i\langle T a^{n}_{ {\bf{k}}^{''} }({\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle} + \Lambda_{ {\bf{k}}^{''} }({\bf{q}}) \frac{ -i\langle T a^{\dagger n}_{ {\bf{k}}^{''} }(-{\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle}]$$ $$(i\omega_{n} + \omega_{ {\bf{k}} }(-{\bf{q}})) \frac{ -i\langle T a^{\dagger n}_{ {\bf{k}} }(-{\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle}$$ $$= -(\frac{ v_{ {\bf{q}} } }{V}) \Lambda_{ {\bf{k}} }({\bf{q}}) \sum_{ {\bf{k}}^{''} }[\Lambda_{ {\bf{k}}^{''} }({\bf{q}}) \frac{ -i\langle T a^{\dagger n}_{ {\bf{k}}^{''} }(-{\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle} + \Lambda_{ {\bf{k}}^{''} }(-{\bf{q}}) \frac{ -i\langle T a^{n}_{ {\bf{k}}^{''} }({\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle} ]$$ Define, $$\sum_{ {\bf{k}} } \Lambda_{ {\bf{k}} } (-{\bf{q}}) \frac{ -i\langle T a^{n}_{ {\bf{k}} }({\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle} = G_{1}( {\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n)$$ $$\sum_{ {\bf{k}} } \Lambda_{ {\bf{k}} } ({\bf{q}}) \frac{ -i\langle T a^{\dagger n}_{ {\bf{k}} }(-{\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle} = G_{2}( {\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n)$$ Multiplying the above equations with $ \Lambda_{ {\bf{k}} }(-{\bf{q}}) $ and summing over $ {\bf{k}} $ one arrives at simple formulas for $ G_{1} $ and $ G_{2} $. $$G_{1}( {\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) = \Lambda_{ {\bf{k}}^{'} }(-{\bf{q}}) \frac{ \delta_{ {\bf{q}}, {\bf{q}}^{'} } } { -i \mbox{ }\beta( i\omega_{n} - \omega_{ {\bf{k}}^{'} }({\bf{q}}) ) }$$ $$+ f_{n}({\bf{q}})[G_{1}( {\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) + G_{2}( {\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) ]$$ and, $$G_{2}( {\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) = f^{}_{n}(-{\bf{q}})[ G_{1}( {\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) + G_{2}( {\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) ]$$ and, $$f_{n}({\bf{q}}) = (\frac{ v_{ {\bf{q}} } }{V}) \sum_{ {\bf{k}} }\frac{ \Lambda_{ {\bf{k}} }(-{\bf{q}}) } { ( i\omega_{n} - \omega_{ {\bf{k}} }({\bf{q}}) ) }$$ In 1D and 3D an analytical solution for $ f_{n}({\bf{q}}) $ is possible. In 1D we have, $$f_{n}(q) = v_{q}(-\frac{m}{q})(\frac{1}{2\pi}) ln ( \frac{ k_{f} - \frac{ m \mbox{ }i\omega_{n} }{q} + \frac{q}{2} } { -k_{f} - \frac{ m \mbox{ }i\omega_{n} }{q} + \frac{q}{2} } ) - \theta(b-a)\mbox{ }v_{q}\mbox{ }(\frac{1}{2\pi})(-\frac{m}{q}) ln( \frac{ b - \frac{ m \mbox{ }i\omega_{n} }{q} + \frac{q}{2} } { a - \frac{ m \mbox{ }i\omega_{n} }{q} + \frac{q}{2} } )$$ where, $ b = min(k_{f}, k_{f} - q) $ and $ a = max(-k_{f}, -k_{f}-q) $. The logarithm is to be interpreted as the principal value, that is, $ Im(ln(z)) = Arg(z) = tan^{-1}(y/x) $, $ x = Re(z), y = Im(z) $, and $ Re(ln(z)) = ln(r) $ and, $ r = (x^{2} + y^{2})^{\frac{1}{2}} $. The nice thing about this definition (not shared by non-principal value defns.) is that $ (ln(z))^{} = ln(z^{}) $. This is because $ -\pi \leq Arg(z) = tan^{-1}(y/x) \leq \pi $. This may easily be solved and the final formulas for the Bose propagators may be written down as, $$G_{ 2 }({\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) = \frac{ f^{}_{n}(-{\bf{q}}) }{(1- f^{}_{n}(-{\bf{q}}))} G_{1} ({\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n)$$ $$G_{1}({\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) + G_{2}({\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) = G_{1}({\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) /( 1 - f^{}_{n}(-{\bf{q}}) )$$ $$G_{1}({\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) = (\frac{1}{-i \mbox{ }\beta}) \frac{ (1-f_{n}^{}(-{\bf{q}}) ) \Lambda_{ {\bf{k}}^{'} }(-{\bf{q}}) \delta_{ {\bf{q}}, {\bf{q}}^{'} } } {(1- f_{n}^{}(-{\bf{q}}) - f_{n}({\bf{q}}) ) ( i\omega_{n} - \omega_{ {\bf{k}}^{'} }({\bf{q}}) ) }$$ $$G_{2}({\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) = (\frac{1}{-i \mbox{ }\beta}) \frac{ f_{n}^{}(-{\bf{q}}) \Lambda_{ {\bf{k}}^{'} }(-{\bf{q}}) \delta_{ {\bf{q}}, {\bf{q}}^{'} } } {(1- f_{n}^{}(-{\bf{q}}) - f_{n}({\bf{q}}) ) ( i\omega_{n} - \omega_{ {\bf{k}}^{'} }({\bf{q}}) ) }$$ $$G_{1}({\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) + G_{2}({\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) = (\frac{1}{-i \mbox{ }\beta})\frac{ \Lambda_{ {\bf{k}}^{'} }(-{\bf{q}}) \delta_{ {\bf{q}}, {\bf{q}}^{'} } } {(1- f_{n}^{}(-{\bf{q}}) - f_{n}({\bf{q}}) ) ( i\omega_{n} - \omega_{ {\bf{k}}^{'} }({\bf{q}}) ) }$$ $$\frac{ -i\langle T a^{n}_{ {\bf{k}} }({\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{ \langle T 1 \rangle } = \frac{ \delta_{ {\bf{k}}, {\bf{k}}^{'} } \delta_{ {\bf{q}}, {\bf{q}}^{'} } } {-i\mbox{ }\beta( i\omega_{n} - \omega_{ {\bf{k}} }({\bf{q}}) )}$$ $$+ (\frac{1}{-i \mbox{ }\beta}) ( \frac{ v_{ {\bf{q}} } }{V} )\frac{ \Lambda_{ {\bf{k}} }(-{\bf{q}}) } {(i\omega_{ n } - \omega_{ {\bf{k}} }({\bf{q}}) ) } \frac{ \Lambda_{ {\bf{k}}^{'} }(-{\bf{q}}) \delta_{ {\bf{q}}, {\bf{q}}^{'} } } {(1- f_{n}^{}(-{\bf{q}}) - f_{n}({\bf{q}}) ) ( i\omega_{n} - \omega_{ {\bf{k}}^{'} }({\bf{q}}) ) }$$ also, $$\frac{ -i\langle T a^{\dagger n}_{ {\bf{k}} }(-{\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{ \langle T 1 \rangle } = -(\frac{1}{-i \mbox{ } \beta}) ( \frac{ v_{ {\bf{q}} } }{V} ) \frac{ \Lambda_{ {\bf{k}} }({\bf{q}}) } {(i\omega_{ n } + \omega_{ {\bf{k}} }(-{\bf{q}}) ) } \frac{ \Lambda_{ {\bf{k}}^{'} }(-{\bf{q}}) \delta_{ {\bf{q}}, {\bf{q}}^{'} } } {(1- f_{n}^{}(-{\bf{q}}) - f_{n}({\bf{q}}) ) ( i\omega_{n} - \omega_{ {\bf{k}}^{'} }({\bf{q}}) ) }$$ The zero temperature correlation function of significance here is, $$-i\langle a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) a_{ {\bf{k}} }({\bf{q}}) \rangle$$ This may be obtained from the above formulas as, $$-i\langle a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) a_{ {\bf{k}} }({\bf{q}}) \rangle = -(\frac{v_{ {\bf{q}} } }{V}) \Lambda_{ {\bf{k}} }(-{\bf{q}})\Lambda_{ {\bf{k}}^{'} }(-{\bf{q}}) \delta_{ {\bf{q}}, {\bf{q}}^{'} } \int_{C} \frac{d\omega}{2\pi \mbox{ }i} \frac{1}{( i\omega - \omega_{ {\bf{k}} }({\bf{q}}) ) ( i\omega - \omega_{ {\bf{k}}^{'} }({\bf{q}}) ) ( 1 - f_{n}^{}(-{\bf{q}}) - f_{n}({\bf{q}}) ) }$$ where $ C $ is the positively oriented contour that encloses the upper half plane. Thus the problem now boils down to computing all the zeros of $ ( 1 - f_{n}^{}(-{\bf{q}}) - f_{n}({\bf{q}}) ) $ that have positive imaginary parts. In 1D, we may proceed as follows, $$1 - f^{}_{n}(-q) - f_{n}(q) = 1 - v_{q}(-\frac{m}{q})(\frac{1}{2\pi}) ln[\frac{k_{f} - \frac{m \mbox{ }i\omega}{q} + \frac{q}{2} } { -k_{f} - \frac{m \mbox{ }i\omega}{q} + \frac{q}{2} }]$$ $$+ \theta(b-a) v_{q}(\frac{1}{2\pi}) (-\frac{m}{q})ln[\frac{b - \frac{m \mbox{ }i\omega}{q} + \frac{q}{2} } { a - \frac{m \mbox{ }i\omega}{q} + \frac{q}{2} }]$$ $$- v_{q}(\frac{m}{q})(\frac{1}{2\pi}) ln[\frac{k_{f} - \frac{m \mbox{ }i\omega}{q} - \frac{q}{2} } { -k_{f} - \frac{m \mbox{ }i\omega}{q} - \frac{q}{2} }]$$ $$+ \theta(b^{'}-a^{'}) v_{q}(\frac{1}{2\pi}) (\frac{m}{q})ln[\frac{b^{'} - \frac{m \mbox{ }i\omega}{q} - \frac{q}{2} } { a^{'} - \frac{m \mbox{ }i\omega}{q} - \frac{q}{2} }]$$ where $ a = max(- k_{f}, - k_{f} - q) $, $ b = min(k_{f}, k_{f} - q) $ and $ a^{'} = max(- k_{f}, - k_{f} + q) $, $ b^{'} = min(k_{f}, k_{f} + q) $. There are several regions of interest. (A) $ 0 \leq q \leq k_{f} $ This means, $ 0 \geq -q \geq -k_{f} $ or $ k_{f} \geq k_{f}-q \geq 0 $, $ -k_{f} \geq -k_{f}-q \geq -2k_{f} $, $ k_{f} \leq k_{f} + q \leq 2k_{f} $, $ -k_{f} \leq -k_{f} + q \leq 0 $ Thus $ a = -k_{f} $, $ b = k_{f} - q $, $ a^{'} = -k_{f} + q $, $ b^{'} = k_{f} $. $ \theta(b-a) = 1 $, $ \theta(b^{'}-a^{'}) = 1 $ Therefore, $$1 - f^{}_{n}(-q) - f_{n}(q) = 1 + v_{q}(\frac{m}{q})(\frac{1}{2\pi}) ln[\frac{k_{f} - \frac{m \mbox{ }i\omega}{q} + \frac{q}{2} } { k_{f} - \frac{m \mbox{ }i\omega}{q} - \frac{q}{2} }]$$ $$+ v_{q}(\frac{1}{2\pi}) (\frac{m}{q})ln[\frac{ -k_{f} - \frac{m \mbox{ }i\omega}{q} - \frac{q}{2} } { - k_{f} - \frac{m \mbox{ }i\omega}{q} + \frac{q}{2} }]$$ $$1 - f^{}_{n}(-q) - f_{n}(q) = 1 + v_{q}(\frac{1}{2\pi}) (\frac{m}{q})ln[ \frac{ (k_{f}+q/2)^{2} + (\frac{m \mbox{ }\omega}{q})^{2} } { (k_{f} - q/2)^{2} + (\frac{m \mbox{ }\omega}{q})^{2} } ] = 0$$ $$ln[ \frac{ (k_{f}+q/2)^{2} + (\frac{m \mbox{ }\omega}{q})^{2} } { (k_{f} - q/2)^{2} + (\frac{m \mbox{ }\omega}{q})^{2} } ] = -(\frac{ 2\mbox{ }\pi\mbox{ }q }{m})(\frac{1}{v_{q}})$$ $$\frac{ (k_{f}+q/2)^{2} + (\frac{m \mbox{ }\omega}{q})^{2} } { (k_{f} - q/2)^{2} + (\frac{m \mbox{ }\omega}{q})^{2} } = exp(-(\frac{ 2\mbox{ }\pi\mbox{ }q }{m})(\frac{1}{v_{q}}))$$ $$(\frac{m \mbox{ }\omega}{q})^{2} = -[(k_{f}+q/2)^{2} - (k_{f} - q/2)^{2} exp(-(\frac{ 2\mbox{ }\pi\mbox{ }q }{m})(\frac{1}{v_{q}})) ] /[1- exp(-(\frac{ 2\mbox{ }\pi\mbox{ }q }{m})(\frac{1}{v_{q}}))]$$ We want to find a root of this that has a positive imaginary part. $$\omega = i\mbox{ }(\frac{q}{m}) \sqrt{ \frac{ (k_{f}+q/2)^{2} - (k_{f} - q/2)^{2} exp(-(\frac{ 2\mbox{ }\pi\mbox{ }q }{m})(\frac{1}{v_{q}})) } {1- exp(-(\frac{ 2\mbox{ }\pi\mbox{ }q }{m})(\frac{1}{v_{q}}))} }$$ The next case is, (B) $ -k_{f} \leq q \leq 0 $. In this case, $ 0 \leq k_{f} + q \leq k_{f} $, $ -2\mbox{ }k_{f} \leq -k_{f} + q \leq -k_{f} $, $ k_{f} \geq -q \geq 0 $, $ 2\mbox{ }k_{f} \geq k_{f}-q \geq k_{f} $, $ 0 \geq -k_{f}-q \geq -k_{f} $, $ a = -k_{f} - q $, $ b = k_{f} $, $ a^{'} = -k_{f} $, $ b^{'} = k_{f} + q $. Thus, $ \theta(b-a) = 1 $, $ \theta(b^{'} - a^{'}) = 1 $ For case (B) $$1 - f^{}_{n}(-q) - f_{n}(q) = 1 + v_{q}(\frac{m}{q})(\frac{1}{2\pi}) ln[\frac{-k_{f} - \frac{m \mbox{ }i\omega}{q} - \frac{q}{2} } { -k_{f} - \frac{m \mbox{ }i\omega}{q} + \frac{q}{2} }]$$ $$+ v_{q}(\frac{1}{2\pi}) (\frac{m}{q})ln[\frac{k_{f} - \frac{m \mbox{ }i\omega}{q} + \frac{q}{2} } { k_{f} - \frac{m \mbox{ }i\omega}{q} - \frac{q}{2} }]$$ Thus the pole for both cases (A) and (B) is given by, $$\omega = i\mbox{ }(\frac{\|q\|}{m}) \sqrt{ \frac{ (k_{f}+q/2)^{2} - (k_{f} - q/2)^{2} exp(-(\frac{ 2\mbox{ }\pi\mbox{ }q }{m})(\frac{1}{v_{q}})) } { 1- exp(-(\frac{ 2\mbox{ }\pi\mbox{ }q }{m})(\frac{1}{v_{q}})) } } \label{ROOT}$$ The quantity under the square root is positive in both cases. Third, we have the case, (C) $ k_{f} \leq q \leq 2\mbox{ }k_{f} $ , this means, $ -k_{f} \geq -q \geq -2\mbox{ }k_{f} $, also, $ 2\mbox{ }k_{f} \leq k_{f} + q \leq 3\mbox{ }k_{f} $, $ 0 \leq -k_{f} + q \leq k_{f} $, $ 0 \geq k_{f}-q \geq -k_{f} $, $ -2\mbox{ }k_{f} \geq -k_{f}-q \geq -3\mbox{ }k_{f} $. $ a = -k_{f} $, $ b = k_{f} - q $, $ a^{'} = -k_{f} + q $, $ b^{'} = k_{f} $. $ \theta(b-a) = 1 $ and $ \theta(b^{'}-a^{'}) = 1 $ This leads to the same root Eq.( \[ROOT\]). In case (D) $ -2\mbox{ }k_{f} \leq q \leq -k_{f} $, $ -k_{f} \leq k_{f}+ q \leq 0 $, $ -3\mbox{ }k_{f} \leq -k_{f}+q \leq -2\mbox{ }k_{f} $, $ 2\mbox{ }k_{f} \geq -q \geq k_{f} $, $ 3\mbox{ }k_{f} \geq k_{f}-q \geq 2\mbox{ }k_{f} $, $ k_{f} \geq -k_{f}-q \geq 0 $. $ a = -k_{f}-q $, $ b = k_{f} $, $ a^{'} = -k_{f} $, $ b^{'} = k_{f}+ q $. $ \theta(b-a) = 1 $ and $ \theta(b^{'}-a^{'}) = 1 $. In this case also we find the root given by Eq.( \[ROOT\]). (E) $ 2\mbox{ }k_{f} \leq q < \infty $, $ -2\mbox{ }k_{f} \geq -q > -\infty $, $ 3\mbox{ }k_{f} \leq k_{f}+q < \infty $, $ k_{f} \leq -k_{f}+q < \infty $, $ -k_{f} \geq k_{f}-q > -\infty $, $ -3\mbox{ }k_{f} \geq -k_{f}-q > -\infty $. $ a = -k_{f} $, $ b = k_{f}-q $, $ a^{'} = -k_{f}+q $ $ b^{'} = k_{f} $. $ \theta(b-a) = 0 $, $ \theta(b^{'}-a^{'}) = 0 $. This case is somewhat different. $$1 - f^{}_{n}(-q) - f_{n}(q) = 1 + v_{q}(\frac{m}{q})(\frac{1}{2\pi}) ln[\frac{k_{f} - \frac{m \mbox{ }i\omega}{q} + \frac{q}{2} } { -k_{f} - \frac{m \mbox{ }i\omega}{q} + \frac{q}{2} }]$$ $$- v_{q}(\frac{m}{q})(\frac{1}{2\pi}) ln[\frac{k_{f} - \frac{m \mbox{ }i\omega}{q} - \frac{q}{2} } { -k_{f} - \frac{m \mbox{ }i\omega}{q} - \frac{q}{2} }]$$ This when solved gives the same result as Eq.( \[ROOT\]). So too does the final case (F) which is (F) $ -\infty < q \leq -2 \mbox{ }k_{f} $, $ -\infty < k_{f}+q \leq -k_{f} $, $ -\infty < -k_{f}+q \leq -3 \mbox{ }k_{f} $, $ \infty > -q \geq 2 \mbox{ }k_{f} $, $ \infty > k_{f}-q \geq 3 \mbox{ }k_{f} $, $ \infty > -k_{f}-q \geq k_{f} $. This case also leads to the same result namely, Eq.( \[ROOT\]). Therefore the final result may be written as, $$\langle a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'})a_{ {\bf{k}} }({\bf{q}}) \rangle = (\frac{1}{V})\frac { \Lambda_{k}(-q)\Lambda_{k^{'}}(-q) \delta_{ q, q^{'} } } { (\omega_{R}(q)+ \omega_{k}(q))(\omega_{R}(q)+ \omega_{k^{'}}(q)) (\frac{m}{q^{2}})(\frac{1}{2 \pi k_{f}})2(\frac{m}{q})^{2}\omega_{R}(q) (cosh(\lambda(q))-1) }$$ Here we may write, $$\lambda(q) = (\frac{2 \pi q}{m})(\frac{1}{v_{q}})$$ $$\omega_{R}(q) = (\frac{ \|q\| }{m}) \sqrt{ \frac{ (k_{f} + q/2)^{2} - (k_{f} - q/2)^{2}exp(-\lambda(q)) } { 1 - exp(-\lambda(q)) } }$$ In other words, $$\langle c^{\dagger}_{ k }c_{ k } \rangle = n_{F}(k) + (2\pi k_{f}) \int_{-\infty}^{+\infty} \mbox{ }\frac{ dq_{1} }{2\pi}\mbox{ } \frac{ \Lambda_{ k - q_{1}/2 }(-q_{1}) } { 2\omega_{R}(q_{1})(\omega_{R}(q_{1}) + \omega_{k - q_{1}/2}(q_{1}))^{2} (\frac{ m^{3} }{q_{1}^{4}})( cosh(\lambda(q_{1})) - 1 ) }$$ $$- (2\pi k_{f}) \int_{-\infty}^{+\infty} \mbox{ }\frac{ dq_{1} }{2\pi}\mbox{ } \frac{ \Lambda_{ k + q_{1}/2 }(-q_{1}) } { 2\omega_{R}(q_{1})(\omega_{R}(q_{1}) + \omega_{k + q_{1}/2}(q_{1}))^{2} (\frac{ m^{3} }{q_{1}^{4}})( cosh(\lambda(q_{1})) - 1 ) }$$ In order to see how good the present theory is, it is desirable to compare these results with the Calogero-Sutherland model or more specifically with the spin-spin correlation function of the Haldane-Shastry model. This is given by [@Lesage], $ \rho = 1/2 $, $ \alpha = 2 $, $ m = 1 $, $ k_{f} = \pi/2 $. $$\langle 0 \| \Psi^{\dagger}(x,0)\Psi(x^{'},0) \| 0 \rangle = \int_{-\infty}^{+\infty} \mbox{ }\frac{ dk }{2\pi}\mbox{ } exp(i \mbox{ }k\mbox{ }(x^{'}-x)) \langle c^{\dagger}_{k}c_{k} \rangle$$ $$= (\frac{1}{4})\frac{ (\Gamma(3/2))^{2} } { (\Gamma(1/2))^{2} (\Gamma(1))^{2} } \int_{-1}^{+1} \mbox{ } dv_{1} \mbox{ } \int_{-1}^{+1} \mbox{ } dv_{2} \mbox{ } (1-v^{2}_{1})^{ -\frac{1}{2} } (1-v^{2}_{2})^{ -\frac{1}{2} } \|v_{1} - v_{2}\|$$ $$exp(i\mbox{ }\frac{\pi}{2}(x-x^{'})\mbox{ }v_{1}) exp(i\mbox{ }\frac{\pi}{2}(x-x^{'})\mbox{ }v_{2})$$ $$= 4\mbox{ }\frac{ (\Gamma(3/2))^{2} } { (\Gamma(1/2))^{2} (\Gamma(1))^{2} } \sum_{n = 0}(-)^{n}\frac{ \pi^{2n}(x-x^{'})^{2n} }{ (2\mbox{ }n)! } \frac{1}{(2 \mbox{ }n + 1)^{2}}$$ For this we have to use the interaction given by ($ V(x) = \frac{\alpha(\alpha-1)}{x^{2}} $), $$v_{q} = -\alpha(\alpha-1) \mbox{ }(\pi \|q\|)$$ Qualitative Conclusions ------------------------- It is clear that one of the features of the Luttinger liquid that is a result of the Mattis-Leib solution is absent, namely the discontinuous dependence of the momentum distribution on the coupling strength as the latter goes to zero. This may therefore be an artifact of the Luttinger model and not a generic feature of all 1D systems. Also the highly nonperturbative manner in which the momentum distribution of the interacting system approaches the noninteracting one as the coupling goes to zero may be seen quite easily. The other point that may be seen is that $ \langle \rho_{ {q} \neq 0 } \rangle = 0 $ indicating that there is no Wigner crystallization at any density. Furthermore, the vanishing of the quasiparticle residue is seen only for sufficiently large values of the repulsion between the fermions. Thus the conventional view that the fermi surface is destroyed for arbitrarily weak repulsion in case of 1d fermions does not seem to hold up in this case. Momentum Distribution in 3D ---------------------------- For this one has to compute the expressions in 3D: $$f_{n}({\bf{q}}) = \frac{ v_{ {\bf{q}} } }{(2\pi)^{2}} \int_{0}^{k_{f}} \mbox{ }dk \mbox{ }k^{2} \mbox{ } \{ (-\frac{m}{k\|q\|}) ln(\frac{ i\omega_{n} - \epsilon_{ {\bf{q}} } - \frac{ k \|q\| }{m} } { i\omega_{n} - \epsilon_{ {\bf{q}} } + \frac{ k \|q\| }{m} })$$ $$+ (\frac{m}{k\|q\|})\theta(k_{F} - \|q\| - k ) ln(\frac{ i\omega_{n} - \epsilon_{ {\bf{q}} } - \frac{ k \|q\| }{m} } { i\omega_{n} - \epsilon_{ {\bf{q}} } + \frac{ k \|q\| }{m} })$$ $$+ (\frac{m}{k\|q\|})\theta(\|q\| + k - k_{f}) \theta(k_{f}^{2} - (k - \|q\|)^{2}) ln(\frac{ i\omega_{n} - \epsilon_{ k_{f} } + \epsilon_{ {\bf{k}} } } { i\omega_{n} - \epsilon_{ {\bf{q}} } + \frac{ k \|q\| }{m} }) \}$$ This may be split into several cases (A) $ 0 < \|{\bf{q}}\| < k_{f} $ $$f_{n}({\bf{q}}) = \frac{v_{ {\bf{q}} }}{(2\pi)^{2}} (\frac{m}{\|{\bf{q}}\|}) \{ -\int_{0}^{k_{f}}dk \mbox{ }k\mbox{ } ln(\frac{i\omega_{n} - \epsilon_{ {\bf{q}} } - \frac{ k \|{\bf{q}}\| }{m} } { i\omega_{n} - \epsilon_{ {\bf{q}} } + \frac{ k \|{\bf{q}}\| }{m} })$$ $$+ \int_{0}^{k_{f}-\|q\|}dk \mbox{ }k\mbox{ } ln(\frac{i\omega_{n} - \epsilon_{ {\bf{q}} } - \frac{ k \|{\bf{q}}\| }{m} } { i\omega_{n} - \epsilon_{ {\bf{q}} } + \frac{ k \|{\bf{q}}\| }{m} })$$ $$+ \int_{k_{F}-\|{\bf{q}}\|}^{k_{f}}dk \mbox{ }k\mbox{ } ln(\frac{i\omega_{n} - \epsilon_{ k_{F} } + \epsilon_{ {\bf{k}} } } { i\omega_{n} - \epsilon_{ {\bf{q}} } + \frac{ k \|{\bf{q}}\| }{m} }) \}$$ (B) $ k_{f} < \|q\| < 2\mbox{ }k_{f} $ $$f_{n}({\bf{q}}) = \frac{ v_{ {\bf{q}} } }{(2\pi)^{2}} (\frac{m}{\|{\bf{q}}\|}) \{ -\int_{0}^{k_{f}} dk \mbox{ }k \mbox{ } ln(\frac{ i\omega_{n} - \epsilon_{ {\bf{q}} } - \frac{ k\|q\|}{m} } { i\omega_{n} - \epsilon_{ {\bf{q}} } + \frac{ k\|q\|}{m} })$$ $$+ \int_{\|q\|-k_{f}}^{k_{f}} dk \mbox{ }k \mbox{ } ln(\frac{ i\omega_{n} - \epsilon_{ k_{f} } + \epsilon_{ {\bf{k}} } } { i\omega_{n} - \epsilon_{ {\bf{q}} } + \frac{ k\|q\|}{m} }) \}$$ (C) $ \|q\| > 2\mbox{ }k_{f} $ $$f_{n}({\bf{q}}) = -\frac{ v_{ {\bf{q}} } }{(2\pi)^{2}} (\frac{m}{\|{\bf{q}}\|}) \int_{0}^{k_{f}} dk \mbox{ }k \mbox{ } ln(\frac{ i\omega_{n} - \epsilon_{ {\bf{q}} } - \frac{ k\|q\|}{m} } { i\omega_{n} - \epsilon_{ {\bf{q}} } + \frac{ k\|q\|}{m} })$$ (A) $$f_{n}({\bf{q}}) = \frac{ v_{ {\bf{q}} } }{(2\pi)^{2}} (\frac{m}{\|q\|}) [-\frac{1}{2}k_{f}^{2} ln(i\omega_{n} - \epsilon_{ {\bf{q}} } - \frac{k_{f}\|q\|}{m}) + \frac{1}{2}(k_{f}-\|q\|)^{2} ln(i\omega_{n} + \epsilon_{ {\bf{q}} } - \frac{k_{f}\|q\|}{m})$$ $$+ (\frac{m^{2}}{2\mbox{ }q^{2}}) \{ \frac{1}{2}(i\omega_{n} - \epsilon_{ {\bf{q}} } - \frac{ k_{f} \|q\| }{m})^{2} - \frac{1}{2}(i\omega_{n} + \epsilon_{ {\bf{q}} } - \frac{ k_{f} \|q\| }{m})^{2}$$ $$+ (i\omega_{n} - \epsilon_{ {\bf{q}} })^{2} ln(\frac{ i\omega_{n} - \epsilon_{ {\bf{q}} } - \frac{ k_{f} \|q\| }{m} } { i\omega_{n} + \epsilon_{ {\bf{q}} } - \frac{ k_{f} \|q\| }{m} })$$ $$+ (\frac{2\|q\|^{2}}{m}) (i\omega_{n}-\epsilon_{ {\bf{q}} }) \}$$ $$+ m\mbox{ }(i\omega_{n})ln(i\omega_{n}) - m\mbox{ }i\omega_{n} - m(\epsilon_{ {\bf{q}} } - \frac{k_{f} \|q\|}{m} + i\omega_{n}) ln(\epsilon_{ {\bf{q}} } - \frac{k_{f} \|q\|}{m} + i\omega_{n}) + m(\epsilon_{ {\bf{q}} } - \frac{k_{f} \|q\|}{m} + i\omega_{n}) ]$$ (B) $$f_{n}({\bf{q}}) = \frac{ v_{ {\bf{q}} } }{(2\pi)^{2}} (\frac{m}{\|q\|}) [-\frac{1}{2}k_{f}^{2} ln(i\omega_{n} - \epsilon_{ {\bf{q}} } - \frac{k_{f}\|q\|}{m}) + \frac{1}{2}(k_{f}-\|q\|)^{2} ln(i\omega_{n} + \epsilon_{ {\bf{q}} } - \frac{k_{f}\|q\|}{m})$$ $$+ (\frac{m^{2}}{2\mbox{ }q^{2}}) \{ \frac{1}{2}(i\omega_{n} - \epsilon_{ {\bf{q}} } - \frac{ k_{f} \|q\| }{m})^{2} - \frac{1}{2}(i\omega_{n} + \epsilon_{ {\bf{q}} } - \frac{ k_{f} \|q\| }{m})^{2}$$ $$+ (i\omega_{n} - \epsilon_{ {\bf{q}} })^{2} ln(\frac{ i\omega_{n} - \epsilon_{ {\bf{q}} } - \frac{ k_{f} \|q\| }{m} } { i\omega_{n} + \epsilon_{ {\bf{q}} } - \frac{ k_{f} \|q\| }{m} })$$ $$+ (\frac{2\|q\|^{2}}{m}) (i\omega_{n}-\epsilon_{ {\bf{q}} }) \}$$ $$+ m\mbox{ }(i\omega_{n})ln(i\omega_{n}) - m\mbox{ }i\omega_{n} - m(\epsilon_{ {\bf{q}} } - \frac{k_{f} \|q\|}{m} + i\omega_{n}) ln(\epsilon_{ {\bf{q}} } - \frac{k_{f} \|q\|}{m} + i\omega_{n}) + m(\epsilon_{ {\bf{q}} } - \frac{k_{f} \|q\|}{m} + i\omega_{n}) ]$$ (C) $$f_{n}({\bf{q}}) = \frac{ v_{ {\bf{q}} } }{(2\pi)^{2}} (\frac{m}{\|q\|}) [ \frac{1}{2}k_{f}^{2} ln(\frac{i\omega_{n} - \epsilon_{ {\bf{q}} } + \frac{ k_{f} \|q\| }{m} } { i\omega_{n} - \epsilon_{ {\bf{q}} } - \frac{ k_{f} \|q\| }{m} })$$ $$-\frac{m^{2}}{2\|q\|^{2}} \{ \frac{1}{2}(i\omega_{n} - \epsilon_{ {\bf{q}} } + \frac{ k_{f} \|q\|}{m})^{2} - \frac{1}{2}(i\omega_{n} - \epsilon_{ {\bf{q}} } - \frac{ k_{f} \|q\|}{m})^{2}$$ $$+ (i\omega_{n} - \epsilon_{ {\bf{q}} })^{2} ln(\frac{ i\omega_{n} - \epsilon_{ {\bf{q}} } + \frac{ k_{f} \|q\|}{m} } { i\omega_{n} - \epsilon_{ {\bf{q}} } - \frac{ k_{f} \|q\|}{m} } ) - \frac{4\mbox{ }k_{f}\|q\|}{m}(i\omega_{n} - \epsilon_{ {\bf{q}} }) \} ]$$ Fermions at Finite Temperature ============================== It should be clear from the previous sections that fermions at finite temp. is a major headache. A straightforward generalisation is not working out for unknown reasons. In this section we shall not distinguish between $ N $ and $ \langle N \rangle $, things are complicated enough as it is ! Take for example, the expectation value of the number density at finite temerature, $$\langle c^{\dagger}_{ {\bf{k}} }c_{ {\bf{k}} } \rangle = n_{F}({\bf{k}}) + \sum_{ {\bf{q}}_{1} } \Lambda_{ {\bf{k}} - {\bf{q}}_{1}/2 }(-{\bf{q}}_{1}) \langle a^{\dagger}_{ {\bf{k}} - {\bf{q}}_{1}/2 }({\bf{q}}_{1}) a_{ {\bf{k}} - {\bf{q}}_{1}/2 }({\bf{q}}_{1}) \rangle$$ $$- \sum_{ {\bf{q}}_{1} } \Lambda_{ {\bf{k}} + {\bf{q}}_{1}/2 }(-{\bf{q}}_{1}) \langle a^{\dagger}_{ {\bf{k}} + {\bf{q}}_{1}/2 }({\bf{q}}_{1}) a_{ {\bf{k}} + {\bf{q}}_{1}/2 }({\bf{q}}_{1}) \rangle$$ Assume the $ n_{F}({\bf{k}}) $ are all evaluated at zero temp. as in the previous sections. Now, taken at face value, one is obliged to compute the thermodynamic expectation values of the Bose occupation probabilities assuming the chemical potential for the Bosons is zero (this means that we are allowed to create and destroy any number of Bosons). Now such a calculation yields an infinite answer for $ \langle c^{\dagger}_{ {\bf{k}} }c_{ {\bf{k}} } \rangle $ as the sum over all $ {\bf{q}}_{1} $ diverges (is proportional to the total number of fermions). This is the reason why a more nefarious approach may be necessary in dealing with fermions at finite temperature. For this the only guide is that all the finite temp. dynamical correlation functions involving the number conserving object $ c^{\dagger}_{ {\bf{k+q/2}} } c_{ {\bf{k-q/2}} } $ should be correctly reproduced. Hopefully the commutation rules involving these number conserving products are not damaged in the bargain. For this let us start with the simplest case namely, $ \langle c^{\dagger}_{ {\bf{k+q/2}} } c_{ {\bf{k-q/2}} } \rangle $ . We know what the answer should be, that is, $$\langle c^{\dagger}_{ {\bf{k+q/2}} } c_{ {\bf{k-q/2}} } \rangle = \delta_{ {\bf{q = 0}} } n_{F, \beta}({\bf{k}})$$ where, $$n_{F, \beta}({\bf{k}}) = [ exp(\beta(\epsilon_{ {\bf{k}} }-\mu)) + 1 ]^{-1}$$ For this to happen we have to do the following, (1) Treat the bosons as before assuming that they are always at zero temperature and with zero chemical potential. (2) Assume that all finite temperature effects are lumped into the coefficients. (3) Fix the coefficients as before by demanding that the finite temp. dynamical moments of the number-conserving products come out right. For the dynamical four-point function to come out right we must ensure that, $$\Lambda^{\beta}_{ {\bf{k}} }({\bf{q}}) = \sqrt{ n_{F, \beta}({\bf{k+q/2}})(1 - n_{F, \beta}({\bf{k-q/2}})) }$$ Unfortunately, the coefficient $ \Lambda^{\beta}_{ {\bf{k}} }({\bf{q}}) $ is no longer either zero or one. This renders matters even more difficult. But the dynamical six-point function is still the same and has to be correctly given as in the zero-temp case. $$I = \langle c^{\dagger}_{ {\bf{k+q/2}} }c_{ {\bf{k-q/2}} } c^{\dagger}_{ {\bf{k^{'}+q^{'}/2}} }c_{ {\bf{k^{'}-q^{'}/2}} } c^{\dagger}_{ {\bf{k^{''}+q^{''}/2}} }c_{ {\bf{k^{''}-q^{''}/2}} } \rangle$$ $$= [(1 - n_{F, \beta}({\bf{k-q/2}}))(1 - n_{F, \beta}({\bf{k^{'}-q^{'}/2}})) n_{F, \beta}({\bf{k+q/2}})\delta_{ {\bf{k+q/2}}, {\bf{k^{''} - q^{''}/2}} } \delta_{ {\bf{k-q/2}}, {\bf{k^{'} + q^{'}/2}} } \delta_{ {\bf{k^{'}-q^{'}/2}}, {\bf{k^{''} + q^{''}/2}} }$$ $$- (1 - n_{F, \beta}({\bf{k-q/2}}))n_{F, \beta}({\bf{k^{'}+q^{'}/2}}) n_{F, \beta}({\bf{k+q/2}}) \delta_{ {\bf{k-q/2}}, {\bf{k^{''} + q^{''}/2}} } \delta_{ {\bf{k+q/2}}, {\bf{k^{'} - q^{'}/2}} } \delta_{ {\bf{k^{'}+q^{'}/2}}, {\bf{k^{''} - q^{''}/2}} } ]$$ In terms of the Bose fields we have, $$I = \Lambda^{\beta}_{ {\bf{k}} }({\bf{q}}) \Lambda^{\beta}_{ {\bf{k}}^{''} }(-{\bf{q}}^{''}) \Gamma_{ {\bf{k}}, {\bf{k}}^{''} }^{ -{\bf{q}}, {\bf{q}}^{''} } ({\bf{k}}^{'}, {\bf{q}}^{'})$$ Equating these two we arrive at a formula for the coefficient $ \Gamma $. It is somewhat different from the zero temp case. But we can remedy that by choosing, $$\Gamma_{ {\bf{k}}_{1}, {\bf{k}}_{2} }^{ {\bf{q}}_{1}, {\bf{q}}_{2} } ({\bf{k}}, {\bf{q}}) = F(\Lambda^{\beta}_{ {\bf{k}}_{1} }(-{\bf{q}}_{1}) \Lambda^{\beta}_{ {\bf{k}}_{2} }(-{\bf{q}}_{2})) [\sqrt{ (1 - n_{F, \beta}({\bf{k}}_{1}+{\bf{q}}_{1}/2)) (1 - n_{F, \beta}({\bf{k}}_{2}+{\bf{q}}_{2}/2)) }$$ $$\delta_{ {\bf{k_{1}-q_{1}/2}}, {\bf{k_{2}-q_{2}/2}} } \delta_{ {\bf{k_{1}+q_{1}/2}}, {\bf{k+q/2}} } \delta_{ {\bf{k-q/2}}, {\bf{k_{2}+q_{2}/2}} }$$ $$- \sqrt{ n_{F, \beta}({\bf{k}}_{1}-{\bf{q}}_{1}/2) n_{F, \beta}({\bf{k}}_{2}-{\bf{q}}_{2}/2) }$$ $$\delta_{ {\bf{k_{1}+q_{1}/2}}, {\bf{k_{2}+q_{2}/2}} } \delta_{ {\bf{k_{1}-q_{1}/2}}, {\bf{k-q/2}} } \delta_{ {\bf{k+q/2}}, {\bf{k_{2}-q_{2}/2}} } ]$$ and, $$F(x) = 1\mbox{ }if\mbox{ }x \neq 0\mbox{ } and\mbox{ }F(x) = 0\mbox{ }for\mbox{ }x = 0$$ For any temperature above zero $ F(x) = 1 $ always. But for exactly at zero temp, the original case is recovered. But more importantly, all the right sort of dynamical correlation functions are recovered if we choose in addition, the following formula for the kinetic energy operator, $$K = \sum_{ {\bf{k}}, {\bf{q}} }\omega_{ {\bf{k}} }({\bf{q}}) a^{\dagger}_{ {\bf{k}} }({\bf{q}})a_{ {\bf{k}} }({\bf{q}})$$ and, $$\omega_{ {\bf{k}} }({\bf{q}}) = F(\Lambda^{\beta}_{ {\bf{k}} }(-{\bf{q}})) (\frac{ {\bf{k.q}} }{m})$$ These choices ensure that all the finite temp. dynamical moments of $ c^{\dagger}_{ {\bf{k+q/2}} }c_{ {\bf{k-q/2}} } $ are correctly recovered. Demonstration of Emergence of Superconductivity ------------------------------------------------- Consider two interactions, one the coulomb repulsion and the other electron-phonon interaction : $$H_{I} = \sum_{ {\bf{q}} \neq 0 }\frac{ v_{ {\bf{q}} } }{2V} \sum_{ {\bf{k}}, {\bf{k}}^{'} } [\Lambda^{\beta}_{ {\bf{k}} }({\bf{q}})a_{ {\bf{k}} }(-{\bf{q}}) + \Lambda^{\beta}_{ {\bf{k}} }(-{\bf{q}})a^{\dagger}_{ {\bf{k}} }({\bf{q}})] [\Lambda^{\beta}_{ {\bf{k}}^{'} }(-{\bf{q}})a_{ {\bf{k}}^{'} }({\bf{q}}) + \Lambda^{\beta}_{ {\bf{k}}^{'} }({\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }(-{\bf{q}})]$$ $$H_{e-phon} = \sum_{ {\bf{q}} \neq 0 }\frac{ M_{ {\bf{q}} } }{ V^{\frac{1}{2}} } \sum_{ {\bf{k}} } [\Lambda^{\beta}_{ {\bf{k}} }({\bf{q}})a_{ {\bf{k}} }(-{\bf{q}}) + \Lambda^{\beta}_{ {\bf{k}} }(-{\bf{q}})a^{\dagger}_{ {\bf{k}} }({\bf{q}})] [b_{ {\bf{q}} } + b^{\dagger}_{ -{\bf{q}} }]$$ The displacement operators are, $$P_{ {\bf{q}} } = b_{ {\bf{q}} } + b^{\dagger}_{ -{\bf{q}} }$$ and, $$X_{ {\bf{q}} } = (\frac{i}{2}) (b_{ -{\bf{q}} } - b^{\dagger}_{ {\bf{q}} })$$ The equations of motion for the Bose fields are, $$i \mbox{ }\frac{\partial}{ \partial t }a^{t}_{ {\bf{k}} }({\bf{q}}) = \omega_{ {\bf{k}} }({\bf{q}})a^{t}_{ {\bf{k}} }({\bf{q}}) + (\frac{ v_{ {\bf{q}} } }{V}) \Lambda^{\beta}_{ {\bf{k}} }(-{\bf{q}})\sum_{ {\bf{k}}^{'} }[ \Lambda^{\beta}_{ {\bf{k}}^{'} }(-{\bf{q}})a^{t}_{ {\bf{k}}^{'} }({\bf{q}}) + \Lambda^{\beta}_{ {\bf{k}}^{'} }({\bf{q}}) a^{\dagger t}_{ {\bf{k}}^{'} }(-{\bf{q}}) ]$$ $$+ \frac{ M_{ {\bf{q}} } }{ V^{\frac{1}{2}} } \Lambda^{\beta}_{ {\bf{k}} }(-{\bf{q}}) [ b_{ {\bf{q}} } + b^{\dagger}_{ -{\bf{q}} } ]$$ $$i\frac{ \partial }{\partial t}X_{ {\bf{q}} } = \frac{ i\mbox{ }M_{ {\bf{q}} } }{ V^{\frac{1}{2}} } \sum_{ {\bf{k}}^{'} } [\Lambda_{ {\bf{k}}^{'} }({\bf{q}})a_{ {\bf{k}}^{'} }(-{\bf{q}}) + \Lambda_{ {\bf{k}}^{'} }(-{\bf{q}})a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}) ] + \frac{ i\Omega_{ {\bf{q}} } }{2}P_{ -{\bf{q}} }$$ $$i\frac{ \partial }{\partial t}P_{ {\bf{q}} } = \frac{ 2\mbox{ }\Omega_{ {\bf{q}} } }{i}X_{ -{\bf{q}} }$$ $$i \mbox{ }\frac{\partial}{ \partial t }a^{\dagger t}_{ {\bf{k}} }(-{\bf{q}}) = -\omega_{ {\bf{k}} }(-{\bf{q}})a^{\dagger t}_{ {\bf{k}} }(-{\bf{q}}) - (\frac{ v_{ {\bf{q}} } }{V}) \Lambda^{\beta}_{ {\bf{k}} }({\bf{q}})\sum_{ {\bf{k}}^{'} }[ \Lambda^{\beta}_{ {\bf{k}}^{'} }(-{\bf{q}})a^{t}_{ {\bf{k}}^{'} }({\bf{q}}) + \Lambda^{\beta}_{ {\bf{k}}^{'} }({\bf{q}})a^{\dagger t} _{ {\bf{k}}^{'} }(-{\bf{q}}) ]$$ $$- \frac{ M_{ {\bf{q}} } }{ V^{\frac{1}{2}} } \Lambda^{\beta}_{ {\bf{k}} }({\bf{q}}) [ b_{ {\bf{q}} } + b^{\dagger}_{ -{\bf{q}} } ]$$ $$(i\frac{\partial}{\partial t} - \omega_{ {\bf{k}} }({\bf{q}})) \frac{ -i\langle T a^{t}_{ {\bf{k}} }({\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle} = \delta_{ {\bf{k}}, {\bf{k}}^{'} } \delta_{ {\bf{q}}, {\bf{q}}^{'} }\delta(t)$$ $$+ (\frac{ v_{ {\bf{q}} } }{V}) \Lambda^{\beta}_{ {\bf{k}} }(-{\bf{q}}) \sum_{ {\bf{k}}^{''} }[\Lambda^{\beta}_{ {\bf{k}}^{''} }(-{\bf{q}}) \frac{ -i\langle T \mbox{ }a^{t}_{ {\bf{k}}^{''} }({\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle} + \Lambda^{\beta}_{ {\bf{k}}^{''} }({\bf{q}}) \frac{ -i\langle T \mbox{ }a^{\dagger t}_{ {\bf{k}}^{''} }(-{\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle}]$$ $$+ \frac{ M_{ {\bf{q}} } }{ V^{\frac{1}{2}} } \Lambda^{\beta}_{ {\bf{k}} }(-{\bf{q}}) \frac{ -i \langle T \mbox{ }P^{t}_{ {\bf{q}} } a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle } {\langle T1 \rangle}$$ $$(i\frac{\partial}{\partial t} + \omega_{ {\bf{k}} }(-{\bf{q}})) \frac{ -i\langle T \mbox{ }a^{\dagger t}_{ {\bf{k}} }(-{\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle} =$$ $$- (\frac{ v_{ {\bf{q}} } }{V}) \Lambda^{\beta}_{ {\bf{k}} }({\bf{q}}) \sum_{ {\bf{k}}^{''} }[\Lambda^{\beta}_{ {\bf{k}}^{''} }(-{\bf{q}}) \frac{ -i\langle T \mbox{ }a^{t}_{ {\bf{k}}^{''} }({\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle} + \Lambda^{\beta}_{ {\bf{k}}^{''} }({\bf{q}}) \frac{ -i\langle T \mbox{ }a^{\dagger t}_{ {\bf{k}}^{''} }(-{\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle}]$$ $$- \frac{ M_{ {\bf{q}} } }{ V^{\frac{1}{2}} } \Lambda^{\beta}_{ {\bf{k}} }({\bf{q}}) \frac{ -i \langle T \mbox{ }P^{t}_{ {\bf{q}} } \mbox{ } a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle } {\langle T1 \rangle}$$ The corresponding propagators with phonons are, $$i \frac{ \partial }{ \partial t } \frac{ -i \langle T \mbox{ }P^{t}_{ {\bf{q}} }\mbox{ } a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{ \langle T 1 \rangle } = (\frac{ 2 \Omega_{ {\bf{q}} } }{i}) \frac{ -i \langle T \mbox{ }X^{t}_{ -{\bf{q}} } \mbox{ }a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{ \langle T 1 \rangle }$$ $$i \frac{ \partial }{ \partial t } \frac{ -i \langle T \mbox{ }X^{t}_{ -{\bf{q}} } \mbox{ }a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{ \langle T 1 \rangle } = (\frac{ i\mbox{ }M_{ -{\bf{q}} } }{ V^{\frac{1}{2}} }) \sum_{ {\bf{k}}^{''} }[ \Lambda^{\beta}_{ {\bf{k}}^{''} }(-{\bf{q}}) \frac{ -i \langle T \mbox{ }a^{t}_{ {\bf{k}}^{''} }({\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{ \langle T 1 \rangle } + \Lambda^{\beta}_{ {\bf{k}}^{''} }({\bf{q}}) \frac{ -i \langle T \mbox{ }a^{\dagger t}_{ {\bf{k}}^{''} }(-{\bf{q}}) a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{ \langle T 1 \rangle } ]$$ $$+ (\frac{ i \mbox{ }\Omega_{ {\bf{q}} } }{2}) \frac{ -i\langle T \mbox{ }P^{t}_{ {\bf{q}} } \mbox{ }a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle } { \langle T 1 \rangle }$$ Transform to frequency domain and we have the corresponding equations : $$G_{1}( {\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) = \Lambda_{ {\bf{k}}^{'} }(-{\bf{q}}) \frac{ \delta_{ {\bf{q}}, {\bf{q}}^{'} } } {-i\mbox{ } \beta( i\omega_{n} - \omega_{ {\bf{k}}^{'} }({\bf{q}}) ) }$$ $$+ f_{n}({\bf{q}})[G_{1}( {\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) + G_{2}( {\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) ] + g_{n}({\bf{q}})G_{3}( {\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n)$$ $$G_{2}( {\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) = f^{}_{n}(-{\bf{q}}) [G_{1}( {\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) + G_{2}( {\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n)] + g^{}_{n}(-{\bf{q}})G_{3}( {\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n)$$ $$i\mbox{ }\omega_{n}\mbox{ }G_{3}( {\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) = (\frac{2 \mbox{ }\Omega_{ {\bf{q}} } }{i}) G_{4}( {\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n)$$ $$i\mbox{ }\omega_{n}\mbox{ }G_{4}( {\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) = \mbox{ }i\mbox{ }[G_{1}( {\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) + G_{2}( {\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) ] + (\frac{ i \mbox{ }\Omega_{ {\bf{q}} } }{2}) G_{3}( {\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n)$$ The solutions may be written down as, $$G_{3}({\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) = -(\frac{ 2 \mbox{ }\Omega_{ {\bf{q}} } } { \Omega^{2}_{ {\bf{q}} } + \omega_{n}^{2} }) [G_{1}( {\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) + G_{2}( {\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n)]$$ $$G_{2}({\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) = (\frac{ \Gamma_{n}({\bf{q}}) }{1 - \Gamma_{n}({\bf{q}})}) G_{1}({\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n)$$ $$\Gamma_{n}({\bf{q}}) = f^{}_{n}(-{\bf{q}}) - \frac{ 2 \mbox{ }\Omega_{ {\bf{q}} } } { \Omega^{2}_{ {\bf{q}} } + \omega_{n}^{2} } g^{}_{n}(-{\bf{q}})$$ and finally, $$G_{1}({\bf{q}}, {\bf{k}}^{'}, {\bf{q}}^{'}; n) = \delta_{ {\bf{q}}, {\bf{q}}^{'} } \frac{ F_{n}({\bf{q}}) \Lambda_{ {\bf{k}}^{'} }(-{\bf{q}}) } {-i\mbox{ } \beta(i \mbox{ }\omega_{n} - \omega_{ {\bf{k}}^{'} }({\bf{q}}) ) }$$ and, $$F_{n}({\bf{q}}) = [ 1 - \frac{ f_{n}({\bf{q}}) }{ 1 - \Gamma_{n}({\bf{q}}) } + (\frac{ 2 \mbox{ }\Omega_{ {\bf{q}} } } { \Omega^{2}_{ {\bf{q}} } + \omega^{2}_{n} }) (\frac{ g_{n}({\bf{q}}) }{1 - \Gamma_{n}({\bf{q}}) }) ]^{-1}$$ here, $$f_{n}({\bf{q}}) = (\frac{ v_{ {\bf{q}} } }{V}) \sum_{ {\bf{k}} } \frac{ ( \Lambda^{\beta}_{ {\bf{k}} }(-{\bf{q}}) )^{2} } { ( i\omega_{n} - \omega_{ {\bf{k}} }({\bf{q}}) ) }$$ $$g_{n}({\bf{q}}) = (\frac{ \|M_{ {\bf{q}} }\|^{2} }{ V }) \sum_{ {\bf{k}} } \frac{ ( \Lambda^{\beta}_{ {\bf{k}} }(-{\bf{q}}) )^{2} } { ( i\omega_{n} - \omega_{ {\bf{k}} }({\bf{q}}) ) }$$ For the four-point function we have to proceed as follows, the quantity of interest is(with phonons and coulomb), $$\frac{1}{i^{2}}\langle T (a^{t_{1}}_{ {\bf{k}}_{1} }({\bf{q}}_{1}) \mbox{ }a^{t_{2}}_{ {\bf{k}}_{2} }({\bf{q}}_{2}) a^{\dagger t^{'}_{2}}_{ {\bf{k}}^{'}_{2} }({\bf{q}}^{'}_{2}) a^{\dagger t^{'}_{1}}_{ {\bf{k}}^{'}_{1} }({\bf{q}}^{'}_{1})) \rangle$$ for this we as usual decompose as follows, $$\frac{1}{i^{2}}\langle T (a^{t_{1}}_{ {\bf{k}}_{1} }({\bf{q}}_{1}) \mbox{ }a^{t_{2}}_{ {\bf{k}}_{2} }({\bf{q}}_{2}) a^{\dagger t^{'}_{2}}_{ {\bf{k}}^{'}_{2} }({\bf{q}}^{'}_{2}) a^{\dagger t^{'}_{1}}_{ {\bf{k}}^{'}_{1} }({\bf{q}}^{'}_{1})) \rangle$$ $$= \sum_{ n }\mbox{ }exp(\omega_{n}\mbox{ }t_{1}) \frac{1}{i^{2}}\langle T (a^{n}_{ {\bf{k}}_{1} }({\bf{q}}_{1}) \mbox{ }a^{t_{2}}_{ {\bf{k}}_{2} }({\bf{q}}_{2}) a^{\dagger t^{'}_{2}}_{ {\bf{k}}^{'}_{2} }({\bf{q}}^{'}_{2}) a^{\dagger t^{'}_{1}}_{ {\bf{k}}^{'}_{1} }({\bf{q}}^{'}_{1})) \rangle$$ $$(i \mbox{ }\omega_{n} - \omega_{ {\bf{k}}_{1} }({\bf{q}}_{1}) ) \frac{1}{i^{2}} \langle T(a^{n}_{ {\bf{k}}_{1} }({\bf{q}}_{1}) a^{t_{2}}_{ {\bf{k}}_{2} }({\bf{q}}_{2}) a^{\dagger t^{'}_{2}}_{ {\bf{k}}^{'}_{2} }({\bf{q}}^{'}_{2}) a^{\dagger t^{'}_{1}}_{ {\bf{k}}^{'}_{1} }({\bf{q}}^{'}_{1}) ) \rangle$$ $$= F_{1}({\bf{k}}_{1}\mbox{ }{\bf{q}}_{1}\mbox{ }n, {\bf{k}}_{2}\mbox{ }{\bf{q}}_{2}\mbox{ }t_{2}, {\bf{k}}^{'}_{2}\mbox{ }{\bf{q}}^{'}_{2}\mbox{ }t^{'}_{2}, {\bf{k}}^{'}_{1}\mbox{ }{\bf{q}}^{'}_{1}\mbox{ }t^{'}_{1})$$ $$+ (\frac{ \Lambda_{ {\bf{k}}_{1} }(-{\bf{q}}_{1}) }{V}) [v_{ {\bf{q}}_{1} } - (\frac{ 2 \mbox{ }\Omega_{ {\bf{q}}_{1} } } { \omega^{2}_{n} + \Omega^{2}_{ {\bf{q}}_{1} } }) \|M_{ {\bf{q}}_{1} }\|^{2}]$$ $$\times [{\tilde{F}}_{1}({\bf{q}}_{1}n, {\bf{k}}_{2}\mbox{ }{\bf{q}}_{2}\mbox{ }t_{2}, {\bf{k}}^{'}_{2}\mbox{ }{\bf{q}}^{'}_{2}\mbox{ }t^{'}_{2}, {\bf{k}}^{'}_{1}\mbox{ }{\bf{q}}^{'}_{1}\mbox{ }t^{'}_{1}) + {\tilde{F}}_{2}({\bf{q}}_{1}n, {\bf{k}}_{2}\mbox{ }{\bf{q}}_{2}\mbox{ }t_{2}, {\bf{k}}^{'}_{2}\mbox{ }{\bf{q}}^{'}_{2}\mbox{ }t^{'}_{2}, {\bf{k}}^{'}_{1}\mbox{ }{\bf{q}}^{'}_{1}\mbox{ }t^{'}_{1}) ]/ (1 - \Gamma_{n}({\bf{q}}_{1}) - \Gamma^{}_{n}(-{\bf{q}}_{1}))$$ $${\tilde{F}}_{1}({\bf{q}}_{1}n, {\bf{k}}_{2}\mbox{ }{\bf{q}}_{2}\mbox{ }t_{2}, {\bf{k}}^{'}_{2}\mbox{ }{\bf{q}}^{'}_{2}\mbox{ }t^{'}_{2}, {\bf{k}}^{'}_{1}\mbox{ }{\bf{q}}^{'}_{1}\mbox{ }t^{'}_{1})$$ $$= \sum_{ {\bf{k}}_{1} } \frac{ \Lambda_{ {\bf{k}}_{1} }(-{\bf{q}}_{1}) }{ ( i\omega_{n} - \omega_{ {\bf{k}}_{1} }({\bf{q}}_{1}) ) } F_{1}({\bf{k}}_{1}{\bf{q}}_{1}n, {\bf{k}}_{2}\mbox{ }{\bf{q}}_{2}\mbox{ }t_{2}, {\bf{k}}^{'}_{2}\mbox{ }{\bf{q}}^{'}_{2}\mbox{ }t^{'}_{2}, {\bf{k}}^{'}_{1}\mbox{ }{\bf{q}}^{'}_{1}\mbox{ }t^{'}_{1})$$ $${\tilde{F}}_{2}({\bf{q}}_{1}n, {\bf{k}}_{2}\mbox{ }{\bf{q}}_{2}\mbox{ }t_{2}, {\bf{k}}^{'}_{2}\mbox{ }{\bf{q}}^{'}_{2}\mbox{ }t^{'}_{2}, {\bf{k}}^{'}_{1}\mbox{ }{\bf{q}}^{'}_{1}\mbox{ }t^{'}_{1})$$ $$= \sum_{ {\bf{k}}_{1} } \frac{ \Lambda_{ {\bf{k}}_{1} }({\bf{q}}_{1}) }{ ( i\omega_{n} + \omega_{ {\bf{k}}_{1} }(-{\bf{q}}_{1}) ) } F_{2}({\bf{k}}_{1}{\bf{q}}_{1}n, {\bf{k}}_{2}\mbox{ }{\bf{q}}_{2}\mbox{ }t_{2}, {\bf{k}}^{'}_{2}\mbox{ }{\bf{q}}^{'}_{2}\mbox{ }t^{'}_{2}, {\bf{k}}^{'}_{1}\mbox{ }{\bf{q}}^{'}_{1}\mbox{ }t^{'}_{1})$$ Bosons at zero temperature ---------------------------- In this subsection we do what we did earlier except that here we are bosonizing the bosons. Let $ b_{ {\bf{k}} } $ and $ b^{\dagger}_{ {\bf{k}} } $ be the Bose fields in question. Analogous to Fermi sea displacements we introduce Bose condensate displacements. $$b^{\dagger}_{ {\bf{k+q/2}} } b_{ {\bf{k-q/2}} } = N \mbox{ }\delta_{ {\bf{k}}, 0 } \delta_{ {\bf{q}}, 0 } + (\sqrt{N}) [\delta_{ {\bf{k+q/2}}, 0 }\mbox{ }d_{ {\bf{k}} }(-{\bf{q}}) + \delta_{ {\bf{k-q/2}}, 0 }\mbox{ }d^{\dagger}_{ {\bf{k}} }({\bf{q}})]$$ $$+ \sum_{ {\bf{q}}_{1} }T_{1}({\bf{k}}, {\bf{q}}, {\bf{q}}_{1}) d^{\dagger}_{ {\bf{k+q/2-q_{1}/2}} }({\bf{q}}_{1}) d_{ {\bf{k-q_{1}/2}} }(-{\bf{q}}+{\bf{q}}_{1})$$ $$+ \sum_{ {\bf{q}}_{1} }T_{2}({\bf{k}}, {\bf{q}}, {\bf{q}}_{1}) d^{\dagger}_{ {\bf{k-q/2+q_{1}/2}} }({\bf{q}}_{1}) d_{ {\bf{k+q_{1}/2}} }(-{\bf{q}}+{\bf{q}}_{1})$$ To make all the dynamical moments of $ b^{\dagger}_{ {\bf{k+q/2}} } b_{ {\bf{k-q/2}} } $ come out right and the commutation rules amongst them also come out right provided we choose $ T_{1} $ and $ T_{2} $ such that, $$(\sqrt{N})^{2}\delta_{ {\bf{k+q/2}}, 0 } \delta_{ {\bf{k^{''}-q^{''}/2}}, 0 } \sum_{ {\bf{q}}_{1} }T_{1}({\bf{k}}^{'}, {\bf{q}}^{'}, {\bf{q}}_{1}) \langle d_{ {\bf{k}} }(-{\bf{q}}) d^{\dagger}_{ {\bf{k}}^{'} + {\bf{q}}^{'}/2 - {\bf{q}}_{1}/2 } ({\bf{q}}_{1})d_{ {\bf{k}}^{'} - {\bf{q}}_{1}/2 } (-{\bf{q}}^{'} + {\bf{q}}_{1}) d^{\dagger}_{ {\bf{k}}^{''} }({\bf{q}}^{''}) \rangle$$ $$+(\sqrt{N})^{2}\delta_{ {\bf{k+q/2}}, 0 } \delta_{ {\bf{k^{''}-q^{''}/2}}, 0 } \sum_{ {\bf{q}}_{1} }T_{2}({\bf{k}}^{'}, {\bf{q}}^{'}, {\bf{q}}_{1}) \langle d_{ {\bf{k}} }(-{\bf{q}}) d^{\dagger}_{ {\bf{k}}^{'} - {\bf{q}}^{'}/2 + {\bf{q}}_{1}/2 } ({\bf{q}}_{1})d_{ {\bf{k}}^{'} + {\bf{q}}_{1}/2 } (-{\bf{q}}^{'} + {\bf{q}}_{1}) d^{\dagger}_{ {\bf{k}}^{''} }({\bf{q}}^{''}) \rangle$$ = $$N\mbox{ }\delta_{ {\bf{k+q/2}}, 0 } \delta_{ {\bf{k+q/2}}, {\bf{k^{''}-q^{''}/2}} } \delta_{ {\bf{k-q/2}}, {\bf{k^{'}+q^{'}/2}} } \delta_{ {\bf{k^{'}-q^{'}/2}}, {\bf{k^{''}+q^{''}/2}} }$$ This means, $$T_{1}( -{\bf{q}}-{\bf{q}}^{'}/2, {\bf{q}}^{'}, -{\bf{q}} ) = 1; \mbox{ }{\bf{q}} \neq 0; \mbox{ }{\bf{q}}^{'} \neq 0$$ $$T_{2}( -{\bf{q}}^{'}/2, {\bf{q}}^{'}, -{\bf{q}} ) = 0; \mbox{ }{\bf{q}} \neq 0; \mbox{ }{\bf{q}}^{'} \neq 0$$ in order for the kinetic energy operator to have the form, $$K = \sum_{ {\bf{k}} }\epsilon_{ {\bf{k}} } \mbox{ } d^{\dagger}_{ (1/2){\bf{k}} }({\bf{k}}) \mbox{ }d_{ (1/2){\bf{k}} }({\bf{k}})$$ we must have, $$\epsilon_{ {\bf{k+q_{1}/2}} }T_{1}( {\bf{k+q_{1}/2}}, {\bf{0}}, {\bf{q}}_{1}) + \epsilon_{ {\bf{k-q_{1}/2}} }T_{2}( {\bf{k-q_{1}/2}}, {\bf{0}}, {\bf{q}}_{1}) = \delta_{ {\bf{k}}, {\bf{q}}_{1}/2 }\epsilon_{ {\bf{q}}_{1} } \label{EQ1}$$ In order for, $$\sum_{ {\bf{k}} }b^{ \dagger }_{ {\bf{k}} }b_{ {\bf{k}} } = N$$ we must have, $$T_{1}({\bf{k+q_{1}/2}}, {\bf{0}}, {\bf{q_{1}}}) + T_{2}({\bf{k-q_{1}/2}}, {\bf{0}}, {\bf{q_{1}}}) = 0 \label{EQ2}$$ In order for the commutation rules amongst the $ b^{\dagger}_{ {\bf{k+q/2}} } b_{ {\bf{k-q/2}} } $ to come out right, we must have in addition to all the above relations, $$T_{2}({\bf{q}}/2, {\bf{q}}, -{\bf{q}}^{'}) = 0$$ $$T_{2}(-{\bf{q}}/2, {\bf{q}}, {\bf{q}}+{\bf{q}}^{'}) = 0$$ Any choice of $ T_{1} $ and $ T_{2} $ consistent with the above relations should suffice. From the above relations (Eq.( \[EQ1\]) and Eq.( \[EQ2\])), we have quite unambiguously, $$T_{1}({\bf{k+q_{1}/2}}, {\bf{0}}, {\bf{q}}_{1}) = \delta_{ {\bf{k}}, {\bf{q}}_{1}/2 }$$ $$T_{2}({\bf{k-q_{1}/2}}, {\bf{0}}, {\bf{q}}_{1}) = -\delta_{ {\bf{k}}, {\bf{q}}_{1}/2 }$$ $$T_{1}(-{\bf{q}}-{\bf{q}}^{'}/2, {\bf{q}}^{'}, -{\bf{q}}) = 1$$ and, $$T_{1}({\bf{k}}, {\bf{q}}, {\bf{q}}_{1}) = 0; \mbox{ }otherwise$$ $$T_{2}({\bf{k}}, {\bf{q}}, {\bf{q}}_{1}) = 0; \mbox{ }otherwise$$ This means that we may rewrite the formula for $ b^{\dagger}_{ {\bf{k+q/2}} }b_{ {\bf{k-q/2}} } $ as follows, $$b^{\dagger}_{ {\bf{k+q/2}} }b_{ {\bf{k-q/2}} } = N \delta_{ {\bf{k}}, 0 }\delta_{ {\bf{q}}, 0 } + (\sqrt{N}) [\delta_{ {\bf{k+q/2}}, 0 }d_{ {\bf{k}} }(-{\bf{q}}) + \delta_{ {\bf{k-q/2}}, 0 }d^{\dagger}_{ {\bf{k}} }({\bf{q}})]$$ $$+ d^{\dagger}_{ (1/2){\bf{k+q/2}} }({\bf{k+q/2}}) d_{ (1/2){\bf{k-q/2}} }({\bf{k-q/2}})$$ $$-\delta_{ {\bf{k}}, 0 }\delta_{ {\bf{q}}, 0 } \sum_{ {\bf{q}}_{1} }d^{\dagger}_{ {\bf{q}}_{1}/2 }({\bf{q}}_{1}) d_{ {\bf{q}}_{1}/2 }({\bf{q}}_{1})$$ Consider an interaction of the type, $$H_{I} = (\frac{\rho_{0}}{2}) \sum_{ {\bf{q}} \neq 0 } v_{ {\bf{q}} }\sum_{ {\bf{k}}, {\bf{k}}^{'} } [\delta_{ {\bf{k+q/2}}, 0 }\mbox{ }d_{ {\bf{k}} }(-{\bf{q}}) + \delta_{ {\bf{k-q/2}}, 0 }\mbox{ }d^{\dagger}_{ {\bf{k}} }({\bf{q}})] [\delta_{ {\bf{k^{'}-q/2}}, 0 }\mbox{ }d_{ {\bf{k}}^{'} }({\bf{q}}) + \delta_{ {\bf{k^{'}+q/2}}, 0 }\mbox{ } d^{\dagger}_{ {\bf{k}}^{'} }(-{\bf{q}})]$$ or, $$H_{I} = (\frac{\rho_{0}}{2}) \sum_{ {\bf{q}} \neq 0 } v_{ {\bf{q}} } [d_{ -{\bf{q}}/2 }(-{\bf{q}}) + d^{\dagger}_{ {\bf{q}}/2 }({\bf{q}})] [d_{ {\bf{q}}/2 }({\bf{q}}) + d^{\dagger}_{ -{\bf{q}}/2 }(-{\bf{q}})]$$ and the free case is given by, $$H_{0} = \sum_{ {\bf{q}} }\epsilon_{ {\bf{q}} } d^{\dagger}_{ (1/2){\bf{q}} }({\bf{q}})d_{ (1/2){\bf{q}} }({\bf{q}})$$ The full hamiltonian may be diagonalised as follows, $$H = \sum_{ {\bf{q}} }\omega_{ {\bf{q}} } f^{\dagger}_{ {\bf{q}} }f_{ {\bf{q}} }$$ and, $$f_{ {\bf{q}} } = (\frac{ \omega_{ {\bf{q}} } + \epsilon_{ {\bf{q}} } + \rho_{0}v_{ {\bf{q}} } } { 2 \mbox{ }\omega_{ {\bf{q}} } })^{\frac{1}{2}} d_{ {\bf{q}}/2 }({\bf{q}}) + (\frac{ -\omega_{ {\bf{q}} } + \epsilon_{ {\bf{q}} } + \rho_{0}v_{ {\bf{q}} } } { 2 \mbox{ }\omega_{ {\bf{q}} } })^{\frac{1}{2}} d^{\dagger}_{ -{\bf{q}}/2 }(-{\bf{q}})$$ $$f^{\dagger}_{ -{\bf{q}} } = (\frac{ -\omega_{ {\bf{q}} } + \epsilon_{ {\bf{q}} } + \rho_{0}v_{ {\bf{q}} } } { 2 \mbox{ }\omega_{ {\bf{q}} } })^{\frac{1}{2}} d_{ {\bf{q}}/2 }({\bf{q}}) + (\frac{ \omega_{ {\bf{q}} } + \epsilon_{ {\bf{q}} } + \rho_{0}v_{ {\bf{q}} } } { 2 \mbox{ }\omega_{ {\bf{q}} } })^{\frac{1}{2}} d^{\dagger}_{ -{\bf{q}}/2 }(-{\bf{q}})$$ $$d_{ {\bf{q}}/2 }({\bf{q}}) = (\frac{ \omega_{ {\bf{q}} } + \epsilon_{ {\bf{q}} } + \rho_{0}v_{ {\bf{q}} } } { 2 \mbox{ }\omega_{ {\bf{q}} } })^{\frac{1}{2}}f_{ {\bf{q}} } - (\frac{ -\omega_{ {\bf{q}} } + \epsilon_{ {\bf{q}} } + \rho_{0}v_{ {\bf{q}} } } { 2 \mbox{ }\omega_{ {\bf{q}} } })^{\frac{1}{2}}f^{\dagger}_{ -{\bf{q}} }$$ $$d^{\dagger}_{ -{\bf{q}}/2 }(-{\bf{q}}) = (\frac{ \omega_{ {\bf{q}} } + \epsilon_{ {\bf{q}} } + \rho_{0}v_{ {\bf{q}} } } { 2 \mbox{ }\omega_{ {\bf{q}} } })^{\frac{1}{2}}f^{\dagger}_{ -{\bf{q}} } - (\frac{ -\omega_{ {\bf{q}} } + \epsilon_{ {\bf{q}} } + \rho_{0}v_{ {\bf{q}} } } { 2 \mbox{ }\omega_{ {\bf{q}} } })^{\frac{1}{2}}f_{ {\bf{q}} }$$ $$\omega_{ {\bf{q}} } = \sqrt{ \epsilon^{2}_{ {\bf{q}} } + 2 \rho_{0}v_{ {\bf{q}} }\epsilon_{ {\bf{q}} } }$$ From this we may deduce, $$\langle d^{\dagger}_{ (1/2){\bf{q}} }({\bf{q}}) d_{ (1/2){\bf{q}} }({\bf{q}}) \rangle = \frac{ -\omega_{ {\bf{q}} } + \epsilon_{ {\bf{q}} } + \rho_{0}v_{ {\bf{q}} } }{ 2\mbox{ }\omega_{ {\bf{q}} } }$$ From this it is possible to write down the filling fraction. $$f_{0} = N_{0}/N = 1 - (1/N)\sum_{ {\bf{q}} }\langle d^{\dagger}_{ (1/2){\bf{q}} }({\bf{q}}) d_{ (1/2){\bf{q}} }({\bf{q}}) \rangle$$ or, $$f_{0} = N_{0}/N = 1 - (1/2\pi^{2}\rho_{0})\int_{q_{min}}^{\infty} \mbox{ }dq\mbox{ } q^{2} ( \frac{ -\omega_{ {\bf{q}} } + \epsilon_{ {\bf{q}} } + \rho_{0}v_{ {\bf{q}} } }{ 2\mbox{ }\omega_{ {\bf{q}} } } )$$ Here the lower limit is necessary to ensure that the state $$d_{ (1/2){\bf{q}} }({\bf{q}}) \|G\rangle$$ has positive norm. and the value of $ q_{min} $ is given by, $$-\omega_{ q_{min} } + \epsilon_{ q_{min} }+ \rho_{0}v_{ q_{min} } = 0$$ First define, $$A_{ {\bf{q}} } = (\frac{ \omega_{ {\bf{q}} } + \epsilon_{ {\bf{q}} } + \rho_{0}v_{ {\bf{q}} } }{2 \mbox{ }\omega_{ {\bf{q}} } })^{\frac{1}{2}}$$ $$B_{ {\bf{q}} } = (\frac{ -\omega_{ {\bf{q}} } + \epsilon_{ {\bf{q}} } + \rho_{0}v_{ {\bf{q}} } }{2 \mbox{ }\omega_{ {\bf{q}} } })^{\frac{1}{2}}$$ The density operator is, $$\rho_{ {\bf{q}} }(t) = \sqrt{N}[d_{ -(1/2){\bf{q}} }(-{\bf{q}})(t) + d^{\dagger}_{ (1/2){\bf{q}} }({\bf{q}})(t)] + \sum_{ {\bf{k}} }d^{\dagger}_{ (1/2){\bf{k+q/2}} }({\bf{k+q/2}})(t) d_{ (1/2){\bf{k-q/2}} }({\bf{k-q/2}})(t)$$ $$d_{ -(1/2){\bf{q}} }(-{\bf{q}})(t) = A_{ {\bf{q}} }f_{ -{\bf{q}} }e^{-i\mbox{ }\omega_{ {\bf{q}} }t} - B_{ {\bf{q}} }f^{\dagger}_{ {\bf{q}} }e^{i\mbox{ }\omega_{ {\bf{q}} }t}$$ $$d^{\dagger}_{ (1/2){\bf{q}} }({\bf{q}})(t) = A_{ {\bf{q}} }f^{\dagger}_{ {\bf{q}} }e^{i\mbox{ }\omega_{ {\bf{q}} }t} - B_{ {\bf{q}} }f_{ -{\bf{q}} }e^{-i\mbox{ }\omega_{ {\bf{q}} }t}$$ $$d_{ (1/2){\bf{k-q/2}} }({\bf{k-q/2}})(t) = A_{ {\bf{k-q/2}} }f_{ {\bf{k-q/2}} }e^{-i\mbox{ }\omega_{ {\bf{k-q/2}} }t} - B_{ {\bf{k-q/2}} }f^{\dagger}_{ {\bf{-k+q/2}} } e^{i\mbox{ }\omega_{ {\bf{k-q/2}} }t}$$ $$d^{\dagger}_{ (1/2){\bf{k+q/2}} }({\bf{k+q/2}})(t) = A_{ {\bf{k+q/2}} }f^{\dagger}_{ {\bf{k+q/2}} } e^{i\mbox{ }\omega_{ {\bf{k+q/2}} }t} - B_{ {\bf{k+q/2}} }f_{ {\bf{-k-q/2}} }e^{-i\mbox{ }\omega_{ {\bf{k+q/2}} }t}$$ Now define, $$S^{>}({\bf{q}}t) = \langle \rho_{ {\bf{q}} }(t) \rho_{ -{\bf{q}} }(0)\rangle = N\langle [ d_{ -(1/2){\bf{q}} }(-{\bf{q}})(t) + d^{\dagger}_{ (1/2){\bf{q}} }({\bf{q}})(t) ] [ d_{ (1/2){\bf{q}} }({\bf{q}})(0) + d^{\dagger}_{ -(1/2){\bf{q}} }(-{\bf{q}})(0) ] \rangle$$ $$\sum_{ {\bf{k}}, {\bf{k}}^{'} } \langle d^{\dagger}_{ (1/2){\bf{k+q/2}} }({\bf{k+q/2}})(t) d_{ (1/2){\bf{k-q/2}} }({\bf{k-q/2}})(t) d^{\dagger}_{ (1/2){\bf{k^{'}-q/2}} }({\bf{k^{'}-q/2}})(0) d_{ (1/2){\bf{k^{'}+q/2}} }({\bf{k^{'}+q/2}})(0) \rangle$$ $$= N (\frac{ \epsilon_{ {\bf{q}} } }{\omega_{ {\bf{q}} } }) \mbox{ }exp(-i \mbox{ }\omega_{ {\bf{q}} }t)$$ $$+ \sum_{ {\bf{k}}, {\bf{k}}^{'} } \langle B_{ {\bf{k+q/2}} } f_{ {\bf{-k-q/2}} }e^{-i\omega_{ {\bf{k+q/2}} }t} A_{ {\bf{k-q/2}} }f_{ {\bf{k-q/2}} }e^{-i\omega_{ {\bf{k-q/2}} }t} A_{ {\bf{k^{'}-q/2}} }f^{\dagger}_{ {\bf{k^{'}-q/2}} } B_{ {\bf{k^{'}+q/2}} }f^{\dagger}_{ {\bf{-k^{'}-q/2}} } \rangle$$ $$+ \sum_{ {\bf{k}}, {\bf{k}}^{'} } \langle B_{ {\bf{k+q/2}} } f_{ {\bf{-k-q/2}} }e^{-i\omega_{ {\bf{k+q/2}} }t} B_{ {\bf{k-q/2}} }f^{\dagger}_{ {\bf{-k+q/2}} }e^{i\omega_{ {\bf{k-q/2}} }t} B_{ {\bf{k^{'}-q/2}} }f_{ {\bf{-k^{'}+q/2}} } B_{ {\bf{k^{'}+q/2}} }f^{\dagger}_{ {\bf{-k^{'}-q/2}} } \rangle$$ $$S^{>}({\bf{q}},t) = N (\frac{ \epsilon_{ {\bf{q}} } }{\omega_{ {\bf{q}} } }) \mbox{ }exp(-i \mbox{ }\omega_{ {\bf{q}} }t)$$ $$+ \sum_{ {\bf{k}} } exp(-i \mbox{ }(\omega_{ {\bf{k+q/2}} }+\omega_{ {\bf{k-q/2}} })t) [ B^{2}_{ {\bf{k+q/2}} }A^{2}_{ {\bf{k-q/2}} } + B_{ {\bf{k+q/2}} }B_{ {\bf{-k+q/2}} }A_{ {\bf{k-q/2}} } A_{ {\bf{k+q/2}} } ]$$ $$S^{<}({\bf{q}},t) = N (\frac{ \epsilon_{ {\bf{q}} } }{\omega_{ {\bf{q}} } }) \mbox{ }exp(i \mbox{ }\omega_{ {\bf{q}} }t) + \sum_{ {\bf{k}} } exp(i \mbox{ }(\omega_{ {\bf{k+q/2}} }+\omega_{ {\bf{k-q/2}} })t) [ B^{2}_{ {\bf{k-q/2}} }A^{2}_{ {\bf{k+q/2}} } + B_{ {\bf{k-q/2}} }B_{ {\bf{-k-q/2}} }A_{ {\bf{k+q/2}} } A_{ {\bf{k-q/2}} } ]$$ and, $$S^{>}({\bf{q}},t) = \langle \rho_{ {\bf{q}} }(t) \rho_{ -{\bf{q}} }(0) \rangle$$ $$S^{<}({\bf{q}},t) = \langle \rho_{ -{\bf{q}} }(0) \rho_{ {\bf{q}} }(t) \rangle$$ From the above equations, it is easy to see that there is a coherent part corresponding to the Bogoliubov spectrum and an incoherent part which is due to correlations and is responsible (hopefully) for the roton minimum. 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--- author: - 'Sergios Petridis, Theodoros Giannakopoulos and Costantine D. Spyropoulos' bibliography: - 'low\_cost\_pupilometry.bib' title: Unobtrusive Low Cost Pupil Size Measurements using Web cameras --- Motivation ========== Unobtrusive every day health monitoring can be of important use for the elderly population. In particular, pupil size may be a valuable source of information, since, apart from pathological cases, it can reveal the emotional state, the fatigue and the ageing. To allow for unobtrusive monitoring to gain acceptance, one should seek for efficient methods of monitoring using common low-cost hardware. A low cost camera that monitors the user while in front of a laptop or behind a mirror [@poh2011medical] falls into this scenario. Detecting pupils and pupil sizes in this context is of great importance. Namely, pupil sizes may be a valuable source of information, since, apart from pathological cases, it can reveal the emotional state [@Partala2003185], the fatigue [@doi:10.1076/0271-3683(200007)2111-ZFT535] and the ageing [@winn1994factors] of the subject under monitoring. Towards this end, this work presents a method for detecting iris and pupils, including both their centers and sizes, from low resolution visible-spectrum images, using a robust unsupervised filter-based approach. Iris detection performance outperforms most state of the art methods compared, and is competitive to few others. With respect to pupil detection, to our knowledge, detecting pupil sizes detection is not reported elsewhere in the related literature. Using a dataset compiled in particular for this purpose, we show that our method is accurate enough to provide significant information for everyday long-term monitoring. Relevant Work ============= The task of detecting eyes in images or videos is crucial and challenging in many computer vision applications. First, eye detection is a vital component of most face recognition systems, where eyes are used for feature extraction, alignment, face normalization, etc.. In addition, eye tracking is widely used in human computer interaction (gaze tracking). Eye detection systems can be categorized according to the adopted data acquisition method in (a) visible imaging and (b) infrared imaging. According to the first [@2001_hausdorffDistance; @2004_eyedetection_projection_functions; @2006_eyeDetectionEdge_asteriadis; @2011_eyeDetection; @2008valentiaccurate; @2004Cristinacceamulti-stage; @2005hamouzfeature], ambient light reflected from the eye area is captured, hence the task is rather difficult, due to the fact that captured information can contain multiple specular and diffuse components [@2006-MS-Dongheng-Li]. On the other hand, infrared-based approaches [@2006-MS-Dongheng-Li; @2005_StarburstEyeTracking; @2009_GeometrickEyeTracking] manage to eliminate specular reflections and lead to a better and accurate pupil detection. Another discrimination between eye detection approaches is based on the distance of the recording device: (a) head-mounted and (b) remote systems. Needless to say, head-mounted approaches can lead to more accurate systems. However, under particular requirements of low cost and low level of obstruction, remote sensing is the only acceptable solution. Method ====== Preprpocessing {#ssec:preprocessing} -------------- The overall scheme of the proposed method is presented in Figure \[fig:method\]. ![The overall architecture of the proposed method.[]{data-label="fig:method"}](EyeDetection.jpg){width="7.0cm"} At a first stage, the face is detected. Face detection is a well studied problem in machine vision [@viola2001rapid] and there exist now several commercial tools that achieve high accuracy with high speed. For our purposes, we have used SHORE^TM^[^1] which achieves face detection at a frame rate greater than 50fps. The same engine, also provides directly as a rough estimate of the two eyes area, which we have used to initiate iris and pupil detection. ![Sample eye area image[]{data-label="fig:eye"}](single_eye.jpg){width="30.00000%"} Sclera and Iris detection {#ssec:iris_detection} ------------------------- The sclera/iris detection method aims at determining the coordinates $( e^L_x, E^L_y), (e^R_x, e^R_y)$ and the radii $e^L_r, e^R_r$ of the left and right irises, considered as circular disks[^2]. Detection is done independently in each eye and is achieved by maximizing that output of a scoring process, while applying a specialized bank of linear filters parametrized by the radius of the iris, within the rough area of the eye. Therefore, this process results in both the estimation of the center of iris and its radius. The overall score is evaluated as a sum of three scores, based on luminosity, saturation and symmetry, whose definitions are given below. ![Mask for iris and sclera detection: The figure indicates with different shades the the three regions of the mask. The actual values depend on the criterion used and on the normalisation factor.[]{data-label="fig:mask"}](eye_mask.jpg){width="30.00000%"} ### Detection based on luminosity Denoting the luminosity pixels of the eye rough area as $I_L$ and the set of applied masks as $\{M^L_r\}$, the luminosity score is defined as: $$l( e_x, e_y, e_r ) = I_L[x,y,r] \cdot M^L_r$$ where $\cdot$ above denotes element by element multiplication and $I_L[x,y,r]$ is the luminosity values of an image region centered at $(x,y)$ and with size equal to the size of mask $M^L_r$. The motivation here is that the iris can be located as a region darker than the surrounding sclera. To that end, we have used a mask with three regions, as depicted in Figure \[fig:mask\], where the elements of each region all share the same value. In particular: iris : a circle centered at the center of mask, where elements have the same negative value sclera : a region defined as the difference of the above circle and the co-centered ellipse of equal radius along the vertical axis and double radius along the horizontal axis, where elements have the same positive value skin : a region define as the difference of the above ellipse and a co-center rectangular region, where the elements have zero value The mask values are normalized such that they sum up to zero. ### Detection based on saturation To detect the sclera and the iris, it is useful to observe that the sclera is typically much less saturated than both the iris and the surrounding skin. To that end we apply the same method as above using a set of masks $M^S_r$. These masks are similar to the ones used for detection based on luminosity, in that they are composed of the same regions. The difference lie in the value of pixel within each region. Namely, for the iris and skin elements share the same positive value whereas for the sclera the same negative value. As for luminosity, the mask values are normalized such that they sum up to zero. The score based on saturation is then evaluated as $$s( e_x, e_y, e_r ) = I_S[x,y,r] \cdot M^S_r$$ where $I_S$ is the saturation values of the eye rough area considered. Note that, as a practical approximation, our method actually uses the $V$ channel of the $YUV$ format to approximate saturation. ### Detection based on symmetry A further observation to boost the accuracy of sclera and iris detection is that these regions show a significant symmetry. In particular, we have experimentally found that checking for horizontal symmetry inside the iris and sclera regions significantly removes false eye detections. In particular, by denoting with superscript H an image region obtain by horizontal flipping, we evaluate the symmetry score as $$\begin{split} h( e_x, e_y, e_r ) &= \bigl(\| I_L[x,y,r] - I^H_L[x,y,r] \| \\ &+ \| I_S[x,y,r] - I^H_S[x,y,r] \| \bigr) \cdot M^H_r \end{split}$$ Note that both luminance and saturation values are used here. The mask $M^H_r$ is of similar structure to $M^L_r$, though with negative elements inside the sclera and iris and zero elements within the skin area. ### Overall Score The overall score for each candidate iris center and radius is evaluated as a sum over the respective luminance, saturation and symmetry scores. $$c( e_x, e_y, e_r ) = l( e_x, e_y, e_r ) + s( e_x, e_y, e_r ) + h( e_x, e_y, e_r ) \label{eq:irisScore}$$ and the final choice is made by exhaustively searching over the rough eye area found during preprocessing. $$( \hat e_x, \hat e_y, \hat e_r ) = \operatorname{arg\,max}_{e_x, e_y, e_r} c( e_x, e_y, e_r )$$ Pupil Detection and measurement ------------------------------- As soon as the disk that defines the iris area for each eye has been estimated, the pupil is detected as a circle within the iris that optimal satisfies a gradient-based criterion. Candidate pupils are circles with center $(p_x,p_y)$ in the close neighborhood of the iris center $(e_x,e_y)$, with radius such that they fall strictly inside the iris area. The gradient criterion is evaluated as the difference between the average luminosity of the pixel defining the perimeter of the candidate circle with the average luminosity of the immediate outer pixels of the circle. The greater the difference, the greater the possibility that the circle corresponds to the pupil. Note that the sign of the difference is important here. Though this approach is conceptually simple, it has shown that it is quite accurate, even in the presence of light reflections, which may degrade the performance of a non-gradient method. Optimal Frame ------------- The procedure outlined for detecting the iris and the pupil is repeated for every frame obtained from the camera, for both eyes. Since the ultimate goal is to measure the pupil size, and given that pupil size does not change from frame to frame[^3], it is not needed to measure the pupil on each frame, but rather on a frame where it can be measured with higher confidence. To that end, we describe now a method that evaluates the optimal frame based on which detecting the pupil and measuring its size can be attempted. Namely, for every frame, we compute an overall confidence score, as the product of the following measures: - left eye iris detection score - right eye iris detection score - $e^L_r$ and $e^R_r$ equality based score - $e^L_y$ and $e^R_y$ equality based score - $p^L_y$ and $p^R_y$ equality based score Regarding the first two items of the above list are directly given through Eq \[eq:irisScore\]. The equality based scores are evaluated based on the generic formula $$s = \frac{\| l - r \|}{\max \{l,r\}}$$ where $l$ (respectively $r$) is a measured obtained from the left (respectively right) eye. Therefore, as the video is streaming, we evaluate the overall confidence score and compare it to the one that has been obtained so far. In case it is greater, the less confident value is discarded and the new one is kept as the optimal one. In this way, the latest results are always based on the more confident frames. The procedure is repeated until the person under monitoring is stopped from being tracked, or after the end of a predefined time duration. In both case, the confidence score is reset and the procedure is repeated again. Complexity Analysis =================== A main concern in the development of the proposed method has been to keep low the overall complexity. This has been important for two reasons. The first one relates to low resources that the method should be needing. Either running as a background process on a tablet, or as a process on a dedicated hardware, detecting and measuring pupils should take as few resources as possible, given that the same hardware may be hosting other processes too. The second one relates to the speed of execution. Even though pupil size measurement is not critical, and therefore latency is affordable, the need of video recording should nevertheless be avoided to address users privacy concerns. Overall, our goal has been to measuring pupils in at least real time given limited processing resources and no storage. One-pass iteration ------------------ The method that has been described in this paper does achieve this goal. In particular, a significant speed up has been achieved by allowing scores involving iterations over pixels to be computed with a single pass over the respective image region. This has been possible, since all masks share the same structure and therefore scores are simultaneously updated by iterating over the region pixels. Moreover, since mask elements have no more than three values for each mask, the computation requires a number of additions equal to the number of pixel in the region. Multiplications are only constant with respect to image region size. ### Pupil Scores In the same direction, we also stress that candidate pupil scores are evaluated within the same iteration. To achieve this, the value of the mask element within the iris is used. In particular, notice that while the sign of the elements can be used to identify that they belong to the iris, the value is of non particular importance when evaluating the luminance and saturation values, since, as noted above, a plain summation of the values within the iris is performed, followed by a normalization using a pre-calculated normalization factor. Therefore, the value of the elements have been used to tag the distance from the mask center, as depicted in Figure \[fig:actualMask\]. In this way, while iterating over the mask elements, an array indexed by the distance from the center, progressively accumulates the sum of the values with the same distance from the center. After iterating over all elements, this array will contain, in each element (index), the sum of luminance values for the given index. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - - - - 8 8 8 7 8 8 8 - - - - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - - - - - 8 7 7 7 6 7 7 7 8 - - - - - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - - - - - - 8 7 6 6 6 5 6 6 6 7 8 - - - - - - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - - - - - - 8 7 6 6 5 5 4 5 5 6 6 7 8 - - - - - - 0 0 0 0 0 0 0 0 0 0 0 0 0 - - - - - - - 8 7 6 5 4 4 3 4 4 5 6 7 8 - - - - - - - 0 0 0 0 0 0 0 0 0 0 0 0 - - - - - - - 8 7 6 5 4 3 2 3 4 5 6 7 8 - - - - - - - 0 0 0 0 0 0 0 0 0 0 0 - - - - - - - 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 - - - - - - - 0 0 0 0 0 0 0 0 0 0 0 - - - - - - - 8 7 6 5 4 3 2 3 4 5 6 7 8 - - - - - - - 0 0 0 0 0 0 0 0 0 0 0 0 - - - - - - - 8 7 6 5 4 4 3 4 4 5 6 7 8 - - - - - - - 0 0 0 0 0 0 0 0 0 0 0 0 0 - - - - - - 8 7 6 6 5 5 4 5 5 6 6 7 8 - - - - - - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - - - - - - 8 7 6 6 6 5 6 6 6 7 8 - - - - - - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - - - - - 8 7 7 7 6 7 7 7 8 - - - - - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - - - - 8 8 8 7 8 8 8 - - - - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ### Parallelization Note also that our method can be graciously parallelized in many cores — one instruction processing architecture which would allow a further speedup on the execution. Such a solution is highly desirable in cases which one wishes to make the most out of a dedicated hardware including both CPU and GPU. Actually, the authors are currently implementing the method using the CUDA programming language, such that it can be executed on a low energy consumption nettop (Zotac Z-BOX ID84 PLUS) featuring a Intel Atom D2550 1.86 GHz Dual-Core CPU and NVIDIA GeForce GT 520M (512 MB) GPU. ### Tuning the frame rate We further notice that in case the method needs to be implemented in a lower processing capabilities hardware, real time analysis can be guaranteed by lowering down the video frame rate per second. Of course, in this case, a lower accuracy may be noticed, given that frames containing clearer pupil sizes may have been missed. The frame rate achieved in the Zotac Z-BOX ID84 PLUS using only CPU has been 5 frames per second, whereas a frame rate above 30 has been achieve for a PC feature an Intel Core i5-2500 CPU @ 3.30GHz. Results ======= Our evaluation had two purposes. First, to evaluate the performance of the iris detection module. In this case, we are only interested in estimating the iris center (not its size), since most of the related publicly available datasets only have center annotations (e.g., [@bioID]). Second, to evaluate the performance of pupil detection. Here, we are interested in estimating the exact pupil area (center and radius). Towards this end, we have built a dataset with pupil-related annotations. Iris center localization performance {#ssec:eyeCenter} ------------------------------------ The proposed method has been evaluated against nine state of the art methods. Two of them are provided by the MATLAB Vision Toolbox [@cartMatlab; @CastrillonDGH07], namely (a) CART [@cartMatlab] and (b) HAAR [@CastrillonDGH07], and allowed an in depth comparison using several performance measures. The others have been compared using reported results in [@2001_hausdorffDistance; @2004_eyedetection_projection_functions; @2006_eyeDetectionEdge_asteriadis; @2011_eyeDetection; @2008valentiaccurate; @2004Cristinacceamulti-stage; @2005hamouzfeature]. Furthermore, a “Rough” estimation has been used as baseline based on setting the estimated iris center as the center of the initial individual eye areas, which are extracted as explained in Section \[ssec:preprocessing\]. In some cases only the results of MATLAB-related methods are shown, since these are reproducible, while all compared methods are only shown for the case of Table \[tbl:results\_tol\]. We compared the methods against the widely used BioID dataset [@bioID], using all available samples. BioID test cases include a larger variety of illuminations conditions. The performance measures involved in this evaluation are defined as follows. First, let $d_l$ (respectively $d_r$ be the euclidean distance between the detected and manually annotated left (respectively right) iris centers. Also, let $d_{lr}$ be the distance between the manually annotated left and right iris centers. The relative errors* for the two detected irises are evaluated as $ e_l = \frac{d_l}{d_{lr}} \text{ and } e_r = \frac{d_r}{d_{lr}} $ whereas the relative error over both eyes as $e = (e_l + e_r ) / 2$. The respective error measures over the dataset are naturally defined as average errors over all dataset samples: $E_l = \frac{1}{N}\sum_{i=1}^{N} e_l(i)$, $E_r = \frac{1}{N}\sum_{i=1}^{N} e_r(i)$ and $E = \frac{1}{2}(E_l + E_r)$, where $N$ is the total number of samples in the testing set. Table \[tbl:results\_avg\] shows the average iris detection errors for the compared MATLAB-related methods. The proposed method outperforms all compared methods. ----------------------------------------------------------------------- Method $E_l$ $E_r$ $E$ ----------------------------------------------- ------- ------- ------- HAAR 0.052 0.040 0.046 CART 0.060 0.057 0.058 Rough 0.054 0.053 0.053 [Proposed]{} & 0.035 & 0.021 & 0.028\ ******** ----------------------------------------------------------------------- : Iris Detection Average Relative Error (left, right and overall) results (MATLAB-related methods compared)[]{data-label="tbl:results_avg"} Furthermore, we have compared our method against an tolerance-based accuracy measure widely used in the literature [@2001_hausdorffDistance; @2004_eyedetection_projection_functions; @2006_eyeDetectionEdge_asteriadis; @2011_eyeDetection; @2008valentiaccurate; @2004Cristinacceamulti-stage; @2005hamouzfeature]. Namely, given an error tolerance $T$, an eye detection result is considered as successful if both errors $e_l$ and $e_r$ are less than $T$: $$A_T = \frac{\sum_{i: max(e_l(i), e_r(i)) \leq T} 1 }{N}$$ Using this measure, it has been possible to compare the proposed method against [@2001_hausdorffDistance; @2004_eyedetection_projection_functions; @2006_eyeDetectionEdge_asteriadis; @2011_eyeDetection; @2008valentiaccurate; @2004Cristinacceamulti-stage; @2005hamouzfeature] as well, using the corresponding reported results. A typical value for the threshold is $T=0.25$, because it corresponds to an accuracy of about half the width of an eye in the image, while $T=0.1$ has also been used [@2004_eyedetection_projection_functions; @bioID; @2006_eyeDetectionEdge_asteriadis]. In Table \[tbl:results\_tol\] the tolerance-based accuracy for three different tolerance thresholds is presented. The proposed method outperforms most of the compared methods, except for the case of $T=0.05$. ------------------------------------------- ------ ----- ------ Method 0.05 0.1 0.25 HAAR 22 75 98 CART 8 68 99 Rough 15 68 98 [@2001_hausdorffDistance] 40 80 91 [@2004_eyedetection_projection_functions] - - 95 [@2006_eyeDetectionEdge_asteriadis] 50 82 98 [@2011_eyeDetection] 45 85 95 [@2008valentiaccurate] 84 91 99 [@2004Cristinacceamulti-stage] 56 96 98 [@2005hamouzfeature] 59 77 93 [Proposed]{} & 47 & 92 & 99\ ------------------------------------------- ------ ----- ------ : Iris detection tolerance-based accuracy results for three different tolerance values (all methods compared)[]{data-label="tbl:results_tol"} Pupil size estimation performance {#ssec:pupilPerformance} --------------------------------- Pupils size estimation evaluation needs a different setting from the one described above. To begin with, annotations of the pupil’s whole area, not just its center, is needed. To our knowledge, there is no available dataset with such annotations. Therefore, we have built a manually annotated dataset of pupil area. Three humans annotated the irises and pupils, to estimate an inter-annotator agreement level. Final pupil annotations have been considered s the average ares of all annotators. The compiled dataset consists of 50 1280x960 resolution face images of 10 different humans. With respect to the performance measures, we have used the recall, precision and F1 measures over the pupil areas. In particular, let $A_a$ be the area of the manually annotated pupil, $A_e$ be the area of the estimated pupil and $A_c$ be the area of their intersection. The recall, precision and $F_1$ rates are defined as $R = \frac{A_c}{A_a}$, $P = \frac{A_c}{A_e}$ and $F1 = \frac{2\cdot R \cdot P}{R + P}$ respectively. These same measures have also been used for evaluating the inter-annotator agreement. The results of the evaluation are displayed in Table \[tbl:results\_pupil\]. Note that the inter-annotation agreement scores are relatively low, revealing the difficulty even for a human in annotating the pupil area, especially for dark and brown eyes. In terms of average $F_1$ measure, the proposed pupil detection method is $12\%$ less accurate than the human annotation performance. In terms of pupil diameter size estimation, our method achieves on average $85\%$ accuracy. This means that for an average 6mm pupil diameter, average error is 0.9mm, which is much lower than the threshold of 2mm indicated for significant pupil differentiations [@Partala2003185; @winn1994factors]. A specific benchmark dataset containing such cases would nevertheless be needed to explicitly verify this conclusion. Precision Recall F1 ----------------- ----------- -------- ---- i-a.a 82 84 79 Proposed Method 66 68 67 : Pupil detection performance and comparison to inter-annotator agreement (i-a.a).[]{data-label="tbl:results_pupil"} Discussion ========== A method for iris and pupil detection, including their sizes, has been presented, based on a robust unsupervised recursive filtering technique. Evaluation on iris center detection has shown that the proposed method outperforms most of related algorithms. Pupil size estimation was evaluated on a separate dataset which also contains annotations regarding not only pupils position but their sizes as well. The final pupil detection performance results showed that the proposed method’s accuracy is accurate enough to be considered as a low cot pupillometry for long-term monitoring. Further results on video should be presented to show the utility of the confidence of the result based on frame sequences. Acknowledgments {#acknowledgments .unnumbered} =============== The research leading to these results has been funded by the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agree- ment no 288532. For more details, please see <http://www.usefil.eu.> [^1]: SHORE^TM^:Sophisticated High-speed Object Recognition Engine, Fraunhofer IIS [^2]: The left (respectively right) eye is denoted by subscript L (respectively R) [^3]: we assume here that lighting conditions stay the same
--- abstract: 'We study in detail various aspects of the renormalization of the spin-1 resonance propagator in the effective field theory framework. First, we briefly review the formalisms for the description of spin-1 resonances in the path integral formulation with the stress on the issue of propagating degrees of freedom. Then we calculate the one-loop $1^{--}$ meson self-energy within the Resonance chiral theory in the chiral limit using different methods for the description of spin-one particles, namely the Proca field, antisymmetric tensor field and the first order formalisms. We discuss in detail technical aspects of the renormalization procedure which are inherent to the power-counting non-renormalizable theory and give a formal prescription for the organization of both the counterterms and one-particle irreducible graphs. We also construct the corresponding propagators and investigate their properties. We show that the additional poles corresponding to the additional one-particle states are generated by loop corrections, some of which are negative norm ghosts or tachyons. We count the number of such additional poles and briefly discuss their physical meaning.' --- =1 LU TP 09-33 [Renormalization and additional degrees of freedom within\ the chiral effective theory for spin-1 resonances]{}\ [Karol Kampf$^{\,1,2}$, Jiri Novotny$^{\,2}$ and Jaroslav Trnka$^{\,2,3}$]{}\ $\ ^{1}$[Department of Theoretical Physics, Lund University, Sölvegatan 14A, SE 223-62 Lund, Sweden.]{}\ $\ ^{2}$[Institute of Particle and Nuclear Physics, Faculty of Mathematics and Physics,\ Charles University in Prague, V Holešovičkách 2, 18000 Prague, Czech Republic.]{}\ $\ ^{3}$[Department of Physics, Princeton University, 08540 Princeton, NJ, USA.]{} Introduction ============ As is well known, in the low energy region the dynamical degrees of freedom of QCD are not quarks and gluons but the low lying hadronic states and, as a consequence, a non-perturbative description of the their dynamics is inevitable. An approach using effective Lagrangians appears to be very efficient for this purpose and it has made a considerable progress recently. In the very low energy region ($E\ll \Lambda _H\sim 1{\rm GeV}$), the octet of the lightest pseudoscalar mesons ($\pi $, $K$, $\eta $) represents the only relevant part of the QCD spectrum. The Chiral Perturbation Theory ($% \chi $PT) [@Weinberg; @Gasser1; @Gasser2] based on the spontaneously broken chiral symmetry $SU(3)_L\times SU(3)_R$ grew into a very successful model-independent tool for the description of the Green functions (GF) of the quark currents and related low-energy phenomenology. The pseudoscalar octet is treated as the octet of pseudo-Goldstone bosons (PGB) and $\chi $PT is organized according to the Weinberg power-counting formula [@Weinberg] as a rigorously defined simultaneous perturbative expansion in small momenta and the light quark masses. Recently, the calculations are performed at the next-to-next-to-leading order $O(p^6)$ (for a comprehensive review and further references see [@Bijnens:2006zp]). In the intermediate energy region ($\Lambda _H\leq E<2{\rm GeV}$), where the set of relevant degrees of freedom includes also the low lying resonances, the situation is less satisfactory. This region is not separated by a mass gap from the rest of the spectrum and, as a consequence, there is no appropriate scale playing the role analogous to that of $\Lambda _H$ in $\chi PT$. Therefore, the effective theory in this region cannot be constructed as a straightforward extension of the $\chi $PT low energy expansion by means of introducing resonances e.g. as homogenously (but nonlinearly) transformed matter fields in the sense of [@Coleman:1969sm], [Callan:1969sn]{} and pushing the scale $\Lambda _H$ to $2 {\rm GeV}$. In order to introduce another type of effective Lagrangian description, the considerations based on the large $N_C$ expansion together with the high-energy constraints derived from perturbative QCD and OPE appear to be particularly useful. In the limit $N_C\rightarrow \infty $, the chiral symmetry is enlarged to $U(3)_L\times U(3)_R$ and the spectrum relevant for the correlators of the quark bilinears consists of an infinite tower of free stable mesonic resonaces exchanged in each channel and classified according to the symmetry group $U(3)_V$. An appropriate description should therefore require an infinite number of resonance fields entering the $U(3)_L\times U(3)_R$ symmetric effective Lagrangian. Because the quasi-classical expansion is correlated with the large $N_C$ expansion, the interaction vertices are suppressed by an appropriate power of $\,N_C^{-1/2}$ according to the number of the meson legs. At the leading order only the tree graphs have to be taken into account . An approximation to this general picture where we limit the number of the resonance fields to one in each channel and matching the resulting theory in the high energy region with OPE is known as the Resonance Chiral Theory (R$\chi $T) (it was introduced in seminal papers [@Ecker1; @Ecker2]). Integrating out the resonance fields from the Lagrangian of R$\chi $T in the low energy region and the subsequent matching with $\chi $PT has become very successful tool for the estimates of the resonance contribution to the values of the $O(p^4)$ [Ecker1]{} and $O(p^6)$ [@Cirigliano:2006hb; @Kampf:2006bn] low energy constants (LEC) entering the $\chi $PT Lagrangian. Therefore, studying R$% \chi $T can help us to understand not only the dynamics of resonances but also the origin of LECs in $\chi $PT. However, even when restricting to the case of the matter field formalism, it is known from the very beginning [@Ecker2] that the form of the R$\chi $T Lagrangian is not determined uniquely. The reason is that the resonances with a given spin can be described in many ways using fields with different Lorentz structure. For example, for the spin-one resonances one can use * i.a.* the Proca vector field or the antisymmetric tensor field or both (within the first order formalism [@Bruns:2004tj; @FO1]). Though the theories based on different types of fields with Lagrangians which contain only finite number of operators are not strictly equivalent already on the tree level (in general, it is necessary to include nonlocal interaction or infinite number of operators and contact terms to ensure the complete equivalence, see [@FO1]), we can always ensure a weak equivalence of all three formalisms up to a given fixed chiral order (this was established to $O(p^4)$ in [@Ecker2] and enlarged to $O(p^6)$ in [@FO1]). As we have mentioned above, the lack of the mass gap (which could provide us with a scale playing the role analogous to $\Lambda _{H}$) prevents us from using a straightforward extension of the Weinberg power-counting formula [@Weinberg] taking the resonance masses and momenta of the order $O(p)$ on the same footing as for PGB. Also the usual chiral power counting which takes the resonance masses as an additional heavy scale (which is counted as $O(1)$) fails within the R$\chi $T in a way analogous to the $\chi PT$ with baryons [@Gasser:1987rb]. Nevertheless, it seems to be fully legitimate to go beyond the tree level R$\chi $T and calculate the loops [@vecform; @SS; @SS2; @VV; @SanzCillero:2009pt; @Rosell:2009yb; @SanzCillero:2007ib; @SanzCillero:2009ap; @Rosell:2005ai]. Being suppressed by one power of $1/N_{C}$, the loops allow to encompass such NLO effects in the $1/N_{C}$ expansion as resonance widths, resonance cuts and the final state interaction and (by means matching with $\chi $PT) to determine the NLO resonance contribution to LEC (and their running with renormalization scale). However, we can expect both technical and conceptual complications connected with the renormalization of the effective theory for which no natural organization of the expansion (other than the $1/N_C$ counting) exists. Especially, because there is no natural analog of the Weinberg power counting in R$\chi $T, we can expect mixing of the naive chiral orders in the process of the renormalization (e.g the loops renormalize the $O(p^2)$ LEC and also counterterms of unusually high chiral orders are needed). Also a straightforward construction of the propagator from the self-energy using the Dyson re-summation can bring about the appearance of new poles in the GF. Because the spin-one particles are described using fields transforming under the reducible representation of the rotation group and due to the lack of an appropriate protective symmetry, some of these additional poles can correspond to new degrees of freedom, which are frozen at the tree level. The latter might be felt as a pathological artefact of the not carefully enough formulated theory, particularly because these extra poles might be negative norm ghosts or tachyons [@Slovak]. On the other hand, however, we could also try to take an advantage of this feature and to adjust the poles in such a way that they correspond to the well established resonance states [@Kampf:2008xp]. Let us note, that similar problems are generic for the description of the higher spin particles in terms of quantum field theory. As an example we can mention e.g. the problem with the renormalization of quantum gravity which is trying to be cured by imposing additional symmetry or by introducing a non-perturbative quantization believing that UV divergences are only artefact of a perturbative theory. In the context of the extensions of the $\chi PT$, this has been studied in connection with introducing of the spin-$3/2$ isospin-$3/2~\ \Delta (1232)$ resonance in the baryonic sector (for a review see [@Pascalutsa:2006up] and references therein). The Rarita-Schwinger field commonly used for its description contains along with the spin-$3/2$ sector also spin-$1/2$ sector, which is frozen at the tree level due to the form of the free equations of motion. These provides the necessary constraints reducing the number of propagating spin degrees of freedom to four corresponding to spin $% 3/2$ particles. However, these constraints are generally not present in the interacting theory and negative norm ghost [@Johnson:1960vt] and/or tachyonic [@Velo:1969bt] poles might appear beyond the tree level. The appearance of these extra unphysical degrees of freedom can be avoided by means of the requirement of additional protective gauge symmetry under which the interaction Lagrangian has to be invariant. Such a symmetry, which is also a symmetry of the kinetic term (but not of the mass term), is an analog of the $U(1)$ gauge symmetry of the electromagnetic field and its role is also similar. As it has been shown by means of path integral formalism, it leads to the same constraints as in the noninteracting theory and prevent therefore the extra spin-$1/2$ states from propagating. On the other hand, it has been proved, that the most general interaction Lagrangian at most bilinear in Rarita-Schwinger field (i.e. without the protective gauge symmetry) is on shell equivalent to the gauge invariant one [@Krebs:2008zb] . The latter is, however, nonlocal (or equivalently it contains an infinite number of terms). Also the above protective gauge symmetry is, as a rule, in a conflict with chiral symmetry, and has therefore to be implemented with a care. Though there are efficient methods how to handle this obstacles in concrete loop calculations [@Pascalutsa:2006up], [@Krebs:2008zb], the problem still has not been solved completely. In the following, we would like to discuss these problems in more detail. As an explicit example we use the one-loop renormalization of the propagator corresponding to the fields which originally describe $1^{--}$ vector resonance ($\rho $ meson) at the tree level within the Proca field, the antisymmetric tensor field and within the first order formalism in the chiral limit. The situation here is quite similar to the case of spin-3/2 resonances discussed above. In addition, to the spin-1 degrees of freedom, there are extra sectors that are frozen at the tree level. There exists a protective gauge symmetry which prevents these modes from propagation. The kinetic term is invariant with respect to this symmetry while the mass term is not. By means of an explicit calculation we will show that (unlike the ordinary $\chi PT$) the one-loop corrections to the self-energy need counterterms with a number of derivatives ranging from zero up to six and also that a new kinetic counterterm with two derivatives (which was not present in the tree level Lagrangian) is necessary. We will also demonstrate that the corresponding propagator obtained by means of Dyson re-summation of the one-particle irreducible self-energy insertions has unavoidably additional poles. Due to the unusual higher order growth of the self-energy in the UV region some of them are inevitably pathological (with a negative norm or a negative mass squared). Though these additional poles are decoupled in the limit $N_C\rightarrow \infty $, for reasonable concrete values of the parameters of the Lagrangian they might appear near or even inside the region for which $R\chi T$ was originally designed. We also discuss briefly within the antisymmetric tensor formalism a possible interpretation of some of the non-pathological poles as a manifestation of the dynamical generation of various types of additional 1+- states. We will also show that the appropriate adjustment of coupling constants in the antisymmetric tensor case allows us (at least in principle) to generate in this way the one which could be identified e.g. with the $b_1(1235)$ meson [@Kampf:2008xp]. Such a mechanism is analogous to the model [@Boglione:2002vv] for the dynamical generation of the scalar resonances from the bare quark-antiquark ”seed”, the propagator of which develops (after dressing with pseudoscalar meson loops) additional poles identified e.g. as $a_0(980)$ (cf. also [@Tornqvist:1995kr],[Tornqvist:1995ay]{}). The paper organized as follows. In Section \[Propagators and poles\] we remind the basic facts about the propagators and briefly discuss the issue of the additional degrees of freedom in all three formalisms for the description of spin-one resonances. We use the path integral formulation where the protective symmetry analogous to the Rarita-Schwinger case is manifest. In Section \[Section\_power\_counting\] we discuss the power counting. We try to formulate here a formal self-consistent organization of the counterterms and one-particle irreducible graphs, which sorts the operators in the Lagrangian according to the number of derivatives as well as number of the resonance fields and which is useful for the proof of renormalizability of the $R\chi T$ as an effective theory. In Section \[ch4\] we present the results of the explicit calculation of the self-energies. Then we give a list of counterterms and briefly discuss the renormalization prescription. Section \[Section\_propagators\] is devoted to the construction of the propagators and to the discussion of their poles. Because the basic ideas are similar within all three formalisms, we concentrate here on the antisymmetric tensor case. Section \[Section\_summary\] contains summary and conclusions. Some of the long formulae are postponed to the appendices: the explicit form of the renormalization scale independent parameters of the self-energies are collected in Appendix \[Appendix\_alpha\_beta\], namely for the Proca field in \[appendix Proca\], for the antisymmetric tensor field in \[appendix tensor\] and for the first order formalism in \[appendix first order\]. In Appendix \[Appendix\_positivity\] we give a proof of the positivity of the spectral functions for the antisymmetric tensor propagator. Propagators and poles\[Propagators and poles\] ============================================== In this section, we collect the basic properties of the propagators and the corresponding self-energies within the Proca field, the antisymmetric tensor field and the first order formalisms. The discussion will be as general as possible without explicit references to $R\chi T$, which can be assumed as the special example of the general case. Proca formalism --------------- ### General properties of the propagator We start our discussion with a standard textbook example of the interacting Proca field. Let us write the Lagrangian in the form $$\mathcal{L}=\mathcal{L}_{0}+\mathcal{L}_{int},$$ where the free part of the Lagrangian is $$\mathcal{L}_{0}=-\frac{1}{4}\widehat{V}_{\mu \nu }\widehat{V}^{\mu \nu }+% \frac{1}{2}M^{2}V_{\mu }V^{\mu }$$ with $$\widehat{V}_{\mu \nu }=\partial _{\mu }V_{\nu }-\partial _{\nu }V_{\mu }.$$ Without any additional assumptions on the form and symmetries of the interaction part of the Lagrangian $\mathcal{L}_{int}$, we can expect the following general structure of the full two-point one-particle irreducible (1PI) Green function $$\Gamma _{\mu \nu }^{(2)}(p)=(M^{2}-p^{2}+\Sigma ^{T}(p^{2}))P_{\mu \nu }^{T}+(M^{2}+\Sigma ^{L}(p^{2}))P_{\mu \nu }^{L}. \label{vector_gamma_2}$$ Here $$\begin{aligned} P_{\mu \nu }^{L} &=&\frac{p_{\mu }p_{\nu }}{p^{2}} \label{P_L} \\ P_{\mu \nu }^{T} &=&g_{\mu \nu }-\frac{p_{\mu }p_{\nu }}{p^{2}} \label{P_T}\end{aligned}$$ are the usual longitudinal and transverse projectors and $\Sigma ^{T,L}$ are the corresponding transverse and longitudinal self-energies, which vanish in the free field limit. Inverting (\[vector\_gamma\_2\]) we get for the full propagator $$\Delta _{\mu \nu }(p)=-\frac{1}{p^{2}-M^{2}-\Sigma ^{T}(p^{2})}P_{\mu \nu }^{T}+\frac{1}{M^{2}+\Sigma ^{L}(p^{2})}P_{\mu \nu }^{L}.$$ The possible (generally complex) poles of such a propagator are of two types; either at $p^{2}=s_{V}$, where $s_{V}$ is given by the solutions of $$s_{V}-M^{2}-\Sigma ^{T}(s_{V})=0, \label{V_pole_spin_1}$$ or at $p^{2}=s_{S}$ where $s_{S}$ is the solution of $$M^{2}+\Sigma ^{L}(s_{S})=0. \label{V_pole_spin_0}$$ Let us first discuss the poles of the first type. Assuming that ([V\_pole\_spin\_1]{}) is satisfied for $s_V=M_V^2>0$, then for $p^2\rightarrow $ $% M_V^2$ $$\begin{aligned} \Delta _{\mu \nu }(p) &=&\frac{Z_V}{p^2-M_V^2}\left( -g_{\mu \nu }+\frac{% p_\mu p_\nu }{M^2}\right) +O(1) \notag \\ &=&\frac{Z_V}{p^2-M_V^2}\sum_\lambda \varepsilon _\mu ^{(\lambda )}(p)\varepsilon _\nu ^{(\lambda )}(p)+O(1)\end{aligned}$$ where $$Z_V=\frac 1{1-\Sigma ^{^{\prime }T}(M_V^2)}$$ and where $\varepsilon _\mu ^{(\lambda )}(p)$ are the usual spin-one polarization vectors. Under the condition $Z_V>0$ the poles of this type correspond to spin-one one particle states $\|p,\lambda ,V\rangle $ which couple to the Proca field as $$\langle 0\|V_\mu (0)\|p,\lambda ,V\rangle =Z_V{}^{1/2}\varepsilon _\mu ^{(\lambda )}(p).$$ At least one of these states is expected to be perturbative in the sense that its mass and coupling to $V_\mu $ can be written as $$\begin{aligned} M_V^2 &=&M^2+\delta M_V^2 \\ Z_V &=&1+\delta Z_V,\end{aligned}$$ where $\delta M_V^2$ and $\delta Z_V$ are small corrections vanishing in the free field limit. This solution corresponds to the original degree of freedom described by the free part of the Lagrangian $\mathcal{L}_0$. The additional one particle states corresponding to the other possible (non-perturbative) solutions of (\[V\_pole\_spin\_1\]) decouple in the free field limit. The second type of poles is given by (intrinsically nonperturbative) solutions of (\[V\_pole\_spin\_0\]). Suppose that this condition is satisfied by $s_{S}=M_{S}^{2}>0$. For $p^{2}\rightarrow $ $M_{S}^{2}$ $$\Delta _{\mu \nu }(p)=\frac{Z_{S}}{p^{2}-M_{S}^{2}}\frac{p_{\mu }p_{\nu }}{% M_{S}^{2}}+O(1)$$ where $$Z_{S}=\frac{1}{\Sigma ^{^{\prime }L}(M_{S}^{2})}.$$ Assuming $Z_{S}>0$ this pole corresponds to the spin-zero one particle state $\|p,S\rangle $ which couples to $V_{\mu }$ as $$\langle 0\|V_{\mu }(0)\|p,S\rangle =\mathrm{i}p_{\mu }\frac{Z_{S}{}^{1/2}}{% M_{S}}.$$ For the free field this scalar mode is frozen and does not propagate according to the special form of the Proca field Lagrangian. Therefore, in the limit of vanishing interaction the extra scalar state decouples. Without any additional assumptions on the symmetries of the interaction Lagrangian we can therefore expect the appearance of additional dynamically generated degrees of freedom. The general picture is, however, more subtle. Note that, the interpretation of the above additional spin-one and spin-zero poles as physical one-particle asymptotic states depends on the proper positive sign of the corresponding residues $Z_V,\,Z_S>0$, otherwise the norm of these states is negative and the poles correspond to the negative norm ghosts. Similarly, also poles with $M_{V,S}^2<0$ can be generated, which correspond to the tachyonic states. Let us illustrate this feature using a toy example. Suppose, that the only interaction terms are of the form $$\mathcal{L}_{int}\equiv \mathcal{L}_{ct}=-\frac \alpha 4\widehat{V}_{\mu \nu }\widehat{V}^{\mu \nu }-\frac \beta 2(\partial _\mu V^\mu )^2+\frac \gamma {2M^2}(\partial _\mu \widehat{V}^{\mu \nu })(\partial ^\rho \widehat{V}% _{\rho \nu })+\frac \delta {2M^2}(\partial _\mu \partial _\rho V^\rho )(\partial ^\mu \partial _\sigma V^\sigma ). \label{L_V_toy}$$ Such a Lagrangian can be typically produced by radiative corrections in an effective field theory with Proca field, which does not couple to other fields in a $U(1)$ gauge invariant way, and can provide us with counterterms necessary to renormalize the loops contributing to the $V$ field self-energy. $\mathcal{L}_{ct}$ gives rise to the following contributions to $\Sigma ^T(p^2)$ and $\Sigma ^L(p^2)$ $$\begin{aligned} \Sigma ^T(p^2) &=&-\alpha p^2+\gamma \frac{p^4}{M^2} \label{Sigma_T_vector} \\ \Sigma ^L(p^2) &=&-\beta p^2+\delta \frac{p^4}{M^2}. \label{Sigma_L_vector}\end{aligned}$$ As a result, we have two spin-one and two spin-zero one-particle states. The masses and residue of the spin-one states are then $$\begin{aligned} M_{V\pm }^2 &=&M^2\left( 1+\frac{1+\alpha -2\gamma \mp \sqrt{(1+\alpha )^2-4\gamma }}{2\gamma }\right) \label{MV} \\ 1-\Sigma ^{^{\prime }T}(M_{V\pm }^2) &=&\pm \sqrt{(1+\alpha )^2-4\gamma }, \label{ResV}\end{aligned}$$ which are real for for $(1+\alpha )^2-4\gamma >0$. In the limit $\alpha ,\,\gamma \rightarrow 0$, $\alpha /\gamma =const$ we get either the perturbative solution with mass $M_{V+}$ or (for $\gamma >0$) an additional spin-one ghost with mass $M_{V-}$ (for $1+\alpha >0$ and $\gamma <0$ this pole is tachyonic). Similarly for the spin-zero states $$\begin{aligned} M_{S\pm }^2 &=&M^2\left( \frac{\beta \mp \sqrt{\beta ^2-4\delta }}{2\delta }% \right) \label{MS} \\ \Sigma ^{^{\prime }L}(M_{S\pm }^2) &=&\mp \sqrt{\beta ^2-4\delta }. \label{ResS}\end{aligned}$$ The poles are real for $\beta ^2>4\delta $ and e.g.* for $\beta ,\delta >0$ one of the poles is spin-zero ghost. In both cases for appropriate values of the parameters we can get also two tachyons or even the complex Lee-Wick pair of ghosts. These features are of course well known in the connection with the higher derivative regularization (as well as with the properties of the gauge-fixing term). ### Additional degrees of freedom in the path integral formalism$ \label{path integral vector}$ The additional degrees of freedom discussed in the previous subsection can be made manifest in the path integral formalism. Let us start with the generating functional for the interacting Proca field $$Z[J]=\int \mathcal{D}V\exp \left( \mathrm{i}\int \mathrm{d}^4x\left( -\frac 14\widehat{V}_{\mu \nu }\widehat{V}^{\mu \nu }+\frac 12M^2V_\mu V^\mu +% \mathcal{L}_{int}(V,J,\ldots )\right) \right) ,$$ where the external sources are denoted collectively by $J$. In order to separate the transverse and longitudinal degrees of freedom of the field $% V_\mu $ within the path integral we can use the standard Faddeev-Popov trick with respect to the $U(1)$ gauge transformation of the field $V_\mu $$$V_\mu \rightarrow V_\mu +\partial _\mu \Lambda . \label{V_gauge}$$ As a result, we get the generating functional in the form $$Z[J]=\int \mathcal{D}V_{\perp }\mathcal{D}\Lambda \exp \left( \mathrm{i}\int \mathrm{d}^4x\left( \frac 12V_{\perp }^\mu \square V_{\perp \mu }+\frac 12M^2V_{\perp }^\mu V_{\perp \mu }+\frac 12M^2\partial _\mu \Lambda \partial ^\mu \Lambda +\mathcal{L}_{int}(V_{\perp }-\partial \Lambda ,J,\ldots )\right) \right) . \label{FP_trick}$$ Here $\mathcal{D}V_{\perp }=\mathcal{D}V\delta (\partial _\mu V^\mu )$ and $$V_{\perp }^\mu =\left( g^{\mu \nu }-\frac{\partial ^\mu \partial ^\nu }{% \square }\right) V_\nu$$ is the transverse part of the vector field $V^\mu $, the longitudinal part of which corresponds to the scalar field $\Lambda $,* i.e.* $$V^\mu =V_{\perp }^\mu +\partial ^\mu \Lambda . \label{V_components}$$ The free propagators of the fields $V_{\perp }^\mu $ and $\Lambda $ are $$\begin{aligned} \Delta _{\perp }^{\mu \nu }(p) &=&-\frac{P^{T\,\mu \nu }}{p^2-M^2} \label{propVtrans} \\ \Delta _\Lambda (p) &=&\frac 1{M^2}\frac 1{p^2}. \label{propLambda}\end{aligned}$$ Both these propagators have spurious poles at $p^2=0$, however, the only necessary combination which matters in the Feynman graphs is $$\Delta _0^{\mu \nu }(p)=\Delta _{\perp }^{\mu \nu }(p)+p^\mu p^\nu \Delta _\Lambda (p),$$ which coincides with the original free propagator of the field $V^\mu $ and the spurious poles cancel each other. Note that, provided the interaction Lagrangian $\mathcal{L}_{int}$ is symmetric under the $U(1)$ gauge transformation (\[V\_gauge\]), the spin-zero field $\Lambda $ completely decouples and can be integrated out . The theory can then be formulated solely in terms of the field $V_{\perp }^{\mu }$. The $U(1)$ invariant form of the interaction allows to simplify the propagator $\Delta _{\perp }^{\mu \nu }(p)$$$\Delta _{\perp }^{\mu \nu }(p)\rightarrow -\frac{g_{\mu \nu }}{p^{2}-M^{2}}$$ within the Feynman graphs and the spurious pole $p^{2}=0$ in ([propVtrans]{}) becomes harmless. In this case, the scalar one-particle states cannot be dynamically generated. On the other hand, in the case when $% \mathcal{L}_{int}$ is not invariant with respect to (\[V\_gauge\]), we cannot forget the longitudinal component of $V^{\mu }$ which has now nontrivial interactions and, as a result, contributions to $\Sigma ^{L}$ can be generated. Let us now return to the illustrative example discussed in the previous subsection. Suppose that the interaction Lagrangian has the form $$\mathcal{L}_{int}=\mathcal{L}_{ct}+\mathcal{L}_{int}^{^{\prime }}$$ where $\mathcal{L}_{ct}$ is the toy interaction Lagrangian (\[L\_V\_toy\]) and we assume $\alpha >-1\,$and $\delta >0$ in what follows. Then it is possible to transform $Z[J]$ to the form of the path integral with all the additional degrees of freedom represented explicitly in the Lagrangian and the integration measure. In terms of the transverse and longitudinal degrees of freedom we get $$\begin{aligned} \mathcal{L}_{int}(V_{\perp }-\partial \Lambda ,J,\ldots ) &=&\mathcal{L}% _{ct}(V_{\perp }-\partial \Lambda ,J,\ldots )+\mathcal{L}_{int}^{^{\prime }}(V_{\perp }-\partial \Lambda ,J,\ldots ) \notag \\ &=&\frac \alpha 2V_{\perp }^\mu \Box V_{\perp \mu }-\frac \beta 2(\Box \Lambda )^2+\frac \gamma {2M^2}(\Box V_{\perp }^\mu )(\Box V_{\perp \mu })+\frac \delta {2M^2}(\partial _\mu \square \Lambda )(\partial ^\mu \Box \Lambda ) \notag \\ &&+\mathcal{L}_{int}^{^{\prime }}(V_{\perp }-\partial \Lambda ,J\ldots ).\end{aligned}$$ In order to lower the number of derivatives in the kinetic terms we integrate in auxiliary scalar fields $\chi $, $\rho $, $\pi $, $\sigma $ and auxiliary transverse vector field $B_{\perp \mu }$ writing e.g. $$\exp \left( -i\int \mathrm{d}^4x\frac \beta 2(\Box \Lambda )^2\right) =\int \mathcal{D}\chi \exp \left( i\int \mathrm{d}^4x\left( \frac 1{2\beta }\chi ^2-\partial _\mu \chi \partial ^\mu \Lambda \right) \right)$$ and similarly for other higher derivative terms. After the superfluous degrees of freedom are identified and integrated out, the fields are re-scaled and then the resulting mass matrix can be diagonalized by means of two symplectic rotations with angles $\theta _V$ and $\theta _S$ (the technical details are postponed to the Appendix \[PI\_Proca\]). Finally we get (under the conditions $(1+\alpha )^2>4\gamma $ and $\beta ^2>4\delta $) $$Z[J]=\int \mathcal{D}V_{\perp }\mathcal{D}B_{\perp }\mathcal{D}\Lambda \mathcal{D}\chi \mathcal{D}\sigma \exp \left( i\int \mathrm{d}^4x\mathcal{L}% (V_{\perp },B_{\perp },\Lambda ,\chi ,\sigma ,J,\ldots )\right)$$ where $$\begin{aligned} \mathcal{L}(V_{\perp },B_{\perp },\Lambda ,\chi ,\sigma ,J,\ldots ) &=&\frac 12V_{\perp }^\mu \square V_{\perp \mu }+\frac 12M_{V+}^2V_{\perp }^\mu V_{\perp \mu }-\frac 12B_{\perp }^\mu \square B_{\perp }^\mu +\frac 12M_{V-}^2B_{\perp }^\mu B_{\perp \mu } \notag \\ &&+\frac 12\partial _\mu \sigma \partial ^\mu \sigma -\frac 12M_{S+}^2\sigma ^2-\frac 12\partial _\mu \chi \partial ^\mu \chi -\frac 12M_{S-}^2\chi ^2+\frac 12M^2\partial _\mu \Lambda \partial ^\mu \Lambda \notag \\ &&+\mathcal{L}_{int}^{^{\prime }}(\overline{V}^{(\theta )},J,\ldots ). \notag \\ &&\end{aligned}$$ and $$\overline{V}^{(\theta )}=\frac{\exp \theta _V}{(1+\alpha )^{1/2}}(V_{\perp }+B_{\perp })-\partial \chi \cosh \theta _S-\partial \sigma \sinh \theta _S-\partial \Lambda$$ and where $M_{V\pm }^2$, $M_{S\pm }^2$ are the mass eigenvalues (\[MV\]) and (\[MS\]). The theory is now formulated in terms of two spin one and two spin zero fields, whereas two of them, namely $B_{\perp }^\mu $ and $% \chi $ have a wrong sign of the kinetic terms and are therefore negative norm ghosts. As above, the field $\Lambda $ does not correspond to any dynamical degree of freedom, its role is merely to cancel the spurious poles of the free propagators of the transverse fields $V_{\perp }$ and $B_{\perp } $ at $p^2=0$. Antisymmetric tensor formalism ------------------------------ For the antisymmetric tensor field in the formalism [@Ecker1; @Ecker2] the situation is quite analogous to the Proca field case so our discussion will be parallel to the previous subsection. Let us write the Lagrangian in the form $$\mathcal{L}=\mathcal{L}_0+\mathcal{L}_{int}.$$ where the free part is $$\mathcal{L}_0=-\frac 12(\partial _\mu R^{\mu \nu })(\partial ^\rho R_{\rho \nu })+\frac 14M^2R_{\mu \nu }R^{\mu \nu },$$ and introduce the transverse and longitudinal projectors $$\begin{aligned} \Pi _{\mu \nu \alpha \beta }^T &=&\frac 12\left( P_{\mu \alpha }^TP_{\nu \beta }^T-P_{\nu \alpha }^TP_{\mu \beta }^T\right) \\ \Pi _{\mu \nu \alpha \beta }^L &=&\frac 12\left( g_{\mu \alpha }g_{\nu \beta }-g_{\nu \alpha }g_{\mu \beta }\right) -\Pi _{\mu \nu \alpha \beta }^T\end{aligned}$$ with $P_{\mu \alpha }^T$ given by (\[P\_T\]). Again, in analogy with ([vector\_gamma\_2]{}), for completely general $\mathcal{L}_{int}$ we can expect the following general form of the full two-point 1PI Green function $$\Gamma _{\mu \nu \alpha \beta }^{(2)}(p)=\frac 12(M^2+\Sigma ^T(p^2))\Pi _{\mu \nu \alpha \beta }^T+\frac 12(M^2-p^2+\Sigma ^L(p^2))\Pi _{\mu \nu \alpha \beta }^L \label{tensor_gamma_2}$$ where $\Sigma ^{T,L}$ are the corresponding self-energies. The full propagator is then obtained by means of the inversion of $\Gamma _{\mu \nu \alpha \beta }^{(2)}$ in the form $$\Delta _{\mu \nu \alpha \beta }(p)=-\frac 2{p^2-M^2-\Sigma ^L(p^2)}\Pi _{\mu \nu \alpha \beta }^L+\frac 2{M^2+\Sigma ^T(p^2)}\Pi _{\mu \nu \alpha \beta }^T. \label{tensor_Delta_2}$$ This propagator has two types of poles analogous to (\[V\_pole\_spin\_1\]) and (\[V\_pole\_spin\_0\]), either at $p^2=s_V,$ satisfying $$s_V-M^2-\Sigma ^L(s_V)=0, \label{T_pole_parity+}$$ or at $p^2=s_{\widetilde{V}}$ where $$M^2+\Sigma ^T(s_{\widetilde{V}})=0. \label{T_pole_parity-}$$ Assuming that the solution of (\[T\_pole\_parity+\]) satisfies $s_V=$ $% M_V^2>0,$ the propagator behaves at this pole as $$\begin{aligned} \Delta _{\mu \nu \alpha \beta }(p) &=&\frac{Z_V}{p^2-M_V^2}\frac{p_\mu g_{\nu \alpha }p_\beta -p_\nu g_{\mu \alpha }p_\beta -(\alpha \leftrightarrow \beta )}{M_V^2}+O(1) \notag \\ &=&\frac{Z_V}{p^2-M_V^2}\sum_\lambda u_{\mu \nu }^{(\lambda )}(p)u_{\alpha \beta }^{(\lambda )}(p)^{}+O(1)\end{aligned}$$ where $$Z_V=\frac 1{1-\Sigma ^{^{\prime }L}(M_V^2)}$$ and the wave function $u_{\mu \nu }^{(\lambda )}(p)$ can be expressed in terms of the spin-one polarization vectors $\varepsilon _\nu ^{(\lambda )}(p) $ as $$u_{\mu \nu }^{(\lambda )}(p)=\frac{\mathrm{i}}{M_V}\left( p_\mu \varepsilon _\nu ^{(\lambda )}(p)-p_\nu \varepsilon _\mu ^{(\lambda )}(p)\right) . \label{wave_function}$$ For $Z_V>0$ the pole of this type corresponds therefore to the spin-one state $\|p,\lambda ,V\rangle $ which couples to $R_{\mu \nu }$ as $$\langle 0\|R_{\mu \nu }(0)\|p,\lambda ,V\rangle =Z_V{}^{1/2}u_{\mu \nu }^{(\lambda )}(p). \label{u_function}$$ Analogously to the Proca case, at least one of these poles is expected to be perturbative and corresponds to the original degree of freedom described by the free Lagrangian $\mathcal{L}_0$. This means $$\begin{aligned} M_V^2 &=&M^2+\delta M_V^2 \\ Z_V &=&1+\delta Z_V\end{aligned}$$ with small corrections $\delta M_V^2$ and $\delta Z_V$ vanishing in the free field limit. The other possible nonperturbative solutions of ([T\_pole\_parity+]{}) decouple in this limit. Provided there exists a solution of (\[T\_pole\_parity-\]) for which $s_{% \widetilde{V}}=M_{\widetilde{V}}^2>0$, we get at this pole $$\begin{aligned} \Delta _{\mu \nu \alpha \beta }(p) &=&\frac{Z_{\widetilde{V}}}{p^2-M_{% \widetilde{V}}^2}\left( g_{\mu \alpha }g_{\nu \beta }+\frac{p_\mu g_{\nu \alpha }p_\beta -p_\mu g_{\nu \beta }p_\alpha }{M_A^2}-(\mu \leftrightarrow \nu )\right) +O(1) \notag \\ &=&\frac{Z_{\widetilde{V}}}{p^2-M_{\widetilde{V}}^2}\sum_\lambda w_{\mu \nu }^{(\lambda )}(p)w_{\alpha \beta }^{(\lambda )}(p)^{}+O(1)\end{aligned}$$ where $$Z_{\widetilde{V}}=\frac 1{\Sigma ^{^{\prime }T}(M_{\widetilde{V}}^2)}$$ and the wave function is dual to the wave function (\[wave\_function\]) $$w_{\mu \nu }^{(\lambda )}(p)=\widetilde{u}_{\mu \nu }^{(\lambda )}(p)=\frac 12\varepsilon _{\mu \nu \alpha \beta }u^{(\lambda )\alpha \beta }(p).$$ Provided $Z_{\widetilde{V}}>0$, the poles of this type correspond to the spin-one particle states $\|p,\lambda ,\widetilde{V}\rangle $ with the opposite intrinsic parity in comparison with $\|p,\lambda ,V\rangle $, which couple to the antisymmetric tensor field as $$\langle 0\|R_{\mu \nu }(0)\|p,\lambda ,\widetilde{V}\rangle =Z_{\widetilde{V}% }{}^{1/2}w_{\mu \nu }^{(\lambda )}(p). \label{w_function}$$ This degree of freedom is frozen in the free propagator due to the specific form of the free Lagrangian and it decouples in the limit of the vanishing interaction. As in the Proca field case, we can therefore generally expect dynamically generated additional degrees of freedom, which can be either regular asymptotic states ($M_{V,\widetilde{V}}^{2},\,Z_{V,\widetilde{V}}>0$) or negative norm ghosts ($M_{V,\widetilde{V}}^{2}>0,\,Z_{V,\widetilde{V}}<0$) or tachyons ($M_{V,\widetilde{V}}^{2}<0$). Complex poles on the unphysical sheets can be then interpreted as resonances. As the toy illustration of these possibilities, let us take the interaction Lagrangian similar to (\[L\_V\_toy\]) in the Proca field case e.g. in the form $$\begin{aligned} \mathcal{L}_{int} &\equiv &\mathcal{L}_{ct}=-\frac{\alpha -\beta }2(\partial _\mu R^{\mu \nu })(\partial ^\rho R_{\rho \nu })-\frac \beta 4(\partial _\mu R^{\alpha \beta })(\partial ^\mu R_{\alpha \beta }) \notag \\ &&+\frac{\gamma -\delta }{2M^2}(\partial _\alpha \partial _\mu R^{\mu \nu })(\partial ^\alpha \partial ^\rho R_{\rho \nu })+\frac \delta {4M^2}(\partial _\rho \partial _\mu R^{\alpha \beta })(\partial ^\rho \partial ^\mu R_{\alpha \beta }). \label{L_R_toy}\end{aligned}$$ We get then the following contributions to the longitudinal and transverse self-energies $$\begin{aligned} \Sigma ^L(p^2) &=&-\alpha p^2+\gamma \frac{p^4}{M^2} \\ \Sigma ^T(p^2) &=&-\beta p^2+\delta \frac{p^4}{M^2}.\end{aligned}$$ These are exactly the same as (\[Sigma\_L\_vector\]) and ([Sigma\_T\_vector]{}) (with the identification $\Sigma ^{T,L}\leftrightarrow \Sigma ^{L,T}$). Therefore, provided we further identify $M_{S\pm }^2\leftrightarrow M_{\widetilde{V}\pm }^2$, the properties of the poles and residues are the same as in the previous subsection (see the discussion after (\[Sigma\_L\_vector\]) and (\[Sigma\_T\_vector\])), with the only exception that instead of the extra spin-zero states with the mass (\[MS\]) we have now extra spin-one states with the same mass (\[MS\]) but with the opposite parity in comparison with the original degrees of freedom described by the free lagrangian $\mathcal{L}_0$. ### Path integral formulation\[path integral tensor\] We can again made the additional degrees of freedom manifest within the path integral approach in the way parallel to subsection \[path integral vector\]. An analog of the $U(1)$ gauge symmetry used in the case of the Proca field formalism in order to separate the transverse and longitudinal components of the field $V_\mu $ is here the following transformation with a pseudovector[^1] parameter $\Lambda _\alpha $ $$R^{\mu \nu }\rightarrow R^{\mu \nu }+\frac 12\varepsilon ^{\mu \nu \alpha \beta }\widehat{\Lambda }_{\alpha \beta }, \label{R_gauge}$$ where $$\widehat{\Lambda }_{\alpha \beta }=\partial _\alpha \Lambda _\beta -\partial _\beta \Lambda _\alpha .$$ This leaves the kinetic term invariant, while the mass term is changed. Note, that the transformation with the parameters $\Lambda _\alpha $ and $% \Lambda _\alpha ^\lambda $ where $$\Lambda _\alpha ^\lambda =\Lambda _\alpha +\partial _\alpha \lambda \label{Lambda_gauge}$$ are the same. This residual gauge invariance has to be taken into account when using the Faddeev-Popov trick in order to isolate the longitudinal and transverse degrees of freedom of the field $R_{\mu \nu }$. Analog of the formula (\[V\_components\]) is now $$R^{\mu \nu }=R_{\parallel }^{\mu \nu }+\frac 12\varepsilon ^{\mu \nu \alpha \beta }\widehat{\Lambda }_{\alpha \beta }$$ where $R_{\parallel }^{\mu \nu }$ is the longitudinal component of $R_{\mu \nu }$. Its transverse component is described with the transverse component $% \Lambda _{\perp }^\mu $ of the field $\Lambda ^\mu $ where $$\Lambda ^\mu =\Lambda _{\perp }^\mu +\partial ^\mu \lambda .$$ Starting with the path integral representation of the generating functional[^2] $$Z[J]=\int \mathcal{D}R\exp \left( \mathrm{i}\int \mathrm{d}^4x\left( -\frac 12(\partial _\mu R^{\mu \nu })(\partial ^\rho R_{\rho \nu })+\frac 14M^2R_{\mu \nu }R^{\mu \nu }+\mathcal{L}_{int}(R^{\mu \nu },J,\ldots )\right) \right)$$ and using the Faddeev-Popov trick twice with respect to the transformations (\[R\_gauge\]) and (\[Lambda\_gauge\]) we finally find for $Z[J]$ the following representation $$Z[J]=\int \mathcal{D}R_{\parallel }\mathcal{D}\Lambda _{\perp }\exp \left( \mathrm{i}\int \mathrm{d}^4x\mathcal{L}(R_{\parallel }^{\mu \nu },\Lambda _{\perp }^\mu ,\ldots )\right)$$ where the integral measure is $$\mathcal{D}R_{\parallel }\mathcal{D}\Lambda _{\perp }=\mathcal{D}R\mathcal{D}% \Lambda \delta (\partial _\alpha R_{\mu \nu }+\partial _\nu R_{\alpha \mu }+\partial _\mu R_{\nu \alpha })\delta (\partial _\mu \Lambda ^\mu )$$ and $$\begin{aligned} R_{\parallel }^{\mu \nu } &=&-\frac 1{2\square }(\partial ^\mu g^{\nu \alpha }\partial ^\beta +\partial ^\nu g^{\mu \beta }\partial ^\alpha -(\mu \leftrightarrow \nu ))R_{\alpha \beta } \\ \Lambda _{\perp }^\mu &=&\left( g^{\mu \nu }-\frac{\partial ^\mu \partial ^\nu }{\square }\right) \Lambda _\nu .\end{aligned}$$ are the longitudinal part of the tensor field $R^{\mu \nu }$and the transverse part of the vector field $\Lambda ^\mu $ (describing the transverse part of the tensor field $R^{\mu \nu }$) respectively[^3]. The Lagrangian expressed in these variables reads $$\mathcal{L}(R_{\parallel }^{\mu \nu },\Lambda _{\perp }^\mu ,J,\ldots )=\frac 14R_{\parallel }^{\mu \nu }\square R_{\parallel \,\mu \nu }+\frac 14M^2R_{\parallel }^{\mu \nu }R_{\parallel \,\mu \nu }+\frac 12M^2\Lambda _{\perp }^\mu \square \Lambda _{\perp \mu }+\mathcal{L}_{int}(R_{\parallel }^{\mu \nu }-\frac 12\varepsilon ^{\mu \nu \alpha \beta }\widehat{\Lambda }% _{\alpha \beta },J,\ldots ).$$ The free propagators of the fields $R_{\parallel }^{\mu \nu }$ and $\Lambda _{\perp }^\mu $ are therefore $$\begin{aligned} \Delta _{\parallel }^{\mu \nu \alpha \beta }(p) &=&-\frac 2{p^2-M^2}\Pi ^{L\,\mu \nu \alpha \beta } \\ \Delta _{\perp }^{\mu \nu }(p) &=&-\frac 1{M^2}\frac 1{p^2}P^{T\,\mu \nu }\end{aligned}$$ and, similarly to the case of the Proca field, they have spurious poles at $% p^2=0$. Due to the form of the interaction, however, only the combination $$\begin{aligned} \Delta _0^{\mu \nu \alpha \beta }(p) &=&\Delta _{\parallel }^{\mu \nu \alpha \beta }(p)+\varepsilon ^{\mu \nu \rho \sigma }\varepsilon ^{\alpha \beta \kappa \lambda }p_\rho p_\kappa \Delta _{\perp \,\sigma \lambda }(p) \notag \\ &=&-\frac 2{p^2-M^2}\Pi ^{L\,\mu \nu \alpha \beta }+\frac 2{M^2}\Pi ^{T\,\mu \nu \alpha \beta }\end{aligned}$$ corresponding to the free propagator of the original tensor field $R^{\mu \nu }$ is relevant within the Feynman graphs and the spurious poles cancel. By analogy with the Proca field case, for the interaction Lagrangian invariant with respect to the transformation (\[R\_gauge\]) the field $% \Lambda _{\perp }^\mu $ completely decouples and can be integrated out. Such a form of the interaction also allows to modify the propagator $\Delta _{\parallel }^{\mu \nu \alpha \beta }(p)$ within the Feynman graphs $$\Delta _{\parallel }^{\mu \nu \alpha \beta }(p)\rightarrow -\frac{g_{\mu \alpha }g_{\nu \beta }-g_{\mu \beta }g_{\nu \alpha }}{p^2-M^2}$$ and no spurious pole at $p^2=0\,$effectively appears. In this case the opposite parity spin-one states discussed in the previous subsection cannot be dynamically generated. In order to illustrate the appearance of the additional degrees of freedom connected with the interaction Lagrangian (\[L\_R\_toy\]) within the path integral formalism, we can make the same exercise with the interaction Lagrangian (\[L\_R\_toy\]) as we did in the previous subsection with ([L\_V\_toy]{}). Our aim is again to make the additional degrees of freedom explicit in the path integral representation of $Z[J]$. The procedure is almost one-to-one to the case of the Proca fields so that we will be more concise. The technical details can be found in the Appendix [PI\_antisymmetric]{}. We assume the interaction Lagrangian to be of the form $$\mathcal{L}_{int}=\mathcal{L}_{ct}+\mathcal{L}_{int}^{^{\prime }},$$ where $\mathcal{L}_{ct}$ is given by (\[L\_R\_toy\]) and we assume $\alpha >-1\,$and $\delta >0$ as above. $\mathcal{L}_{int}$ can be then re-express it in terms of the longitudinal and transverse components of the original field $R_{\mu \nu }$ $$\mathcal{L}_{int}(R_{\parallel }^{\mu \nu }-\frac 12\varepsilon ^{\mu \nu \alpha \beta }\widehat{\Lambda }_{\alpha \beta },J,\ldots )=\mathcal{L}% _{ct}(R_{\parallel }^{\mu \nu }-\frac 12\varepsilon ^{\mu \nu \alpha \beta }% \widehat{\Lambda }_{\alpha \beta },J,\ldots )+\mathcal{L}_{int}^{^{\prime }}(R_{\parallel }^{\mu \nu }-\frac 12\varepsilon ^{\mu \nu \alpha \beta }% \widehat{\Lambda }_{\alpha \beta },J,\ldots )$$ where $$\begin{aligned} \mathcal{L}_{ct}(R^{\mu \nu }-\frac 12\varepsilon ^{\mu \nu \alpha \beta }% \widehat{\Lambda }_{\alpha \beta }J,\ldots ) &=&\frac \alpha 4R_{\parallel }^{\mu \nu }\square R_{\parallel \,\mu \nu }+\frac \gamma {4M^2}(\square R_{\parallel }^{\mu \nu })(\square R_{\parallel \,\mu \nu }) \notag \\ &&+\frac \beta 2(\square \Lambda _{\perp }^\mu )(\square \Lambda _{\perp \mu })-\frac \delta {2M^2}(\partial ^\alpha \square \Lambda _{\perp }^\mu )(\partial _\alpha \square \Lambda _{\perp \mu }).\end{aligned}$$ We then introduce the auxiliary (longitudinal) antisymmetric tensor field $% B_{\parallel }^{\mu \nu }$ and (transverse) vector fields $\chi _{\perp }^\mu $, $\rho _{\perp }^\mu $, $\sigma _{\perp }^\mu $ and $\pi _{\perp }^\mu $ in order to avoid the higher derivative terms in a complete analogy with the Proca field case. Again, not all the fields correspond to propagating degrees of freedom and such redundant fields can be integrated out. After rescaling the fields and diagonalization of the resulting mass terms by means of two symplectic rotations with angles $\theta _V$ and $% \theta _{\widetilde{V}}$ exactly as in the case of the Proca fields (see the Appendix \[PI\_antisymmetric\] for details) we end up with $$Z[J]=\int \mathcal{D}R_{\parallel }\mathcal{D}B_{\parallel }\mathcal{D}% \Lambda _{\perp }\mathcal{D}\chi _{\perp }\mathcal{D}\rho _{\perp }\mathcal{D% }\sigma _{\perp }\mathcal{D}\pi _{\perp }\exp \left( \mathrm{i}\int \mathrm{d% }^4x\mathcal{L}(R_{\parallel },B_{\parallel },\Lambda _{\perp },\chi _{\perp },\rho _{\perp },\sigma _{\perp },\pi _{\perp },J,\ldots )\right)$$ with (cf. (\[integrated\_L\_V\])) $$\begin{aligned} \mathcal{L} &=&\frac 14R_{\parallel }^{\mu \nu }\square R_{\parallel \,\mu \nu }+\frac 14M_{V+}^2R_{\parallel }^{\mu \nu }R_{\parallel \,\mu \nu } \notag \\ &&-\frac 14B_{\parallel }^{\mu \nu }\square B_{\parallel \,\mu \nu }+\frac 14M_{V-}^2B_{\parallel }^{\mu \nu }B_{\parallel \,\mu \nu } \notag \\ &&+\frac 12M^2\Lambda _{\perp }^\mu \square \Lambda _{\perp \mu } \notag \\ &&-\frac 12\chi _{\perp }^\mu \square \chi _{\perp \mu }+\frac 12M_{% \widetilde{V}-}^2\chi _{\perp }^\mu \chi _{\perp \mu }+\frac 12\sigma _{\perp }^\mu \square \sigma _{\perp \mu }+\frac 12M_{\widetilde{V}% +}^2\sigma _{\perp }^\mu \sigma _{\perp \mu } \notag \\ &&+\mathcal{L}_{int}(\overline{R}^{(\theta )},J,\ldots )\end{aligned}$$ where $$\overline{R}^{(\theta )\mu \nu }=\frac{\exp \theta _V}{(1+\alpha )^{1/2}}% (R_{\parallel }^{\mu \nu }+B_{\parallel }^{\mu \nu })-\frac 12\varepsilon ^{\mu \nu \alpha \beta }\left( \widehat{\Lambda }_{\alpha \beta }+\widehat{% \sigma }_{\perp \alpha \beta }\sinh \theta _{\widetilde{V}}+\widehat{\chi }% _{\perp \alpha \beta }\cosh \theta _{\widetilde{V}}\right)$$ and with the diagonal mass terms corresponding to the eigenvalues (\[MV\], \[MS\]) (with identification $M_{\widetilde{V}\pm }^2\rightarrow M_{S\pm }^2$). Again we have two pairs of fields with the opposite signs of the kinetic terms, namely $(R_{\parallel }^{\mu \nu },B_{\parallel }^{\mu \nu })$ and $(\chi _{\perp }^\mu ,\sigma _{\perp }^\mu )$ respectively. As a result we have found four spin-one states, two of them being negative norm ghosts, namely $B_{\parallel }^{\mu \nu }$ and $\sigma _{\perp }^\mu $ and two of them with the opposite parity, namely $\chi _{\perp }^\mu $ and $\sigma _{\perp }^\mu $. As in the Proca field case, the field $\Lambda _{\perp }^\mu $ effectively compensates the spurious $p^2=0$ poles in the $% R_{\parallel }^{\mu \nu }$ and $B_{\parallel }^{\mu \nu }$ propagators within Feynman graphs. First order formalism --------------------- The first order formalism is a natural alternative to the previous two (for the motivation and details of the quantization see [@FO1], cf. also [Bruns:2004tj]{}). It introduces both vector and antisymmetric tensor fields into the Lagrangian, therefore the analysis is a little bit more complex in comparison with previous two cases. In this case, the Lagrangian is of the form $$\mathcal{L}=\mathcal{L}_0+\mathcal{L}_{int}$$ where now the free part is $$\mathcal{L}_0=MV_\nu \partial _\mu R^{\mu \nu }+\frac 12M^2V_\mu V^\mu +\frac 14M^2R_{\mu \nu }R^{\mu \nu }.$$ Instead of just one one-particle irreducible two point Green function we have a matrix $$\Gamma ^{(2)}(p)=\left( \begin{array}{ll} \Gamma _{VV}^{(2)}(p)_{\mu \nu } & \Gamma _{VR}^{(2)}(p)_{\alpha \mu \nu } \\ \Gamma _{RV}^{(2)}(p)_{\mu \nu \alpha } & \Gamma _{RR}^{(2)}(p)_{\mu \nu \alpha \beta }% \end{array} \right) \label{matrix_gamma}$$ where (without any additional assumptions on the form of $\mathcal{L}_{int}$) the matrix elements have the following general form (cf. ([vector\_gamma\_2]{}) and (\[tensor\_gamma\_2\])) $$\begin{aligned} \Gamma _{RR}^{(2)}(p)_{\mu \nu \alpha \beta } &=&\frac 12(M^2+\Sigma _{RR}^T(p^2))\Pi _{\mu \nu \alpha \beta }^T+\frac 12(M^2+\Sigma _{RR}^L(p^2))\Pi _{\mu \nu \alpha \beta }^L \\ \Gamma _{VV}^{(2)}(p)_{\mu \nu } &=&(M^2+\Sigma _{VV}^T(p^2))P_{\mu \nu }^T+(M^2+\Sigma _{VV}^L(p^2))P_{\mu \nu }^L \\ \Gamma _{RV}^{(2)}(p)_{\mu \nu \alpha } &=&\frac{\mathrm{i}}2\left( M+\Sigma _{RV}(p^2)\right) \Lambda _{\mu \nu \alpha } \\ \Gamma _{VR}^{(2)}(p)_{\alpha \mu \nu } &=&\frac{\mathrm{i}}2\left( M+\Sigma _{VR}(p^2)\right) \Lambda _{\alpha \mu \nu }^t.\end{aligned}$$ Here $\Sigma _{RR}^{T,L}(p^2)$, $\Sigma _{VV}^{T,L}(p^2)$ and $\Sigma _{RV}(p^2)=\Sigma _{VR}(p^2)$ are corresponding self-energies and the off-diagonal tensor structures are $$\Lambda _{\mu \nu \alpha }=-\Lambda _{\alpha \mu \nu }^t=p_\mu g_{\nu \alpha }-p_\nu g_{\mu \alpha }.$$ This matrix of propagators $$\Delta (p)=\left( \begin{array}{ll} \Delta _{VV}(p)_{\mu \nu } & \Delta _{VR}(p)_{\alpha \mu \nu } \\ \Delta _{RV}(p)_{\mu \nu \alpha } & \Delta _{RR}(p)_{\mu \nu \alpha \beta }% \end{array} \right)$$ can be obtained by means of the inversion of the matrix (\[matrix\_gamma\]) with the result $$\begin{aligned} \Delta _{RR}(p)_{\mu \nu \alpha \beta } &=&\frac 2{M^2+\Sigma _{RR}^T(p^2)}\Pi _{\mu \nu \alpha \beta }^T+2\frac{M^2+\Sigma _{VV}^T(p^2)}{% D(p^2)}\Pi _{\mu \nu \alpha \beta }^L \\ \Delta _{VV}(p)_{\mu \nu } &=&\frac 1{M^2+\Sigma _{VV}^L(p^2)}P_{\mu \nu }^L+% \frac{M^2+\Sigma _{RR}^L(p^2)}{D(p^2)}P_{\mu \nu }^T \\ \Delta _{RV}(p)_{\mu \nu \alpha } &=&-\mathrm{i}\frac{M+\Sigma _{RV}(p^2)}{% D(p^2)}\Lambda _{\mu \nu \alpha } \\ \Delta _{VR}(p)_{\alpha \mu \nu } &=&-\mathrm{i}\frac{M+\Sigma _{VR}(p^2)}{% D(p^2)}\Lambda _{\alpha \mu \nu }^t,\end{aligned}$$ where $$D(p^2)=(M^2+\Sigma _{RR}^L(p^2))(M^2+\Sigma _{VV}^T(p^2))-p^2(M+\Sigma _{RV}(p^2))(M+\Sigma _{VR}(p^2)).$$ Let us now discuss the structure of the poles, which is now richer than in previous two cases. We have three possible types of poles, namely $s_V$, $s_{% \widetilde{V}}$ and $s_S$, being solutions of $$\begin{aligned} D(s_V) &=&0 \notag \\ M^2+\Sigma _{RR}^T(s_{\widetilde{V}}) &=&0 \notag \\ M^2+\Sigma _{VV}^L(s_S) &=&0 \label{RV_zeros}\end{aligned}$$ respectively. As far as the pole $s_V$ is concerned, let us assume $% s_V=M_V^2>0$. We get then at this pole (see also previous two subsections) $$\begin{aligned} \Delta _{RR}(p)_{\mu \nu \alpha \beta } &=&\frac{Z_{RR}}{p^2-M_V^2}% \sum_\lambda u_{\mu \nu }^{(\lambda )}(p)u_{\alpha \beta }^{(\lambda )}(p)^{}+O(1) \\ \Delta _{VV}(p)_{\mu \nu } &=&\frac{Z_{VV}}{p^2-M_V^2}\sum_\lambda \varepsilon _\mu ^{(\lambda )}(p)\varepsilon _\nu ^{(\lambda )}(p)+O(1) \\ \Delta _{RV}(p)_{\mu \nu \alpha } &=&\frac{Z_{RV}}{p^2-M_V^2}\sum_\lambda u_{\mu \nu }^{(\lambda )}(p)\varepsilon _\alpha ^{(\lambda )}(p)^{}+O(1) \\ \Delta _{VR}(p)_{\alpha \mu \nu } &=&\frac{Z_{VR}}{p^2-M_V^2}\sum_\lambda \varepsilon _\alpha ^{(\lambda )}(p)u_{\mu \nu }^{(\lambda )}(p)+O(1)\end{aligned}$$ where $u_{\mu \nu }^{(\lambda )}(p)$ is given by (\[wave\_function\]) and the residue are $$\begin{aligned} Z_{RR} &=&\frac{M^2+\Sigma _{VV}^T(M_V^2)}{D^{^{\prime }}(M_V^2)} \\ Z_{VV} &=&\frac{M^2+\Sigma _{RR}^L(M_V^2)}{D^{^{\prime }}(M_V^2)} \\ Z_{RV} &=&\frac{M+\Sigma _{RV}(M_V^2)}{D^{^{\prime }}(M_V^2)}M_V=Z_{VR}=% \frac{M+\Sigma _{VR}(M_V^2)}{D^{^{\prime }}(M_V^2)}M_V.\end{aligned}$$ Note that, as a consequence of (\[RV\_zeros\]) we get the following relation $$Z_{RR}Z_{VV}=Z_{RV}^2=Z_{VR}^2,$$ (remember $\Sigma _{RV}(p^2)=\Sigma _{VR}(p^2)$), therefore assuming $% Z_{RR},Z_{VV}>0$ the pole $p^2=M_V^2>0$ corresponds to the spin-one one-particle state $\|p,\lambda ,V\rangle $ which couples to the fields as $$\begin{aligned} \langle 0\|R_{\mu \nu }(0)\|p,\lambda ,V\rangle &=&Z_{RR}{}^{1/2}u_{\mu \nu }^{(\lambda )}(p) \\ \langle 0\|V_\mu (0)\|p,\lambda ,V\rangle &=&Z_{VV}{}^{1/2}\varepsilon _\mu ^{(\lambda )}(p).\end{aligned}$$ Again at least one of such states is expected to be perturbative as above and it correspond to the original degree of freedom described by $\mathcal{L}% _0$; the others decouple when the interactions is switched off. The other possible poles, $s_S=M_S^2$ and $s_{\widetilde{V}}=M_{\widetilde{V}}^2$ are analogical to the spin-zero and spin-one (opposite parity) states discussed in detail in the previous two subsections; they correspond to the modes which are frozen at the leading order and decouple in the free field limit. As we have already discussed, without further restriction on the form of the interaction, all the additional states can be also negative norm ghosts or tachyons. Let us illustrate the general case using a toy interaction Lagrangian of the form $$\begin{aligned} \mathcal{L}_{ct} &=&-\frac{\alpha _{V}}{4}\widehat{V}_{\mu \nu }\widehat{V}% ^{\mu \nu }-\frac{\beta _{V}}{2}(\partial _{\mu }V^{\mu })^{2} \notag \\ &&-\frac{\alpha _{R}-\beta _{R}}{2}(\partial _{\mu }R^{\mu \nu })(\partial ^{\rho }R_{\rho \nu })-\frac{\beta _{R}}{4}(\partial _{\mu }R^{\alpha \beta })(\partial ^{\mu }R_{\alpha \beta }). \label{L_RV_toy}\end{aligned}$$This gives $$\begin{aligned} \Sigma _{RR}^{L}(p^{2}) &=&-\alpha _{R}p^{2} \notag \\ \Sigma _{RR}^{T}(p^{2}) &=&-\beta _{R}p^{2} \notag \\ \Sigma _{VV}^{T}(p^{2}) &=&-\alpha _{V}p^{2} \notag \\ \Sigma _{VV}^{L}(p^{2}) &=&-\beta _{V}p^{2} \notag \\ \Sigma _{RV}(p^{2}) &=&\Sigma _{VR}(p^{2})=0\end{aligned}$$and for $\beta _{V,R}>0$ the spectrum of one-particle states consists of one spin-zero ghost, one spin-one ghost with opposite parity. Their masses and residue are $$\begin{aligned} M_{S}^{2} &=&\frac{M^{2}}{\beta _{V}},\,\,\,\,\,Z_{S}=-\frac{1}{\beta _{V}} \\ M_{\widetilde{V}}^{2} &=&\frac{M^{2}}{\beta _{R}},\,\,\,\,Z_{\widetilde{V}}=-% \frac{1}{\beta _{R}}\end{aligned}$$(provided $\beta _{R}<0$ or $\beta _{V}<0$ the corresponding states are tachyons) and two spin-one states with masses $$\begin{aligned} M_{V\pm }^{2} &=&M^{2}\frac{1+\alpha _{R}+\alpha _{V}\pm \sqrt{\mathcal{D}}}{% 2\alpha _{R}\alpha _{V}} \notag \\ \mathcal{D} &=&(1+\alpha _{R}+\alpha _{V})^{2}-4\alpha _{R}\alpha _{V}.\end{aligned}$$To get both $M_{V\pm }^{2}>0$ we need $\mathcal{D}>0$, $\alpha _{V}\alpha _{R}>0$ and $1+\alpha _{R}+\alpha _{V}>0$; in this case we get for the residue $Z_{RR}^{(\pm )}$ and $Z_{VV}^{(\pm )}$ at poles $M_{V\pm }^{2}$ $$\alpha _{R}Z_{RR}^{(+)}Z_{RR}^{(-)}=\alpha _{V}Z_{VV}^{(+)}Z_{VV}^{(-)}=% \frac{1}{\mathcal{D}}>0$$Assuming $Z_{RR}^{(-)},\,Z_{VV}^{(-)}>0$ (note that, for small couplings $% M_{V-}^{2}=M^{2}(1+O(\alpha _{R},\alpha _{V}))$ with $% Z_{RR}^{(-)},Z_{VV}^{(-)}=1+O(\alpha _{R},\alpha _{V})$ corresponds to the perturbative solution), the additional spin one-state is either positive norm state for $\alpha _{V,R}>0$ or ghost for $\alpha _{V,R}<0$ (in this latter case the extra kinetic terms in $\mathcal{L}_{ct}$ have wrong signs). Also in this case the propagating degrees of freedom can be made manifest within the path integral formalism. The corresponding discussion is in a sense synthesis of subsections \[path integral vector\] and \[path integral tensor\] and is postponed to Appendix \[PI\_first\_order\]. Organization of the counterterms\[Section\_power\_counting\] ============================================================ Let us now return to the concrete case of $R\chi T$. Our aim is to calculate the one loop self-energies defined in the previous section in all three formalisms discussed there. In the process of the loop calculation we are lead to the problem of performing a classification of the countertems, which have to be introduced in order to renormalize infinities. For this purpose, it is convenient to have a scheme, which allows us to assign to each operator in the Lagrangian and to each Feynman graph an appropriate expansion index. Indices of the counterterms, which are necessary in order to cancel the divergences of the given Feynman graph, should be then correlated with the indices of the vertices of the graph as well as with the number of the loops. When we restrict ourselves to the (one-particle irreducible) graphs with a given index, the number of the allowed operators contributing to the graph as well as that of necessary counterterms should be finite. There are several possibilities how to do it, some of them being quite efficient but purely formal and unphysical, some of them having good physical meaning, but not very useful in practise. In the literature, several attempts to organize the individual terms of the $% R\chi T$ Lagrangian can be found. Let us briefly comment on some of them from the point of view of its applicability to our purpose. The first one is intimately connected with the effective chiral Lagrangian $% \mathcal{L}_{\chi ,\mathrm{res}}$ which appears as a result of the (tree-level) integrating out of the resonances from the $R\chi T$. Such a counting assigns to each operator of the resonance part of the $R\chi T$ Lagrangian $\mathcal{L}_{\mathrm{res}}$ a chiral order according to the minimal chiral order of the coupling (LEC) of the effective chiral Lagrangian $\mathcal{L}_{\chi ,\mathrm{res}}$ to which the corresponding operator contributes [@Kampf:2006bn], [@Cirigliano:2006hb]. More generally, in this scheme the chiral order of the operators from $\mathcal{L}% _{\mathrm{res}}$ refers to the minimal chiral order of its contribution to the generating functional of the currents $Z[v,a,p,s]=% \sum_nZ^{(2n)}[v,a,p,s] $. The loop expansion of $Z[v,a,p,s]$ formally corresponds to the expansion around the classical fields which are solutions of the classical equation of motion. The formal chiral order of the resonance fields corresponds then to the chiral order of the leading term of the expansion of the classical resonance fields in powers of $p$ and external sources according to the standard chiral power counting,* i.e.* $$V^\mu =O(p^3),\,\,\,\,\,R^{\mu \nu }=O(p^2).\, \label{fields order I}$$ At the same time, for the resonance mass (which plays a role of the hadronic scale within the standard power counting) we take $$\,M=O(1), \label{mass order I}$$ and for the external sources as usual $$v^\mu ,a^\mu =O(p),\,\,\,\,\,\chi ,\chi ^{+}=O(p^2). \label{source order I}$$ The resonance propagators are then of the (minimal) order $O(1)$ and the order of the operators which contain the resonance fields is at least $% O(p^4) $. This formal power counting therefore restricts both the number of the resonance fields in the generic operator as well as the number of the derivatives. When combined with the large $N_C$ arguments, it allows for the construction of the complete operator basis necessary for the saturation of the LEC’s in the chiral Lagrangian at a given chiral order and a leading order in the $1/N_C$ expansion [@Cirigliano:2006hb]. Originally this type of power counting was designed for the leading order (tree-level) matching of $R\chi T$ and $\chi PT$ within the large $N_C$ expansion and there is no straightforward extension to the general graph $% \Gamma $ with $L$ loops. The reason is that the above power counting of the resonance propagators inside the loops does not reproduce correctly the standard chiral order of the graph. As a result, the loop graphs violate the naive chiral power counting in a way analogous to the $\chi PT$ with baryons [@Gasser:1987rb] . The second possibility applicable to loops is to generalize the Weinberg [@Weinberg] power counting scheme and formally arrange the computation as an expansion in the power of the momenta and the resonance masses [@Lutz:2008km] (though there is no mass gap and no natural scale which would give to such a formal power counting a reasonable physical meaning[^4]). Nevertheless, provided we make a following assignment to the resonance field and to the resonance mass $M$$$V^\mu ,R^{\mu \nu }=O(1),\,\,\,M=O(p)$$ we get for the kinetic and mass term of the resonance field $$\mathcal{L}_{kin},\mathcal{L}_{mass}=O(p^2)$$ i.e. the same order as for the lowest order chiral Lagrangian, which allows the same power counting of the resonance propagators as for PGB within the pure $\chi PT$. As a result, the Weinberg formula for the order $D_\Gamma $ of a given graph $\Gamma $ with $L$ loops built from the vertices with the order $D_V$, $$D_\Gamma =2+2L+\sum_V(D_V-2), \label{weinberg}$$ remains valid also within $R\chi T$. Note however, that now $p^2/M^2=O(1)$ and therefore the counterterms needed for renormalization of the graph with chiral order $D_\Gamma $ might contain more than $D_\Gamma $ derivatives (this feature is typical for graphs with resonances inside the loops because of the nontrivial numerator of the resonance propagator). Therefore this type of power counting is less useful for the classification of the counterterms than in the case of the pure $\chi PT$, where $D_\Gamma $ gives an upper bound on the number of derivatives of the counterterms needed to renormalize $\Gamma $. There are also some other complications, which depreciate this counting in the case of $R\chi T$. First note that the interaction vertices with the resonance fields can carry a chiral order smaller than two. This applies e.g. to the trilinear vertex in the antisymmetric tensor representation $$\mathcal{O}^{RRR}=\mathrm{i}g_{\rho \sigma }\langle R_{\mu \nu }R^{\mu \rho }R^{\nu \sigma }\rangle \label{RRR_vertex}$$ or to the odd intrinsic parity vertex mixing the vector and rge antisymmetric tensor field in the first order formalism $$\mathcal{O}^{RV}=\varepsilon _{\alpha \beta \mu \nu }\langle \{V^\alpha ,R^{\mu \nu }\}u^\beta \rangle . \label{VRGB_vertex}$$ Therefore, increasing number of such vertices will decrease the formal chiral order causing again a mismatch between the chiral counting and the loop expansion. Furthermore, such a naive scheme unlike the previous one does not restrict the number of the resonance fields in a general operator because only the number of derivatives, the resonance masses and the external sources score. The former drawback can be formally cured by adding an artificial power of $M$ in front of such operators[^5] (or equivalently counting the corresponding couplings as $O(p^2)$ and $O(p)$ respectively) in order to increase artificially their chiral order and preserve the validity of the Weinberg formula, which now can serve as a formal tool for the classification of the counterterms. How to treat the latter drawback we will discuss further bellow. Let us, however, stress once again, that there is no physical content in such a classification scheme, though it might be technically useful. Third possibility how to assign an index to the given interaction terms and to the general graphs, independent of the previous two, is offered by the large $N_{C}$ expansion. In the $N_{C}\rightarrow \infty $ limit, the amplitude of the interaction of the $n$ mesonic resonances is suppressed at least by the factor $O(N_{C}^{1-n/2})$ and, more generally, the matrix element of arbitrary number of quark currents and $n$ mesons in the initial and final states has the same leading order behavior; e.g. for the GB decay constant we get $F=O(N_{C}^{1/2})$. Because within the chiral building blocks the GB fields always go with the factor $1/F$, we can treat the coupling $c_{\mathcal{O}}\,$corresponding to the operator $% \mathcal{O}$ of the $R\chi T$ Lagrangian as $c_{\mathcal{O}}=O(N_{C}^{\omega _{\mathcal{O}}})$, where $$\omega _{\mathcal{O}}=1-\frac{n_{R}^{\mathcal{O}}}{2}-s_{\mathcal{O}}, \label{NC_index}$$$n_{R}^{\mathcal{O}}$ is the number of the resonance fields contained in $% \mathcal{O}$ and $s_{\mathcal{O}}$ is a possible additional suppression coming e.g. from multiple flavor traces or from the fact, that this coupling appears as a counterterm renomalizing the loop divergences[^6]. From such an operator, generally the infinite number of vertices $V\,$with increasing number $n_{GB}^{V}$ of GB legs can be derived, each accompanied with a factor $c_{\mathcal{O}}F^{-n_{GB}/2}$ and therefore, suppressed as $O(N_{C}^{\omega _{{V}}})$, where the index $\omega _{V}$ is given by[^7] $$\omega _{V}=1-\frac{n_{R}^{\mathcal{O}}}{2}-\frac{n_{GB}^{V}}{2}-s_{\mathcal{% O}}.$$For a given graph, we have the large $N_{C}$ behavior $O(N_{C}^{\omega _{{% \Gamma }}})$ where[^8] $$\omega _{\Gamma }=\sum_{V}\omega _{V}=1-\frac{1}{2}E-L-\sum_{\mathcal{O}}s_{% \mathcal{O}}, \label{large_N_C}$$where $L$ is number of the loops, $E$ is the number of external mesonic lines and we have used the identities $$\begin{aligned} \sum_{V}(n_{R}^{V}+n_{GB}^{V}) &=&2I_{R}+2I_{GB}+E \notag \\ I_{R}+I_{GB} &=&L+V-1\end{aligned}$$relating $L$ and $E$ with the number of resonance and GB internal lines $% I_{R}$ and $I_{GB}$. The loop expansion is therefore correlated with the large $N_{C}$ expansion; higher loops need additionally $N_{C}-$suppressed counterterms $\mathcal{O}_{ct}$ with higher $s_{\mathcal{O}_{ct}}$: $$s_{\mathcal{O}_{ct}}=\left( 1-\frac{1}{2}E\right) -\omega _{\Gamma }=L+\sum_{% \mathcal{O}}s_{\mathcal{O}} \label{large_N_C_s}$$Though the formula (\[large\_N\_C\]) refers seemingly to individual vertices, reformulated in in the form (\[large\_N\_C\_s\]) it points to the members of the chiral symmetric operator basis of the $R\chi T$ Lagrangian. However, as it stays, it does not suit for our purpose because the large $% N_{C}$ counting rules give no restriction for the number of derivatives as well as to the number of resonance fields (once the couplings respect the leading order large $N_{C}$ behavior described above). The formula ([large\_N\_C\_s]{}) expresses merely the the fact that the large $N_{C}$ expansion coincide with the loop one. Let us now describe another useful technical way how to classify the couterterms, which could overcome the problems with the above schemes and is in a sense a combination of them. Let us start with the familiar formula for the degree of superficial divergence $d_\Gamma \,$of a given one particle irreducible graph $\Gamma $, which provides us with the upper bound on the number of derivatives $d_{\mathcal{O}_{ct}}$ in a counterterm $% \mathcal{O}_{ct}$ needed for the renormalization of $\Gamma $. Because in the Proca and antisymmetric tensor formalisms the spin $1$ resonance propagator behaves as[^9] $O(1)$ for $% p\rightarrow \infty $, we get $$d_{\mathcal{O}_{ct}}\leq d_\Gamma =4L-2I_{GB}+\sum_{\mathcal{O}}d_{\mathcal{O% }} \label{d_ct}$$ where $d_{\mathcal{O}}$ means the number of derivatives of the vertex $V$ derived from the operator $\mathcal{O}$. Eliminating $I_{GB}$ in favour of $% L $ and $I_R$ and using the identity $$\sum_{\mathcal{O}}n_R^{\mathcal{O}}=2I_R+E_R,$$ relating $I_R$ with the number of external resonance lines $E_R$, we get eventually $$d_{\mathcal{O}_{ct}}\leq d_\Gamma =2+2L+\sum_{\mathcal{O}}(d_{\mathcal{O}% }+n_R^{\mathcal{O}}-2)-E_R.$$ Adding further to both sides $\sum_{\mathcal{O}}(2n_s^{\mathcal{O}}+2n_p^{% \mathcal{O}}+n_v^{\mathcal{O}}+n_a^{\mathcal{O}})$, the total number of insertions of the external $v$, $a$, $p$ and $s$ sources weighted with its chiral order, we have $$D_{\mathcal{O}_{ct}}+n_R^{ct}-2\leq 2L+\sum_{\mathcal{O}}(D_{\mathcal{O}% }+n_R^{\mathcal{O}}-2)$$ where $D_{\mathcal{O}}$ is the usual chiral order (as in pure $\chi PT$) of generic operator $\mathcal{O}$. Therefore, introducing an index $i_{\mathcal{% O}}$ of a general operator $\mathcal{O}$ as follows[^10] $$i_{\mathcal{O}}=D_{\mathcal{O}}+n_R^{\mathcal{O}}-2 \label{index_O}$$ we get analog of the Weinberg formula[^11], now in the form of an upper bound $$i_{\mathcal{O}_{ct}}\leq i_\Gamma =2L+\sum_{\mathcal{O}}i_{\mathcal{O}}. \label{power_counting}$$ Let us now discuss its properties more closely. First, the number of operators with given $i_{\mathcal{O}}\leq i_{\max }$ is finite, because this requirement limits both the number of derivatives as well as the number of resonance fields. Second, note that, for general operator $\mathcal{O}$ the index $i_{\mathcal{O}}\geq 0$. We have $i_{\mathcal{O}}=0$ for the leading order $\chi PT$ Lagrangian, for the resonance mass (counter)terms as well as for the resonance-GB mixing term $\langle A^\mu u_\mu \rangle $ possible for $1^{+-}$ resonances in the Proca field formalism[^12]. The usual interaction terms with one resonance field and $% O(p^2)$ building blocks correspond to the sector $i_{\mathcal{O}}=1$, the same is true for the trilinear resonance vertex (\[RRR\_vertex\]) as well as for the “mixed” vertex (\[VRGB\_vertex\]), while the two resonance vertices with $O(p^2)$ building blocks correspond to the sector $i_{\mathcal{% O}}=2$, etc. Therefore, according to the formula (\[power\_counting\]), the loop expansion is correlated with the organization of the operators and loop graphs according to the indices $i_{\mathcal{O}}$ and $i_\Gamma $ respectively analogously to the pure $\chi PT$, with the only exception that also lower sectors of the Lagrangian w.r.t. $i_{\mathcal{O}}$ are renormalized at each step. Therefore, we get the renormalizability provided we limit ourselves to the graphs composed from one-particle ireducible building blocs for which the RHS of (\[power\_counting\]) is smaller or equal to $i_{\max }$. The counting rules can be summarized as follows $$R_{\mu \nu },\,V_\mu =O(p),\,M=O(1)$$ and for the external sources as usual $$v^\mu ,a^\mu =O(p),\,\,\chi ,\chi ^{+}=O(p^2).$$ Note also that, the index $i_{\mathcal{O}}$ can be rewritten as $$i_{\mathcal{O}}=D_{\mathcal{O}}-2\left( 1-\frac{n_R^{\mathcal{O}}}2\right)$$ and in the last bracket we recognize the exponent controlling the leading large $N_C\,$ behavior of the coupling constant in front of the operator $% \mathcal{O}$. Remember, however, that the loop induced counterterms have an additional $1/N_C$ suppression for each loop (cf. (\[large\_N\_C\])). Therefore it is natural to modify the index $i_{\mathcal{O}}$ and $i_\Gamma $ as follows (the coefficient $1/2$ is a matter of convenience, see bellow) $$\begin{aligned} \widehat{i}_{\mathcal{O}} &=&\frac{i_{\mathcal{O}}}2+s_{\mathcal{O}}=\frac 12D_{\mathcal{O}}-\left( 1-\frac{n_R^{\mathcal{O}}}2-s_{\mathcal{O}}\right) =\frac 12D_{\mathcal{O}}-\omega _{\mathcal{O}} \notag \\ \widehat{i}_\Gamma &=&\frac{i_\Gamma }2+s_\Gamma =L+\sum_{\mathcal{O}}\frac{% i_{\mathcal{O}}}2+s_\Gamma =2L+\sum_{\mathcal{O}}\widehat{i}_{\mathcal{O}}\end{aligned}$$ where $\omega _{\mathcal{O}}$ is given by (\[NC\_index\]) and we have used (\[large\_N\_C\_s\]) in the last line. With such a modified indices $\widehat{i% }_{\mathcal{O}}$, $\widehat{i}_\Gamma $ the formula (\[power\_counting\]) has the form $$\widehat{i}_{{\mathcal{O}_{ct}}}\leq \widehat{i}_\Gamma =2L+\sum_{\mathcal{O}% }\widehat{i}_{\mathcal{O}} \label{power_counting_hat}$$ The content of this redefinition of $i_{\mathcal{O}}$ is evident: the operators are now classified according to the combined derivative and large $% N_C$ expansion according to the counting rules (for pure $\chi PT$ introduced in [@Moussallam:1994xp], [@Kaiser:1998ds], [Kaiser:2000gs]{}) $$p=O(\delta ^{1/2}),\,\,v,a=O(\delta ^{1/2}),\,\,\chi ,\chi ^{+}=O(\delta ),\,\,\frac 1{N_C}=O(\delta )$$ In what follows we shall use for the classification of the counterterms and for the organization of our calculation the index $i_{\mathcal{O}}$ given by (\[index\_O\]) and (\[power\_counting\]). Note however, that these formulae similarly to the previous cases, do not have much of physical content and serve only as a formal tool for the proof of the renormalizability and for the ordering of the counterterms. Namely, the index $i_\Gamma $ which is by construction related to the superficial degree of the divergence (and which applies to one-particle irreducible graphs only) does not reflect the infrared behavior of the (one-particle irreducible) graph $\Gamma $, rather it refers to its ultraviolet properties. Note also, that the hierarchy of the contributions to the GF by means of fixing $i_\Gamma $ for one-particle irreducible building blocks[^13] might appear to be unusual. For instance, let us assume the antisymmetric tensor formalism. Taking then $% i_\Gamma =0$ allows only the tree graphs with vertices from pure $O(p^2)$ chiral Lagrangian with resonaces completely decoupled (the only $i_{\mathcal{% O}}=0$ relevant term with resonance fields is the resonance mass terms) and such a case is therefore equivalent to the LO $\chi PT$. When fixing $% i_\Gamma \leq 1$, also the terms linear in the resonance fields (at least in the antisymmetric tensor formalism, where the linear sources start at $% O(p^2) $) can be used as the one-particle irreducible building blocks and again only the tree graphs are in the game. However, the resonance propagator is still derived from the mass terms only. Therefore, summing up all the tree graphs with resonance internal lines leads then effectively to the contributions equivalent to those of the pure $O(p^4)$ $\chi PT$ operators with $O(p^4)$ LEC saturated with the resonances in the usual way[^14]. Because the resonance kinetic term has $% i_{\mathcal{O}}=2$, the resonances start to propagate only when we take $% i_\Gamma \leq 2$. At this level we recover the complete NLO $\chi PT$ as a part of the theory (including the loop graphs) supplemented with tree graphs built from the free resonance propagators and vertices with $i_{\mathcal{O}% }\leq 2$. As far as the resonance part of the Lagrangian is concerned, these vertices coincide with the $O(p^6)$ vertices in the first type of power counting we have considered in the beginning of this section (where we assumed $R_{\mu \nu }=O(p^2)$, see (\[fields order I\])) but also the four resonance term without derivatives is allowed. The resonance loops start to contribute at $i_\Gamma \leq 3$ (with the resonance tadpoles) and $i_\Gamma \leq 4$ (with the pure resonance bubbles). In order to renormalize the corresponding divergences, plethora of new counterterms with increasing number of resonances as well as increasing order of the chiral building blocks is needed. In what follows we will encounter graphs with $i_\Gamma =6$ (the mixed GB and resonance bubbles) for which we will need counterterms up to the index $i_{\mathcal{O}}\leq 6$. The self-energies at one loop\[ch4\] ==================================== In this section we present the main result of our paper, namely the one-loop self-energies within all three formalisms discussed in the Section [Propagators and poles]{} in the chiral limit. In what follows, the loops are calculated within the dimensional regularization scheme. In order to avoid complications with the $d-$dimensional Levi-Civita tensor, we use its simplest variant known as Dimensional reduction, i.e. we perform the four-dimensional tensor algebra first in order to reduce the tensor integrals to scalar ones and only then we continue to $d$ dimensions. The Proca field case -------------------- Our starting point is the following Lagrangian for $1^{--}$ resonances [Prades:1993ys]{} (see also [@Knecht:2001xc]) $$\begin{aligned} \mathcal{L}_{V} &=&-\frac{1}{4}\langle \widehat{V}_{\mu \nu }\widehat{V}% ^{\mu \nu }\rangle +\frac{1}{2}M^{2}\langle V_{\mu }V^{\mu }\rangle \notag \\ &&-\frac{\mathrm{i}}{2\sqrt{2}}g_{V}\langle \widehat{V}^{\mu \nu }[u_{\mu },u_{\nu }]\rangle +\frac{1}{2}\sigma _{V}\varepsilon _{\alpha \beta \mu \nu }\langle \{V^{\alpha },\widehat{V}^{\mu \nu }\}u^{\beta }\rangle +\ldots \label{Lagrangian V}\end{aligned}$$where we have written down explicitly only the terms contributing to the self-energy. Originally it was constructed to encompass terms up to the order $O(p^{6})$ within the chiral power-counting (\[fields order I\], [mass order I]{}). In the large $N_{C}$ limit the couplings behave as $% g_{V}=O(N_{C}^{1/2})$ and $\sigma _{V}=O(N_{C}^{-1/2})$. This suggests that the odd intrinsic parity terms are of higher order, however the vertices relevant for our calculations have the same order $O(N_{C}^{-1})$ in both cases due to the presence of the factor $1/F=O(N_{C}^{-1/2})$ which accompanies each Goldstone bosons field. In the above Lagrangian the operators shown explicitly have no more than two derivatives and two resonance fields. Therefore, because the interaction terms are $O(p^{2})$ we would expect (by analogy with the $\chi $PT power counting) the counterterms necessary to cancel the divergencies of the one-loop graphs to have four derivatives at most. However, the nontrivial structure of the free resonance propagator (namely the presence of the $P_{L}$ part) results in the failure of this naive expectation. In fact, according to (\[index\_O\]) and ([power\_counting]{}), the operators in (\[Lagrangian V\]) have index up to $% i_{\mathcal{O}}\leq 2$, whereas the Feynman graphs corresponding to the self-energies $\Sigma _{L,T}$ (depicted in Fig. \[vectorg\]) ![The one-loop graphs contributing to the self-energy of the Proca field. The dotted and full lines corresponds to the Goldstone boson and resonance propagators respectively. Both one-loop graphs have $i_\Gamma=6$[]{data-label="vectorg"}](11vector.pdf) $i_{\Gamma }=6$. In order to cancel the infinite part of the loops we have therefore to introduce a set of counterterms with two resonance fields and indices[^15] $i_{\mathcal{O}}\leq 6$, namely $$\begin{aligned} \mathcal{L}_{V}^{ct} &=&\frac{1}{2}M^{2}Z_{M}\langle V_{\mu }V^{\mu }\rangle \mathcal{+}\frac{Z_{V}}{4}\langle \hat{V}_{\mu \nu }\hat{V}^{\mu \nu }\rangle -\frac{Y_{V}}{2}\langle (D_{\mu }V^{\mu })^{2}\rangle \notag \\ &&+\frac{X_{V1}}{4}\langle \{D_{\alpha },D_{\beta }\}V_{\mu }\{D^{\alpha },D^{\beta }\}V^{\mu }\rangle +\frac{X_{V2}}{4}\langle \{D_{\alpha },D_{\beta }\}V_{\mu }\{D^{\alpha },D^{\mu }\}V^{\beta }\rangle \notag \\ &&+\frac{X_{V3}}{4}\langle \{D_{\alpha },D_{\beta }\}V^{\beta }\{D^{\alpha },D^{\mu }\}V_{\mu }\rangle +\frac{X_{V4}}{2}\langle D^{2}V_{\mu }\{D^{\mu },D^{\beta }\}V_{\beta }\rangle +X_{V5}\langle D^{2}V_{\mu }D^{2}V^{\mu }\rangle \notag \\ &&+\mathcal{L}_{V}^{ct(6)}. \label{ctV}\end{aligned}$$ Here the last term accumulates the operators with six derivatives ($i_{% \mathcal{O}}=6$), which we do not write down explicitly. The bare couplings are split into a finite part renormalized at a scale ${\mu }$ and a divergent part. The infinite parts of the bare couplings are fixed according to $$\begin{aligned} Z_{M} &=&Z_{M}^{r}(\mu ) \\ Z_{V} &=&Z_{V}^{r}(\mu )+\frac{80}{3}\left( \frac{M}{F}\right) ^{2}\sigma _{V}^{2}\lambda _{\infty } \\ X_{V} &=&X_{V}^{r}(\mu )-\frac{80}{9}\left( \frac{M}{F}\right) ^{2}\sigma _{V}^{2}\frac{1}{M^{2}}\lambda _{\infty } \\ Y_{V} &=&Y_{V}^{r}(\mu ) \\ X_{V}^{^{\prime }} &=&X_{V}^{^{\prime }r}(\mu )\end{aligned}$$where $$\begin{aligned} X_{V}^{r}(\mu ) &=&X_{V1}^{r}(\mu )+X_{V5}^{r}(\mu ) \\ X_{V}^{^{\prime }r}(\mu ) &=&X_{V1}^{r}(\mu )+X_{V2}^{r}(\mu )+X_{V3}^{r}(\mu )+X_{V4}^{r}(\mu )+X_{V5}^{r}(\mu ),\end{aligned}$$and $$\lambda _{\infty }=\frac{{\mu }^{d-4}}{(4\pi )^{2}}\left( \frac{1}{d-4}-% \frac{1}{2}(\ln 4\pi -\gamma +1)\right) .$$The result can be written in the form (in the following formulae $x=s/M^{2}$) $$\begin{aligned} \Sigma _{T}^{r}(s) &=&M^{2}\left( \frac{M}{4\pi F}\right) ^{2}\left[ \sum_{i=0}^{3}{\alpha }_{i}x^{i}-\frac{1}{2}g_{V}^{2}\left( \frac{M}{F}% \right) ^{2}x^{3}\widehat{B}(x)-\frac{40}{9}\sigma _{V}^{2}(x-1)^{2}x% \widehat{J}(x)\right] \\ \Sigma _{L}^{r}(s) &=&M^{2}\left( \frac{M}{4\pi F}\right) ^{2}\sum_{i=0}^{3}{% \beta }_{i}x^{i}\end{aligned}$$In the above formulae ${\alpha }_{i}$ and ${\beta }_{i}$ can be expressed in terms of the renormalization scale independent combinations of the counterterm couplings and $\chi $logs. The explicit formulae are collected in the Appendix \[appendix Proca\]. The functions $\widehat{B}(x)$ and $% \widehat{J}(x)$ correspond to the vacuum bubbles with two Goldstone boson lines or with one Goldstone boson and one resonance line respectively. On the first (physical) sheet, $$\begin{aligned} \widehat{B}(x) &=&\widehat{B}^{I}(x)=1-\ln (-x) \notag \\ \widehat{J}(x) &=&\widehat{J}^{I}(x)=\frac{1}{x}\left[ 1-\left( 1-\frac{1}{x}% \right) \ln (1-x)\right], \label{loop_functions}\end{aligned}$$where we take the principal branch of the logarithm ($-\pi <\mathrm{{Im}\ln x\leq \pi }$) with cut for $x<0$. On the second sheet we have then $\widehat{% B}^{II}(x-\mathrm{i}0)=\widehat{B}^{I}(x+\mathrm{i}0)=\widehat{B}^{I}(x-% \mathrm{i}0)+2\pi \mathrm{i}$ and similarly for $\widehat{J}(x)$, therefore $$\begin{aligned} \widehat{B}^{II}(x) &=&\widehat{B}^{I}(x)+2\pi \mathrm{i} \notag \\ \widehat{J}^{II}(x) &=&\widehat{J}^{I}(x)+\frac{2\pi \mathrm{i}}{x}\left( 1-% \frac{1}{x}\right) . \label{loop_functions_II}\end{aligned}$$The equation for the pole in the $1^{--}$ channel $$s-M^{2}-\Sigma _{T}(s)=0$$has a perturbative solution corresponding to the original $1^{--}$ vector resonance, which develops a mass correction and a finite width of the order $% O(1/N_{C})$ due to the loops. This solution can be written in the form $% \overline{s}=M_{\mathrm{phys}}^{2}-\mathrm{i}M_{\mathrm{phys}}\Gamma _{% \mathrm{phys}}$ where $$\begin{aligned} M_{\mathrm{phys}}^{2} &=&M^{2}+\mathrm{Re}\Sigma _{T}(M^{2})=M^{2}\left[ 1+\left( \frac{M}{4\pi F}\right) ^{2}\left( \sum_{i=0}^{3}{\alpha }_{i}-% \frac{1}{2}g_{V}^{2}\left( \frac{M}{F}\right) ^{2}\right) \right] \\ M_{\mathrm{phys}}\Gamma _{\mathrm{phys}} &=&-\mathrm{Im}\Sigma _{T}(M^{2})=M^{2}\left( \frac{M}{4\pi F}\right) ^{2}\frac{1}{2}% g_{V}^{2}\left( \frac{M}{F}\right) ^{2}\pi\end{aligned}$$which gives a constraint on the values of ${\alpha }_{i}$’s $$M_{\mathrm{phys}}^{2}+\frac{1}{\pi }M_{\mathrm{phys}}\Gamma _{\mathrm{phys}% }=M^{2}\left[ 1+\left( \frac{M}{4\pi F}\right) ^{2}\sum_{i=0}^{3}{\alpha }% _{i}\right]$$and in terms of the physical mass and the width we have then $$\begin{aligned} \Sigma _{T}^{r}(s) &=&M_{\mathrm{phys}}^{2}\left( \frac{M_{\mathrm{phys}}}{% 4\pi F}\right) ^{2}\left[ \sum_{i=0}^{3}{\alpha }_{i}x^{i}-\frac{40}{9}% \sigma _{V}^{2}(x-1)^{2}x\widehat{J}(x)\right] -\frac{1}{\pi }M_{\mathrm{phys% }}\Gamma _{\mathrm{phys}}x^{3}\widehat{B}(x) \\ \Sigma _{L}^{r}(s) &=&M_{\mathrm{phys}}^{2}\left( \frac{M_{\mathrm{phys}}}{% 4\pi F}\right) ^{2}\sum_{i=0}^{3}{\beta }_{i}x^{i}.\end{aligned}$$For further numerical estimates it is convenient to adopt the on shell renormalization prescription demanding $M^{2}=M_{\mathrm{phys}}^{2}$ and also to identify $F$ with $F_{\pi }$ (because $F=F_{\pi }$ at the leading order). This gives $$\frac{1}{\pi }\frac{\Gamma _{\mathrm{phys}}}{M_{\mathrm{phys}}}=\left( \frac{% M}{4\pi F}\right) ^{2}\sum_{i=0}^{3}{\alpha }_{i}$$and, introducing parameters $a_{i}$, $b_{i}$ with natural size $O(1)$$$\begin{aligned} a_{i} &=&\pi \frac{M_{\mathrm{phys}}}{\Gamma _{\mathrm{phys}}}\left( \frac{% M_{\mathrm{phys}}}{4\pi F_{\pi }}\right) ^{2}{\alpha }_{i}\sim O(1) \\ b_{i} &=&\pi \frac{M_{\mathrm{phys}}}{\Gamma _{\mathrm{phys}}}\left( \frac{% M_{\mathrm{phys}}}{4\pi F_{\pi }}\right) ^{2}{\beta }_{i}\sim O(1)\end{aligned}$$we get in this scheme for $\sigma _{T,L}^{r}(x)=M_{\mathrm{phys}}^{-2}\Sigma _{T,L}^{r}(M_{\mathrm{phys}}^{2}x)$ $$\begin{aligned} \sigma _{T}^{r}(x) &=&\frac{1}{\pi }\frac{\Gamma _{\mathrm{phys}}}{M_{% \mathrm{phys}}}\left( 1+\sum_{i=1}^{3}a_{i}(x^{i}-1)-x^{3}\widehat{B}% (x)\right) -\frac{40}{9}\left( \frac{M_{\mathrm{phys}}}{4\pi F_{\pi }}% \right) ^{2}\sigma _{V}^{2}(x-1)^{2}x\widehat{J}(x) \\ \sigma _{L}^{r}(x) &=&\frac{1}{\pi }\frac{\Gamma _{\mathrm{phys}}}{M_{% \mathrm{phys}}}\sum_{i=0}^{3}b_{i}x^{i}.\end{aligned}$$ The antisymmetric tensor case ----------------------------- We start with the following Lagrangian for $1^{--}$ resonances (here only the terms relevant for the one-loop selfenergy are shown explicitly) $$\begin{aligned} \mathcal{L}_{R} &=&-\frac{1}{2}\langle D_{\mu }R^{\mu \nu }D^{\alpha }R_{\alpha \nu }\rangle +\frac{1}{4}M^{2}\langle R^{\mu \nu }R_{\mu \nu }\rangle \notag \\ &+&\frac{iG_{V}}{2\sqrt{2}}\langle R^{\mu \nu }[u_{\mu },u_{\nu }]\rangle +d_{1}\varepsilon _{\mu \nu \alpha \sigma }\langle D_{\beta }u^{\sigma }\{R^{\mu \nu },R^{\alpha \beta }\}\rangle \notag \\ &+&d_{3}\varepsilon _{\rho \sigma \mu \lambda }\langle u^{\lambda }\{D_{\nu }R^{\mu \nu },R^{\rho \sigma }\}\rangle +d_{4}\varepsilon _{\rho \sigma \mu \alpha }\langle u_{\nu }\{D^{\alpha }R^{\mu \nu },R^{\rho \sigma }\}\rangle \notag \\ &+&\mathrm{i}\lambda ^{VVV}\langle R_{\mu \nu }R^{\mu \rho }R^{\nu \sigma }\rangle +\ldots \label{Lagrangian T}\end{aligned}$$Note that, in the large $N_{C}$ limit the coupling $G_{V}$ behaves as $% G_{V}=O(N_{C}^{1/2})$, whereas $d_{i}=O(1)$ and $\lambda ^{VVV}=O(N_{C}^{-1/2}) $. Apparently the intrinsic parity odd part and the trilinear resonance coupling are thus of higher order. However, the trilinear vertices contributing to the one-loop self-energies are $% O(N_{C}^{-1/2})$ in both cases due to the appropriate power of $% 1/F=O(N_{C}^{-1/2})$ accompanying $u_{\alpha }$. Therefore, the operators with two and three resonance fields cannot be got rid of using the large $% N_{C}$ arguments. Also nonzero $d_{i}$ are required in order to satisfy the OPE constraints for VVP GF at the LO; especially for $d_{3}$ we get [VVP]{} $$d_{3}=-\frac{N_{C}}{64\pi ^{2}}\left( \frac{M}{F_{V}}\right) ^{2}+\frac{1}{8}% \left( \frac{F}{F_{V}}\right) ^{2} \label{d_3}$$where $F_{V}$ is the strength of the resonance coupling to the vector current. The Lagrangian (\[Lagrangian T\]) includes terms up to the index $i_{% \mathcal{O}}\leq 2$. The one-loop Feynman graphs contributing to the self-energy are depicted in Fig. \[tensor\_graphs\]. The first two bubbles include only interaction vertices with $i_{\cal{O}}=1$ and therefore they have indices $i_{\Gamma }=4$ while the third one is built from vertices with $i_{\cal{O}}=2$ and has the index $i_{\Gamma }=6$. In order to cancel the infinite part of the loops we have then to add counterterms with indices $i_{\mathcal{O}}\leq 6$, namely the following set $$\begin{aligned} \mathcal{L}_{R}^{ct} &=&\frac{1}{4}M^{2}Z_{M}\langle R^{\mu \nu }R_{\mu \nu }\rangle +\frac{1}{2}Z_{R}\langle D_{\alpha }R^{{\alpha }{\mu }}D^{\beta }R_{% {\beta }{\mu }}\rangle +\frac{1}{4}Y_{R}\langle D_{\alpha }R^{{\mu }{\nu }% }D^{\alpha }R_{{\mu }{\nu }}\rangle \notag \\ &&+\frac{1}{4}X_{R1}\langle D^{2}R^{{\mu }{\nu }}\{D_{\nu },D^{\sigma }\}R_{{% \mu }{\sigma }}\rangle +\frac{1}{8}X_{R2}\langle \{D_{\nu },D_{\alpha }\}R^{{% \mu }{\nu }}\{D^{\sigma },D^{\alpha }\}R_{{\mu }{\sigma }}\rangle \notag \\ &&+\frac{1}{8}X_{R3}\langle \{D^{\sigma },D^{\alpha }\}R^{{\mu }{\nu }% }\{D_{\nu },D_{\alpha }\}R_{{\mu }{\sigma }}\rangle \notag \\ &&+\frac{1}{4}W_{R1}\langle D^{2}R^{{\mu }{\nu }}D^{2}R_{{\mu }{\nu }% }\rangle +\frac{1}{16}W_{R2}\langle \{D^{{\alpha }},D^{\beta }\}R^{{\mu }{% \nu }}\{D_{{\alpha }},D_{\beta }\}R_{{\mu }{\nu }}\rangle \notag \\ &&+\mathcal{L}_{R}^{ct(6)}, \label{ctR}\end{aligned}$$where the last term accumulates the operators with six derivatives ($i_{% \mathcal{O}}=6)$, which we do not write down explicitly. The infinite parts of the bare couplings are fixed as $$\begin{aligned} Z_{M} &=&Z_{M}^{r}({\mu })+\frac{80}{3}\left( \frac{M}{F}\right) ^{2}d_{1}^{2}\lambda _{\infty }-60\left( \frac{\lambda ^{VVV}}{M}\right) ^{2}\lambda _{\infty } \\ Z_{R} &=&Z_{R}^{r}({\mu })+\frac{40}{9}\left( \frac{M}{F}\right) ^{2}\frac{1% }{M^{2}}(12d_{1}(d_{3}+d_{4})-d_{3}^{2}-9d_{4}^{2}+6d_{3}d_{4})\lambda _{\infty } \\ &&+80\left( \frac{\lambda ^{VVV}}{M}\right) ^{2}\frac{1}{M^{2}}\lambda _{\infty } \\ Y_{R} &=&Y_{R}^{r}({\mu })+\frac{40}{9}\left( \frac{M}{F}\right) ^{2}\frac{1% }{M^{2}}(6d_{1}^{2}-12d_{1}(d_{3}+d_{4})+5d_{3}^{2}+9d_{4}^{2}-6d_{3}d_{4})% \lambda _{\infty } \\ &&-40\left( \frac{\lambda ^{VVV}}{M}\right) ^{2}\frac{1}{M^{2}}\lambda _{\infty } \\ X_{R} &=&X_{R}^{r}({\mu })+\frac{40}{9}\left( \frac{M}{F}\right) ^{2}\frac{1% }{M^{2}}(d_{3}^{2}-6d_{3}d_{4}+5d_{4}^{2})\lambda _{\infty }-\left( \frac{% G_{V}}{F}\right) ^{2}\frac{1}{M^{2}}\lambda _{\infty } \\ W_{R} &=&W_{R}^{r}({\mu })+\frac{40}{9}\left( \frac{M}{F}\right) ^{2}\frac{1% }{M^{4}}(d_{3}^{2}+6d_{3}d_{4}-5d_{4}^{2})\lambda _{\infty }-10\left( \frac{% \lambda ^{VVV}}{M}\right) ^{2}\frac{1}{M^{4}}\lambda _{\infty }\end{aligned}$$where $$\begin{aligned} X_{R}^{r}({\mu }) &=&X_{R1}^{r}({\mu })+X_{R2}^{r}({\mu })+X_{R3}^{r}({\mu }) \\ W_{R}^{r}({\mu }) &=&W_{R1}^{r}({\mu })+W_{R2}^{r}({\mu }).\end{aligned}$$An explicit calculation gives for the renormalized self-energies (in the following formulae $x=s/M^{2}$) $$\begin{aligned} \Sigma _{L}^{r}(s) &=&M^{2}\left( \frac{M}{4\pi F}\right) ^{2}\left[ \sum_{i=0}^{3}{\alpha }_{i}x^{i}-\left( \frac{1}{2}\left( \frac{G_{V}}{F}% \right) ^{2}x^{2}\widehat{B}(x)+\frac{40}{9}d_{3}^{2}(x^{2}-1)^{2}\widehat{J}% (x)\right) \right] \\ &&-5\left( \frac{\lambda ^{VVV}}{4\pi }\right) ^{2}(x-4)(x+2)\overline{J}(x) \\ \Sigma _{T}^{r}(s) &=&M^{2}\left( \frac{M}{4\pi F}\right) ^{2}\left[ \sum_{i=0}^{3}{\beta }_{i}x^{i}+\frac{20}{9}\left( 2d_{3}^{2}+(d_{3}^{2}+6d_{3}d_{4}+d_{4}^{2})x+2d_{4}^{2}x^{2}\right) (x-1)^{2}\widehat{J}(x)\right] \\ &&+5\left( \frac{\lambda ^{VVV}}{4\pi }\right) ^{2}(x^{2}-2x+4)\overline{J}% (x).\end{aligned}$$Here the functions $\widehat{B}(x)$ and $\widehat{J}(x)$ are same as in the previous subsection and $\overline{J}(x)$ is given on the physical sheet by $$\overline{J}(x)=\overline{J}^{I}(x)=2+\sqrt{1-\frac{4}{x}}\ln \frac{\sqrt{1-% \frac{4}{x}}-1}{\sqrt{1-\frac{4}{x}}+1}.$$with the same branch of the logarithm as before. On the second sheet we have $\overline{J}^{II}(x-\mathrm{i}0)=\overline{J}^{I}(x+\mathrm{i}0)=\overline{J% }^{I}(x-\mathrm{i}0)+2\mathrm{i}\pi \sqrt{1-4/x}$ and therefore $$\overline{J}^{II}(x)=\overline{J}^{I}(x)+2\mathrm{i}\pi \sqrt{1-\frac{4}{x}}.$$The explicit dependence of the renormalization scale invariant polynomial parameters ${\alpha }_{i}$ and ${\beta }_{i}$ on the counterterm couplings and $\chi $logs are given in the Appendix \[appendix tensor\]. In order to simplify the following discussion we put $\lambda ^{VVV}=0$ in the rest of this subsection. This is in accord with the fact, that the corresponding trilinear interaction term can be effectively removed by resonance field redefinition [@Cirigliano:2006hb]. Also, the two-resonance cut starts at $x=4$ which is far from the region we are interested in. Here the effect of the resonance bubble can be effectively absorbed to the polynomial part of the self-energies. The equation for the propagator poles in the $1^{--}$ channel $$s-M^{2}-\Sigma _{L}(s)=0$$has an approximative perturbative solution corresponding to the original $% 1^{--}$ vector resonance, which develops a mass correction and a finite width of the order $O(1/N_{C})$ due to the loops. This solution can be written in the form $\overline{s}=M_{\mathrm{phys}}^{2}-\mathrm{i}M_{\mathrm{% phys}}\Gamma _{\mathrm{phys}}$ where $$\begin{aligned} M_{\mathrm{phys}}^{2} &=&M^{2}+\mathrm{Re}\Sigma _{L}(M^{2})=M^{2}\left[ 1+\left( \frac{M}{4\pi F}\right) ^{2}\left( \sum_{i=0}^{3}{\alpha }_{i}-% \frac{1}{2}\left( \frac{G_{V}}{F}\right) ^{2}\right) \right] \\ M_{\mathrm{phys}}\Gamma _{\mathrm{phys}} &=&-\mathrm{Im}\Sigma _{L}(M^{2})=M^{2}\left( \frac{M}{4\pi F}\right) ^{2}\frac{1}{2}\left( \frac{% G_{V}}{F}\right) ^{2}\pi \mathrm{,}\end{aligned}$$which gives a constraint on the values of ${\alpha }_{i}$’s $$M_{\mathrm{phys}}^{2}+\frac{1}{\pi }M_{\mathrm{phys}}\Gamma _{\mathrm{phys}% }=M^{2}\left( 1+\frac{1}{(4\pi )^{2}}\left( \frac{M}{F}\right) ^{2}\sum_{i=0}^{3}{\alpha }_{i}\right) .$$This allows us to re-parameterize perturbatively $\Sigma _{L}(s)$ in terms of $% M_{\mathrm{phys}}$ and $\Gamma _{\mathrm{phys}}$ as $$\begin{aligned} \Sigma _{L}^{r}(s) &=&M_{\mathrm{phys}}^{2}\left( \frac{M_{\mathrm{phys}}}{% 4\pi F}\right) ^{2}\left[ \sum_{i=0}^{3}{\alpha }_{i}x^{i}-\frac{40}{9}% d_{3}^{2}(x^{2}-1)^{2}\widehat{J}(x)\right] -\frac{1}{\pi }\Gamma _{\mathrm{% phys}}M_{\mathrm{phys}}x^{2}\widehat{B}(x) \\ \Sigma _{T}^{r}(s) &=&M_{\mathrm{phys}}^{2}\left( \frac{M_{\mathrm{phys}}}{% 4\pi F}\right) ^{2}\left[ \sum_{i=0}^{3}{\beta }_{i}x^{i}+\frac{20}{9}\left( 2d_{3}^{2}+(d_{3}^{2}+6d_{3}d_{4}+d_{4}^{2})x+2d_{4}^{2}x^{2}\right) (x-1)^{2}\widehat{J}(x)\right] .\end{aligned}$$ As for the Proca field case, within the on shell renormalization prescription $M^{2}=M_{\mathrm{phys}}^{2}$ and we get a constraint $$\frac{1}{\pi }\frac{\Gamma _{\mathrm{phys}}}{M_{\mathrm{phys}}}=\left( \frac{% M_{\mathrm{phys}}}{4\pi F_{\pi }}\right) ^{2}\sum_{i=0}^{3}{\alpha }_{i}. \label{alpha_size}$$As a result, we can re-write the self-energy (in the units of $M_{\mathrm{% phys}}^{2}$, i.e. as in the previous section $\sigma _{T,L}^{r}(x)=M_{% \mathrm{phys}}^{-2}\Sigma _{T,L}^{r}(M_{\mathrm{phys}}^{2}x)$ in what follows) in the form $${\sigma }_{L}^{r}(x)=\frac{1}{\pi }\frac{\Gamma _{\mathrm{phys}}}{M_{\mathrm{% phys}}}\left[ 1-x^{2}\widehat{B}(x)+\sum_{i=1}^{3}a_{i}(x^{i}-1)\right] -% \frac{40}{9}\left( \frac{M_{\mathrm{phys}}}{4\pi F_{\pi }}\right) ^{2}d_{3}^{2}(x^{2}-1)^{2}\widehat{J}(x)$$using the re-scaled parameters $a_{i}$ with a natural size $O(1)$ $$a_{i}=\pi \frac{M_{\mathrm{phys}}}{\Gamma _{\mathrm{phys}}}\left( \frac{M_{% \mathrm{phys}}}{4\pi F_{\pi }}\right) ^{2}{\alpha }_{i}\sim O(1).$$So that the $\Sigma _{L}^{r}(s)$ has four independent parameters ${\alpha }% _{i}$, $i=1,2,3$ and $d_{3}$. Similarly, $\Sigma _{T}^{r}(s)$ can be written in this scheme in terms of six independent dimensionless parameters $b_{i}$ , $d_{3}$ and $\gamma $ $$\begin{aligned} b_{i} &=&\pi \frac{M_{\mathrm{phys}}}{\Gamma _{\mathrm{phys}}}\left( \frac{% M_{\mathrm{phys}}}{4\pi F_{\pi }}\right) ^{2}{\beta }_{i}\sim O(1) \\ \gamma &=&d_{4}/d_{3}\sim O(1)\end{aligned}$$as $${\sigma }_{T}^{r}(x)=\frac{1}{\pi }\frac{\Gamma _{\mathrm{phys}}}{M_{\mathrm{% phys}}}\sum_{i=0}^{3}b_{i}x^{i}+\frac{20}{9}\left( \frac{M_{\mathrm{phys}}}{% 4\pi F_{\pi }}\right) ^{2}d_{3}^{2}\left( 2+(1+6\gamma +\gamma ^{2})x+2\gamma ^{2}x^{2}\right) (x-1)^{2}\widehat{J}(x).$$In order to satisfy the OPE constraints for $VVP$ correlator [@VVP], we have to put further (according to (\[d\_3\])) $$d_{3}=-\frac{3}{4}\left( \frac{M_{\mathrm{phys}}}{4\pi F_{\pi }}\right) ^{2}\left( \frac{F_{\pi }}{F_{V}}\right) ^{2}\left[ 1-\frac{1}{6}\left( \frac{4\pi F_{\pi }}{M_{\mathrm{phys}}}\right) ^{2}\right] \label{d_3_1}$$which reduces the number of the independent parameters for ${\sigma }% _{L}^{r}(x)$ and ${\sigma }_{T}^{r}(x)$ to three and five respectively. The first order formalism ------------------------- In this case, the interaction part of the Lagrangian describing $1^{--}$ resonances collects all the terms from the previous two formalisms. It contains also one extra term which mixes the the fields $R_{\mu \nu }$ and $% V_{\alpha }$ $$\begin{aligned} \mathcal{L}_{RV} &=&\frac{1}{2}M^{2}\langle V_{\mu }V^{\mu }\rangle +\frac{1% }{4}M^{2}\langle R^{\mu \nu }R_{\mu \nu }\rangle -\frac{1}{2}\langle R^{\mu \nu }\widehat{V}_{\mu \nu }\rangle \\ &&-\frac{\mathrm{i}}{2\sqrt{2}}g_{V}\langle \widehat{V}^{\mu \nu }[u_{\mu },u_{\nu }]\rangle +\frac{1}{2}\sigma _{V}\varepsilon _{\alpha \beta \mu \nu }\langle \{V^{\alpha },\widehat{V}^{\mu \nu }\}u^{\beta }\rangle \\ &&+\frac{iG_{V}}{2\sqrt{2}}\langle R^{\mu \nu }[u_{\mu },u_{\nu }]\rangle +d_{1}\varepsilon _{\mu \nu \alpha \sigma }\langle D_{\beta }u^{\sigma }\{R^{\mu \nu },R^{\alpha \beta }\}\rangle \\ &&+d_{3}\varepsilon _{\rho \sigma \mu \lambda }\langle u^{\lambda }\{D_{\nu }R^{\mu \nu },R^{\rho \sigma }\}\rangle +d_{4}\varepsilon _{\rho \sigma \mu \alpha }\langle u_{\nu }\{D^{\alpha }R^{\mu \nu },R^{\rho \sigma }\}\rangle \\ &&+\frac{1}{2}M\sigma _{RV}\varepsilon _{\alpha \beta \mu \nu }\langle \{V^{\alpha },R^{\mu \nu }\}u^{\beta }\rangle +\mathrm{i}\lambda ^{VVV}\langle R_{\mu \nu }R^{\mu \rho }R^{\nu \sigma }\rangle +\ldots\end{aligned}$$Because the free diagonal propagators are the same as in the pure Proca or antisymmetric tensor cases, all the graphs depicted in the Figs. \[vectorg\], \[tensor\_graphs\] contribute also here to the diagonal self-energies $\Sigma _{RR}$ and $\Sigma _{VV}$. The mixed vertex and mixed propagator generate additional graphs contributing to $\Sigma _{RR}$, $% \Sigma _{VV}$ and $\Sigma _{RV}$ which are depicted in the Figs. \[tensor\_graphs vector\], \[tensor\_graphs tensor\] and \[tensor\_graphs mixed\] respectively (in the latter case also the GB bubble contributes). Similarly, the set of counterterms necessary to renormalize the infinities includes all the terms (\[ctV\]) and (\[ctR\]) and additional mixed terms $$\begin{aligned} \mathcal{L}_{RV}^{ct} &=&\frac{1}{2}M^{2}Z_{MV}\langle V_{\mu }V^{\mu }\rangle \mathcal{+}\frac{Z_{V}}{4}\langle \hat{V}_{\mu \nu }\hat{V}^{\mu \nu }\rangle -\frac{Y_{V}}{2}\langle (D_{\mu }V^{\mu })^{2}\rangle \\ &&+\frac{X_{V1}}{4}\langle \{D_{\alpha },D_{\beta }\}V_{\mu }\{D^{\alpha },D^{\beta }\}V^{\mu }\rangle +\frac{X_{V2}}{4}\langle \{D_{\alpha },D_{\beta }\}V_{\mu }\{D^{\alpha },D^{\mu }\}V^{\beta }\rangle \\ &&+\frac{X_{V3}}{4}\langle \{D_{\alpha },D_{\beta }\}V^{\beta }\{D^{\alpha },D^{\mu }\}V_{\mu }\rangle +\frac{X_{V4}}{2}\langle D^{2}V_{\mu }\{D^{\mu },D^{\beta }\}V_{\beta }\rangle +X_{V5}\langle D^{2}V_{\mu }D^{2}V^{\mu }\rangle \\ &&+\frac{1}{4}M^{2}Z_{MR}\langle R^{\mu \nu }R_{\mu \nu }\rangle +\frac{1}{2}% Z_{R}\langle D_{\alpha }R^{{\alpha }{\mu }}D^{\beta }R_{{\beta }{\mu }% }\rangle +\frac{1}{4}Y_{R}\langle D_{\alpha }R^{{\mu }{\nu }}D^{\alpha }R_{{% \mu }{\nu }}\rangle \\ &&+\frac{1}{4}X_{R1}\langle D^{2}R^{{\mu }{\nu }}\{D_{\nu },D^{\sigma }\}R_{{% \mu }{\sigma }}\rangle +\frac{1}{8}X_{R2}\langle \{D_{\nu },D_{\alpha }\}R^{{% \mu }{\nu }}\{D^{\sigma },D^{\alpha }\}R_{{\mu }{\sigma }}\rangle \\ &&+\frac{1}{8}X_{R3}\langle \{D^{\sigma },D^{\alpha }\}R^{{\mu }{\nu }% }\{D_{\nu },D_{\alpha }\}R_{{\mu }{\sigma }}\rangle \\ &&+\frac{1}{4}W_{R1}\langle D^{2}R^{{\mu }{\nu }}D^{2}R_{{\mu }{\nu }% }\rangle +\frac{1}{16}W_{R2}\langle \{D^{{\alpha }},D^{\beta }\}R^{{\mu }{% \nu }}\{D_{{\alpha }},D_{\beta }\}R_{{\mu }{\nu }} \\ &&-\frac{1}{2}Z_{RV}M\langle R^{\mu \nu }\widehat{V}_{\mu \nu }\rangle +% \frac{1}{2}X_{RV1}M\langle D^{\alpha }R^{\mu \nu }D_{\alpha }\widehat{V}% _{\mu \nu }\rangle +\frac{1}{2}X_{RV2}M\langle D_{\mu }R^{\mu \nu }D^{\sigma }\widehat{V}_{\sigma \nu } \\ &&+\mathcal{L}_{RV}^{ct(6)}\label{ctRV}\end{aligned}$$ Now the infinite parts of the bare couplings have to be fixed as follows $$\begin{aligned} Z_{RV} &=&Z_{RV}^{r}(\mu )-\frac{20}{3}\left( \frac{M}{F}\right) ^{2}(\sigma _{RV}+2\sigma _{V})(2d_{1}-\sigma _{RV})\lambda _{\infty } \\ X_{RV} &=&X_{RV}^{r}(\mu )-\frac{20}{9}\left( \frac{M}{F}\right) ^{2}\frac{1% }{M^{2}}(\sigma _{RV}+2\sigma _{V})(4d_{3}+\sigma _{RV})\lambda _{\infty } \\ Z_{MV} &=&Z_{MV}^{r}(\mu ) \\ Z_{V} &=&Z_{V}^{r}(\mu )+\frac{20}{3}\left( \frac{M}{F}\right) ^{2}\left( \sigma _{RV}(\sigma _{RV}+2\sigma _{V})+4\sigma _{V}^{2}\right) \lambda _{\infty } \\ X_{V} &=&X_{V}^{r}(\mu )-\frac{20}{9}\left( \frac{M}{F}\right) ^{2}\frac{1}{% M^{2}}\left( \sigma _{RV}(\sigma _{RV}+2\sigma _{V})+4\sigma _{V}^{2}\right) \lambda _{\infty } \\ Y_{V} &=&Y_{V}^{r}(\mu ) \\ X_{V}^{^{\prime }} &=&X_{V}^{^{\prime }r}(\mu ) \\ Z_{MR} &=&Z_{MR}^{r}({\mu })+\frac{20}{3}\left( \frac{M}{F}\right) ^{2}(4d_{1}^{2}-\sigma _{RV}(\sigma _{RV}-2d_{1}))\lambda _{\infty } \\ Z_{R} &=&Z_{R}^{r}({\mu })+\frac{40}{9}\left( \frac{M}{F}\right) ^{2}(12d_{1}(d_{3}+d_{4})-d_{3}^{2}-9d_{4}^{2}+6d_{3}d_{4})\lambda _{\infty }+\frac{10}{9}\sigma _{RV}(10d_{3}+18d_{4}+\sigma _{RV})\lambda _{\infty } \\ Y_{R} &=&Y_{R}^{r}({\mu })+\frac{10}{9}\left( \frac{M}{F}\right) ^{2}\frac{1% }{M^{2}}(24d_{1}^{2}-48d_{1}(d_{3}+d_{4})+20d_{3}^{2}+36d_{4}^{2} \\ &&-24d_{3}d_{4}-\sigma _{RV}^{2}+2\sigma _{RV}(d_{3}+3d_{4}))\lambda _{\infty } \\ X_{R} &=&X_{R}^{r}({\mu })+\frac{40}{9}\left( \frac{M}{F}\right) ^{2}\frac{1% }{M^{2}}(d_{3}^{2}-6d_{3}d_{4}+5d_{4}^{2})\lambda _{\infty } \\ &&-\frac{10}{9}\frac{1}{M^{2}}\sigma _{RV}(6(d_{3}+d_{4})-\sigma _{RV})\lambda _{\infty }-\left( \frac{G_{V}}{F}\right) ^{2}\frac{1}{M^{2}}% \lambda _{\infty } \\ W_{R} &=&W_{R}^{r}({\mu })+\frac{40}{9}\left( \frac{M}{F}\right) ^{2}\frac{1% }{M^{4}}(d_{3}^{2}+6d_{3}d_{4}-5d_{4}^{2})\lambda _{\infty }+\frac{10}{9}% \sigma _{RV}(\sigma _{RV}-2(d_{3}+3d_{4}))\lambda _{\infty }\end{aligned}$$where $$\begin{aligned} X_{V}^{r}(\mu ) &=&X_{V1}^{r}(\mu )+X_{V5}^{r}(\mu ) \\ X_{V}^{^{\prime }r}(\mu ) &=&X_{V1}^{r}(\mu )+X_{V2}^{r}(\mu )+X_{V3}^{r}(\mu )+X_{V4}^{r}(\mu )+X_{V5}^{r}(\mu ) \\ X_{R}^{r}({\mu }) &=&X_{R1}^{r}({\mu })+X_{R2}^{r}({\mu })+X_{R3}^{r}({\mu }) \\ W_{R}^{r}({\mu }) &=&W_{R1}^{r}({\mu })+W_{R2}^{r}({\mu }) \\ X_{RV}^{r}({\mu }) &=&X_{RV1}^{r}({\mu })+X_{RV2}^{r}({\mu })\end{aligned}$$The renormalized self-energies can be then written in the form $$\begin{aligned} \Sigma _{RV}(s)^{r} &=&M\left( \frac{M}{4\pi F}\right) ^{2}\left[ \sum_{i=0}^{2}{\alpha }_{i}^{RV}x^{i}+\frac{1}{2}\frac{g_{V}G_{V}}{M}\left( \frac{M}{F}\right) ^{2}x^{2}\widehat{B}(x)\right. \\ &&\left. +\frac{10}{9}(\sigma _{RV}+2\sigma _{V})(2d_{3}x+2d_{3}-\sigma _{RV})(x-1)^{2}\widehat{J}(x)\right] =\Sigma _{VR}(s)^{r} \\ \Sigma _{VV}^{T}(s)^{r} &=&M^{2}\left( \frac{M}{4\pi F}\right) ^{2}\left[ \sum_{i=0}^{3}{\alpha }_{i}^{VV}x^{i}-\frac{1}{2}g_{V}^{2}\left( \frac{M}{F}% \right) ^{2}x^{3}\widehat{B}(x)\right. \\ &&\left. -\frac{10}{9}\left( \sigma _{RV}(\sigma _{RV}+2\sigma _{V})+4\sigma _{V}^{2}\right) (x-1)^{2}x\widehat{J}(x)\right] \\ \Sigma _{VV}^{L}(s)^{r} &=&M^{2}\left( \frac{M}{4\pi F}\right) ^{2}\sum_{i=0}^{3}{\beta }_{i}^{VV}x^{i} \\ \Sigma _{RR}^{L}(s)^{r} &=&M^{2}\left( \frac{M}{4\pi F}\right) ^{2}\left[ \sum_{i=0}^{3}{\alpha }_{i}^{RR}x^{i}-\frac{1}{2}\left( \frac{G_{V}}{F}% \right) ^{2}x^{2}\widehat{B}(x)\right. \\ &&\left. -\frac{10}{9}(4d_{3}^{2}(x+1)^{2}-2d_{3}\sigma _{RV}(x+1)+\sigma _{RV}^{2})(x-1)^{2}\widehat{J}(x)\right] \\ \Sigma _{RR}^{T}(s)^{r} &=&M^{2}\left( \frac{M}{4\pi F}\right) ^{2}\left[ \sum_{i=0}^{3}{\beta }_{i}^{RR}x^{i}+\frac{5}{9}(8d_{3}^{2}-4\sigma _{RV}d_{3}+2\sigma _{RV}^{2}\right. \\ &&\left. +(4d_{3}^{2}-2\sigma _{RV}d_{3}+\sigma _{RV}^{2}+24d_{3}d_{4}+4d_{4}^{2}-6\sigma _{RV}d_{4})x+8d_{4}^{2}x^{2})(x-1)^{2}\widehat{J}(x)\right] .\end{aligned}$$Here again the renormalization scale independent coefficients of the polynomial parts of the self-energies are expressed in terms of the couplings and chiral logs; the explicit formulae can be found in the Appendix \[appendix first order\]. The equation for the poles in the $1^{--}$ channel $$D(s)=(M^2+\Sigma _{RR}^L(s))(M^2+\Sigma _{VV}^T(s))-s(M+\Sigma _{RV}(s))(M+\Sigma _{VR}(s))=0$$ can be solved perturbatively writing the solution in the form $\overline{s}% =M_{\mathrm{phys}}^2-\mathrm{i}M_{\mathrm{phys}}\Gamma _{\mathrm{phys}% }=M^2+\Delta $. To the first order in $\Delta $ and the self-energies we get then $$\overline{s}=M^2+\Sigma _{RR}^L(M^2)+\Sigma _{VV}^T(M^2)-M(\Sigma _{RV}(M^2)+\Sigma _{VR}(M^2))$$ and therefore $$\begin{aligned} M_{\mathrm{phys}}^2 &=&M^2+\mathrm{Re}\left[ \Sigma _{RR}^L(M^2)+\Sigma _{VV}^T(M^2)-M(\Sigma _{RV}(M^2)+\Sigma _{VR}(M^2))\right] \\ &=&M^2\left[ 1+\left( \frac M{4\pi F}\right) ^2\left( \sum_{i=0}^3({\alpha }% _i^{RR}+{\alpha }_i^{VV})-2\sum_{i=0}^2{\alpha }_i^{RV}-\frac 12\left( \frac MF\right) ^2\left( g_V+\frac{G_V}M\right) ^2\right) \right] \\ M_{\mathrm{phys}}\Gamma _{\mathrm{phys}} &=&-\mathrm{Im}\left[ \Sigma _{RR}^L(M^2)+\Sigma _{VV}^T(M^2)-M(\Sigma _{RV}(M^2)+\Sigma _{VR}(M^2))% \right] \\ &=&\pi M^2\left( \frac M{4\pi F}\right) ^2\frac 12\left( \frac MF\right) ^2\left( g_V+\frac{G_V}M\right) ^2\end{aligned}$$ which yield the constraint $$M_{\mathrm{phys}}^2+\frac 1\pi M_{\mathrm{phys}}\Gamma _{\mathrm{phys}% }=M^2\left( 1+\left( \frac M{4\pi F}\right) ^2\left( \sum_{i=0}^3({\alpha }% _i^{RR}+{\alpha }_i^{VV})-2\sum_{i=0}^2{\alpha }_i^{RV}\right) \right) .$$ In the on-shell scheme $M^2=M_{\mathrm{phys}}^2$we get further $$\frac 1\pi \frac{\Gamma _{\mathrm{phys}}}{M_{\mathrm{phys}}}=\left( \frac{M_{% \mathrm{phys}}}{4\pi F_\pi }\right) ^2\left( \sum_{i=0}^3({\alpha }_i^{RR}+{% \alpha }_i^{VV})-2\sum_{i=0}^2{\alpha }_i^{RV}\right)$$ On the contrary to the previous two cases, this allows to exclude both the constants $g_V$ and $G_V$ in favor of the physical observables only for the combination $$\begin{aligned} \sigma (x) &\equiv &x\sigma _{RR}^L(x)+\sigma _{VV}^T(x)-x(\sigma _{RV}(x)+\sigma _{VR}(x)) \\ &=&\frac 1\pi \frac{\Gamma _{\mathrm{phys}}}{M_{\mathrm{phys}}}\left( 1+\sum_{i=0}^4a_i(x-1)^i-x^3\widehat{B}(x)\right) \\ &&-\frac{20}9\left( \frac{M_{\mathrm{phys}}}{4\pi F_\pi }\right) ^2x(x-1)^2% \widehat{J}(x)\left[ d_3(x+1)(2d_3(x+1)+\sigma _{RV}+4\sigma _V)+\sigma _V(\sigma _{RV}-2\sigma _V)\right]\end{aligned}$$ (here $\Sigma _{RR}^{Lr}=M^2\sigma _{RR}^L$, $\Sigma _{RV}^r=M\sigma _{RV}$ etc.), where $$a_i=\pi \frac{M_{\mathrm{phys}}}{\Gamma _{\mathrm{phys}}}\left( \frac{M_{% \mathrm{phys}}}{4\pi F_\pi }\right) ^2\left( {\alpha }_{i-1}^{RR}+{\alpha }% _i^{VV}-2{\alpha }_{i-1}^{RV}\right)$$ with ${\alpha }_{-1}^{RR}={\alpha }_{-1}^{RV}=0$ are parameters of order $% O(1)$, From the OPE constraints applied to $VVP$ correlator within the first order formalism we get further $$\begin{aligned} d_3 &=&-\frac{N_C}{64\pi ^2}\left( \frac M{F_V}\right) ^2+\frac 18\left( \frac F{F_V}\right) ^2+\frac 12(\sigma _{RV}+\sigma _V) \\ &=&-\frac 34\left( \frac{M_{\mathrm{phys}}}{4\pi F_\pi }\right) ^2\left( \frac{F_\pi }{F_V}\right) ^2\left[ 1-\frac 16\left( \frac{4\pi F_\pi }{M_{% \mathrm{phys}}}\right) ^2\right] +\frac 12(\sigma _{RV}+\sigma _V)\end{aligned}$$ Using dimensionless variables, we can write the condition for the poles in the form $$(1+\sigma _{RR}^L(x))(1+\sigma _{VV}^T(x))-x(1+\sigma _{RV}(x))(1+\sigma _{VR}(x))=0$$ in the $1^{--}$ channel and $$\begin{aligned} 1+\sigma _{RR}^T(x) &=&0 \\ 1+\sigma _{VV}^L(x) &=&0\end{aligned}$$ in the $1^{+-}$ and $0^{+-}$ channels respectively. Within the on-shell scheme $$\begin{aligned} \sigma _{RV}(s)^r &=&\frac 1\pi \frac{\Gamma _{\mathrm{phys}}}{M_{\mathrm{% phys}}}\left( \sum_{i=0}^2a_i^{RV}x^i-(1-C)Cx^2\widehat{B}(x)\right) \\ &&+\frac{10}9\left( \frac{M_{\mathrm{phys}}}{4\pi F_\pi }\right) ^2\left[ (\sigma _{RV}+2\sigma _V)(2d_3x+2d_3-\sigma _{RV})(x-1)^2\widehat{J}(x)% \right] \\ \sigma _{VV}^T(s)^r &=&\frac 1\pi \frac{\Gamma _{\mathrm{phys}}}{M_{\mathrm{% phys}}}\left( \sum_{i=0}^3a_i^{VV}x^i+C^2x^3\widehat{B}(x)\right) \\ &&-\frac{10}9\left( \frac{M_{\mathrm{phys}}}{4\pi F_\pi }\right) ^2\left[ \left( \sigma _{RV}(\sigma _{RV}+2\sigma _V)+4\sigma _V^2\right) (x-1)^2x% \widehat{J}(x)\right] \\ \sigma _{VV}^L(s)^r &=&\frac 1\pi \frac{\Gamma _{\mathrm{phys}}}{M_{\mathrm{% phys}}}\sum_{i=0}^3b_i^{VV}x^i \\ \sigma _{RR}^L(s)^r &=&\frac 1\pi \frac{\Gamma _{\mathrm{phys}}}{M_{\mathrm{% phys}}}\left( \sum_{i=0}^3a_i^{VV}x^i+(1-C)^2x^3\widehat{B}(x)\right) \\ &&-\frac{10}9\left( \frac{M_{\mathrm{phys}}}{4\pi F_\pi }\right) ^2\left[ (4d_3^2(x+1)^2-2d_3\sigma _{RV}(x+1)+\sigma _{RV}^2)(x-1)^2\widehat{J}(x)% \right] \\ \sigma _{RR}^T(s)^r &=&\frac 1\pi \frac{\Gamma _{\mathrm{phys}}}{M_{\mathrm{% phys}}}\sum_{i=0}^3b_i^{RR}x^i+\frac 59\left( \frac{M_{\mathrm{phys}}}{4\pi F_\pi }\right) ^2\left[ (8d_3^2-4\sigma _{RV}d_3+2\sigma _{RV}^2\right. \\ &&\left. +(4d_3^2-2\sigma _{RV}d_3+\sigma _{RV}^2+24d_3d_4+4d_4^2-6\sigma _{RV}d_4)x+8d_4^2x^2)(x-1)^2\widehat{J}(x)\right] .\end{aligned}$$ and $$\frac 1\pi \frac{\Gamma _{\mathrm{phys}}}{M_{\mathrm{phys}}}C^2=\frac 12g_V^2\left( \frac{M_{\mathrm{phys}}}{F_\pi }\right) ^2M_{\mathrm{phys}% }\left( \frac{M_{\mathrm{phys}}}{4\pi F_\pi }\right) ^2$$ and the other parameters are of natural size $O(1)$ with the constraint $$\sum_{i=0}^3(a_i^{RR}+a_i^{VV})-2\sum_{i=0}^2a_i^{RV}=\sum_{i=0}^4a_i=1.$$ Note on the counterterms ------------------------ Let us note, that the counterterm Lagrangians (\[ctV\]), (\[ctR\]) and (\[ctRV\]) might be further simplified using the leading order equations of motion (EOM) in order to eliminate the terms with more then two derivatives as it has been done e.g. in [@SS2]. However, this does not mean, that we do not need to introduce such counterterms at all. As we have proved by means of the above explicit calculations, without the higher derivative counterterms (or equivalently without the couterterms proportional to the EOM) we would not have the off-shell self-energies finite. In fact, the infinities originating in the missing EOM-proportional counterterms are not always dangerous. Note e.g., that such infinities are in fact harmless, provided we restrict our treatment to strict one-loop contribution to the GF of quark bilinears or to the corresponding on-shell S-matrix elements. Namely, in this case, the one-loop generating functional of the GF is obtained by means of the Gaussian functional integration of the quantum fluctuations around the solution of the lowest order EOM. As a result, the EOM can be safely used to simplify the infinite part of the one-loop generating functional. On the strict one-loop level the infinite parts of the self-energy subgraphs corresponding to the missing EOM-proportional counterterms cancel with similar infinities stemming from the vertex corrections. Nevertheless, already at the one-loop level these counterterms might be necessary under some conditions. Namely, near the resonance poles we can (and in fact have to) go beyond the strict one-loop expansion e.g. by means of the Dyson resumation of the one-loop self-energy contributions to the propagator. This will generally destroy such a compensation of infinities. This is the reason why we keep the counterterm Lagrangian in the general form (\[ctV\]), (\[ctR\]) and (\[ctRV\]). From self-energies to propagators\[Section\_propagators\] ========================================================= In the previous sections we have given the explicit form of the self-energies in a given approximation within all three formalisms for the description of the spin-1 resonances. Here we would like to discuss interpretation of these results and the construction of the corresponding propagators. We will concentrate on the most frequently used antisymmetric tensor representation, where all the characteristic features of other approaches are visible without unsubstantial technical complications. The remaining two cases can be discussed along the same lines with similar results. Let us remind the form of the self-energies for the antisymmetric tensor case $$\begin{aligned} {\sigma }_{L}^{r}(x) &=&\frac{1}{\pi }\frac{\Gamma _{\mathrm{phys}}}{M_{% \mathrm{phys}}}\left[ 1-x^{2}\widehat{B}(x)+\sum_{i=1}^{3}a_{i}(x^{i}-1)% \right] -\frac{40}{9}\left( \frac{M_{\mathrm{phys}}}{4\pi F_{\pi }}\right) ^{2}d_{3}^{2}(x^{2}-1)^{2}\widehat{J}(x) \label{selfL} \\ {\sigma }_{T}^{r}(x) &=&\frac{1}{\pi }\frac{\Gamma _{\mathrm{phys}}}{M_{% \mathrm{phys}}}\sum_{i=0}^{3}b_{i}x^{i}+\frac{20}{9}\left( \frac{M_{\mathrm{% phys}}}{4\pi F_{\pi }}\right) ^{2}d_{3}^{2}\left( 2+(1+6\gamma +\gamma ^{2})x+2\gamma ^{2}x^{2}\right) (x-1)^{2}\widehat{J}(x),\nonumber \\ \label{selfT}\end{aligned}$$ where $d_{3}$ is given by (\[d\_3\]) and where we have already re-parametrized the general result in terms of the parameters of the perturbative solution of the pole equation in the $1^{--}$ channel (which we have identified with the original degree of freedom). In doing that we have tacitly assumed the validity of the general relation between the self-energies and the propagator (\[tensor\_Delta\_2\]). The equations determining the additional poles of the propagators are then $$\begin{aligned} f_{L}(x) &\equiv &x-1-{\sigma }_{L}^{r}(x)=0 \label{poleL} \\ f_{T}(x) &\equiv &1+{\sigma }_{T}^{r}(x)=0. \label{poleT}\end{aligned}$$ In what follows we shall discuss these equations in more detail. We will find a lower and upper bound on the number of their solutions and give a proof, that the corresponding lover bounds are greater than one on both sheets. We will also briefly discuss the compatibility of the relation (\[tensor\_Delta\_2\]) with the Källén-Lehman representation and show, that at least one of the roots of (\[poleL\]) and (\[poleT\]) corresponds inevitably either to the negative norm ghost or the tachyon. The number of poles using Argument principle -------------------------------------------- Let us first briefly discuss a determination of the number of solution of the equations (\[poleL\]) and (\[poleT\]). This can be made using the theorem known as Argument principle (see e.g. [@Ablowitz]). According to this theorem, for a meromorphic function $f(z)$ with no zeros or poles on a simple closed contour $C$, the difference between the number of zeros $N$ and poles $P$ (counted according to their multiplicity) inside $C$ is given as $$N-P=\frac{1}{2\pi}[\arg f(z)]_{C}.$$ Here $[\arg f(z)]_{C}$ is the change of the argument of $f(z)$ along $C$. Using this theorem we will show, that in both cases (\[poleL\]) and (\[poleT\]) there is a nonzero lower bound on the number of solutions on the first and the second sheet, which correspond to the poles of the propagator (\[tensor\_Delta\_2\]). We will also give conditions for the saturation of these lower bounds. Let us start with (\[poleT\]). The left hand side of the pole equation $% f_{T}(z)=1+{\sigma }_{T}^{r}(z)$ is analytic on the first sheet (and meromorphic on the second sheet) of the cut complex plane with cut from $z=1$ to $z=+\infty $. Let us choose contour $C=C_{+}+C_{R}-C_{-}+C_{\varepsilon }$ which is usually used for the proof of the dispersive representation for the self-energy, namely the one consisting of the infinitesimal circle $% C_{\varepsilon }$ encircling the point $z=1$ clockwise, two straight lines $% C_{\pm }$ infinitesimally above and bellow the real axis going from $z=1$ to $z=R$ and a circle $C_{R}$ corresponding to $z=R\,\mathrm{e}^{\mathrm{i}% \theta }$, $0<\theta <2\pi $, and take the limit with $\varepsilon \rightarrow 0$, $R\rightarrow \infty $ in the end. According to the argument principle, the total change of the phase of $\ $ the function $% f_{T}^{I,II}(z)$ along this contour gives the number of zeros (with their multiplicities) $n^{I}$ of $f(z)$ on the first sheet and $n^{II}-2$, where $% n^{II}$ is the number of zeros of $f(z)$ on the second sheet (note that $% f_{T}^{II}(z)$ has pole of the second order at $z=0$) lying inside the contour $C$, i.e. $$\begin{aligned} n^{I} &=&\frac{1}{2\pi }[\arg f_{T}^{I}(z)]_{C} \\ n^{II} &=&\frac{1}{2\pi }[\arg f_{T}^{II}(z)]_{C}+2.\end{aligned}$$ Let us assume the contour $C_{\varepsilon }$ first. Suppose that $x=1$ is not a solution of the equation $f_{T}(z)=0$. As a consequence, $[\arg f_{T}^{I,II}(z)]_{C_{\varepsilon }}$ vanishes [^16]. On the contour $C_R$, [[i.e.]{}]{} for $% z=R\,\mathrm{e}^{\mathrm{i}\theta }$ we get for $b_{3}\neq 0$$$f_{T}^{I,II}(R\,\mathrm{e}^{\mathrm{i}\theta })=R^{3}\mathrm{e}^{3\mathrm{i}% \theta }\left( \frac{1}{\pi }\frac{\Gamma _{\mathrm{phys}}}{M_{\mathrm{phys}}% }b_{3}+\frac{20}{9}\left( \frac{M_{\mathrm{phys}}}{4\pi F_{\pi }}\right) ^{2}d_{3}^{2}2\gamma ^{2}\left[ 1-\ln R+\mathrm{i}(2\pi -\theta \mp \pi )% \right] +O\left( \frac{1}{R},\frac{\ln R}{R}\right) \right)$$and therefore, for $R\rightarrow \infty $, $[\arg f_{T}^{I,II}(z)]_{C_{R}}\rightarrow 6\pi $. The same is valid also for $% b_{3}=0$ with $\gamma \neq 0$. However, for $b_{3}=\gamma =0$ we get $$f_{T}^{I,II}(R\,\mathrm{e}^{\mathrm{i}\theta })=R^{2}\mathrm{e}^{2\mathrm{i}% \theta }\left( \frac{1}{\pi }\frac{\Gamma _{\mathrm{phys}}}{M_{\mathrm{phys}}% }b_{2}+\frac{20}{9}\left( \frac{M_{\mathrm{phys}}}{4\pi F_{\pi }}\right) ^{2}d_{3}^{2}\left[ 1-\ln R+\mathrm{i}(2\pi -\theta \mp \pi )\right] +O\left( \frac{1}{R},\frac{\ln R}{R}\right) \right) .$$In this case $[\arg f_{T}^{I,II}(z)]_{C_{R}}\rightarrow 4\pi $ and because $% d_{3}\neq 0$ (unless we are in a conflict with OPE for the tree level $VVP$ correlator[^17]), this gives also the lower bound for $[\arg f_{T}^{I,II}(z)]_{C_{R}}$. Finally let us discuss the lines $C_{\pm}$. Because $\mathrm{% Im}$ $f_{T}^{I}(x\pm \mathrm{i}0)=\mathrm{Im}{\sigma }_{T}^{r}(x\pm \mathrm{i% }0)\gtrless 0$ (and $f_{T}^{I}$ is real analytic), $\mathrm{Im}$ $% f_{T}^{II}(x\pm \mathrm{i}0)>0$ for $x>1$, and $\mathrm{Re}f_{T}^{I,II}(R\pm \mathrm{i}0\,)\rightarrow -\infty $ for $R\rightarrow \infty $, we can easily conclude that in this limit $[\arg f_{T}^{I,II}(z)]_{C_{+}}=0$ unless $f_{T}^{I,II}(1)>0$, in the latter case $[\arg f_{T}^{I,II}(z)]_{C_{+}}=\pi $ and in both cases $[\arg f_{T}^{I,II}(z)]_{C_{-}}=\pm \lbrack \arg f_{T}^{I,II}(z)]_{C_{+}}$. Putting all pieces together we get under the assumption $f_{T}^{I,II}(1)\neq 0$ the following bound $$\lbrack \arg f_{T}^{I,II}(z)]_{C}\geq 4\pi$$and therefore for the number of zeros in the cut complex plane we get$$\begin{aligned} 2 &\leq &n^{I}\leq 4 \\ 4 &\leq &n^{II}\leq 5\end{aligned}$$where the lower bound is saturated for $f_{T}^{I,II}(1)<0$, $b_{3}=\gamma =0$ and the upper bound for $f_{T}^{I,II}(1)>0$ and either $b_{3}\neq 0$ or $% \gamma \neq 0$. For $f_{T}^{I,II}(1)=0$ (provided we include also this zero with its multiplicity into $n^{I,II}$) the these bounds are valid too[^18]. An analogous simple analysis for $f_{L}(z)=z-1-$ ${\sigma }_{L}^{r}(z)$ in the cut complex plane with the cut from $z=0$ to $z=+\infty $ gives [^19] for $% f_{L}^{I,II}(0)\neq 0$ $$\begin{aligned} 3 &\leq &n^{I}\leq 4 \\ n^{II} &=&5\end{aligned}$$where the lower bounds are saturated for $f_{L}^{I,II}(0)<0$ otherwise $n^{I}$ equals to the upper bound. We can not therefore avoid in any way the generation of the additional poles (some of them might even be of the higher order) in both $1^{--}$ and $1^{+-% \text{ }}$channels of the propagator only by means of an appropriate choice of the free parameters $a_{i}$, $b_{i}$ and $\gamma $. The minimal number of the additional poles (with their orders) on the second sheet is the same for both channels (note that, one pole in $1^{--}$ channel has to correspond to the perturbative solution describing the original degrees of freedom we have started with). The conditions for the saturation of the lower bounds in the $% 1^{--}$ and $1^{+-}$ channels are $$-\frac{1}{\pi }\frac{\Gamma _{\mathrm{phys}}}{M_{\mathrm{phys}}}\left[ 1-\sum_{i=1}^{3}a_{i}\right] +\frac{20}{9}\left( \frac{M_{\mathrm{phys}}}{% 4\pi F_{\pi }}\right) ^{2}d_{3}^{2}<1$$and $$\begin{aligned} b_{3} &=&\gamma =0 \\ -\frac{1}{\pi }\frac{\Gamma _{\mathrm{phys}}}{M_{\mathrm{phys}}}% \sum_{i=0}^{3}b_{i} &>&1\end{aligned}$$respectively. Note that, while the first condition is in accord with the large $N_{C}$ counting, the last one is not. Let us now discuss the physical relevance of such additional poles. The Källén-Lehman representation and nature of the poles -------------------------------------------------------- In this subsection, we will show that the the propagator ([tensor\_Delta\_2]{}) with self-energies (\[selfL\]) and (\[selfT\]) is incompatible with the Källén-Lehman representation with the positive spectral function. Moreover, at least one of the solutions of both equations (\[poleL\]) and (\[poleT\]) is pathological and corresponds to the negative norm ghost or the tachyonic pole. Let us first briefly remind the Källén-Lehman representation of the antisymmetric tensor field propagator. According to the Lorentz structure we can write the following spectral representation of the full propagator (modulo generally non-covariant contact terms) $$\Delta _{\mu \nu \alpha \beta }(p)=p^{2}\Pi _{\mu \nu \alpha \beta }^{T}(p)\Delta _{T}(p^{2})-p^{2}\Pi _{\mu \nu \alpha \beta }^{L}(p)\Delta _{L}(p^{2})+\Delta _{\mu \nu \alpha \beta }^{\mathrm{contact}}(p)$$where (up to the necessary subtractions) $$\Delta _{L,T}(p^{2})=\int_{0}^{\infty }\mathrm{d}\mu ^{2}\frac{\rho _{L,T}(\mu ^{2})}{p^{2}-\mu ^{2}+\mathrm{i}0}$$and where the spectral functions $\rho _{T,L}(p^{2})$ are given in terms of the sum over the intermediate states as $$(2\pi )^{-3}\theta (p^{0})\left[ \rho _{T}(p^{2})p^{2}\Pi _{\mu \nu \alpha \beta }^{T}(p)-\rho _{L}(p^{2})p^{2}\Pi _{\mu \nu \alpha \beta }^{L}(p)% \right] =\sum\limits_{N}\delta ^{(4)}(p-p_{N})\langle 0\|R_{\mu \nu }(0)\|N\rangle \langle N\|R_{\alpha \beta }(0)\|0\rangle . \label{spectral}$$Note that, in the above formula we assume all the states $\|N\rangle $ to have a positive norm; the spectral functions $\rho _{L,T}(p^{2})$ are then positive (for the proof see the Appendix \[Appendix\_positivity\]). For the one particle spin-one bound stated states $\|p,\lambda \rangle $ with mass $M$ either $$\langle 0\|R_{\mu \nu }(0)\|p,\lambda \rangle =Z_{L}^{1/2}u_{\mu \nu }^{(\lambda )}(p)$$or $$\langle 0\|R_{\mu \nu }(0)\|p,\lambda \rangle =Z_{T}^{1/2}w_{\mu \nu }^{(\lambda )}(p)$$according to its parity (cf. (\[u\_function\]) and (\[w\_function\])). Therefore (using the formulae from the Appendix \[Appendix\_positivity\]), the corresponding one particle contribution to $\rho _{L,T}(\mu ^{2})$ is $$\rho _{L,T}^{\text{one-particle}}(\mu ^{2})=\frac{2}{M^{2}}Z_{L,T}\delta (\mu ^{2}-M^{2}).$$Positivity $\rho _{L,T}(\mu ^{2})$ implies $Z_{L,T}>0$ in the above one-particle contributions. For free fields with mass $M$ we get $$\begin{aligned} \rho _{L}^{free}(\mu ^{2}) &=&\frac{2}{M^{2}}\left( \delta (\mu ^{2}-M^{2})-\delta (\mu ^{2})\right) \notag \\ \rho _{T}^{free}(\mu ^{2}) &=&\frac{2}{M^{2}}\delta (\mu ^{2}). \label{free_spectral}\end{aligned}$$Note the kinematical poles in $\Delta _{L,T}(p^{2})$ at $\ p^{2}=0$, which do not correspond to any one-particle intermediate state and which sum up to the contact terms of the form $$\Delta _{\mu \nu \alpha \beta }^{free,\mathrm{contact}}(p)=\frac{1}{M^{2}}% \left( g_{\mu \alpha }g_{\beta \nu }-g_{\mu \beta }g_{\nu \alpha }\right) .$$ Let us now define for complex $z$ by means of the analytic continuation (up to the possible subtractions) $$\Delta _{L,T}(z)=\int_{0}^{\infty }\mathrm{d}s\frac{\rho _{L,T}(s)}{s-z}~,~~~ \label{dispersiv_kallan}$$Within the perturbation theory however, the primary quantities are the self-energies, which we define as (cf. (\[tensor\_Delta\_2\])) $$\begin{aligned} \Delta _{T}(s) &=&\frac{1}{s}\frac{2}{M^{2}+\Sigma _{T}(s)} \notag \\ \Delta _{L}(s) &=&\frac{1}{s}\frac{2}{s-M^{2}-\Sigma _{L}(s^{2})}. \label{simas_definition}\end{aligned}$$The poles at $s=0$ are of the kinematical origin and in analogy with the free propagator they sum up into the contact terms provided $\Sigma _{T}(0)=\Sigma _{L}(0)$. The formulae (\[simas\_definition\]) can be understood as the Dyson re-summation of the $1PI$ self-energy insertions to the propagator or as an inversion of the $1PI$ two-point function. Due to the positivity of $\rho _{L,T}(s)$, we get for the imaginary parts of $\ \Sigma _{L,T}$ the following positivity (negativity) constraints: $$\begin{aligned} \mathrm{Im}\Sigma _{L}(s+i0) &=&\frac{1}{2}\theta (s)s~\mathrm{Im}\Delta _{L}(s+i0)\|s-M^{2}-\Sigma _{L}(s+i0)\|^{2}\leq 0 \notag \\ \mathrm{Im}\Sigma _{T}(s+i0) &=&-\frac{1}{2}\theta (s)s~\mathrm{Im}\Delta _{T}(s+i0)\|M^{2}+\Sigma _{L}(s+i0)\|^{2}\geq 0. \label{positiv_negativ}\end{aligned}$$ Let us now turn to the $R\chi T$ -like effective theories and try to demonstrate their possible limitations. In such a framework the self-energies $\Sigma _{L,T}$ are given by a sum of the 1PI graphs organized according to some counting rule (for $R\chi T$ e.g. by the index $% i_{\Gamma \text{ }}$, cf. (\[power\_counting\])). Up to a fixed given order (which we assume to be fixed from now on) we have the asymptotic behavior $% \Sigma _{L,T}(z)=O(z^{n}\ln ^{k}z))$ for $z\rightarrow -\infty $ according to the Weinberg theorem. Here $n$ corresponds to the maximal degree of divergence of the contributing (sub)graphs and therefore, it grows with the number of loops as well as with the index of the vertices (cf. ([index\_O]{}) ). Such a grow of the inverse propagator is known to lead to problems. Suppose e.g., that we can organize the result of the calculation of the 1PI graphs in the form of a dispersive representation for the functions $\Sigma _{L,T}(z)$ on the first sheet[^20] $$\Sigma _{L,T}^{I}(z)=P_{n}^{L,T}(z)+\frac{Q_{n+1}^{L,T}(z)}{\pi }% \int_{x_{t}}^{\infty }\frac{dx}{Q_{n+1}^{L,T}(x)}\frac{\mathrm{Im}\Sigma _{L,T}(x+i0)}{x-z} \label{dispersiv_sigma}$$where $x_{t}$ $\geq 0$ is the lowest multi-particle threshold, $% P_{n}^{L,T}(z)$ and $Q_{n+1}^{L,T}(z)$ (we suppose $Q_{n+1}^{L,T}(x)>0$ for $% x>0$) are renormalization scale independent real polynomials of the order $n$ and $n+1$ respectively and $\mathrm{Im}\Sigma _{L,T}(x+i0)$ can be obtained using the Cutkosky rules. The contributions to $P_{n}^{L,T}(z)$ stem from the counterterms necessary to renormalize the superficial divergences of the contributing 1PI graphs as well as from the loops ($\chi $logs)[^21]. As a consequence, the functions $z^{k}\Delta _{L,T}(z)$ where $0\leq k\leq n$ and where $\Delta _{L,T}(s)$ is naively defined by (\[simas\_definition\]) are analytic (up to the finite number of complex poles $z_{j}$ generally different for $\Delta _{L}$ and $\Delta _{T}$ and a kinematical pole at $z=0$ - see bellow) in the cut complex plane. As far as the number of poles $z_{j}$ are concerned, provided $\mathrm{Im}\Sigma _{L,T}(x+i0)\lessgtr 0$ as suggested by (\[positiv\_negativ\]), we can almost literally repeat the analysis from the previous subsection based on the argument principle. The change of a phase of the inverse propagator along the path $C_{R}$ is now $% [\arg \Delta ^{-1}_{L,T}(z)]_{C_{R}}\rightarrow 2\pi n$ (for $R\rightarrow \infty $), while the absolute value of the $[\arg \Delta ^{-1}_{L,T}(z)]_{C_{\pm }}$ is bounded by $\pi ~\ $due to the positivity (negativity) of $\mathrm{Im}\Sigma _{L,T}(x\pm i0)$. Provided $\Delta ^{-1}_{L,T}(x_{t})\neq 0$, we can therefore conclude $$\begin{aligned} n-1 &\leq &n^{I} \\ n &\leq &n^{II}-p^{II}\end{aligned}$$where $n^{I,II}$ is the number of the solutions of the equation $\Delta ^{-1}_{L,T}(z)=0$ on the first and second sheet respectively and $p^{II}$ is the number of the poles (weighted with their order) of $% \Sigma _{L,T}(z)$ on the second sheet[^22]. Therefore, because $z^{k}\Delta _{L,T}(z)=O(z^{k-n-1})$, we can write for $% 0<k\leq n$ an unsubtracted dispersion relation (cf. (\[dispersiv\_kallan\]), we will omit the subscript $L,~T$ in the following formulae for brevity and write simply $\Delta (z)$, $\rho (s)$ etc.) $$z^{k}\Delta (z)=\sum_{j>0}\frac{R_{j}z_{j}^{k}}{z-z_{j}}+\frac{1}{\pi }% \int_{x_{t}}^{\infty }\mathrm{d}x\frac{x^{k}\mathrm{disc}\Delta (x)}{x-z}$$or $$\Delta (z)=\frac{1}{z^{k}}\sum_{j>0}\frac{R_{j}z_{j}^{k}}{z-z_{j}}+\frac{1}{% \pi z^{k}}\int_{x_{t}}^{\infty }\mathrm{d}x\frac{x^{k}\mathrm{disc}\Delta (x)% }{x-z}.$$and for $k=0$ (note the kinematical pole at $z=0$) $$\Delta (z)=\frac{R_{0}}{z}+\sum_{j>0}\frac{R_{j}}{z-z_{j}}+\frac{1}{\pi }% \int_{x_{t}}^{\infty }\mathrm{d}x\frac{\mathrm{disc}\Delta (x)}{x-z}$$Due to the asymptotic fall off $\Delta (z)=O(z^{-n-1})$ the discontinuity $% \mathrm{disc}\Delta (x)$ has to satisfy the following sum rules $$\begin{aligned} -\frac{1}{\pi }\int_{x_{t}}^{\infty }\mathrm{d}xx^{k}\mathrm{disc}\Delta (x)+\sum_{j}R_{j}z_{j}^{k} &=&0,~~~0<k\leq n-1. \label{sum_rule} \\ -\frac{1}{\pi }\int_{x_{t}}^{\infty }\mathrm{d}x\mathrm{disc}\Delta (x)+\sum_{j}R_{j}+R_{0} &=&0\end{aligned}$$Suppose on the other hand validity of the dispersive representation ([dispersiv\_kallan]{}). Then all the poles have to be real, and we can identify $$\rho (s)=-\frac{1}{\pi }\mathrm{disc}\Delta (s)+\sum_{j}R_{j}\delta (s-z_{j})+R_{0}\delta (s).$$ However, the sum rules (\[sum\_rule\]) are generally inconsistent with the spectral representation (\[dispersiv\_kallan\]). The validity of some of them might require either an appearance of the states with the negative norm in the spectrum, i.e. we are in a conflict with the positivity of the spectral function $\rho (s)\geq 0$ or an appearance of physically non-acceptable tachyon poles leading to the acausality. For instance, suppose $\mathrm{disc}\Delta (s)\leq 0$, then for $R_{0}\geq 0$ at least one of the poles has to correspond to a negative norm one-particle state (ghost). On the other hand, for $\mathrm{disc}\Delta (s)\leq 0$, $R_{j}>0$ we can still satisfy the $k=0$ sum rule with negative $R_{0}$, however, from the $k=1$ sum rule we need at least one pole to be negative (tachyon) (in this case, however, the sum rules with even $k$ cannot be satisfied)[^23]. These considerations illustrate the known fact that the representation of the propagator based on the formulas (\[simas\_definition\]) has limited range of validity within the fixed order of the perturbation theory and has to be taken with some care. One point of view might be that the range of applicability of the formulae (\[simas\_definition\]) is $\|z\|<\Lambda _{\mathrm{max}}=\mathrm{min}% \{\|z_{j}\|\}$ where $\{z_{j}\}$ is the set of unwanted poles. Provided there exists a genuine expansion parameter $\alpha $ applicable to the organization of the perturbative series, according to which $\Sigma _{L,T}$ $% =\sum_{i>0}\alpha ^{i}\Sigma _{L,T}^{(i)}$ (e.g. expanding in powers of $\alpha =1/N_{C}$ in $R\chi T$), one can expect the additional (generally pathological) poles of $\Delta _{L,T}(z)$ to decouple ( i.e. $\Lambda _{\mathrm{max}}\rightarrow \infty $ for $\alpha \rightarrow 0$). In such a case we could argue that they are in fact harmless. However, the size of $% \Lambda _{\mathrm{max}}$ for actual value of $\ \alpha $ need not to be far from $M$ which could invalidate this approach to the theory in the region for which it was originally designed. Alternatively, instead of using the (partial) Dyson re-summation, we can expand directly $\Delta _{\mu \nu \alpha \beta }(p)$ to the fixed finite order $n$ which leads to $$\begin{aligned} \Delta _{L}(s) &=&\frac{2}{s}\left( \frac{1}{s-M^{2}}+\alpha \frac{1}{s-M^{2}% }\Sigma _{L}^{(1)}(s^{2})\frac{1}{s-M^{2}}+\ldots +\alpha ^{n}\frac{1}{% s-M^{2}}\Sigma _{L}^{(n)}(s^{2})\frac{1}{s-M^{2}}\right) \\ \Delta _{T}(s) &=&\frac{2}{s}\left( \frac{1}{M^{2}}+\alpha \frac{1}{M^{2}}% \Sigma _{T}^{(1)}(s^{2})\frac{1}{M^{2}}+\ldots +\alpha ^{n}\frac{1}{M^{2}}% \Sigma _{T}^{(n)}(s^{2})\frac{1}{M^{2}}\right) .\end{aligned}$$This expansion (which does not give rise to the additional poles of the propagator) might be useful for $s\ll M^{2}$, however, in this case a higher-order pole at $s=M^{2}$ is generated, which is not correct physically in the resonance region $s\sim M^{2}$. Here we instead expect a single pole on the second sheet of $\Delta _{L}(z)$, where $z=M_{\mathrm{phys}}^{2}-iM_{% \mathrm{phys}}\Gamma _{\mathrm{phys}}$ (where the mass $M_{\mathrm{phys}% }^{2}=M^{2}+O(\alpha )$ and the width $\Gamma _{\mathrm{phys}}=O(\alpha )$) corresponding to the original degree of freedom of the free Lagrangian. Therefore, the Dyson re-summation (i.e. the application of the formulae (\[simas\_definition\])) suplemented with some other more sophisticated approaches (e.g. the Redmond and Bogolyubov method [Redmond, Bogolyubov]{} consisting of the subtraction[^24] of the additional unwanted poles from the propagator, or diagonal Padé approximation method [@Lambert1973]) seems to be inevitable for $s\sim M^{2}$. However, in the concrete case of our calculations of the antisymmetric tensor field propagator, the plain Dyson re-summation might produce various types of poles some of which we illustrate in the next subsection. Examples of the poles --------------------- The additional poles of the propagator can have different nature. Let us assume the $1^{--}$ channel first. By construction for any values of the constants $a_i$ we have one pole on the second sheet (which is directly accessible from the physical sheet by means of the crossing of the cut for $0<z<1$) which corresponds to the physical resonance ($\rho$ meson) we have started with at the tree level. On the first sheet we get then a typical resonance peak. These two structures are illustrated in the Fig. \[poleL\], where the square of the modulus of the propagator function, namely [i.e.]{} $\|z-1-\sigma_L(z)\|^{-2}$, is plotted[^25] on the first and the second sheet for $a_i=0$. In this case, no additional pole appears in the region of assumed applicability of $R\chi T$. However, for another set of parameters we can get also pathological poles not far from this region ([e.g.]{} tachyon as it is illustrated in analogous Fig. \[poleLT\], now for $a_0=a_1=a_2=10$, $a_3=0$). In the $1^{+-}$ channel, there is no tree-level pole in the propagator. The structure of the poles of the Dyson resumed propagator is strongly dependent on the parameters $b_i$ and $\gamma$ in this case. Let us illustrate this briefly. Note e.g. that, the equation (\[poleT\]) can have (exact) solution $x=1$ on the first sheet provided the parameters $b_{i}$ satisfy the following constraint $$\sum_{i=0}^{3}b_{i}=-\pi \frac{M_{\mathrm{phys}}}{\Gamma _{\mathrm{phys}}}% \sim -16 \label{bconstraint1}$$where the numerical estimate corresponds to $(M_{\mathrm{phys}},\Gamma _{% \mathrm{phys}})\sim (M_{\rho },\Gamma _{\rho })$. In order to interpret this solution as a $1^{+-\text{ }}$bound state pole we need the residuum $Z_{A}$ at this pole to be positive,* i.e.* $$Z_{A}^{-1}={\sigma }_{T}^{r}(x)^{^{\prime }}\|_{x=1}=\frac{1}{\pi }\frac{% \Gamma _{\mathrm{phys}}}{M_{\mathrm{phys}}}\sum_{j=1}^{3}jb_{j}>0 \label{bconstraint2}$$otherwise the pole is a negative norm ghost state. Of course, from the phenomenological point of view, both these possibilities are meaningless. Note also that, the constraints (\[bconstraint1\]) and (\[bconstraint2\]) require unnatural large values of the parameters $b_{i}$ and it is also in a conflict with the large $N_{C}$ counting[^26]. For $\gamma =0$, a pathological tachyonic solution of (\[poleT\]) exists for $x=-2$ provided $$\sum_{i=0}^{3}(-2)^{i}b_{i}=-\pi \frac{M_{\mathrm{phys}}}{\Gamma _{\mathrm{% phys}}}$$which might be satisfied with more reasonable values of the parameters $% b_{i} $ than in the previous case. More generally, we can have pathological poles $x=x_{\gamma }$ where $x_{\gamma }$ is a solution of $$2+(1+6\gamma +\gamma ^{2})x_{\gamma }+2\gamma ^{2}x_{\gamma }^{2}=0.$$This $x_{\gamma }$ is a pole of the propagator on both physical and unphysical sheets under the conditions that the following constraint on the parameters $b_{i}$$$\sum_{i=0}^{3}x_{\gamma }^{i}b_{i}=-\pi \frac{M_{\mathrm{phys}}}{\Gamma _{% \mathrm{phys}}}$$is satisfied. Here $x_{\gamma }$ is real (and negative) for $\|\gamma +5\|>2% \sqrt{6}$ and it represents therefore a physically unacceptable tachyonic pole. Outside of this region of $\gamma $ we get pair of complex conjugate poles on the physical sheet with $\mathrm{Re}x_{\gamma }>0$ when $-3+2\sqrt{2% }>\gamma >-3-2\sqrt{2}$. However, we can easily get a more realistic situation and ensure that the position of the complex pole $z_{R}=x_{R}-\mathrm{i}y_{R}$ on the second sheet in the $1^{+-\text{ }}$channel corresponds e.g. to a resonance $% b_{1}(1235)$. In this case, two conditions for $b_{i}$, and $\gamma $ have to be satisfied, which correspond to the real and imaginary part of the pole equation $1+{\sigma }_{T}^{r}(z_{R})=0$. This allows us to eliminate two of the five independent parameters in favor of the mass and the width of the desired resonance[^27]. However, it might be difficult to eliminate additional pathological poles in the assumed region of applicability of $R\chi T$. We illustrate this in the Fig. \[pole\_b1\], where the the square of the modulus of the propagator function $\|1+\sigma_T(z)\|^{-2}$ on the first and the second sheet for $b_0=-2.16$, $b_1=-3.66$, $b_2=-4.45$, $b_3=1.47$ and $\gamma=0$ is plotted on the first and the second sheet. In addition to the desired $b_1(1235)$ pole on the second sheet we get also four additional poles on the second sheet which is difficult to interpret physically as well as two additional structures the first sheet one of which can be interpreted as an tachyonic pole. In general it is not so straightforward to formulate the conditions for $a_{i}$, $b_{i}$, and $\gamma $ under which there are no additional poles on the real axis in the antisymmetric tensor field propagator. Because $\mathrm{Im}\sigma _{L}^{r}(x+i0)$ is negative for $x>0$ (and similarly $\mathrm{Im}\sigma _{T}^{r}(x+i0)$ is positive for $x>1$), we can clearly conclude, that there is no real pole in these regions on the first and the second sheet. As far as the regions of $x<0$ (for $\sigma _{L}^{r}$) and $x<1$ (for $\sigma _{T}^{r}$) are concerned, we can proceed as follows. Note, that we can write for the functions $\widehat{J}(x)$ and $% \widehat{B}(x)$ the following dispersive representation $$\begin{aligned} \widehat{B}(x) &=&2+x+(x+1)^{2}\int_{0}^{\infty }\frac{dx^{^{\prime }}}{% (x^{^{\prime }}+1)^{2}}\frac{1}{x^{^{\prime }}-x} \\ &\equiv &2+x+b(x) \\ \widehat{J}(x) &=&\int_{1}^{\infty }\frac{dx^{^{\prime }}}{x^{^{\prime }}}% \left( 1-\frac{1}{x^{^{\prime }}}\right) \frac{1}{x^{^{\prime }}-x},\end{aligned}$$from which the representation (\[dispersiv\_sigma\]) for $\Sigma _{L}$ with desired properties easily follows. From this we can see that on the first sheet $b(x)$, $\widehat{J}(x)>0$ for $x<0$ and $x<1$ respectively. Similarly, for $\Sigma _{T}$ we can write $$\left( 2+(1+6\gamma +\gamma ^{2})x+2\gamma ^{2}x^{2}\right) \widehat{J}(x)=1+% \frac{1}{6}\left( 3\gamma ^{2}+18\gamma +5\right) x+j(x)$$where $$j(x)=x^{2}\int_{1}^{\infty }\frac{dx^{^{\prime }}}{x^{^{\prime }3}}\left( 1-% \frac{1}{x^{^{\prime }}}\right) \left( 2+(1+6\gamma +\gamma ^{2})x^{^{\prime }}+2\gamma ^{2}x^{^{\prime }2}\right) \frac{1}{x^{^{\prime }}-x}$$and $j(x)>0$ for $x<1$. The equations (\[P\_L\]) and (\[poleT\]) have therefore the following structure $$\begin{aligned} p_{L}(x) &=&-\frac{1}{\pi }\frac{\Gamma _{\mathrm{phys}}}{M_{\mathrm{phys}}}% x^{2}\left( b(x)+2\right) -\frac{40}{9}\left( \frac{M_{\mathrm{phys}}}{4\pi F_{\pi }}\right) ^{2}d_{3}^{2}(x^{2}-1)^{2}\widehat{J}(x) \label{eq_L} \\ p_{T}(x) &=&-\frac{20}{9}\left( \frac{M_{\mathrm{phys}}}{4\pi F_{\pi }}% \right) ^{2}d_{3}^{2}(x-1)^{2}\left( j(x)+1\right), \label{eq_T}\end{aligned}$$where $p_{L,T}(x)$ are the following polynomials of the third order $$\begin{aligned} p_{L}(x) &=&x-1-\frac{1}{\pi }\frac{\Gamma _{\mathrm{phys}}}{M_{\mathrm{phys}% }}\left[ 1-x^{3}+\sum_{i=1}^{3}a_{i}(x^{i}-1)\right] =(x-1)q_{L}(x) \\ p_{T}(x) &=&1+\frac{1}{\pi }\frac{\Gamma _{\mathrm{phys}}}{M_{\mathrm{phys}}}% \sum_{i=0}^{3}b_{i}x^{i}+\frac{10}{27}\left( \frac{M_{\mathrm{phys}}}{4\pi F_{\pi }}\right) ^{2}d_{3}^{2}\left( 3\gamma ^{2}+18\gamma +5\right) x(x-1)^{2}.\end{aligned}$$where $$q_{L}(x)=1-\frac{1}{\pi }\frac{\Gamma _{\mathrm{phys}}}{M_{\mathrm{phys}}}% \left( (1+x+x^{2})(a_{3}-1)+a_{1}+a_{2}(x+1)\right)$$Because the right hand sides of the equations (\[eq\_L\]) and (\[eq\_T\]) are negative in the regions of interest, the sufficient (but not necessary) condition of the absence of the poles in these regions is $q_{L}(x)<0$ for $% x<0$ and $p_{T}(x)>0$ for $x<1$. For $q_{L}(x)$ this can be achieved in many ways, e.g. for $$\begin{aligned} a_{3} &\geq &1 \\ q_{L}(0) &=&1-\frac{1}{\pi }\frac{\Gamma _{\mathrm{phys}}}{M_{\mathrm{phys}}}% \left( a_{1}+a_{2}+a_{3}-1\right) <0 \\ q_{L}^{^{\prime }}(0) &=&-\frac{1}{\pi }\frac{\Gamma _{\mathrm{phys}}}{M_{% \mathrm{phys}}}(a_{3}+a_{2}-1)>0\end{aligned}$$i.e. $$a_{1}>\pi \frac{M_{\mathrm{phys}}}{\Gamma _{\mathrm{phys}}}% ,\,\,a_{2}<0,\,\,a_{3}\geq 1.\,\,$$Note however, that such a condition for $a_{1}$ requires unnatural value for this parameter and is in a conflict with the large $N_{C}$ counting. Similarly, the condition $p_{T}(x)>0$ can be ensured e.g. when the coefficients at the third power of $x$ vanish identically, i.e. $$b_{3}=-\frac{10}{27}\left( \frac{M_{\mathrm{phys}}}{4\pi F_{\pi }}\right) ^{2}\pi \frac{M_{\mathrm{phys}}}{\Gamma _{\mathrm{phys}}}d_{3}^{2}\left( 3\gamma ^{2}+18\gamma +5\right) ,$$the coefficients at the second power of $x$ are positive, i.e $$b_{2}>\frac{20}{27}\left( \frac{M_{\mathrm{phys}}}{4\pi F_{\pi }}\right) ^{2}\pi \frac{M_{\mathrm{phys}}}{\Gamma _{\mathrm{phys}}}d_{3}^{2}\left( 3\gamma ^{2}+18\gamma +5\right) ,$$and $$\begin{aligned} p_{T}(1) &=&1+\frac{1}{\pi }\frac{\Gamma _{\mathrm{phys}}}{M_{\mathrm{phys}}}% \sum_{i=0}^{3}b_{i}>0 \\ p_{T}^{^{\prime }}(1) &=&\frac{1}{\pi }\frac{\Gamma _{\mathrm{phys}}}{M_{% \mathrm{phys}}}\sum_{j=0}^{3}b_{j}j>0.\end{aligned}$$On the contrary to the previous case, these conditions respect the large $% N_{C}$ counting. Therefore without any detailed information about the actual value of the $a_{i}$ and $b_{i}$ it seems to be quite natural to have tachyonic pole in the $1^{--}$ channel and no bound states or tachyon poles in the $1^{+-}$ channel of the propagator. Summary and discussion\[Section\_summary\] ========================================== In this paper we have studied and illustrated various aspects of the renormalization procedure of the Resonance Chiral Theory using the spin-one resonance self-energy and the corresponding propagator as a concrete example. The explicit calculation of the one-loop self-energies within three possible formalisms for the description of the spin-one resonances, namely the Proca filed, antisymmetric tensor field and the first order formalism is the main result of our article. Because the theory is non-renormalizable and the loop corrections break the ordinary chiral power counting, we had presumed an accurence of problems of several types which have proved to be true within our explicit example. The first sort of problems concerned the technical aspects of the process of renormalization, namely the organization of the loop corrections and the counterterms and the mixing of the ordinary chiral orders by the loops. In order to organize our calculations we have proposed a self-consistent scheme for classification of the one-particle irreducible graphs $\Gamma $ and corresponding counterterms $\mathcal{O}_{i}$ which renormalize its superficial divergences. The classification is according to the indices $% i_{\Gamma }$ and $i_{\mathcal{O}_{i}}$ assigned to graph $\Gamma $ and operator $\mathcal{O}_{i}$ respectively. Though the scheme based on $i_{% \mathcal{O}}$ restricts both the chiral order of the chiral building blocs (number of derivatives and external sources) as well as the number of resonance fields in the operators in the $R\chi T$ Lagrangian at each fixed order and can be understood as a combination of the chiral and $1/N_{C}$ counting, it is however not possible to assign to $i_{\Gamma }$ a clear physical meaning connected with the infrared characteristics of the graphs $% \Gamma $. Nevertheless the scheme works at least formally and can be used for the proof of the renormalizability of $R\chi T$ to given order $% i_{\Gamma },i_{\mathcal{O}_{i}}\leq $ $i_{\max }$. We have used it at the level $i_{\max }\leq 6$ and proved that the complete set of counterterms from zero up to six derivatives is necessary to renormalize the divergences of the one-loop self-energies in the contrary to the naive expectations based on the usual chiral powercounting. The last aspect, namely that the complete set of counterterms including also those with two derivatives (i.e. the kinetic terms) is necessary, is connected to the second sort of problems. The tree level Lagrangian is constructed using just one of such a kinetic term in order to ensure the propagation of just three degrees of freedom corresponding to the spin-one particle state. If we would include all possible kinetic terms with two derivatives into the free Lagrangian, we would get (according to the formalism used) additional poles in the free propagator corresponding to the additional one-particle states some of them being necessarily either negative norm ghost or tachyon. This was the first signal of the problems with unphysical degrees of freedom connected with the one-loop corrections to the self-energies. The higher derivative kinetic terms further increase the number of these extra degrees of freedom. We have studied this feature also using the path integral representation and integrated in additional fields which appear to be responsible for the additional propagator poles. The problems with additional degrees of freedom are also connected with the well known fact that the propagator obtained by means of the Dyson re-summation of the perturbative one-particle irreducible self-enery insertions might be incompatible with the Källén-Lehman spectral representation even in the case of the renormalizable theories [@Symanzik]. As is well known, in this case tachyonic or negative norm ghost state can appear as an additional pole. Such an extra pole is usually harmless because it is very far from the energy range where the theory is applicable. In the power-counting non-renormalizable effective theories like $R\chi T$ such problems are much stronger either because of the worse UV behavior of the self-energies (which increases the number of additional poles) or because the additional pathological poles might lie near the region where the theory was assumed to be valid. The nontrivial Lorentz structure of the fields describing spin-one resonances further complicates this delineation because some of the additional poles might have different quantum numbers than the original tree-level degrees of freedom. As far as this type of poles is concerned, we have demonstrated using the path integral formalism that it can be eliminated by means of the requirement of additional protective symmetry of the interaction Lagrangian, which is an analog of $U(1)$ gauge transformation known for the Proca and Rarita-Schwinger fields. However, these symmetries are in general in conflict with chiral symmetry, though individual interaction vertices can posses such a symmetry accidentally. The results of our calculations proved to fit this general picture. Using the explicit example of the one-loop antisymmetric tensor self-energy we have shown that the Dyson re-summed propagator has always (ie. irrespectively to the actual values of the couterterm couplings) at least three additional poles on the first sheet in the $1^{--}$ channel, just five such poles on the second sheet (one of them corresponding to the original degree of freedom) and at least two additional poles on the first sheet in the $1^{+-}$ channel and at least four such poles on the second sheet. As we have seen in explicit analysis of the pole equations, without any additional information about the size of the counterterm couplings and consequently about the actual values of the renormalization scale invariant parameters entering the polynomial part of the self-energies, a rich variety of poles in the propagator is possible. Some of the poles might be unphysical (complex conjugated pairs of poles on the first sheet and tachyonic or negative norm ghosts on both sheets) and some of them even can be situated near or inside the assumed applicability region of $R\chi T$. It might be argued that the additional poles are just artifacts of the inappropriate treatment of the theory and that the one-loop one-particle irreducible self-energy insertion cannot be re-summed in order to construct a reliable approximation of the full resonance propagator. However, the mere truncation of the Dyson series keeping only first two terms (corresponding to tree-level contribution and to the strict one-loop correction to the propagator respectively) generates double poles at $s=M^{2}$ on both sheets and is therefore in contradiction with the expected analytic structure of the full propagator. Though this might be an useful approximation of the full propagator for $s\ll M^{2}$, it cannot be correct in the resonance region. Therefore provided we would like to use $R\chi T$ at one-loop also for $s\sim M^{2}$, the construction the propagator using some sort of re-summation (i.e. the Dyson one or its modifications like e.g. the Redmond and Bogolyubov procedure or Padé approximation) might be inevitable. The actual position of the additional poles (if there are any within the chosen procedure) might be then understood as a bound limiting the range of applicability of the theory. In the most optimistic scenario all the additional poles are far form the region of interest and $R\chi T$ can be treated as a consistent effective theory describing just the degrees of freedom we start with at the tree level. The less satisfactory case when only the pathological poles are far-distant, we can either abandon the theory as inconsistent or alternatively we can try to interpret the non-pathological poles as a prediction of the theory corresponding to the dynamical generation of higher resonances. Such a treatment was used in the case of scalar resonances in [@Boglione:2002vv] (see also [@Tornqvist:1995kr; @Tornqvist:1995ay]). Eventually in the case when all the additional poles lie near $s\sim M^{2}$, either the approximative construction of the propagator or one-loop $R\chi T$ itself might be problematic. Which scenario actually turns up depends on the values of the couplings in the $R\chi T$ Lagrangian. Acknowledgement {#acknowledgement .unnumbered} =============== We would like to thank A. Pich, J. J. Sanz-Cillero and M. Zdrahal for useful discussions and valuable comments on the manuscript. We also thank M. Chizhov for drawing our attention to an error in previous version of the manuscript. This work is supported in part by the Center for Particle Physics (Project no. LC 527), GAUK (Project no.6908; 114-10/258002), GACR \[202/07/P249\] and the EU Contract No. MRTN-CT-2006-035482, “FLAVIAnet”. J. T. is also supported by the U.S. Department of State (International Fulbright S&T award). Additional degrees of freedom in the path integral - the Proca field \[PI\_Proca\] ================================================================================== Suppose that the interaction Lagrangian has the form $$\mathcal{L}_{int}=\mathcal{L}_{ct}+\mathcal{L}_{int}^{^{\prime }}$$ where $\mathcal{L}_{ct}$ is the toy interaction Lagrangian (\[L\_V\_toy\]). Our aim will be to transform $Z[J]$ to the form of the path integral with all the additional degrees of freedom represented explicitly in the Lagrangian and the integration measure. In terms of the transverse and longitudinal degrees of freedom we get $$\begin{aligned} \mathcal{L}_{int}(V_{\perp }-\partial \Lambda ,J,\ldots ) &=&\mathcal{L}% _{ct}(V_{\perp }-\partial \Lambda ,J,\ldots )+\mathcal{L}_{int}^{^{\prime }}(V_{\perp }-\partial \Lambda ,J,\ldots ) \notag \\ &=&\frac{\alpha }{2}V_{\perp }^{\mu }\Box V_{\perp \mu }-\frac{\beta }{2}% (\Box \Lambda )^{2}+\frac{\gamma }{2M^{2}}(\Box V_{\perp }^{\mu })(\Box V_{\perp \mu })+\frac{\delta }{2M^{2}}(\partial _{\mu }\square \Lambda )(\partial ^{\mu }\Box \Lambda ) \notag \\ &&+\mathcal{L}_{int}^{^{\prime }}(V_{\perp }-\partial \Lambda ,J\ldots ).\end{aligned}$$ In order to lower the number of derivatives in the kinetic terms we integrate in auxiliary scalar fields $\chi $, $\rho $, $\pi $, $\sigma $ and auxiliary transverse vector field $B_{\perp \mu }$. Writing $$\exp \left( -i\int \mathrm{d}^{4}x\frac{\beta }{2}(\Box \Lambda )^{2}\right) =\int \mathcal{D}\chi \exp \left( i\int \mathrm{d}^{4}x\left( \frac{1}{% 2\beta }\chi ^{2}-\partial _{\mu }\chi \partial ^{\mu }\Lambda \right) \right)$$ and similarly for other higher derivative terms we can finally formulate the theory as $$Z[J]=\int \mathcal{D}V_{\perp }\mathcal{D}B_{\perp }\mathcal{D}\Lambda \mathcal{D}\chi \mathcal{D}\rho \mathcal{D}\sigma \mathcal{D}\pi \exp \left( i\int \mathrm{d}^{4}x\mathcal{L}(V_{\perp },B_{\perp },\Lambda ,\chi ,\rho ,\sigma ,\pi ,J,\ldots )\right)$$ with $$\begin{aligned} \mathcal{D}B_{\perp } &=&\mathcal{D}B\delta (\partial _{\mu }B^{\mu }) \\ B_{\perp }^{\mu } &=&\left( g^{\mu \nu }-\frac{\partial ^{\mu }\partial ^{\nu }}{\square }\right) B_{\nu }.\end{aligned}$$ and $$\begin{aligned} \mathcal{L}(V_{\perp },B_{\perp },\Lambda ,\chi ,\rho ,\sigma ,\pi ,J,\ldots ) &=&\frac{1}{2}(1+\alpha )V_{\perp }^{\mu }\square V_{\perp \mu }+\frac{1}{2% }M^{2}V_{\perp }^{\mu }V_{\perp \mu }-\frac{1}{2\gamma }M^{2}B_{\perp }^{\mu }B_{\perp \mu }-B_{\perp }^{\mu }\square V_{\perp }^{\mu } \notag \\ &&+\frac{1}{2}M^{2}\partial _{\mu }\Lambda \partial ^{\mu }\Lambda +\frac{1}{% 2\beta }\chi ^{2}-\partial _{\mu }\chi \partial ^{\mu }\Lambda \notag \\ &&-\frac{1}{2\delta }M^{2}\partial _{\mu }\rho \partial ^{\mu }\rho -\partial _{\mu }\rho \partial ^{\mu }\sigma -\partial _{\mu }\pi \partial ^{\mu }\Lambda -\pi \sigma \notag \\ &&+\mathcal{L}_{int}^{^{\prime }}(V_{\perp }-\partial \Lambda ,J,\ldots ) \label{trans_L_V}\end{aligned}$$ In this formulation the kinetic terms have no more than two derivatives, however, the number of fields is higher than the actual number of degrees of freedom. We therefore have to integrate out the redundant variables. As a first step we diagonalize the kinetic terms performing the shifts $$\begin{aligned} V_{\perp }^{\mu } &\rightarrow &V_{\perp }^{\mu }+\frac{1}{1+\alpha }% B_{\perp }^{\mu } \notag \\ \Lambda &\rightarrow &\Lambda +\frac{1}{M^{2}}\chi +\frac{1}{M^{2}}\pi \notag \\ \rho &\rightarrow &\rho -\frac{\delta }{M^{2}}\sigma \notag \\ \chi &\rightarrow &\chi -\pi \label{shift_L_V}\end{aligned}$$ respectively to the form $$\begin{aligned} \mathcal{L}(V_{\perp },B_{\perp },\Lambda ,\chi ,\rho ,\sigma ,\pi ,J,\ldots ) &=&\frac{1}{2}(1+\alpha )V_{\perp }^{\mu }\square V_{\perp \mu }+\frac{1}{2% }M^{2}V_{\perp }^{\mu }V_{\perp \mu } \notag \\ &&-\frac{1}{2}(1+\alpha )^{-1}B_{\perp }^{\mu }\square B_{\perp }^{\mu }+% \frac{1}{2}M^{2}\left( (1+\alpha )^{-2}-\gamma ^{-1}\right) B_{\perp }^{\mu }B_{\perp \mu } \notag \\ &&+M^{2}(1+\alpha )^{-1}V_{\perp }^{\mu }B_{\perp \mu } \notag \\ &&+\frac{1}{2}M^{2}\partial _{\mu }\Lambda \partial ^{\mu }\Lambda -\frac{1}{% 2M^{2}}\partial _{\mu }\chi \partial ^{\mu }\chi +\frac{1}{2\beta }(\chi -\pi )^{2} \notag \\ &&-\frac{1}{2\delta }M^{2}\partial _{\mu }\rho \partial ^{\mu }\rho +\frac{% \delta }{2M^{2}}\partial _{\mu }\sigma \partial ^{\mu }\sigma -\pi \sigma \notag \\ &&+\mathcal{L}_{int}^{^{\prime }}(\overline{V},J,\ldots ). \label{shifted_L_V}\end{aligned}$$ where $$\overline{V}=V_{\perp }+\frac{1}{1+\alpha }B_{\perp }-\partial \Lambda -% \frac{1}{M^{2}}\partial \chi$$ Now the superfluous degrees of freedom are easily identified. Namely, the fields $\rho $ and $\sigma $ decouple and moreover $\pi $ has no kinetic term. Both of them can be therefore easily integrated out. As a result of the gaussian integration we get $$Z[J]=\int \mathcal{D}V_{\perp }\mathcal{D}B_{\perp }\mathcal{D}\Lambda \mathcal{D}\chi \mathcal{D}\sigma \exp \left( i\int \mathrm{d}^{4}x\mathcal{L% }(V_{\perp },B_{\perp },\Lambda ,\chi ,\sigma ,J,\ldots )\right)$$ where $$\begin{aligned} \mathcal{L}(V_{\perp },B_{\perp },\Lambda ,\chi ,\sigma ,J,\ldots ) &=&\frac{% 1}{2}(1+\alpha )V_{\perp }^{\mu }\square V_{\perp \mu }+\frac{1}{2}% M^{2}V_{\perp }^{\mu }V_{\perp \mu } \notag \\ &&-\frac{1}{2}(1+\alpha )^{-1}B_{\perp }^{\mu }\square B_{\perp }^{\mu }+% \frac{1}{2}M^{2}\left( (1+\alpha )^{-2}-\gamma ^{-1}\right) B_{\perp }^{\mu }B_{\perp \mu } \notag \\ &&+M^{2}(1+\alpha )^{-1}V_{\perp }^{\mu }B_{\perp \mu } \notag \\ &&-\frac{1}{2M^{2}}\partial _{\mu }\chi \partial ^{\mu }\chi +\frac{\delta }{% 2M^{2}}\partial _{\mu }\sigma \partial ^{\mu }\sigma -\frac{1}{2}\beta \sigma ^{2}-\chi \sigma \notag \\ &&+\frac{1}{2}M^{2}\partial _{\mu }\Lambda \partial ^{\mu }\Lambda \notag \\ &&+\mathcal{L}_{int}^{^{\prime }}(\overline{V},J,\ldots ). \label{integrated_L_V}\end{aligned}$$ Let us assume $\alpha >-1\,$and $\delta >0$ in what follows. Note that, in this case the fields $B_{\perp }^{\mu }$ and $\chi $ have opposite minus sign at their kinetic terms. This is a signal of the appearence of the negative norm ghosts in the spectrum of the theory. The dangerous fields $B_{\perp }^{\mu }$ and $\chi $ mix with the fields $V_{\perp }^{\mu }$ and $\sigma $ respectively. In order to identify the mass eigenstates we further rescale the fields $$\begin{aligned} V_{\perp }^{\mu } &\rightarrow &(1+\alpha )^{-1/2}V_{\perp }^{\mu } \notag \\ B_{\perp }^{\mu } &\rightarrow &(1+\alpha )^{1/2}B_{\perp }^{\mu } \notag \\ \chi &\rightarrow &M\chi \notag \\ \sigma &\rightarrow &\delta ^{-1/2}M\sigma\end{aligned}$$ and afterwards we diagonalize the mass terms $$\begin{aligned} \mathcal{L}_{mass} &=&\frac{1}{2}\frac{M^{2}}{1+\alpha }\left( V_{\perp }^{\mu }V_{\perp \mu }+\left( 1-\frac{(1+\alpha )^{2}}{\gamma }\right) B_{\perp }^{\mu }B_{\perp \mu }\right) \notag \\ &&-\frac{1}{2}M^{2}\left( \beta \sigma ^{2}+\delta ^{-1/2}\chi \sigma \right) \label{L_V_mass}\end{aligned}$$ by means of an appropriate $Sp(2)$ symplectic rotation of the fields $% V_{\perp }^{\mu }$, $B_{\perp }^{\mu }$ and $\chi $, $\sigma $ $$\begin{aligned} V_{\perp }^{\mu } &\rightarrow &V_{\perp }^{\mu }\cosh \theta _{V}+B_{\perp }^{\mu }\sinh \theta _{V} \notag \\ B_{\perp }^{\mu } &\rightarrow &V_{\perp }^{\mu }\sinh \theta _{V}+B_{\perp }^{\mu }\cosh \theta _{V} \notag \\ \chi &\rightarrow &\chi \cosh \theta _{S}+\sigma \sinh \theta _{S} \notag \\ \sigma &\rightarrow &\chi \sinh \theta _{S}+\sigma \cosh \theta _{S}.\end{aligned}$$ This is possible for $(1+\alpha )^{2}>4\gamma $ and $\beta ^{2}>4\delta $, when the off-diagonal elements of the mass matrix vanish for $$\begin{aligned} \tanh \theta _{V} &=&\frac{(1+\alpha )^{2}-2\gamma -(1+\alpha )\sqrt{% (1+\alpha )^{2}-4\gamma }}{2\gamma } \notag \\ \tanh \theta _{S} &=&\frac{\sqrt{\beta ^{2}-4\delta }-\beta }{2\delta ^{1/2}}% .\end{aligned}$$ We get finally for the generating functional $$Z[J]=\int \mathcal{D}V_{\perp }\mathcal{D}B_{\perp }\mathcal{D}\Lambda \mathcal{D}\chi \mathcal{D}\sigma \exp \left( i\int \mathrm{d}^{4}x\mathcal{L% }(V_{\perp },B_{\perp },\Lambda ,\chi ,\sigma ,J,\ldots )\right)$$ where $$\begin{aligned} \mathcal{L}(V_{\perp },B_{\perp },\Lambda ,\chi ,\sigma ,J,\ldots ) &=&\frac{% 1}{2}V_{\perp }^{\mu }\square V_{\perp \mu }+\frac{1}{2}M_{V+}^{2}V_{\perp }^{\mu }V_{\perp \mu }-\frac{1}{2}B_{\perp }^{\mu }\square B_{\perp }^{\mu }+% \frac{1}{2}M_{V-}^{2}B_{\perp }^{\mu }B_{\perp \mu } \notag \\ &&+\frac{1}{2}\partial _{\mu }\sigma \partial ^{\mu }\sigma -\frac{1}{2}% M_{S+}^{2}\sigma ^{2}-\frac{1}{2}\partial _{\mu }\chi \partial ^{\mu }\chi -% \frac{1}{2}M_{S-}^{2}\chi ^{2}+\frac{1}{2}M^{2}\partial _{\mu }\Lambda \partial ^{\mu }\Lambda \notag \\ &&+\mathcal{L}_{int}^{^{\prime }}(\overline{V}^{(\theta )},J,\ldots ). \notag \\ &&\end{aligned}$$ where now $$\overline{V}^{(\theta )}=\frac{\exp \theta _{V}}{(1+\alpha )^{1/2}}(V_{\perp }+B_{\perp })-\partial \chi \cosh \theta _{S}-\partial \sigma \sinh \theta _{S}-\partial \Lambda$$ and where $M_{V\pm }^{2}$, $M_{S\pm }^{2}$ are the mass eigenvalues ([MV]{}) and (\[MS\]). The theory is now formulated in terms of two spin one and two spin zero fields, whereas two of them, namely $B_{\perp }^{\mu }$ and $\chi $, are negative norm ghosts. The field $\Lambda $ do not correspond to any dynamical degree of freedom, its role is merely to cancel the spurious poles of the free propagators of the transverse fields $% V_{\perp }$ and $B_{\perp }$ at $p^{2}=0$. The additional degrees of freedom in the path integral-the antisymmetric tensor case\[PI\_antisymmetric\] ========================================================================================================= We assume the interaction Lagrangian to be of the form $$\mathcal{L}_{int}=\mathcal{L}_{ct}+\mathcal{L}_{int}^{^{\prime }},$$ where $\mathcal{L}_{ct}$ is given by (\[L\_R\_toy\]) and re-express it in the terms of the longitudinal and transverse components of the original field $R_{\mu \nu }$ $$\mathcal{L}_{int}(R_{\parallel }^{\mu \nu }-\frac{1}{2}\varepsilon ^{\mu \nu \alpha \beta }\widehat{\Lambda }_{\alpha \beta },J,\ldots )=\mathcal{L}% _{ct}(R_{\parallel }^{\mu \nu }-\frac{1}{2}\varepsilon ^{\mu \nu \alpha \beta }\widehat{\Lambda }_{\alpha \beta },J,\ldots )+\mathcal{L}% _{int}^{^{\prime }}(R_{\parallel }^{\mu \nu }-\frac{1}{2}\varepsilon ^{\mu \nu \alpha \beta }\widehat{\Lambda }_{\alpha \beta },J,\ldots )$$ where $$\begin{aligned} \mathcal{L}_{ct}(R^{\mu \nu }-\frac{1}{2}\varepsilon ^{\mu \nu \alpha \beta }% \widehat{\Lambda }_{\alpha \beta }J,\ldots ) &=&\frac{\alpha }{4}% R_{\parallel }^{\mu \nu }\square R_{\parallel \,\mu \nu }+\frac{\gamma }{% 4M^{2}}(\square R_{\parallel }^{\mu \nu })(\square R_{\parallel \,\mu \nu }) \notag \\ &&+\frac{\beta }{2}(\square \Lambda _{\perp }^{\mu })(\square \Lambda _{\perp \mu })-\frac{\delta }{2M^{2}}(\partial ^{\alpha }\square \Lambda _{\perp }^{\mu })(\partial _{\alpha }\square \Lambda _{\perp \mu }).\end{aligned}$$ We can introduce the auxiliary (longitudinal) antisymmetric tensor field $% B_{\parallel }^{\mu \nu }$ and (transverse) vector fields $\chi _{\perp }^{\mu }$, $\rho _{\perp }^{\mu }$, $\sigma _{\perp }^{\mu }$ and $\pi _{\perp }^{\mu }$ in order to avoid the higher derivative terms and write in complete analogy with the Proca field case $$Z[J]=\int \mathcal{D}R_{\parallel }\mathcal{D}B_{\parallel }\mathcal{D}% \Lambda _{\perp }\mathcal{D}\chi _{\perp }\mathcal{D}\rho _{\perp }\mathcal{D% }\sigma _{\perp }\mathcal{D}\pi _{\perp }\exp \left( \mathrm{i}\int \mathrm{d% }^{4}x\mathcal{L}(R_{\parallel },B_{\parallel },\Lambda _{\perp },\chi _{\perp },\rho _{\perp },\sigma _{\perp },\pi _{\perp },J,\ldots )\right)$$ where the measures and fields are $$\begin{aligned} \mathcal{D}B_{\parallel } &=&\mathcal{D}B\delta (\partial _{\alpha }B_{\mu \nu }+\partial _{\nu }B_{\alpha \mu }+\partial _{\mu }B_{\nu \alpha }) \\ B_{\parallel }^{\mu \nu } &=&-\frac{1}{2\square }(\partial ^{\mu }g^{\nu \alpha }\partial ^{\beta }+\partial ^{\nu }g^{\mu \beta }\partial ^{\alpha }-(\mu \leftrightarrow \nu ))B_{\alpha \beta }\end{aligned}$$ and for $\phi ^{\mu }=\chi ^{\mu }$, $\rho ^{\mu }$, $\sigma ^{\mu }$ and $% \pi ^{\mu }$ $$\begin{aligned} \mathcal{D}\phi _{\perp } &=&\mathcal{D}\phi \delta (\partial _{\mu }\phi ^{\mu }) \\ \phi _{\perp }^{\mu } &=&\left( g^{\mu \nu }-\frac{\partial ^{\mu }\partial ^{\nu }}{\square }\right) \phi _{\perp \nu }.\end{aligned}$$ The Lagrangian is then $$\begin{aligned} \mathcal{L} &=&\frac{1+\alpha }{4}R_{\parallel }^{\mu \nu }\square R_{\parallel \,\mu \nu }+\frac{1}{4}M^{2}R_{\parallel }^{\mu \nu }R_{\parallel \,\mu \nu } \notag \\ &&-\frac{1}{\gamma }M^{2}B_{\parallel }^{\mu \nu }B_{\parallel \,\mu \nu }+B_{\parallel }^{\mu \nu }\square R_{\parallel \,\mu \nu } \notag \\ &&+\frac{1}{2}M^{2}\Lambda _{\perp }^{\mu }\square \Lambda _{\perp \mu }-% \frac{1}{2\beta }\chi _{\perp }^{\mu }\chi _{\perp \mu }-\chi _{\perp }^{\mu }\square \Lambda _{\perp \mu } \notag \\ &&+\frac{1}{2\delta }M^{2}\partial ^{\alpha }\rho _{\perp }^{\mu }\partial _{\alpha }\rho _{\perp \mu }-\partial ^{\alpha }\rho _{\perp }^{\mu }\partial _{\alpha }\sigma _{\perp \mu }-\partial ^{\alpha }\Lambda _{\perp }^{\mu }\partial _{\alpha }\pi _{\perp \mu }-\pi _{\perp }^{\mu }\sigma _{\perp \mu } \notag \\ &&+\mathcal{L}_{int}\left( R^{\mu \nu }-\frac{1}{2}\varepsilon ^{\mu \nu \alpha \beta }\widehat{\Lambda }_{\alpha \beta },J,\ldots \right) . \label{trans_L_R}\end{aligned}$$ Note that, the fields $\chi ^{\mu }$, $\rho ^{\mu }$, $\sigma ^{\mu }$ and $% \pi ^{\mu }$ mix with $\Lambda ^{\mu }$ and are therefore pseudovectors. The Lagrangian (\[trans\_L\_R\]) is completely analogical to (\[trans\_L\_V\]) up to the more Lorentz indices, so will be brief in the next steps. First we identify the redundant degrees of freedom diagonalizing the kinetic terms by means of the following sequence of shifts (cf. (\[shift\_L\_V\])) $$\begin{aligned} R_{\parallel }^{\mu \nu } &\rightarrow &R_{\parallel }^{\mu \nu }-2(1+\alpha )^{-1}B_{\parallel }^{\mu \nu } \notag \\ \Lambda _{\perp }^{\mu } &\rightarrow &\Lambda _{\perp }^{\mu }+\frac{1}{% M^{2}}\chi _{\perp }^{\mu }-\frac{1}{M^{2}}\pi _{\perp }^{\mu } \notag \\ \rho _{\perp }^{\mu } &\rightarrow &\rho _{\perp }^{\mu }+\frac{\delta }{% M^{2}}\sigma _{\perp }^{\mu } \notag \\ \chi _{\perp }^{\mu } &\rightarrow &\chi _{\perp }^{\mu }+\pi _{\perp }^{\mu }.\end{aligned}$$ As a result we get the Lagrangian in the form (cf. (\[shifted\_L\_V\])) $$\begin{aligned} \mathcal{L} &=&\frac{1}{4}(1+\alpha )R_{\parallel }^{\mu \nu }\square R_{\parallel \,\mu \nu }+\frac{1}{4}M^{2}R_{\parallel }^{\mu \nu }R_{\parallel \,\mu \nu } \notag \\ &&-(1+\alpha )^{-1}B_{\parallel }^{\mu \nu }\square B_{\parallel \,\mu \nu }+(1+\alpha )^{-2}M^{2}B_{\parallel }^{\mu \nu }B_{\parallel \,\mu \nu }-% \frac{1}{\gamma }M^{2}B_{\parallel }^{\mu \nu }B_{\parallel \,\mu \nu } \notag \\ &&-(1+\alpha )^{-1}M^{2}R_{\parallel }^{\mu \nu }B_{\parallel \,\mu \nu } \notag \\ &&+\frac{1}{2}M^{2}\Lambda _{\perp }^{\mu }\square \Lambda _{\perp \mu }-% \frac{1}{2M^{2}}\chi _{\perp }^{\mu }\square \chi _{\perp \mu }-\frac{1}{% 2\beta }(\chi _{\perp }^{\mu }+\pi _{\perp }^{\mu })(\chi _{\perp \mu }+\pi _{\perp \mu }) \notag \\ &&+\frac{1}{2\delta }M^{2}\partial ^{\alpha }\rho _{\perp }^{\mu }\partial _{\alpha }\rho _{\perp \mu }-\frac{\delta }{2M^{2}}\partial ^{\alpha }\sigma _{\perp }^{\mu }\partial _{\alpha }\sigma _{\perp \mu }-\pi _{\perp }^{\mu }\sigma _{\perp \mu } \notag \\ &&+\mathcal{L}_{int}\left( \overline{R},J,\ldots \right) ,\end{aligned}$$ where $$\overline{R}^{\mu \nu }=R_{\parallel }^{\mu \nu }-2(1+\alpha )^{-1}B_{\parallel }^{\mu \nu }-\frac{1}{2}\varepsilon ^{\mu \nu \alpha \beta }(\widehat{\Lambda }_{\alpha \beta }+\frac{1}{M^{2}}\widehat{\chi }% _{\perp \alpha \beta }).$$ Integrating out the superfluous fields $\rho _{\perp \mu }$ and $\pi _{\perp \mu }$ which are decoupled from the interaction we get $$Z[J]=\int \mathcal{D}R_{\parallel }\mathcal{D}B_{\parallel }\mathcal{D}% \Lambda _{\perp }\mathcal{D}\chi _{\perp }\mathcal{D}\rho _{\perp }\mathcal{D% }\sigma _{\perp }\mathcal{D}\pi _{\perp }\exp \left( \mathrm{i}\int \mathrm{d% }^{4}x\mathcal{L}(R_{\parallel },B_{\parallel },\Lambda _{\perp },\chi _{\perp },\rho _{\perp },\sigma _{\perp },\pi _{\perp },J,\ldots )\right)$$ with (cf. (\[integrated\_L\_V\])) $$\begin{aligned} \mathcal{L} &=&\frac{1}{4}(1+\alpha )R_{\parallel }^{\mu \nu }\square R_{\parallel \,\mu \nu }+\frac{1}{4}M^{2}R_{\parallel }^{\mu \nu }R_{\parallel \,\mu \nu } \notag \\ &&-(1+\alpha )^{-1}B_{\parallel }^{\mu \nu }\square B_{\parallel \,\mu \nu }+(1+\alpha )^{-2}M^{2}B_{\parallel }^{\mu \nu }B_{\parallel \,\mu \nu }-% \frac{1}{\gamma }M^{2}B_{\parallel }^{\mu \nu }B_{\parallel \,\mu \nu } \notag \\ &&-(1+\alpha )^{-1}M^{2}R_{\parallel }^{\mu \nu }B_{\parallel \,\mu \nu } \notag \\ &&+\frac{1}{2}M^{2}\Lambda _{\perp }^{\mu }\square \Lambda _{\perp \mu } \notag \\ &&-\frac{1}{2M^{2}}\chi _{\perp }^{\mu }\square \chi _{\perp \mu }+\frac{% \delta }{2M^{2}}\sigma _{\perp }^{\mu }\square \sigma _{\perp \mu }+\frac{1}{% 2}\beta \sigma _{\perp }^{\mu }\sigma _{\perp \mu }+\chi _{\perp }^{\mu }\sigma _{\perp \mu } \notag \\ &&+\mathcal{L}_{int}(S,J,\ldots )\end{aligned}$$ Again, assuming $\alpha >-1$ and $\delta >0$ we have two pairs of fields with opposite signs of the kinetic terms, namely $(R_{\parallel }^{\mu \nu },B_{\parallel }^{\mu \nu })$ and $(\chi _{\perp }^{\mu },\sigma _{\perp }^{\mu })$ respectively.The fields within both of these these pairs mix. After re-scaling $$\begin{aligned} R_{\parallel }^{\mu \nu } &\rightarrow &(1+\alpha )^{-1/2}R_{\parallel }^{\mu \nu } \\ B_{\parallel }^{\mu \nu } &\rightarrow &\frac{1}{2}(1+\alpha )^{1/2}B_{\parallel }^{\mu \nu } \notag \\ \chi _{\perp }^{\mu } &\rightarrow &M\chi _{\perp }^{\mu } \notag \\ \sigma _{\perp }^{\mu } &\rightarrow &\frac{M}{\sqrt{\delta }}\sigma _{\perp }^{\mu } \notag\end{aligned}$$ the form of the mass matrix becomes identical to that of (\[L\_V\_mass\]) (with obvious identifications) and we can therefore perform the same symplectic rotations as in the Proca field case and under the same assumptions to get diagonal mass terms corresponding to the eigenvalues ([MV]{}, \[MS\]). As a result we have found four spin-one states, two of them being negative norm ghosts, namely $B_{\parallel }^{\mu \nu }$ and $\sigma _{\perp }^{\mu }$ and two of them with opposite parity, namely $\chi _{\perp }^{\mu }$ and $\sigma _{\perp }^{\mu }$. As in the Proca field case, the field $\Lambda _{\perp }^{\mu }$ effectively compensates for the spurious $% p^{2}=0$ poles in the $R_{\parallel }^{\mu \nu }$ and $B_{\parallel }^{\mu \nu }$ propagators within Feynman graphs. Path integral formulation of the first order formalism[PI\_first\_order]{} ========================================================================== Within the first order formalism, the path integral formulation is merely a generalization of the previous two cases, so we will be as brief as possible in what follows. Note that, now the kinetic term is invariant with respect to the both transformations (\[V\_gauge\]) and (\[R\_gauge\]), therefore the manifestation of the degrees of freedom within the the path integral formalism can be done in analogy with the previous two cases. Using triple Faddeev-Popov trick in the path integral $$Z[J]=\int \mathcal{D}R\exp \left( \mathrm{i}\int \mathrm{d}^{4}x\left( MV_{\nu }\partial _{\mu }R^{\mu \nu }+\frac{1}{2}M^{2}V_{\mu }V^{\mu }+\frac{% 1}{4}M^{2}R_{\mu \nu }R^{\mu \nu }+\mathcal{L}_{int}(V^{\alpha },R^{\mu \nu },J,\ldots )\right) \right)$$ we get $$Z[J]=\int \mathcal{D}R_{\parallel }\mathcal{D}\Lambda _{\perp }\mathcal{D}% V_{\perp }\mathcal{D}\Lambda \exp \left( \mathrm{i}\int \mathrm{d}^{4}x% \mathcal{L}(R_{\parallel }^{\mu \nu },\Lambda _{\perp }^{\rho },V_{\perp }^{\alpha }\ldots ,\Lambda ,J,\ldots )\right)$$ where $$\begin{aligned} \mathcal{L}(R_{\parallel }^{\mu \nu },\Lambda _{\perp }^{\rho },V_{\perp }^{\alpha }\ldots ,\Lambda ,J,\ldots ) &=&MV_{\perp \nu }\partial _{\mu }R_{\parallel }^{\mu \nu }+\frac{1}{2}M^{2}V_{\perp \mu }V_{\perp }^{\mu }+% \frac{1}{4}M^{2}R_{\parallel \mu \nu }R_{\parallel }^{\mu \nu } \notag \\ &&+\frac{1}{2}M^{2}\Lambda _{\perp }^{\mu }\square \Lambda _{\perp \mu }+% \frac{1}{2}M^{2}\partial _{\mu }\Lambda \partial ^{\mu }\Lambda \notag \\ &&+\mathcal{L}_{int}(R_{\parallel }^{\mu \nu }-\frac{1}{2}\varepsilon ^{\mu \nu \alpha \beta }\widehat{\Lambda }_{\alpha \beta },V_{\perp }^{\alpha }-\partial ^{\alpha }\Lambda ,J,) \label{FO_phys}\end{aligned}$$ and, as in the previous subsections $$\begin{aligned} \mathcal{D}R_{\parallel } &=&\mathcal{D}R\delta (\partial_{\alpha } R_{\mu \nu }+\partial _{\nu }R_{ \alpha \mu}+\partial _{\mu }R_{\nu \alpha }) \notag \\ \mathcal{D}\Lambda _{\perp } &=&\mathcal{D}\Lambda \delta (\partial _{\mu }\Lambda ^{\mu }) \notag \\ \mathcal{D}V_{\perp } &=&\mathcal{D}V\delta (\partial _{\mu }\Lambda ^{\mu }) \notag \\ R_{\parallel }^{\mu \nu } &=&-\frac{1}{2\square }(\partial ^{\mu }g^{\nu \alpha }\partial ^{\beta }+\partial ^{\nu }g^{\mu \beta }\partial ^{\alpha }-(\mu \leftrightarrow \nu ))R_{\alpha \beta } \notag \\ \Lambda _{\perp }^{\mu } &=&\left( g^{\mu \nu }-\frac{\partial ^{\mu }\partial ^{\nu }}{\square }\right) \Lambda _{\nu } \notag \\ V_{\perp }^{\mu } &=&\left( g^{\mu \nu }-\frac{\partial ^{\mu }\partial ^{\nu }}{\square }\right) V_{\nu }.\end{aligned}$$ In order to diagonalize the kinetic terms we perform a shift $$V_{\perp }^{\mu }\rightarrow V_{\perp }^{\mu }-\frac{1}{M}\partial _{\nu }R_{\parallel }^{\nu \mu }$$ and get $$\begin{aligned} \mathcal{L}(R_{\parallel }^{\mu \nu },\Lambda _{\perp }^{\rho },V_{\perp }^{\alpha }\ldots ,\Lambda ,J,\ldots ) &=&\frac{1}{4}R_{\parallel }^{\mu \nu }\square R_{\parallel \,\mu \nu }+\frac{1}{4}M^{2}R_{\parallel \mu \nu }R_{\parallel }^{\mu \nu }+\frac{1}{2}M^{2}V_{\perp \mu }V_{\perp }^{\mu } \notag \\ &&+\frac{1}{2}M^{2}\Lambda _{\perp }^{\mu }\square \Lambda _{\perp \mu }+% \frac{1}{2}M^{2}\partial _{\mu }\Lambda \partial ^{\mu }\Lambda \notag \\ &&+\mathcal{L}_{int}(R_{\parallel }^{\mu \nu }-\frac{1}{2}\varepsilon ^{\mu \nu \alpha \beta }\widehat{\Lambda }_{\alpha \beta },V_{\perp }^{\alpha }-% \frac{1}{M}\partial _{\nu }R_{\parallel }^{\nu \mu }-\partial ^{\alpha }\Lambda ,J,). \notag \\ &&\end{aligned}$$ The discussion of the role of the field $R_{\parallel }^{\mu \nu }$ and the $% \Lambda _{\perp }^{\mu }$ is the same as in the antisymmetric tensor case. The extra fields $V_{\perp }^{\mu }$ and $\Lambda $ do not correspond to the original degree of freedom, their free propagators are $$\begin{aligned} \Delta _{V\perp }^{\mu \nu }(p) &=&\frac{P^{T\,\mu \nu }}{M^{2}} \\ \Delta _{\Lambda }(p) &=&\frac{1}{M^{2}}\frac{1}{p^{2}}\end{aligned}$$ with spurious poles at $p^{2}=0$. According to the form of the interaction, only the combination with spurious poles cancelled, namely $$\Delta _{V\perp }^{\mu \nu }(p)+p^{\mu }p^{\nu }\Delta _{\Lambda }(p)+\frac{1% }{M^{2}}p_{\alpha }p_{\beta }\Delta _{\parallel }^{\alpha \mu \beta \nu }(p)=-\frac{P^{T\,\mu \nu }}{p^{2}-M^{2}}+\frac{P^{L\,\,\mu \nu }}{M^{2}}$$ enters the Feynman graphs. Alternatively, we could make in (\[FO\_phys\]) the following shift $$R_{\parallel }^{\mu \nu }\rightarrow R_{\parallel }^{\mu \nu }+\frac{1}{M}% \left( \partial ^{\mu }V_{\perp }^{\nu }-\partial ^{\nu }V_{\perp }^{\mu }\right)$$ leading to $$\begin{aligned} \mathcal{L}(R_{\parallel }^{\mu \nu },\Lambda _{\perp }^{\rho },V_{\perp }^{\alpha }\ldots ,\Lambda ,J,\ldots ) &=&\frac{1}{2}V_{\perp \mu }\square V_{\perp }^{\mu }+\frac{1}{2}M^{2}V_{\perp \mu }V_{\perp }^{\mu }+\frac{1}{4}% M^{2}R_{\parallel \mu \nu }R_{\parallel }^{\mu \nu } \notag \\ &&+\frac{1}{2}M^{2}\Lambda _{\perp }^{\mu }\square \Lambda _{\perp \mu }+% \frac{1}{2}M^{2}\partial _{\mu }\Lambda \partial ^{\mu }\Lambda \notag \\ &&+\mathcal{L}_{int}(R_{\parallel }^{\mu \nu }+\frac{1}{M}\left( \partial ^{\mu }V_{\perp }^{\nu }-\partial ^{\nu }V_{\perp }^{\mu }\right) -\frac{1}{2% }\varepsilon ^{\mu \nu \alpha \beta }\widehat{\Lambda }_{\alpha \beta },V_{\perp }^{\alpha }-\partial ^{\alpha }\Lambda ,J,\ldots ). \notag \\ &&\end{aligned}$$ In this formulation, the role of the fields $V_{\perp }^{\mu }$ and the field $\Lambda $ is the same as in the Proca field case. $R_{\parallel }^{\mu \nu }$ does not correspond to the original degree of freedom and, as in the previous formulation, it serves together with $\Lambda _{\perp \mu }$ to cancel the spurious $p^{2}=0$ poles. Let us end up this subsection with the path integral treatment of the toy quadratic interaction Lagrangian (\[L\_RV\_toy\]). Using the same transformations as before we get $$\begin{aligned} \mathcal{L}(R_{\parallel }^{\mu \nu },\Lambda _{\perp }^{\rho },V_{\perp }^{\alpha }\ldots ,\Lambda ,J,\ldots ) &=&MV_{\perp \nu }\partial _{\mu }R_{\parallel }^{\mu \nu }+\frac{1}{2}M^{2}V_{\perp \mu }V_{\perp }^{\mu }+% \frac{1}{4}M^{2}R_{\parallel \mu \nu }R_{\parallel }^{\mu \nu } \notag \\ &&+\frac{1}{2}M^{2}\Lambda _{\perp }^{\mu }\square \Lambda _{\perp \mu }+% \frac{1}{2}M^{2}\partial _{\mu }\Lambda \partial ^{\mu }\Lambda \notag \\ &&+\frac{\alpha _{V}}{2}V_{\perp \mu }\square V_{\perp }^{\mu }-\frac{\beta _{V}}{2}(\square \Lambda )^{2}+\frac{\alpha _{R}}{4}R_{\parallel \mu \nu }\square R_{\parallel }^{\mu \nu }+\frac{\beta _{R}}{4}\square \Lambda _{\perp }^{\mu }\square \Lambda _{\perp \mu } \notag \\ &&+\mathcal{L}_{int}^{^{\prime }}(R_{\parallel }^{\mu \nu }-\frac{1}{2}% \varepsilon ^{\mu \nu \alpha \beta }\widehat{\Lambda }_{\perp \alpha \beta },V_{\perp }^{\alpha }-\partial ^{\alpha }\Lambda ,J,\ldots )\end{aligned}$$ Introducing the auxiliary fields analogous to the previous two examples, we have $$\begin{aligned} \mathcal{L}(R_{\parallel }^{\mu \nu },\Lambda _{\perp }^{\mu },\chi ,\chi _{\perp }^{\mu },\sigma _{\perp }^{\mu },\pi _{\perp }^{\mu },J,\ldots ) &=&% \frac{\alpha _{V}}{2}V_{\perp \mu }\square V_{\perp }^{\mu }+\frac{1}{2}% M^{2}V_{\perp \mu }V_{\perp }^{\mu } \notag \\ &&+\frac{\alpha _{R}}{4}R_{\parallel \mu \nu }\square R_{\parallel }^{\mu \nu }+\frac{1}{4}M^{2}R_{\parallel \mu \nu }R_{\parallel }^{\mu \nu } \notag \\ &&+MV_{\perp \nu }\partial _{\mu }R_{\parallel }^{\mu \nu } \notag \\ &&+\frac{1}{2}M^{2}\Lambda _{\perp }^{\mu }\square \Lambda _{\perp \mu }-% \frac{1}{2}M^{2}\Lambda \square \Lambda \notag \\ &&+\frac{1}{2\beta _{V}}\chi ^{2}+\chi \square \Lambda -\frac{1}{2\beta _{R}}% \chi _{\perp }^{\mu }\chi _{\perp \mu }-\chi _{\perp }^{\mu }\square \Lambda _{\perp \mu } \notag \\ &&+\mathcal{L}_{int}^{^{\prime }}(R_{\parallel }^{\mu \nu }-\frac{1}{2}% \varepsilon ^{\mu \nu \alpha \beta }\widehat{\Lambda }_{\perp \alpha \beta },V_{\perp }^{\alpha }-\partial ^{\alpha }\Lambda ,J,\ldots )\end{aligned}$$ The kinetic terms can be diagonalized now by means of the shifts $$\begin{aligned} \Lambda _{\perp }^{\mu } &\rightarrow &\Lambda _{\perp }^{\mu }+\frac{1}{% M^{2}}\chi _{\perp }^{\mu } \\ \Lambda &\rightarrow &\Lambda +\frac{1}{M^{2}}\chi\end{aligned}$$ to the form $$\begin{aligned} \mathcal{L}(R_{\parallel }^{\mu \nu },\Lambda _{\perp }^{\mu },\chi ,\chi _{\perp }^{\mu },J,\ldots ) &=&\frac{\alpha _{V}}{2}V_{\perp \mu }\square V_{\perp }^{\mu }+\frac{1}{2}M^{2}V_{\perp \mu }V_{\perp }^{\mu } \notag \\ &&+\frac{\alpha _{R}}{4}R_{\parallel \mu \nu }\square R_{\parallel }^{\mu \nu }+\frac{1}{4}M^{2}R_{\parallel \mu \nu }R_{\parallel }^{\mu \nu } \notag \\ &&+MV_{\perp \nu }\partial _{\mu }R_{\parallel }^{\mu \nu } \notag \\ &&+\frac{1}{2}M^{2}\Lambda _{\perp }^{\mu }\square \Lambda _{\perp \mu }-% \frac{1}{2}M^{2}\Lambda \square \Lambda \notag \\ &&-\frac{1}{2M^{2}}\chi _{\perp }^{\mu }\square \chi _{\perp \mu }-\frac{1}{% 2\beta _{R}}\chi _{\perp }^{\mu }\chi _{\perp \mu } \notag \\ &&+\frac{1}{2M^{2}}\chi \square \chi +\frac{1}{2\beta _{V}}\chi ^{2} \notag \\ &&+\mathcal{L}_{int}^{^{\prime }}(S,W,J,\dots ), \notag \\ && \label{L_FO_final}\end{aligned}$$ where $$\begin{aligned} \overline{R}^{\mu \nu } &=&R_{\parallel }^{\mu \nu }-\frac{1}{2}\varepsilon ^{\mu \nu \alpha \beta }\widehat{\Lambda }_{\perp \alpha \beta }-\frac{1}{% 2M^{2}}\varepsilon ^{\mu \nu \alpha \beta }\widehat{\chi }_{\perp \alpha \beta } \\ \overline{V}^{\mu } &=&V_{\perp }^{\alpha }-\partial ^{\alpha }\Lambda -% \frac{1}{M^{2}}\partial ^{\alpha }\chi .\end{aligned}$$ In the formula (\[L\_FO\_final\]) the scalar and axial-vector ghost field as well as two propagating dynamically mixed spin-1 degrees of freedom are explicit. The parameters $\protect\alpha _{i}$ and $\protect\beta _{i}$ in terms of LECs\[Appendix\_alpha\_beta\] ======================================================================================================= In this appendix we present the expressions for the renormalization scale independent polynomial parameters entering the self-energies (cf. Section \[ch4\]). The Proca field case\[appendix Proca\] -------------------------------------- $$\begin{aligned} \alpha _0 &=&\left( \frac{4\pi F}M\right) ^2Z_M^r(\mu ) \\ \alpha _1 &=&\left( \frac{4\pi F}M\right) ^2Z_V^r(\mu )-\frac{40}3\sigma _V^2\left( \ln \frac{M^2}{\mu ^2}+\frac 13\right) \\ \alpha _2 &=&\left( \frac{4\pi F}M\right) ^2M^2X_V^r(\mu )+\frac{40}9\sigma _V^2\left( \ln \frac{M^2}{\mu ^2}+\frac 13\right) \\ \alpha _3 &=&\left( \frac{4\pi F}M\right) ^2M^4U_V^r(\mu )+g_V^2\left( \frac MF\right) ^2\left( \ln \frac{M^2}{\mu ^2}-\frac 23\right) \\ \beta _0 &=&\left( \frac{4\pi F}M\right) ^2Z_M^r(\mu )=\alpha _0 \\ \beta _1 &=&\left( \frac{4\pi F}M\right) ^2Y_V^r(\mu ) \\ \beta _2 &=&\left( \frac{4\pi F}M\right) ^2M^2X_V^{^{\prime }r}(\mu ) \\ \beta _3 &=&\left( \frac{4\pi F}M\right) ^2M^4V_V^r(\mu ).\end{aligned}$$ Here $U_V$ and $V_V$ are certain linear combinations of the couplings of $% \mathcal{L}_V^{ct(6)}$ renormalized as $$\begin{aligned} U_V &=&U_V^r(\mu )-2g_V^2\left( \frac MF\right) ^4\frac 1{M^4}\lambda _\infty \\ V_V &=&V_V^r(\mu )\end{aligned}$$ The antisymmetric tensor case\[appendix tensor\] ------------------------------------------------ $$\begin{aligned} {\alpha }_0 &=&\left( \frac{4\pi F}M\right) ^2Z_M^r({\mu })-\frac{40}% 3d_1^2\ln \frac{M^2}{{\mu }^2}-\frac{20}9(3d_1^2-d_3^2)-5\left( \frac{% \lambda ^{VVV}}M\right) ^2\left( \frac FM\right) ^2\left( 7-6\ln \frac{M^2}{{% \mu }^2}\right) \\ {\alpha }_1 &=&\left( \frac{4\pi F}M\right) ^2(Z_R^r({\mu })+Y_R^r({\mu }))-% \frac{40}9(3d_1^2+2d_3^2)\ln \frac{M^2}{{\mu }^2}-\frac{20}3\left( d_1^2+\frac 19d_2^2\right) \\ &&+\frac{10}3\left( \frac{\lambda ^{VVV}}M\right) ^2\left( \frac FM\right) ^2\left( 7-6\ln \frac{M^2}{{\mu }^2}\right) \\ {\alpha }_2 &=&\left( \frac{4\pi F}M\right) ^2M^2(X_R^r({\mu })+W_R^r({\mu }% ))-\frac{40}9d_3^2\left( \ln \frac{M^2}{{\mu }^2}+\frac 13\right) +\frac 12\left( \frac{G_V}F\right) ^2\left( \ln \frac{M^2}{{\mu }^2}-\frac 23\right) \\ &&-\frac 53\left( \frac{\lambda ^{VVV}}M\right) ^2\left( \frac FM\right) ^2\left( 2-3\ln \frac{M^2}{{\mu }^2}\right) \\ {\alpha }_3 &=&\left( \frac{4\pi F}M\right) ^2M^4U_R^r({\mu })+\frac{40}% 9d_3^2\left( \ln \frac{M^2}{{\mu }^2}+\frac 13\right) \\ {\beta }_0 &=&\left( \frac{4\pi F}M\right) ^2Z_M^r({\mu })-\frac{40}% 3d_1^2\ln \frac{M^2}{{\mu }^2}-\frac{20}9(3d_1^2+d_3^2)-\frac 53\left( \frac{% \lambda ^{VVV}}M\right) ^2\left( \frac FM\right) ^2\left( 11-6\ln \frac{M^2}{% {\mu }^2}\right) \\ {\beta }_1 &=&\left( \frac{4\pi F}M\right) ^2Y_R^r({\mu })-\frac{20}% 9(6d_1^2-12d_1(d_3+d_4)+5d_3^2+9d_4^2-6d_3d_4)\ln \frac{M^2}{{\mu }^2} \\ &&-\frac{20}{27}(9d_1^2-18d_1(d_3+d_4)-7d_3^2-12d_4^2+18d_3d_4) \\ &&\frac{20}3\left( \frac{\lambda ^{VVV}}M\right) ^2\left( \frac FM\right) ^2\left( 7+3\ln \frac{M^2}{{\mu }^2}\right) \\ {\beta }_2 &=&\left( \frac{4\pi F}M\right) ^2M^2W_R^r({\mu })-\frac{20}% 9(d_3^2+6d_3d_4-5d_4^2)\ln \frac{M^2}{{\mu }^2}-\frac{80}{27}(d_3^2+4d_4^2) \\ &&-\frac 53\left( \frac{\lambda ^{VVV}}M\right) ^2\left( \frac FM\right) ^2\left( 4-3\ln \frac{M^2}{{\mu }^2}\right) \\ {\beta }_3 &=&\left( \frac{4\pi F}M\right) ^2M^4V_R^r({\mu })-\frac{40}% 9d_4^2\left( \ln \frac{M^2}{{\mu }^2}-\frac 23\right) .\end{aligned}$$ Here $U_R$ and $V_R$ are certain linear combinations of the couplings of $% \mathcal{L}_R^{ct(6)}$ with the infinite parts fixed as $$\begin{aligned} U_R &=&U_R^r({\mu })-\frac{80}9\left( \frac MF\right) ^2\frac 1{M^4}d_3^2\lambda _\infty \\ V_R &=&V_R^r({\mu })+\frac{80}9\left( \frac MF\right) ^2\frac 1{M^4}d_4^2\lambda _\infty .\end{aligned}$$ The first order formalism\[appendix first order\] ------------------------------------------------- $$\begin{aligned} {\alpha }_{0}^{RV} &=&\left( \frac{4\pi F}{M}\right) ^{2}Z_{RV}^{r}(\mu )+% \frac{10}{9}(\sigma _{RV}+2\sigma _{V})\left[ (d_{1}-d_{3})+3(2d_{1}-\sigma _{RV})\left( \ln \frac{M^{2}}{\mu ^{2}}+\frac{1}{3}\right) \right] \\ {\alpha }_{1}^{RV} &=&\left( \frac{4\pi F}{M}\right) ^{2}M^{2}X_{RV}^{r}(\mu )+\frac{10}{9}(\sigma _{RV}+2\sigma _{V})(4d_{3}+\sigma _{RV})\left( \ln \frac{M^{2}}{\mu ^{2}}+\frac{1}{3}\right) \\ {\alpha }_{2}^{RV} &=&\left( \frac{4\pi F}{M}\right) ^{2}M^{4}Y_{RV}^{r}(\mu )-\frac{20}{9}(\sigma _{RV}+2\sigma _{V})d_{3}\left( \ln \frac{M^{2}}{\mu ^{2}}+\frac{1}{3}\right) \\ &&-\frac{1}{2}\frac{g_{V}G_{V}}{M}\left( \frac{M}{F}\right) ^{2}\left( \ln \frac{M^{2}}{\mu ^{2}}-\frac{2}{3}\right) \\ {\alpha }_{0}^{VV} &=&\left( \frac{4\pi F}{M}\right) ^{2}Z_{MV}^{r}(\mu ) \\ {\alpha }_{1}^{VV} &=&\left( \frac{4\pi F}{M}\right) ^{22}Z_{V}^{r}(\mu )-% \frac{10}{3}\left( \sigma _{RV}(\sigma _{RV}+2\sigma _{V})+4\sigma _{V}^{2}\right) \left( \ln \frac{M^{2}}{\mu ^{2}}+\frac{1}{3}\right) \\ {\alpha }_{2}^{VV} &=&\left( \frac{4\pi F}{M}\right) ^{2}M^{2}X_{V}^{r}(\mu )+\frac{10}{9}\left( \sigma _{RV}(\sigma _{RV}+2\sigma _{V})+4\sigma _{V}^{2}\right) \left( \ln \frac{M^{2}}{\mu ^{2}}+\frac{1}{3}\right) \\ {\alpha }_{3}^{VV} &=&\left( \frac{4\pi F}{M}\right) ^{2}M^{4}U_{V}^{r}(\mu )+g_{V}^{2}\left( \frac{M}{F}\right) ^{2}\left( \ln \frac{M^{2}}{\mu ^{2}}-% \frac{2}{3}\right) \\ \beta _{0}^{VV} &=&\left( \frac{4\pi F}{M}\right) ^{2}Z_{MV}^{r}(\mu )=\alpha _{0}^{VV} \\ \beta _{1}^{VV} &=&\left( \frac{4\pi F}{M}\right) ^{2}Y_{V}^{r}(\mu ) \\ \beta _{2}^{VV} &=&\left( \frac{4\pi F}{M}\right) ^{2}M^{2}X_{V}^{^{\prime }r}(\mu ) \\ \beta _{3}^{VV} &=&\left( \frac{4\pi F}{M}\right) ^{2}M^{4}V_{V}^{r}(\mu ) \\ {\alpha }_{0}^{RR} &=&\left( \frac{4\pi F}{M}\right) ^{2}Z_{MR}^{r}({\mu })+% \frac{10}{3}\left( \sigma _{RV}(2d_{1}-\sigma _{RV})-4d_{1}^{2}\right) \left( \ln \frac{M^{2}}{{\mu }^{2}}+\frac{1}{3}\right) \\ &&-\frac{10}{9}\left( d_{1}-d_{3}\right) \left( 2d_{1}+2d_{3}-\sigma _{RV}\right) \\ {\alpha }_{1}^{RR} &=&\left( \frac{4\pi F}{M}\right) ^{2}(Z_{R}^{r}({\mu }% )+Y_{R}^{r}({\mu }))-\frac{40}{9}(3d_{1}^{2}+2d_{3}^{2})\ln \frac{M^{2}}{{% \mu }^{2}}-\frac{20}{3}\left( d_{1}^{2}+\frac{1}{9}d_{2}^{2}\right) \\ &&+\frac{10}{9}\left( \ln \frac{M^{2}}{{\mu }^{2}}+\frac{1}{3}\right) \sigma _{RV}(4d_{3}+\sigma _{RV}) \\ {\alpha }_{2}^{RR} &=&\left( \frac{4\pi F}{M}\right) ^{2}M^{2}(X_{R}^{r}({% \mu })+W_{R}^{r}({\mu }))-\frac{20}{9}d_{3}(2d_{3}+\sigma _{RV})\left( \ln \frac{M^{2}}{{\mu }^{2}}+\frac{1}{3}\right) \\ &&+\frac{1}{2}\left( \frac{G_{V}}{F}\right) ^{2}\left( \ln \frac{M^{2}}{{\mu }^{2}}-\frac{2}{3}\right) \\ {\alpha }_{3}^{RR} &=&\left( \frac{4\pi F}{M}\right) ^{2}M^{4}U_{R}^{r}({\mu })+\frac{40}{9}d_{3}^{2}\left( \ln \frac{M^{2}}{{\mu }^{2}}+\frac{1}{3}% \right) \\ {\beta }_{0}^{RR} &=&\left( \frac{4\pi F}{M}\right) ^{2}Z_{MR}^{r}({\mu })+% \frac{10}{3}(\sigma _{RV}(2d_{1}-\sigma _{RV})-4d_{1}^{2})\left( \ln \frac{% M^{2}}{{\mu }^{2}}+\frac{1}{3}\right) \\ &&-\frac{20}{9}(d_{1}^{2}+d_{3}^{2})+\frac{10}{9}\sigma _{RV}\left( d_{1}+d_{3}-\sigma _{RV}\right) \\ &=&{\alpha }_{0}^{RR}-\frac{40}{9}d_{3}^{2}+\frac{10}{9}\sigma _{RV}\left( 2d_{3}-\sigma _{RV}\right) \\ {\beta }_{1}^{RR} &=&\left( \frac{4\pi F}{M}\right) ^{2}Y_{R}^{r}({\mu })-% \frac{20}{9}% (6d_{1}^{2}-12d_{1}(d_{3}+d_{4})+5d_{3}^{2}+9d_{4}^{2}-6d_{3}d_{4})\ln \frac{% M^{2}}{{\mu }^{2}} \\ &&-\frac{20}{27}% (9d_{1}^{2}-18d_{1}(d_{3}+d_{4})-7d_{3}^{2}-12d_{4}^{2}+18d_{3}d_{4}) \\ &&-\frac{5}{27}\sigma _{RV}\left( 32d_{3}+6(d_{3}+9d_{4})\ln \frac{M^{2}}{{% \mu }^{2}}-3\sigma _{RV}\left( \ln \frac{M^{2}}{{\mu }^{2}}-\frac{2}{3}% \right) \right) \\ {\beta }_{2}^{RR} &=&\left( \frac{4\pi F}{M}\right) ^{2}M^{2}W_{R}^{r}({\mu }% )-\frac{20}{9}(d_{3}^{2}+6d_{3}d_{4}-5d_{4}^{2})\ln \frac{M^{2}}{{\mu }^{2}}-% \frac{80}{27}(d_{3}^{2}+4d_{4}^{2}) \\ &&+\frac{5}{27}\sigma _{RV}\left( 8d_{3}+6(d_{3}+3d_{4})\ln \frac{M^{2}}{{% \mu }^{2}}-3\sigma _{RV}\left( \ln \frac{M^{2}}{{\mu }^{2}}-\frac{2}{3}% \right) \right) \\ {\beta }_{3}^{RR} &=&\left( \frac{4\pi F}{M}\right) ^{2}M^{4}V_{R}^{r}({\mu }% )-\frac{40}{9}d_{4}^{2}\left( \ln \frac{M^{2}}{{\mu }^{2}}-\frac{2}{3}% \right) .\end{aligned}$$ Here $U_{V}$, $V_{V}$, $U_{R}$, $V_{R}$ and $Y_{RV}$ are certain linear combination of the couplings from $\mathcal{L}_{RV}^{ct(6)}$ with infinite parts fixed according to $$\begin{aligned} U_{V} &=&U_{V}^{r}(\mu )-2g_{V}^{2}\left( \frac{M}{F}\right) ^{4}\frac{1}{% M^{4}}\lambda _{\infty } \\ V_{V} &=&V_{V}^{r}(\mu ) \\ U_{R} &=&U_{R}^{r}({\mu })-\frac{80}{9}\left( \frac{M}{F}\right) ^{2}\frac{1% }{M^{4}}d_{3}^{2}\lambda _{\infty } \\ V_{R} &=&V_{R}^{r}({\mu })+\frac{80}{9}\left( \frac{M}{F}\right) ^{2}\frac{1% }{M^{4}}d_{4}^{2}\lambda _{\infty } \\ Y_{RV} &=&Y_{RV}^{r}(\mu )+\frac{40}{9}\left( \frac{M}{F}\right) ^{2}\frac{1% }{M^{4}}(\sigma _{RV}+2\sigma _{V})d_{3}\lambda _{\infty }+\frac{g_{V}G_{V}}{% M}\left( \frac{M}{F}\right) ^{2}\frac{1}{M^{4}}\lambda _{\infty }\end{aligned}$$ Proof of the positivity of the spectral functions[Appendix\_positivity]{} ========================================================================= Here we prove the positivity of the spectral functios $\rho _{L,T}(\mu ^{2})$ defined as $$(2\pi )^{-3}\theta (p^{0})\left[ \rho _{T}(p^{2})p^{2}\Pi _{\mu \nu \alpha \beta }^{T}(p)-\rho _{L}(p^{2})p^{2}\Pi _{\mu \nu \alpha \beta }^{L}(p)% \right] =\sum\limits_{N}\delta ^{(4)}(p-p_{N})\langle 0\|R_{\mu \nu }(0)\|N\rangle \langle N\|R_{\alpha \beta }(0)\|0\rangle .$$ Let us define for $p^{2}>0$$$\begin{aligned} u_{\mu \nu }^{(\lambda )}(p) &=&\frac{\mathrm{i}}{\sqrt{p^{2}}}\left( p_{\mu }\varepsilon _{\nu }^{(\lambda )}(p)-p_{\nu }\varepsilon _{\mu }^{(\lambda )}(p)\right) \\ w_{\mu \nu }^{(\lambda )}(p) &=&\frac{1}{2}\varepsilon _{\mu \nu }^{% \phantom{\mu\nu}\alpha \beta }u_{\alpha \beta }^{(\lambda )}(p)\end{aligned}$$ where $\varepsilon _{\mu }^{(\lambda )}(p)$ are the usual spin-one polarization vectors corresponding to the mass $\sqrt{p^{2}}$. Then for $% p^{2}>0$ we get the following orthogonality relations $$\begin{aligned} u_{\mu \nu }^{(\lambda )}(p)u^{(\lambda ^{^{\prime }})\mu \nu }(p)^{\ast } &=&-2\delta ^{\lambda \lambda ^{^{\prime }}} \\ w_{\mu \nu }^{(\lambda )}(p)w^{(\lambda ^{^{\prime }})\mu \nu }(p)^{\ast } &=&2\delta ^{\lambda \lambda ^{^{\prime }}} \\ u_{\mu \nu }^{(\lambda )}(p)w^{(\lambda ^{^{\prime }})\mu \nu }(p)^{\ast } &=&0\end{aligned}$$ and the projectors can be written for $p^2>0$ in terms of the polarization sums as $$\begin{aligned} \Pi _{\mu \nu \alpha \beta }^{L}(p) &=&-\frac{1}{2}\sum_{\lambda }u_{\mu \nu }^{(\lambda )}(p)u_{\alpha \beta }^{(\lambda )}(p)^{\ast } \\ \Pi _{\mu \nu \alpha \beta }^{T}(p) &=&\frac{1}{2}\sum_{\lambda }w_{\mu \nu }^{(\lambda )}(p)w_{\alpha \beta }^{(\lambda )}(p)^{\ast }.\end{aligned}$$ Multiplying (\[spectral\]) by $u_{\mu \nu }^{(\lambda )}(p)^{\ast }u_{\alpha \beta }^{(\lambda )}(p)$ and $w_{\mu \nu }^{(\lambda )}(p)^{\ast }w_{\alpha \beta }^{(\lambda )}(p)$ respectively we get the positivity constraints for the spectral functions $$\begin{aligned} 0 &\leq &\sum\limits_{N}\delta ^{(4)}(p-p_{N})\|\langle 0\|R_{\mu \nu }(0)\|N\rangle u^{(\lambda ^{^{\prime }})\mu \nu }(p)^{\ast }\|^{2}=2(2\pi )^{-3}\theta (p^{0})\rho _{L}(p^{2})p^{2} \\ 0 &\leq &\sum\limits_{N}\delta ^{(4)}(p-p_{N})\|\langle 0\|R_{\mu \nu }(0)\|N\rangle w^{(\lambda ^{^{\prime }})\mu \nu }(p)^{\ast }\|^{2}=2(2\pi )^{-3}\theta (p^{0})\rho _{T}(p^{2})p^{2}.\end{aligned}$$ [99]{} S. 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[^2]: Here $J$ are the external sources, cf. previous subsecrion. [^3]: Note again that, the field $\Lambda _{\mu }$ has opposite parity than the field $R_{\mu \nu }$ (being pseudovector for proper tensor field $R_{\mu \nu }$ and vice versa). [^4]: Sometimes it is argued [@Lutz:2008km],[@Harada:2003jx], that such a counting can be used within the large $N_{C}$ limit, due to the fact that the natural $\chi PT$ scale $\Lambda _{\chi PT}=4\pi F=O(\sqrt{N_{C}})$ grows with $N_{C}$ while the masses of the resonances behave as $O(1)$. In fact this results only in the suppression of the loops but generally not in the suppression of the counterterm contributions. In the latter case the expansion is rather controlled by the scale $\Lambda _{H}\sim M_{R}=O(1)$, where $M_{R}$ is the typical mass of the higher resonance in the considered channel not included in truncated Lagrangian corresponding to minimal hadronic ansatz. [^5]: In the case of $\mathcal{O}^{RV}$ it seems to be natural from the dimensional reason. [^6]: Note that, each additional mesonic loop yields a further suppression $1/N_{C} $, see also bellow. [^7]: Here and in what follows we use subscript $\mathcal{O}$ when referring to the operator, while the superscript $V$ corresponds to the concrete vertex derived from the operator $\mathcal{O}$. [^8]: Here and in what follows, the sum over $\mathcal{O}$ include all the operators from which the individual vertices entering the graph $\Gamma $ are derived with necessary multiplicity.\[note\] [^9]: In the case of the first order formalism, the mixed propagator behaves as $% O(p^{-1})$. In this case, $d_{\Gamma }=4L-2I_{GB}-I_{RV}+\sum_{\mathcal{O}% }d_{\mathcal{O}}$ where $I_{RV}$ is number of the internal mixed lines. In the following considerations we can take the r.h.s. of (\[d\_ct\]) as an upper bound on $d_{\Gamma }$ with the conclusions unchanged. [^10]: Analogous assignment of the chiral order to the interaction terms with at least two resonance fields is proposed in [@Lutz:2008km], note however, that in this reference it is used by means of substitution $D_V\rightarrow i_{% \mathcal{O}}$ in the Weinberg formula (\[weinberg\]) with counting $M=O(p)$. [^11]: This can be recovered for $n_R^{\mathcal{O}}=0$, when the inequality changes to the equality. [^12]: Note however, that this term can be removed by means of the field redefinition. [^13]: That means at a given level $i_{\mathrm{max}}$ we allow for all the graphs with one-particle irreducible building blocks satisfying $i_\Gamma \leq i_{% \mathrm{max}}$. This point of view is crucial in order to preserve the symmetric properties of the corresponding GF. [^14]: Here we tacitly assume that the trilinear term without derivatives has been removed by means of field redefinition, cf. [Cirigliano:2006hb,Kampf:2006bn]{}. [^15]: Note that, for these counterterms the index $i_{\mathcal{O}}$ coincides with the usual chiral order $D_{\mathcal{O}}$. [^16]: In the case $f_{T}^{I,II}(x)\rightarrow 0$ for $x\rightarrow 1$ when $% f_{T}^{I,II}(x)=(x-1)^{k}$ $g_{T}^{I,II}(x)$ where $k\leq 3$ and when $% g_{T}^{I,II}(x)$ (which has the branching point at $x=1$) has a finite nonzero limit at $% x=1$) we get $[\arg f_{T}^{I,II}(z)]_{C_{\varepsilon }}=-2\pi k$.\[footnote\_zero\] [^17]: Note however, that the requirement that the tree level conditions for OPE are satisfied might be modified by loop corrections. [^18]: In this case the point $x=1$ is solution of $f_{T}^{I,II}(x)=0$ and provided $f_{T}^{I,II}(x)=(x-1)^{k}$ $g_{T}^{I,II}(x)$ (zero with multiplicity $k\le 3$) we have according to the footnote \[footnote\_zero\] the phase deficit $-2\pi k$ ([i.e.]{} the number of the poles different from $z=1$ is then reduced by $k$) in comparison with the case $f_{T}^{I,II}(x)\neq 0$. [^19]: Note, that in this case, $$f_{L}^{I,II}(R\,\mathrm{e}^{\mathrm{i}\theta })=R^{3}\mathrm{e}^{3\mathrm{i}% \theta }\left( -\frac{1}{\pi }\frac{\Gamma _{\mathrm{phys}}}{M_{\mathrm{phys}% }}a_{3}+\frac{40}{9}\left( \frac{M_{\mathrm{phys}}}{4\pi F_{\pi }}\right) ^{2}d_{3}^{2}\left[ 1-\ln R+\mathrm{i}(2\pi -\theta \mp \pi )\right] +O\left( \frac{1}{R},\frac{\ln R}{R}\right) \right)$$and therefore $[\arg f_{L}^{I,II}(z)]_{C_{R}}=6\pi $. [^20]: Here we do not assume the existence of any CDD poles [@Castillejo] for simplicity. In general case, provided the spectral representation of $\Delta _{L,T}$ is valid in the form (\[dispersiv\_kallan\]), and $\mathrm{Im}\Delta _{L,T}^{-1}(s)=O(s^{n})$ for $s\rightarrow \infty $ we formally get $$\Delta _{L,T}^{-1}(z)=P_{n}(z)+Q_{n+1}^{L,T}(z)\left( \frac{1}{\pi }% \int_{x_{t}}^{\infty }\frac{dx}{Q_{n+1}^{L,T}(x)}\frac{\mathrm{Im}\Delta _{L,T}^{-1}(x)}{x-z}-\sum_{i}\frac{C_{i}}{z-z_{0i}}\right)$$where $C_{i}>0$ and $0<z_{0i}<x_{t}$ correspond to the CDD poles. [^21]: In what follows we give such an representation of our one-loop $i_{\Gamma }\leq 6$ result explicitly. [^22]: Note that, the case $n=1$ is in some sense exceptional. In this case it is possible to get a realistic resonance propagator compatible with the Källén-Lehman representation with no pole on the first sheet and one pole on the unphysical sheet. Such a propagator has been obtained in [Achasov:2004uq]{} for scalar resonances. Cf. also [@Giacosa:2007bn]. [^23]: An analogous discussion can be done for the second sheet. Concrete examples of various types of poles will be given in the next section. [^24]: Note that, in order to perform this on the lagrangian level, nonperturbative and nonlocal counterterms would have to be added to the theory. However the status of such a counterterms is not clear, cf. [@Caro:1996ex]. [^25]: We have used the following numerical inputs: $M_{\rm{phys}}=770MeV$, $\Gamma_{\rm{phys}}=150MeV$, $F=93.2MeV$, $F_V=154MeV$. [^26]: While $b_{i}=O(1)$ in the large $N_{C}$ limit, the right hand side of ([bconstraint1]{}) begaves as $O(N_{C})$. [^27]: Similar conditions we get in the $1^{--}$ channel, provided we demand to generate e.g. $\rho (1450)$ dynamically.
--- abstract: 'Debye temperature, $\Theta_D$, of Fe-rich Fe$_{100-x}$Cr$_x$ disordered alloys with $0\le x \le 22.3$ was determined from the temperature dependence of the central shift of Mössbauer spectra recorded in the temperature range of 60 – 300 K. Its compositional dependence shows a maximum at $x \approx 5$ with a relative increase of $\sim 30$% compared to a pure iron. The composition at which the effect occurs correlates well with that at which several other quantities, e. g. the Curie temperature and the spin-wave stiffness coefficient, $D_0$, show their maxima, but the enhancement of $\Theta_D$ is significantly greater and comparable with the enhancement of the hyperfine field (spin-density of itinerant $s$-like electrons) in the studied system. The results suggest that the electron-phonon interaction is important in this alloy system.' author: - 'B. F. O. Costa' - 'J. Cieslak' - 'S. M. Dubiel' title: 'Anomalous behavior of the Debye temperature in Fe-rich Fe-Cr alloys' --- Fe$_{100-x}$Cr$_x$ alloys are both of scientific and technological interests. The former follows, among other, from the fact that they can be regarded as a model system for studying various magnetic properties and testing appropriate theoretical models. The latter is related to the fact that the alloys form a matrix for a production of chromium steels that, due to their excellent properties, find a wide application in industry [@Hishinuma02]. For example, the steels containing 2–20 at% Cr are regarded as good candidates for the design of structural components in advanced nuclear energy installations such as Generation IV and fusion reactors. In that range of composition, the alloys show an anomalous behaviour in that several physical quantities exhibit extreme values. However, their position and the relative value depend on the quantity. For example, the Curie temperature, $T_C$, has its maximum value at $x \approx 5$ at%, which is paralleled by a maximum in the neutron value of the spin-wave stiffness coefficient, $D_0$. However, the relative increase in $T_C$ from the pure Fe is only $\sim$1% [@Adcock31] compared with the $\sim$10% effect in the $D_0$ value [@Lowde65]. An enhancement was also found in the hyperfine (hf) field as measured at $^{57}$Fe nuclei [@Dubiel76] as well as at $^{119}$Sn nuclei [@Dubiel80]. In both cases the maximum was for $x \approx 10$ at% with the relative increase of $\sim$4% for the former and $\sim$15% for the latter probe nuclei. Hyperfine field is usually positively correlated with the magnetic moment, $\mu$, hence in the light of the above mentioned results, one should expect a paralleled behavior for the latter. Indeed, an increased value of $\mu$ localized at Fe site was revealed from neutron diffraction experiments [@Aldred76; @Lander71]. In this case the maximum occurs at $x \approx 20$ at% and its relative enhancement is equal to $\sim$7%. Increment of $\mu$ was predicted theoretically to exist at $x \approx 15$ at% with the relative effect of $\sim$17% [@Frollani75],as well as at $x \approx 5$ at% with the relative effect of $\sim$3%[@Olsson06], and, that of $T_C$ at $x\approx 15$ at% and with the effect of $\sim$25% [@Kakehashi87]. The phenomenon in the latter was explained by the enhancement of Fe-Fe exchange coupling due to the alloying effect. The importance of magnetism in the understanding of these alloys, in general, and Fe-rich ones, in particular, also seems to be crucial in the light of recent theoretical calculations that predict a negative sign of the heat of formation of Fe-Cr alloys with Cr content less than 10–12 at% [@Olsson06; @Klaver06] as well as those obtained with ab initio calculations combined with synergic synchrotron x-ray absorption experiments showing an anomaly at $x \approx 13$ at% [@Froideval07]. Also a drastic decrease in the corrosion rate with chromium content increase occurs within a concentration range of 9-13 at % [@Wranglen85], but its reason is not fully understood. Recent calculations based on first-principles quantum-mechanical theory shed some light on the issue. In particular, they demonstrate that within this concentration range there is a transition between two surface regimes; for bulk Cr content greater than $\sim$10 at% the Cr-rich surfaces become favorable while for a lower concentration the Fe-rich ones prevail [@Ropo07]. According to the authors of Ref. , they are related with two competing magnetic effects: the magnetically induced immiscibility gap in bulk Fe-Cr alloys and the stability of magnetic surfaces. In addition, the authors show that other theoretical bulk and surface properties of Fe-rich ferromagnetic Fe-Cr alloys have their minima viz. effective chemical potential at $x \approx 15$ at%, and the mixing enthalpy at $x \approx 4$ at% [@Ropo07]. The latter concentration coincides well with that at which a change in the sulphidation preference takes place [@Cieslak93]. For the complete, or at least, a better understanding of all theses effects and their relationship, further theoretical and experimental studies are, however, necessary. Towards the latter end, we have carried out measurements of the Debye temperature, $\Theta_D$, for $0 \le x \le 22.3$ in order to verify whether or not this quantity related to the dynamics of the lattice exhibits an enhancement of its value. Though it is widely believed that the possible effect of the electron-phonon interaction on the magnetism of metallic system is of no significant importance [@Herring66], because the modification of the exchange interaction or magnetization caused by the interaction should be of the order of $10^{-2}$. However, calculations by Kim [@Kim82; @Kim88] suggest that in an itinerant-electron ferromagnet the electron-phonon interaction might be important. Including exchange interaction between electrons enhances the effect by a factor of $\sim$10 to $\sim$100 [@Kim82; @Kim88]. If these calculations are correct, then a correlation between magnetic and dynamic properties should be experimentally observed. Although in the literature the are already some data on the Debye temperature in the Fe-Cr system, they are not conclusive in that respect as they were measured in a very narrow and low temperature range viz. 1.4 – 4.2 K [@Cheng60], and, in the concentration range of interest, there are only three data points. A more detailed study of the issue was also prompted by an anomalous behavior of $\Theta_D$ found for the Cr-rich samples, and, in particular, by the smallest value of $\Theta_D$ for the most Cr-rich alloy. ![Room temperature Mössbauer spectra recorded on Fe$_{100-x}$Cr$_x$ samples for various $x$-values (0) $x$ = 0, (1) $x$ = 1.3, (4) $x$ = 3.9, (5) $x$ = 4.85, (10) $x$ = 10.25 and (22) $x$ = 22.3. The solid lines are the best-fit to experimental data. []{data-label="fig1"}](FIG01.eps){width=".42\textwidth"} ![Mössbauer spectra recorded on Fe$_{95.15}$Cr$_{4.85}$ sample at different temperatures (in Kelvin) shown. Solid lines represent the best-fits to the experimental spectra. []{data-label="fig2"}](FIG02.eps){width=".42\textwidth"} For the present study previous samples were used [@Dubiel76; @Dubiel81]. The Debye temperature was measured by means of the Mössbauer spectroscopy. For that purpose a series of Mössbauer spectra was recorded in a transmission geometry for each sample in the temperature range of 60 – 300 K using a standard spectrometer and a $^{57}$Co/Rh source of 14.4 keV gamma rays. Temperature of the samples which were kept in a cryostat were stabilized with an accuracy of $\sim$0.2 K during measurements. Examples of the recorded spectra are shown in Figs. 1 and 2. They were fitted in two ways, I and II, to get an average value of the central shift, $<CS>$, which is the quantity of merit for determining $\Theta_D$. In way I, each spectrum was fitted assuming it consists of a number of six-line patterns subspectra, each of them corresponding to a particular atomic configuration around the probe $^{57}$Fe nucleis, $(m,n)$, where $m$ is a number of Cr atoms in the first neighbor shell (NN), and $n$ is a number of Cr atoms in the next neighbor shell (NNN). It was further assumed that the effect of neighboring Cr atoms on spectral parameters (hyperfine field, and central shift) was additive. Using this procedure, which is described elsewhere in detail [@Dubiel76; @Dubiel80; @Dubiel81], the average central shift, $<CS>$, could have been calculated. In way II, each spectrum was fitted in terms of the hyperfine field distribution method [@LeCaer79]. Following the experimental result [@Dubiel81], a linear correlation between the hf. field and the isomer shift was assumed in the fitting procedure. The Debye temperature was determined from the temperature dependence of $<CS>$: $$<CS(T)> = IS(T) + SODS(T) \label{equ1}$$ where $IS(T)$ is the isomer shift and it is related to the charge density at the probe nucleus and has a weak temperature dependence [@Willigeroth84], so it is usually approximated by a constant term, $IS(0)$, which is eventually composition dependent. $SODS$ is the so-called second-order Doppler shift which shows a strong temperature dependence. Assuming the whole temperature dependence of $<CS>$ goes via $SODS$ term and using the Debye model for the phonon spectrum one arrives at the following formula: $$CS(T) =IS(0)-{3kT \over 2mc}\left[{3\Theta_D \over 8T}+ 3\left({T \over \Theta_D}\right)^3 \int^{\Theta_D \over T}_0 {x^3 \over e^x-1} dx \right] \label{equ2}$$ ![Dependence of the average central shift, $<CS>$, on temperature for Fe$_{95.15}$Cr$_{4.85}$ alloy. The solid line represents the best-fit to the experimental data in terms of equation (\[equ2\]).[]{data-label="fig3"}](FIG03.eps){width=".45\textwidth"} where $m$ is the mass of the $^{57}$Fe nucleus, $k$ is the Boltzmann constant, $c$ is the velocity of light. Fitting equation (\[equ2\]) to the $<CS(T)>$ - values determined in the ways I and II (whose typical temperature behavior is illustrated in Fig. 3), enabled determination of the $\Theta_D$ - values. It has turned out that within the error limit they agree well with each other, hence, for final considerations, the average, $<CS> = 0.5 (<CS>_I + <CS>_{II})$, between each corresponding pair has been taken. Its normalized ($\Theta_D$ = 429 K was found for a pure Fe - see the Table \[table1\]) dependence on Cr content, $x$, is shown in Fig. 4, where, for comparison, normalized values of other physical quantities that have maximum in the their value are shown. $x$ 0.0 1.2 3.9 4.8 8.6 10.2 15.0 22.3 ------------------ ---------- ------- -------- -------- -------- -------- -------- -------- -------- $\Theta_D$ \[K\] 429 493 557 548 569 519 524 463 $\Delta\Theta_D$ \[K\] 14 11 15 11 18 18 33 29 $IS(0)$ \[mm/s\] 0.006 -0.024 -0.030 -0.014 -0.027 -0.020 -0.033 -0.031 : \[table1\] Determined $\Theta_D$ values of Fe-rich Fe$_{100-x}$Cr$_x$ disordered alloys. $\Delta\Theta_D$ stands for the $\Theta_D$-error. Values of $IS(0)$ relative to Co/Rh source are also displayed. From the $IS(0)$-data shown in Table I it is clear that $IS(0)$ hardly depends on $x$, hence the anomaly observed in $\Theta_D(x)$ has its full origin in $SODS$. The concentration at which the maximum occurs is similar or very close to that at which several other quantities or phenomena show anomalies, and, in particular, (a) the Curie temperature [@Adcock31] and (b) the neutron value of the spin-wave stiffness coefficient, $D_0$ [@Lowde65]. The increase of the former can be explained in terms of the increase of the magnetic bonding between Fe atoms [@Kakehashi87]. However, no quantitative agreement between the enhancement factor of $T_c$ and $Theta_D$ exists. That of the latter is significantly greater and it is rather close to the enhancement of the $^{119}$Sn-site hyperfine field, $H(0,0)$, but not to the one of the $^{57}$Fe-site hyperfine field, $H(0,0)$. As the former mostly originates from the polarization of the conduction (itinerant) electrons and the latter from the polarization of the core (localized) electrons, one is tempted to conclude that for the enhancement of $\Theta_D$ revealed in this study the itinerant electros are predominantly responsible. ![(Color online) Normalized values of various physical quantities, $A/A(0)$, as measured (presented with various symbols connected by lines to guide the eye) or calculated (for the magnetic moment $\mu_1$ illustrated by a dotted line [@Frollani75] and for $\mu_2$ by dashed-dotted line [@Olsson06] as well as the Curie point by a dashed line [@Kakehashi87]) for Fe$_{100-x}$Cr$_x$ alloys that show a maximum in the concentration range of $0 \le x \le 26$.[]{data-label="fig4"}](FIG04.eps){width=".45\textwidth"} In other words, the change of the Debye temperature of Fe-Cr alloys in the investigated range of their composition reflects underlying change in the polarization of the conduction electrons. This in turn, can be taken as evidence that in the investigated alloy system there is a coupling between spin-polarization and the phonon spectrum. A lack of a quantitative agreement as measured in terms of the position of the maximum and the amplitude of the enhancement means that the coupling is not linear. The results presented in this paper were obtained in the frame of the bilateral Polish-Portuguese project 2007/2008. [58]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , , **, (). , , (). , , , , (). , , (). , , (). , , , , (). , , (). , , , (). , , , (). , , (). , , , (). , , , , , , , , (). , , (, ). , , , , , , **, (). , , (). , , , , (, **, ). , , (). , , (). , , , (). , , (). , , (). , , **, ().
--- abstract: \| We consider the probability distributions of values in the complex plane attained by Fourier sums of the form $\frac{1}{\sqrt{n}}\sum_{j=1}^n a_j e^{-2\pi i j \nu}$ when the frequency $\nu$ is drawn uniformly at random from an interval of length $1$. If the coefficients $a_j$ are i.i.d. drawn with finite third moment, the distance of these distributions to an isotropic two-dimensional Gaussian on ${{\mathbb C}}$ converges in probability to zero for any pseudometric on the set of distributions for which the distance between empirical distributions and the underlying distribution converges to zero in probability. author: - 'Dominik Janzing [^1]' - Naji Shajarisales - Michel Besserve date: 'July 21, 2017' title: A central limit like theorem for Fourier sums --- The classical version of the central limit theorem states that for a series of real-valued independent identically distributed (iid) random variables $X_1,X_2,\dots$ with ${\mathbf{E}}[X_j]=0$ and finite variance $\sigma^2$ the sequence $$\frac{1}{\sqrt{n}} \sum_{j=1}^n X_j$$ converges in distribution to a Gaussian random variable with zero mean and variance $\sigma^2$ [@billingsley1995 p.357]. Formulations for random vectors $X_j$ state convergence to multi-variate Gaussians [@vaart1998]. Other well-known generalizations drop the assumption ‘identically distributed’ and replace it, for instance, with the Lyapunov condition $$\lim_{n\to \infty} \frac{1}{s_n^{2+\delta}} \sum_{i=1}^n {\mathbf{E}}[X_j^{2+\delta}] =0,$$ for some $\delta>0$, where $s_n$ denotes the sum of all variances of $X_1,\dots,X_n$ [@billingsley1995 p.362]. Then $$\frac{1}{\sqrt{s_n}} \sum_{j=1}^n X_j$$ converges in distribution to a standard Gaussian. It is also known that the independence assumption can be replaced with appropriate notions of weak dependence, e.g. [@billingsley1995 Theorem 27.4]. However, significantly more general scenarios yield Gaussians as limiting distributions. Here we consider sequences of Fourier sums of the form $$\label{eq:ahat} \hat{a}^n(\nu) :=\frac{1}{\sqrt{n}} \sum_{j=1}^n a_j e^{-2\pi i \nu j},$$ and show that sampling from random frequencies yields asymptotically a Gaussian – in a sense to be specified below – if the coefficients $a_j$ are i.i.d. drawn. More precisely, let $A_1,A_2,\dots$ be a sequence of real-valued i.i.d. variables on a probability space $(\Omega,\Sigma,P_\Omega)$. Then we first define for each frequency $\nu$ the sequence $(\hat{A}^n_\nu)_{n\in {{\mathbb N}}}$ of random variables via $$\hat{A}^n_\nu := \frac{1}{\sqrt{n}} \sum_{j=1}^n A_j e^{-2\pi i \nu j}.$$ Using known vector-valued central limit theorems one can easily show that, under some technical condition of Lyapunov type detailed below, $\hat{A}^n_\nu$ converges to a Gaussian on the complex plane for each $\nu$ since it is obtained by a sum of the independent (but not identically distributed) complex-valued random variables $X_j:=A_j e^{-2\pi i \nu j}$. Here, however, we define for each $\omega \in \Omega$ the sequence $(\hat{A}^n_.(\omega))_{n\in {{\mathbb N}}}$ of random variables on the probability space $([-1/2,1/2],{{\cal B}},\lambda)$, with ${{\cal B}}$ denoting the Borel sigma algebra and $\lambda$ the Lebesque measure (to formalize the random choice of a frequency), via $$\hat{A}^n_. (\omega) :\quad \nu \mapsto \hat{A}^n_\nu(\omega)= \emph{•}\frac{1}{\sqrt{n}} \sum_{j=1}^n A_j(\omega) e^{-2\pi i \nu j}.$$ The problem is motivated by Ref. [@shajarisales2015telling] which considers linear time invariant filters whose coefficients are randomly chosen. The question arising there was how the filter’s frequency response behaves in the limit $n\to\infty$ for ‘typical’ choices of filter coefficients $a_j$ when the latter are randomly drawn. Note that the problem would become simple if we were to consider $\hat{a}^n(\nu)$ at the discrete frequencies $\nu_l:=j/n$ for $l=1,\dots,n/2$ (in signal processing $\|\hat{a}^n(\nu_l)\|^2$ is also known as the periodogram of a signal [@fay2001]) and $a_j$ were assumed to be drawn from independent Gaussians. For Gaussian $A_j$, the random variables $\hat{A}^n_{\nu_l}$ defined on the probability space $(\Omega,\Sigma,P_\Omega)$ are also independent Gaussians for these different discrete frequencies, which can easily checked by computing the covariances. The question changes drastically when we consider the full continuum of frequencies, because $\hat{A}^n_\nu$ and $\hat{A}^n_{\nu'}$ are not in general independent for $\nu \neq \nu'$. We will show, however, that they become asymptotically independent, which then results in an appropriate limit theorem for $\hat{A}^n_.(\omega)$. Crucial to phrase our limit theorem is the following type of distance measures on probability distributions: \[def:well\] For an arbitrary sequence $Z_1,Z_2,\dots$ of i.i.d. random variables on the probability space $(\Omega',\Sigma',P_{\Omega'})$ let $P_Z$ denote the distribution of each $Z_j$ and $\hat{P}_{Z_1(\omega'),\dots,Z_k(\omega')}$ denote the empirical distribution after the first $k$ samples. Let ${\cal M}_l$ denote the set of probability measures on the Borel-measurable subsets of ${{\mathbb R}}^l$. Then a pseudometric $d:{{\cal M}}_l \times {{\cal M}}_l\rightarrow {{\mathbb R}}^+_0$ is called ‘well-behaved’ if the distance between $P_Z$ and $\hat{P}_{Z_1(\omega'),\dots,Z_k(\omega')}$ converges in probability to zero uniformly over all i.i.d. sequences. More precisely, for every $\epsilon,\delta>0$ there is a $k_0$ such that for all $k\geq k_0$ $$P_{\Omega'} \left\{ d(P_Z,\hat{P}_{Z_1(\omega'),\dots,Z_k(\omega')}) \geq \epsilon \right\} \leq \delta,$$ holds for all sequences $Z_1,Z_2,\dots$ and probability spaces $(\Omega',\Sigma',P_{\Omega'})$. For distributions $Q,R$ on ${{\mathbb R}}$ with cumulative distribution functions $F_Q$ and $F_R$, respectively, $d(Q,R):=\\|F_Q-F_R\\|_\infty$ provides a simple example of a well-behaved distance since $$\\|F_{\hat{Q}_k} -F_Q\\|_\infty \geq \epsilon,$$ occurs with probability at most $2e^{-2k \epsilon^2}$ due to Massart’s formulation [@Massart90] of the Dvoretzky-Kiefer-Wolfowith (DKW) inequality. Another example is given by $d(Q,R):=\sup_B \|Q(B)-R(B)\|$ where $B$ runs over some set of sets whose indicator functions have finite VC-dimension. This follows from Vapnik and Chervonenkis’ uniform bound on the deviation of empirical frequencies of events from the corresponding probabilities [@vapnik98]. Using Reproducing Kernel Hilbert Spaces (RKHS) one can construct a further example: the so-called kernel mean embedding [@smola2007] represents distributions as vectors in a Hilbert space. Then the Hilbert space distance is a well-behaved metric. This follows easily from the uniform consistency result in [@gretton2006 Theorem 4] for the empirical estimator of this distance[^2]. The purpose of this article is to show the following result: \[thm:main\] Let $P_{\hat{A}^n_.(\omega)}$ denote the distribution of $\hat{A}^n_.(\omega)$ and $G$ the distribution on ${{\mathbb C}}$ for which real and imaginary parts are independent Gaussians with mean zero and variance $1/2$. Then the distance between $P_{\hat{A}^n_.(\omega)}$ and $G$ converges to zero in probability for every well-behaved pseudometric $d$. More, precisely, the random variable $$\omega \mapsto d(P_{\hat{A}^n_.(\omega)},G)$$ converges to zero in probability. Obviously, the interval $[-1/2,1/2]$ can be replaced by any interval of length $1$, as stated in the abstract. The first step of the proof will be to investigate the asymptotics of the variances and covariances of real and imaginary part of $\hat{A}^n_\nu$ and the covariances between real and imaginary parts $\hat{A}^n_\nu$ and $\hat{A}^n_{\nu'}$ for different frequencies $\nu,\nu'$. To this end, we represent complex numbers as vectors in ${{\mathbb R}}^2$ and obtain the following result: \[lem:covasym\] Let $\nu_1,\dots,\nu_k$ be some arbitrary non-zero frequencies in $(-1/2,1/2)$ with $\|\nu_i\|\neq \|\nu_j\|$ for $i\neq j$. Let $C^{(n)}$ denote the covariance matrix of the random vector $\frac{1}{\sqrt{n}}\sum_{j=1}^n S_j$ with $$S_j := A_j \Big(\cos(2\pi \nu_1 j),\sin(2\pi \nu_1 j), \cos(2\pi \nu_2 j),\sin(2\pi \nu_2 j),\dots, \cos(2\pi \nu_k j),\sin(2\pi \nu_k j)\Big)^T.$$ Then $$\lim_{n\to\infty} C^{(n)} = \frac{1}{2} {\bf 1},$$ where ${\bf 1}$ denotes the identity in $2k$ dimensions. Proof: We first introduce the vector $c:=(1,0)^T$ and the rotation matrix $$D_\nu := \left(\begin{array}{cc} \cos(2\pi \nu ) & -\sin (2 \pi \nu) \\ \sin (2\pi \nu ) & \cos (2\pi \nu) \end{array} \right).$$ Using powers of these rotations, we can write the random vector $S_j$ as the direct sum $$S_j := A_j \left[ D^j_{\nu_1} c \oplus D^j_{\nu_2} c \oplus \cdots \oplus D^j_{\nu_k} c\right].$$ Its covariance matrix reads $$C_j := \underbrace{\left(\begin{array}{cccc} D_{\nu_1}^j & & & \\ & D_{\nu_2}^j & & \\ & & \ddots & \\ & & & D_{\nu_k}^j \end{array}\right)}_{D^j} \underbrace{ \left(\begin{array}{cccc} c c^T & cc^T & \cdots & ccT \\ cc^T & \ddots & & \vdots \\ \vdots & & & \\ cc^T & \cdots & & cc^T \end{array}\right) }_{C_0} \underbrace{ \left(\begin{array}{cccc} D_{\nu_1}^{-j} & & & \\ & D_{\nu_2}^{-j} & & \\ & & \ddots & \\ & & & D_{\nu_k}^{-j} \end{array}\right)}_{D^{-j}}.$$ Since the random vectors $S_1,\dots,S_n$ are uncorrelated (because the variables $A_j$ are independent and thus uncorrelated), the weighted sum $$S^{(n)}:=\frac{1}{\sqrt{n}} \sum_{j=1}^n S_j$$ has the covariance matrix $$C^{(n)} :=\frac{1}{n} \sum_{j=1}^n D^j C_0 D^{-j}.$$ Block $ll'$ within the $k\times k$ block matrices of format $2\times 2$ reads $$C^{(n)}_{ll'} = \frac{1}{n} \sum_{j=1}^n F_{ll'}^j (cc^T),$$ where $F^j_{ll'}$ denotes the $j$th power of the map $F_{ll'}$ on the space ${\cal M}_2({{\mathbb C}})$ of complex-valued $2\times 2$-matrices defined via $$F_{ll'}(M):= D_{\nu_l} M D^{-1}_{\nu_{l'}}.$$ Note that $F_{ll'}$ is a unitary map on ${\cal M}_2({{\mathbb C}})$ with respect to the inner product $\langle A, B\rangle:={\rm tr}(B^\dagger A)$, where $\dagger$ denotes the Hermitian conjugate. Therefore, von Neumann’s mean ergodic theorem [@ReedSimon80] implies $$\frac{1}{n} \sum_{j=1}^n F_{ll'}^j (cc^T) = Q_{ll'} (cc^T),$$ where $Q_{ll'}$ denotes the orthogonal projection[^3] onto the $F_{ll'}$-invariant subspace of ${\cal M}_2({{\mathbb C}})$. If $r_1:=(1,i)^T$ and $r_2:=(1,-i)^T$ denote the joint eigenvectors of all $D_\nu$ with eigenvalues $e^{\pm 2\pi \nu}$ then $F_{ll'}$ has the $4$ eigenvectors $r_{j} r_l^\dagger $ with $j,l=1,2$ and eigenvalues $e^{i 2\pi (\pm \nu_l \pm \nu_{l'})}, e^{i 2\pi (\pm \nu_l \mp \nu_{l'})} $. For $l\neq l'$ the $F_{ll'}$-invariant subspace is $0$ because all eigenvalues differ from $1$ due to $0\neq \|\nu_l\|\neq \|\nu_{l'}\|\neq 0$. Hence, the non-diagonal blocks of $C^{(n)}$ vanish in the limit. To consider the diagonal blocks, note that $F_{ll}$ is then just the adjoint map of $D_{\nu_l}$, which can be restricted to the space of real-valued symmetric matrices ${\cal M}^{sym}_2({{\mathbb R}})$. We then conclude $$\lim_{n \to \infty} \frac{1}{n} \sum_{j=1}^n F^j_{ll} (M) = Q_{ll}(M) = \frac{1}{2} {\rm tr} (M) {\bf 1} \quad \forall M \in {\cal M}^{sym}_2({{\mathbb R}}).$$ This is since multiples of the identity are the only real symmetric matrices that commute with $D_\nu$ for $\nu \in (-1/2,1/2) \setminus\{0\}$ and because $F_{ll}$ preserves the trace. $\Box$ We now state the following central limit theorem for random vectors [@bentkus05]: \[lem:bentkus\] Let $X_1,\dots,X_n$ be independent random vectors in ${{\mathbb R}}^d$ such that ${\mathbf{E}}[X_j]=0$ for all $j$. Write $S:=\sum_{j=1}^n X_j$ and assume that the covariance matrix $C_S$ of $S$ is invertible. Let $Z$ be a centered Gaussian random vector with covariance matrix $C_S$. Let ${\cal C}$ denote the set of convex subsets of ${{\mathbb R}}^d$. Then, $$\label{eq:boundbentkus} \sup_{B \in {\cal C}} \left\|P\{ S\in B \} - P\{Z \in B\}\right\| \leq \eta d^{1/4} \sum_{j=1}^n \beta_j,$$ with $$\beta_j:= {\mathbf{E}}\left[\left\\| \sqrt{C_S}^{-1} X_j\right\\|^3\right],$$ for some $\eta>0$. Since we will not use the explicit bound we derive a simpler asymptotic statement as implication: \[lem:derived\] Let $Y_j$ with $j\in {{\mathbb N}}^$ independent random vectors with covariance matrices $C_j$ such that $C^{(n)}:=\frac{1}{n} \sum_{j=1}^n C_j$ converges to some invertible matrix $C$ with respect to any matrix norm. Assume, moreover, that there exists a constant $b<\infty$ such that ${\mathbf{E}}[\\|Y_j\\|^3] \leq b$ for all $j$. Then $$S^{(n)}:=\frac{1}{\sqrt{n}} \sum_{j=1}^n Y_j$$ converges in distribution to a multivariate Gaussian with covariance matrix $C$. Proof: Applying Lemma \[lem:bentkus\] to the variables $X_j:=\frac{1}{\sqrt{n}} Y_j$ yields $$\beta_j = \frac{1}{n^{3/2}}{\mathbf{E}}\left[\left\\|\sqrt{C^{(n)}}^{-1} Y_j\right\\|^3\right] \leq \frac{1}{n^{3/2}} \left\\|\sqrt{C^{(n)}}^{-1}\right\\|^3 b,$$ where $\\|. \\|$ denotes the operator norm. Here we have assumed that $C^{(n)}$ is invertible, which is certainly true for sufficiently large $n$ since $C$ is invertible. Since $\left\\|\sqrt{C^{(n)}}^{-1}\right\\|$ converges to the constant $\gamma:=\\|\sqrt{C}^{-1}\\|$, we can bound $\sum_{j=1}^n \beta_j$ for all $n\geq n_0$ for sufficiently large $n_0$ by $(\gamma+\epsilon) b/\sqrt{n}$ with some fixed $\epsilon$. Let $Z_n$ be a Gaussian with covariance matrix $C^{(n)}$ and $Z$ be a Gaussian with covariance matrix $C$. Since the right hand side of converges to zero, we have $$\label{eq:convA} \sup_{B \in {\cal C}} \|P\{ S^{(n)}\in B \} - P\{Z_n \in B\}\| \rightarrow 0.$$ Since $P\{ Z_n \in B \}$ converges to $P\{ Z \in B\}$ uniformly in $B$ (this is because the mapping of the covariance matrix to its Gaussian density is continuous at $C$ for the uniform norm topology on the mapping’s codomain) remains true when $Z_n$ is replaced with $Z$. Hence, $S^{(n)}$ converges in distribution to $Z$. $\Box$ We now combine Lemma \[lem:derived\] and Lemma \[lem:covasym\] and obtain: \[lem:indepGauss\] Given $k$ frequencies $\nu_1,\nu_2,\dots,\nu_k \in [-1/2,1/2]\setminus \{0\} $ with $\|\nu_j\|\neq \|\nu_{j'}\|$ for $j\neq j'$ and assume ${\mathbf{E}}[\|A_j\|^3]$ to be finite. Then the sequence of random vectors $$(\hat{A}^n_{\nu_1},\hat{A}^n_{\nu_2},\dots,\hat{A}^n_{\nu_k}) \in {{\mathbb C}}^k$$ converges in distribution to $(W_1,\dots,W_k)$ where $W_l$ are i.i.d. random variables with distribution $G$ as in Theorem \[thm:main\]. The proof is immediate after representing real and imaginary parts of each $\hat{A}^n_{\nu_l}$ by an ${{\mathbb R}}^2$-valued random variable as in Lemma \[lem:covasym\] and applying Lemma \[lem:derived\] to the random vector in ${{\mathbb R}}^{2k}$. A uniform bound for ${\mathbf{E}}[\\|S_j\\|^3]$ follows easily from finiteness of ${\mathbf{E}}[\|A_j\|^3]$. The fact that different $\hat{A}^n_\nu$ are asymptotically independent and identically distributed for different $\nu$ has a very intuitive consequence: computing the Fourier sum $\hat{a}^n(\nu)$ for different frequencies and one fixed instance $a=a_1,a_2,\dots$ resembles the distribution of $\hat{A}^n_\nu$ for fixed $\nu$. This suggests that the distribution of $\hat{A}^n_\nu(\omega)$ with $\nu$ uniformly chosen from $[-1/2,1/2]$ and fixed $\omega$ yields asymptotically also a Gaussian. To formally phrase this idea we first need the following result: \[lem:paths\] Let $(\tau,\Sigma_\tau,P_\tau)$ denote a probability space and for each $t\in \tau$, let $(X^n_t)$ be a sequence of random vectors on the probability space $(\Omega,\Sigma,P_\Omega)$. Further, assume that the map $$X^n_.(\omega): t\mapsto X^n_t(\omega)$$ is $\Sigma_\tau$-measurable for all $n\in {{\mathbb N}}$ and $\omega \in \Omega$ and thus defines a random variable on $(\tau,\Sigma_\tau,P_\tau)$ whose distribution we denote by $P_{X^n_.(\omega)}$. For every $k\in {{\mathbb N}}$ and $P_\tau^k$-almost all $k$-tuples $(t_1,\dots,t_k)$ let the sequence of random vectors $(X^n_{t_1},\dots,X^n_{t_k})$ converge in distribution to $(Z_1,\dots,Z_k)$ for $n\to \infty$ where $Z_1,\dots,Z_k$ are i.i.d. random variables on $(\Omega',\Sigma',P_{\Omega'})$ with distribution $P_Z$. Then the distance $d(X^n_.(\omega),P_Z)$ converges in probability to zero for any well-behaved pseudometric $d$. More precisely, the random variable $$\omega \mapsto d(X^n_.(\omega),P_Z)$$ on $(\Omega,\Sigma,P_\Omega)$ converges to zero in probability. Proof: We have to show that for every $\epsilon,\delta >0$ there is an $n_0$ such that $$P_\Omega \left\{ \omega\,\big\|\, d(P_{X^n_.(\omega)}, P_Z) \geq \epsilon \right\} \leq \delta,$$ for all $n\geq n_0$. For any $k\in {{\mathbb N}}$, let $t_1,\dots,t_k$ be i.i.d. drawn from $P_\tau$. Since $d$ is well-behaved, the distance between $\hat{P}_{X^n_{t_1}(\omega),\dots,X^n_{t_k}(\omega)}$ and $P_{X^n_.(\omega)}$ converges to zero in probability uniformly in $n$. Thus, we can choose $k$ such that for all $n$ $$\label{eq:firsterm} d(P_{X^n_.(\omega)},\hat{P}_{X^n_{t_1}(\omega),\dots,X^n_{t_k}(\omega)}) \leq \epsilon/2$$ holds with probability at least $1-\delta/3$, and that, at the same time, $$\label{eq:Zemp} d(\hat{P}_{Z_1,\dots,Z_k},P_Z) \leq \epsilon/2$$ also holds with probability at least $1-\delta/3$. Using the triangle inequality for $d$ we obtain $$\label{eq:2terms} d(P_{X^n_.(\omega)},P_Z) \leq d(P_{X^n_.(\omega)}, \hat{P}_{X^n_{t_1}(\omega),\dots,X^n_{t_k}(\omega)}) + d(\hat{P}_{X^n_{t_1}(\omega),\dots,X^n_{t_k}(\omega)}, P_Z).$$ The sequence of random vectors $(X_{t_1}^n,\dots,X_{t_k}^n)$ converge in distribution to $(Z_1,\dots,Z_k)$ for each fixed $k$-tuple $(t_1,\dots,t_k)$. In other words, for any measurable set $B$ in ${{\mathbb R}}^k$ we have $$\label{eq:Bconv} \lim_{n\to \infty} P_\Omega\{\omega \,\|\, (X_1^n(\omega),\dots,X^n_k(\omega)) \in B \} = P_{\Omega'} \{ \omega'\,\|\,(Z_1(\omega'),\dots,Z_k(\omega')) \in B\} .$$ Setting $$B:= \{z_1,\dots,z_k\,\| d(\hat{P}_{z_1,\dots,z_k},P_Z) \leq \epsilon/2 \},$$ we conclude from that we can find an $n_0$ such that $$\label{eq:distreplace} P_\Omega \{ \omega\,\| d(\hat{P}_{X^n_{t_1}(\omega),\dots,X^n_{t_k}(\omega)},P_Z) \leq \epsilon/2 \} \geq P_{\Omega'} \{ \omega'\,\| d(\hat{P}_{Z_1(\omega'),\dots,Z_k(\omega')},P_Z) \leq \epsilon/2\} -\delta/3,$$ for all $n\geq n_0$. Using and we can thus choose $n_0$ such that for all $n\geq n_0$ $$\label{eq:secondterm} d(\hat{P}_{X^n_{t_1}(\omega),\dots,X^n_{t_k}(\omega)} , P_Z) \leq \epsilon/2$$ with probability at most $1- 2 \delta/3$. Combining with we have thus ensured that the right hand side of is smaller than $\epsilon$ with probability at least $1-\delta$. $\Box$ We are now able to prove Theorem \[thm:main\] via Lemma \[lem:paths\]. To this end, let $(\tau,\Sigma_\tau,P_\tau)= ([-1/2/1/2],{{\cal B}},\lambda)$ and set $$X^n_. (\omega) := ({\rm Re} \hat{A}^n_.(\omega), {\rm Im} \hat{A}^n_.(\omega)),$$ where ${\rm Re}$ and ${\rm Im}$ denote real and imaginary part, respectively. If we then draw $d$ frequencies $\nu_1,\dots,\nu_d$ for arbitrarily large $d$, we satisfy $P^k_\tau$ -almost surely the condition $0 \neq \|\nu_j\| \neq \|\nu_{j'}\| \neq 0$ required by Lemma \[lem:indepGauss\]. Thus, the sequence of random vectors $(\hat{A}^n_{\nu_1},\dots,\hat{A}^n_{\nu_k})$ converges in distribution to $(W_1,\dots,W_k)$ where $W_j$ are distributed according to $G$. Therefore, the random variable $$\omega \mapsto d(P_{\hat{A}^n_.(\omega)},G)$$ converges to zero in probability due to Lemma \[lem:paths\]. [10]{} P. Billingsley. . Series in Probability and Mathematical Statistics. John Wiley & Sons, New York$ $, 1995. A. Van der Vaart. . Cambridge University Press, New Tork, 1998. N. Shajarisales, D. Janzing, B. Schölkopf, and M. Besserve. Telling cause from effect in deterministic linear dynamical systems. In [Proceedings of the 32nd International Conference on Machine Learning (ICML-15)]{}, pages 285–294, 2015. G. Fay and P. Soulier. The periodogram of an i.i.d. sequence. , pages 315–343, 2001. P. Massart. . , 18(3):1269–1283, 1990. V. Vapnik. . John Wileys & Sons, New York, 1998. A. Smola, A. Gretton, L. Song, and B. Schölkopf. . , 4755:40–41, 2007. A. Gretton, K. Borgwardt, M. Rasch, B. Schölkopf, and A. Smola. A kernel method for the two-sample-problem. In [Proceedings of the 19th International Conference on Neural Information Processing Systems*]{}, NIPS’06, pages 513–520, 2006. M. Reed and B. Simon. . Academic Press, San Diego, California, 1980. V. Bentkus. . , 49(2):311–323, 2005. [^1]: corresponding author: dominik.janzing@tuebingen.mpg.de [^2]: If both samples in [@gretton2006 Theorem 4] are drawn from the same distribution and one of the sample sizes tends to infinity the bound describes the distance between empirical and true distribution. [^3]: We could have also applied the mean ergodic theorem to unitary map $C_0\mapsto D C_0 D^{-1}$ instead of applying it to each block separately, but finally this would not have simplified the analysis.
--- abstract: 'The limits of applicability of the Lang-Kobayashi (LK) model for a semiconductor laser with optical feedback are analyzed. The model equations, equipped with realistic values of the parameters, are investigated below solitary laser threshold where Low Frequency Fluctuations (LFF) are usually observed. The numerical findings are compared with experimental data obtained for the selected polarization mode from a Vertical Cavity Surface Laser (VCSEL) subject to polarization selective external feedback. The comparison reveals the bounds within which the dynamics of the LK can be considered as realistic. In particular, it clearly demonstrates that the deterministic LK, for realistic values of the linewidth enhancement factor $\alpha$, reproduces the LFF only as a transient dynamics towards one of the stationary modes with maximal gain. A reasonable reproduction of real data from VCSEL can be obtained only by considering noisy LK or alternatively deterministic LK for extremely high $\alpha$-values.' author: - 'Alessandro Torcini$^{(a,d)}$, Stephane Barland$^{(b)}$, Giovanni Giacomelli$^{(a)}$, and Francesco Marin$^{(c,d)}$' title: \| Low frequency fluctuations in a Vertical Cavity Lasers:\ experiments versus Lang-Kobayashi dynamics --- Introduction {#sec:0} ============ The dynamics of semiconductor lasers with optical feedback is studied both experimentally and theoretically since almost 30 years ([@primolff], for a review see e.g. [@book_lff] ). The interest for such configuration, commonly encountered in many applications (e.g, communication in optical fibers, optical data storage, sensing etc) arises from the rich phenomenology observed, ranging from multistability, bursting, intermittency, irregular and rare drops of the intensity (low frequency fluctuations (LFF)) and transition to developed chaos (coherence collapse (CC)). A complete understanding of the physical mechanisms at the basis of such complex behavior is, however, still lacking. In particular, the origin of the LFF regime is under debate since the very first observations and yet this puzzling problem has not been solved. Their origin was ascribed to stochastic effects [@hk; @hohl] or to deterministic but chaotic dynamics [@sano], and more recently even to the interplay between regular periodic and quasi-periodic solutions [@david]. The LFF dynamics has been investigated by using several type of emitters, mainly edge-emitting: ranging from longitudinal multimode [@multi_exp1; @excit; @multi_exp2; @multi_teo1; @multi_teo2; @multi_teo3; @solari] to single-mode DFB [@dfb] semiconductor lasers. From the experimental point of view, a complete characterization of the LFF dynamics is quite difficult, because of the very different time-scales involved [@romanelli]. Indeed, fast oscillations on the 10-ps range have been observed with streak camera measurements [@streak1; @streak2], representing the fundamental scale on which the system evolves. On the other hand, the duration of such fast pulsing regime between LFF events can be as long as hundreds of nanoseconds or even microseconds. In literature, the LFF dynamics has been experimentally characterized in several manners, starting from a relatively simple statistical analysis of the time separation T between LFF[@sukov; @stat1; @dfb] (relating the average $\langle T \rangle$ between LFF with the pump current) to Hurst exponents for the laser phase dynamics [@lam]. A widely used theoretical description of the system is the Lang-Kobayashi model [@LK], introduced in 1980 in the effort to provide a simplified but effective analysis of an edge-emitting semiconductor laser optically coupled with a distant reflector. In the model, both the multiple reflections from the mirror (low coupling) and the possible multimodal structure of the laser were neglected. The opportunity to include such effects has been discussed in several papers [@balle_josab; @solari; @model_gen; @multi_teo1; @multi_teo2; @multi_teo3], but the model still remains presented as the standard theoretical approach to the system. While most of the phenomenology observed in the different experiments is grab by the model, quite often a more precise or quantitative comparison is obtained at the expense of a choice of parameters far from those actually measured or even not physically plausible. Recently, a new configuration has been proposed and studied, based on a Vertical Cavity Surface Laser (VCSEL) with a polarized optical feedback [@romanelli]. Such a laser is longitudinal single-mode (due to the very short cavity) but may support different, high order transverse modes for strong enough pumping current (see e.g.[@book_vcsel]). The simmetry of the cavity allows also for the possible laser action on two different, linear polarizations selected by the crystal axis. The dynamics of the VCSEL with isotropical optical feedback has been examined experimentally in [@balle_josab; @naumenko03] and theoretically in [@masoller], while the role played by polarized optical feedback has been discussed in [@besnard; @loiko; @loiko01]. In particular, the setup used in [@romanelli] employed a polarizer in the feedback arm, in order to couple back only the radiation of one polarizations; moreover, a suitable range of pump current was chosen, to assure single transverse mode behavior. In such configuration, the appearance of LFF was reported and characterized. The possibility to control the role of the laser modes in the dynamics in this setup allows for a consistent description via the LK model and therefore for an effective test of its predictions, at variance with similar setup where instead the polarization selection were not used [@balle_josab; @naumenko03]. Our aim in the present paper is to clarify the origin of the LFF dynamics by comparing numerical results obtained by solving the LK equations with experimental measurements done on a VCSEL. In particular, the parameters employed for the integration of the LK rate equations have been carefully derived from the analysis of the same VCSEL employed for the experiments [@bar05]. In Sec. II we describe our experimental setup, reporting the main phenomenology observed in the range of variation of the more relevant parameters of the system, namely, the pump current and the phase of feedback. The LK model is introduced and commented in Sec. III, together with the numerical methods employed for its integration and the choice of the parameter values derived from the experiment. In Sec. IV the properties of the stationary solutions are discussed, while in Sec. V a careful characterization of the deterministic model is given, detailing the transient phenomena and the Lyapunov analysis. In Sec. VI the effect of noise is introduced and analyzed, discussing also its possible importance in the experiment. A detailed comparison of the numerical results with the experimental measurements is given in Sec. VII, with particular regard to the distribution of the intensity and of the inter-event times for different parameter choice, including the alpha-factor and the acquisition bandwidths. Finally, we draw our conclusions in Sec. VIII. Experimental Setup and Settings {#sec:1} =============================== ![(a) Average power output versus the input current for the solitary laser (dots) and for the laser with feedback (stars). (Color online) (b) Polarization modes: power outputs as a function of time for the VCSEL with feedback at $I=2.64$ mA. Upper trace (black): main polarization; lower trace (red): secondary polarization. []{data-label="fig:exp1"}](f1a "fig:") ![(a) Average power output versus the input current for the solitary laser (dots) and for the laser with feedback (stars). (Color online) (b) Polarization modes: power outputs as a function of time for the VCSEL with feedback at $I=2.64$ mA. Upper trace (black): main polarization; lower trace (red): secondary polarization. []{data-label="fig:exp1"}](f1b "fig:") The experimental measurements are performed using a VCSEL semiconductor laser with moderate polarized optical feedback. In particular, our analysis is limited to a regime of pumping below the solitary threshold $I_{th} \sim 2.76$ mA, where the VCSEL emits light on a single linearly polarized transverse mode. Longitudinal modes are not allowed by the cavity, and other transverse modes are not present up to currents $\sim 6.5$ mA. The solitary laser emission remains well polarized up to roughly the same current (see Fig. \[fig:exp1\] (b)). The technical details of the source are the following. The laser is an air-post VCSEL made by CSEM [@gulden], operating around 770 nm. The mesa diameter is 9.4 $\mu$m, a ring contact defines the ouput window with a diameter of 5 $\mu$m, and the active medium is composed of three 8 nm quantum wells. The temperature of the laser case is stabilized within 1 mK, the pump current is controlled by a home made battery operated power supply whose current noise is below 40 pA/Hz$^{-1/2}$ in the frequency range from 1kHz to 3 MHz. The feedback is applied on the polarization direction of the solitary laser emission. The external cavity includes collimation optics, two polarizers, a variable attenuator, and the feedback mirrors that is mounted on a piezoelectric transducer at about 50 cm from the laser. The output radiation, after optical isolators, is detected by an avalanche photodiode with a bandwidth of about 2 GHz, whose signal, sometimes after low-pass filtering, is recorded by a 4 GHz bandwidth digital scope. More details on the experiment can be found in Refs [@romanelli; @Soriano2004]. Optical feedback results in a reduced threshold $I_{th}^{red} \sim 2.42$ mA, as shown in Fig. \[fig:exp1\] (a). We have examined the dynamic behavior of the output intensity for various pump currents, both above and below $I_{th}$, with a particular attention to possible effects of the feedback phase $\Delta\phi$ which is varied by acting on the external mirror piezoelectric transducer. As shown in Fig. \[fig:exp2\], we can identify several regimes: (I) at $I < I_{th}$ one observes a single mode LFF dynamics (i.e. the main polarization exhibits LFFs, while the secondary polarization remains off, see Fig. \[fig:exp1\] (b)); (II) $ I_{th} < I < 3.5$ mA: in this regime the LFF dynamics of the main polarization is accompanied by a synchronized spiking behavior in the secondary polarization (coupled modes LFF). This regime was analyzed in Ref. [@romanelli] and, in more details, in Ref. [@Soriano2004], where it is proposed that the dynamics of the secondary polarization is driven by the main polarization, whose behavior is not influenced by the orthogonal polarization mode. For larger pump current one begins to observe Coherence Collapse (regime (III) in Fig. \[fig:exp2\]) and moreover the feedback phase begins to play a fundamental role. In particular, for increasing $I$ in larger and larger portion of the phase interval the laser is stationary (regime (IV) in Fig. \[fig:exp2\]). This phenomenon is not yet understood and it will be subject of a future analysis [@loiko]. Anyway, in the present paper we will limit our analysis to regime (I) (i.e. for $I < I_{th}$), where the VCSEL has a single-mode LFF dynamics and the phase delay of the feedback does not play any role. We remark that this statement is suggested by the experiment, where the phase stability would be enough to discriminate such effects, as shown in the analysis of regimes (III-IV). ![Phase diagram of the VCSEL with feedback: phase of the feedback $\Delta \phi$ as a function of the pump current $I$. The roman numbers denote regions of different dynamical regimes: (I) single-mode LFF, (II) two-mode LFF, (III) coherence collapse and (IV) stable emission. The vertical dashed line indicates (from low to high current) successive current thresholds: the reduced one $I^{red}_{th}$, the solitary laser one $I_{th}$ and the current value separating LFF from coherence collapse. The dots represent the maximal $\Delta \phi$ for which the laser emission remains stable, while the solid line is a guide for eyes to distinguish regions (III) from (IV).[]{data-label="fig:exp2"}](f2) Numerical model and methods {#sec:2} =========================== The dynamics of the VCSEL for $I < I_{th}$ is a purely single mode dynamics and therefore we expect that it could be reproduced by employing the Lang-Kobayashi (LK) [@LK] rate equations for the complex field $E(t)$ and the carrier density $n(t)$. In order to achieve an accurate and reliable comparison of the numerical results with the experimental ones we will employ for our simulations the laser working parameters reported in Table \[parameter\]. These parameters have been determined via a series of suitable experiments for exactly the same VCSEL employed to obtain the measurements examine in this paper [@bar05]. The only lacking parameter that was not possible to obtain in the previous characterization is the feedback strength, which is however determined from the threshold reduction. In this article we use the rate equations derived in Ref. [@bar05] for the single-mode solitary laser, modified to include the feedback. Defining the deviation $\Delta n$ of the carrier density from transparency normalized to have unitary value at threshold, i.e. $$\Delta n = \frac{n-1}{n_{th}-1} \qquad , \label{carrier}$$ the following form is obtained [@bar05]: $$\begin{aligned} \tau_n \dot {\Delta n} &=& - \Delta n + 1 + \eta(\mu-1) -\Delta n\|E\|^2 \nonumber \\ \dot E &=& \frac{1 + i \alpha}{2 \tau_p} [\Delta n-1]E + \frac{k}{\tau_p} {\rm e}^{-i \omega \tau} E(t - \tau) +\sqrt{R_0} \tilde \xi(t) \label{model}\end{aligned}$$ where $\tilde \xi(t)= \xi_R(t) + i \xi_I(t)$ is a complex Gaussian noise term with zero mean and correlation given by $<\xi_R(t) \xi_R(0)> = <\xi_I(t) \xi_I(0)>= \delta(t)$ and $<\xi_R(t) \xi_I(0)>= 0$. The noise variance $R_0=(n/n_{th})^2 R_{sp}$ represents a multiplicative noise term proportional to the square of the reduced carrier density $n/n_{th}$ ($n_{th}$ being the threshold carrier density) and to the variance of the spontaneous emission noise $R_{sp}$. The parameter $\mu=I/I_{th}$ is the pump current rescaled to unity at threshold, $\tau_n$ and $\tau_p$ are carrier and photon lifetimes, respectively, $\tau$ is the delay (or external roundtrip time), $\alpha$ is the linewidth enhancement factor and $\eta$ the reduced gain (for the exact definitions of these quantities in terms of the laser parameters and for the approximations employed to derive (\[model\]) see Ref. [@bar05]). By reexpressing the time scale in terms of the photon life-time ($\tau_p$) the equations assume the usual form for the LK model and read as $$\begin{aligned} T {\dot \Delta n} &=& -\Delta n + p -\Delta n \|E\|^2 \nonumber \\ \dot E &=& \frac{1 + i \alpha}{2} [\Delta n -1] E + k {\rm e}^{-i \omega \tau} E(t - \tau) +\sqrt{R} \tilde \xi(t) \label{model1}\end{aligned}$$ where $T=\tau_n / \tau_p$, $p=1+\eta (\mu -1)$ and $R = (n/n_{th})^2 R_{sp} \times \tau_p$. The complex field can be expressed as $E(t) = A(t) +i B(t) = \rho(t) \exp{i \psi(t)}$ and the equations can be rewritten as: $$\begin{aligned} \dot A(t) &=& \left(\frac{\Delta n(t) -1}{2}\right) \left(A(t) +\alpha B(t) \right) + k [A(t -\tau) \cos{\phi} +B(t-\tau)\sin{\phi}] +\sqrt{R/2} \xi_R(t) \nonumber \\ \dot B(t) &=& \left(\frac{\Delta n(t) -1}{2} \right)\left(B(t) - \alpha A(t) \right) + k [B(t-\tau)\cos{\phi} -A(t -\tau) \sin{\phi} ] +\sqrt{R/2} \xi_I(t) \nonumber \\ T {\Delta \dot n}(t)&=& p -\Delta n (t) \left[1+A^2(t)+B^2(t) \right] \label{model2}\end{aligned}$$ where $\phi= \omega \tau$. Moreover, by assuming that $n \sim n_{th}$ the noise terms become additive with an adimensional variance $R = 2.76 \times 10^{-3}$, the other quantities entering in (\[model2\]), expressed in $\tau_p$ units, are $\tau = 302.5$, $ \omega \times \tau=8.743 \times 10^6$ and $T = 30.8333$, the numerical values have been obtained by employing the parameters values in Table \[parameter\]. In order to reproduce the power-current response curve for two different experimental data sets we have chosen feedback strengths $k =0.25$ and 0.35, while typically we considered pump currents and linewidth enhancement factors in the range $0.9 \le \mu \le 1.20$, and $3 \le \alpha \le 5$, respectively. The deterministic equations (\[model2\]) with $R\equiv0$ have been integrated by employing the method introduced by Farmer in 1982 [@farmer] equipped with a standard 4th order Runge-Kutta scheme, while for integrating the equations with the stochastic terms we have employed an Heun integration scheme [@toral]. The simulations have been performed by integrating the field variables with time steps of duration $\Delta t = \tau/(N-1)$, with $N =1,000 - 10,000$. The dynamical properties of the system can be estimated in terms of the associated Lyapunov spectrum, that fully characterizes the linear instabilities of infinitesimal perturbations of the reference system. By following the approach reported in [@farmer], we have estimated the Lyapunov spectrum $\{ \lambda_k \} \quad (k=1,\dots,2N+1)$ by integrating the linearized dynamics associated to eqs (\[model2\]) in the tangent space and by performing periodic Gram-Schmidt ortho-normalizations according to the method reported in [@benettin]. The Lyapunov eigenvalues $\lambda_k$ are real numbers ordered from the the largest to the smallest, a positive maximal Lyapunov $\lambda_1$ is a indication that the dynamics of the system is chaotic. Moreover, from the knowledge of the Lyapunov spectrum it is possible to obtain an estimation of the number of degrees of freedom actively involved in the chaotic dynamics in terms of the Kaplan-Yorke dimension [@ky]: $$D_{KY} = j + \frac{\sum_{k=1}^j \lambda_k}{\|\lambda_{j+1}\|} \label{d_ky}$$ where $j$ is the maximal index for which $\sum_{k=1}^j \lambda_k \ge 0$. Description Symbol Value -------------------------------------------- ---------- ------------------------------------------------ Linewidth enhancement factor $\alpha$ $3.2 \pm 0.1$ Photon lifetime in the cavity $\tau_p$ $12 \pm 1 \enskip ps$ Carrier lifetime $\tau_n$ $0.37 \pm 0.02 \enskip ns$ External roundtrip time $\tau$ $3.63 \enskip ns$ Variance of the spontaneous emission noise $R_{sp}$ $(2.3 \pm 0.6) \times 10^{-4} \enskip ps^{-1}$ Reduced gain $\eta$ $5.8 \pm 0.6$ : Experimental values of the parameters entering in the model (\[model\]) (from [@bar05]). \[parameter\] Stationary Solutions {#sec:3} ==================== A first characterization of the phase space of the LK system can be achieved by individuating the corresponding stationary solutions and by analyzing their stability properties. The stationary solutions of the above set of equations can be found by setting $\dot \rho = \Delta \dot n = 0$ and ${\dot \psi} = \Omega$, i.e. by looking for solutions of the form $$E_S(t) = \rho_S {\rm e}^{i \Omega t} \qquad {\rm and} \qquad \Delta n(t) = \Delta n_S \quad . \label{staz}$$ The stationary solutions can be parametrized in terms of the variable $\theta= \phi + \Omega \tau$, and by setting $J = (p-1)/2$ they can be expressed as follows [@yanchuk] $$X_S = \frac{(\Delta n_S-1)}{2} = - k \cos(\theta)$$ $$\rho_S^2 = 2 \frac{J-X_S}{2 X_S +1} \ge 0$$ and $$\Omega = \frac{\theta -\phi}{\tau} = - k \sqrt{1+\alpha^2} \sin(\theta + \theta_0)$$ where $\theta_0 = {\rm atan}({\alpha})$ and $-\pi < \theta \le\pi$. The stationary solutions are usually termed external cavity modes (ECMs) and correspond to stationary lasing states, in the limit of long delay they are densely covering the following ellipse: $$k^2 = X_S^2 +(\Omega -\alpha X_S)^2 \quad . \label{ellisse}$$ The stability properties of these solutions can be obtained by estimating the associated Floquet spectrum $\{\Lambda_n\}$, these eigenvalues are typically complex and their number is infinite, due to the delay term present in the LK equations. However, the stability properties of the ECMs are determined by the eigenvalues with the largest real part, in particular by the maximal one $\Lambda^M = ({\rm Re} \enskip \Lambda^M, {\rm Im} \enskip \Lambda^M)$. These can be easily determined by solving the characteristic equation obtained by linearizing around a certain ECM: $$\begin{aligned} & &Z^2_n [\Lambda_n + \varepsilon(1+\rho_S^2)] +2 Z_n [\Lambda_n \cos(\theta)( \Lambda_n+\varepsilon(1+\rho_S^2) + \varepsilon \rho_S^2(1- 2k \cos(\theta))(\cos(\theta)-\alpha \sin(\theta))/2] \nonumber \\ &+& \Lambda_n^2[\Lambda_n +\varepsilon(1+\rho_S^2)] + \Lambda_n \varepsilon \rho_S^2 (1-2k \cos (\theta)) \equiv a_n Z^2_n + 2b_n Z_n + c_n = 0 \quad . \label{charact}\end{aligned}$$ where $Z_n = k (1 -{\rm e}^{- \Lambda_n \tau})$ and $\varepsilon = 1/T$. The [pseudo-continous]{} spectrum [@ya2] can be obtained by solving eq. (\[charact\]) in terms of $Z_n$ and by considering $\Lambda_n$ as a parameter. In this case the solutions of the corresponding second order equation are $$\Lambda_n^{\pm} = - \frac{1}{\tau} \log\left[ 1 + \frac{b_n \mp \sqrt{b_n^2 -a_n c_n}}{k a_n} \right ] + i 2 \pi n \label{sol}$$ and by self-consistently solve the above equation one can find all the eigenvalues, that are arrangend in two branches. However, the spectrum contains also isolated eigenvalues, that can be obtained by solving directly Eq. (\[charact\]) in terms of $\Lambda_n$ and by considering $Z_n$ as a parameter. In this case the equation is a cubic and by solving self-consistely its expression one finds (up to) three distinct eigenvalues. While the pseudo-continuos spectrum emerges due to the presence of the delay in the system, the isolated eigenvalues originate from those characterizing the three dimensional single laser rate equations in absence of the delay, i.e., the Eq. (\[model1\]) with $\tau=0$ [@ya2]. Typically, depending on their linear stability properties ECM are divided into [modes]{} and [antimodes]{} [@levine]. Antimodes are characterized by a positive real eigenvalue and are therefore unstable. For the modes instead the maximal real eigenvalue is zero and they are unstable whenever the associated spectrum crosses the imaginary axis. Various types of instabilities can be observed for these delayed systems and they can be classified for analogy with the spatially extended systems as for example [modulational-type]{} or [Turing-type]{} instabilities [@yanchuk; @ya2]. Examples of the unstable branch for the spectra $\{\Lambda_k\}$ associated to Turing-type and modulationali-type instabilities are reported in Fig. \[fig:spectra\], for a classification of the possible instabilities of equilibria for delay-differential equations see [@ya2]. ![(Color online) Unstable branch $\Lambda$ associated to specific modes of the LK. The mode $\theta=-0.32$ for $\alpha=3.20$ reveals a sort of modulational instability (red squares), while the mode $\theta=-0.08$ for $\alpha=3.22$ shows a Turing-like instability (blue circles). The results refer to $\mu=0.93$. Due to the symmetry of the spectra only the part corresponding to $Im \Lambda > 0$ is displayed. []{data-label="fig:spectra"}](f3) In particular, the modes correspond to $\theta$-values located within the interval $[\theta_2 : \theta_1]$, where $\theta_1 \sim {\rm atan} (1/\alpha)$ and $\theta_2 \sim -\pi +{\rm atan} (1/\alpha)$. Moreover, stationary solutions are acceptable only if they correspond to positive intensities $\rho^2_S \ge 0$, i.e. if they are situated within the interval $[-\theta_R:\theta_R]$ with $\theta_R = {\rm acos}{(-J/k)}$. In the range of parameters examined in the present paper the number of stable modes (SMs) is always between two and four and they are located in a narrow interval of $\theta$ located around zero (i.e. around the so called Maximum Gain Mode (MGM)). As we will report in the following section, by integrating the deterministic version of the model (\[model2\]) for moderate $\alpha$- and $\mu$-values, we always observe a relaxation towards one of the SMs. In particular for $3 < \alpha < 5$ the dynamics seem to relax always towards one of the SMs located in proximity of the MGM (corresponding to $\theta \equiv 0$) (see Fig. \[fig:modes\]). It should be noticed that the MGM is a solution of the system only for appropriate choices of $\phi$. Two examples of typical ECMs are reported in Fig. \[fig:modes\] for the present system for $\alpha=3.2$ and 5 and $\mu=0.93$, in both cases two stable attracting modes have been identified. Recently, a similar coexistence of two stable solutions located in proximity of the MGM has been reported experimentally for an edge-emitter laser with a low level of optical feedback [@mendez] ![Intensities versus $\theta$ for ECMs of eq. (\[model2\]) with $\mu=0.93$ and for $\alpha=3.2$ (a) and $5.0$ (b). The solid line indicates $\theta_1$. The ECMs with $\theta < \theta_1$ are modes, while the others are antimodes. The empty circles indicates the SMs towards which we numerically observed relaxation of the system by considering up to 100 different random initial conditions. []{data-label="fig:modes"}](f4a.eps "fig:") ![Intensities versus $\theta$ for ECMs of eq. (\[model2\]) with $\mu=0.93$ and for $\alpha=3.2$ (a) and $5.0$ (b). The solid line indicates $\theta_1$. The ECMs with $\theta < \theta_1$ are modes, while the others are antimodes. The empty circles indicates the SMs towards which we numerically observed relaxation of the system by considering up to 100 different random initial conditions. []{data-label="fig:modes"}](f4b.eps "fig:") As reported in [@yanchuk] the stability properties of the MGM do not depend on $\alpha$, therefore it is reasonable to expect that some SM will be always present in a narrow window around $\theta$ for any chosen linewidth enhancement factor [@levine] and that they will coexist with the chaotic dynamics, as observed experimentally in [@heil]. Deterministic Dynamics {#sec:4} ====================== An open problem concerning the deterministic LK equations is if and for which range of parameters these equations faithfully reproduce the experimentally observed dynamics. Particular interest is usually focused on the $\alpha$-value to be employed to obtain a realistic behaviour of the model. LFFs as a transient phenomenon {#sec:4.1} ------------------------------ To answer this question we consider the deterministic verson of the model (\[model2\]) (i.e. by assuming $R\equiv 0$) equipped with the parameter values deduced by the experimental data [@bar05] apart for $\mu$ and $\alpha$ that will be varied. In particular, we would like to understand if the system shows a LFF behaviour and if such dynamics is statistically stationary or not. In order to verify it, we initialize randomly the amplitude $\rho(t=0)$, the phase $\phi(t=0)$ and the excess carrier density $\Delta n(t=0)$, then we follow the dynamics and we examine the evolution of the field intensity $\rho^2(t)$. If the dynamics end up on a stationary solution we register the time $T_s$ necessary to reach it and we average this time over many ($M$) different initial conditions. In order to measure $T_s$, we estimate the time needed to the standard deviation of the intensity (evaluated over sub-windows of time duration $t_w$) to decrease below a chosen threshold $\Gamma$. The average times $<T_s>$ are reported in Fig. \[fig:times\] ![(Color online) Logarithm of the transient times $<T_s>$ as a function of $\alpha$ for various $\mu$-values: namely, $\mu=0.93$ (black filled circles), $0.95$ (red empty triangles), $0.97$ (blue asteriskes) and $0.99$ (green empty squares). The data refers to $k=0.25$, $\Gamma=10^{-5}$, $t_w = 1000 \tau = 3.630 \mu s$ and $M \sim 10-100$, and they have been obtained by employing an integration time step $\Delta t=3.63 ps$. The time scale of the figure is $\tau_0 = 1 \enskip$ ns. The bars reported for each measured value indicate the range of variability of $<T_s>$ (within a $\alpha$-interval of amplitude 0.1) due to its finer structure shown in the inset. In the inset are displayed the transient times for $\mu=0.93$ reported at a higher resolution in $\alpha$, namely for a resolution of $0.02$. []{data-label="fig:times"}](f5.eps) The main result shown in Fig. \[fig:times\] is that the system after a transitory phase (shorter or longer) settles down to a SM and that the duration of the transient increases for increasing $\alpha$ or $\mu$-values. Preliminary indications in this direction have been previously reported in [@dfb]. These results clearly indicate that the LFF dynamics is just a transient phenomenon for the deterministic LK equation for commonly employed $\alpha$-values (i.e., for $\alpha \sim 3.0-3.5$). For each fixed $\mu$ it seems that $<T_s>$ diverge above some “critical” $\alpha$-value, however we cannot exclude that the system will also in this case finally converge to a SM. At least we can consider the reported critical values as lower bounds above which LFF could occur as a stationary phenomenon and not as a transitory state. With the chosen numerical accuracy and due to the available computational resources, for any pratical purpose a transient longer than $0.1 -1$ s can be considered as an infinite time. An interesting feature is that the variability of $<T_s>$ with $\alpha$ is quite wild and it reflects the stability of the modes in proximity of the MGM, as shown in Fig. \[fig:times\_stablemodes\]. However the average values $<T_s>$ (averaged also over $\alpha$-intervals of width $0.1$) still indicates a clear trend of the transient times to increase with $\alpha$. We expect that the time scale associated to the convergence to a stable mode is ruled by the eigenvalue with the maximal real part (apart the eigenvalue zero, that is always present due to the phase invariance of the LK equations). In particular we expect that the intensity of the signal will converge towards the stable mode as $$\rho^2(t) \sim \rho^2(0) {\rm e}^{\rm 2 Re \enskip \Lambda^M \enskip t} cos(\rm 2 Im \enskip \Lambda^M \enskip t) + \rho_S^2$$ where $\Lambda^M$ is the eigenvalue with maximal (non zero) real part associated to the considered stable solution. For small $\mu$ we have observed that the dynamics can collapse to different stable solutions (typically, from 1 to 3). By estimating the probability $P_m=N_m/M$ to end up in one of these states (being $N_m$ the number of initial conditions converging to the $m$-mode) and by indicating the corresponding eigenvalue with maximal real part as $\Lambda^M (m)$, a reasonable estimate of $<T_s>$ is given by $$T_{est} = \frac{\ln{\Gamma}}{2} \sum_m \frac{P_m} {\|\rm Re \Lambda^M (m)\|} \quad , \label{test}$$ where $\Gamma$ is the employed threshold. As it can be seen in Fig. \[fig:times\_stablemodes\], the estimation is quite good and the periodicity of the two quantities is identical in the examined range of $\alpha$-values and for $\mu = 0.93$. The expression (\[test\]) always gives a good estimate of $<T_s>$ for $\alpha <4$ and for $\mu < 1$, but the agreement worsens for increasing $\alpha$-values. However, even a more rough estimate is capable to give a reasonable approximation of $<T_s>$ and in particular to capture its periodicity. This estimate is simply given by $$T_{1} = \frac{\ln{\Gamma}}{2} \frac{1} {\|\rm Re \bar \Lambda^M\|} \quad , \label{max}$$ where $\bar \Lambda^M$ is the eigenvalue with maximal non zero real part associated to the stable mode with higher $\theta$ (i.e. the first stable mode located in proximity of the anti-mode boundary). From the present analysis it emerges that the stable modes play a relevant role for the (transient) dynamics of the deterministic LK equations at least for $\alpha < 4$. Moreover the observed strongly fluctuating behaviour of $<T_s>$ as a function of $\alpha$ indicates that the choice of this parameter is quite critical, since a small variation can lead to an increase of an order of magnitude of the transient time[^1] ![(Color online) Transient times $<T_s>$ (black filled circles), $T_{est}$ (red empty triangles) and $T_{1}$ (blue empty square) in nanoseconds as a function of $\alpha$. The results have been obtained by examining the decay of $\rho^2(t)$ with $\Gamma=10^{-5}$ and $t_w = 1000 \tau = 3.630 \mu s$ and $M \sim 100-500$. The employed integration time step is $\Delta t=0.363$ ps and the data refer to $k=0.25$ and $\mu=0.93$. []{data-label="fig:times_stablemodes"}](f6.eps) Lyapunov Analysis {#sec:4.2} ----------------- We have characterized the transient dynamics preceeding the collapse in the stationary state in terms of the maximal Lyapunov $\lambda_1$ and of the associated Kaplan-Yorke (or Lyapunov) dimension $D_{KY}$. In particular, these quantities reported in Fig. \[fig:lyap\] have been estimated by integrating the linearized dynamics for a sufficiently long time period $T_{int}$ and by averaging over $M$ different initial realizations. It is clear from the figures that the average maximal Lyapunov exponent $<\lambda_1>$ is definetely not zero for all the considered situations and that it increases (almost steadily) with the parameter $\alpha$ as well as with the pump parameter. Moreover, also this indicator reflects the stability properties of the SMs located in proximity of the MGM, by exhibiting large oscillations as a function of $\alpha$, as shown in the inset of \[fig:lyap\](a). The values of $<D_{KY}>$ reported in \[fig:lyap\](b) clearly indicate that the system cannot be described as low dimensional, even during the transient and even below the solitary laser threshold. As a matter of fact the number of active degrees of freedom ranges between $10$ and $50$. It should be noticed that $<D_{KY}>$ is determined by the instability properties not only of anti-modes and but also of modes that have bifurcated (via Hopf instabilities) becoming unstable for increasing $\alpha$. ![(Color online) (a) Average maximal Lyapunov exponents $<\lambda_1>$ as a function of $\alpha$ and for various $\mu$-values below threshold: namely, $\mu=0.93$ (black filled circles), $0.95$ (red empty triangles), $0.97$ (blue asterisks) and $0.99$ (green empty squares). The bars reported for each measured value indicates the range of variability of $<\lambda_1>$ due to its finer structure as measured within a $\alpha$-interval of width 0.1. In the inset the data for $<\lambda_1>$ are reported for a higher resolution in $\alpha$ (namely 0.02) for $\mu=0.93$. (b) Average Kaplan-Yorke dimensions $<D_{KY}>$ as a function of $\mu$ for $\alpha=4$ (black filled circles) and 5 (red empty squares). All the data refer to $k=0.25$, for the $<\lambda_1>$ estimation $\Delta t=0.363$ ps, $M=500$, and $T_{int}=3.63$ ms while for the $<D_{KY}>$ evaluation $\Delta t=3.63$ ps, $M=20$ and $T_{int}=0.14$ ms. []{data-label="fig:lyap"}](f7a.eps "fig:") ![(Color online) (a) Average maximal Lyapunov exponents $<\lambda_1>$ as a function of $\alpha$ and for various $\mu$-values below threshold: namely, $\mu=0.93$ (black filled circles), $0.95$ (red empty triangles), $0.97$ (blue asterisks) and $0.99$ (green empty squares). The bars reported for each measured value indicates the range of variability of $<\lambda_1>$ due to its finer structure as measured within a $\alpha$-interval of width 0.1. In the inset the data for $<\lambda_1>$ are reported for a higher resolution in $\alpha$ (namely 0.02) for $\mu=0.93$. (b) Average Kaplan-Yorke dimensions $<D_{KY}>$ as a function of $\mu$ for $\alpha=4$ (black filled circles) and 5 (red empty squares). All the data refer to $k=0.25$, for the $<\lambda_1>$ estimation $\Delta t=0.363$ ps, $M=500$, and $T_{int}=3.63$ ms while for the $<D_{KY}>$ evaluation $\Delta t=3.63$ ps, $M=20$ and $T_{int}=0.14$ ms. []{data-label="fig:lyap"}](f7b.eps "fig:") Noisy Dynamics {#sec:5} ============== We have examined the dynamics (\[model\]) for increasing level of noise, namely for $10^{-6} < R < 10^{-2}$. Also in this case, for $\alpha=3.3$ and for noise levels smaller than $10^{-3}$ the LFF dynamics only occurs during a transient. However for increasing $R$ values we observed a transition to sustained LFF and the transition region was characterized by an intermittent behaviour. These behaviours are exemplified in Fig. \[fig:interm\] for $\alpha=3.3$ and $\mu=0.97$. As shown in Fig. \[fig:interm\](a) the orbit spends long times in proximity of one of the SM and then, due to noise fluctuations, escapes from the attraction basin associated to the stable solution and exhibit LFFs before being newly reattracted by the SM. This intermittent dynamics can be interpreted as an activated escape process induced by noise fluctuations, and therefore the average residence time $<T_{res}>$ in the attraction basin of the SM can be expressed in the following way $$<T_{res}> \propto \exp{[W/R]} \label{kramers}$$ where $W$ represent a barrier that the orbit should overcome in order to escape from the SM valley. As shown in Fig. \[fig:escape\] the process can be indeed interpreted in terms of the Kramers expression (\[kramers\]) for $ 2\times 10^{-4} < R < 7 \times 10^{-4}$. It means that for $ R < W$ one should expect an intermittent behaviour, while for $ R > W$ the dynamics of the orbit will be essentially diffusive, since the noise fluctuations are sufficient to drive the orbit always out of the SM valley. These indications suggest that in order to observe a “non transient” or “non intermittent” LFF dynamics the amount of noise present in the system should be larger than $W$. Also in [@dfb] it has been clearly stated that the LK-equations with parameters tuned to reproduce the dynamics of a DFB laser with $\alpha =3.4$ can give rise to stationary LFF only in presence of noise. It is important to remark that for $\alpha=3.3$ the experimentally measured variance of the noise $R=2.76 \times 10^{-3}$ is above the barrier $W=1.87 \times 10^{-3}$ found from the fit of the numerical $<T_{res}>$ with expression (\[kramers\]), performed in the small noise range (see Fig. \[fig:escape\]). Moreover in the experiments we never observed relaxation of the dynamics towards a SM. ![Intensity of the field $\rho^2$ as a function of the time for the noisy LK equations. The data have been filtered with a low-pass filter at $80$ MHz and refer to $\mu=0.97$, $\alpha=3.3$. The results reported in (a) correspond to a noise variance $R=3 \times 10^{-4}$, while those in (b) are relative to $R=3 \times 10^{-3}$. In (a) a typical time of residence $T_{res}$ around one of the SMs is indicated. The inset in (b) is an enlargement of the actual dynamics. []{data-label="fig:interm"}](f8a.eps "fig:") ![Intensity of the field $\rho^2$ as a function of the time for the noisy LK equations. The data have been filtered with a low-pass filter at $80$ MHz and refer to $\mu=0.97$, $\alpha=3.3$. The results reported in (a) correspond to a noise variance $R=3 \times 10^{-4}$, while those in (b) are relative to $R=3 \times 10^{-3}$. In (a) a typical time of residence $T_{res}$ around one of the SMs is indicated. The inset in (b) is an enlargement of the actual dynamics. []{data-label="fig:interm"}](f8b.eps "fig:") ![Average residence times in the SM as a function of the inverse of the variance of the additive noise to the LK equations. The dashed line is a exponential fit $\propto \exp{[W/R]}$ to the numerical data, the fitted exponential slope is $W=0.00187$. The data refer to $\mu=0.97$ and $\alpha=3.3$. []{data-label="fig:escape"}](f9.eps) Lyapunov Analysis {#sec:5.1} ----------------- Also in the noisy case we have examined the degree of chaoticity in the system by estimating the maximal Lyapunov exponent along noisy orbits of the system. The role of noise is fundamental in destabilizing the dynamics of the system and in rendering the asymptotic dynamics chaotic. ![(color online) Maximal Lyapunov exponents $\lambda_1$ as a function of $\alpha$ for $\mu=0.93$ (a) and $\mu=0.97$ (b) and for various noise amplitude $R$: namely, the data for $R=3 \times 10^{-5}$ are indicated by green asterisxs, those for $R=3 \times 10^{-4}$ by blue filled triangles, and the ones corresponding to $R=3 \times 10^{-3}$ by red filled circles. The values estimated during the transient dynamics in absence of noise are indicated by black empty squares, while the asymptotic values for $R=0$ by black filled squares. All the data refer to $k=0.25$, for the estimation of $\lambda_1$ in the noisy case one orbit has been followed for a time $t=3.63$ ms with time step $\Delta t=0.363$ ps, while in the deterministic case the asymptotic results have also been averaged over $M=10$ different initial conditions. For details on the estimation of the transient Lyapunov exponents see the previous section \[sec:4.2\]. []{data-label="fig:lyapnoise"}](f10a.eps "fig:") ![(color online) Maximal Lyapunov exponents $\lambda_1$ as a function of $\alpha$ for $\mu=0.93$ (a) and $\mu=0.97$ (b) and for various noise amplitude $R$: namely, the data for $R=3 \times 10^{-5}$ are indicated by green asterisxs, those for $R=3 \times 10^{-4}$ by blue filled triangles, and the ones corresponding to $R=3 \times 10^{-3}$ by red filled circles. The values estimated during the transient dynamics in absence of noise are indicated by black empty squares, while the asymptotic values for $R=0$ by black filled squares. All the data refer to $k=0.25$, for the estimation of $\lambda_1$ in the noisy case one orbit has been followed for a time $t=3.63$ ms with time step $\Delta t=0.363$ ps, while in the deterministic case the asymptotic results have also been averaged over $M=10$ different initial conditions. For details on the estimation of the transient Lyapunov exponents see the previous section \[sec:4.2\]. []{data-label="fig:lyapnoise"}](f10b.eps "fig:") In particular, as shown in Fig. \[fig:lyapnoise\](a) we observe that at $\mu=0.93$ and $k=0.25$ the deterministic dynamics ($R=0$) is asymptotically stable in the range $\alpha \le 4.4$, while the noisy dynamics becomes more and more chaotic for increasing $R$. For the value $R=3.3 \times 10^{-3}$, close to the experimental one, the dynamics is completely destabilized in the whole examined range $2.5 \le \alpha \le 4.5$, while for smaller $R$-values the range of destabilization is reduced. These results confirm the role of the noise in rendering the LFF an asymptotic phenomenon. Moreover the maximal Lyapunov increeases steadily with $\alpha$ at $R=3.3 \times 10^{-3}$. Similar findings apply in the case $\mu =0.97$ (see Fig. \[fig:lyapnoise\](b)). The wild oscillations in the $<\lambda_1>$ values observable at level of noise $ R < 10^{-4}$ reflect the stability properties of the SMs attracting the asymptotic dynamics. Comparison between experimental and numerical data {#sec:6} ================================================== This Section will be devoted to a detailed comparison of numerical versus experimental results with the aim to clarify if the deterministic or noisy LK eqs are indeed able to reproduce the experimental findings. Distributions of the field intensities {#sec:6.1} -------------------------------------- ![Field intensities distributions $P(\rho^2)$ for the experimental signal filtered at 200 MHz. The experimental intensities have been arbitrarly rescaled to match the corresponding average intensities obtained from the simulation of the noisy LK eqs at $\alpha=3.3$ and $k=0.35$ with noise variance $R=3.3\times 10^{-3}$. The data refer from left to rigth to $I=2.48$ mA, 2.50, 2.54, 2.58, 2.64, 2.70, and 2.75. Since $I_{th}=2.76$ these data correspond to $ 0.9 < \mu < 1.0$. The first distribution has a large contribution from the Gaussian electronic noise, which also explains the negative $\rho^2$ values. []{data-label="fig:pdf_exp"}](f11.eps) As a first indicator we have considered the distribution of the field intensities $P(\rho^2)$, in particular in order to match the experimental findings we consider the signal filtered at 200 MHz. The experimental results for the probability distribution functions (PDFs) of the field intensities $\rho^2$ are reported in Fig. \[fig:pdf\_exp\] for various currents below the solitary threshold value. It should be noticed that the amplitudes $\rho^2$ have been rescaled in order to match the corresponding numerical values for the noisy LK eqs with $\alpha=3.3$ and noise variance $R=3.3\times 10^{-3}$, but that no arbitrarly shift have been applied to the data. A peculiar characteristic of these data is that for increasing pump current the PDFs become more and more asymmetric, revealing a peak at large intensities that shifts towards higher and higher $\rho^2$ values and a sort of plateau at smaller intensities. The corresponding PDFs are reported in Figs. \[fig:pdf\_sim.noise\] and \[fig:pdf\_sim.det\] for data obtained from the integration of the noisy and deterministic LK eqs, respectively. A better agreement between numerical and experimental findings is found for the noisy dynamics with $\alpha \sim 3.3 - 4.0$ and $k=0.35$, with a noise variance similar to the experimental one (namely, $R=3\times 10^{-3}$). For the deterministic case (reported in Fig. \[fig:pdf\_sim.det\] for $\alpha=5.0$ and $k=0.35$) a non zero tail at $\rho \sim 0$ is observed even for $\mu \sim 1.0$, contrary to what observed for the experimental data. These results indicate that it is necessary to include the noise in the LK eqs to obtain a reasonable agreement with the experiment, at least at the level of the intensities PDFs. However, the numerical data seem unable to reproduce the narrow peak present in the experimental ones at large intensities and for $\mu \to 1$. In the next sub-section a further comparison will be performed to validate these preliminary indications. ![Field intensities distributions $P(\rho^2)$ for the numerical data obtained by the integration of a noisy LK eqs. filtered at 200 MHz. The data refer from left to rigth to $\mu=0.90$, 0.91, 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, and 0.99 for (a) $\alpha=3.3$ and (b) $\alpha=4.0$. Both the sets of data correspond to $k=0.35$ and noise variance $R=3\times 10^{-3}$. []{data-label="fig:pdf_sim.noise"}](f12a.eps "fig:") ![Field intensities distributions $P(\rho^2)$ for the numerical data obtained by the integration of a noisy LK eqs. filtered at 200 MHz. The data refer from left to rigth to $\mu=0.90$, 0.91, 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, and 0.99 for (a) $\alpha=3.3$ and (b) $\alpha=4.0$. Both the sets of data correspond to $k=0.35$ and noise variance $R=3\times 10^{-3}$. []{data-label="fig:pdf_sim.noise"}](f12b.eps "fig:") ![Field intensities distributions $P(\rho^2)$ for the numerical data obtained by the integration of a deterministic LK eqs. filtered at 200 MHz. The data refer from left to rigth to $\mu=0.91$, 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, and 0.99 for $\alpha=5.0$ and $k=0.35$. []{data-label="fig:pdf_sim.det"}](f13.eps) Average values of the LFF times {#sec:6.2} ------------------------------- We will first compare the experimental and numerical measurements of the average times between two consecutive drops of the field intensities $<T_{LFF}>$. They have been evaluated in two (consistent) ways: both from a direct measurement of periods between threshold crossing, and from the Fourier power spectrum of the temporal signal $\rho^2(t)$. The direct measurements of the $T_{LFF}$ from the time trace of $\rho^2(t)$ have been performed by defining two thresholds $\Gamma_1 < \Gamma_2$ and by identifing two consecutive time crossing of $\Gamma_1$, provided that in the intermediate time the signal has overcome the threshold $\Gamma_2$ at least once. The thresholds have been defined as $\Gamma_1= <\rho^2> - 2STD$ and $\Gamma_2= <\rho^2> + STD/2$, where $< \cdot> $ and STD indicate the average and the standard deviation of the signal itself. The measurement of $<T_{LFF}>$ in terms of the power spectrum has been obtained by considering the power spectrum $S(\omega)$ of $\rho^2(t)$ and by evaluating the position $\omega_{M}$ of the peak with the highest frequency, then $<T_{LFF}> = 2 \pi / \omega_{M}$. As already mentioned the two estimations are generally in very good agreement. ![(Color online) Average LFF times $<T_{LFF}>$ as a function of the pump parameter $\mu - \mu^{red}$: black filled circles refer to experimental data, while the other symbols to the results obtained from the integration of the LK eqs. In the two figures are reported two different sets of experimental measures and the associated numerical data refer to $k=0.25$ (a) and $k=0.35$ (b). In particular, red empty triangles correspond to the evolution of the noisy LK at $\alpha=3.3$ (with (a) $\mu^{red}=0.914$ and (b) $\mu^{red}=$0.880) and blue empty squares to $\alpha=4.0$ (with $\mu^{red}=0.913$ and 0.878, resp. for (a) and (b)). In both cases $R=3\times 10^{-3}$. The green stars denote the data of the deterministic LK eqs for $\alpha=5.0$ (in this case $\mu^{red}=0.915$ and 0.882, resp. for (a) and (b)). For the experimental measures $\mu^{red}=0.916$ in (a) and 0.875 in (b). The vertical dashed lines indicates the position of the solitary threshold for the experimental data, while the dash-dotted lines represent the decay $c/(\mu -\mu^{red})$, with $c=6.2$ ns and 11 ns in (a) and (b), respectively. $\mu^{red}$ is defined as the ratio $I^{red}_{th}/I_{th}$. []{data-label="fig:T_LFF"}](f14a.eps "fig:") ![(Color online) Average LFF times $<T_{LFF}>$ as a function of the pump parameter $\mu - \mu^{red}$: black filled circles refer to experimental data, while the other symbols to the results obtained from the integration of the LK eqs. In the two figures are reported two different sets of experimental measures and the associated numerical data refer to $k=0.25$ (a) and $k=0.35$ (b). In particular, red empty triangles correspond to the evolution of the noisy LK at $\alpha=3.3$ (with (a) $\mu^{red}=0.914$ and (b) $\mu^{red}=$0.880) and blue empty squares to $\alpha=4.0$ (with $\mu^{red}=0.913$ and 0.878, resp. for (a) and (b)). In both cases $R=3\times 10^{-3}$. The green stars denote the data of the deterministic LK eqs for $\alpha=5.0$ (in this case $\mu^{red}=0.915$ and 0.882, resp. for (a) and (b)). For the experimental measures $\mu^{red}=0.916$ in (a) and 0.875 in (b). The vertical dashed lines indicates the position of the solitary threshold for the experimental data, while the dash-dotted lines represent the decay $c/(\mu -\mu^{red})$, with $c=6.2$ ns and 11 ns in (a) and (b), respectively. $\mu^{red}$ is defined as the ratio $I^{red}_{th}/I_{th}$. []{data-label="fig:T_LFF"}](f14b.eps "fig:") In Fig. \[fig:T\_LFF\] the average times $<T_{LFF}>$ are reported for two different sets of experimental measurements as a function of the pump parameter $\mu$ and compared with numerical data. In Fig. \[fig:T\_LFF\] (a) are reported the experimental findings already shown in [@romanelli], the estimation of $T_{LFF}$ have been performed both by direct inspection of the signal and via the first zero of the autocorrelation function (this second method corresponds to an evaluation from the Fourier power spectrum). The numerical data have been obtained with the 2 methods outlined above for $k=0.25$ for both noisy and deterministic LK eqs. In Fig. \[fig:T\_LFF\] (b) a new set of experimental data is reported and compared with simulation results for $k=0.35$, in this case all the data have been obtained by the method of the thresholds. From the figures it is clear that a reasonably good agreement between experimental and numerical data is observed for the deterministic case only for $\alpha=5$ (results for smaller $\alpha$-values, obtained during the transient preceeding the stable phase, are not shown but they exhibits a worst agreement with experimental findings) and for the noisy dynamics for $\alpha=4.0$. ![(Color online) Standard deviation of the LFF times $V$ as a function of the pump parameter $\mu -\mu^{red}$: the symbols are the same as those reported in Fig. \[fig:T\_LFF\] (b). The dash-dotted line indicates the power-law decay $1/(\mu -\mu^{red})^{3/2}$. []{data-label="fig:T_V"}](f14nuova.eps) A more detailed analysis can be obtained by considering not only the average values of the LFF times, but also the associated standard deviation $V$. This quantity reported in Fig. \[fig:T\_V\] exibits a clear decrease with $\mu$ by approaching the solitary treshold, indicating a modification of the observed dynamics that tends to be more “regular”. Also in this case the comparison of experimental and numerical data suggests that the best agreement is again attained with the noisy dynamics at $\alpha=4.0$. At this stage of the comparison we can sketch some preliminary conclusions: the LK eqs are able to reproduce reasonably well the experimental data for the VCSEL below the solitary threshold both in the deterministic case and in the noisy situation. However in the deterministic case a quite large value of the linewidth enhancement factor (with respect to the experimentally measured one) is required. A more detailed comparison will be possible by considering the PDFs of the $T_{LFF}$. Distributions of the LFF times {#sec:6.3} ------------------------------ In this sub-section we will examine the whole distribution of the $T_{LFF}$ in more details. Considering the experimental data, we observe that all the measured PDFs obtained for different pump currents reveal an exponential-like tail at long times and a rapid drop at short times (as shown in Fig. \[fig:pdf\_LFF\]). These results are in agreement with those reported in [@sukov] for a single-transverse-mode semiconductor laser in proximity of $I_{th}$. ![Probability density distributions of the $T_{LFF}$. Solid lines refer to experimental data, the dashed ones to the Inverse Gaussian Distribution (\[inv\_gau\]) with the average and standard deviation corresponding to the experimental ones. (a) $I=2.56$ mA, (b) $I=2.64$ mA, and (c) $I=2.70$ mA []{data-label="fig:pdf_LFF"}](f15a "fig:") ![Probability density distributions of the $T_{LFF}$. Solid lines refer to experimental data, the dashed ones to the Inverse Gaussian Distribution (\[inv\_gau\]) with the average and standard deviation corresponding to the experimental ones. (a) $I=2.56$ mA, (b) $I=2.64$ mA, and (c) $I=2.70$ mA []{data-label="fig:pdf_LFF"}](f15b "fig:") ![Probability density distributions of the $T_{LFF}$. Solid lines refer to experimental data, the dashed ones to the Inverse Gaussian Distribution (\[inv\_gau\]) with the average and standard deviation corresponding to the experimental ones. (a) $I=2.56$ mA, (b) $I=2.64$ mA, and (c) $I=2.70$ mA []{data-label="fig:pdf_LFF"}](f15c "fig:") The typical dynamics corresponding to a LFF can be summarized as follows: a sudden drop of the intensity is followed by a steady increase of $\rho^2$, associated to fluctuations of the intensity, until a certain threshold is reached and the intensity is reset to its initial value and restart with the same “sysiphus cycle” [@sano]. This behaviour and the observed shapes of the PDFs suggest that the dynamics of the intensities can be modelized in terms of a Brownian motion plus drift. In other words, by denoting with $x(t)$ the intensity, an effective equation of the following type can be written to reproduce its dynamical behaviour: $$\dot x(t) = \eta + \sigma \xi(t) \label{bm_drift}$$ with initial condition $x(0)=x_0$, where $\xi(t)$ is a Gaussian noise term with zero average and unitary variance, $\eta$ represents the drift, $\sigma$ is the noise strength. Within this framework the average first passage time to reach a fixed threshold $\Gamma$ is simply given by $\tau= (\Gamma - x_0)/\eta$ , while the corresponding standard deviation is $V = [\sqrt{(\Gamma - x_0)} \sigma] / \eta^{3/2}$ [@tuckwell]. A resonable assumption would be that $\eta$ is directly proportional to the pump parameter $(\mu - \mu^{red})$, (where $\mu^{red}=I^{red}_{th}/I_{th}$ is the rescaled pump current value at the reduced threshold) and by further assuming that the threshold $\Gamma$ is independent on the pump current this would imply that $$<T_{LFF}> = \frac{c}{(\mu - \mu^{red})} \qquad, \quad {\rm and} \quad V_{LFF} = \frac{c}{(\mu - \mu^{red})^{3/2}} \label{teo_estim}$$ these dependences are indeed quite well verified for the experimental data above the solitary threshold as shown in Figs \[fig:T\_LFF\] and \[fig:T\_V\]. For the simple model introduced by eq. (\[bm\_drift\]), the PDF of the first passage times is the so-called Inverse Gaussian Distribution [@gaussiana]: $$P(T)=\frac{\tau}{\sqrt{2 \pi \gamma T^3}} {\rm e}^{-(T-\tau)^2/(2 \gamma T)} \label{inv_gau}$$ where $\gamma= V^2/\tau$. A comparison of this expression with the experimentally measured $P(T_{LFF})$ is reported in Fig. \[fig:pdf\_LFF\]. The good agreement suggests that the “sysiphus cycles” can be due to few elementary ingredients: a stochastic motion subjected to a drift plus a reset mechanism once the intensity has overcome a certain threshold. A way of rewriting the distribution (\[inv\_gau\]) in a more compact form as a function of only one parameter, the so-called coefficient of variation $\delta=V/\tau$ (i.e., the ratio between the standard deviation $V$ and the mean $\tau$), is to rescale the time as $z = (T- \tau)/V$ and the PDFs as $g(z)=VP(T)$. This procedure leads to the following expression: $$g(z)=\frac{1}{\sqrt{2 \pi (\delta z+1)^3}} {\rm e}^{-z^2/2(\delta z+1)} \quad . \label{inv_gau_res}$$ It is clear that all the PDFs will coincide, once rescaled in this way, if the coefficient of variation $\delta$ has the same value for all the considered pump currents. However this is not the case and indeed we measured values of $\delta$ in the range $[0.28:0.66]$ for $ I < I_{th}$, nonetheless if we report in a single graph all these curves the overall matching is very good, as shown in Fig. \[fig:rescaled\_exp\]. Let us finally compare these distributions $g(z)$ with the corresponding ones obtained from direct simulations of the LK eqs.. As one can see from Fig. \[fig:rescaled\_sim\] the agreement is good for the data obtained from the simulation of the noisy LK eqs at $\alpha=4.0$, while it is worse for the deterministic LK eqs at $\alpha=5.0$. Conclusions {#sec:7} =========== We presented a detailed experimental and numerical study of a semiconductor laser with optical feedback. The choice of a Vertical Cavity Laser pumped close to its threshold, together with a polarized optical feedback, assures a great control over the possibility of lasing action of other orders longitudinal and/or transverse modes than the fundamental one, and of the activation of the other polarization. In such a way, the description of the system using the Lang-Kobayashi model is well justified and it allows for a meaningful comparison with the experimental data. The analysis has been performed with particular regard to the LFF regime, where the model has been numerically integrated using parameters carefully measured in the laser sample used for the measurements. The comparison of the the measurements carried out in the VCSEL with polarized optical feedback with the predictions of the deterministic LK model suggest that in the examined range of parameters the dynamics of the model is characterized by a chaotic transient leading to stable ECMs with high gain. The transient duration increases (and possibly diverges) with increasing values of the rescaled pump current $\mu$ and of the linewidth enhancement factor $\alpha$. We have not found evidence of periodic or quasi-periodic asymptotic attractors, as instead reported in [@david], this can be due to the $\alpha$-range examined in the present paper (namely $2.4 \le \alpha \le 5.5$), since these solutions become relevant for the dynamics only for $\alpha > 5$ (as stated in [@david]). However, a stationary LFF dynamics with characteristics similar to those measured experimentally, can be obtained for realistic values of the $\alpha$ parameter (namely, $\alpha \sim 3 - 4$) only via the introduction of an additive noise term in the LK equations. 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--- abstract: 'In the early 1990s, A. Bezdek and W. Kuperberg used a relatively simple argument to show a surprising result: The maximum packing density of circular cylinders of infinite length in $\mathbb{R}^3$ is exactly $\pi/\sqrt{12}$, the planar packing density of the circle. This paper modifies their method to prove a bound on the packing density of finite length circular cylinders. In fact, the maximum packing density for unit radius cylinders of length $t$ in $\mathbb{R}^3$ is bounded above by $\pi/\sqrt{12} + 10/t$.' author: - \| Wöden Kusner\ Department of Mathematics\ University of Pittsburgh bibliography: - 'RodPaperArXiv.bib' title: Upper bounds on packing density for circular cylinders with high aspect ratio --- [^1] Introduction ============ The problem of computing upper bounds for the packing density of a specific body in $\mathbb{R}^3$ can be difficult. Some known or partially understood non-trivial classes of objects are based on spheres [@hales2005proof], bi-infinite circular cylinders [@bezdek1990maximum], truncated rhombic dodecahedra [@bezdek1994remark] and tetrahedra [@gravel2011upper]. This paper proves an upper bound for the packing density of congruent capped circular cylinders in $\mathbb{R}^3$. The methods are elementary, but can be used to prove non-trivial upper bounds for packings by congruent circular cylinders and related objects, as well as the sharp bound for half-infinite circular cylinders. Synopsis -------- The density bound of A. Bezdek and W. Kuperberg for bi-infinite cylinders is proved in three steps. Given a packing of $\mathbb{R}^3$ by congruent bi-infinite cylinders, first decompose space into regions closer to a particular axis than to any other. Such a region contains the associated cylinder, so density may be determined with respect to a generic region. Finally, this region can be sliced perpendicular to the particular axis and the area of these slices estimated: This area is always large compared to the cross-section of the cylinder. In the case of a packing of $\mathbb{R}^3$ by congruent finite-length cylinders, this method fails. The ends of a cylinder may force some slice of a region to have small area. For example, if a cylinder were to abut another, a region in the decomposition might not even wholly contain a cylinder. To overcome this, one shows that these potentially small area slices are always associated to a small neighborhood of the end of a cylinder. For a packing by cylinders of a relatively high aspect ratio, neighborhoods of the end of a cylinder are relatively rare. By quantifying the rarity of cylinder ends in a packing, and bounding the error contributed by any particular cylinder end, the upper density bound is obtained. Objects considered ------------------ Define a $t$-cylinder to be a closed solid circular cylinder in $\mathbb{R}^3$ with unit radius and length $t.$ Define a capped $t$-cylinder (Figure \[fig1\]) to be a closed set in $\mathbb{R}^3$ composed of a $t$-cylinder with solid unit hemispherical caps. A capped $t$-cylinder $C$ decomposes into the $t$-cylinder body $C^0$ and two caps $C^1$ and $C^2\negthinspace.$ The axis of the capped $t$-cylinder $C$ is the line segment of length $t$ forming the axis of $C^0\negthinspace.$ The capped $t$-cylinder $C$ is also the set of points at most 1 unit from its axis. Packings and densities ---------------------- A $packing$ of $X\subseteq \mathbb{R}^3$ by capped $t$-cylinders is a countable family $\mathscr{C} = \{C_i\}_{i \in I}$ of congruent capped $t$-cylinders $C_i$ with mutually disjoint interiors and $C_i \subseteq X$. For a packing $\mathscr{C}$ of $\mathbb{R}^3$, the $restriction$ of $\mathscr{C}$ to $X\subseteq \mathbb{R}^3$ is defined to be a packing of $\mathbb{R}^3$ by capped $t$-cylinders $\{C_i: C_i \subseteq X\}.$ Let $B(R)$ be the closed ball of radius $R$ centered at $0$. In general, let $B_x(R)$ be the closed ball of radius $R$ centered at $x$. The density $\rho(\mathscr{C}, R, R')$ of a packing $\mathscr{C}$ of $\mathbb{R}^3$ by capped $t$-cylinders with $R\le R'$ is defined as $$\rho(\mathscr{C},R, R') \hspace{2pt}=\hspace{-7pt} \sum_{C_i\subseteq B(R)}\frac{\operatorname{Vol}(C_i)}{\operatorname{Vol}(B(R'))}.$$ The upper density $\rho^+$ of a packing $\mathscr{C}$ of $\mathbb{R}^3$ by capped $t$-cylinders is defined as $$\rho^+(\mathscr{C}) = \limsup_{R\rightarrow \infty} \rho(\mathscr{C},R, R).$$ ![A capped $t$-cylinder with body $C^0$, axis $a$ and caps $C^1$ and $C^2$.[]{data-label="fig1"}](figure1.png){width="50.00000%"} In general, a $packing$ of $X\subseteq \mathbb{R}^3$ by a convex, compact object $K$ is a countable family $\mathscr{K} = \{K_i\}_{i \in I}$ of congruent copies of $K$ with mutually disjoint interiors and $K_i \subseteq X$. Restrictions and densities of packings by $K$ are defined in an analogous fashion to those of packings by capped $t$-cylinders. The Main Results ================ Let $t_0 = \frac{4}{3}(\frac{4}{\sqrt{3}}+1)^3 = 48.3266786\dots$ for the remainder of the paper. This value comes out of the error analysis in Section \[part2\]. \[main\] Fix $t \ge 2t_0.$ Fix $R \ge 2/\sqrt{3}.$ Fix a packing $\mathscr{C}$ of $\mathbb{R}^3$ by capped $t$-cylinders. Then $$\rho(\mathscr{C},R-2/\sqrt{3},R) \le \frac{t+\frac{4}{3}}{\frac{\sqrt{12}}{\pi}(t-2t_0) + (2t_0)+\frac{4}{3}}.$$ This is the content of Sections \[setup\], \[part1\], \[part2\]. Notice that for $t \le 2t_0$, the only upper bound provided is the trivial one. Fix $t\ge 2t_0.$ The upper density of a packing $\mathscr{C}$ of $\mathbb{R}^3$ by capped $t$-cylinders satisfies the inequality $$\rho^+(\mathscr{C}) \le \frac{t+\frac{4}{3}}{\frac{\sqrt{12}}{\pi}(t-2t_0) + (2t_0)+\frac{4}{3}}.$$ Let $V_R$ and $W_R$ be subsets of the index set $I$, with $V_R = \{i : C_i \subseteq B(R)\}$ and $W_R= \{i : C_i \subseteq B(R-2/\sqrt{3})\}.$ By definition, $$\rho^+(\mathscr{C}) = \limsup_{R\rightarrow \infty} \left( \sum_{W_R} \frac{\operatorname{Vol}(C_i)}{\operatorname{Vol}(B(R))} \hspace{2pt} + \sum_{V_R\smallsetminus W_R} \hspace{-2pt}\frac{\operatorname{Vol}(C_i)}{\operatorname{Vol}(B(R))} \right) .$$ As $R$ grows, the term $\sum_{V_R\smallsetminus W_R} \operatorname{Vol}(C_i)/\operatorname{Vol}(B(R))$ tends to $0$. Further analysis of the right-hand side yields $$\rho^+(\mathscr{C}) = \limsup_{R\rightarrow \infty}\rho(\mathscr{C},R-2/\sqrt{3}, R).$$ By Theorem \[main\], the stated inequality holds. \[nestdensity\] Given a packing of $t$-cylinders with density $\rho$ where $t$ is at least 2, there is a packing of capped $(t-2)$-cylinders with packing density $(\frac{ t - \frac{2}{3} }{t})\cdot\rho.$ ![Nesting capped $(t-2)$-cylinders in $t$-cylinders.[]{data-label="fig3"}](figure2.png){width="50.00000%"} From the given packing of $t$-cylinders, construct a packing by capped $(t-2)$-cylinders by nesting as illustrated in Figure \[fig3\]. By comparing volumes, this packing of capped $(t-2)$-cylinders has the required density. Fix $t \ge 2t_0+2$. The upper density of a packing $\mathscr{Z}$ of $\hspace{2pt} \mathbb{R}^3$ by $t$-cylinders satisfies the inequality $$\rho^+(\mathscr{Z}) \le \frac{t}{\frac{\sqrt{12}}{\pi}(t-2-2t_0) + (2t_0)+\frac{4}{3}}.$$ Assume there exists a packing by $t$-cylinders exceeding the stated bound. Then Lemma \[nestdensity\] gives a packing of capped $(t-2)$-cylinders with density greater than $$\left(\frac{ t - \frac{2}{3} }{t}\right)\cdot \left(\frac{t}{\frac{\sqrt{12}}{\pi}(t-2-2t_0) + (2t_0)+\frac{4}{3}}\right) = \left(\frac{t-2+\frac{4}{3}}{\frac{\sqrt{12}}{\pi}(t-2-2t_0) + (2t_0)+\frac{4}{3}}\right).$$ This contradicts the density bound of Theorem \[main\] for capped $(t-2)$-cylinders. Set Up {#setup} ====== For the remainder of the paper, fix the notation $\mathscr{C}^$ to be the restriction of $\mathscr{C}$ to $B(R-2/\sqrt{3})$, indexed by $I^$. To bound the density $\rho(\mathscr{C}^,R-2/\sqrt{3},R)$ for a fixed packing $\mathscr{C}$ and a fixed $R \ge 2/\sqrt{3}$, decompose $B(R)$ into regions $D_i$ with disjoint interiors such that $C_i \subseteq D_i$ for all $i$ in $I^$. For such a packing $\mathscr{C}^$ with fixed $R$, define the Dirichlet cell* $D_i$ of a capped $t$-cylinder $C_i$ to be the set of points in $B(R)$ no further from the axis $a_i$ of $C_i$ than from any other axis $a_j$ of $C_j$. For any point $x$ on axis $a_i$, define a plane $P_x$ normal to $a_i$ and containing $x$. Define the Dirichlet slice $d_x$ be the set $D_i \cap P_x.$ For a fixed Dirichlet slice $d_x$, define $S_x(r)$ to be the circle of radius $r$ centered at $x$ in the plane $P_x$. Important circles are $S_x(1)$, which coincides with the cross section of the boundary of the cylinder, and $S_x(2/\sqrt{3})$, which circumscribes the regular hexagon in which $S_x(1)$ is inscribed. An $end$ of the capped $t$-cylinder $C_i$ refers to an endpoint of the axis $a_i.$ ![A capped cylinder $C$ and the slab $L$.[]{data-label="slab"}](figure3.png){width="50.00000%"} Define the slab $L_i$ to be the closed region of $\mathbb{R}^3$ bounded by the normal planes to $a_i$ through the endpoints of $a_i$ and containing $C_i^0$ (Figure \[slab\]). The Dirichlet cell $D_i$ decomposes into the region $D_i^0 = D_i \cap L_i$ containing $C_i^0$ and complementary regions $D_i^1$ and $D_i^2$ containing the caps $C_i^1$ and $C_i^2$ respectively (Figure \[Dirdecomp\]). ![Decomposing a Dirichlet cell.[]{data-label="Dirdecomp"}](figure4.png){width="50.00000%"} Aside from a few degenerate cases, the set of points equidistant from a point $x$ and line segment $a$ in the affine hull of $x$ and $a$ form a $parabolic$ $spline$ (Figure \[spline\]). A parabolic spline is a parabolic arc extending in a $C^1$ fashion to rays at the points equidistant to both the point $x$ and an endpoint of the line segment $a$. Call the points where the parabolic arc meets the rays the Type points of the curve. A $parabolic$ $spline$ $cylinder$ is a surface that is the cylinder over a parabolic spline. ![Parabolic spline associated with point $x$ and segment $a_j$.[]{data-label="spline"}](figure5.png){width="50.00000%"} Qualified Points {#part1} ================ Fix a packing $\mathscr{C}$ of $\mathbb{R}^3$ by capped $t$-cylinders. Fix $R\ge 2/\sqrt{3}$ and restrict to $\mathscr{C}^$. A point $x$ on an axis is qualified* if the Dirichlet slice $d_x$ has area greater than $\sqrt{12}$, the area of the regular hexagon in which $S_x(1)$ is inscribed. \[qual\] Fix a packing $\mathscr{C}$ of $\mathbb{R}^3$ by capped $t$-cylinders. Fix $R\ge 2/\sqrt{3}$ and restrict to $\mathscr{C}^$. Let $x$ be a point on an axis $a_i$, where $i$ is a fixed element of $I^\hspace{-3pt}.$ If $\hspace{2pt} B_x(4/\sqrt{3})$ contains no ends of $\mathcal{C}^$, then $x$ is qualified. The proof of this proposition will modify the Main Lemma of [@bezdek1990maximum]. A series of lemmas allow for the truncation and rearrangement of the Dirichlet slice. The goal is to construct from $d_x$ a subset $d_x^{}$ of $P_x$ with the following properties: - $d_x^{}$ contains $S_x(1)$. - The boundary of $d_x^{}$ is composed of line segments and parabolic arcs with apexes touching $S_x(1).$ - The non-analytic points of the boundary of $d_x^{}$ lie on $S_x(2/\sqrt{3}).$ - The area of $d_x^{}$ is less than the area of $d_x.$ Then the computations of [@bezdek1990maximum §6] apply. If a point $x$ satisfies the conditions of Proposition \[qual\], then the Dirichlet slice $d_x$ is a bounded convex planar region, the boundary of which is a simple closed curve consisting of a finite union of parabolic arcs, line segments and circular arcs. Without loss of generality, fix a point $x$ on $a_i$. For each $j \ne i$ in $I^\negthinspace,$ let $d^j$ be the set of points in $P_x$ no further from $a_i$ than from $a_j$. The Dirichlet slice $d_x$ is the intersection of $B(R)$ with $d^j$ for all $j \ne i$ in $I^\negthinspace.$ The boundary of $d^j$ is the set of points in $P_x$ that are equidistant from $a_i$ and $a_j$. As $P_x$ is perpendicular to $a_i$ at $x$, the boundary of $d^j$ is also the set of points in $P_x$ equidistant from $x$ and $a_j$. This is the intersection of the plane $P_x$ with the set of points in $\mathbb{R}^3$ equidistant from $x$ and $a_j$. The set of points in $\mathbb{R}^3$ equidistant from $x$ and $a_j$ is a parabolic spline cylinder perpendicular to the affine hull of $x$ and $a_j$. Therefore the set of points equidistant from $x$ and $a_j$ in $P_x$ is also a parabolic spline, with $x$ on the convex side. In the degenerate cases where $x$ is in the affine hull of $a_j$ or $P_x$ is parallel to $a_j$, the set of points equidistant from $x$ and $a_j$ in $P_x$ are lines or is empty. The region $d_x$ is clearly bounded as it is contained in $B(R).$ The point $x$ lies in the convex side of the parabolic spline so each region $d^j$ is convex. The set $B(R)$ contains $x$ and is convex, so $d_x$ is convex. This is a finite intersection of regions bounded by parabolic arcs, lines and a circle, so the rest of the lemma follows. To apply the results of [@bezdek1990maximum], the non-analytic points of the boundary of the Dirichlet slice $d_x$ must be controlled. From the construction of $d_x$ as a finite intersection, the non-analytic points of the boundary of $d_x$ fall into three non-disjoint classes of points: the Type points of a parabolic spline that forms a boundary arc of $d_x$, Type points defined to be points on the boundary of $d_x$ that are also on the boundary of $B(R)$, and Type points, defined to be points on the boundary of $d_x$ that are equidistant from three or more axes. Type III points are the points on the boundary of $d_x$ where the parabolic spline boundaries of various $d^j$ intersect. \[key1\] If a point $x$ satisfies the conditions of Proposition \[qual\], then no non-analytic points of the boundary of $d_x$ are in $\operatorname{int}(\operatorname{Conv}(S_x(2/\sqrt{3})))$, where the interior is with respect to the subspace topology of $P_x$ and $\operatorname{Conv}(\cdot)$ is the convex hull. It is enough to show there are no Type , Type , or Type points in $\operatorname{int}(\operatorname{Conv}(S_x(2/\sqrt{3}))).$ By hypothesis, $B_x(4/\sqrt{3})$ contains no ends. The Type points are equidistant from $x$ and an end. As there are no ends contained in $B_x(4/\sqrt{3})$, there are no Type points in $\operatorname{int}(\operatorname{Conv}(S_x(2/\sqrt{3})))$. By hypothesis, $x$ is in $B(R-2/\sqrt{3})$. Therefore there are no points on the boundary of $B(R)$ in $\operatorname{int}(\operatorname{Conv}(S_x(2/\sqrt{3})))$ and therefore no Type points in $\operatorname{int}(\operatorname{Conv}(S_x(2/\sqrt{3})))$. As a Type point is equidistant from three or more axes, at some distance $\ell$, it is the center of a ball tangent to three unit balls. This is because a capped $t$-cylinder contains a unit ball which meets the ball of radius $\ell$ centered at the Type point. These balls do not overlap as the interiors of the capped $t$-cylinders have empty intersection. Lemma $3$ of [@kuperberg1991placing] states that if a ball of radius $\ell$ intersects three non-overlapping unit balls in $\mathbb{R}^3$, then $\ell \ge 2/\sqrt{3} -1.$ It follows that there are no Type points in $\operatorname{int}(\operatorname{Conv}(S_x(2/\sqrt{3})))$. Fix a packing $\mathscr{C}$. Then for for all $i\ne j$ and $i,j \in I^\negthinspace,$ there is a supporting hyperplane $Q$ of $\operatorname{int}(C_i)$ that is parallel to $a_i$ and separating $\operatorname{int}(C_i \cap L_i)$ from $\operatorname{int}(C_j \cap L_i).$ We can extend $C_i\cap L_i$ to an infinite cylinder $\bar C_i$ where $C_j \cap L_i$ and $\bar C_i$ have disjoint interiors. The sets $C_j \cap L_i$ and $\bar C_i$ are convex, so the Minkowski hyperplane separation theorem gives the existence of a hyperplane separating $\operatorname{int}(C_j \cap L_i)$ and $\operatorname{int}(\bar C_i)$. This hyperplane is parallel to the axis $a_i$ by construction. We may take $Q$ to be the parallel translation to a supporting hyperplane of $\operatorname{int}(C_i)$ that still separates $\operatorname{int}(C_i \cap L_i)$ from $\operatorname{int}(C_j \cap L_i).$ See Figure \[seper\] for an example. ![The hyperplane $Q$ separates $\operatorname{int}(C_i \cap L_i)$ from $\operatorname{int}(C_j \cap L_i).$[]{data-label="seper"}](figure6.png){width="50.00000%"} \[key2\] Fix a packing $\mathscr{C}.$ Fix a point $x$ on the axis $a_i$ of $C_i$ such that $B_x(4/\sqrt{3})$ contains no ends. Let $y$ and $z$ be points on the circle $S_x(2/\sqrt{3})$. If each of $y$ and $z$ is equidistant from $C_i$ and $C_j$, then the angle $yxz$ is smaller than or equal to $2 \arccos (\sqrt{3}-1) := \alpha_0,$ which is approximately $85.88^\circ.$ By hypothesis, $B_x(4/\sqrt{3})$ contains no ends, including the end of the axis $a_i$. Therefore any points of $C_j$ that are not in $L_i$ are at a distance greater than $4/\sqrt{3}$ from $x$. The points of $C_i$ and $C_j$ that $y$ and $z$ are equidistant from must be in the slab $L_i$, so it is enough to consider $y$ and $z$ equidistant from $C_i$ and $C_j \cap L_i.$ By construction, the hyperplane $Q$ separates all points of $C_j \cap L_i$ from $x$. Let $k$ be the line of intersection between $P_x$ and $Q$. As $y$ and $z$ are at a distance of $2/\sqrt{3} -1$ from both $C_j\cap L_i$ and $C_i$, they are at most that distance from $Q$. They are also at most that distance from $k$. The largest possible angle $yxz$ occurs when $y$ and $z$ are on the $x$ side of $k$ in $P_x$, each at exactly the distance $2/\sqrt{3} -1$ from $k$ as illustrated in Figure \[fig7\] . This angle is exactly $2 \arccos (\sqrt{3}-1) := \alpha_0.$ ![An extremal configuration for the angle $\alpha_0.$[]{data-label="fig7"}](figure7.png){width="50.00000%"} The following lemma is proved in [@bezdek1990maximum]. \[key3\] Let $y$ and $z$ be points on $S_x(2/\sqrt{3})$ such that $60^\circ < yxz < \alpha_0.$ For every parabola $p$ passing through $y$ and $z$ and having $S_x(1)$ on its convex side, let $xypzx$ denote the region bounded by segments $xy$, $xz$, and the parabola $p$. Let $p_0$ denote the parabola passing through $y$ and $z$ and tangent to $S_x(1)$ at its apex. $$\operatorname{Area}(xyp_0zx) \le \operatorname{Area}(xypzx).$$ Truncating and rearranging -------------------------- Consider the Dirichlet slice $d_x$ of a point $x$ satisfying the conditions of Proposition \[qual\]. The following steps produce a region with no greater area than that of $d_x$. Step 1: The boundary of $d_x$ intersects $S_x(2/\sqrt{3})$ in a finite number of points. Label them $y_1, y_2, \dots y_n, y_{n+1} = y_1$ in clockwise order. Intersect $d_x$ with $S_x(2/\sqrt{3})$ and call the new region $d_x^$. By Lemma \[key1\], this is a region bounded by arcs of $S_x(2/\sqrt{3})$, parabolic arcs and line segments, with non-analytic points on $S_x(2/\sqrt{3})$. Step 2: For $i = 1, 2, \dots , n$ if $y_ixy_{i+1} > 60^\circ$ and if the boundary of $d_x^$ between $y_i$ and $y_{i+1}$ is a circular arc of $S_x(2/\sqrt{3})$, then introduce additional vertices $z_{i_1}, z_{i_2}, \dots z_{i_k}$ on the circular arc $y_iy_{i+1}$ so that the polygonal arc $y_iz_{i_1} z_{i_2} \dots z_{i_k}y_{i+1}$ does not intersect $S_x(1)$. Relabel the set of vertices $v_1, v_2, \dots v_m, v_{m+1} = v_1$ in clockwise order. Step 3: If $v_ixv_{i+1} \le 60^\circ$ then truncate $d_x^$ along the line segment $v_iv_{i+1}$ keeping the part of $d_x^$ which contains $S_x(1)$. This does not increase area by construction. If $v_ixv_{i+1} > 60^\circ$ then $v_iv_{i+1}$ is a parabolic arc. Replace it by the parabolic arc through $v_i$ and $v_{i+1}$ touching $S_x(1)$ at its apex. This does not increase area by Lemma \[key2\]. This new region $d_x^{}$ has smaller area than $d_x,$ contains $S_x(1),$ and bounded by line segments and parabolic arcs touching $S_x(1)$ at their apexes, with all non-analytic points of the boundary on $S_x(2/\sqrt{3})$. If consecutive non-analytic points on the boundary have interior angle no greater than $60^\circ$, they are joined by line segments. If they have interior angle between $60^\circ\negthinspace$ and $\alpha_0$, they are joined by a parabolic arc. The following lemma is a consequence of [@bezdek1990maximum §6], which determines the minimum area of such a region. The region $d_x^{}$ has area at least $\sqrt{12}.$ Proposition \[qual\] follows. Decomposition of $B(R)$ and Density Calculation {#part2} =============================================== Decomposition ------------- Fix a packing $\mathscr{C}$. Fix $R \ge 2/\sqrt{3}$ and restrict to $\mathscr{C}^\hspace{-3pt}.$ Let the set $A$ be the union of the axes $a_i$ over $I^\hspace{-3pt}.$ Let $\mathrm{d}\mu$ be the 1-dimensional Hausdorff measure on $A$. Let $X$ be the subset of qualified points of $A$. Let $Y$ be the subset of $A$ given by $\{x \in A : B_x(\frac{4}{\sqrt{3}}) \textrm{ contains no ends}\}.$ Let $Z$ be the subset of $A$ given by $\{x\in A : B_x(\frac{4}{\sqrt{3}}) \textrm{ contains an end}\}.$ Notice that $Y\subseteq X \subseteq A$ from Proposition \[qual\] and $Z = A-Y$ by definition. The sets are $A$, $X$, $Y$, and $Z$ are measurable. The set $A$ is just a finite disjoint union of lines in $\mathbb{R}^3$. The area of the Dirichlet slice $d_x$ is piecewise continuous on $A$, so $X$ is a Borel subset of $A$. Similarly the conditions defining $Y$ and $Z$ make them Borel subsets of $A$. The ball $B(R)$ is finite volume, so $I^$ has some finite cardinality $n$. Decompose $B(R)$ into the regions $\{D^0_i\}_{i=1}^n$, $\{D^1_i\}_{i=1}^n$ and $\{D^2_i\}_{i=1}^n$. Further decompose the regions $\{D^0_i\}_{i=1}^n$ into Dirichlet slices $d_x$, where $x$ is an element of $A.$ Density computation ------------------- From the definition of density, $$\begin{aligned} &\rho(\mathscr{C},R-2/\sqrt{3},R) = \frac{\sum_{I^}\operatorname{Vol}(C_i^0)+\sum_{I^}\operatorname{Vol}(C_i^1)+\sum_{I^}\operatorname{Vol}(C_i^2)}{\sum_{I^}\operatorname{Vol}(D_i^0)+\sum_{I^}\operatorname{Vol}(D_i^1)+\sum_{I^}\operatorname{Vol}(D_i^2)}\end{aligned}$$ as $C_i^j \subseteq D_i^j$, and $\operatorname{Vol}(C_i^0) = t\pi$, and $\operatorname{Vol}(C_i^1)=\operatorname{Vol}(C_i^2) = \frac{2}{3}\pi$, it follows that $$\label{inequality} \rho(\mathscr{C},R-2/\sqrt{3},R) \le \frac{nt\pi+n\frac{4}{3}\pi}{\sum_{I^}\operatorname{Vol}(D_i^0)+n\frac{4}{3}\pi}.$$ Finally, $\rho(\mathscr{C},R-2/\sqrt{3},R)$ is explicitly bounded by the following lemma. \[bound\] For $t \ge 2t_0$, $$\sum_{I^} \operatorname{Vol}(D_i^0) \ge n(\sqrt{12}(t-2t_0) + \pi(2t_0)).$$ The sum $\sum_{I^}\hspace{-2pt}\operatorname{Vol}(D_i^0)$ may be written as an integral of the area of the Dirichlet slices $d_x $ over $A$ $$\sum_{I^*} \operatorname{Vol}(D_i^0) \hspace{2pt}= \int\limits_A \operatorname{Area}(d_x) \,\mathrm{d}\mu.$$ Using the area estimates from Proposition \[qual\], there is an inequality $$\int\limits_A \operatorname{Area}(d_x) \,\mathrm{d}\mu \ge \int\limits_{X} \sqrt{12}\,\mathrm{d}\mu +\hspace{-7pt} \int\limits_{A-X}\hspace{-4pt} \pi \,\mathrm{d}\mu.$$ As $\sqrt{12} > \pi$ and the integration is over a region $A$ with $\mu(A)< \infty$, passing to the subset $Y \subseteq X$ gives $$\int\limits_{X} \sqrt{12}\,\mathrm{d}\mu +\hspace{-7pt} \int\limits_{A-X} \hspace{-4pt}\pi \,\mathrm{d}\mu \ge \int\limits_{Y} \sqrt{12}\,\mathrm{d}\mu +\hspace{-7pt} \int\limits_{A-Y} \hspace{-4pt}\pi \,\mathrm{d}\mu \hspace{3pt}=\hspace{-2pt} \int\limits_{A-Z} \sqrt{12}\,\mathrm{d}\mu+\hspace{-4pt} \int\limits_{Z} \pi\,\mathrm{d}\mu.$$ The measure of $Z$ is the measure of the subset of $A$ that is contained in all the balls of radius $4/\sqrt{3}$ about all the ends of all the cylinders in the packing. This is bounded from above by considering the volume of cylinders contained in balls of radius $4/\sqrt{3}+1.$ If the cylinders completely filled the ball, they would contain at most axis length $\frac{4}{3}(\frac{4}{\sqrt{3}}+1)^3 = t_0$. As each cylinder has two ends, there are at worst $2n$ disjoint balls to consider. Therefore $2nt_0 \ge \mu(Z).$ Provided $t\ge 2t_0$, we have the inequality $$\int\limits_{A-Z} \sqrt{12}\,\mathrm{d}\mu+\hspace{-4pt} \int\limits_{Z} \pi\,\mathrm{d}\mu \ge (nt-2nt_0)\sqrt{12} + 2n(t_0)\pi.$$ By combining inequality (\[inequality\]) with Lemma \[bound\] and simplifying, it follows that $$\begin{aligned} \rho(\mathscr{C},R-2/\sqrt{3},R) \le \frac{t+\frac{4}{3}}{\frac{\sqrt{12}}{\pi}(t-2t_0) + (2t_0)+\frac{4}{3}}\end{aligned}$$ for an arbitrary packing $\mathscr{C}$ of $\mathbb{R}^3$ by capped congruent $t$-cylinders. Conclusions, applications, further questions ============================================ A rule of thumb, a dominating hyperbola --------------------------------------- For $t \ge 0$, the upper bounds for the density of packings by capped and uncapped $t$-cylinders are dominated by curves of the form $\frac{\pi}{\sqrt{12}} + N/t$. Numerically, one finds a useful rule of thumb: The upper density $\rho^+$ of a packing $\mathscr{C}$ of $\mathbb{R}^3$ by capped $t$-cylinders satisfies $$\rho^+(\mathscr{C}) \le \frac{\pi}{\sqrt{12}} + \frac{10}{t}.$$ The upper density $\rho^+$ of a packing $\mathscr{C}$ of $\mathbb{R}^3$ by $t$-cylinders satisfies $$\rho^+(\mathscr{C}) \le \frac{\pi}{\sqrt{12}} + \frac{10}{t}.$$ Examples -------- While the requirement that $t$ be greater than $2t_0$ for a non-trivial upper bound is inconvenient, the upper bound converges rapidly to $\pi/\sqrt{12} = 0.9069\dots$ and is nontrivial for tangible objects, as illustrated in the table below.\ \ Item Length Diameter t Density $\le$ -- ----------------- ---------- ---------- ------------------------------------------- --------------- -- -- Broomstick 1371.6mm 25.4mm 108 0.9956... 20’ PVC Pipe 6096mm 38.1mm 320 0.9353... Capellini 300mm 1mm 600 0.9219... Carbon Nanotube - - $2.64 \times 10^8$ [@wang2009fabrication] 0.9069... \ \ Some further results -------------------- There are other conclusions to be drawn. For example: Consider the density of a packing $\mathscr{C} = \{C_i\}_{i \in I}$ of $\mathbb{R}^3$ by congruent unit radius circular cylinders $C_i$, possibly of infinite length. The upper density $\gamma^+$ of such a packing may be written $$\gamma^+(\mathscr{C}) = \limsup_{r \rightarrow \infty} \sum_I\frac{\operatorname{Vol}(C_i\cap B_0(r))}{\operatorname{Vol}(B_0(r))}$$ and coincides with $\rho^+(\mathscr{C})$ when the lengths of $C_i$ are uniformly bounded. The upper density $\gamma$ of half-infinite cylinders is exactly $\pi/\sqrt{12}.$ ![Packing cylinders with high density.[]{data-label="obvi"}](figure8.png){width="50.00000%"} The lower bound is given by the obvious packing $\mathscr{C}'$ with parallel axes (Figure \[obvi\]) and $\gamma^+(\mathscr{C}') =\pi/\sqrt{12}$. Since a packing $\mathscr{C}(\infty)$ of $\mathbb{R}^3$ by half-infinite cylinders also gives a packing $\mathscr{C}(t)$ of $\mathbb{R}^3$ by $t$-cylinders, the inequality $$\frac{t}{\frac{\sqrt{12}}{\pi}(t-2-2t_0) + (2t_0)+\frac{4}{3}} \ge \rho^+(\mathscr{C}(t)) = \gamma^+(\mathscr{C}(t)) \ge \gamma^+(\mathscr{C}(\infty))$$ holds for all $t \ge 2t_0.$ Given a packing $\mathscr{C} = \{C_i\}_{i \in I}$ by non-congruent capped unit cylinders with lengths constrained to be between $2t_0$ and some uniform upper bound $M$, the density satisfies the inequality $$\rho^+(\mathscr{C}) \le \frac{t+\frac{4}{3}}{\frac{\sqrt{12}}{\pi}(t-2t_0) + (2t_0)+\frac{4}{3}}$$ where $t$ is the average cylinder length given by $\liminf_{r \rightarrow \infty} \mu(a_i) / \|J\|$, where $J$ is the set $\{i\in I: C_i \subseteq B(r)\}$. None of the qualification conditions require a uniform length $t$. Inequality \[inequality\] may be considered with respect to the total length of $A$ rather than $nt$. It may be easier to compute a bound using the following Given a packing $\mathscr{C} = \{C_i\}_{i \in I}$ by non-congruent capped unit cylinders with lengths constrained to be between $2t_0$ and some uniform upper bound $M$, the density satisfies the inequality $$\rho^+(\mathscr{C}) \le \frac{t+\frac{4}{3}}{\frac{\sqrt{12}}{\pi}(t-2t_0) + (2t_0)+\frac{4}{3}}$$ where $t$ is the infimum of cylinder length. The stated bound is a decreasing function in $t$, so this follows from the previous theorem. Questions and Remarks --------------------- Similar but much weaker results can be obtained for the packing density of curved tubes by realizing them as containers for cylinders. Better bounds on tubes would come from better bounds on $t$-cylinders for $t$ small. There is the conjecture of Wilker, where the expected packing density of congruent unit radius circular cylinders of $any$ length is $exactly$ the planar packing density of the circle, but that is certainly beyond the techniques of this paper. A more tractable extension of this might be to parametrize the densities for capped $t$-cylinders from the sphere to the infinite cylinder by controlling the ends. In this paper, the analysis assumes anything in a neighborhood of an end packs with density 1, whereas it is expected that the ends and nearby sections of tubes would pack with density closer to $\pi/\sqrt{18}.$ $\hspace{3pt}$ In [@torquato2012organizing], it is conjectured that the densest packing is obtained from extending a dense sphere packing, giving a density bound of $$\rho^+(\mathscr{C}(t)) =\frac{\pi}{\sqrt{12}} \hspace{4pt} \frac{t + \frac{4}{3} } {t + \frac {2\sqrt{6}}{3} }.$$\ The structure of high density cylinder packings is also unclear. For infinite circular cylinders, there are nonparallel packings with positive density [@kuperberg1990nonparallel]. In the case of finite length $t$-cylinders, there exist nonparallel packings with density close to $\pi/\sqrt{12}$, obtained by laminating large uniform cubes packed with parallel cylinders, shrinking the cylinders and perturbing their axes. It is unclear how or if the alignment of cylinders is correlated with density. Finally, as the upper bound presented is not sharp, it is still not useable to control the defects of packings achieving the maximal density. A conjecture is that, for a packing of $\mathbb{R}^3$ by $t$-cylinders to achieve a density of $\pi/\sqrt{12}$, the packing must contain arbitrarily large regions of $t$-cylinders with axes arbitrarily close to parallel. [^1]: Research partially supported by NSF grant 1104102.
--- address: \| Department of Basic Science, University of Tokyo\ 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan author: - Daijiro Yoshioka date: 19 November 1998 title: 'Comment on “Effective Mass and g-Factor of Four Flux Quanta Composite Fermions"' --- \#1[[$\backslash$\#1]{}]{} \#1\#2\#3\#4[[\#1]{} [\#2]{} (\#4) \#3]{} In a recent Letter, Yeh et al.[@yeh] have shown beautiful experimental results which indicate that the composite fermions with four flux quanta ($^4$CF) behave as fermions with mass and spin just like those with two flux quanta. They observed the collapse of the fractional quantum Hall gaps when the following condition is satisfied with some integer $j$, g\^\\_[B]{}B\_[tot]{} = j \_[c]{}\^\, where $g^$ and $\omega_{\rm c}^$ are the g-factor and the cyclotron frequency of the $^4$CF, respectively. They argued that level crossing and hence the collapse occur when eq.(1) is satisfied. However, as they have pointed out, level crossing always occurs away from the Fermi level, and gap at the Fermi level remains finite in their interpretation. Thus the reason of the collapse was left as a mystery. Here I will show that part of the mystery is resolved by considering the electron-hole symmetry properly. The experiment was done around $\nu=3/4$, where $^4$CF quantum Hall effect occurs at the electron filling factor $\nu=(3p+1)/(4p+1)$, $p= \pm 1, \pm 2...$. Yeh et al. considered these fillings as electron-hole symmetric states of filling factor $\nu=p/(4p+1)$. However, this treatment is not appropriate. We should consider the spin freedom of the $^4$CF as coming from the actual spin of the original electrons. Therefore, the electron-hole symmetric state of the filling factor $\nu=(3p+1)/(4p+1)$ is realized at $\nu = 1 + p/(4p+1)$. Now we need to establish a rule for the composite fermion transformation in this situation. For that purpose it is helpful to consider the case of two flux quanta composite fermion ($^2$CF) around $\nu=1/2$. The electron-hole symmetric state of this case is realized around $\nu=1+1/2$. Requiring that the condition for the collapse of the quantum Hall state has the electron-hole symmetry, we get a set of rules. We explain these rules by applying them to the present case of the filling factor $\nu=1 + p/(4p+1)$. To make the discussion concrete, we consider a finite size system with $N_e$ electrons and the Landau level degeneracy $N_0$, i.e. $\nu=1 + p/(4p+1) = N_e/N_0$. This system is converted into that of $^4$CF by the following rules. (1) All the electrons are changed into CF’s. (2) However, each electrononic states in the down spin Landau level gives four flux quanta with opposite sign, thus the effective flux quanta in the system is $N_{0,{\rm eff}} = N_0 - 4(N_e-N_0)$. Then the effective magnetic field for the CF’s is reduced from $B_{\perp}$ to $B_{\rm eff} = B_{\perp}/(4p+1)$. The remaining rules are that (3) the maximum allowed CF filling factor for each spin state is $\|4p+1\|$,[@comment1] and (4) the Zeeman splitting of these levels are $g^\mu_{\rm B} B_{\rm tot}$. Therefore in the limit of infinite $g^$ the down spin CF Landau levels are fully occupied up to $\nu_\downarrow = \|4p+1\|$, and up spin levels are occupied with filling factor $\nu_\uparrow =\|p\|$.[@comment] Now we can investigate the condition for the collapse of the quantum Hall state. In the course of reducing $g^$ from infinity, the collapse of the gap at the Fermi level occurs when the highest occupied Landau level (HOLL) of the down spin CF coincide with the lowest unoccupied Landau level(LULL) of the up spin CF. When $\nu_\uparrow = \|p\| + k$, and $\nu_\downarrow = \|4p+1\| -k$, with integer $k$, the energies of the down spin HOLL and up spin LULL are at E\_[,]{} = \_c\^\ (\_[,]{} ) g\^\\_[B]{} B\_[tot]{}, respectively. Therefore the condition for the collapse is g\^\\_[B]{} B\_[tot]{} = (\|4p+1\|-\|p\|-2k-1) \_c\^\* j \_c\^\. For negative $p$ this condition completely explains the strong collapse of the experiment. For example at $p=-2$ or $\nu=5/7$, the collapse is expected at $j=0$, 2, and 4, which agrees with the experiment where among $j=1$, 2, and 3 only $j=2$ shows the collapse. On the other hand, for positive value of $p$ the condition above, $j=3p-2k$, gives only weak collapse, and stronger collapses are observed at $j=3p-2k-1$. This lack of symmetry between positive and negative values of $p$ has not been observed for the quantum Hall states of $^2$CF around $\nu=3/2$, where every strong collapse is associated with the collapse of the energy gap at the Fermi level.[@du] The present asymmetry may be related to the fact that the positive $p$ gives electron filling factor close to unity: Around $\nu=1$ skyrmion excitation that mixes the up and down spin states are known to exists. Thus I have shown that the strong collapse is related to the closing of the energy gap at the Fermi level of the $^4$CF as long as the opposite spin CFs are independent of each other. I thank Dan Tsui who showed me the experimental results prior to publication while we stayed at Aspen Center for Physics, where main part of this work was done. A.S. Yeh, H.L. Stormer, D.C. Tsui, L.N. Pfeiffer, K.W. Baldwin, and K.W. West, Phys. Rev. Lett. [82]{}, 592 (1999). We assume that [electrons]{} occupy only the lowest Landau levels for each spin. Notice that the large $g^$ limit, where $\nu$ and $1+\nu$ states should be equivalent, is recovered correctly by these rules. R.R. Du, A.S. Yeh, H.L. Stormer, D.C. Tsui, L.N. Pfeiffer, and K.W. West, Phys. Rev. Lett. [75]{}, 3926 (1995).
--- abstract: 'We provide further information on an effective band-dependent tight-binding model that allows for a simple understanding of the block-like transitions between $P_s$ regimes, on the analysis of multifractality of single-particle states, on the special case of $a=1$, on the relation between inverse participation ratio and survival probability, on the convergence of the GAAH model towards the AAH model when $a\to\infty$, and on localization and ergodic-to-multifractal transitions.' author: - 'X. Deng' - 'S. Ray' - 'S. Sinha' - 'G. V. Shlyapnikov' - 'L. Santos' title: \| Supplementary material of\ “One-dimensional quasicrystals with power-law hopping” --- Effective band-dependent tight-binding model ============================================ The abrupt block-like transitions between $P_s$ regimes may be understood from the analysis of the dispersion of the subbands. We study the GAAH model: $$\begin{aligned} H&=&-J\sum_j \sum_{s=1}^\infty \frac{1}{s^a} \left (c_j^\dag c_{j+s} +{\mathrm H. c. } \right ) \nonumber \\ &-& \Delta \sum_j \cos \left ( \beta (2\pi j + \phi) \right )c_j^\dag c_j, \label{eq:H}\end{aligned}$$ We consider $\beta = \frac{1}{p+\beta_1}$, with integer $p\gg 1$, and $\beta_1\ll 1$. We consider first the case $\beta=1/p$. In that case the system presents a period $p$, and hence the tight-binding band splits into $p$ subbands. By solving the eigensystem $$\begin{aligned} E \psi(j) &=& -J\sum_j \sum_{r=1}^\infty \frac{1}{r^a} \left ( e^{i\kappa r/n} \psi(j-r) + {\mathrm c.c.} \right ) \nonumber \\ &-&\Delta \sum_j \cos \left ( \beta (2\pi j + \phi) \right ) \psi(j),\end{aligned}$$ we obtain the dispersion of the subbands as a function of the quasimomentum $-\pi < \kappa \leq \pi$, and the phase shift $-\pi<\phi\leq \pi$. Like in renormalization group calculations for the AAH model [@Suslov1982; @Wilkinson1984; @Szabo2018], our numerical calculations show that the $\kappa$ and $\phi$ dependences remain uncoupled, and the subband dispersion acquires the form: $$E_m(\kappa,\phi) = E_{m,0} - 2 J_{m} {\cal R} \left ( Li_{a_{m}}(e^{i\kappa}) \right ) - \Delta_m \cos(\phi)$$ where $m$ denotes the band index, $E_{m,0}$ is the central energy of the $m$-th band, $Li_a(x)=\sum_{s=1}^\infty \frac{x^s}{s^a}$ is the polylogarithm function, and ${\cal R}$ indicates the real part. The quasimomentum dependence is the one that is expected for a model with power-law hopping with power $a_m$. Indeed, introducing the transformation $c_j=\sum_\kappa e^{i\kappa j}b_\kappa$, the hopping term in Eq. becomes $-2J\sum_{\kappa} {\cal R}(Li_a(e^{i\kappa})) b_\kappa^\dag b_\kappa$. For $a\gg 1$, ${\cal R}(Li_a(e^{i\kappa}))\simeq \cos(\kappa)$ and the dispersion of a tight-binding model with nearest-neighbor hopping is retrieved. Hence, the subbands present the dispersion expected for a power-law Hamiltonian, but with regularized band-dependent hopping amplitude $J_m$ and power $a_m$. In this effective model, the Wannier functions are those of a superlattice with a unit cell which is $p$ times larger than the original one. Since $\beta=1/(p+\beta_1)$, there is a slowly varying $\phi=2\pi\beta_1 x$ in the new lattice which leads to a quasi-disorder term with an effective band-dependent strength $\Delta_m$. The effective Hamiltonian for the $m$-th band is hence: $$\begin{aligned} H_m&=&-J_m\sum_j \sum_{r=1}^\infty \frac{1}{r^{a_m}} \left (c_j^\dag c_{j+r} + {\mathrm H. c. } \right )\nonumber \\ &-&\Delta_m \sum_j \cos \left ( \beta_1 (2\pi j + \phi') \right ) c_j^\dag c_j, \label{eq:Hm}\end{aligned}$$ with an effective displacement $\phi'$. We consider below $p=10$, but similar reasonings apply as long as $p$ is sufficiently large. The evaluation of the most energetic subbands for $a>2$ shows that they present a dispersion with $a_m>6$. For these $a_m$ values, the $\kappa$ dependence of the dispersion is practically indistinguishable from a cosine, $-2J_m\cos(\kappa)$, i.e. the dependence of a model with nearest-neighbor hopping. As a result, the subband behaves effectively as an AAH model [@footnote-2]. Hence, when $\Delta_m/J_m=2$ we expect that the whole subband localizes. Since the relation between $\Delta$ and $\Delta_m/J_m$ is band-dependent, the different subbands localize for different values of $\Delta$, starting with the most energetic band $m=p$. This results in the hierarchy of $P_s$ phases discussed in the main text. The $P_s$ phase is characterized by the localization of the highest $s$ bands, and hence the mobility edge would be at a fraction $1-s/p$ of the whole spectrum. For $s/p \ll 1$, the edge may be approximated as $(1-\beta)^s$, which is the numerically observed mobility edge for $\beta=1/(p+\beta)$, with $p>1$. From the effective model discussed above we can evaluate the boundaries of the $P_s$ phases. We depict in Fig \[fig:SM1\] the case of $\beta=1/(10+\beta)=\sqrt{26}-5$, showing that there is an excellent agreement between the results obtained from the spectral analysis of the GAAH model, and those obtained from the effective band model. The reasoning is however significantly more involved for $a<2$ or for the lowest subbands (i.e. when approaching the boundary of the AL phase), since the subbands retain a significant power law character, i.e. the $\kappa$ dependence of the subband dispersion cannot be well approximated by a cosine. In that case a more careful renormalization analysis, beyond the scope of this paper, is necessary to describe the $P_s$ phases. Multifractality =============== In the main text we have monitored the localized, ergodic, or multifractal character of the single-particle states (SPS) by means of the analysis of $D_2$ and the level spacings. As briefly discussed in this section, we have complemented this study with the analysis of the multifractal spectrum (MFS) and fractal dimensions $D_{q>2}$. For a given eigenstate $\|\psi_n \rangle=\sum_j \psi_n(j) \|j\rangle$, we may evaluate the moments $I_q(n)=\sum_j \|\psi_n(j)\|^{2q}\propto L^{-\tau(q)}$, with $L$ the number of sites. The Legendre transform $\tau(q)=q\alpha-f(\alpha)$ defines the MFS $f(\alpha)$, which characterizes the Hausdorff dimension of the manifold of sites where $\|\psi_{n}(j)\|^2=L^{-\alpha}$ [@Evers2008; @DeLuca2014; @Deng2016]. Normalization, $\sum_{j}\|\psi_{n}(j)\|^2=1$, requires $\alpha\geq 0$. The MFS is evaluated as discussed in Refs. [@DeLuca2014; @Deng2016]. Figure \[fig:SM2\] shows our results for $\beta=(\sqrt{5}-1)/2$ for $a=0.1$ and $1.5$ evaluated for values of $\Delta$ within the $P_2$ phase. We monitor the eigenstates with energies $E_{\beta^2 L<n\le L}$. In the $P_2$ phase those states are localized for $a>1$ and multifractal for $a<1$. Algebraically localized states, $\|\psi_{n}(j)\|^2\propto 1/\|j-j_{0}\|^{\gamma}$, present a triangular MFS $f(\alpha)$, where $f(\alpha)=k\alpha$ for $0<\alpha<1/k$ with $k=1/\gamma$ [@DeLuca2014; @Deng2016]. We obtain such a dependence for $a=1.5$. On the contrary, for $a=0.1$ the MFS presents the characteristic dependence of multifractal states [@DeLuca2014; @Deng2016], in particular the expected symmetry $f(1+\alpha) = f(1-\alpha)+\alpha$ [@Mirlin2006]. The multifractal character of the non-ergodic states for $a<1$ is further proved by the analysis of fractal dimensions $D_{q>2}$. Ergodic (localized) states are characterized by $D_q=1$ ($0$) for all $q$, whereas the non-ergodic states for $a<1$ are characterized by a $q$-dependent $D_q$, as expected for multifractal states (see Fig. \[fig:SM3\]). \[fig:SM3\] Critical line at $a=1$ ====================== We discuss at this point in some more detail the case of $a=1$, since its numerical evaluation is particularly involved. We may understand this case following the ideas of Ref. [@Monthus2018]. In the limit of $J=0$ all eigenstates $\|\psi^{(0)}_n\rangle=\|n\rangle$ are localized, and the corresponding eigenvalue is given by the on-site quasi-periodic potential, $E_n$. For finite hopping $J/\Delta\ll 1$, the eigenstates may be evaluated to first-order in perturbation theory, $\|\psi^{(1)}_n\rangle\simeq \|n\rangle+\sum_{n' \ne n}\|n'\rangle \langle{n'}\|\psi^{(1)}_n\rangle$, with $\|\langle{n'}\|\psi^{(1)}_n\rangle\|^2=\|\frac{\frac{1}{\|n-n'\|^a}}{E_n-E_{n'}}\|^2$. The hopping radius scales as $1/L^a$, whereas the level spacings do it as $1/L$. Hence, when $a>1$ the wavefunction is localized, while $a=1$ is critical. As a result, for $a=1$ the logarithmic correction, $I_q = L^{-\tau(q)}+b(\ln{L})^{\eta(q)}$, plays an important role [@Monthus2018], and must be carefully considered in our fitting of the inverse participation ratio. For $a\to 1$ and a large number of sample states, we obtain the best fit of our numerical results for $\eta(2)\approx -1$, which is consistent with Ref. [@Monthus2018], and $\tau(2)=0$ (i.e. $D_2=0$). In contrast, for $a=0.8$ we find $b\simeq 0$, and a finite $0<D_2<1$, characteristic of multifractal states. Inverse participation ratio and survival probability ==================================================== The value of $D_2$ may fluctuate within the block of multifractal states. As illustrated in Fig. \[fig:SM4\], we have observed that for $a<1$ the fractal dimension $D_2$ does not present a dependence on the energy (the error bars in Fig. 5(c) of the main text indicate this uncertainty). Only in the vicinity of $a=1$ fluctuations of $D_2$ for different multifractal eigenstates become significant. This is, on the other hand, expected due to the above mentioned critical nature of the $a=1$ case. For the determination of $D_2$ close to $a=1$ we evaluate it with the highest accuracy over the multifractal states. From our numerics, we obtain the best fit $D_2(a)\approx \frac{1}{3}(1-a)^{1/2}$ for $0<a\leq 1$ (for $a=1$, as mentioned above, we obtain $I_2\propto 1/\ln L$). Furthermore, in our numerics, for $a<1$ we do not find any localized state up to very large quasi-disorder strengths $\Delta=10$. Indeed, we have checked that $D_2$ has no significant $\Delta$ dependence when evaluated for a given $a<1$. The survival probability for an infinite system is expected to decay with a power law $l(t)\propto t^{-\gamma}$ for $t\to\infty$ [@Ketzmerick1992]. We have evaluated the exponent $\gamma$ by averaging the function $l(t)$ for up to $100$ different realizations of the displacement $\phi$ (we have checked that averaging over further realizations does not change the results) for lattices with up to $L=28657$ sites with open boundary conditions. After an initial stage characterized by an exponential decay of $l(t)$, the Loschmidt amplitude is well fitted as $l(t)\sim t^{-\gamma}$. The observed small oscillations, that we associate to spectral rigidity [@DeTomasi2018], do not compromise the fitting. We have checked this by considering the survival probability in a small but finite central region, $F(R=10,t)$. This averaging removes the small oscillations, leaving the bare overall decay $t^{-\gamma}$. Fitting $F(R=10,t)$ provides almost identical results as those obtained with the bare $l(t)$. Hence, in the numerical results depicted in the main text we have employed the fit of $l(t)$. The small error bars in the determination of $\gamma$, depicted in Fig. 5(c) of the main text, result from the uncertainty of this fitting procedure. For the AAH model, we obtain $\gamma=1$, $0$, and $\simeq 0.26$ (after averaging over disorder samples) for $\Delta<2$ (ergodic), $\Delta>2$ (localized), and $\Delta=2$ (multifractal), respectively [@Zhong1995]. For the GAAH model the long-time evolution of the survival probability is again well fitted by a power-law dependence. In the AE (AL) regime, we recover $\gamma=1$ ($0$). In the $P_s$ regimes, for $a>1$ the long-time evolution is dominated by the localized states, and hence $\gamma=0$. On the contrary, for $a<1$ the long-time dependence of the survival probability is dominated by the (slowly expanding) multifractal states. We hence obtain $0<\gamma<1$, with a best fit $\gamma\simeq D_2/(2-a)$ (Fig. 5(c) of the main text). As for the dependence of $D_2$ on the system size, the long-time dynamics of $l(t)$ is also peculiar for $a=1$, for which we observe a decay with a power law in $\ln t$. Convergence of the GAAH model towards the AAH model for $a\to\infty$ ==================================================================== As discussed in the main text, for $a>1$ there are two distinct blocks of eigenstates within the $P_s$ regimes: ergodic and localized states, which for $\beta=(\sqrt{5}-1)/2$ are, respectively, those with energies $E_{n\leq\beta^s L}$ and $E_{n>\beta^s L}$. On the other hand, it is known that for the AAH model all eigenstates remain ergodic (localized) for $\Delta<2$ ($\Delta>2$), whereas for $\Delta=2$ all states are multifractal. As shown in Fig. 1 of the main text, for $a\gg 1$ the $P_s$ regions squeeze into the vicinity of $\Delta=2$. However, the question remains on how the GAAH model converges towards the AAH model when $a \to \infty$, i.e. how the multifractal nature of $\Delta=2$ in the AAH model is retrieved. Figure \[fig:SM5\] illustrates this convergence. For $\Delta=2$ within the $P_2$ regime, we evaluate the states with eigenenergy $E_n$, at indices $n=1$ (ergodic) and $\beta L$ (localized), for various values of $a$. Multifractal states appear in the thermodynamic limit only for $a\to\infty$. However, for finite-size systems there are multifractal states for a sufficiently large $a$. This may be understood from the comparison of the energy difference between two neighboring levels, $s_L\propto 1/L$, with the next-to-nearest-neighbor hopping, $J_2 \propto 1/2^a$. If $J_2 \ll s_L$, the GAAH model may be approximated by the AAH model and multifractal states appear for $\Delta=2$. However, when $J_2 \gg s_L$, the GAAH model has not yet converged to the AAH one, and the system presents only ergodic and localized states. Hence, only when $a \gg log_2L$ the well-known multifractality for $\Delta=2$ in the AAH model is retrieved. These unusual delocalized states which are ergodic at large scales but non-ergodic at small scales are reminiscent of the scaling of Anderson transition in random graphs [@GarciaMata2017]. Localization and ergodic-to-multifractal transitions ==================================================== In this section we briefly discuss the transition of eigenstates from ergodic to localized (multifractal) when increasing $\Delta$ for $a>1$ ($<1$). We have analyzed these transitions for systems sizes up to $L=75025$ and with periodic boundary conditions, performing a finite-size scaling of the transition point. This analysis is illustrated in Fig. \[fig:SM6\], where we consider the case of $a=0.5$, and $0.5<\Delta<1$. In this regime the fractal dimension $D_2$ jumps from $1$ to $\approx 0.24$ for the eigenstates $E_{\beta^2<n\le \beta L}$ (boundary between the $P_1$ and $P_2$ regimes). Using the odd-even and even-odd energy spacings introduced in the main text, we define the level gap as $\delta s = \ln(s^{o-e})-\ln(s^{e-o})$. We observe that $\delta{s}\propto \ln{L}$. Therefore, using this relation we can extrapolate it to the thermodynamic limit. In general, $\delta s$ is finite for ergodic states and vanishes for non-ergodic ones. Figure \[fig:SM6\] hence reveals the transition between ergodic and multifractal states. We obtain similar results when analyzing the localization transition for $a>1$. These results show that the boundaries in Fig. 1 of the main text have very little dependence on the system size. [99]{} I. M. Suslov, Sov. Phys. JETP [56]{}, 612 (1982). M. Wilkinson, Proc. R. Soc. Lond. A [391]{}, 305 (1984). A. Szabó and U. Schneider, Phys. Rev. B [98]{}, 134201 (2018). The fact that $a_m$, although large, remains finite means that the transition between $P_s$ regimes is not infinitely abrupt, but rather splits into a tightly-close self-similar set of transitions. However, for all practical purposes the transition between $P_s$ regimes can be considered a single abrupt transiition. F. Evers and A. D. Mirlin, Rev. Mod. Phys. [80]{}, 1355 (2008). A. De Luca, B. L. Altshuler, V. E. Kravtsov, and A. Scardicchio, Phys. Rev. Lett. [113]{}, 046806 (2014). X. Deng, B. L. Altshuler, G. V. Shlyapnikov, and L. Santos, Phys. Rev. Lett. [117]{}, 020401 (2016). A. D. Mirlin, Y. V. Fyodorov, A. Mildenberger, and F. Evers, Phys. Rev. Lett. [97]{}, 046803 (2006). C. Monthus, Fractals, doi: /10.1142/S0218348X19500075; arXiv:1706.04099. R. Ketzmerick, G. Petschel and T. Geisel, Phys. Rev. Lett. [69]{}, 695 (1992). G. De Tomasi, M. Amini, S. Bera, I. M. Khaymovich, and V. E. Kravtsov, arXiv:1805.06472. J. X. Zhong and R. Mosseri, J. Phys: Condens. 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--- abstract: 'The earlier-developed master equation approach and kinetic cluster methods are applied to study kinetics of L1$_0$-type orderings in alloys, including the formation of twinned structures characteristic of cubic-tetragonal-type phase transitions. A microscopical model of interatomic deformational interactions is suggested which generalizes a similar model of Khachaturyan for dilute alloys to the physically interesting case of concentrated alloys. The model is used to simulate A1$\to$L1$_0$ transformations after a quench of an alloy from the disordered A1 phase to the single-phase L1$_0$ state for a number of alloy models with different chemical interactions, temperatures, concentrations, and tetragonal distortions. We find a number of peculiar features in both transient microstructures and transformation kinetics, many of them agreeng well with experimental data. The simulations also demonstrate a phenomenon of an interaction-dependent alignment of antiphase boundaries in nearly-equilibrium twinned bands which seems to be observed in some experiments.' address: 'Russian Research Centre ‘Kurchatov Institute’, Moscow 123182, Russia' author: - 'K. D. Belashchenko[@Ames], I. R. Pankratov, G. D. Samolyuk[@Ames] and V. G. Vaks' title: 'Kinetics of formation of twinned structures under L1$_0$-type orderings in alloys' --- Introduction {#intro} ============ Studies of microstructural evolution under alloy phase transformations from the disordered FCC phase (A1 phase) to the CuAu I-type ordered tetragonal phase (L1$_0$ phase) attract interest from both fundamental and applied points of view. A characteristic feature of such transitions is the formation in the ordered phase of peculiar ‘polytwinned’ structures consisting of arrays of ordered bands separated by the antiphase boundaries (APBs) lying in the (110)-type planes, while the tetragonal axes of antiphase-ordered domains (APDs) in the adjacent bands have ‘twin-related’ (100) and (010)-type orientations [@Leroux; @Zhang-91; @Zhang-92; @Yanar; @Tanaka; @Oshima; @Syutkina]. Transformation A1$\to$L1$_0$ includes a number of intermediate stages, including the ‘tweed’ stage discussed below. These transformations are inherent, in particular, to many alloy systems with outstanding magnetic characteristics, such as Co–Pt, Fe–Pt, Fe–Pd and similar alloys, and studies of their microstructural features, for example, properties and evolution of APBs, are interesting for applications of these systems in various magnetic devices for which the structure and the distribution of APBs can be very important [@Zhang-91; @Zhang-92; @Yanar]. The physical reason for the formation of twinned structures was discussed by a number of authors [@Roitburd; @Khach-Shat; @Khach-book; @Vaks-01], and it is explained by the elimination of the volume-dependent part of elastic energy for such structures. However, theoretical treatments of the kinetics of A1$\to$L1$_0$ transformation seem to be rather scarce as yet. Khachaturyan and coworkers [@CWK] discussed kinetics of tweed and twin formation using a 2D model in a square lattice with a number of simplifying approximations: a mean-field-type kinetic equation; a phenomenological description of interaction between elastic strains and local order parameters; an isotropic elasticity; an unrealistic interatomic interaction model (with the nearest-neighbour interaction being by an order of magnitude weaker than more distant interactions), etc. In spite of all these assumptions, some features of evolution found by Khachaturyan and coworkers [@CWK] agree qualitatively with experimental observations [@Zhang-91; @Zhang-92; @Yanar]. It may illustrate a low sensitivity of these features to the real structure and interactions in an alloy. However, such an oversimplified approach is evidently insufficient to study the details of evolution and their dependence on the characteristics of an alloy, such as the type of interatomic interaction, concentration, temperature, etc, which seems to be most interesting for both applications and physical studies of the problem. In this work we investigate kinetics of the A1$\to$L1$_0$ transition using the microscopical master equation approach and the kinetic cluster field method [@Vaks-96; @BV-98]. Earlier this method was used to study A1$\to$L1$_2$-type transformations [@BDPSV] as well as early stages of the A1$\to$L1$_0$ transition when the deformational interaction $H_d$ due to the tetragonal distortion of the L1$_0$ phase is still insignificant for the evolution [@PV]. Here we consider all stages of this transition, including the tweed and twin stages when the interaction $H_d$ becomes important. To this end we first derive a microscopical model for $H_d$ which generalizes the analogous model of Khachaturyan for dilute alloys [@Khach-book] to the physically interesting case of concentrated alloys. Then we employ the kinetic cluster field method to simulate A1$\to$L1$_0$ transformation in the presence of deformational interaction $H_d$ for a number of alloy models with both short-range and extended-range chemical interactions at different temperatures, concentrations and tetragonal deformations. The simulations reveal a number of interesting microstructural features, many of them agreeing well with experimental observations [@Zhang-91; @Zhang-92; @Yanar]. We observe, in particular, a peculiar phenomenon of an interaction-dependent alignment of orientations of APBs within twin bands which was earlier discussed phenomenologically [@Vaks-01]. The simulations also show that the type of microstructural evolution strongly depends on the interaction type as well as on the concentration $c$ and temperature $T$. In particular, drastic, phase-transiton-like changes in morphology of APBs within twin bands can occur under variation of $c$ or $T$ in the short-range-interaction systems. The paper is organized as follows. In section 2 we derive a microscopical expression for the deformational interaction $H_d$ in concentrated alloys. In section 3 we describe our methods of simulation of A1$\to$L1$_0$ transition which are similar to those used earlier [@BDPSV; @PV]. In section 4 we investigate the transformation kinetics for the alloy systems with an extended or intermediate interaction range, and in section 5, that for the short-range-interaction systems. Our main conclusions are summarized in section 6. Model for deformational interaction in concentrated alloys {#Deform-int} ========================================================== We consider a binary substitutional alloy A$_c$B$_{1-c}$. Various distributions of atoms over lattice sites $i$ are described by the sets of occupation numbers $\{n_i\}$ where the operator $n_{i}=n_{{\rm A}i}$ is unity when the site $i$ is occupied by atom A and zero otherwise. The effective Hamiltonian $H_{\rm eff}$ describing the energy of these distributions has the form $$H_{\rm eff}=\sum_{i>j}v_{ij}n_in_j+ \sum_{i>j>k}v_{ijk}n_in_jn_k+\ldots \label{H_{eff}}$$ where $v_{i\ldots j}$ are effective interactions. The interactions $v_{i\ldots j}$ include the ‘chemical’ contributions $v_{i\ldots j}^c$ which describe the energy changes under the substitution of some atoms A by atoms B in the rigid lattice, and the ‘deformational’ interactions $v_{i\ldots j}^d$ related to the difference in the lattice deformation under such a substitution. A microscopical model for $v^d$ in dilute alloys was suggested by Khachaturyan [@Khach-book]. The deformational interaction in concentrated alloys can lead to some new effects that are absent in the dilute alloys, in particular, to the lattice symmetry changes under phase transformations, such as the tetragonal distortion under L1$_0$ ordering. Earlier these effects were treated only phenomenologically [@CWK]. Below we describe a microscopical model for calculations of $v^d$ which generalizes the Khachaturyan’s approach [@Khach-book] to the case of concentrated alloys. Let us denote the position of site $k$ in the disordered ‘averaged’ crystal as ${\bf r}_k$. Because of the randomness of a real disordered or partially ordered alloy the actual atomic position (averaged over thermal vibrations) is not ${\bf r}_k$ but ${\bf r}_k+{\bf u}_k$ where ${\bf u}_k$ is the ‘static displacement’. Supposing this displacement to be small we can expand the ‘adiabatic’ (averaged over rapid phonon motion) alloy energy $H=H\{n_i,{\bf u}_k\}$ to second order in ${\bf u}_k$: $$H=H_c\{n_i\}-\sum_k u_{\alpha k}F^{\alpha k}+{1\over 2}\sum_{k,l} u_{\alpha k}u_{\beta l} A_{\alpha k,\beta l} \label{H_u}$$ where $\alpha$ and $\beta$ are Cartesian indices and the summation over repeated Greek indices is implied here and below. The term $H_c\{n_i\}$ in (\[H\_u\]) describes interactions in the undistorted average crystal lattice, i. e. chemical interactions $v_{i\ldots j}^c$ mentioned above; $F^{\alpha k}$ can be called the generalized Kanzaki force; and $A_{\alpha k,\beta l}$ is the force constant matrix. Both quantities $F^{\alpha k}$ and $A_{\alpha k,\beta l}$ are certain functions of occupation numbers $n_i$, and their evaluation needs some further approximations. Below we consider ordering phase transitions at the fixed mean concentration $c$. Changes of elastic constants and phonon spectra under such transitions are usually small[@Rouchy]. Therefore, the force constant matrix $A_{\alpha k,\beta l}$ can be reasonably well approximated with the simple ‘average crystal’ approximation: $A_{\alpha k,\beta l}\{n_i\}\rightarrow A_{\alpha k,\beta l}\{c\}\equiv\overline A_{\alpha k,\beta l}$. To approximate the Kanzaki force $F^{\alpha k}$ we first formally write it as a series in the occupation numbers $n_i$: $$F^{\alpha k}\{n_i\}=F^{\alpha k}_0+ \sum_iF^{\alpha k,i}_1n_i+ \sum_{i>j}F^{\alpha k,ij}_2n_in_j+\ldots \label{F_k}$$ Equilibrium values of displacements ${\bf u}_k={\bf u}_k^e\{n_i\}$ at the given distribution $\{n_i\}$ are determined by the minimization of energy (\[H\_u\]) over ${\bf u}_k$, and the constant $F^{\alpha k}_0$ in (\[F\_k\]) affects only the reference point ${\bf u}_k^e\{0\}$ in the function ${\bf u}_k^e\{n_i\}$. This constant can be determined, for example, from the condition of vanishing of mean static displacements in the averaged crystal at some $c=c_0$, which implies the relation: $\langle F^{\alpha k}\{n_i\}\rangle_{c=c_0}=0$ where the symbol $\langle \ldots\rangle$ means the statistical averaging over an alloy. The constants $F^{\alpha k}_0$ are insignificant for what follows, and below they are omitted to simplify formulas. In writing an explicit expression for the contribution $H_K$ (to be called for brevity the ‘Kanzaki term’) of the occupation-dependent Kanzaki forces in energy (\[H\_u\]) one should consider that due to the translation invariance it can include only differences of displacements $({\bf u}_k-{\bf u}_i)$, $({\bf u}_k-{\bf u}_j)$, etc. Therefore, this term should have the form $$H_K=\sum_{k,i}({\bf u}_i-{\bf u}_k){\bf f}^{k,i}_1n_i+ \sum_{k,ij}({\bf u}_i-{\bf u}_k){\bf f}^{k,ij}_2n_in_j+\ldots \label{H_K}$$ where ${\bf f}^{k,i_1\ldots i_m}_m\equiv {\bf f}^{k}_m$ are some parameters describing interaction of lattice deformations with site occupations. Representation (\[H\_K\]) for $H_K$ as a sum of contributions of $m$-site ‘clusters’ proportional to products $n_{i_1}\ldots n_{i_m}$ is analogous to similar cluster expansions for the ‘chemical’ Hamiltonian $H_c\{n_i\}$ in (\[H\_u\]). These expansions have been widely discussed, in particular, in connection with first-principle calculations of chemical interactions , see e. g. [@Zunger]. The calculations have shown that the values of $m$-site interactions $v^{c,m}$ in most alloys rapidly decrease with an increase of $m$, and the pairwise interaction $v^{c,2}$ is usually dominant. It is natural to expect that a similar rapid convergence is also typical for the expansion (\[H\_K\]). Therefore, below we omit many-site interactions ${\bf f}^{k}_m$ with $m>2$ in Eq. (\[H\_K\]). At the same time, in estimates of parameters ${\bf f}^{k}_m$ for real alloys below we combine some model assumptions about ${\bf f}^{k}_m$ with using of available experimental data about the variations of lattice deformations with concentration and orderings, and such estimates may also implicitly include the contributions of many-site interactions ${\bf f}^{k}_m$. For what follows it is convenient to proceed from functions ${\bf u}_k={\bf u}({\bf r}_k)$, $n_i=n({\bf r}_i)$, ${\bf f}_1^{k,i}={\bf f}_1({\bf r}_k-{\bf r}_i)$, ${\bf f}_2^{k,ij}={\bf f}_2({\bf r}_k-{\bf r}_i,{\bf r}_j-{\bf r}_i)$ and $\overline A_{\alpha k,\beta l}=\overline A_{\alpha\beta}({\bf r}_k-{\bf r}_l)$ in Eqs. (\[H\_u\]) and (\[H\_K\]) to their Fourier components in the average crystal lattice. Then the energy (\[H\_u\]) takes the form: $$H=H_c\{n_i\}+{\frac{1}{N}}\sum_{\bf k}{\bf u}_{-{\bf k}} \left(n_{\bf k}{\bf f}_{1{\bf k}}+ \sum_{\bf R}\sigma_{\bf k}^{\bf R}{\bf f}_{2{\bf k}}^{\bf R}\right) +{\frac{1}{2N}}\sum_{\bf k} u_{-{\bf k}}^{\alpha}\overline A_{\bf k}^{\alpha\beta}u_{{\bf k}}^{\beta}. \label{H_uk}$$ Here $N$ is the total number of crystal cells, the summation over $\bf k$ goes within the Brillouin zone of the averaged crystal, and we use the following notation: $$\begin{aligned} \fl {\bf u}_{\bf k}=\sum_{\bf r}{\bf u}({\bf r})e^{-i\bf{kr}};\qquad \hspace{6mm} n_{\bf k}=\sum_{\bf r}n({\bf r})e^{-i\bf{kr}};\qquad \sigma_{\bf k}^{\bf R}=\sum_{\bf r}n({\bf r})n({\bf r}+{\bf R})e^{-i\bf{kr}};\nonumber\\ \fl {\bf f}_{1{\bf k}}=\sum_{\bf r}{\bf f}_{1}({\bf r})(1-e^{-i\bf{kr}});\quad {\bf f}_{2{\bf k}}^{\bf R}=\sum_{\bf r}{\bf f}_{2}({\bf r,\bf R})(1-e^{-i\bf{kr}});\quad \overline A_{\bf k}^{\alpha\beta}=\sum_{\bf r}\overline A_{\alpha\beta}({\bf r})e^{-i\bf{kr}}. \label{u_k}\end{aligned}$$ If one adopts a commonly used model of ‘central’ Kanzaki forces in which forces ${\bf f}_1^{k,i}$ and ${\bf f}_2^{k,ij}$ in (\[H\_K\]) are supposed to be proportional to the vector ${\bf r}_{ki}=({\bf r}_k-{\bf r}_i)$, the vector functions ${\bf f}_{1{\bf k}}$ and ${\bf f}_{2{\bf k}}$ in (\[u\_k\]) can be expressed via two scalar functions, $\varphi_1$ and $\varphi_2$: $${\bf f}_{1{\bf k}}=\sum_{\bf r}{\bf r}\,\varphi_1 (r)\,(1-e^{-i\bf{kr}}),\quad {\bf f}_{2{\bf k}}^{\bf R}=\sum_{\bf r}{\bf r}\,\varphi_2({\bf r,\bf R})\,(1-e^{-i\bf{kr}}). \label{varphi}$$ The functions $\varphi_1$ and $\varphi_2$ in (\[varphi\]) determine the dependence of equilibrium lattice parameters on concentration or ordering. To show it we first note that the homogeneous deformation $\overline u_{\alpha\beta}$ is described by Fourier-components ${\bf u}_{\bf k}$ with small ${\bf k}\to 0$, while functions ${\bf f}_{1{\bf k}}$ and ${\bf f}_{2{\bf k}}$ in Eqs. (\[H\_uk\]) and (\[u\_k\]) at small ${\bf k}$ are linear in ${\bf k}$. Thus the contribution of homogeneous deformations to the Kanzaki term in (\[H\_uk\]) is proportional to Fourier-components $u^{\alpha\beta}_{\bf k}$ of the elastic strain $u_{\alpha\beta}= (\partial u_{\alpha}/\partial x_{\beta}+\partial u_{\beta}/\partial x_{\alpha})/2$ at ${\bf k}\to 0$ and, according to first equation (\[u\_k\]), these components are related to $\overline u_{\alpha\beta}$ as $$u^{\alpha\beta}_{\bf k}\|_{{\bf k}\to 0}=i(k_{\beta}u_{\bf k}^{\alpha}+ k_{\alpha}u_{\bf k}^{\beta})\|_{{\bf k}\to 0}=N\overline u_{\alpha\beta}. \label{u_strain}$$ At small $\bf k$ the force constant matrix $\overline A_{\bf k}^{\alpha\beta}$ in (\[H\_uk\]) is bilinear in $\bf k$, and the last term of (\[H\_uk\]) corresponds to the standard expression for the elastic energy bilinear in $\overline u_{\alpha\beta}$ and linear in the elastic constants $c_{\alpha\beta\gamma\delta}$, see e.g. [@Khach-book]. Therefore, the total contribution of terms with the homogeneous elastic strain $\overline u_{\alpha\beta}$ to energy (\[H\_uk\]) (to be called ‘the elastic strain energy’ $E_{el}$) can be written as $$E_{el}=-\overline u_{\alpha\beta}\left(A_1^{\alpha\beta}n_0+ \sum_{\bf R}A_{2{\bf R}}^{\alpha\beta}\sigma_0^{\bf R}\right)+ {\frac{1}{2}}N\Omega\,c_{\alpha\beta\gamma\delta} \overline u_{\alpha\beta}\overline u_{\gamma\delta}. \label{E_el0}$$ Here $\Omega$ is the volume per atom in the average crystal; quantities $A_1^{\alpha\beta}$ and $A_{2{\bf R}}^{\alpha\beta}$ are expressed via functions $\varphi_1$ and $\varphi_2$ in (\[varphi\]) as: $$A_1^{\alpha\beta}=\sum_{\bf r}x_{\alpha}x_{\beta}\varphi_1({\bf r}), \qquad A_{2{\bf R}}^{\alpha\beta}=\sum_{\bf r}x_{\alpha}x_{\beta}\varphi_2({\bf r},{\bf R}), \label{A_1,2}$$ where $x_{\alpha}$ is the Cartesian component of vector ${\bf r}=(x_1,x_2,x_3)$; and $n_0$ or $\sigma_0^{\bf R}$ is the Fourier component, $n_{\bf k}$ or $\sigma_{\bf k}^{\bf R}$, at ${\bf k}=0$. According to Eq. (\[u\_k\]), the operator $n_0$ or $\sigma_0^{\bf R}$ is the sum of a macroscopically large number $N$ of similar terms. Thus within the statistical accuracy each of these operators can be substituted by its average value: $$n_0=N\langle n({\bf r})\rangle=Nc; \qquad \sigma_0^{\bf R}=N\langle n({\bf r})\,n({\bf r}+{\bf R})\rangle. \label{sigma_0}$$ The last average in (\[sigma\_0\]) can be expressed via mean occupations of sites and their correlators. In an ordered alloy there exist several non-equivalent sublattices $s$ with the lattice vectors ${\bf r}_s$ and mean occupations $c_s=\langle n({\bf r}_s)\rangle$, and so the last average in (\[sigma\_0\]) includes averaging over all sublattices $s$: $$\langle n({\bf r})\,n({\bf r}+{\bf R})\rangle =\sum_s\nu_s\left(c_s\,c_{s{\bf R}}+ K_{s{\bf R}}\right). \label{nn_average}$$ Here $c_{s{\bf R}}$ is the mean occupation $\langle n({\bf r})\rangle$ for ${\bf r}={\bf r}_s+{\bf R}$; $\nu_s=N_s/N$ is the relative number of sites in the sublattice $s$; and $K_{s{\bf R}}$ is the correlator of occupations of sites located at ${\bf r}={\bf r}_s$ and at ${\bf r}={\bf r}_s+{\bf R}$: $$K_{s{\bf R}}=\langle \left[n({\bf r}_s)-c_s\right]\left[n({\bf r}_s+{\bf R})-c_{s{\bf R}}\right]\rangle \label{K_R}.$$ In a disordered alloy all sites are equivalent, thus $c_s=c_{s{\bf R}}=c$; $\nu_s=1$; and both index $s$ and the summation over $s$ in (\[nn\_average\]) are omitted. Using Eqs. (\[sigma\_0\]) and (\[nn\_average\]) one can rewrite the elastic strain energy (\[E\_el0\]) as $$\hspace{-15mm} E_{el}=-N\overline u_{\alpha\beta}\left[A_1^{\alpha\beta}\,c+ \sum_{\bf R}\sum_s\nu_s\left(c_s\,c_{s{\bf R}}+ K_{s{\bf R}}\right)A_{2{\bf R}}^{\alpha\beta}\right]+ {\frac{1}{2}}N\Omega\,c_{\alpha\beta\gamma\delta} \overline u_{\alpha\beta}\overline u_{\gamma\delta}. \label{E_elK}$$ The correlator $K_{s{\bf R}}$ in Eq. (\[E\_elK\]) can be calculated using that or another method of statistical theory. However, for most alloy systems of practical interest, in particular, at $c$ and $T$ values not close to the thermodynamic instabilty points $T_s$, the correlators $K_{s{\bf R}}$ are small and can be neglected. Then equation (\[E\_elK\]) is simplified: $$E_{el}=-\overline u_{\alpha\beta}\left[NA_1^{\alpha\beta}\,c+ \sum_{{\bf r},{\bf R}}c({\bf r})c({\bf r}+{\bf R})A_{2{\bf R}}^{\alpha\beta}\right]+ {\frac{1}{2}}N\Omega\,c_{\alpha\beta\gamma\delta} \overline u_{\alpha\beta}\overline u_{\gamma\delta}. \label{E_el}$$ Equilibrium values of $\overline u_{\alpha\beta}$ in the absence of applied stress are determined by the minimization of energy $E_{el}$ with respect to $\overline u_{\alpha\beta}$ which gives: $$\Omega\,c_{\alpha\beta\gamma\delta}\overline u_{\gamma\delta}= A_1^{\alpha\beta}\,c+ {\frac{1}{N}}\sum_{{\bf r},{\bf R}}c({\bf r})c({\bf r}+{\bf R})A_{2{\bf R}}^{\alpha\beta}. \label{u_eq}$$ Eq. (\[u\_eq\]) enables one to express the equilibrium strain $\overline u_{\alpha\beta}$ via the concentration, order parameters, and the interaction parameters $A_1^{\alpha\beta}$ and $A_{2{\bf R}}^{\alpha\beta}$, and it can also be used to estimate these interaction parameters from experimental data on $\overline u_{\alpha\beta}(c,T)$. Let us consider Eqs. (\[E\_el\]) and (\[u\_eq\]) in particular cases. For a disordered phase with $c({\bf r})=c$, Eq. (\[u\_eq\]) takes the form $$\Omega\,c_{\alpha\beta\gamma\delta}\overline u_{\gamma\delta}= A_1^{\alpha\beta}\,c+A_2^{\alpha\beta}\,c^2 \label{A_12c}$$ where $A_2^{\alpha\beta}=\sum_{\bf R}A_{2{\bf R}}^{\alpha\beta}$. If the disordered phase has a cubic symmetry (as for the FCC or BCC alloys), quantities $A_1^{\alpha\beta}$ and $A_2^{\alpha\beta}$ are proportional to the Kronecker symbol $\delta_{\alpha\beta}$, and Eq. (\[A\_12c\]) determines the concentrational dilatation $u(c)=\overline u_{\alpha\alpha}(c)-u_{\alpha\alpha}(0)$: $$u(c)=(A_1c+A_2c^2)/\Omega B. \label{u_c}$$ Here $B=(c_{11}+2c_{12})/3$ is the bulk modulus; $c_{ij}$ are the elastic constants in Voigt’s notation; and coefficients $A_1$ and $A_2$ are expressed via functions $\varphi_1$ and $\varphi_2$ in (\[varphi\]), (\[A\_1,2\]): $$A_1=\sum_{\bf r}\varphi_1({\bf r})r^2/3;\qquad A_2=\sum_{{\bf r},{\bf R}}\varphi_2({\bf r},{\bf R})r^2/3. \label{A_1}$$ The linear in $c$ term in (\[u\_c\]) corresponds to the Vegard law while the term with $A_2$ describes the non-linear deviations from this law. Such deviations were observed for many alloys, and these data can be used to estimate $A_2$ values, but in these estimates one should also take into consideration a possible concentration dependence of the bulk modulus $B$. For the ordered phase, the mean occupation $c({\bf r})$ can be written as a superposition of concentration waves corresponding to certain superstructure vectors ${\bf k}_p$ [@Khach-book]: $$c({\bf r})=c+{1\over 2}\sum_p\left[\eta_p\exp(i{\bf k}_p{\bf r})+ \eta_p^\exp(-i{\bf k}_p{\bf r})\right], \label{cr_CW}$$ and amplitudes $\eta_p$ can be considered as order parameters. After the substitution of expressions (\[cr\_CW\]) for $c({\bf r})$ and $c({\bf r}+{\bf R})$ in Eq. (\[E\_el\]) the linear in $\eta_p$ terms vanish due to the crystal symmetry, and the first term of (\[E\_el\]) becomes the sum of the ordering-independent term and the term bilinear in order parameters: $$E_{el}=-N\overline u_{\alpha\beta}\left(A_1^{\alpha\beta}\,c+ A_2^{\alpha\beta}\,c^2+\sum_pq_{\alpha\beta pp}\|\eta_p\|^2\right)+ {\frac{1}{2}}N\Omega\,c_{\alpha\beta\gamma\delta} \overline u_{\alpha\beta}\overline u_{\gamma\delta}. \label{E_el-q}$$ Here quantities $q_{\alpha\beta pp}$ have a different form in the cases ($a$) when the superstructure vector ${\bf k}_p$ is half of some reciprocal lattice vector ${\bf g}$ and thus both the order parameter $\eta_p$ and all factors $\exp (i{\bf k}_p{\bf r})$ in (\[cr\_CW\]) are real, and ($b$) when ${\bf k}_p\neq {\bf g}/2$: $$(a)\qquad {\bf k}_p={\bf g}/2:\hspace{10mm} q_{\alpha\beta pp}=\sum_{{\bf r},{\bf R}}x_{\alpha}x_{\beta} \varphi_2({\bf r},{\bf R})\exp\, (i{\bf k}_p{\bf r}); \label{q_a}$$ $$(b)\qquad {\bf k}_p\neq {\bf g}/2:\hspace{10mm} q_{\alpha\beta pp}={1\over 2} \sum_{{\bf r},{\bf R}}x_{\alpha}x_{\beta} \varphi_2({\bf r},{\bf R})\cos\, ({\bf k}_p{\bf r}). \label{q_b}$$ The coefficients $q_{\alpha\beta pp}$ in (\[E\_el-q\]) (to be called the ‘striction’ coefficients, in an analogy with the terminology used in the ferroelectricity or magnetism theory) are commonly used in phenomenological theories of lattice distortions under orderings [@Roitburd; @Khach-Shat; @Khach-book; @Vaks-01; @CWK]. Eqs. (\[q\_a\]), (\[q\_b\]) and (\[A\_1,2\]) provide the microscopic expression for these coefficients via the function $\varphi_2$ describing non-pairwise Kanzaki forces in Eqs. (\[H\_uk\])–(\[varphi\]). Let us apply Eqs. (\[cr\_CW\])–(\[q\_a\]) to the case of L1$_0$ or L1$_2$ ordering in FCC alloys which are described by three real order parameters $\eta_{\alpha}$ [@Khach-book; @BDPSV]. Eqs. (\[cr\_CW\]) here take the form $$c({\bf r})=c+\eta_1\exp (i{\bf k}_1{\bf r})+\eta_2\exp (i{\bf k}_2{\bf r}) +\eta_3\exp (i{\bf k}_3{\bf r}), \label{c_r}$$ where ${\bf k}_{\alpha}={\bf g}_{\alpha}/2$ is the superstructure vector corresponding to $\eta_{\alpha}$: $${\bf k_1}=[100]2\pi/a,\qquad {\bf k_2}=[010]2\pi/a,\qquad{\bf k_3}=[001]2\pi/a. \label{k_alpha}$$ In the cubic L1$_2$ structure one has: $\|\eta_1\|=\|\eta_2\|=\|\eta_3\|$, $\eta_1\eta_2\eta_3>0$, and four types of ordered domains are possible. In the L1$_0$-ordered structure with the tetragonal axis $\alpha$ a single parameter $\eta_{\alpha}$ is present which is either positive or negative, and so six types of ordered domains are possible. The striction coefficients for L1$_0$ or L1$_2$ ordering are determined by Eq. (\[q\_a\]). Due to the cubic symmetry of the ‘average’ FCC crystal, there are only two different striction coefficients, $q_{1111}$ and $q_{1122}$ (and those obtained from them by the cubic symmetry operations), which for brevity will be denoted as $q_{11}$ and $q_{12}$, respectively: $$q_{11}=\sum_{{\bf r},{\bf R}}x_1^2\varphi_2({\bf r},{\bf R})\exp (i{\bf k}_1{\bf R});\quad q_{12}=\sum_{{\bf r},{\bf R}}x_1^2\varphi_2({\bf r},{\bf R})\exp (i{\bf k}_2{\bf R}). \label{q_12}$$ Variation of elastic constants $c_{\alpha\beta\gamma\delta}$ with ordering is usually small[@Rouchy], and for simplicity it will be neglected. Then minimizing energy (\[E\_el-q\]) with respect to $\overline u_{\alpha\beta}$ we obtain the expressions for lattice deformations induced by ordering (\[c\_r\]): $$%\fl \hspace{-20mm}\overline u =q_+(\eta_1^2+\eta_2^2+\eta_3^2)/\Omega c_+,\quad \overline\varepsilon =q_-[\eta_1^2-(\eta_2^2+\eta_3^2)/2]/\Omega c_-,\quad \zeta =q_-(\eta_2^2-\eta_3^2)/\Omega c_-. \label{overline_u}$$ Here $\overline u=\overline u_{11}+\overline u_{22}+\overline u_{33}$ describes the volume change; $\overline\varepsilon =\overline u_{11}-(\overline u_{22}+\overline u_{33})/2$ is the tetragonal distortion; $\zeta =\overline u_{22}-\overline u_{33}$ is the shear deformation; and $q_{\pm}$ or $c_{\pm}$ are linear combinations of striction or elastic constants: $$%\fl \hspace{-10mm} q_-=q_{11}-q_{12}; \quad c_-=c_{11}-c_{12};\quad q_+=q_{11}+2q_{12}; \quad c_+=c_{11}+2c_{12}. \label{q_pm}$$ For the L1$_2$ ordering, values $\|\eta_1\|=\|\eta_2\|=\|\eta_3\|=\eta $ are the same, so just the volume striction $\overline u=3q_+\eta^2/\Omega c_+$ is present, while in the L1$_0$-ordered domain with $\eta_2=\eta_3=0$ one has both the volume and the tetragonal striction: $$\overline u=q_+\eta_1^2/\Omega c_+,\qquad \overline\varepsilon =q_-\eta_1^2/\Omega c_-. \label{overline_u-eta1}$$ Therefore, using experimental data about the lattice distortions and order parameters under L1$_2$ and L1$_0$ orderings one can estimate the striction coefficients $q_{11}$ and $q_{12}$ and thus the non-pairwise Kanzaki interaction $\varphi_2$ in Eqs. (\[q\_12\]). Below we suppose for simplicity the interaction $\varphi_2({\bf r},{\bf R})$ to be short-ranged, i. e. significant only when each of three relative distances $r$, $R$ and $\|{\bf r}-{\bf R}\|$ does not exceed the nearest-neighbour distance $\rho =a/\sqrt{2}$. Then this function can be written as $$\varphi_2({\bf r},{\bf R})=\delta_{r,\rho}\delta_{R,\rho}\left(\varphi_{a}\,\delta_{\|{\bf r}-{\bf R}\|,0} +\varphi_{b}\,\delta_{\|{\bf r}-{\bf R}\|,\rho}\right) \label{varphi_ab}$$ where $\delta_{r,\rho}$ is the Kronecker symbol equal to unity when $r=\rho$ and zero otherwise while $\varphi_{a}$ and $\varphi_{b}$ are the interaction parameters. The assumption (\[varphi\_ab\]) is analogous to that used by Khachaturyan [@Khach-book] for the pairwise Kanzaki interaction $\varphi_1({\bf r})$ in (\[varphi\]): $$\varphi_1({\bf r})=\varphi_1\delta_{r,\rho} \label{varphi_1}$$ where the constant $\varphi_1$ is estimated from experimental data on concentrational dilatation. First-principle estimates of lattice distortions in dilute alloys [@BSVZ] seem to imply that the assumption (\[varphi\_1\]) yields the correct order of magnitude of $\varphi_1({\bf r})$. Therefore, the analogous assumption (\[varphi\_ab\]) for $\varphi_2({\bf r},{\bf R})$ can be reasonable, too. Substituting Eq. (\[varphi\_ab\]) into (\[q\_12\]) we obtain the explicit expression for coefficients $q_{ik}$ via parameters $\varphi_{a}$ and $\varphi_{b}$ in (\[varphi\_ab\]): $$q_{11}=-2a^2\varphi_{a};\qquad q_{12}=-4a^2\varphi_{b}. \label{q_varphi}$$ The coefficient ${\bf f}_{1{\bf k}}$ in Eqs. (\[H\_uk\]) and (\[u\_k\]) for model (\[varphi\_1\]) has the form [@Khach-book]: $${\bf f}_{1{\bf k}}=4\varphi_1i\sum_{\alpha =1}^3{\bf a}_{\alpha}\sin\,({\bf ka}_{\alpha}) \sum_{\beta\neq\alpha}\cos\,({\bf ka}_{\beta}) \label{f_1k}$$ where ${\bf a}_{\alpha}$ is ${\bf e}_{\alpha}a/2$ and ${\bf e}_{\alpha}$ is the unit vector along the main crystal axis $\alpha$. The function ${\bf f}_{2{\bf k}}^{\bf R}$ in (\[H\_uk\]), (\[u\_k\]) for model (\[varphi\_ab\]) is the sum of two terms: $${\bf f}_{2{\bf k}}^{\bf R}={\bf f}_{a{\bf k}}^{\bf R}+{\bf f}_{b{\bf k}}^{\bf R}. \label{f_2k}$$ Here, ${\bf f}_{a{\bf k}}^{\bf R}$ is $\varphi_a\delta_{R,\rho}{\bf R}\left(1-e^{-i{\bf kR}}\right)$, while the function ${\bf f}_{b{\bf k}}^{\bf R}$ for ${\bf R}$ equal to ${\bf R}_{\alpha}+ {\bf R}_{\beta}$ (where ${\bf R}_{\alpha}$ is ${\bf a}_{\alpha}$ or $(-{\bf a}_{\alpha})$, ${\bf R}_{\beta}$ is ${\bf a}_{\beta}$ or $(-{\bf a}_{\beta})$, and $\beta\neq\alpha$) can be written as $$\fl {\bf f}_{b{\bf k}}^{\bf R}=2\varphi_b%\delta_{R,\rho} \left[{\bf R}+\left(i{\bf a'}\sin {\bf ka'}-{\bf R}_{\alpha}\cos\,{\bf ka'}\right)e^{-i{\bf kR}_{\alpha}} +\left(i{\bf a'}\sin {\bf ka'}-{\bf R}_{\beta}\cos\,{\bf ka'}\right)e^{-i{\bf kR}_{\beta}}\right] \label{f_bk}$$ where ${\bf a'}$ is $[{\bf e}_{\alpha}{\bf e}_{\beta}]a/2$. Relations (\[varphi\]), (\[varphi\_ab\])–(\[f\_bk\]) together with (\[u\_c\]) and (\[q\_varphi\]) provide a simplified model for the Kanzaki term $H_K$ in Eqs. (\[H\_K\]) and (\[H\_uk\]). This model will be used below in simulations of A1$\to$L1$_0$ transitions. To get an idea about the actual scale of parameters of this model, let us estimate quantities $q_{ik}$, $\varphi_a$ and $\varphi_b$ in Eqs. (\[varphi\_ab\]) and (\[q\_varphi\]) for the alloys Co–Pt for which detailed data about the lattice distortion under L1$_2$ and L1$_0$ orderings are available [@Berg-Cohen; @Leroux-88]. The volume change $\overline u$ under both the L1$_2$ ordering in CoPt$_3$ and L1$_0$ ordering in CoPt appears to be very small [@Berg-Cohen; @Leroux-88; @Leroux]: $\overline u\lesssim 10^{-3}$. According to Eqs. (\[overline\_u\]) and (\[overline\_u-eta1\]) it implies the relation: $q_{12}\simeq q_{11}/2$. The value $q_-=(q_{11}-q_{12})$ for CoPt can be estimated from second equation (\[overline\_u-eta1\]) using data of Ref. [@Leroux-88] for $\eta_1$ and $\overline\varepsilon $ at $T=0.84\,T_c$: $\eta_1\simeq 0.4$; $\varepsilon\simeq -0.04$ (with the thermal expansion effect subtracted); and for the atomic volume: $\Omega =\Omega (T_c+0)\simeq 13.8{\rm\AA}^3$. Using also for the elastic constant $c_-=(c_{11}-c_{12})$ its value for the FCC platinum, $c_-\simeq 0.97$ Mbar [@Ducastelle], we obtain: $q_-\simeq 2.6\cdot 10^4$ K. Combining it with the above-mentioned relation $q_{12}\simeq q_{11}/2$ and using Eq. (\[overline\_u\]) we find: $\varphi_a\simeq 2\cdot 10^4$ K$/a^2$, and $\varphi_b\simeq 5\cdot 10^3$ K$/a^2$. Let us also note that the ordering-induced elastic energy per atom $\varepsilon _{el}^{ord}$ in the CoPt alloy is small: $\varepsilon _{el}^{ord}\simeq \Omega c_-\overline\varepsilon ^2/6\simeq$ , which is much less than the L1$_0$ ordering temperature $T_c\simeq 1100$ K. The equilibrium values of displacements ${\bf u}_{\bf k}^e={\bf u}_{\bf k}^e(n_i)$ are found by the minimization of energy (\[H\_uk\]) over ${\bf u_k}$. Substituting these ${\bf u}_{\bf k}^e$ into Eq. (\[H\_uk\]) we obtain the effective Hamiltonian $H=H_c+H_d$ where the deformational interaction $H_d$ can be written as $$\fl H_d=-{\frac{1}{2N}}\sum_{\bf k} \left(n_{-\bf k}{\bf f}_{1{\bf k}}^+ \sum_{\bf R}\sigma_{-\bf k}^{\bf R}{\bf f}_{2{\bf k}}^{{\bf R}}\right) {\bf G_k} \left(n_{\bf k}{\bf f}_{1{\bf k}}+ \sum_{\bf R}\sigma_{\bf k}^{\bf R}{\bf f}_{2{\bf k}}^{\bf R}\right)=H_{d2}+H_{d3}+H_{d4}. \label{H_d}$$ Here the matrix ${\bf G_k}=G_{\bf k}^{\alpha\beta}$ is inverse to the force constant matrix $\overline A_{\bf k}^{\alpha\beta}$, and the matrix product $\bf aBc$ means the sum $a_{\alpha}B_{\alpha\beta}b_{\beta}$. The term $H_{d2}$, $H_{d3}$ and $H_{d4}$ in (\[H\_d\]) describes the pairwise, three-particle and four-particle deformational interaction, respectively: $$\fl H_{d2}={\frac{1}{2}}\sum_{\bf r, r'} n({\bf r})\Phi_2({\bf r}-{\bf r'})n({\bf r'}),\qquad H_{d3}= {\frac{1}{2}}\sum_{\bf r, r'}\sum_{\bf R} n({\bf r})\Phi_3^{\bf R}({\bf r}-{\bf r'})n({\bf r'})n({\bf r'}+{\bf R}), \label{H_d2-3}$$ $$H_{d4}={\frac{1}{2}}\sum_{\bf r, r'}\sum_{\bf ,R,R'}n({\bf r})n({\bf r}+{\bf R}) \Phi_4^{\bf R,R'}({\bf r}-{\bf r'})n({\bf r'})n({\bf r'}+{\bf R'}), \label{H_d4}$$ where the potential $\Phi_2$, $\Phi_3^{\bf R}$ or $\Phi_4^{\bf R,R'}$ is given by the expression: $$\fl \left\{\Phi_2({\bf r}); \Phi_3^{\bf R}({\bf r}); \Phi_4^{\bf R,R'}({\bf r})\right\} =-{\frac{1}{N}}\sum_{\bf k}e^{i{\bf kr}}\left\{{\bf f}_{1{\bf k}}^{\bf G_k}{\bf f}_{1{\bf k}};\ {\bf f}_{1{\bf k}}^{\bf G_k}{\bf f}_{2{\bf k}}^{{\bf R}}+ {\bf f}_{2{\bf k}}^{{\bf R}}{\bf G_k}{\bf f}_{1{\bf k}};\ {\bf f}_{2{\bf k}}^{{\bf R}}{\bf G_k}{\bf f}_{2{\bf k}}^{{\bf R'}}\right\}. \label{Phi_234}$$ As the matrix ${\bf G_k}$ in (\[Phi\_234\]) at small $k$ includes the well-known ‘elastic singularity’ [@Khach-book]: ${\bf G_k}\sim 1/k^2$, each of terms $H_{d2}$, $H_{d3}$ and $H_{d4}$ in (\[H\_d\]) includes the long-ranged elastic interaction. The formation of twinned structures discussed below is determined by the four-particle interaction $H_{d4}$. The rest deformational interactions, $H_{d2}$ and $H_{d3}$, for the single-phase L1$_0$ ordering under consideration lead just to some quantitative renormalizations of chemical interaction $H_c$ in (\[H\_uk\]) which are usually small and insignificant. Therefore, below we retain in the deformational interaction (\[H\_d\]) only the last term $H_{d4}$. Let us also note that each term in the sum (\[H\_d4\]) for $H_{d4}$ at fixed $\bf r$ and $\bf r'$ has the order of magnitude of the above-mentioned ordering-induced elastic energy $\varepsilon_{el}^{ord}$ which usually is small. Thus the interaction $H_{d4}$ can be significant only because of ‘coherent’ contributions of many sites $\bf r$ and $\bf r'$ due to the long-ranged elastic interaction. Therefore, local fluctuations of occupations $n({\bf r})$ in the interaction $H_{d4}$ are insignificant, and it can be treated in the ‘kinetic mean-field approximation’ (KMFA) [@Vaks-96; @BV-98; @BDPSV] which neglects such fluctuations and corresponds to the substitution in (\[H\_d4\]) of each occupation operator $n({\bf r})$ by its mean value $c({\bf r})=\langle n({\bf r})\rangle$ where $\langle \ldots\rangle$ means averaging over the space- and time-dependent distribution function [@Vaks-96; @BV-98; @BDPSV]. Therefore, in considerations of A1$\to$L1$_0$ transformations below we approximate the total effective Hamiltonian $H$ in (\[H\_uk\]) by the following expression: $$\hspace{-20mm}H=H_c+H_{d4}=H_c\{n({\bf r})\}+ {\frac{1}{2}}\sum_{\bf r, r',R,R'}c({\bf r})c({\bf r}+{\bf R}) \Phi_4^{\bf R,R'}({\bf r}-{\bf r'})c({\bf r'})c({\bf r'}+{\bf R'}) \label{H_cd}$$ where the potential $\Phi_4^{\bf R,R'}({\bf r})$ is given by the last equation (\[Phi\_234\]). Models and methods of simulation ================================ To simulate A1$\to$L1$_0$ transformations in an alloy with the Hamiltonian (\[H\_cd\]) we use the methods described in Refs. [@BDPSV] and [@PV] to be referred to as I and II, respectively. Evolution of atomic distributions is described by the kinetic tetrahedron cluster field method [@BDPSV] in which mean occupations $c_i=c({\bf r}_i)=\langle n({\bf r}_i)\rangle$ averaged over the space- and time-dependent distribution function obey the kinetic equation (I.10): $$dc_i/dt=2\sum_jM_{ij} \sinh [\beta (\lambda_j-\lambda_i)/2]. \label{c_i(t)}$$ Here $\beta =1/T$ is the inverse temperature; $M_{ij}$ is the generalized mobility proportional to the configurationally independent factor $\gamma_{nn}$ in the probability of an inter-site atomic exchange A$i\leftrightarrow {\rm B}j$ between neighbouring sites $i$ and $j$ per unit time; and $\lambda_i=\lambda_i\{c_j\}$ is the local chemical potential equal to the derivative of the generalized free energy $F\{c_i\}$ defined in Refs. [@Vaks-96; @BV-98] with respect to $c_i$: $\lambda_i=\partial F/\partial c_i$. The expression for $M_{ij}=M_{ij}\{c_k\}$ employed in our simulations is given by Eq. (I.12) with the asymmetrical potential $u_i$ taken zero for simplicity, while the local chemical potential $\lambda_i$ now is the sum of the chemical and the deformational term, $\lambda_i^c$ and $\lambda_i^d$. The microscopical expressions for $\lambda_i^c$ are given by equations (I.13) – (I.16) which include only chemical interactions $v_{ij}=v_{ij}^c$, while the deformational contribution $\lambda_i^d= \lambda_i^d({\bf r}_i)$ is the variational derivative of the second term in (\[H\_cd\]) with respect to $c_i=c({\bf r}_i)$: $$\hspace{-5mm} \lambda_i^d({\bf r})=\delta H_{d4}/\delta c({\bf r})= 2\sum_{\bf r',R,R'}c({\bf r}+{\bf R}) \Phi_4^{\bf R,R'}({\bf r}-{\bf r'})c({\bf r'})c({\bf r'}+{\bf R'}). \label{lambda_d}$$ For the chemical interaction $v_{ij}^c$ we employ the five alloy models used in I and II: 1\. The second-neighbour interaction model with the nearest-neighbour interaction $v_1=1000$K (in the Boltzmann constant $k_B$ units) and $v_2/v_1=\epsilon =-0.125$. 2\. The same model with $\epsilon =-0.25$. 3\. The same model with $\epsilon =-0.5$. 4\. The fourth-neighbour interaction model with $v_n$ estimated by Chassagne [@Chassagne] from their experimental data for disordered Ni–Al alloys: $v_1=1680\,\mathrm{K}$, $v_2=-210\,\mathrm{K}$, $v_3=35\,\mathrm{K}$, and $v_4=-207\,\mathrm{K}$. 5\. The fourth-neighbour interaction model with $v_1=1000$K, $v_2/v_1=-0.5$, $v_3/v_1=0.25$, and $v_4/v_1=-0.125$. The effective interaction range $R_{int}$ for these models monotonously increases with the model number. Therefore, a comparison of the simulation results for these models enables one to study the influence of $R_{int}$ on the microstructural evolution. The critical temperature $T_c$ for the phase transition A1$\to$L1$_0$ in the absence of deformational interaction $H_{d4}$ (which seems to have little effect on $T_c$ in our simulations) for model 1, 2, 3, 4 and 5 is 614, 840, 1290, 1950 and 2280 K, respectively [@VS]. For the Kanzaki force ${\bf f}_{2{\bf k}}^{\bf R}$ entering the expression (\[Phi\_234\]) for the potential $\Phi_4^{\bf R,R'}({\bf r})$ in (\[lambda\_d\]) we use Eqs. (\[f\_2k\]) and (\[f\_bk\]). The interaction parameters $\varphi_a$ and $\varphi_b$ in these equations can be expressed via spontaneous deformations $\overline u$ and $\overline\varepsilon$ using Eqs. (\[overline\_u-eta1\]) and (\[q\_varphi\]). For simplicity we assume the volume striction to be small (as it is for the Co–Pt alloys mentioned above): $\overline u\simeq 0$, while the tetragonal distortion will be characterized by its maximum value $\varepsilon_m$ in a stoichiometric alloy, i. e. by the value $\overline\varepsilon$ in (\[overline\_u-eta1\]) at $\eta_1=0.5$. Therefore, interactions $\varphi_a$ and $\varphi_b$ in our simulations are determined by the relations: $$\varphi_a=-a(c_{11}-c_{12})\varepsilon_m /3;\qquad \varphi_b=a(c_{11}-c_{12})\varepsilon_m /12. \label{varphi_epsilon}$$ For the lattice constant $a$ in (\[varphi\_epsilon\]) we take a typical value $a\simeq 4$ Å, and for the elastic constant $(c_{11}-c_{12})$, the value $0.97$ Mbar corresponding to FCC platinum [@Ducastelle]. For the force constant matrix $\overline {\bf A}_{\bf k}$ (which determines the matrix ${\bf G_k}=(\overline {\bf A}_{\bf k})^{-1}$ in Eq. (\[Phi\_234\])) we use the model described in Refs. [@BSV; @BDPSV]. It corresponds to a Born-von Karman model with the first- and second-neighbour force constants only, and the second-neighbour constants are supposed to correspond to a spherically symmetrical interaction. This model includes three independent force constants which are expressed in terms of elastic constants $c_{ik}$, and these constants were chosen equal to those of the FCC platinum [@Ducastelle]: $c_{11}=3.47$ Mbar, $c_{12}=2.5$ Mbar, and $c_{44}=0.77$ Mbar. As it was discussed in I and II, the transient partially ordered alloy states can be described using either mean occupations $c_i=c({\bf r}_i)$ or local order parameters $\eta_{\alpha i}^2$ and local concentrations $\overline c_i$ defined by Eqs. (I.24) and (I.25). The simulation results below are usually presented as the distributions of quantities $\eta_i^2=\eta_{1i}^2+\eta_{2i}^2+\eta_{3i}^2$, to be called the ‘$\eta^2$–representation’, and these distributions are similar to those observed in the experimental transmission electron microscopy (TEM) images [@BDPSV]. Our simulations were performed in the FCC simulation boxes of sizes $V_b=L^2\times H$ (where $L$ and $H$ are given in the lattice constant $a$ units) with periodic boundary conditions. We used both quasi-2D simulations with $H=1$ and 3D simulation with $H\sim L$. For the given coordinate $z=na$ (with $n=0$ for 2D simulation) each of figures below shows all FCC lattice sites lying in two adjacent planes, $z=na$ and $z=(n+1/2)a$. The point $(x,y)$ with $(x/a, y/a)$ equal to $(l,m)$, $(l+1/2,m)$, $(l+1/2,m+1/2)$ or $(l,m+1/2)$ in the figures corresponds to the lattice site with $(x/a, y/a, z/a)$ equal to $(l,m,n)$, $(l+1/2,m,n+1/2)$, $(l+1/2,m+1/2,n)$ or $(l,m+1/2,n+1/2)$, respectively. Therefore, at $V_b=L^2\times H$ the figure shows $4L^2$ lattice sites. The simulation methods were the same as in I and II. In simulations of A1$\to$L1$_0$ transformation the initial as-quenched distribution $c_i(0)$ was characterized by its mean value $c$ and small random fluctuations $\delta c_i$; usually we used $\delta c_i=\pm 0.01$. The distribution of initial fluctuations $\delta c_i$ for the given simulation box volume $V_b$ was identical for all models and the same as that used in II. The sensitivity of simulation results to variations of these initial fluctuations $\delta c_i$ was discussed in II and was found to be insignificant for the features of evolution discussed below. Kinetics of A1$\to$L1$_0$ transformations in systems with an extended or intermediate interaction range ======================================================================================================= As discussed in I, II and below, features of microstructural evolution under A1$\to$L1$_0$ and A1$\to$L1$_2$ transitions sharply depend on the effective interaction range $R_{int}$ in an alloy. In this section we discuss A1$\to$L1$_0$ transitions for the systems with an extended or intermediate interaction range, such as our models 5 and 4, while the short-range-interaction systems are considered in the next section. Some results of our simulations are presented in figures 1–8. The symbol A or $\overline {\rm A}$ in these figures corresponds to an L1$_0$-ordered domain with the tetragonal axis $c$ along (100) and the positive or negative value, respectively, of the order parameter $\eta_{1}$; the symbol B or $\overline {\rm B}$, to that for the $c$-axis along (010) and the order parameter $\eta_{2}$; and the symbol C or $\overline {\rm C}$, to that for the $c$-axis along (001) and the order parameter $\eta_{3}$. Figure \[m4s-3D\] shown in the $c$-representation illustrates the occupation of lattice sites for each domain type. The APB separating two APDs with the same tetragonal axis (i. e. APDs A and $\overline {\rm A}$, B and $\overline {\rm B}$ or C and $\overline {\rm C}$) will be for brevity called the ‘shift-APB’, and the APB separating the APDs with perpendicular tetragonal axes will be called the ‘flip-APB’. Before discussing figures 1–8 we remind the general ideas about the formation of twinned structures \[1–11\]. To avoid discussing the problems of nucleation, in this work we consider the transformation temperatures $T$ lower than the ordering spinodal temperature $T_s$. Then the evolution under A1$\to$L1$_0$ transition includes the following stages [@Zhang-91; @Zhang-92; @Yanar; @Tanaka; @Oshima]: \(i) The initial stage of the formation of finest L1$_0$-ordered domains when their tetragonal distortion makes still little effect on the evolution and all six types of APD are present in microstructures in the same proportion. It corresponds to the so-called ‘mottled’ contrast in TEM images [@Tanaka; @Oshima]. \(ii) The next, intermediate stage which corresponds to the so-called ‘tweed’ contrast in TEM images. The tetragonal deformation of the L1$_0$-ordered APDs here leads to the predominance of the (110)-type orientations of flip-APBs, but all six types of APD (i. e. APDs with all three orientations of the tetragonal axis $c$) are still present in microstructures in comparable proportions [@Zhang-91; @Zhang-92; @Yanar]. \(iii) The final, polytwinned stage when the tetragonal distortion of the L1$_0$-ordered APDs becomes the main factor of evolution and leads to the formation of (110)-type oriented twin bands. Each band includes only two types of APD with the same $c$ axis, and these axes in the adjacent bands are ‘twin’ related, have the alternate (100) and (010) orientations for the given set of the (110)-oriented bands [@Zhang-91; @Zhang-92; @Yanar]. The thermodynamic driving force for the (110)-type orientation of flip-APBs is the gain in the elastic energy of adjacent APDs: at other orientations this energy increases under the growth of an APD proportionally to its volume [@Roitburd; @Khach-Shat; @Khach-book; @Vaks-01]. For an APD with the characteristic size $l$ and the surface $S_d$, this elastic energy $E_{el}^v\sim c_-\overline\varepsilon^2S_dl$ begins to affect the microstructural evolution when it becomes comparable with the surface energy $E_s\sim\sigma S_d$ where $\sigma$ is the APB surface tension. The ‘tweed’ stage (ii) corresponds to the relation $E_{el}^v\sim E_s$ or to the characteristic APD size $$l_0\sim \sigma/c_-\overline\varepsilon^2, \label{l_0}$$ and so this size sharply increases under decreasing distortion $\overline\varepsilon$. Figures 1–7 illustrate quasi-2D simulations for which microstructures include only edge-on APBs normal to the (001) plane. The elimination of the volume-dependent elastic energy mentioned above is here possible only for the (100) and (010)-oriented APDs separated by the (110) or (1$\bar 1$0)-oriented APBs, while in the (001)-oriented APDs C and $\overline {\rm C}$ this elastic energy is always present. Therefore, the tweed stage (ii) in these simulations corresponds to both the predominance of (110) or (1$\bar 1$0)-oriented APBs separating domains A or $\overline {\rm A}$ from B or $\overline {\rm B}$ and the decrease of the portion of domains C and $\overline {\rm C}$ in the microstructures. In the 3D case each of three posible types of a polytwin, that without (001), (100), or (010)-oriented APDs, can be formed in the given part of an alloy stochastically due to the local fluctuations of composition \[1–7\]. It is illustrated, in particular, by 3D simulation shown in figure \[m4s-3D\], while quasi-2D simulations describe the formation of only one polytwin type mentioned above. The distortion parameter $\|\varepsilon_m \|=0.1$ for the simulations shown in figures \[m5s\_e10\]–\[m4s\_e10-1800\] was chosen so that the APD size $l_0$ in Eq. (\[l\_0\]) characteristic for manifestations of elastic effects has the scale typical for real CoPt-type alloys. In particular, if we take a conventional assumption that the APB energy $\sigma$ is proportional to the transition temperature $T_c$: $\sigma\sim T_cf(T')$ where $f$ is some function of the reduced temperature $T'=T/T_c$, then using the relation $\varepsilon_m =\overline\varepsilon /4\eta_1^2$ and the parameters $\overline\varepsilon$, $\eta_1$, and $T_c$ for CoPt and for our models mentioned above we find that the right-hand side of Eq. (\[l\_0\]) for models 5 and 4 at $\|\varepsilon_m \|=0.1$ is close to that for the CoPt alloy at similar $T'$ values within about ten percent. Therefore, the microstructures at both the initial stage (i) and the tweed stage (ii) can be reproduced by figures \[m5s\_e10\]–\[m4s\_e10-1800\] with no significant distortion of scales. Under a furher growth of an APD its size $l$ becomes comparable with the simulation box size $L$, and the periodic boundary conditions begin to significantly affect the evolution. Therefore, the later stages of transformation can be more adequately simulated if we reduce the characteristic size $l_0$ in Eq. (\[l\_0\]) using the larger values of the parameter $\varepsilon_m$, such as $\|\varepsilon_m \|=0.15-0.2$ used in the simulations shown in figures \[m5s\_e15\]–\[m4s-3D\]. Let us first discuss figures \[m5s\_e10\]–\[m4s\_e10-1800\] corresponding to a ‘realistic’ distortion parameter $\|\varepsilon_m \|=0.1$. The initial stage (i) in these figures corresponds to frames \[m5s\_e10\](a)–\[m5s\_e10\](b), , and \[m4s\_e10-1800\](a)–\[m4s\_e10-1800\](b); the tweed stage (ii), to frames \[m5s\_e10\](c)–\[m5s\_e10\](d), \[m4s\_e10-1300\](c)–\[m4s\_e10-1300\](e), and \[m4s\_e10-1800\](c); and the twin stage (iii), to frames \[m5s\_e10\](e)–\[m5s\_e10\](f), \[m4s\_e10-1300\](f), and \[m4s\_e10-1800\](d). The detailed consideration of the initial stage for models 4 and 5 neglecting the deformational effects [@PV] revealed the following features of evolution: \(a) The presence of abundant processes of fusion of in-phase domains which are one of main mechanisms of domain growth at this stage. \(b) The presence of peculiar long-living configurations, the quadruple junctions of APDs (4-junctions) of the type A$_1$A$_2$$\overline{\rm A_1}$A$_3$ where A$_2$ and A$_3$ can correspond to any two of four types of APD different from A$_1$ and $\overline{\rm A_1}$. \(c) The presence of many processes of ‘splitting’ of a shift-APB into two flip-APBs which lead to either a fusion of in-phase domains mentioned in point (a) ($s\to f$ process) or a formation of a 4-junction mentioned in point (b) ($s\to 4j$ process). Figures \[m5s\_e10\]–\[m4s\_e10-1800\] show that all these microstructural features are also present when the deformational effects are taken into account, and not only at the initial stage (i) but also at the tweed stage (ii). In particular, the beginning and the end of an $s\to f$ process (marked by the single and the thick arrow, respectively) can be followed in frames \[m5s\_e10\](a) and \[m5s\_e10\](b); \[m5s\_e10\](c) and \[m5s\_e10\](d) ; \[m5s\_e10\](d) and \[m5s\_e10\]e; \[m4s\_e10-1300\](b) and \[m4s\_e10-1300\](c); \[m4s\_e10-1300\](c) and \[m4s\_e10-1300\](d); and \[m4s\_e10-1800\](a) and \[m4s\_e10-1800\](b). The fusion with the disappearance of an intermediate APD which initially separates two in-phase domains to be fused [@PV] can be followed in the lower right part of frames \[m5s\_e10\](a) and \[m5s\_e10\](b) and in the upper right part of frames \[m4s\_e10-1300\](b) and \[m4s\_e10-1300\](c) (which is marked by a thick arrow in frames \[m5s\_e10\](b) and \[m4s\_e10-1300\](c), respectively). A number of long-living 4-junctions marked by thin arrows are seen in frames \[m5s\_e10\](a) –\[m5s\_e10\](d) , \[m4s\_e10-1300\](a)–\[m4s\_e10-1300\](c), and \[m4s\_e10-1800\](a). An $s\to 4j$ process can be followed in the lower right part of frames \[m5s\_e10\](a)–\[m5s\_e10\](c) . The processes and configurations (a), (b) and (c) can also be seen in figures 4–7 discussed below. Frames \[m4s\_e10-1300\](a)–\[m4s\_e10-1300\](e) also display some (100)-oriented and thin conservative APBs. As discussed in [@PV] and below, such APBs are most typical of the short-range-interaction systems where they have a low surface energy (being zero for the stoichiometric nearest-neighbor interaction model) unlike other, non-conservative APBs. Under an increase of the interaction range, as well as temperature or the deviation from stoichiometric composition, the anisotropy in the APB surface energy sharply decreases [@PV]. Therefore, in figure \[m4s\_e10-1300\] (and figure \[m4s\_e15\] below) corresponding to the intermediate-range-interaction model 4 the conservative APBs are few but observable, while for the extended-range-interaction model 5 in figure \[m5s\_e10\], as well as at elevated $T$ or significant ‘non-stoichiometry’ $\delta c=(0.5-c)$ in figures \[m4s\_e10-1800\] or \[m4ns\_e15\] for model 4, such APBs are absent entirely. Comparison of figures \[m4s\_e10-1300\] and \[m4s\_e10-1800\] illustrates the sharp dependence of microstructural evolution on the transformation temperature $T$. Under elevating this temperature to values near the critical temperature $T_c$: $(T_c-T)\lesssim 0.1\,T_c$, both flip and shift-APBs notably thicken, the anisotropy in their surface energy falls off, while the characteristic size of initial APDs (formed after a rapid quench A1$\to$L1$_0$) increases. The latter is related to an increase at $T\to T_c$ of the characteristic wavelength for the ordering instability which is due to the narrowing of the interval of effective wavenumbers ${\bf q}={\bf k}-{\bf k}_s$ near the superstructure vector ${\bf k}_s$ for which the ordering concentration waves are unstable at $T<T_c$. Frames \[m5s\_e10\](c)–\[m5s\_e10\](d), \[m4s\_e10-1300\](c)–\[m4s\_e10-1300\](e), and \[m4s\_e10-1800\](c) show evolution at the tweed stage. They illustrate, in particular, kinetics of the (110)-type alignment of APBs between APDs A or $\overline {\rm A}$ and B or $\overline {\rm B}$ at this stage, as well as a “dying out” of (100)-oriented APDs C and $\overline{\rm C}$. These frames also show that in the simulation with a realistic distortion parameter $\|\varepsilon_m\|=0.1$ (fitted to the structural data for CoPt) the APD size $l_0$ (\[l\_0\]) characteristic of the tweed stage is about $(20-40)\,a$. It agrees with the order of magnitude of this size observed in the CoPt-type alloys FePt and FePd [@Zhang-91; @Zhang-92; @Yanar]. As mentioned, the final, twin stage of the transformation can be more adequately simulated with the larger values of parameter $\|\varepsilon_m\|$ which are employed in the simulations shown in figures \[m5s\_e15\]–\[m4s-3D\]. Before discussing these figures we note some typical configurations observed in experimental studies of transient twinned microstructures [@Zhang-91; @Zhang-92; @Yanar] seen, for example, in figures 5, 6, 9 and 2 in Refs. [@Zhang-91] [@Zhang-92], [@Yanar] and [@CWK], respectively: \(1) semi-loop-like shift-APBs adjacent to the twin band boundaries; \(2) ‘S-shaped’ shift-APBs stretching across the twin band; and \(3) short and narrow twin bands (for brevity to be called ‘microtwins’) which lie within the larger twin bands and usually have one or two shift-APBs near their edges. Comparing our results with experiments one should consider that due to the limited size of the simulation box the twin band width $d$ in our simulations has the same order of magnitude as the APD size $l_0$ (\[l\_0\]) characteristic of the tweed stage, while in experiments $d$ usually much exceeds $l_0$ [@Zhang-91; @Zhang-92; @Yanar]. Therefore, the distribution of shift-APBs within twin bands in our simulation is usually much more close to equilibrium than in experiments. In spite of this difference, the simulations reproduce all characteristic transient configurations (1)–(3) and elucidate the mechanisms of their formation. In particular, both the semi-loop and the S-shaped shift-APBs are formed from regular-shaped approximately quadrangular APDs (characteristic of the beginning of the twin stage) due to the disappearance of adjacent APDs which are ‘wrongly-oriented’ with respect to the given twin band. The formation of semi-loop configurations is illustrated by frames \[m5s\_e10\](d)–\[m5s\_e10\](f), \[m4s\_e15\](d)–\[m4s\_e15\](e), and \[m5s\_e20\](b)–\[m5s\_e20\](c); while the formation of S-shaped APBs can be seen in frames \[m4s\_e10-1300\](d)–\[m4s\_e10-1300\](f), \[m5s\_e15\](d)–\[m5s\_e15\](f), \[m4s\_e15\](d)–\[m4s\_e15\](e), and \[m5s\_e20\](b)–\[m5s\_e20\](c). The formation and evolution of microtwins is illustrated by frames \[m5s\_e15\](c)–\[m5s\_e15\](d) and \[m4s\_e15\](c)–\[m4s\_e15\](d). These frames show that the microtwin is actually a small and narrow APD for which deformational effects are strong enough to align its flip-APBs along the (110)-type directions. However, the standard mechanism of coarsening via the growth of larger APDs at the expence of smaller ones leads to the shrinking and eventually to the disappearance of a microtwin which is usually accompanied by the formation of S-shaped and/or semi-loop shift-APBs. The latter is illustrated by frames \[m5s\_e15\](d)–\[m5s\_e15\](f) and \[m4s\_e15\](d)–\[m4s\_e15\](e). Let us also note that the microtwin configuration shown in frame \[m5s\_e15\](d) is strikingly similar to that seen in the central part of experimental figure 2 in Ref. [@CWK]. Let us now discuss the final, ‘nearly equilibrium’ microstructures shown in last frames of figures \[m5s\_e10\]–\[m4s-3D\]. A characteristic feature of these microstructures is a peculiar alignment of shift-APBs: within a (100)-oriented twin band in a (110)-type polytwin the APBs tend to align normally to some direction ${\bf n}=(\cos\alpha, \sin\alpha, 0)$ characterized by a ‘tilting’ angle $\alpha$ between the band orientation and the APB plane. Figures \[m5s\_e10\]–\[m4s-3D\] show that this tilting angle is not very sensitive to the variations of temperature or concentration but it sharply depends on the interaction type, particularly on the interaction range. For the extended-range-interaction model 5 this angle is close to $\pi/4$ (slightly exceeding this value), while for the intermediate-range-interaction model 4 it is notably less than $\pi/4$. A similar alignment of APBs for the short-range interaction systems is illustrated by figures \[m1s\_e15\]–\[m2\_e10\] where the tilting angle is close to zero. A phenomenological theory of this interaction-dependent tilting of APBs within nearly-equilibrium twin bands has been suggested in [@Vaks-01]. The tilting is explained by the competition between the anisotropy of the APB surface tension $\sigma$ and a tendency to minimize the total APB area within the given twin band which corresponds to $\alpha =\pi/4$. For the alloy systems with both the intermediate and the short interaction range the surface tension $\sigma (\alpha)$ is minimal at $\alpha =0$ [@Vaks-01; @PV]. Thus for such alloy systems the tilting angle is less than $\pi/4$, and it decreases with the decrease of the interaction range. For the extended-range-interaction systems the anisotropy of the APB surface tension is weak [@BDPSV; @PV], and so the tilting angle is close to $\pi/4$. Therefore, the comparison of experimental tilting angles with theoretical calculations [@Vaks-01] can provide both qualitative and quantitative information on the effective chemical interactions in an alloy. The alignment of shift-APBs discussed above seems to be clearly seen in the experimental microstructure for CoPt shown in figure 5 of Ref. [@Leroux] where the tilting angle is notably less than $\pi/4$. It can indicate that the effective interactions in CoPt have an intermediate interaction range. This agrees with the usual estimates of these interactions for transition metal alloys, see e.g. [@Zunger; @Chassagne; @Turchi]. Comparison of figures \[m5s\_e15\] and \[m5s\_e20\] illustrates the influence of temperature $T$ on the evolution. Under elevating $T$ we again observe a thickening of APBs, as well as a coarsening of initial APDs. Frames \[m5s\_e20\](d)–\[m5s\_e20\](f) also illustrate a process of ‘transverse coarsening’ of twin bands via a shrinkage and disappearance of some microtwinned bands. Such transverse coarsening appears to be seen in a number of experimental microstructures, for example, in figure 6 in [@Zhang-92] or figure 2 in [@CWK]. Frames \[m5s\_e20\](d)–\[m5s\_e20\](f) show that the thermodynamic driving force for such transverse coarsening is mainly the gain in the surface energy of shift-APBs due the decrease of their total area under this process. Figures \[m4s\_e15\] and \[m4ns\_e15\] illustrate the concentration dependence of the evolution. The non-stoichiometry $\delta c =(0.5-c)$ affects the evolution similarly to temperature $T$: under an increase of both $\delta c$ and $T$ all APBs thicken, while shift-APBs become less stable with respect to flip-APBs [@PV]. The latter leads to an enhancement of processes of splitting of shift-APBs as well as of the transverse coarsening mentioned above; it is illustrated by frames \[m4ns\_e15\](b)– \[m4ns\_e15\](f). Some results of a 3D simulation with $V_b=52^2\times 30$ are presented in figure \[m4s-3D\]. In this figure we employ the $c$-representation (described in the caption) in which the regions containing the vertical or horizontal lines (that is, the vertical or horizontal crystal planes filled by A atoms) correspond to the APDs with the (100) or (010)-oriented tetragonal axis, respectively, while the checkered regions correspond to the APDs with the tetragonal axis normal to the plane of figure. This simulation aimed mainly to complement 2D simulations with an illustration of geometrical features of 3D microstructures. Figure \[m4s-3D\] illustrates, in particular, a stochastic formation of different polytwin sets with three possible types of orientation mentioned above. A limited size of the simulation box prevent us from a detailed consideration of evolution with this 3D simulation. Therefore, below we discuss only the problem of a 3D orientation of tilted shift-APBs in final, ‘nearly-equilibrium’ microstructures. Let us consider a (100)-oriented twin band in the form of a plate of height $h$, length $l$, and width $d$ in the direction (001), (110), and (1$\bar 1$0), respectively, with $d\lesssim h\ll l$ (which is a typical form of twin bands observed in TEM experiments [@Leroux; @Zhang-91; @Zhang-92; @Yanar; @Tanaka; @Oshima; @Syutkina]). The equilibrium orientation of a plane shift-APB in this band corresponds to the minimum of its energy $E_s=\sigma S$ where $S$ is the APB area and $\sigma$ is the surface tension determined mainly by the angle $\alpha$ between the APB orientation ${\bf n}=(\sin\alpha, \cos\alpha\cos\varphi, \cos\alpha\sin\varphi)$ and the band orientation ${\bf n}_0=(100)$ [@Vaks-01]. For the ‘needle-shaped’ twin band under consideration the upper and the lower boundary of a shift-APB usually lies at the top and the bottom edge of this band, respectively. Minimization of energy $E_s$ in this case yields: $\varphi =0$, i. e. the APB is normal to the (001) plane, and its orientation ${\bf n}=(\sin\alpha, \cos\alpha, 0)$ is determined by the interaction-dependent tilting angle $\alpha$ defined in [@Vaks-01]. This conclusion seems to be supported by the present 3D simulation: the lower and the upper tilted shift-APB within the (010)-oriented twin band below the main diagonal of frame \[m4s-3D\](c) corresponds to the grey line stretching across the checkered region in frame \[m4s-3D\](e) and \[m4s-3D\](f), respectively, and both these lines are approximately normal to the (001) plane. Kinetic features of A1$\to$L1$_0$ transformations in the short-range interaction systems ======================================================================================== As mentioned, transient microstructures under L1$_0$ ordering for the short-range interaction systems include many conservative APBs. Such APBs are virtually immobile, and so the evolution is realized via motion of other, non-conservative APBs which results in a number of peculiar kinetic features [@BDPSV; @PV]. The initial stage of the A1$\to$L1$_0$ transformation for the short-range interaction systems was discussed in detail in Ref. [@PV]. In this section we consider the tweed and twin stages of such transformations and note the differences with the case of systems with the larger interaction range. Some results of our simulations for the short-range-interaction systems are presented in figures \[m1s\_e15\]–\[m2\_e10\]. In these simulations we used sufficiently high temperatures $T'\gtrsim 0.9-0.8$ to accelerate evolution to final, ‘nearly-equilibrium’ configurations as the presence of conservative APBs slowes down (or even ‘freezes’) this evolution, particularly at low $T'$. Figure \[m1s\_e15\] illustrates the evolution for model 1; as discussed in [@BDPSV], this model seems to correspond to the Cu–Au-type alloys. A distinctive feature of microstructures shown in figure \[m1s\_e15\] is a predominance of the above-mentioned conservative APBs with the (100)-type orientation. Frames \[m1s\_e15\](a)–\[m1s\_e15\](c) show both the conservative shift-APBs and the conservative flip-APBs also illustrating their orientational properties [@PV]. For quasi-2D microstructures with edge-on APBs shown in figure \[m1s\_e15\], c-shift APBs separating APDs A and (c-APBs are horizontal; c-APBs are vertical; and c-APBs can be both horizontal and vertical; c-flip APBs – (which separate APDs A or $\overline{\rm A}$ from C or $\overline{\rm C}$) are horizontal; c-APBs – are vertical; and c-APBs – should lie in the plane of figure and thus they are not seen in figure \[m1s\_e15\]. Figure \[m1s\_e15\] also shows that the conservative APBs are notably thinner than non-conservative ones, particularly so for c-flip APBs. Frames \[m1s\_e15\](a)–\[m1s\_e15\](c) show that at first stages of evolution the portion of conservative APBs with respect to non-conservative ones increases, due to the lower surface energy of the c-APBs. Later on, with the beginning of the tweed stage, the deformational effects become important leading to a dying out of both APDs C and $\overline{\rm C}$ and their . However, the conservative shift-APBs within twin bands survive, and in the final frame \[m1s\_e15\]d they are mostly ‘step-like’ consisting of (100)-type oriented conservative segments and small non-conservative ledges. These step-like APBs can be viewed as a ‘facetted’ version of tilted APBs discussed above and seen in figures \[m5s\_e10\]–\[m4s-3D\]. Such step-like APBs were observed under the L1$_0$ ordering of CuAu and some CuAu-based alloys [@Syutkina], and they are also similar to those observed under the L1$_2$ ordering in both simulations [@BDPSV] and experiments for the Cu$_3$Au alloy [@Potez]. As it was repeatedly noted in Ref. [@PV] and above, an increase of non-stoichiometry $\delta c= (0.5-c)$ or temperature $T$ leads to a sharp decrease of both the anisotropy of the APB energy and the energy preference of conservative APBs with respect to non-conservative ones. Therefore, under an increase of $\delta c$ or $T$ the portion of conservative APBs in transient microstructures falls off, and at sufficiently high $\delta c$ or $T$ such APBs are not formed under the transformation at all. It results in drastic microstructural changes of evolution, including sharp, phase-transition-like changes in morphology of aligned shift-APBs within twin bands, from the ‘faceting’ to the ‘tilting’. This is illustrated by figure \[m1ns\_e15\] which shows the evolution of model 1 at a significant non-stoichiometry $\delta c =0.06$, and this evolution is qualitatively different with that for a stoichiometric alloy shown in figure \[m1s\_e15\]. Figure \[m2\_e10\] illustrates the transition from the ‘facetted’ to the ‘tilted’ morphology of shift-APBs within nearly-equilibrium twin bands under variations of $T$ or $\delta c$ for model 2. An examination of intermediate stages of transformations illustrated by this figure shows that the morphological changes are realized via some local bends of facetted APBs. It is also illustrated by a comparison of frames \[m2\_e10\](a), \[m2\_e10\](c) and \[m2\_e10\](d) with each other. Therefore, the ‘morphological phase transition’ mentioned above is actually smeared over some interval of temperature or concentration. However, frames \[m2\_e10\](a)–\[m2\_e10\](d) show that the ‘intervals of smearing’ of such transitions can be relatively narrow. Conclusion ========== Let us summarize the main results of this work. The earlier-described master equation approach [@Vaks-96; @BV-98] is used to study the microstructural evolution under L1$_0$-type orderings in alloys, including the formation of twinned structures due to the spontaneous tetragonal deformation inherent to such orderings. To this end we first derive a microscopical model for the effective interatomic deformational interaction which arise due to the so-called Kanzaki forces describing interaction of lattice deformations with site occupations. This model generalizes an analogous model of Khachaturyan for dilute alloys [@Khach-book] to the physically interesting case of concentrated alloys. We take into account the non-pairwise contribution to Kanzaki forces, and the resulting effective interaction $H_d$ is non-pairwise, too, unlike the case of dilute alloys. This effective interaction describes, in particular, the lattice symmetry change effects under phase transformations, such as the tetragonal distortion mentioned above. Assuming the non-pairwise Kanzaki forces to be short-ranged, we can express the deformational interaction $H_d$ in terms of two microscopical parameters which can be estimated from the experimental data about the lattice distortions under phase transformations. We present these estimates for alloys Co–Pt for which such structural data are available [@Leroux-88]. Then we employ the kinetic cluster field method [@BDPSV; @PV] to simulate A1$\to$L1$_0$ transformation after a quench of an alloy from the disordered A1 phase to the single-phase L1$_0$ field of the phase diagram in the presence of deformational interaction $H_d$. We consider five alloy models with different types of chemical interaction, from the short-range-interaction model 1 to the extended-range-interaction model 5, at different temperatures $T$, concentrations $c$, and spontaneous tetragonal distortions $\overline\varepsilon$. We use both 2D and 3D simulations, and all significant features of microstructural evolution in both types of simulation were found to be similar. The evolution under A1$\to$L1$_0$ transition can be divided into three stages, in accordance with an increasing importance of the deformational interaction $H_d$: the ‘initial’, ‘tweed’ and ‘twin’ stage. For the initial stage (discussed in detail previously [@PV]), the deformational effects are insignificant. For the tweed stage, the effects of $H_d$ become comparable with those of chemical interaction $H_c$ and lead to the formation of specific microstructures discussed in section 4. For the final, twin stage the tetragonal distortion of L1$_0$-ordered antiphase domains (APDs) becomes the main factor of the evolution and leads to the formation of (110)-type oriented twin bands. Each band includes only two types of APD with the same tetragonal axis, and these axes in the adjacent bands are ‘twin’ related, have the alternate (100) and (010) orientations for the given set of (110)-type oriented bands. The microstructural evolution strongly depends on the interaction type, particularly on the interaction range $R_{int}$. For the systems with an extended or intermediate $R_{int}$ at both the initial and the tweed stage we observe the following features (mentioned previously [@PV] for the initial stage): (a) abundant processes of fusion of in-phase domains; (b) a great number of peculiar long-living configurations, the quadruple junctions of APDs described in section 4; and (c) numerous processes of ‘splitting’ of an antiphase boundary separating the APDs with the same tetragonal axis (‘shift-APB’) into two APBs separating the APDs with perpendicular tetragonal axis (flip-APBs). The simulations also illustrate a sharp temperature dependence of the evolution, in particular, a notable increase of both the width of APBs and the characteristic size of initial APDs under elevating $T$. The deviation from stoichiometry affects the evolution similarly to temperature: under an increase of both non-stoichiometry $\delta c =(0.5-c)$ and $T$ all APBs thicken, while shift-APBs become less stable with respect to flip-APBs. For the twin stage, our simulations reveal the following typical features of transient microstructures: (1) semi-loop-like shift-APBs adjacent to the twin band boundaries; (2) ‘S-shaped’ shift-APBs stretching across the twin band; (3) short and narrow twin bands (‘microtwins’) lying within the larger twin bands; and (4) processes of ‘transverse coarsening’ of twinned structures via a shrinkage and disappearance of some microtwins. All these features agree with experimental observations [@Zhang-91; @Zhang-92; @Yanar]. For the final, nearly-equilibrium twin bands the simulations demonstrate a peculiar alignment of shift-APBs with a certain tilting angle between the band orientation and the APB plane, and this tilting angle sharply depends on the interaction type, particularly on the interaction range $R_{int}$. Such alignment of APBs seems to be observed in the CoPt alloy [@Leroux], and a comparison of experimental tilting angles with theoretical calculations [@Vaks-01] can provide information about the effective interactions in an alloy. A distinctive feature of evolution for the short-range-interaction systems is the presence of many conservative APBs with the (100)-type orientation. The conservative flip-APBs disappear in the course of the evolution, but the conservative shift-APBs survive and are present in the final twinned microstructures. Such ‘nearly equilibrium’ shift-APBs are mostly ’step-like’ consisting of (100)-type oriented conservative segments and small non-conservative ledges, which can be viewed as a ‘facetted’ version of tilted APBs mentioned above. This (100)-type alignment of shift-APBs within twin bands seems to agree with available experimental observations for the CuAu alloy [@Syutkina] for which chemical interactions are supposed to be short-ranged [@Potez; @BDPSV]. Under an increase of non-stoichiometry $\delta c$ or temperature $T$ the energy preference of conservative APBs with respect to non-conservative ones decreases, and the portion of conservative APBs in the microstructures falls off. It results in drastic microstructural changes, including sharp, phase-transition-like changes in morphology of aligned shift-APBs within twin bands, from their ‘faceting’ to the ‘tilting’. Such ‘morphological phase transitions’ are actually smeared over some intervals of temperature or concentration, but the simulations show that the intervals of smearing can be narrow. Finally, let us make a general remark about kinetics of multivariant orderings in alloys, such as the L1$_2$, L1$_0$ and D0$_3$ orderings discussed in Refs. [@BDPSV; @PV; @BSV] and in this work. It is known that the thermodynamic behavior of different systems under various phase transitions reveals features of universality and insensitivity to the microscopical details of structure, particularly in the critical region near thermodynamic instability points. The results of this and other studies of multivariant orderings show that such universality does not seem to hold for their phase transformation kinetics, at least outside the critical region (which for such orderings is usually either quite narrow or absent at all). The microstructural evolution reveals a great variety of peculiar features, the detailed form of which sharply depends on the type of interatomic interaction, the type of the crystal structure and ordering, the degree of non-stoichiometry, and other ‘non-universal’ characteristics. Acknowledgments {#acknowledgments .unnumbered} =============== The authors are much indebted to V. Yu. Dobretsov for the help in this work; to N. N. Syutkin and V. I. Syutkina, for the valuable information about details of experiments [@Syutkina]; and to Georges Martin, for numerous stimulating discussions. The work was supported by the Russian Fund of Basic Research under Grants No 00-02-17692 and 00-15-96709. [99]{} Present address: Ames Laboratory, Ames, IA 50011, USA. Leroux C, Loiseau A, Broddin D and Van Tendeloo G 1991 [Phil. Mag.]{} B [64]{} 57 Zhang B, Lelovic M and Soffa W A 1991 [Scripta Met.]{} [25]{} 1577 Zhang B and Soffa W A 1992 [Phys. Stat. Sol.]{} (a) [131]{} 707 Yanar C, Wiezorek J M K and Soffa W A 2000 [Phase Transformations and Evolution in Materials]{} ed P E A Turchi and A Gonis (Warrendale: The MMM Society) p 39 Tanaka Y, Udoh K-I, Hisatsune K and Yasuda K 1994 [Phil. Mag. A]{} [69]{} 925 Oshima R, Yamashita M, Matsumoto K and Hiraga K 1994 [Solid–Solid Phase Transformations]{} ed W C Johnson (Warrendale: The MMM Society) p 407 Syutkina V I, Abdulov R Z, Zemtsova N D and Yasyreva L P 1984 [Fiz. Metal. Metalloved.]{} [58]{} 745 Roitburd A L 1968 [Fiz. Tverd. Tela]{} [10]{} 3619 (Engl. Transl. 1969 [Sov. Phys. Solid State]{} [10]{} 2870) Khachaturyan A G and Shatalov G A 1969 [Zh. Eksp. Teor. Fiz.]{} [56]{} 1037 (Engl. Transl. 1969 [Sov. Phys.–JETP]{} [10]{} 557) Khachaturyan A G 1983 [Theory of Structural Phase Transformations in Solids]{} (New York: Wiley) Vaks V G 2001 [Pis. Zh. Eksp. Teor. Fiz.]{} [73]{} 274 (Engl. Transl. 2001 [JETP Lett.]{} [73]{} 237) Chen L-Q, Wang Y Z and Khachaturyan A G 1992 [Phil. Mag. 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--- abstract: 'We consider a surface that admits a ${\mathbb{Q}}$-Gorenstein degeneration to a cyclic quotient singularity $\frac{1}{dn^2}(1,dna-1)$. Under several technical assumptions, we construct $d$ exceptional vector bundles of rank $n$ which are orthogonal to each other.' address: 'Korea Institute for Advanced Study, 85 Hoegiro Dongdaemun-gu, Seoul 02455, Republic of Korea' author: - 'Cho, Yonghwa' bibliography: - 'arXiv\_OrthogonalCollection.bib' title: 'Orthogonal exceptional collections from ${\mathbb{Q}}$-Gorenstein degeneration of surfaces' --- Introduction ============ Let $\mathcal X / (0 \in \Delta)$ be a one parameter deformation of a complex normal projective surface $\mathcal X_0$ with at worst quotient singularities. The deformation $\mathcal X / (0 \in \Delta)$ is said to be ${\mathbb{Q}}$-Gorenstein if $K_\mathcal X$ is ${\mathbb{Q}}$-Cartier. Locally, a quotient singularity admitting a ${\mathbb{Q}}$-Gorenstein smoothing is either a rational double point or a cyclic quotient singularity $\frac{1}{dn^2}(1,dna-1)$ where $d, n, a>0$ are integers with $n>a>0$ and $\op{gcd}(n,a)=1$. The singularity in the latter is called a singularity of class $T$. In the case $d=1$ we call $\frac{1}{n^2}(1,na-1)$ a Wahl singularity. On a surface with a Wahl singularity and $p_g=q=0$, ${\mathbb{Q}}$-Gorenstein smoothing of the Wahl singularity gives rise to an exceptional vector bundle(a vector bundle $E$ such that $\bigoplus_p {\operatorname{Ext}}^p(E,E)={\mathbb{C}}$) on the general fiber(Hacking[@Hacking:ExceptionalVectorBundle]). A natural question follows: > What can be said if one considers a ${\mathbb{Q}}$-Gorenstein smoothing of $\frac{1}{dn^2}(1,dna-1)$ with $d>1$? The article concerns an answer to this question. \[thm: Main thm Summary\] Let $X$ be a normal projective surface with $H^1(\mathcal O_X)=H^2(\mathcal O_X)=0$, let $(P\in X) \simeq \frac{1}{dn^2}(1,dna-1)$ be a singularity of class T, and let $\mathcal X / (0 \in \Delta)$ be a ${\mathbb{Q}}$-Gorenstein smoothing of $X$. Assume there exists a Weil divisor $D \in {\operatorname{Cl}}X$ such that $D$ is Cartier except at $P$ and the image of $D$ along ${\operatorname{Cl}}X \to H_2(X;{\mathbb{Z}}) \to H_1(L;{\mathbb{Z}})$, where $L$ is the link of $(P \in X)$, generates $H_1(L;{\mathbb{Z}})/n^2$. After a finite base change $(0 \in \Delta') \to (0 \in \Delta)$, there exist exceptional vector bundles $E_1,\ldots,E_d$ on the general fiber $S := \mathcal X_t$ such that $\op{rank} E_k=n$ and ${\operatorname{Ext}}_S^p(E_k,E_\ell)=0$ for each $p,\ k\neq \ell$. The theorem is motivated from comparing degenerations of del Pezzo surfaces and three block collections([@KarpovNogin:3block]). Suppose $X$ is a normal projective surface with quotient singularities admitting ${\mathbb{Q}}$-Gorenstein smoothing to ${\mathbb P}^2$. Then, $X$ is isomorphic to either ${\mathbb P}^2(a^2,b^2,c^2)$, where $(a,b,c)$ is a solution to Markov equation $a^2+b^2+c^2=3abc$, or a partial smoothing of one of these weighted projective planes(Manetti[@Manetti:NormalDegenerationOfPlane], Hacking-Prokhorov[@HackingProkorov:DegenerationOfDelPezzo]). It can be shown that ${\mathbb P}(a^2,b^2,c^2)$ has only Wahl singularities. We may apply the construction method in [@Hacking:ExceptionalVectorBundle] to produce three exceptional vector bundles of respective ranks $a,b,c$ on ${\mathbb P}^2$, which form a full exceptional collection in ${\operatorname{D}^{\sf b}}({\mathbb P}^2)$. Conversely, every full exceptional collection in ${\operatorname{D}^{\sf b}}({\mathbb P}^2)$ arises in this way(cf. Gorodentsev-Rudakov[@GorodenstevRudakov:ExceptionalBundleOnPlane], Rudakov[@Rudakov:MorkovNumberAndExceptional]). In general, it is difficult to find equations classifying full exceptional collections in del Pezzo surfaces. We narrow down our interests into three block collections, namely, the full exceptional collections of the form $$\langle\, E^1_1,\,\ldots,\,E^1_{d_1},\ \ E^2_1,\,\ldots,\, E^2_{d_2},\ \ E^3_1,\,\ldots,\,E^3_{d_3} \, \rangle.$$ The term “block” refers to each subcollection $\langle\, E_1^k,\ldots,E^k_{d_k}\, \rangle$. In each block, the objects are vector bundles of the same rank, and ${\operatorname{Ext}}^p(E^k_i, E^k_j)=0$ for each $p,k$ and $i\neq j$. In three block collections, one may associate Markov type equations as follows. Let $r_k$ be the rank of the object $E^k_i$. Then, the equation $$\label{eq: Markov type equations} d_1 r_1^2 + d_2 r_2^2 + d_3 r_3^2 = \lambda r_1r_2r_3,\quad \lambda = \sqrt{K_X^2 d_1d_2d_3}$$ holds. These equations play a central role in the classification problem of three block collections(Karpov-Nogin[@KarpovNogin:3block]). The equations (\[eq: Markov type equations\]) also emerge in the classification problem of ${\mathbb{Q}}$-Gorenstein degenerations of del Pezzo surfaces. One of the main results of [@HackingProkorov:DegenerationOfDelPezzo] says the following. Let $X$ be a normal projective toric surface with Picard number one that admits a ${\mathbb{Q}}$-Gorenstein smoothing to a del Pezzo surface. Then, $X$ is one of the (fake) weighted projective spaces having three singular points $\frac{1}{d_ir_i^2}(1,d_ir_ia_i-1)$. Comparing with the result of [@Hacking:ExceptionalVectorBundle], it is natural to expect that each singularity contributes each block in the three block collection. Notations and conventions {#notations-and-conventions .unnumbered} ------------------------- [ ]{} - We always work over the field of complex numbers. - For a complex analytic space $\mathcal X$ and a point $P \in \mathcal X$, $(P \in \mathcal X)$ denotes the germ of analytic neighborhoods at $P$. The readers who prefer algebraic language may replace $\mathcal X$ by an algebraic ${\mathbb{C}}$-scheme, $(P \in \mathcal X)$ by a germ of étale neighborhoods, and $(0 \in \Delta)$($\Delta$ a complex disk) by $(0 \in T)$($T$ an algebraic curve smooth at $0$). - Let $\mu_r$ be the cyclic group generated by the $r$th root of unity $\zeta_r = \exp( 2\pi\sqrt{-1} / r)$. For an integer $a_1,\ldots,a_m$ with $\op{gcd}(r,m)=1$, we define the action $$\mu_r \times {\mathbb{C}}^m \to {\mathbb{C}}^m,\quad \zeta_r \cdot (z_1,\,\ldots,\, z_m) = (\zeta_r^{a_1} z_1,\,\ldots, \zeta_r^{a_m} z_m ).$$ The singularity $(0 \in {\mathbb{C}}^m / \mu_r)$ is denoted by $\bigl(0 \in {\mathbb{C}}^m / \frac{1}{r}(a_1,\ldots,a_m)\bigr)$, or more simply, $\frac{1}{r}(a_1,\ldots,a_m)$. - The Weil divisor class group is denoted by ${\operatorname{Cl}}X$, and $\mathcal O_X(D)$ is the reflexive sheaf of rank one associated to $D \in {\operatorname{Cl}}X$. - Given a family $\mathcal X / (0 \in \Delta)$ and a sheaf $\mathcal E$ over $\mathcal X$, the restriction $\mathcal E\big\vert_{\mathcal X_t}$ is simply denoted by $\mathcal E_t$. - For a smooth projective variety $X$, ${\operatorname{D}^{\sf b}}(X) = {\operatorname{D}^{\sf b}}(\mathop{\bf Coh} X)$ is the bounded derived category of coherent sheaves on $X$. Given a set of objects $A \subset {\operatorname{D}^{\sf b}}(X)$, $\langle A \rangle$ denotes the smallest ${\mathbb{C}}$-linear full triangulated subcategory containing $A$. Exceptional collections {#sec:Exceptional collection} ======================= Let $V$ be a smooth projective variety, and let ${\operatorname{D}^{\sf b}}(V)$ be the (bounded) derived category of coherent sheaves on $V$. We are interested in the decomposition of ${\operatorname{D}^{\sf b}}(V)$ into smaller pieces. One natural attempts is to consider orthogonal decompositions, namely, the decomposition ${\operatorname{D}^{\sf b}}(V) = \langle \mathcal T_1, \mathcal T_2 \rangle$ such that $${\operatorname{Hom}}_{{\operatorname{D}^{\sf b}}(V)}(T_1,T_2) = {\operatorname{Hom}}_{{\operatorname{D}^{\sf b}}(V)}(T_2,T_1) = 0,\quad \text{for all}\ T_1 \in \mathcal T_1\ \text{and}\ T_2 \in \mathcal T_2.$$ However, there is no orthogonal decomposition unless $V$ is disconnected([@Huybrechts:FourierMukai Proposition 3.10]). Discarding the condition ${\operatorname{Hom}}_{{\operatorname{D}^{\sf b}}(V)}(T_1,T_2)=0$, one obtains the notion of semiorthogonal decomposition, which is more interesting than the orthogonal decompositions. The full triangulated subcategories $\mathcal T_1,\mathcal T_2 \subset {\operatorname{D}^{\sf b}}(V)$ form a semiorthogonal decomposition if the following holds: 1. ${\operatorname{D}^{\sf b}}(V) = \langle \, \mathcal T_1,\, \mathcal T_2 \, \rangle$; 2. ${\operatorname{Hom}}_{{\operatorname{D}^{\sf b}}(V)} ( T_2, T_1) = 0$ for any $T_1 \in \mathcal T_1$, $T_2 \in \mathcal T_2$. The category $\mathcal T_2$ is said to be left orthogonal to $\mathcal T_1$, and is denoted by ${}^\perp \mathcal T_1$. Similarly, $\mathcal T_1 = \mathcal T_2^\perp$ is right orthogonal to $\mathcal T_2$. Once the notion of decomposition is established, one needs to find the natural candidates of the components. The simplest possible component would be the derived category ${\operatorname{D}^{\sf b}}({\operatorname{Spec}}{\mathbb{C}})$ of a point. The structure of ${\operatorname{D}^{\sf b}}({\operatorname{Spec}}{\mathbb{C}})$ is rather simple; indeed, every object is isomorphic to a complex of the form $$\bigoplus_p^{\text{finite}} {\mathbb{C}}^{r_p}[-p] = \bigl( \ldots \to {\mathbb{C}}^{r_{p-1}} {\xrightarrow{\,0\,}} {\mathbb{C}}^{r_p} {\xrightarrow{\,0\,}} {\mathbb{C}}^{r_{p+1}} \to \ldots \bigr).$$ Assume that ${\operatorname{D}^{\sf b}}(V)$ contains ${\operatorname{D}^{\sf b}}({\operatorname{Spec}}{\mathbb{C}})$ as a semiorthogonal component. Then, the embedding $\Phi \colon {\operatorname{D}^{\sf b}}({\operatorname{Spec}}{\mathbb{C}}) \hookrightarrow {\operatorname{D}^{\sf b}}(V)$ is determined by the image of the complex $\underline {\mathbb{C}}= (\ldots \to 0 \to {\mathbb{C}}\to 0 \to \ldots)$ supported at degree $0$. The object $E := \Phi(\underline {\mathbb{C}})$ is called an exceptional object. We may reformulate the definition intrinsically as follows. An object $E \in {\operatorname{D}^{\sf b}}(V)$ is exceptional if $${\operatorname{Hom}}_{{\operatorname{D}^{\sf b}}(V)}(E,E[p]) = \left\{ \begin{array}{ll} {\mathbb{C}}& p=0 \\ 0 & p\neq 0 \end{array} \right.{}.$$ Given an exceptional object $E_1$, one may define the semiorthogonal decompositions ${\operatorname{D}^{\sf b}}(V) = \langle E_1, \mathcal A \rangle$, where $\mathcal A = \langle \{ A \in \mathcal {\operatorname{D}^{\sf b}}(V) : {\operatorname{Hom}}(A, E_1[p])=0,\ \forall p \} \rangle$ is the category left orthogonal to $\langle E_1 \rangle$. In some cases, it is possible to find another semiorthogonal complement ${\operatorname{D}^{\sf b}}({\operatorname{Spec}}{\mathbb{C}}) {\xrightarrow{\,\sim\,}} \langle E_2 \rangle \subset \mathcal A$. Repeating this procedure, we have an ordered set of exceptional objects $E_1,\ldots,E_n \in {\operatorname{D}^{\sf b}}(V)$ satisfying the semiorthogonality condition. This suggests the following definition. \[def: exceptional and orthogonal collection\] 1. An ordered set of exceptional objects $E_1,\ldots,E_n \in {\operatorname{D}^{\sf b}}(V)$ is called an exceptional collection if $${\operatorname{Hom}}_{{\operatorname{D}^{\sf b}}(V)} (E_{i+k},E_i[p])=0,\ \text{for every $p$ and $k > 0$}.$$ An exceptional collection $E_1,\ldots,E_n \in {\operatorname{D}^{\sf b}}(V)$ is said to be full if $${\operatorname{D}^{\sf b}}(V) = \langle E_1,\ldots,E_n \rangle.$$ 2. An orthogonal collection is an ordered set $(E_1,\ldots,E_n)$ of exceptional objects that is completely orthogonal, namely, $${\operatorname{Hom}}_{{\operatorname{D}^{\sf b}}(V)} (E_{i+k},E_i[p])=0,\ \text{for each $p$ and $k \neq 0$}.$$ Singularities of class T and their ${\mathbb{Q}}$-Gorenstein deformations ========================================================================= Let $X \subset {\mathbb P}^N$ be a complex surface, and let $P \in X$ be a normal singularity. We may take a sufficiently small ball $B_{\mathbb P}\subset {\mathbb P}^N $ at $P$. Let $B = B_{\mathbb P}\cap X$. The boundary $L$ of $B$ is called the link of the singularity $(P \in X)$. The link is known to capture various topological nature of $(P \in X)$ \[thm: Mumford TopologyNSS\] Let $\pi \colon Y \to (P \in X)$ be a resolution of $(P \in X)$. Assume that the exceptional divisor $E_1\cup \ldots\cup E_r$ of $\pi$ is simple normal crossing. Let $\alpha_i$ be a loop around $E_i$ oriented by $\displaystyle\int_{\alpha_i} f_i^{-1} df_i = 2\pi \sqrt{-1}$, where $(f_i=0)$ is a local defining equation of $E_i$. Then, the $H_1(L;{\mathbb{Z}})$ has the group presentation: $$H_1(L;{\mathbb{Z}}) = \bigl\langle \alpha_1,\ldots,\alpha_r : \sum_{j=1}^r (E_i.E_j) \alpha_j = 0,\ \forall i=1,\ldots,r\bigr\rangle.$$ The Mayer-Vietoris sequence associated to $X = (X \setminus B) \cup B$ defines the map $H_2(X;{\mathbb{Z}}) \to H_1(L;{\mathbb{Z}})$. Composition with the cycle class map $\op{Cl}X \to H_2(X;{\mathbb{Z}})$ defines $$\label{eq: Div Cycle map} \gamma \colon \op{Cl}X \to H_1(L;{\mathbb{Z}}),\quad D \mapsto \sum_j ( D' . E_j) \alpha_j.$$ where $D' \in {\operatorname{Cl}}Y$ is a divisor such that $\pi_* D' = D$. Thanks to Theorem \[thm: Mumford TopologyNSS\], the right hand side of (\[eq: Div Cycle map\]) does not depend on the choice of $D'$. For instance, if one replaces $D'$ by $D'+E_1$, then $$\begin{aligned} \sum_j ( D' + E_1 \mathbin. E_j) \alpha_j &= \sum_j ( D' . E_j) \alpha_j + \sum_j (E_1.E_j) \alpha_j \\ &= \sum_j (D' . E_j)\alpha_j. \end{aligned}$$ The above observation leads to the following proposition. \[prop: divisors having same description in the link\] Let $D' \in {\operatorname{Cl}}Y$ and $D := \pi_* D' \in {\operatorname{Cl}}X$. There is a one-to-one correspondence $$\bigl\{ D' + \sum_j a_j E_j \in {\operatorname{Cl}}Y \mathrel: a_1,\ldots,a_r \in {\mathbb{Z}}\bigr\} \leftrightarrow \bigl\{ (k_1,\ldots,k_r) \in {\mathbb{Z}}^r : \gamma(D) = \sum_j k_j \alpha_j \bigr\}$$ given by $D'' \mapsto \bigl( (D'' . E_1),\, \ldots, \, (D'' . E_r) \bigr)$. Let $D'' = D' + \sum_j a_j E_j$ and define the column vectors $\mathbf d''$, $\mathbf d'$, $\mathbf k$, $\mathbf a$ having entries as $(D''.E_i)$, $(D'.E_i)$, $k_i$, $a_i$, respectively. Let $A = \bigl( (E_i.E_j) \bigr)_{i,j}$ be the intersection matrix. Then, we have $$\mathbf{d''} = \mathbf{d'} + A \mathbf{a}.$$ Now, bijectivity follows since $A$ is negative definite(see [@Badescu:Surfaces Corollary 2.7]). For a cyclic quotient singularity $\frac1m(1,q)$, the exceptional locus of the minimal resolution forms a chain of smooth rational curves. Let +0.33 /in [1/1,2/2,3/3,4/r]{}[ (A) at (,0); ]{} be the dual intersection graph of the exceptional curves, and let $b_i = -(E_i.E_i) > 0$. The numbers $m,q$ are recovered from the Hirzebruch-Jung continued fraction $$\frac{m}{q} = [b_1,\ldots,b_r] := b_1 - \frac{1}{b_2 - \frac{1}{\ldots - \frac{1}{b_r} }}.$$ \[prop: Link algebra\] Let $A(b_1,\ldots,b_r)$ be the matrix $$\left[ \begin{array}{cccccc} b_1 & -1 & 0 & \ldots & 0 & 0 \\ -1 & b_2 & -1 &\ldots & 0 & 0 \\ \multicolumn{3}{c}{\vdots} & \ddots & \multicolumn{2}{c}{\vdots} \\ 0 & 0 & 0 &\ldots & b_{r-1} & -1 \\ 0 & 0 & 0 &\ldots & -1 & b_r \\ \end{array} \right],$$ and let $G(b_1,\ldots,b_r)$ be the group generated by the symbols $\alpha_1,\ldots,\alpha_r$, subject to the relations $$A(b_1,\ldots,b_r) \left[ \begin{array}{c} \alpha_1 \\ \vdots \\ \alpha_r \end{array} \right]=0.$$ Then, $\alpha_k = \det A(b_1,\ldots,b_{k-1}) \cdot \alpha_1$ for each $k=2,\ldots,r$. Furthermore, $$m = \det A(b_1,\ldots,b_r),\quad q = \det A(b_2,\ldots,b_r).$$ We have a recurrence formula $$A(b_1,\ldots,b_k) = b_k A(b_1,\ldots,b_{k-1}) - A(b_1,\ldots,b_{k-2}),\quad k \geq 2,$$ when we set $\det A({\varnothing}) = 1$ conventionally. Let $n_k \in {\mathbb{Z}}_{>0}$ be the smallest positive integer that fits into the formula $\alpha_k = n_k \cdot \alpha_1$. The relations of $G(b_1,\ldots,b_r)$ reads $n_k = b_{k-1} n_{k-1} - n_{k-2}$. Since $n_1 = 1 = A({\varnothing})$ and $n_2 = b_1 = A(b_1)$, we get $n_k = A(b_1,\ldots,b_{k-1}) $. To prove the remaining identities, let $\alpha_k,\beta_k>0$ be the relatively prime integers such that $$\frac{\alpha_k}{\beta_k} = [b_k,\ldots,b_1].$$ Then, $\beta_{k+1} = \alpha_k$ and $\alpha_{k+1} = b_{k+1}\alpha_k - \alpha_{k-1}$. It follows that $\alpha_k = n_{k+1} = \det A(b_1,\ldots,b_k)$. This shows $m = \det A(b_1,\ldots,b_r)$ and $q' = \det A(b_1,\ldots, b_{r-1})$, where $qq' \equiv 1 {\ \operatorname{mod\,}}m$. Applying the same argument to $[b_1,\ldots,b_k] = \gamma_k/\delta_k$, one finds $q = \det A(b_2,\ldots,b_r)$. Our main interest is the cyclic quotient singularities having special forms, namely, when $$\label{eq: T Sing} (m,q) = (dn^2,\, dna-1)\ \text{where}\ d,n,a \in {\mathbb{Z}}_{>0}\ \text{with}\ \op{gcd}(n,a)=1.$$ These are the cyclic quotient singularities which admit ${\mathbb{Q}}$-Gorenstein smoothing. \[def: QGorDef\] Let $X$ be a normal surface such that $K_X$ is ${\mathbb{Q}}$-Cartier, and let $\mathcal X / (0 \in \Delta)$ be a deformation of $X$. The deformation $\mathcal X / (0 \in \Delta)$ is said to be ${\mathbb{Q}}$-Gorenstein if it is locally equivariant with respect to the deformation of an index one cover. More precisely, for each singular point $P \in X$, there exists a neighborhood $P \in \mathcal V \subset \mathcal X$ which satisfies the following: the index one cover $Z \to \mathcal V_0 := \mathcal V \cap X$ admits a deformation $\mathcal Z / (0 \in \Delta)$ to which the action $Z \to \mathcal V_0$ extends and the corresponding quotient is $\mathcal Z / (0 \in \Delta) \to \mathcal V / (0 \in \Delta)$. Let $u,v \in {\mathbb{C}}^2 / \frac{1}{dn^2}(1,dna-1)$ be the orbifold coordinates. After the base change $x=u^{dn}$, $y=v^{dn}$, $z = uv$, we get $$(0 \in (xy=z^{dn})) \subset {\mathbb{C}}^3_{x,y,z} \Big/ \frac{1}{n}(1,-1,a)$$ The index one cover is simply obtained by replacing the ambient space by ${\mathbb{C}}^3$, hence is an $A_{dn-1}$ singularity. The versal deformation of it is $$(0 \in (xy = z^{dn} + c_{dn-2} z^{dn-2} + \ldots + c_1 z + c_0 ) \subset {\mathbb{C}}_{x,y,z}^3 \times \Delta^{dn-1}_{\underline c}.$$ Thus, the equivariant deformation should be given by gathering the $z^n$ term and its powers: $$\label{eq: versal Q-Gor} \mathcal X^{\sf ver} := (xy = z^{dn} + t_{d-1} z^{(d-1)n} + \ldots + t_1 z^n + t_0 ) \subset {\mathbb{C}}_{x,y,z}^3 \Big/ \frac1n(1,-1,a) \times \Delta^{d}_{\underline t}$$ The deformation $(0 \in \mathcal X^{\sf ver}) / (0 \in \Delta^d_{\underline t})$ is called the versal ${\mathbb{Q}}$-Gorenstein deformation of $\frac{1}{dn^2}(1,dna-1)$. Indeed, every ${\mathbb{Q}}$-Gorenstein deformation of $\frac{1}{dn^2}(1,dna-1)$ is a pullback of the versal ${\mathbb{Q}}$-Gorenstein deformation. Let $(P \in \mathcal X) / (0 \in \Delta)$ be a one parameter ${\mathbb{Q}}$-Gorenstein smoothing of $(P \in X) \simeq \frac{1}{dn^2}(1,dna-1)$. Consider a small 3-dimensional complex ball $\mathcal B \subset \mathcal X$ at $P$. Then the slice over the general fiber, namely, $\mathcal B_t := \mathcal B \cap \mathcal X_t$ for general $t \in \Delta$, is called the Milnor fiber of the smoothing. Since the complement of the Milnor fiber is homeomorphic to $X^\circ := X \setminus \mathcal B_0$, one can construct the underlying topological space of $\mathcal X_t$ by a topological surgery illustrated in Figure \[fig: Topological Surgery\]. node at (0,3); [ (90+45:40pt and 20pt) arc (90+45:360+90-45:40pt and 20pt); (0,0) ellipse \[x radius=40pt, y radius=4pt\]; ]{} (CoordTempA) at (90+45:40pt and 20pt); (CoordTempB) at (360+90-45:40pt and 20pt); [ (, ) arc (180:360: pt and 3pt ); (, ) arc (90+45:90-45: 40 pt and 10pt ); ]{} [ (90+45:40pt and 20pt) arc (90+45:360+90-45:40pt and 20pt); (0,0) ellipse \[x radius=40pt, y radius=4pt\]; ]{} [ [ (-pt,) arc (180:360: pt and 3pt ); (-pt,) arc (90+45:90-45: 40 pt and 10pt ); ]{} (SingularP) at ($(0, 0.35) + (0,\dimHeight)$); (-pt,) .. controls ++(75:0.25) and ++(90+45:0.5) .. (SingularP) .. controls ++(90-45:0.5) and ++(180-75:0.25) .. (pt, ); (SingularP) circle (1pt); ]{} [ (90+45:40pt and 20pt) arc (90+45:360+90-45:40pt and 20pt); (0,0) ellipse \[x radius=40pt, y radius=4pt\]; ]{} [ [ (-pt,) arc (180:360: pt and 3pt ); (-pt,) arc (90+45:90-45: 40 pt and 10pt ); ]{} (SingularP) at ($(0, 0.35) + (0,\dimHeight)$); (-pt,) .. controls ++ (60:0.75) and ++(180-60:0.75) .. (pt, ); ($(0,1.66\dimHeight)+(135: 8pt and 2pt)$) arc (135:360+45: 8pt and 2pt); (0,1.5\) ellipse \[x radius = 6pt, y radius=2pt\]; ; ]{} [ [ (-pt,) arc (180:360: pt and 3pt ); (-pt,) arc (90+45:90-45: 40 pt and 10pt ); ]{} (SingularP) at ($(0, 0.35) + (0,\dimHeight)$); (-pt,) .. controls ++(75:0.25) and ++(90+45:0.5) .. (SingularP) .. controls ++(90-45:0.5) and ++(180-75:0.25) .. (pt, ); (SingularP) circle (1pt); ]{} [ [ (-pt,) arc (180:360: pt and 3pt ); (-pt,) arc (90+45:90-45: 40 pt and 10pt ); ]{} (SingularP) at ($(0, 0.35) + (0,\dimHeight)$); (-pt,) .. controls ++ (60:0.75) and ++(180-60:0.75) .. (pt, ); ($(0,1.66\dimHeight)+(135: 8pt and 2pt)$) arc (135:360+45: 8pt and 2pt); (0,1.5\) ellipse \[x radius = 6pt, y radius=2pt\]; ; ]{} (-0.8,0.8) – ++(180-30:1.6) node \[above, midway,rotate=-30\] [remove $\mathcal B_0$]{}; (0.8,0.8) – ++(30:1.6) node \[above, midway,rotate=30\] [replace by $\mathcal B_t$]{}; (-4,0) – (-2,0); (2,0) – (4,0); node\[anchor=north\] at (-6,-1) [$ X = X^\circ \underset{L}{\cup} \mathcal B_0$]{}; node\[anchor=north\] at (0,-1) [$ X^\circ,\ L = \partial X^\circ$]{}; node\[anchor=north\] at (6,-1) [$ \mathcal X_t = X^\circ \underset{L}{\cup} \mathcal B_t$]{}; node\[red, anchor=south\] at (0,.55) [$L$]{}; The ball $\mathcal B_0$ is contractible, as it is the image of a contractible ball along ${\mathbb{C}}^2 \to {\mathbb{C}}^2/\frac{1}{dn^2}(1,dna-1)$. Thus, $H_2(X,\mathcal B_0\,;{\mathbb{Z}}) \simeq H_2(X;{\mathbb{Z}})$. Moreover, using excision principle, we see $$H_2(\mathcal X_t, \mathcal B_t\,;{\mathbb{Z}}) \simeq H_2(X^\circ, L \,; {\mathbb{Z}}) \simeq H_2(X, \mathcal B_0\,;{\mathbb{Z}}).$$ Consequently, the relative homology sequence of the pair $(\mathcal X_t, \mathcal B_t)$ reads $$\label{eq: Relative Homology Seq} \ldots \to H_2(\mathcal B_t;{\mathbb{Z}}) \to H_2(\mathcal X_t;{\mathbb{Z}}) {\xrightarrow{\,\rm sp\,}} H_2(X;{\mathbb{Z}}) {\xrightarrow{\,\delta\,}} H_1(\mathcal B_t;{\mathbb{Z}}) \to \ldots$$ For a ${\mathbb{Q}}$-Gorenstein smoothing of $\frac{1}{dn^2}(1,dna-1)$, we have $H_2(\mathcal B_t) \simeq {\mathbb{Z}}^{d-1}$ and $H_1(\mathcal B_t) \simeq {\mathbb{Z}}/n{\mathbb{Z}}$(see for instance, [@Manetti:NormalDegenerationOfPlane Proposition 13]). A cycle $\alpha \in H_2(\mathcal X_t;{\mathbb{Z}})$ specializes to $X$ via the map $\mathrm{sp}$, and a cycle $\beta \in H_2(X;{\mathbb{Z}})$ lifts to the general fiber if and only if the image under $H_2(X;{\mathbb{Z}}) \to H_1(\mathcal B_t;{\mathbb{Z}})$ vanishes. Wahl degeneration and exceptional bundles ========================================= Throughout this section, we consider the following situation: $X$ is a projective normal surface with rational singularities at worst, $H^1(\mathcal O_X)=H^2(\mathcal O_X)=0$, and $(P \in X) \simeq \frac{1}{n^2}(1,na-1)$ for $n>a>0$ with $\op{gcd}(n,a)=1$. The latter one is a particular case among singularities of class $T$, which is called a Wahl singularity. Also, we assume that there exists a ${\mathbb{Q}}$-Gorenstein smoothing $\mathcal X / (0 \in \Delta)$. Under these assumptions, the cycle class map $c_1 \colon {\operatorname{Cl}}X \to H_2(X;{\mathbb{Z}})$ is an isomorphism(cf. [@Kollar:Seifert Proposition 4.11]). The map $\delta$ in (\[eq: Relative Homology Seq\]) can be regarded as ${\operatorname{Cl}}X \to H_1(\mathcal B_t;{\mathbb{Z}})$. It factors through $\gamma \colon {\operatorname{Cl}}X \to H_1(L;{\mathbb{Z}})$ in (\[eq: Div Cycle map\]), followed by the map $H_1(L;{\mathbb{Z}}) \to H_1(\mathcal B_t;{\mathbb{Z}})$ induced by the inclusion. Since the latter map can be identified with ${\mathbb{Z}}/n^2{\mathbb{Z}}\twoheadrightarrow {\mathbb{Z}}/n{\mathbb{Z}}$, (\[eq: Relative Homology Seq\]) implies the following. $$\label{eq: Lifting criterion} \text{A divisor $D_0 \in {\operatorname{Cl}}X$ lifts to $D_t \in {\operatorname{Cl}}\mathcal X_t$ if and only if $\gamma(D_0)$ is divisible by $n$.}$$ Thanks to Theorem \[thm: Mumford TopologyNSS\], $\gamma(D_0)$ can be computed if we know how the exceptional curves of the minimal resolution $(E \subset Y) \to (P \in X)$ intersect with the proper transform of $D_0$. However, the sequence (\[eq: Relative Homology Seq\]) is originated from the topological surgery(Figure \[fig: Topological Surgery\]), hence a priori there is no reason for the existence of “geometric” deformation from $D_0$ to $D_t$. The following theorem provides a geometric interpretation via exceptional vector bundles. \[thm: Hacking\] Let $X$, $(P \in X)\simeq \frac{1}{n^2}(1,na-1)$, $\mathcal X / (0 \in \Delta)$ as above. Suppose there exists a divisor $D_0 \in \op{Cl} X$ satisfying 1. $D_0$ is Cartier except at $P$; 2. for a resolution $Y \to (P \in X)$ with the chain $E_1 \cup \ldots \cup E_r$ of exceptional curves, $\gamma(D_0) = \alpha_1$ where $\alpha_i$ is the generator of $H_1(L;{\mathbb{Z}})$ corresponding to $E_i$. Then, after the base change $(0 \in \Delta') \to (0 \in \Delta)$, $t \mapsto t^a$, there is a reflexive sheaf $\mathcal E$ over $\mathcal X' := \mathcal X \times_{\Delta} \Delta'$, locally free except at $P$, such that 1. $(\mathcal E_0)^{\vee\vee} = \mathcal O_X(D_0)^{\oplus n}$; 2. for general $t \in \Delta$, $\mathcal E_t$ is an exceptional vector bundle of rank $n$; 3. $\op{sp}(c_1(\mathcal E_t)) = c_1(nD_0)$; 4. if $\mathcal H/(0 \in \Delta)$ is a relatively ample divisor in $\mathcal X' / ( 0 \in \Delta')$, then $\mathcal E_t$ is slope stable with respect to $\mathcal H_t'$. This theorem is a key ingredient of this paper. To derived proper variations, we need to introduce how to construct $\mathcal E$. First of all, we may assume that $(P \in \mathcal X) / ( 0 \in \Delta)$ is a versal ${\mathbb{Q}}$-Gorenstein smoothing of $\frac{1}{n^2}(1,na-1)$, namely, it is locally isomorphic to $$(xy = z^n + t ) \subset {\mathbb{C}}^3_{x,y,z} \Big / \frac{1}{n}(1,-1,a) \times \Delta_t.$$ After the base change, $\mathcal X'$ is the deformation $$\label{eq: QGor Smoothing after Base change} (xy = z^n + t^a) \subset {\mathbb{C}}^3_{x,y,z} \Big / \frac 1n (1,-1,a) \times \Delta_t.$$ Now we consider a weighted blow up as follows. First, embed the ambient space into the affine toric variety ${\mathbb{C}}^4_{x,y,z,t} / \frac 1n (1,-1,a,n)$. The corresponding fan $\Sigma$ is the orthant spanned by the rays ${\mathbb{R}}_{>0} \cdot e_1,\ldots, {\mathbb{R}}_{>0} \cdot e_4$ in the lattice $N := {\mathbb{Z}}^4 + {\mathbb{Z}}\cdot \frac1n(1,-1,a,n)$. The weighted blow up is given by adding the ray ${\mathbb{R}}_{>0} \cdot (1,na-1,a,n)$(and the subsequent subdivision of cones). Since $\mathcal X' \subset {\mathbb{C}}^4 / \frac 1n (1,-1,a,n)$, we may consider the proper transform of $\mathcal X'$ along this weighted blow up. Let us denote it by $\tilde{\mathcal X}$. The exceptional divisor of $\tilde{\mathcal X}_0 \to \mathcal X_0'$ is isomorphic to $$W:= (XY = Z^n + T^a) \subset {\mathbb P}(1,na-1,a,n).$$ Moreover, the central fiber $\tilde{\mathcal X}_0$ is the union of $\tilde X_0$(the proper transform of $X$), and $W$ where these components intersect scheme-theoretically along the smooth rational curve $C:=(T=0) \subset W$. On the other hand, the component $\tilde X_0$ is a partial resolution of $X$, having an irreducible exceptional locus that corresponds to $E_1$ in the resolution(see Figure \[fig: tilde X\_0\]). The hypotheses on $D_0$ ensures the existence of $\tilde D_0 \in {\operatorname{Pic}}\tilde X_0$ with $(\tilde D_0 . C)=1$(Remark \[rmk: finding desired divisor on partial resolution\]). node at (0,4); (posX) at (-2,-3); (posPartialRes) at (2.25,0.5); (posMinRes) at (-3,1.5); node at (-22.5:2) [$X$]{}; [ ($(0,0)!-0.5!($(75:1.85) + (3,0)$)$) – ++(75:1.85) – ++(3,0) – ++($(0,0)-(75:1.85)$) – cycle; ]{} (0,0) circle (1.5pt); node\[anchor=north\] at (0,0) [$\scriptstyle P$]{}; node at (-22.5:1.75) [$\tilde X_0$]{}; [ ($(0,0)!-0.5!($(75:1.85) + (3,0)$)$) – ++(75:1.85) – ++(16:1.7) – ++($(3,0)-(16:1.7)$) – ++($(0,0)-(75:1.85)$) – ++($(0,0)-(16:1.7)$) – cycle; ]{} (75:1) – (75:-1); node\[anchor=north east\] at (75:1) [$\scriptstyle C$]{}; in [0,1,2]{}[ (:2pt) – (:-2pt); ]{} node\[anchor=west\] at (75:-0.45) [$\scriptstyle Q$]{}; (0,-1) .. controls ++(180+20:0.7) and ++(270-45:0.33) .. (180+40:0.75) .. controls ++(270-45:-0.4) and ++(15:-0.25) .. (0.45,0) .. controls ++(15:1) and ++(180+75:-0.35) .. (45:0.75) .. controls ++(180+75:0.35) and ++(180+25:0.66) .. (1.5,0.65); node\[anchor=east\] at (180+40:0.75) [$\scriptstyle \tilde D_0$]{}; (Anc1) at (1,2); (Anc2) at ($(Anc1) + (180+45:1.75)$); (Anc3) at ($(Anc2) + (270+30:1.25)$); (Anc4) at ($(Anc3) + (180+10:1.65)$); (Anc5) at ($(Anc4) + (90+15:2.05)$); ($(Anc2)!1.15!(Anc1)$) – ($(Anc1)!1.15!(Anc2)$); in [2,4]{}[ ($(Anc\m)!1.15!(Anc\n)$) – ($(Anc\n)!1.15!(Anc\m)$); ]{} ($(Anc3)!0.5!(Anc4)$) circle (15pt) \[insert path=[(3,3) – (3,-3) – (-3,-3) – (-3,3) – (3,3)]{}\]; ($(Anc4)!1.15!(Anc3)$) – ($(Anc3)!1.15!(Anc4)$); node\[shift=[(-0.25,0.25)]{}\] at ($(Anc1)!0.5!(Anc2)$) [$\scriptstyle E_1$]{}; node\[shift=[(0.32,0.06)]{}\] at ($(Anc2)!0.5!(Anc3)$) [$\scriptstyle E_2$]{}; node at ($(Anc3)!0.5!(Anc4)$) [$\ldots$]{}; node\[shift=[(-0.24,0)]{}\] at ($(Anc4)!0.5!(Anc5)$) [$\scriptstyle E_r$]{}; ($(posMinRes)!0.2!(posX)$) – node\[anchor=east\] ------------ minimal resolution ------------ ($(posMinRes)!0.68!(posX)$); ($(posPartialRes)!0.33!(posX)$) – node\[anchor=north west\] [$C \mapsto \{P\}$]{} ($(posPartialRes)!0.6!(posX)$); ($(posMinRes)!0.25!(posPartialRes)$) – node\[anchor=south,sloped\] --------------------------------------- $E_1 \mapsto C$ $E_2\cup\ldots\cup E_r \mapsto \{Q\}$ --------------------------------------- ($(posMinRes)!0.66!(posPartialRes)$); Then in the central fiber $\tilde{\mathcal X}_0 = \tilde X_0 \cup_C W$, one may construct an exceptional vector bundle by glueing $\mathcal O(\tilde D_0)^{\oplus n}$ and a certain exceptional vector bundle $G_W$(called “localized exceptional bundle” in [@Hacking:ExceptionalVectorBundle 5]) on $W$. In particular, the outcome fits into the exact sequence $$0 \to \tilde{\mathcal E}_0 \to \mathcal O_{\tilde X_0}(\tilde D_0)^{\oplus n} \oplus G_W \to \mathcal O_C(1)^{\oplus n} \to 0$$ Using standard arguments in deformation theory of vector bundles, $\tilde{\mathcal E}_0$ extends to a locally free sheaf $\tilde{\mathcal E}$ on $\tilde{\mathcal X}$. The sheaf $\mathcal E$ in the statement is the reflexive hull of the pushforward along $\tilde{\mathcal X} \to \mathcal X'$. \[rmk: finding desired divisor on partial resolution\] Let $D_Y$ be the proper transform of $D_0$ in the resolution $Y \to (P \in X)$. By Proposition \[prop: divisors having same description in the link\] and $\gamma(D_0) = \alpha_1$, there are integers $a_1,\ldots, a_r$ such that $D' := D_Y + a_1 E_1 + \ldots + a_r E_r$ satisfies $(D'.E_1)=1$ and $(D'.E_2)=\ldots = (D'. E_r)=0$. Now, the divisor $\tilde D_0$ is the pushforward of $D'$ along $Y \to \tilde X_0$. More generally, suppose $M_0 \in {\operatorname{Cl}}X$ is a divisor which is Cartier except at $P$ and $\gamma(M_0) = k \alpha_1$. Then, there exists $\tilde M_0 \in {\operatorname{Pic}}\tilde X_0$ such that $(\tilde M_0 . C) = k$. On the other hand, the same technique can be applied to construct line bundles. \[lem: O\_W(na-1)\] Let $\mathcal O_W(na-1)$ be the restriction of $\mathcal O_{{\mathbb P}(1,na-1,a,n)}(na-1)$. Then $\mathcal O_W(na-1)$ is invertible. Let $R$ be the weighted homogeneous coordinate ring of $W$: $$R = {\mathbb{C}}[X,Y,Z,T] / (XY - Z^n - T^a),\quad \deg(X,Y,Z,T) = (1,na-1,a,n).$$ Since $W$ is covered by $(X\neq0) \cup (Y \neq 0) \cup (ZT \neq 0)$, it suffices to show that for each $\mathbf{x} \in \{X,Y,ZT\}$, $$R(na-1)_{(\mathbf{x})} = \Bigl\{ \frac{f}{\mathbf{x}^d} : \deg f - d \deg \mathbf{x} = na-1 \Bigr\}$$ is a free $R_{(\mathbf{x})}$-module of rank one. Indeed, $$R(na-1)_{(X)} \simeq R_{(X)} \cdot X^{na-1},\quad R(na-1)_{(Y)} \simeq R_{(Y)} \cdot Y,\quad R(na-1)_{(ZT)} \simeq R_{(ZT)} \cdot (Z^pT^q)^{na-1}$$ where $p,q$ are integers with $pa+qn=1$. By the intersection theory of weighted projective spaces, one can prove that $\mathcal O_W(na-1)\big\vert_C = \mathcal O_C(n)$. Hence, we may glue $\mathcal O_{\tilde X_0}( n \tilde D_0)$ and $\mathcal O_W(na-1)$ along $\mathcal O_C(n)$ to obtain a line bundle $\tilde {\mathcal L}_0$ on $\tilde{\mathcal X}_0$. It extends to a line bundle $\tilde {\mathcal L}$ on $\tilde{\mathcal X}$. This construction can be generalized to the surfaces with several Wahl singularities. \[prop: Hacking L.B.\] Let $X$ be a projective normal surface with $H^1(\mathcal O_X)=H^2(\mathcal O_X)=0$. Let $M_0 \in {\operatorname{Cl}}X$ be a divisor such that 1. $M_0$ is Cartier except at Wahl singularites $(Q_i \in X) \simeq \frac{1}{n_i^2}(1,na-1)$, $i=1,\ldots,s$; 2. $\gamma_{Q_i}(M_0) \in n_i \cdot H_1(L_i;{\mathbb{Z}})$ where $L_i$ is the link of $(Q_i \in X)$ and $\gamma_{Q_i} \colon {\operatorname{Cl}}X \to H_1(L_i;{\mathbb{Z}})$ is the map in (\[eq: Div Cycle map\]). Let $\mathcal X / (0 \in \Delta)$ be a ${\mathbb{Q}}$-Gorenstein smoothing of $X$. Then, after a finite base change $(0 \in \Delta') \to (0 \in \Delta)$, there exists a reflexive sheaf $\mathcal M$ of rank $1$ over $\mathcal X' := \mathcal X \times_{\Delta} \Delta'$ such that 1. $(\mathcal M_0)^{\vee\vee} = \mathcal O_X(M_0)$; 2. $\mathcal M_t$ is a line bundle on $\mathcal X_t'$; 3. $\op{sp}(c_1(\mathcal M_t)) = c_1(\mathcal M_0)$. We make the base change $(0 \in \Delta') \to (0 \in \Delta)$ such that the ramification index is $a_i$ locally at each $(Q_i \in \mathcal X) / (0 \in \Delta)$. Let $\Phi \colon \tilde{\mathcal X} \to \mathcal X'$ be the proper birational morphism that is obtained by respective weighted blow ups at each $(Q_i \in \mathcal X')$, and let $W_1,\ldots,W_s$ be the corresponding exceptional divisors. Then, the central fiber has the irreducible decomposition $\tilde{\mathcal X}_0 = \tilde X_0 \cup W_1 \cup \ldots \cup W_s$ where $\tilde X_0$ is the proper transform of $X$. Then, by Lemma \[lem: O\_W(na-1)\], we have the following exact sequence of sheaves $$0 \to \tilde{\mathcal M}_0 \to \mathcal O_{\tilde X_0}(\tilde M_0) \oplus \mathcal O_{W_1}(k_1(n_1a_1-1)) \oplus \ldots \oplus \mathcal O_{W_s}(k_s(n_sa_s-1)) \to \mathcal O_{C_1}(k_1n_1) \oplus \ldots \oplus \mathcal O_{C_s}(k_sn_s) \to 0,$$ where $\tilde M_0 \in {\operatorname{Pic}}\tilde X_0$ satisfies $(M_0.C_i) = k_i n_i$ (Remark \[rmk: finding desired divisor on partial resolution\]), and $C_i = \tilde X_0 \cap W_i$. It can be shown that $\tilde{\mathcal M}_0$ is an exceptional line bundle, hence it deforms to $\tilde{\mathcal M} \in {\operatorname{Pic}}\tilde{\mathcal X}$ whose restriction to the general fiber is a line bundle. Combining Theorem \[thm: Hacking\] and Propositin \[prop: Hacking L.B.\], we get the following corollary, which is suitable for our purpose. \[cor: Hacking V.B.\] Let $X$ be a normal projective surface with $H^1(\mathcal O_X) = H^2(\mathcal O_X)=0$. Let $P,Q_1,\ldots,Q_s \in X$ be Wahl singularities of respective indices $(n,a),\, (n_1,a_1),\,\ldots,\,(n_s,a_s)$, and let $L_$ (resp. $\gamma_$) be the link (resp. the map $\gamma$ in (\[eq: Div Cycle map\])) of $* \in \{P,Q_1,\ldots,Q_s\}$. Assume there exists a divisor $D_0 \in {\operatorname{Cl}}X$ such that 1. $D_0$ is Cartier except at $\{ P,Q_1,\ldots,Q_s\}$; 2. for a resolution $(E_1 \cup \ldots \cup E_r \subset Y) \to (P \in X)$ of the singularity, $\gamma_P(D_0) = \alpha_1$ where $\alpha_1$ is the generator corresponding to $E_1$. 3. $\gamma_{Q_i}(D_0) \in n_i \cdot H_1(L_{Q_i};{\mathbb{Z}})$. Let $\mathcal X / (0 \in \Delta)$ be a one parameter ${\mathbb{Q}}$-Gorenstein smoothing. Then, after a finite base change $(0 \in \Delta') \to (0 \in \Delta)$, there exists a reflexive sheaf $\mathcal E$ over $\mathcal X' = \mathcal X \times_{\Delta} \Delta'$, which is locally free over $\mathcal X ' \setminus\{P,Q_1,\ldots,Q_s\}$, satisfying the statements (a–d) in Theorem \[thm: Hacking\] Let $W,W_1,\ldots,W_s$ be the exceptional divisors of the weighted blow up over $P,Q_1,\ldots,Q_s$, respectively, and let $C:=\tilde X_0 \cap W$, $C_i := \tilde X_0 \cap W_i$. We may find $\tilde D_0 \in {\operatorname{Pic}}\tilde X_0$ such that $(\tilde D_0.C)=1$ and $(\tilde D_0 . C_i) = k_i n_i$. Then $\tilde{\mathcal E}_0$ can be obtained by glueing $$\mathcal O_{\tilde X_0}(\tilde D_0)^{\oplus n},\ G_W,\ \mathcal O_{W_1}(k_1(n_1a_1-1))^{\oplus n},\ \ldots,\ \mathcal O_{W_s}(k_s(n_sa_s-1))^{\oplus n}$$ The remaining part is identical to the corresponding part in the proof of Theorem \[thm: Hacking\] or of Proposition \[prop: Hacking L.B.\]. \[rmk: comparing intersection theories\] Assume further that $\op{Sing} X = \{P,Q_1,\ldots,Q_s\}$(that is, no other singularities) in the above corollary. Then, for any line bundle $\mathcal M_t$ on the general fiber, there exists $N \gg 0$ such that $\mathcal M_t^{\otimes N}$ extends to a line bundle over the whole family, i.e. there exists $\mathcal L \in {\operatorname{Pic}}\mathcal X'$ such that $\mathcal L_t = \mathcal M_t^{\otimes N}$. For instance, let $N := n^2 \prod_{i=1}^s n_i^2$, then for any $M_0 \in {\operatorname{Cl}}X$, $\gamma_P(NM_0) = 0$ and $\gamma_{Q_i}(NM_0) = 0\ \forall i$. In particular, $NM_0 \in {\operatorname{Pic}}X$. Proposition \[prop: Hacking L.B.\] yields a reflexive sheaf $\mathcal M$ over $\mathcal X'$ with $\mathcal M_0 = \mathcal O_X(NM_0)$. From the construction, it is obvious that $\mathcal M$ is locally free. One important consequence of this observation is the following: the specialization map induces the isomorphism $H_2(\mathcal X_t';{\mathbb{Q}}) \to H_2(\mathcal X;{\mathbb{Q}})$ preserving the intersection pairing. In \[sec: Main\], we will see that ${\mathbb{Q}}$-Gorenstein smoothing of $\frac{1}{dn^2}(1,dna-1)$ does not have such a property. Orthogonal collections from degenerations {#sec: Main} ========================================= Local analysis {#subsec: Main_ Local analysis} -------------- Let $(P \in X) \simeq \frac{1}{dn^2}(1,dna-1)$ be a singularity of class T. It corresponds to the affine toric variety that is defined by the cone $\sigma \subset N = {\mathbb{Z}}^2$ bounded by two rays $\rho_0 = {\mathbb{R}}_{>0} \cdot (0,1)$ and $\rho_d = {\mathbb{R}}_{>0} \cdot (dn^2,1-dna)$. Let $\Sigma_1$ be the fan which is obtained by subdividing the cone $\sigma$ via $\rho_1 = {\mathbb{R}}_{>0} \cdot (n^2,1-na)$. Then the cones $\sigma_1 := \op{Conv}(\rho_0,\rho_1)$ and $\sigma_2 := \op{Conv}(\rho_1,\rho_d)$ correspond to the singularities $\frac{1}{n^2}(1,na-1)$ and $\frac{1}{(d-1)n^2}(1,(d-1)na-1)$, respectively. A subsequent subdivision by $\{\rho_k = {\mathbb{R}}_{>0}\cdot(kn^2, 1-kna)\}_{k=2,\ldots,d-1}$ gives the fan $\Sigma$ whose associated toric variety has $d \times \frac{1}{n^2}(1,na-1)$. in [0,1]{} (rho0 ) at (0,1); (rhod ) at (4,-3); (0,0) – (rho0 ); (0,0) – (rhod ); node\[anchor=south\] at (rho0 ) [$\scriptstyle (0,1)$]{}; node\[anchor=north\] at (rhod ) [$\scriptstyle (dn^2,1-dna)$]{}; (rho0 ) – (rhod ); in [1]{}[ ; (rho-1) at ($(rho0 1)!\m!(rhod 1)$); (ScopeShift) – (rho-1); ]{} (Cone 0) at ($(rho0 0)!0.38!(rhod 0)$); node\[anchor=south west\] at (Cone 0) [$\scriptstyle \frac{1}{dn^2}(1,dna-1)$]{}; (Cone 1-1) at ($(rho0 1)!0.165!(rhod 1)$); node\[anchor=south west\] at (Cone 1-1) [$\scriptstyle \frac{1}{n^2}(1,na-1)$]{}; (Cone 1-2) at ($(rho0 1)!0.66!(rhod 1)$); node\[anchor=south west\] at (Cone 1-2) [$\scriptstyle \frac{1}{(d-1)n^2}(1,(d-1)na-1)$]{}; node\[anchor=west\] at (rho1-1) [$\scriptstyle (n^2,1-na)$]{}; The versal ${\mathbb{Q}}$-Gorenstein deformation $\mathcal X^{\sf ver} / (0 \in \Delta^d) $ of $(P \in X)$ is given in (\[eq: versal Q-Gor\]). After a suitable base change $(0 \in \Delta^d) \to (0 \in \Delta^d)$, one has $$\mathcal X^{\sf ver}{}' = \bigl( xy = (z^n+t_1) ( z^{(d-1)n} + t_2 z^{(d-2)n} + \ldots + t_{d-1}z^n + t_d )\bigr) \subset {\mathbb{C}}^3 \Big/ \frac1n(1,-1,a) \times \Delta^d.$$ Blowing up at the ideal $(x,z^n+t_1)$, one can deduce the following. \[prop: versal simul M-res\] Let $I = (x,z^n+t_1)$ be the ideal sheaf on $\mathcal X^{\sf ver}{}'$, and let $p \colon \mathcal Z = \op{Bl}_I \mathcal X^{\sf ver}{}' \to \mathcal X^{\sf ver}{}'$ be the blow up. Then, 1. for general $t \in \Delta^d$, $\mathcal Z_t \simeq \mathcal X_t^{\sf ver}{}'$; 2. $p$ is an isomorphism in a neighborhood containing $\mathcal X_0^{\sf ver}{}' \setminus \{P\}$; 3. $\mathcal Z_0$ is isomorphic to the toric variety associated to the fan $\Sigma_1$; 4. the base space $( 0\in \Delta^d) $ admits the decomposition $(0 \in \Delta) \times (0 \in \Delta^{d-1})$ where the first (resp. second) factor parametrizes the versal ${\mathbb{Q}}$-Gorensteing deformation of $\frac{1}{n^2}(1,na-1) \in \mathcal Z_0$ (resp. $\frac{1}{(d-1)n^2}(1,(d-1)na-1) \in \mathcal Z_0$). This is well-known in literature. We leave a reference[@BehnkeChristophersen:MResolution 2], instead of giving a full proof. \[cor: simul M-res\] Let $\mathcal X / (0 \in \Delta)$ be a one parameter ${\mathbb{Q}}$-Gorenstein smoothing of $(P \in X) \simeq \frac{1}{dn^2}(1,dna-1)$. Then after a finite base change $(0 \in \Delta') \to (0 \in \Delta)$, there exists a proper birational morphism $p \colon \bigl(A \subset \mathcal Z\bigr) \to (P \in \mathcal X')$, where $\mathcal X' := \mathcal X \times_\Delta \Delta'$, such that $(A \subset \mathcal Z_0) \to X$ is the map $\op{TV}(\Sigma) \to \op{TV}(\sigma)$ between toric varieties described in the beginning of the section \[subsec: Main\_ Local analysis\]. We have $d\times \frac{1}{n^2}(1,na-1)$ in $\mathcal Z_0$ to which we apply Corollary \[cor: Hacking V.B.\] to construct $d$ exceptional vector bundles on the general fiber. Let $P_1,\ldots,P_d \in \mathcal Z_0$ be the singular points. They correspond to the maximal cones in the fan $\Sigma$. Let $Y \to \mathcal Z_0$ be a resolution of $P_1,\ldots,P_d$ as described in Figure \[fig: min resolution of M-resolution\]. The exceptional locus consists of the curves $\{E_{ij} : 1 \leq i \leq d,\ 1 \leq j \leq r\}$, where each $\{E_{ij} : 1 \leq j \leq r\}$ is the chain of rational curves contracted down to $P_i$. Also, for $k=1,\ldots,d-1$, there are proper transforms $\tilde A_k$ of the curves $A_k := {\operatorname{Div}}\rho_k$ associated to $\rho_k = {\mathbb{R}}_{>0} \cdot (kn^2, 1-kna)$. node at (0,1.6) ; iin [1,2,3,4,5]{}[ in [1,2,3,4]{}[ (coordEi) at (,); ]{} ]{} i/in [1/1,2/2,4/[d-1,]{},5/d]{}[ in [1,2]{}[ node (Ei) at (coordEi) [$\scriptstyle E_{\indexi\j}$]{}; ]{} node (Ei3) at (coordEi3) [\[10pt\]\[3pt\][$\vdots$]{}]{}; node (Ei4) at (coordEi4) [$\scriptstyle E_{\indexi r}$]{}; ]{} node (E31) at (coordE31) [$\cdots$]{}; node (E34) at (coordE34) [$\cdots$]{}; node at ($(coordE31)!0.5!(coordE34)$) [$\cdots$]{}; i/in [1/1,2/2,3/[d-2]{},4/[d-1]{}]{}[ (coordAi) at ($(coordE\i1)!0.5!(coordE\m4)$); node (Ai) at (coordAi) [$\scriptstyle \tilde A_{\indexi}$]{}; (\[xshift=5pt\]Ei4.east) .. controls ($(E\i4)!0.5!(E\m4)$) .. (Ai.south); (Ai.north) .. controls ($(E\i1)!0.5!(E\m1)$) .. (\[xshift=-5pt\]E1.west); ]{} iin [1,2,4,5]{}[ in [1,2,3]{}[ (Ei) – (Ei); ]{} ]{} Let $L_{P_i} \subset \mathcal Z_0$ be the link of $(P_i \in \mathcal Z_0)$. Then $H_1(L_{P_i};{\mathbb{Z}})$ is generated by the loops $\alpha_{i1},\ldots,\alpha_{ir}$ around $E_{i1},\ldots,E_{ir}$, respectively. We consider the product of the maps in (\[eq: Div Cycle map\]): $$\label{eq: M-resolution Cycle map} \gamma \colon {\operatorname{Cl}}\mathcal Z_0 \to \bigoplus_{i=1}^d H_1(L_{P_i};{\mathbb{Z}}).$$ Application to the global case ------------------------------ This subsection treats the proof of the main theorem: \[thm: Main thm\] Let $X$ be a normal projective surface with $H^1(\mathcal O_X)=H^2(\mathcal O_X)=0$, and let $(P \in X)$ be a singular point $\frac{1}{dn^2}(1,dna-1)$ of class T (or $A_{d-1}$ in the case $n=a=1$). Assume that there exists a divisor $D \in {\operatorname{Cl}}X$ such that for each $Q \in \op{Sing} X$ and $\gamma_Q \colon {\operatorname{Cl}}X \to H_1(L_Q;{\mathbb{Z}})$, 1. $\gamma_P(D)$ generates $H_1(L_P;{\mathbb{Z}})/n^2$; 2. $\gamma_Q(D) \in n_Q \cdot H_1(L_Q;{\mathbb{Z}})$ if $Q$ is a Wahl singularity with indices $(n_Q,a_Q)$; 3. $\gamma_Q(D) = 0$ otherwise. Let $\mathcal X / (0 \in \Delta)$ be a one parameter ${\mathbb{Q}}$-Gorenstein smoothing of $X$. Then, after a finite base change $(0 \in \Delta') \to (0 \in \Delta)$, there exist reflexive sheaves $\mathcal E_1,\ldots,\mathcal E_d$ over $\mathcal X' := \mathcal X \times_\Delta \Delta'$ such that 1. $\mathcal E_k$ is a reflexive sheaf of rank $n$, locally free over $\mathcal X' \setminus \op{Sing} \mathcal X_0'$; 2. $\mathcal E_{k,0}^{\vee\vee} \simeq \mathcal O_X( m D )$ for some $m > 0$ independent of $k$; 3. for general $t$, $\mathcal E_{k,t}$ is an exceptional vector bundle of rank $n$; 4. $\langle\, \mathcal E_{1,t},\ldots,\mathcal E_{d,t}\, \rangle \subset {\operatorname{D}^{\sf b}}(\mathcal X_t')$ is an orthogonal collection. After a base change $\mathcal X' = \mathcal X \times_\Delta \Delta'$, we have a map $p \colon \mathcal Z \to \mathcal X'$ which extends $( A_1 \cup \ldots \cup A_{k-1} \subset \mathcal Z_0) \to (P \in \mathcal X')$ in Corollary \[cor: simul M-res\]. Assume the hypotheses of Theorem \[thm: Main thm\]. Let $P_1,\ldots,P_d \in \mathcal Z_0$ be the Wahl singularities $\frac{1}{n^2}(1,na-1)$ lying over $(P \in X)$, let $\gamma \colon {\operatorname{Cl}}\mathcal Z_0 \to \bigoplus_{i=1}^d H_1(L_{P_i};{\mathbb{Z}})$ be as in (\[eq: M-resolution Cycle map\]), and let $D' \in {\operatorname{Cl}}\mathcal Z_0$ be the proper transform of $D$. Then, there exist integers $m,a_1,\ldots,a_{d-1}$ such that $D_0 := mD'+\sum_{k} a_k A_k$ satisfies $$\gamma(D_0) = (\alpha_{11},0,0,\ldots,0),$$ where $\alpha_{11}$ is the generator corresponding to $E_{11}$ in Figure \[fig: min resolution of M-resolution\]. Let $Y \to \mathcal Z_0$ be a resolution as in Figure \[fig: min resolution of M-resolution\], and let $\beta_k \in H_1(L_P;{\mathbb{Z}})$ be the generator corresponding to $\tilde A_{k}$. By Proposition \[prop: Link algebra\], $\beta_1 = n^2 \alpha_{11}$ in $H_1(L_P;{\mathbb{Z}})$. Thus, there exists $c \in {\mathbb{Z}}$ such that $$\gamma_P(mD) = (1 + c n^2) \alpha_{11} = \alpha_{11} + c \beta_1,$$ where $\gamma_P \colon {\operatorname{Cl}}X \to H_1(L_P;{\mathbb{Z}})$. Let $D'' \in {\operatorname{Cl}}Y$ Be the proper transform of $D$ along the composition $Y \to \mathcal Z_0 \to X$. Then, by Proposition \[prop: divisors having same description in the link\], there are integers $\{a_k\}, \{e_{ij}\}$ such that $$D_0'':= mD'' + \sum_k a_k A_k + \sum_{i,j} e_{ij} E_{ij}$$ satisfies $(D_0'' . E_{11})=1$ and $(D_0''.E_{ij})=0$ for $(i,j) \neq (1,1)$. The push-forward of $D_0''$ along $Y \to \mathcal Z_0$ is the desired $D_0$ in the statement. Let $D_k := D_0 + A_1 + \ldots + A_k$($k<d$). Then by Proposition \[prop: Link algebra\] $$\begin{aligned} \gamma(D_i) &= \bigl( \alpha_{11} + \alpha_{1r},\, \ldots,\, \alpha_{k1} + \alpha_{kr},\, \alpha_{k+1,1}, \, 0,\,\ldots,\,0 \bigr) \\ &= \bigl( na\cdot\alpha_{1r},\, \ldots,\,na\cdot\alpha_{kr},\, \alpha_{k+1,1},\, 0,\,\ldots,\,0 \bigr). \end{aligned}$$ Hence, the hypotheses of Corollary \[cor: Hacking V.B.\] are satisfied when one plugs $P \mapsto P_{k+1}$ and $D_0 \mapsto D_k$ into the corollary. This yields an reflexive sheaf $\mathcal F_k$ over $\mathcal Z$(after a base change, but we suppress it in our notation), whose restriction to the general fiber is an exceptional vector bundle $\mathcal F_{k,t}$. We claim that $\mathcal E_k = (p_* \mathcal F_k)^{\vee\vee}$ is the desired one in Theorem \[thm: Main thm\]. We need to verify $$\label{eq: Orthogonality} {\operatorname{Ext}}^p(\mathcal F_{k,t},\, \mathcal F_{\ell,t})=0,\quad \forall p,\ \forall k\neq \ell,$$ for general $t$. $\chi( \mathcal F_{k,t},\, \mathcal F_{\ell,t}) = \sum_p (-1)^p \dim {\operatorname{Ext}}^p(\mathcal F_{k,t},\, \mathcal F_{\ell,t})=0$. Since the major part of the proof involves tedious numerical computations, we give an outline instead of giving all the details. By Riemann-Roch formula, $\chi(\mathcal F_{k,t},\, \mathcal F_{\ell,t})$ is determined by Chern classes of $\mathcal F_{k,t}^\vee \otimes \mathcal F_{\ell,t}$ and their intersection products. By Remark \[rmk: comparing intersection theories\], it suffices to look at their specialization. We have $\op{sp}(c_1(\mathcal F_{i,t})) = n( D_0 + A_1 + \ldots + A_{i-1})$, hence $$\op{sp}(c_1( \mathcal F_{k,t}^\vee \otimes \mathcal F_{\ell,t}) ) = \left\{ \begin{array}{ll} n^2( A_k + \ldots + A_{\ell-1} ) & \text{if }\ell > k \\ -n^2( A_\ell + \ldots + A_{k-1} ) & \text{if } \ell < k. \end{array} \right.$$ We remark that the curves $A_1,\ldots,A_{d-1}$ are exceptional curves of $\mathcal Z_0 \to X$, whose intersection properties are completely determined by the local geometry at $(P \in X)$. Using toric geometry, one may verify $$(A_i)^2 = -\frac{2}{n^2},\ (A_i . A_{i+1}) = \frac{1}{n^2},\ \text{and\ } (K_{\mathcal Z_0} . A_i)=0.$$ This determines $c_1^2$ and $(K_{\mathcal Z_0} . c_1)$ of $\mathcal F_{k,t}^\vee \otimes \mathcal F_{\ell,t}$; indeed, $c_1^2 = -2n^2$, $(K.c_1)=0$. Riemann-Roch formula and $\chi(\mathcal F_{k,t}, \mathcal F_{k,t})=1$ implies $$c_2(\mathcal F_{k,t}) = \frac{n-1}{2n}(c_1^2(\mathcal F_{k,t}) + n^2 + 1). \tag{see \cite[Lemma~5.3]{Hacking:ExceptionalVectorBundle}}$$ By the splitting principle, $c_2(\mathcal F_{k,t}^\vee \otimes \mathcal F_{\ell,t})$ can be expressed in terms of the Chern classes of $\mathcal F_{k,t}$ and $\mathcal F_{\ell,t}$. More specifically, for any vector bundles $\mathcal E$ and $\mathcal G$ of ranks $e,g$ respectively, $$c_2(\mathcal E \otimes \mathcal G) = \frac{g(g-1)}{2} \mathop{c_1^2}(\mathcal E) + (eg-1) \mathop{c_1}(\mathcal E) . \mathop{c_1}(\mathcal G) + \frac{e(e-1)}{2} \mathop{c_1^2}(\mathcal G) + g \mathop{c_2}(\mathcal E) + e \mathop{c_2}(\mathcal G).$$ In particular, we get $c_2(\mathcal F_{k,t}^\vee \otimes \mathcal F_{\ell,t})=0$. Riemann-Roch formula reads $$\chi(\mathcal F_{k,t},\mathcal F_{\ell,t}) = n^2 \chi(\mathcal O_{\mathcal X'_t}) + \frac 12 (c_1^2 - (K.c_1)) - c_2 = 0. \qedhere$$ It remains to prove (\[eq: Orthogonality\]) for $p=0,2$. To prove the case $p=0$, we need to take stability conditions into account. \[prop: Stability\] Let $\mathcal H / (0 \in \Delta)$ be a flat family of ample divisors in $\mathcal X' / (0 \in \Delta')$. For sufficiently large $m > 0$ and $t \neq 0$, $\mathcal F_{k,t}$ is slope stable with respect to $\mathcal A_t := K_{\mathcal X'_t} + (nm)\mathcal H_t$. If $d=1$, namely, $(P \in X) \simeq \frac{1}{n^2}(1,na-1)$, then $\mathcal F_{1,t}$ is slope stable with respect to $\mathcal H_t$([@Hacking:ExceptionalVectorBundle Proposition 4.4]). For $d>1$, let $\mathcal H'$ be the pullback of $\mathcal H$ along $p \colon \mathcal Z \to \mathcal X'$. Then, $\mathcal H'$ is relatively nef over $(0 \in \Delta')$. Since $\op{Nef}(\mathcal Z/ (0 \in \Delta')) = \op{\overline{Ample}}(\mathcal Z/(0 \in \Delta'))$, we may consider a sequence $\{\mathcal H'_i\}_{i \in {\mathbb{Z}}_{>0}}$ of ample ${\mathbb{Q}}$-Cartier divisors which converges to $\mathcal H'$. Using [@Hacking:ExceptionalVectorBundle Proposition 4.4], we see that $\mathcal F_{k,t}$ is slope stable with respect to $\mathcal H_i'$ for $t\neq 0$. Hence, $\mathcal F_{k,t}$ is slope stable with respect to all $\mathcal H_i'$. For any proper quotient $\mathcal F_{k,t} \twoheadrightarrow Q$, the stability conditions says $\mu_{\mathcal H'_{i,t}}(\mathcal F_{k,t}) < \mu_{\mathcal H'_{i,t}}(\mathcal Q)$. Taking limit $i \to \infty$ in both sides, we get $\mu_{\mathcal H'_t}(\mathcal F_{k,t}) \leq \mu_{\mathcal H'_t}(\mathcal Q)$, showing that $\mathcal F_{k,t}$ is slope semistable with respect to $\mathcal H'_t = \mathcal H_t$. This shows that $\mathcal F_{k,t}$ is semistable with respect to $\mathcal H_t$ for $t \neq 0$. Let $\mathcal A = K_{\mathcal X'} + (nm) \mathcal H$ for sufficiently large $m$, so that $\mathcal A_0$ is ample in $X=\mathcal X_0'$. The previous argument can be applied to $\mathcal A$, thus $\mathcal F_{k,t}$ is slope semistable with respect to $\mathcal A_t$. To prove the stability, it suffices to prove that $(c_1(\mathcal F_{k,t}) \mathbin . \mathcal A_t)$ is relatively prime to $n$. By Remark \[rmk: comparing intersection theories\], we may compute the intersection in the central fiber via the specialization map. Since $c_1(\mathcal F_{k,t}) {\mathrel{\mapstochar\xrightarrow[]{\rm sp}}} n( D_0 + A_1 + \ldots + A_{k-1}) $, $(\mathcal A_t \mathbin. c_1(\mathcal F_{k,t}) ) = ( K_{\mathcal Z_0} + nm \mathcal H'_0 \mathbin. nD_0)$. Here, $(\mathcal H'_0 \mathbin. nD_0) = (\mathcal H_t \mathbin. c_1(\mathcal F_{k,t})) \in {\mathbb{Z}}$, so $$(\mathcal A_t \mathbin. c_1(\mathcal F_{k,t})) = ( K_{\mathcal Z_0} \mathbin . nD_0 )\ (\mathrm{mod}\ n)$$ Let $I \in {\mathbb{Z}}/n{\mathbb{Z}}$ be the intersection number $( K_{\mathcal Z_0} \mathbin . nD_0 )$ modulo $n$. Then, $I$ purely depends on the local geometry of $(A_1\cup\ldots\cup A_{k-1} \subset \mathcal Z_0) \to (P \in X)$, hence we may identify $\mathcal Z_0$ with the toric variety associated to the fan $\Sigma$ consisting of the rays $\{ \rho_k := {\mathbb{R}}_{>0} \cdot ( kn^2, 1-kna) : k=0,1,\ldots,d \}$ and the cones $\sigma_k := \op{Cone}(\rho_{k-1},\rho_k)$. In this setup, $A_k\,(k=1,\ldots,d-1)$ are the divisors associated to the rays $\rho_k$. Let $A_0 \subset (A_1\cup \ldots \cup A_{k-1} \subset \mathcal Z_0)$ be the local curve associated to the ray $\rho_0$. Then, $\gamma(D_0) = (\alpha_{11},0,\ldots,0) = \gamma(A_0)$, thus $D_0 - A_0$ is Cartier. This shows that $(D_0 - A_0 \mathbin. K_{\mathcal Z_0} ) \in {\mathbb{Z}}$. Hence, $I = (nA_0 \mathbin. K_{\mathcal Z_0} ) = -a \ (\mathrm{mod}\ n)$(see [@Hacking:ExceptionalVectorBundle p.1192]). Since $(\mathcal A_t \mathbin . c_1(\mathcal F_{k,t})) = I\ (\mathrm{mod}\ n)$, $(\mathcal A_t \mathbin. c_1(\mathcal F_{k,t}))$ is relatively prime to $n$. During the proof, we have shown that $(\mathcal A_t \mathbin. c_1(\mathcal F_{k,t}) ) = (K_{\mathcal Z_0} + nm \mathcal H'_0 \mathbin. nD_0)$ is independent of $k$. This implies: For $k \neq \ell$, ${\operatorname{Hom}}(\mathcal F_{k,t},\, \mathcal F_{\ell,t} ) =0$. For $k \neq \ell$, ${\operatorname{Ext}}^2(\mathcal F_{k,t},\, \mathcal F_{\ell,t} ) = 0$. By Serre duality, it suffices to prove that $H^0( \mathcal F_{k,t} \otimes \mathcal F_{\ell,t}^\vee \otimes \omega_{\mathcal X'_t} ) = 0$. Let $\mathcal G$ be the reflexive hull of $p_* \bigl( \mathcal F_k \otimes \mathcal F_\ell^\vee \otimes \omega_{\mathcal Z / \Delta} \bigr)$(recall: $p \colon \mathcal Z \to \mathcal X'$ is the morphism defined in Corollary \[cor: simul M-res\]). For general $t$, $\mathcal G_t = \mathcal F_{k,t} \otimes \mathcal F_{\ell,t}^\vee \otimes \omega_{\mathcal X'_t}$(cf. [@Conrad:GrothendieckDuality Theorem 3.6.1]), hence our claim is $H^0(\mathcal G_t)=0$. We divide the proof into several steps. 1. We first claim that $H^0(\mathcal G_0) =0$. Let $p_0 \colon \mathcal Z_0 \to X$ be the morphism between central fibers. By Proposition \[prop: versal simul M-res\](b), $p_0$ induces the isomorphism $\mathcal Z_0 \setminus ( A_1 \cup \ldots A_{k-1}) {\xrightarrow{\,\sim\,}} X \setminus \{P\}$. Let us denote $X \setminus\{P\}$ by $X^\circ$. By the construction of $\mathcal F_k$, we have $(p_0{}_* \mathcal F_{k,0})^{\vee\vee} = \mathcal O_X( p_0{}_* D_0)^{\oplus n}$ for all $k=1,\ldots,d$. Hence, $$\begin{aligned} \mathcal G\big\vert_{X^\circ} & \simeq \bigl( \mathcal O_X( p_0{}_* D_0 ) )^{\oplus n} \otimes \mathcal O_X( -p_0{}_* D_0 ) )^{\oplus n} \bigr)^{\vee\vee} \big\vert_{X^\circ} \otimes \omega_{X^\circ} \\ & \simeq \omega_{X^\circ}^{\oplus n^2}. \end{aligned}$$ Since $\mathcal G$ is reflexive, it satisfies the condision $S_2$ of Serre, hence $\mathcal G_0$ satisfies the conditions $S_1$(torsion-freeness). Thus, $\mathcal G_0 \hookrightarrow (\mathcal G_0)^{\vee\vee} = \omega_X^{\oplus n^2}$, and $h^0(\mathcal G_0) \leq n^2\,h^0(\omega_X) = 0$. 2. It is natural to expect that $H^0(\mathcal G_0)=0$ implies $H^0(\mathcal G_t)=0$. However, $\mathcal G$ is not flat over $\Delta'$, so it does not automatically follows from the semicontinuity. Let $\varphi \colon \mathcal X' \to \Delta'$ be the deformation morphism. Then we may define an upper-semicontinuous function $$\label{eq: local rank function} h \colon \Delta' \to {\mathbb{Z}}_{>0},\quad t \mapsto \dim_{\mathbb{C}}\varphi_* \mathcal G \otimes \mathcal O_t.$$ The derived projection formula reads $$\mathbf{R}\varphi_* \mathcal G {\mathbin{\stackrel{\mathbf{L}}{\otimes}}}\mathcal O_t \simeq \mathbf{R}\varphi_* \bigl( \mathcal G {\mathbin{\stackrel{\mathbf{L}}{\otimes}}}\mathcal O_{\mathcal X'_t} \bigr).$$ Note that $\varphi$ is flat, so we do not need to derived $\varphi^* \mathcal O_t = \mathcal O_{\mathcal X'_t}$. The object $\mathcal G {\mathbin{\stackrel{\mathbf{L}}{\otimes}}}\mathcal O_{\mathcal X_t'}$ is isomorphic to the two term complex $\mathcal G \otimes \mathcal O_\mathcal Z(- \mathcal X'_t) \to \mathcal G$. Let $z \in \mathcal O_{\Delta'}$ be a function that vanishes only at $t$ with order $1$. Then, the map $\mathcal G \otimes \mathcal O_\mathcal Z(-\mathcal X'_t) \to \mathcal G$ is given by the multiplication by $s := \varphi^(z)$, hence its kernel consists of the sections whose supports are contained in $(s=0) = \mathcal X'_t$. Since $\mathcal G$ is torsion-free, there are no such sections. This proves that $\mathcal G {\mathbin{\stackrel{\mathbf{L}}{\otimes}}}\mathcal O_t \simeq \mathcal G_t$. Thus, $$\label{eq: aux: isomorphic derived functors} \mathbf{R}\varphi_ \mathcal G {\mathbin{\stackrel{\mathbf{L}}{\otimes}}}\mathcal O_t \simeq \mathbf{R}\varphi_* \mathcal G_t.$$ 3. For a complex $(E^\bullet,\, d^p \colon E^p \to E^{p+1})$, let us denote the $p$th cohomology sheaf by $\mathcal H^p(\mathcal E^\bullet) = \ker d^p / {\operatorname{image}}d^{p-1}$. The $\mathcal H^0$ in the right hand side of (\[eq: aux: isomorphic derived functors\]) is $H^0(\mathcal G_t) \otimes \mathcal O_t$, hence it suffices to prove $\mathcal H^0(\mathbf{R}\varphi_* \mathcal G {\mathbin{\stackrel{\mathbf{L}}{\otimes}}}\mathcal O_t)=0$. We use the spectral sequence for the Tor sheaves ${\mathit{\mathcal T\hskip-2.5pt{}or}}_{-p}(\mathcal A^\bullet, \mathcal B^\bullet) = \mathcal H^p(\mathcal A^\bullet {\mathbin{\stackrel{\mathbf{L}}{\otimes}}}\mathcal B^\bullet)$: $$E_2^{p,q} := {\mathit{\mathcal T\hskip-2.5pt{}or}}_{-p}(\mathbf{R}^q \varphi_* \mathcal G,\, \mathcal O_t) \Rightarrow H^{p+q} := {\mathit{\mathcal T\hskip-2.5pt{}or}}_{-p-q}(\mathbf{R}\varphi_* \mathcal G,\, \mathcal O_t). \tag{\cite[\textsection 3.3]{Huybrechts:FourierMukai}}$$ Since $\mathcal O_t$ admits a locally free resolution $\mathcal O_{\Delta'}(-t) \to \mathcal O_{\Delta'}$, $E_2^{p,q}=0$ unless $p\in \{0,-1\}$. Consequently, $$\mathrm{F}^p H^n / \mathrm{F}^{p+1} H^n \simeq E_\infty^{p,n-p} = E_2^{p,n-p}.$$ The filtration structure induces the short exact sequence $$0 \to E_2^{-1,1} \to H^0 \to E_2^{0,0} \to 0,$$ which can be regarded as an exact sequence of vector spaces over ${\mathbb{C}}(\{t\}) \simeq {\mathbb{C}}$. Hence, $H^0 \simeq E_2^{0,0} \oplus E_2^{-1,1}$. We see that $E_2^{0,0} = \mathcal H^0( \varphi_* \mathcal G {\mathbin{\stackrel{\mathbf{L}}{\otimes}}}\mathcal O_t) = \varphi_* \mathcal G \otimes \mathcal O_t$, and $E_2^{-1,1} = {\mathit{\mathcal T\hskip-2.5pt{}or}}_1( \mathbf{R}^1\varphi_* \mathcal G,\mathcal O_t)$. The last one can be identified with $\mathcal T \otimes \mathcal O_t$, where $\mathcal T$ is the torsion subsheaf of $ \mathbf{R}^1\varphi_* \mathcal G $. Consequently, we get an isomoprhism $$H^0 \simeq E_2^{0,0} \oplus E_2^{-1,1} \simeq (\varphi_* \mathcal G \oplus \mathcal T) \otimes \mathcal O_t.$$ The arguments in [Step 2]{} are also valid for $t=0$, so (\[eq: aux: isomorphic derived functors\]) holds for $t=0$. Looking at $\mathcal H^0$ of both sides, we get $\varphi_\mathcal G \otimes \mathcal O_{0} \simeq H^0(\mathcal G_0) \otimes \mathcal O_0$, and it vanishes by [Step 1]{}. By semicontinuity of (\[eq: local rank function\]), $\varphi_ \mathcal G \otimes \mathcal O_t=0$ for general $t$. Since $\mathcal T$ is torsion sheaf, $\mathcal T \otimes \mathcal O_t=0$ for general $t$. It follows that $H^0 = \mathcal H^0( \mathbf{R}\varphi_* \mathcal G {\mathbin{\stackrel{\mathbf{L}}{\otimes}}}\mathcal O_t)=0$ for general $t$. Example ------- Based on the arguments in [@Hacking:CompactModuliBarcelonaNote 2.5.1], we demonstrate the relation between three block collections and ${\mathbb{Q}}$-Gorenstein degenerations of del Pezzo surfaces. For simplicity, we present a particular toric surface as an explicit example, however, the argument can be generalized to every other toric surfaces listed in [@HackingProkorov:DegenerationOfDelPezzo Theorem 4.1]. (E1) at (8\,-1.75\); (E2) at (-.75\,0.66\); (E4) at (-.75\,-1.75\); (0,0) – (E1); (0,0) – (E2); (0,0) – (E4); (E1) – (E2) – (E4) – cycle; (E3) at ($(E2)!0.5!(E4)$); (0,0) – (E3); in [5,6,7,8,9]{}[ (E) at ($(E4)!\int!(E1)$); (0,0) – (E); ]{} node\[anchor=north west\] at (E1) [$\rho_1$]{}; node\[anchor=south east\] at (E2) [$\rho_2$]{}; node\[anchor=north east\] at (E4) [$\rho_4$]{}; in [3,5,6,7,8,9]{}[ ]{} Let $X$ be the complete toric surface whose fan is spanned by three rays, $\rho_1 = (49,-9)$, $\rho_2 = (-5,1)$, and $\rho_4 = (-5,-9)$. These are the solid line in Figure \[fig: eg: Toric fan\]. According to [@HackingProkorov:DegenerationOfDelPezzo], $X$ is the fake weighted projective plane corresponding to the solution $(r_1,r_2,r_3)=(2,5,9)$ of the Markov type equation $$r_1^2 + 2r_2^2 + 6r_3^2 = 6 r_1r_2r_3.$$ A one-parameter ${\mathbb{Q}}$-Gorenstein smoothing $\mathcal X / (0 \in \Delta)$ has the general fiber $S := \mathcal X_t$ a del Pezzo surface of degree $3$. Since $H^2(\mathcal T_X)=0$([@HackingProkorov:DegenerationOfDelPezzo Proposition 3.1]), there exists a base change $\mathcal X' / (0 \in \Delta')$ such that Corollary \[cor: simul M-res\] can be applied simultaneously to all the singular points of $X$. So, there is a birational morphism $\mathcal Z / (0 \in \Delta') \to \mathcal X' / (0 \in \Delta')$ such that $\mathcal Z_0$ is the fan in Figure \[fig: eg: Toric fan\], and $\mathcal Z_t \simeq \mathcal X'_t$ for general $t$. Let $P_i$ be the torus fixed point corresponding to the cone ${\mathbb{R}}_{\geq0} \cdot \rho_i + {\mathbb{R}}_{\geq0} \cdot \rho_{i+1}$ (regarding $\rho_{10}$ as $\rho_1$). Then, $(P_1 \in \mathcal Z_0) \simeq \frac{1}{2^2}(1,1)$, $(P_2,P_3 \in \mathcal Z_0) \simeq \frac{1}{5^2}(1,4)$, and $(P_4,\ldots,P_9 \in \mathcal Z_0) \simeq \frac{1}{9^2}(1,17)$. Since $\gcd(2,9)=1$, we may choose an integer $c$ such that $c \cdot {\operatorname{Div}}\rho_1$ satisfies the conditions (1–3) of Theorem \[thm: Main thm\]. For instance, $D_1 := 9^2 \cdot {\operatorname{Div}}\rho_1$ satisfies the conditions (1) with respect to $P_1$, and (3) with respect to other $P_k$’s. This produces an exceptional vector bundle $E_{1,t}$ of rank $2$ on $S$. For $k=2,\ldots,9$, let $$D_k = D_{k-1} + {\operatorname{Div}}\rho_k.$$ It can be shown that $D_i$ satisfies the condition (1) with respect to $P_k$, (2) with respect to $P_1,\ldots,P_{k-1}$, and (3) with respect to the remaining $P_k$’s. This yields an exceptional vector bundle $E_{k,t}$ over $S$. Let $\mathcal O^r(\alpha)$ be an exceptional vector bundle on $S$ of rank $r$ and $c_1 = \alpha$ (indeed, exceptional vector bundles on $S$ are determined by $r$ and $c_1$[@Gorodentsev:MovingAnticacnonical Corollary 2.5]). After a suitable choice of $L \in {\operatorname{Pic}}S$, one finds that these bundles form a three block collection $$\begin{aligned} {\operatorname{D}^{\sf b}}(S) &= \langle\, \mathcal E_{1,t} \otimes \mathcal O(L),\, \ldots,\, \mathcal E_{9,t} \otimes \mathcal O(L) \, \rangle \\ &= \langle\, \mathcal O^2(2K+\ell),\quad \mathcal O^5(2\ell) ,\, \mathcal O^5(K+3\ell),\quad \mathcal O^9(K+5\ell-e_1),\,\ldots,\, \mathcal O^9(K+5\ell-e_6)\, \rangle \end{aligned}$$ corresponding to the solution $(r_1,r_2,r_3)=(2,5,9)$ to the Markov type equation. Here, $e_1,\ldots,e_6 \subset S$ is a disjoint set of $(-1)$-curves and $\ell \subset S$ is the pullback of a general line along the contraction $S \to {\mathbb P}^2$ of $e_1,\ldots,e_6$. Acknowledgments {#acknowledgments .unnumbered} --------------- [ ]{} The author is grateful to Mohammad Akhtar, Seung-Jo Jung, Kyoung-Seog Lee, Yongnam Lee, Dongsoo Shin, and Giancarlo Urzúa for valuable discussions from which he took a lot of benefit. A part of this work had been done when the author was a member of Universität Bayreuth – Lehrstuhl Mathematik VIII. This work was supported by ERC Advanced Grant no. 340258 TADMICAMT, and by KIAS Individual Grant no. MG074601 at Korea Institute for Advanced Study.
--- abstract: 'Holistic 3D indoor scene understanding refers to jointly recovering the i) object bounding boxes, ii) room layout, and iii) camera pose, all in 3D. The existing methods either are ineffective or only tackle the problem partially. In this paper, we propose an end-to-end model that simultaneously solves all three tasks in real-time given only a single RGB image. The essence of the proposed method is to improve the prediction by i) parametrizing the targets ([e.g.]{}, 3D boxes) instead of directly estimating the targets, and ii) cooperative training across different modules in contrast to training these modules individually. Specifically, we parametrize the 3D object bounding boxes by the predictions from several modules, [i.e.]{}, 3D camera pose and object attributes. The proposed method provides two major advantages: i) The parametrization helps maintain the consistency between the 2D image and the 3D world, thus largely reducing the prediction variances in 3D coordinates. ii) Constraints can be imposed on the parametrization to train different modules simultaneously. We call these constraints “cooperative losses” as they enable the joint training and inference. We employ three cooperative losses for 3D bounding boxes, 2D projections, and physical constraints to estimate a geometrically consistent and physically plausible 3D scene. Experiments on the SUN RGB-D dataset shows that the proposed method significantly outperforms prior approaches on 3D object detection, 3D layout estimation, 3D camera pose estimation, and holistic scene understanding.' bibliography: - 'nips\_2018.bib' title: 'Cooperative Holistic Scene Understanding: Unifying 3D Object, Layout, and Camera Pose Estimation' --- Introduction ============ Holistic 3D scene understanding from a single RGB image is a fundamental yet challenging computer vision problem, while humans are capable of performing such tasks effortlessly within 200 ms [@potter1975meaning; @potter1976short; @schyns1994blobs; @thorpe1996speed]. The primary difficulty of the holistic 3D scene understanding lies in the vast, but ambiguous 3D information attempted to recover from a single RGB image. Such estimation includes three essential tasks: - The estimation of the 3D camera pose that captures the image. This component helps to maintain the consistency between the 2D image and the 3D world. - The estimation of the 3D room layout. Combining with the estimated 3D camera pose, it recovers a global geometry. - The estimation of the 3D bounding boxes for each object in the scene, recovering the local details. ![Overview of the proposed framework for cooperative holistic scene understanding. (a) We first detect 2D objects and generate their bounding boxes, given a single RGB image as the input, from which (b) we can estimate 3D object bounding boxes, 3D room layout, and 3D camera pose. The blue bounding box is the estimated 3D room layout. (c) We project 3D objects to the image plane with the learned camera pose, forcing the projection from the 3D estimation to be consistent with 2D estimation.[]{data-label="fig:overview"}](framework){width="\linewidth"} Most current methods either are inefficient or only tackle the problem partially. Specifically, - Traditional methods [@gupta2010estimating; @zhao2011image; @zhao2013scene; @choi2013understanding; @schwing2013box; @zhang2014panocontext; @izadinia2016im2cad; @huang2018holistic] apply sampling or optimization methods to infer the geometry and semantics of indoor scenes. However, those methods are computationally expensive; it usually takes a long time to converge and could be easily trapped in an unsatisfactory local minimum, especially for cluttered indoor environments. Thus both stability and scalability become issues. - Recently, researchers attempt to tackle this problem using deep learning. The most straightforward way is to directly predict the desired targets ([e.g.]{}, 3D room layouts or 3D bounding boxes) by training the individual modules separately with isolated losses for each module. Thereby, the prior work [@mousavian20173d; @lee2017roomnet; @kehl2017ssd; @kundu20183d; @zou2018layoutnet; @liu2018planenet] only focuses on the individual tasks or learn these tasks separately rather than jointly inferring all three tasks, or only considers the inherent relations without explicitly modeling the connections among them [@tulsiani2017factoring]. - Another stream of approach takes both an RGB-D image and the camera pose as the input [@lin2013holistic; @song2014sliding; @song2016deep; @song2017semantic; @deng2017amodal; @zou2017complete; @qi2017frustum; @lahoud20172d; @zhang2016deepcontext], which provides sufficient geometric information from the depth images, thereby relying less on the consistency among different modules. In this paper, we aim to address the missing piece in the literature: to recover a geometrically consistent and physically plausible 3D scene and jointly solve all three tasks in an efficient and cooperative way, only from a single RGB image. Specifically, we tackle three important problems: 1. 2D-3D consistencyA good solution to the aforementioned three tasks should maintain a high consistency between the 2D image plane and the 3D world coordinate. How should we design a method to achieve such consistency? 2. CooperationPsychological studies have shown that our biologic perception system is extremely good at rapid scene understanding [@schyns1994blobs], particularly utilizing the fusion of different visual cues [@landy1995measurement; @jacobs2002determines]. Such findings support the necessities of cooperatively solving all the holistic scene tasks together. Can we devise an algorithm such that it can cooperatively solve these tasks, making different modules reinforce each other? 3. Physically PlausibleAs humans, we excel in inferring the physical attributes and dynamics [@kubricht2017intuitive]. Such a deep understanding of the physical environment is imperative, especially for an interactive agent ([e.g.]{}, a robot) to navigate the environment or collaborate with a human agent. How can the model estimate a 3D scene in a physically plausible fashion, or at least have some sense of physics? To address these issues, we propose a novel parametrization of the 3D bounding box as well as a set of cooperative losses. Specifically, we parametrize the 3D boxes by the predicted camera pose and object attributes from individual modules. Hence, we can construct the 3D boxes starting from the 2D box centers to maintain a 2D-3D consistency, rather than predicting 3D coordinates directly or assuming the camera pose is given, which loses the 2D-3D consistency. Cooperative losses are further imposed on the parametrization in addition to the direct losses to enable the joint training of all the individual modules. Specifically, we employ three cooperative losses on the parametrization to constrain the 3D bounding boxes, projected 2D bounding boxes, and physical plausibility, respectively: - The 3D bounding box loss encourages accurate 3D estimation. - The differentiable 2D projection loss measures the consistency between 3D and 2D bounding boxes, which permits our networks to learn the 3D structures with only 2D annotations ([i.e.]{}, no 3D annotations are required). In fact, we can directly supervise the learning process with 2D objects annotations using the common sense of the object sizes. - The physical plausibility loss penalizes the intersection between the reconstructed 3D object boxes and the 3D room layout, which prompts the networks to yield a physically plausible estimation. shows the proposed framework for cooperative holistic scene understanding. Our method starts with the detection of 2D object bounding boxes from a single RGB image. Two branches of convolutional neural networks are employed to learn the 3D scene from both the image and 2D boxes: i) The global geometry network (GGN) learns the global geometry of the scene, predicting both the 3D room layout and the camera pose. ii) The local object network (LON) learns the object attributes, estimating the object pose, size, distance between the 3D box center and camera center, and the 2D offset from the 2D box center to the projected 3D box center on the image plane. The details are discussed in . By combining the camera pose from the GGN and object attributes from the LON, we can parametrize 3D bounding boxes, which grants jointly learning of both GGN and LON with 2D and 3D supervisions. Another benefit of the proposed parametrization is improving the training stability by reducing the variance of the 3D boxes prediction, due to that i) the estimated 2D offset has relatively low variance, and ii) we adopt a hybrid of classification and regression method to estimate the variables of large variances, inspired by [@ren2015faster; @mousavian20173d; @qi2017frustum]. We evaluate our method on SUN RGB-D Dataset [@song2015sun]. The proposed method outperforms previous methods on four tasks, including 3D layout estimation, 3D object detection, 3D camera pose estimation, and holistic scene understanding. Our experiments demonstrate that a cooperative method performing holistic scene understanding tasks can significantly outperform existing methods tackling each task in isolation, further indicating the necessity of joint training. Our contributions are four-fold. i) We formulate an end-to-end model for 3D holistic scene understanding tasks. The essence of the proposed model is to cooperatively estimate 3D room layout, 3D camera pose, and 3D object bounding boxes. ii) We propose a novel parametrization of the 3D bounding boxes and integrate physical constraint, enabling the cooperative training of these tasks. iii) We bridge the gap between the 2D image plane and the 3D world by introducing a differentiable objective function between the 2D and 3D bounding boxes. iv) Our method significantly outperforms the state-of-the-art methods and runs in real-time. Method {#sec:method} ====== ![Illustration of (a) network architecture and (b) parametrization of 3D object bounding box.[]{data-label="fig:architecture"}](architecture){width="\linewidth"} In this section, we describe the parametrization of the 3D bounding boxes and the neural networks designed for the 3D holistic scene understanding. The proposed model consists of two networks, shown in : a global geometric network (GGN) that estimates the 3D room layout and camera pose, and a local object network (LON) that infers the attributes of each object. Based on these two networks, we further formulate differentiable loss functions to train the two networks cooperatively. Parametrization {#sec:param} --------------- #### 3D Objects We use the 3D bounding box $X^W \in \mathbb{R}^{3\times 8}$ as the representation of the estimated 3D object in the world coordinate. The 3D bounding box is described by its 3D center $C^{W} \in \mathbb{R}^3$, size $S^W \in \mathbb{R}^3$, and orientation $R(\theta^W) \in \mathbb{R}^{3 \times 3}$: $X^W = h(C^W, R(\theta^W), S)$, where $\theta$ is the heading angle along the up-axis, and $h(\cdot)$ is the function that composes the 3D bounding box. Without any depth information, estimating 3D object center $C^{W}$ directly from the 2D image may result in a large variance of the 3D bounding box estimation. To alleviate this issue and bridge the gap between 2D and 3D object bounding boxes, we parametrize the 3D center $C^W$ by its corresponding 2D bounding box center $C^{I} \in \mathbb{R}^2$ on the image plane, distance $D$ between the camera center and the 3D object center, the camera intrinsic parameter $K \in \mathbb{R}^{3 \times 3}$, and the camera extrinsic parameters $R(\phi, \psi) \in \mathbb{R}^{3 \times 3}$ and $T \in \mathbb{R}^3$, where $\phi$ and $\psi$ are the camera rotation angles. As illustrated in (b), since each 2D bounding box and its corresponding 3D bounding box are both manually annotated, there is always an offset $\delta^I \in \mathbb{R}^2$ between the 2D box center and the projection of 3D box center. Therefore, the 3D object center $C^W$ can be computed as $$C^W = T + DR(\phi, \psi)^{-1}\frac{K^{-1}\left[C^I + \delta^I, 1\right]^T}{\left\\|K^{-1}\left[C^I + \delta^I, 1\right]^T\right\\|}.$$ Since $T$ becomes $\Vec{0}$ when the data is captured from the first-person view, the above equation could be written as $C^W = p(C^I, \delta^I, D, \phi, \psi, K)$, where $p$ is a differentiable projection function. In this way, the parametrization of the 3D object bounding box unites the 3D object center $C^W$ and 2D object center $C^I$, which helps maintain the 2D-3D consistency and reduces the variance of the 3D bounding box estimation. Moreover, it integrates both object attributes and camera pose, promoting the cooperative training of the two networks. #### 3D Room Layout Similar to 3D objects, we parametrize 3D room layout in the world coordinate as a 3D bounding box $X^L \in \mathbb{R}^{3 \times 8}$, which is represented by its 3D center $C^L \in \mathbb{R}^3$, size $S^L \in \mathbb{R}^3$, and orientation $R(\theta^L) \in \mathbb{R}^{3 \times 3}$, where $\theta^L$ is the rotation angle. In this paper, we estimate the room layout center by predicting the offset from the pre-computed average layout center. Direct Estimations ------------------ As shown in (a), the global geometry network (GGN) takes a single RGB image as the input, and predicts both 3D room layout and 3D camera pose. Such design is driven by the fact that the estimations of both the 3D room layout and 3D camera pose rely on low-level global geometric features. Specifically, GGN estimates the center $C^L$, size $S^L$, and the heading angle $\theta^L$ of the 3D room layout, as well as the two rotation angles $\phi$ and $\psi$ for predicting the camera pose. Meanwhile, the local object network (LON) takes 2D image patches as the input. For each object, LON estimates object attributes including distance $D$, size $S^W$, heading angle $\theta^W$, and the 2D offsets $\delta^I$ between the 2D box center and the projection of the 3D box center. Direct estimations are supervised by two losses $\mathcal{L}_\text{GGN}$ and $\mathcal{L}_\text{LON}$. Specifically, $\mathcal{L}_\text{GGN}$ is defined as $$\mathcal{L}_\text{GGN} = \mathcal{L}_{\phi} + \mathcal{L}_{\psi} + \mathcal{L}_{C^L} + \mathcal{L}_{S^L} + \mathcal{L}_{\theta^L},$$ and $\mathcal{L}_\text{LON}$ is defined as $$\mathcal{L}_\text{LON} = \frac{1}{N} \sum_{j=1}^{N} (\mathcal{L}_{D_j} + \mathcal{L}_{\delta^I_j} + \mathcal{L}_{S_j^W} + \mathcal{L}_{\theta_j^W}),$$ where $N$ is the number of objects in the scene. In practice, directly regressing objects’ attributes ([e.g.]{}, heading angle) may result in a large error. Inspired by [@ren2015faster; @mousavian20173d; @qi2017frustum], we adopt a hybrid method of classification and regression to predict the sizes and heading angles. Specifically, we pre-define several size templates or equally split the space into a set of angle bins. Our model first classifies size and heading angles to those pre-defined categories, and then predicts residual errors within each category. For example, in the case of the rotation angle $\phi$, we define $\mathcal{L}_{\phi} = \mathcal{L}_{\phi-cls} + \mathcal{L}_{\phi-reg}$. Softmax is used for classification and smooth-L1 (Huber) loss is used for regression. Cooperative Estimations ----------------------- Psychological experiments have shown that human perception of the scene often relies on global information instead of local details, known as the gist of the scene [@oliva2005gist; @oliva2006building]. Furthermore, prior studies have demonstrated that human perceptions on specific tasks involve the cooperation from multiple visual cues, [e.g.]{}, on depth perception [@landy1995measurement; @jacobs2002determines]. These crucial observations motivate the idea that the attributes and properties are naturally coupled and tightly bounded, thus should be estimated cooperatively, in which individual component would help to boost each other. Using the parametrization described in , we hope to cooperatively optimize GGN and LON, simultaneously estimating 3D camera pose, 3D room layout, and 3D object bounding boxes, in the sense that the two networks enhance each other and cooperate to make the definitive estimation during the learning process. Specifically, we propose three cooperative losses which jointly provide supervisions and fuse 2D/3D information into a physically plausible estimation. Such cooperation improves the estimation accuracy of 3D bounding boxes, maintains the consistency between 2D and 3D, and generates a physically plausible scene. We further elaborate on these three aspects below. #### 3D Bounding Box Loss As neither GGN or LON is directly optimized for the accuracy of the final estimation of the 3D bounding box, learning directly through GGN and LON is evidently not sufficient, thus requiring additional regularization. Ideally, the estimation of the object attributes and camera pose should be cooperatively optimized, as both contribute to the estimation of the 3D bounding box. To achieve this goal, we propose the 3D bounding box loss with respect to its 8 corners $$\mathcal{L}_{\text{3D}} = \frac{1}{N}\sum_{j = 1}^N \left\\|h(C^W_j, R(\theta_j), S_j) - X_j^{W}\right\\|_2^2,$$ where $X^{W}$ is the ground truth 3D bounding boxes in the world coordinate. @qi2017frustum proposes a similar regularization in which the parametrization of 3D bounding boxes is different. #### 2D Projection Loss In addition to the 3D parametrization of the 3D bounding boxes, we further impose an additional consistency as the 2D projection loss, which maintains the coherence between the 2D bounding boxes in the image plane and the 3D bounding boxes in the world coordinate. Specifically, we formulate the learning objective of the projection from 3D to 2D as $$\mathcal{L}_{\text{PROJ}} = \frac{1}{N}\sum_{j=1}^N \left\\|f(X_j^{W}, R, K) - X_j^{I}\right\\|_2^2,$$ where $f(\cdot)$ denotes a differentiable projection function which projects a 3D bounding box to a 2D bounding box, and $X_j^{I} \in \mathbb{R}^{2 \times 4}$ is the 2D object bounding box (either detected or the ground truth). #### Physical Loss In the physical world, 3D objects and room layout should not intersect with each other. To produce a physically plausible 3D estimation of a scene, we integrate the physical loss that penalizes the physical violations between 3D objects and 3D room layout $$\mathcal{L}_{\text{PHY}} = \frac{1}{N}\sum_{j=1}^N \left(\operatorname{ReLU}(\operatorname{Max}(X_{j}^W) - \operatorname{Max}(X^L)) + \operatorname{ReLU}(\operatorname{Min}(X^L) - \operatorname{Min}(X_j^W))\right),$$ where $\operatorname{ReLU}$ is the activate function, $\operatorname{Max}(\cdot)$ / $\operatorname{Min}(\cdot)$ takes a 3D bounding box as the input and outputs the max/min value along three world axes. By adding the physical constraint loss, the proposed model connects the 3D environments and the 3D objects, resulting in a more natural estimation of both 3D objects and 3D room layout. To summarize, the total loss can be written as $$\mathcal{L}_{\text{Total}} = \mathcal{L}_{\text{GGN}} + \mathcal{L}_{\text{LON}} + \lambda_{\text{COOP}}\left(\mathcal{L}_{\text{3D}} + \mathcal{L}_{\text{PROJ}} + \mathcal{L}_{\text{PHY}}\right),$$ where $\lambda_{\text{COOP}}$ is the trade-off parameter that balances the cooperative losses and the direct losses. Implementation ============== Both the GGN and LON adopt ResNet-34 [@he2016deep] architecture as the encoder, which encodes a 256x256 RGB image into a 2048-D feature vector. As each of the networks consists of multiple output channels, for each channel with an L-dimensional output, we stack two fully connected layers (2048-1024, 1024-L) on top of the encoder to make the prediction. We adopt a two-step training procedure. First, we fine-tune the 2D detector [@dai2017deformable; @bodla2017softnms] with 30 most common object categories to generate 2D bounding boxes. The 2D and 3D bounding box are matched to ensure each 2D bounding box has a corresponding 3D bounding box. Second, we train two 3D estimation networks. To obtain good initial networks, both GGN and LON are first trained individually using the synthetic data (SUNCG dataset [@song2017semantic]) with photo-realistically rendered images [@zhang2017physically]. We then fix six blocks of the encoders of GGN and LON, respectively, and fine-tune the two networks jointly on SUN RGBD dataset [@song2015sun]. To avoid over-fitting, a data augmentation procedure is performed by randomly flipping the images or randomly shifting the 2D bounding boxes with corresponding labels during the cooperative training. We use Adam [@kingma2014adam] for optimization with a batch size of 1 and a learning rate of 0.0001. In practice, we train the two networks cooperatively for ten epochs, which takes about 10 minutes for each epoch. We implement the proposed approach in PyTorch [@paszke2017automatic]. Evaluation ========== ![Qualitative results (top 50%). (Left) Original RGB images. (Middle) Results projected in 2D. (Right) Results in 3D. Note that the depth input is only used to visualize the 3D results.[]{data-label="fig:results"}](results){width="\linewidth"} We evaluate our model on SUN RGB-D dataset [@song2015sun], including 5050 test images and 10335 images in total. The SUN RGB-D dataset has 47 scene categories with high-quality 3D room layout, 3D camera pose, and 3D object bounding boxes annotations. It also provides benchmarks for various 3D scene understanding tasks. Here, we only use the RGB images as the input. shows some qualitative results. We discard the rooms with no detected 2D objects or invalid 3D room layout annotation, resulting in a total of 4783 training images and 4220 test images. More results can be found in the supplementary materials. We evaluate our model on five tasks: i) 3D layout estimation, ii) 3D object detection, iii) 3D box estimation iv) 3D camera pose estimation, and v) holistic scene understanding, all with the test images across all scene categories. For each task, we compare our cooperatively trained model with the settings in which we train GGN and LON individually without the proposed parametrization of 3D object bounding box or cooperative losses. In the individual training setting, LON directly estimates the 3D object centers in the 3D world coordinate. #### 3D Layout Estimation Since SUN RGB-D dataset provides the ground truth of 3D layout with arbitrary numbers of polygon corners, we parametrize each 3D room layout as a 3D bounding box by taking the output of the Manhattan Box baseline from [@song2015sun] with eight layout corners, which serves as the ground truth. We compare the estimation of the proposed model with three previous methods—3DGP [@choi2013understanding], IM2CAD [@izadinia2016im2cad] and HoPR [@huang2018holistic]. Following the evaluation protocol defined in [@song2015sun], we compute the average between the free space of the ground truth and the free space estimated by the proposed method. shows our model outperforms HoPR by 2.0%. The results further show that there is an additional 1.5% performance improvement compared with individual training, demonstrating the efficacy of our method. Note that IM2CAD [@izadinia2016im2cad] manually selected 484 images from 794 test images of living rooms and bedrooms. For fair comparisons, we evaluate our method on the entire set of living room and bedrooms, outperforming IM2CAD by 2.1%. [l\|c\| c c c c c]{} & &\ & IoU & $P_g$ & $R_g$ & $R_r$ & IoU\ 3DGP [@choi2013understanding] & 19.2 & 2.1 & 0.7 & 0.6 & 13.9\ HoPR [@huang2018holistic] & 54.9 & 37.7 & 23.0 & 18.3 & 40.7\ Ours (individual) & 55.4 & 36.8 & 22.4 & 20.1 & 39.6\ Ours (cooperative) & 56.9 & 49.3 & 29.7 & 28.5 & 42.9\ \[tab:holistic\] \[tab:detection\] [c c c c c c c c c c c c]{} & bed & chair & sofa & table & desk & toilet & bin & sink & shelf & lamp & mIoU\ IoU (3D) & 33.1 & 15.7 & 28.0 & 20.8 & 15.6 & 25.1 & 13.2 & 9.9 & 6.9 & 5.9 & 17.4\ IoU (2D) & 75.7 & 68.1 & 74.4 & 71.2 & 70.1 & 72.5 & 69.7 & 59.3 & 62.1 & 63.8 & 68.7\ \[tab:box\_estimation\] #### 3D Object Detection We evaluate our 3D object detection results using the metrics defined in [@song2015sun]. Specifically, the is computed using the 3D between the predicted and the ground truth 3D bounding boxes. In the absence of depth, the threshold of is adjusted from 0.25 (evaluation setting with depth image input) to 0.15 to determine whether two bounding boxes are overlapped. The 3D object detection results are reported in . We report 10 out of 30 object categories here, and the rest are reported in the supplementary materials. The results indicate our method outperforms HoPR by 9.64% on and improves the individual training result by 8.41%. Compared with the model using individual training, the proposed cooperative model makes a significant improvement, especially on small objects such as bins and lamps. The accuracy of the estimation easily influences 3d detection of small objects; oftentimes, it is nearly impossible for prior approaches to detect. In contrast, benefiting from the parametrization method and 2D projection loss, the proposed cooperative model maintains the consistency between 3D and 2D, substantially reducing the estimation variance. Note that although IM2CAD also evaluates the 3D detection, they use a metric related to a specific distance threshold. For fair comparisons, we further conduct experiments on the subset of living rooms and bedrooms, using the same object categories with respect to this particular metric rather than an threshold. We obtain an of 78.8%, 4.2% higher than the results reported in IM2CAD. \[tab:camera\] #### 3D Box Estimation The 3D object detection performance of our model is determined by both the 2D object detection and the 3D bounding box estimation. We first evaluate the accuracy of the 3D bounding box estimation, which reflects the ability to predict 3D boxes from 2D image patches. Instead of using , 3D is directly computed between the ground truth and the estimated 3D boxes for each object category. To evaluate the 2D-3D consistency, the estimated 3D boxes are projected back to 2D, and the 2D is evaluated between the projected and detected 2D boxes. Results using the full model are reported in , which shows 3D estimation is still under satisfactory, despite the efforts to maintain a good 2D-3D consistency. The underlying reason for the gap between 3D and 2D performance is the increased estimation dimension. Another possible reason is due to the lack of context relations among objects. Results for all object categories can be found in the supplementary materials. #### Camera Pose Estimation We evaluate the camera pose by computing the mean absolute error of yaw and roll between the model estimation and ground truth. As shown in , comparing with the traditional geometry-based method [@hedau2009recovering] and previous learning-based method [@huang2018holistic], the proposed cooperative model gains a significant improvement. It also improves the individual training performance with 0.29 degree on yaw and 1.28 degree on roll. #### Holistic Scene Understanding Per definition introduced in [@song2015sun], we further estimate the holistic 3D scene including 3D objects and 3D room layout on SUN RGB-D. Note that the holistic scene understanding task defined in [@song2015sun] misses 3D camera pose estimation compared to the definition in this paper, as the results are evaluated in the world coordinate. Using the metric proposed in [@song2015sun], we evaluate the geometric precision $P_g$, the geometric recall $R_g$, and the semantic recall $R_r$ with the threshold set to 0.15. We also evaluate the between free space (3D voxels inside the room polygon but outside any object bounding box) of the ground truth and the estimation. shows that we improve the previous approaches by a significant margin. Moreover, we further improve the individually trained results by 8.8% on geometric precision, 5.6% on geometric recall, 6.6% on semantic recall, and 3.7% on free space estimation. The performance gain of total scene understanding directly demonstrates that the effectiveness of the proposed parametrization method and cooperative learning process. Discussion ========== In the experiment, the proposed method outperforms the state-of-the-art methods on four tasks. Moreover, our model runs at 2.5 fps (0.4s for 2D detection and 0.02s for 3D estimation) on a single Titan Xp GPU, while other models take significantly much more time; [e.g.]{}, [@izadinia2016im2cad] takes about 5 minutes to estimate one image. Here, we further analyze the effects of different components in the proposed cooperative model, hoping to shed some lights on how parametrization and cooperative training help the model using a set of ablative analysis. \[tab:analysis\] Ablative Analysis ----------------- We compare four variants of our model with the full model trained using $\mathcal{L}_{\text{SUM}}$: 1. The model trained without the supervision on 3D object bounding box corners (w/o $\mathcal{L}_{\text{3D}}$, $S_1$). 2. The model trained without the 2D supervision (w/o $\mathcal{L}_{\text{PROJ}}$, $S_2$). 3. The model trained without the penalty of physical constraint (w/o $\mathcal{L}_{\text{PHY}}$, $S_3$). 4. The model trained in an unsupervised fashion where we only use 2D supervision to estimate the 3D bounding boxes (w/o $\mathcal{L}_{\text{3D}}+ \mathcal{L}_{\text{GGN}}+\mathcal{L}_{\text{LON}}$, $S_4$). Additionally, we compare two variants of training settings: i) the model trained directly on SUN RGB-D without pre-train ($S_5$), and ii) the model trained with 2D bounding boxes projected from ground truth 3D bounding boxes ($S_6$). We conduct the ablative analysis over all the test images on the task of holistic scene understanding. We also compare the 3D mIoU and 2D mIoU of 3D box estimation. summarizes the quantitative results. ![Comparison with two variants of our model.[]{data-label="fig:ablative"}](ablative){width="\linewidth"} #### Experiment $\mathbf{S_1}$ and $\mathbf{S_3}$ Without the supervision on 3D object bounding box corners or physical constraint, the performance of all the tasks decreases since it removes the cooperation between the two networks. #### Experiment $\mathbf{S_2}$ The performance on the 3D detection is improved without the projection loss, while the 2D mIoU decreases by 8.0%. As shown in (b), a possible reason is that the 2D-3D consistency $\mathcal{L}_{\text{PROJ}}$ may hurt the performance on 3D accuracy compared with directly using 3D supervision, while the 2D performance is largely improved thanks to the consistency. #### Experiment $\mathbf{S_4}$ The training entirely in an unsupervised fashion for 3D bounding box estimation would fail since each 2D pixel could correspond to an infinite number of 3D points. Therefore, we integrate some common sense into the unsupervised training by restricting the size of the object close to the average size. As shown in (c), we can still estimate the 3D bounding box without 3D supervision quite well, although the orientations are usually not accurate. #### Experiment $\mathbf{S_5}$ and $\mathbf{S_6}$ $S_5$ demonstrates the efficiency of using a large amount of synthetic training data, and $S_6$ indicates that we can gain almost the same performance even if there are no 2D bounding box annotations. Related Work ------------ #### Single Image Scene Reconstruction Existing 3D scene reconstruction approaches fall into two streams. i) Generative approaches model the reconfigurable graph structures in generative probabilistic models [@zhao2011image; @zhao2013scene; @choi2013understanding; @lin2013holistic; @guo2013support; @zhang2014panocontext; @zou2017complete; @huang2018holistic]. ii) Discriminative approaches [@izadinia2016im2cad; @tulsiani2017factoring; @song2017semantic] reconstruct the 3D scene using the representation of 3D bounding boxes or voxels through direct estimations. Generative approaches are better at modeling and inferring scenes with complex context, but they rely on sampling mechanisms and are always computational ineffective. Compared with prior discriminative approaches, our model focus on establishing cooperation among each scene module. #### Gap between 2D and 3D It is intuitive to constrain the 3D estimation to be consistent with 2D images. Previous research on 3D shape completion and 3D object reconstruction explores this idea by imposing differentiable 2D-3D constraints between the shape and silhouettes [@wu2016single; @rezende2016unsupervised; @yan2016perspective; @tulsiani2015viewpoints; @wu2017marrnet]. @mousavian20173d infers the 3D bounding boxes by matching the projected 2D corners in autonomous driving. In the proposed cooperative model, we introduce the parametrization of the 3D bounding box, together with a differentiable loss function to impose the consistency between 2D-3D bounding boxes for indoor scene understanding. Conclusion ========== Using a single RGB image as the input, we propose an end-to-end model that recovers a 3D indoor scene in real-time, including the 3D room layout, camera pose, and object bounding boxes. A novel parametrization of 3D bounding boxes and a 2D projection loss are introduced to enforce the consistency between 2D and 3D. We also design differentiable cooperative losses which help to train two major modules cooperatively and efficiently. Our method shows significant improvements in various benchmarks while achieving high accuracy and efficiency. Acknowledgement: The work reported herein was supported by DARPA XAI grant N66001-17-2-4029, ONR MURI grant N00014-16-1-2007, ARO grant W911NF-18-1-0296, and an NVIDIA GPU donation grant. We thank Prof. Hongjing Lu from the UCLA Psychology Department for useful discussions on the motivation of this work, and three anonymous reviewers for their constructive comments.
--- abstract: 'Recently there has been a rising interest in training agents, embodied in virtual environments, to perform language-directed tasks by deep reinforcement learning. In this paper, we propose a simple but effective neural language grounding module for embodied agents that can be trained end to end from scratch taking raw pixels, unstructured linguistic commands, and sparse rewards as the inputs. We model the language grounding process as a language-guided transformation of visual features, where latent sentence embeddings are used as the transformation matrices. In several language-directed navigation tasks that feature challenging partial observability and require simple reasoning, our module significantly outperforms the state of the art. We also release <span style="font-variant:small-caps;">xworld3D</span>, an easy-to-customize 3D environment that can potentially be modified to evaluate a variety of embodied agents.' author: - \| Haonan Yu$^{\dagger}$, Xiaochen Lian$^{\dagger}$, Haichao Zhang$^{\dagger}$, and Wei Xu$^{\ddagger}$\ $^{\dagger}$Baidu Research, Sunnyvale CA USA\ $^{\ddagger}$Horizon Robotics, Cupertino CA USA\ `{haonanyu,lianxiaochen,zhanghaichao}@baidu.com,wei.xu@horizon.ai`\ bibliography: - 'corl2018.bib' title: 'Guided Feature Transformation (GFT): A Neural Language Grounding Module for Embodied Agents' --- =1 Introduction ============ This paper examines the idea of building embodied [@Smith2005; @Kiela2016] agents that learn control from linguistic commands and visual inputs. One recent line of work [@Oh2017; @Hermann17; @Chaplot18; @Das2018; @Yu2018] trains such agents situated in simulated environments in an end-to-end fashion, receiving unstructured linguistic commands and raw image pixels as the inputs, and producing navigation actions as the outputs. For successful navigation control, it is crucial for an agent to learn to associate linguistic concepts with visual features, a process known as language grounding [@Harnad1990; @Siskind1994]. To avoid a tremendous amount of labeled data, this line of work trains language grounding oriented by navigation goals via reinforcement learning (RL). Through trials and errors, an agent learns not only to navigate but also to reinforce (or weaken) the connection between visual features and their matched (or unmatched) language tokens. The problem of learning to control alone is quite challenging, especially in an environment with long time horizons and sparse rewards [@Sutton98]. However, in this paper we only concentrate on language grounding as the core of the perception system of an agent, while using an existing RL system design, synchronous advantage actor-critic (ParallelA2C, [@Clemente2017]). Because our model exploits no structural biases of specific tasks, it is possible to plug our language grounding module in many other neural architectures for tasks that combine language and vision. We propose a simple but effective neural language grounding module that models a rich set of language-vision interactions. The module performs a [g]{}uided [f]{}eature [t]{}ransformation (GFT), where latent sentence embeddings computed from the language input are treated as the transformation matrices of visual features. This guided transformation is much more expressive than the existing three categories of language grounding methods for embodied agents: vector concatenation [@Hermann17; @Das2018; @Shu2018], gated networks [@Chaplot18; @Wu2018], and convolutional interaction [@Oh2017; @Yu2018]. In fact, it can be treated as a generalization of the last two categories. GFT is fully differentiable and is embedded in the perception system of our navigation agent that is trained end to end from scratch by RL. In an apples-to-apples comparison, our model significantly outperforms the existing state of the art, over a rich set of navigation tasks that feature challenging partial observability and cluttered background, and require simple reasoning (Figure \[fig:envs\]). Our GFT-powered agent is able to handle both the 2D and 3D environments without any architecture or hyperparameter change between the two scenarios[^1]. This demonstrates the generality and efficacy of GFT as a language grounding module that can potentially benefit a variety of embodied agents for other language-vision tasks. Finally, we will release the <span style="font-variant:small-caps;">xworld3D</span> environment used in the experiments. <span style="font-variant:small-caps;">xworld3D</span> highlights a teacher infrastructure that enables flexible customization of linguistic commands, environment maps, and training curricula. Guided Feature Transformation for Language Grounding {#sec:gfat} ==================================================== The major contribution of this paper is a simple yet effective language grounding module called GFT. We will first describe GFT and then discuss our motivations and its advantages. In a general scenario, given a pair of an image ${\mathbf{o}}$ and a sentence ${\mathbf{l}}$, a language grounding module fuses the two modalities for downstream processings. This is a common process seen in visual question answering (VQA) [@Antol2015], and it also lies at the heart of our agent’s perception system. Method ------ We use a convolutional neural network (CNN) to convert ${\mathbf{o}}$ into a feature cube ${\mathbf{C}}\in\mathbb{R}^{D\times N}$, where $D$ is the number of channels and $N$ is the number of locations in the image spatial domain (collapsed from 2D to 1D for notational simplicity). Suppose that we have an embedding function that converts ${\mathbf{l}}$ to a series of $J$ embedding vectors ${\mathbf{t}}_1,\ldots,{\mathbf{t}}_j,\ldots,{\mathbf{t}}_J$, where each ${\mathbf{t}}_j\in\mathbb{R}^{D(D+1)}$ can be reshaped and treated as a matrix ${\mathbf{T}}_j\in\mathbb{R}^{D \times (D+1)}$. Then we compute a series of $J$ transformations, one followed by another, activated by a nonlinear function $g$: $$\label{eq:staft} {\mathbf{C}}^{[j]}=g({\mathbf{T}}_j \left[ \begin{array}{l} {\mathbf{C}}^{[j-1]}\\ {\mathbf{1}}^{\intercal} \end{array} \right]), \ \ \ 1 \le j \le J,$$ where ${\mathbf{1}}\in \mathbb{R}^{N}$ is an all-one vector and ${\mathbf{C}}^{[0]}={\mathbf{C}}$. This guided transformation yields a feature cube ${\mathbf{C}}^{}={\mathbf{C}}^{[J]}\in\mathbb{R}^{D\times N}$ which is the final grounding result for downstream processings. Overall, it would be expected that the transformation matrices $\mathbf{T}_j$ correctly capture the critical aspects of command semantics, in order for the agent to perform tasks. (Examples of the trained $\mathbf{T}_j$ in a later experiment are visualized in Appendix \[app:visual\].) Despite its simple form, GFT is able to model a rich set of interactions between language and vision, resulting in strong representational power. Indeed, it can be seen as a unifying formulation of two existing language grounding modules, namely, gated networks [@Perez2018; @Chaplot18] and convolutional interaction [@Oh2017; @Yu2018]. Why GFT? -------- [GFT is a stack of generalized* gated networks.*]{} In the following we will ignore the subscript $j$ of ${\mathbf{T}}_j$ for notational simplicity. Let us first write: $${\mathbf{T}}= \left[{\mathbf{T}}'\hspace{6pt}{\mathbf{b}}\right],$$ where ${\mathbf{T}}'\in\mathbb{R}^{D\times D}$, and ${\mathbf{b}}\in\mathbb{R}^D$ is a bias vector for the transformation in Eq. \[eq:staft\]. We would like to investigate what ${\mathbf{T}}'{\mathbf{C}}$ essentially does. Toward this end, we perform a singular value decomposition (SVD) of ${\mathbf{T}}'$: $${\mathbf{T}}'={\mathbf{U}}{\mathbf{\Lambda}}{\mathbf{V}}^{\intercal},$$ where ${\mathbf{U}}\in\mathbb{R}^{D\times D}$ and ${\mathbf{V}}\in\mathbb{R}^{D\times D}$ are both orthogonal matrices. ${\mathbf{\Lambda}}\in\mathbb{R}^{D\times D}$ is a diagonal matrix that contains $D$ values $\lambda_1,\ldots,\lambda_D$ (not necessarily non-negative) on the diagonal. In an extreme case where both ${\mathbf{U}}$ and ${\mathbf{V}}$ are identity matrices, each transformation step of GFT degrades to a gated network. Specifically, let ${\mathbf{c}}_d$ be the $d$-th feature map, we have $$\label{eq:film} {\mathbf{c}}_d^{[j]}=g(\lambda_d {\mathbf{c}}_d^{[j-1]} + b_d)$$ This is exactly the same modulation provided by FiLM [@Perez2018]. Intuitively, Eq. \[eq:film\] performs scaling, thresholding, or negating of the features on the $d$th map of ${\mathbf{C}}^{[j-1]}$, according to the semantics of the input command ${\mathbf{l}}$. A further specialized version was proposed by @Chaplot18 in which they remove $g$ and $b_d$, while activating $\lambda_d$ by sigmoid. Thus when ${\mathbf{U}}$ and ${\mathbf{V}}$ are identities, GFT is a stack of FiLMs. In a general case, ${\mathbf{U}}$ and ${\mathbf{V}}$ are dense and represent general rotations in $\mathbb{R}^D$. Because ${\mathbf{U}}$ and ${\mathbf{V}}$ are computed from the command ${\mathbf{l}}$, they are language-guided. This is a major difference between a transformation step of GFT and that of a gated network. The latter always modulates features in the same original feature space, regardless of the command ${\mathbf{l}}$. As a result, a gated network such as FiLM places more pressure on learning ${\mathbf{C}}$ because a single feature space has to reconcile with a huge number of commands. When the vision is challenging or the language space is huge, modulating only in the original feature space might become a performance bottleneck of the overall agent model. In contrast, a transformation step of GFT can choose to rotate the axes of the feature space (${\mathbf{V}}^{\intercal}$), scale the features in that rotated space (${\mathbf{\Lambda}}{\mathbf{V}}^{\intercal}$), and rotate the scaled feature again (${\mathbf{U}}{\mathbf{\Lambda}}{\mathbf{V}}^{\intercal}$). These choices are determined by the current command. On top of it, GFT performs this “rotate-scale-rotate” operation multiple times in a sequence, when combined together, resulting in high-order nonlinear feature modulation. [GFT performs concept detection over multiple* convolutional steps.]{} An alternative interpretation is possible if we treat Eq. \[eq:staft\] as a $1 \times 1$ convolution with $D$ filters and a stride of one, sharing similar motivations in @Oh2017 and @Yu2018. In such an interpretation, each row of ${\mathbf{T}}$ contains a $1\times 1$ convolutional filter of length $D$ and a scalar bias. The $D$ filters (rows) represent at most $D$ different or complementary aspects of the semantics of the command ${\mathbf{l}}$, and a step of Eq. \[eq:staft\] essentially performs concept detection. For example, a 3D asymmetric object such as a bike has different appearances from different viewing angles. Thus having multiple filters of the sentence “go to bike” improves the representational power and results in a higher chance of finding the corresponding concept when the agent moves around. Overall, GFT performs multi-step concept detection with language-dependent filters at each step. In summary, the simple but general formulation of GFT unifies several existing ideas for grounding language in vision. In the remainder of this paper, we will evaluate it in a challenging language-directed navigation problem. The Navigation Problem {#sec:problem} ====================== We formally introduce our problem as a partially observable Markov decision process (POMDP) [@Kaelbling1998] as follows. The problem is divided into many navigation sessions. At the beginning of each session, a navigation task is sampled by a pre-programmed teacher as $k\sim P(k)$. Given the task, an initial environment state ${\mathbf{s}}^{[1]}$ is sampled by the simulator as ${\mathbf{s}}^{[1]}\sim P({\mathbf{s}}^{[1]}\|k)$, [i.e.,]{} the simulator arranges the scene according to the sampled task. An environment state ${\mathbf{s}}$ contains both the map configuration and the agent’s pose. We assume that the teacher has full access to the environment state, and samples a linguistic command ${\mathbf{l}}\sim P({\mathbf{l}}\|{\mathbf{s}}^{[1]},k)$ which will be used throughout the session. At each time step $t$, the agent only has a partial observation of the environment, computed by a rendering function ${\mathbf{o}}^{[t]}=o({\mathbf{s}}^{[t]})$. We assume that $P(k)$, $P({\mathbf{s}}^{[1]}\|k)$, $P({\mathbf{l}}\|{\mathbf{s}}^{[1]},k)$, and $o(\cdot)$ are all unknown to the agent. Below we use $\theta$ to denote the complete set of the model parameters, and let any function that uses a subset of parameters directly depend on $\theta$ for notational simplicity. The agent takes an action $a^{[t]}$ according to its policy $\pi_{\theta}$, given the current observation ${\mathbf{o}}^{[t]}$, the history ${\mathbf{h}}^{[t-1]}$, and the command ${\mathbf{l}}$: $$a^{[t]} \sim \pi_{\theta}(a^{[t]}\|{\mathbf{o}}^{[t]}, {\mathbf{h}}^{[t-1]}, {\mathbf{l}}),$$ where a latent vector ${\mathbf{h}}^{[t-1]}=h_{\theta}({\mathbf{o}}^{[1:t-1]},a^{[1:t-1]},{\mathbf{l}})$ summarizes all the previous history of the agent at time $t$ in the current session. The teacher observes the action and gives the agent a scalar reward $r^{[t]}=r(a^{[t]},{\mathbf{s}}^{[t]},{\mathbf{l}})$. Note that the reward depends not only on the state and the action, but also on the command. The environment then transitions to a new state ${\mathbf{s}}^{[t+1]}\sim P({\mathbf{s}}^{[t+1]}\|{\mathbf{s}}^{[t]},a^{[t]})$. This act-and-transition iteration goes on until a terminal state or the maximum time $T$ is reached. Our problem is to maximize the expected reward: $$\label{eq:obj} \max\limits_{\theta}\mathbb{E}_{\mathcal{S},\mathcal{A},{\mathbf{l}}}\left[\sum\limits_t\gamma^{t-1}r^{[t]}\right],$$ where $\mathcal{S}=({\mathbf{s}}^{[1]},\ldots,{\mathbf{s}}^{[t]},\ldots)$ and $\mathcal{A}=(a^{[1]},\ldots,a^{[t]},\ldots)$ denote the state and action sequences, respectively. $\gamma\in[0,1]$ is a discount factor. Note that $r(\cdot)$ and $P({\mathbf{s}}^{[t+1]}\|{\mathbf{s}}^{[t]},a^{[t]})$ are also unknown to the agent, namely, the RL agent is model-free. The objective Eq. \[eq:obj\] fits in the standard RL framework and is readily solvable by the actor-critic (AC) algorithm [@Sutton1999]. Specifically, we compute the following policy gradient for any time step $t$: $$\label{eq:gradient} -\mathbb{E}_{{\mathbf{s}}^{[t]},a^{[t]},{\mathbf{l}}} \left[\left(\nabla_{\theta}\log\pi_{\theta}(a^{[t]}\|{\mathbf{o}}^{[t]},{\mathbf{h}}^{[t-1]},{\mathbf{l}}) +\eta\nabla_{\theta}v_{\theta}({\mathbf{o}}^{[t]},{\mathbf{h}}^{[t-1]},{\mathbf{l}})\right)A^{[t]} + \kappa\nabla_{\theta}\mathcal{E}(\pi_{\theta})\right],$$ where $v_{\theta}$ is the estimated value function, $\mathcal{E}(\cdot)$ denotes entropy for encouraging exploration [@Mnih2016], and $\kappa>0$ and $\eta>0$ are constant weights. The advantage $A^{[t]}$ is computed as $$\label{eq:advantage} A^{[t]}=r^{[t]}+\gamma v_{\theta}({\mathbf{o}}^{[t+1]},{\mathbf{h}}^{[t]},{\mathbf{l}})-v_{\theta}({\mathbf{o}}^{[t]},{\mathbf{h}}^{[t-1]},{\mathbf{l}}).$$ Our implementation adopts the ParallelA2C design [@Clemente2017] to aggregate a minibatch of gradients (Eq. \[eq:gradient\]) over multiple identical agents running in parallel (each agent in a separate copy of the environment) over multiple time steps, with $\theta$ synchronized and shared among their models. Note in Eq. \[eq:gradient\] that the inputs of the policy network $\pi_{\theta}$ and the value network $v_{\theta}$ are identical. Thus we share a sub-network between $\pi_{\theta}$ and $v_{\theta}$ for parameter efficiency [@Mnih2016]. The sub-network outputs a latent state representation ${\mathbf{f}}^{[t]}$ and has two stages: $${\mathbf{f}}^{[t]} =f_{\theta}\left(m_{\theta}({\mathbf{o}}^{[t]},{\mathbf{l}}),{\mathbf{h}}^{[t-1]}\right).$$ The first stage $m_{\theta}$ is a multimodal function that grounds language in vision, and the second stage $f_{\theta}$ combines the grounding result with the agent history ${\mathbf{h}}^{[t-1]}$. We instantiate $m_{\theta}$ by using GFT for language grounding, and instantiate $f_{\theta}$ as a gated recurrent unit (GRU) [@Cho2014]. An overview of the agent architecture is illustrated in Figure \[fig:overview\]. We refer the reader to more details in Appendix \[app:ga2c\]. Related Work ============ [Virtual navigation.]{} Prior to this work, there have been several studies demonstrating virtual agents learning to navigate in virtual environments, based on reinforcement signals [@Jaderberg2017; @Mirowski2017]. Despite the impressive results achieved, these studies usually have fixed goals for agents. For example, an agent always learns to pick up apples or avoid enemies with specific appearances. In other words, the agent’s goals are fixed and cannot be changed, unless the rewards are modified followed by retraining. There is no language understanding involved: the perceptual inputs are images only. [Multi-goal virtual navigation.]{} A recent line of work augments the above virtual navigation with multiple non-linguistic goals. These goals are specified in different forms: target images [@Zhu2017a; @Zhu2017b], one-hot or continuous embeddings [@Brahmbhatt2017; @Gupta2017a; @Oh2017; @Savva2017; @Mirowski2018], target poses [@Gupta2017b], etc. In contrast, our focus is on understanding linguistic commands. [Language-directed virtual navigation.]{} Another recent line of work [@Hermann17; @Oh2017; @Chaplot18; @Das2018; @Yu2018; @Anderson2018; @Gordon2018] augments the virtual navigation problem with linguistic inputs, where an agent’s goal always depends on an instructed command. Accordingly, it is crucial for these methods to ground language in vision. Our GFT generalizes and improves some existing language grounding modules (details in Section \[sec:gfat\]), while incurring negligible additional costs in implementation and training time. [Visual question answering.]{} Unlike VQA [@Antol2015; @Lu2016; @Yang2016; @Perez2018; @Das2018; @Gordon2018], our problem does not require the agent to answer questions. Instead, the agent takes movement actions to respond to the teacher. However, both problems require language grounding, the study of which might be transferred from one problem to another. Indeed, in Section \[sec:exp\], the [FiLM]{} comparison module is adapted from @Perez2018, the [CGated]{} and [Concat]{} modules were adopted in some early work on VQA [@Antol2015], and the [Concept]{} comparison module resembles the stacked attention network (SAN) [@Yang2016] when there is only one single attention layer. Although GFT is proposed for embodied agents, we hope that it will also benefit research on VQA. [Grounding language in vision and robotics.]{} Our work is also related to language grounding in realistic images [@Yu2013; @Gao2016; @Rohrbach2016] and robotics navigation under language [@Chen2011; @Tellex2011; @Barrett2017], where static labeled datasets are required. In addition, these methods for language understanding usually employ structural assumptions specifically for their problems. In contrast, our GFT module is general-purpose and could potentially be easily applied to a wide range of problems that require language grounding. Experiments {#sec:exp} =========== We evaluate our agent in two challenging environments: <span style="font-variant:small-caps;">xworld2D</span> and <span style="font-variant:small-caps;">xworld3D</span> (Figure \[fig:envs\]). Both environments host the same set of five types of language-directed navigation tasks described in Table \[tab:tasks\]. A common syntax is shared by the two environments for generating task commands. Except object words, the remaining lexicon including grammatical and spatial-relation words, is also shared. Thus the only differences between <span style="font-variant:small-caps;">xworld2D</span> and <span style="font-variant:small-caps;">xworld3D</span> in our experiments are graphics and objects. Both environments generate random navigation sessions following the problem definition in Section \[sec:problem\]. Despite the huge difference between the visual structures of the two worlds, we apply a structurally identical agent to both of them. This identity includes the same network architecture, the same set of actions (`move_forward`, `move_backward`, `move_left`, `move_right`, `turn_left`, and `turn_right`), and the same set of hyperparameter values ([e.g.,]{} learning rate, batch size, momentum, layer sizes, [etc]{}). Only the model parameters are different and learned separately. Such an experiment setting tests the generalizability, efficacy, and portability of GFT as a language grounding module. \[tab:tasks\] Comparison Methods ------------------ We perform an apples-to-apples comparison with six state-of-the-art language grounding modules for embodied agents. To do so, we make minimal changes to our agent architecture when implementing the comparison methods: only the multimodal function $m_{\theta}$ is changed for each method, with the remaining components unchanged. Regardless of the choice, the output of $m_{\theta}$ is always flattened to a vector as an input to $f_{\theta}$. We assume that the input command is always first encoded into a fixed-length embedding ${\mathbf{l}}_{\text{BoW}}$ by a bag-of-words (BoW) encoder for training efficiency. The six comparison methods are: [@Hermann17; @Das2018; @Shu2018; @Gordon2018] directly concatenates a compact language embedding and a compact visual embedding. We project [^2] both ${\mathbf{l}}_{\text{BoW}}$ and ${\mathbf{C}}$ to the same dimension before the concatenation. [@Chaplot18] weights the feature maps of ${\mathbf{C}}\in \mathbb{R}^{D\times N}$ by a gate vector ${\mathbf{l}}_{\text{gate}}\in [0,1]^{D}$. In our case, ${\mathbf{l}}_{\text{gate}}$ is generated by a two-layer multilayer perceptron (MLP) from ${\mathbf{l}}_{\text{BoW}}$. [@Wu2018] is a variant of [Gated]{}. Instead of weighting feature maps of ${\mathbf{C}}$, they project ${\mathbf{C}}$ down to a visual embedding which is then weighted by ${\mathbf{l}}_{\text{gate}}$, a gate vector projected from ${\mathbf{l}}_{\text{BoW}}$ to the same dimension of the visual embedding. [@Perez2018] follows exactly Eq. \[eq:film\] which can be seen as a special case of our method. All $\lambda_{d}$ and $b_{d}$ are generated by a two-layer MLP from ${\mathbf{l}}_{\text{BoW}}$. [@Yu2018] directly treats ${\mathbf{l}}_{\text{BoW}}$ as a $1 \times 1$ filter. An attention map is obtained by convolving ${\mathbf{C}}$ with the filter. In addition, ${\mathbf{C}}$ is convolved with a $1\times 1$ filter at a stride of one to produce an environment map. Finally, the attention map and the environment map are concatenated. [@Anderson2018] makes several modifications to the original implementation to better suit our problem. First, we train the CNN from scratch. Second, we compute the instruction context directly as ${\mathbf{l}}_{\text{BoW}}$ without using word attention, since our teacher will not issue detailed, long-paragraph commands. Third, we add one additional layer of GRU after the concatenation of the decoder state and the instruction context, to model longer-range temporal dependency. Two variants of our agent are reported: [GFT-1]{} ($J=1$) and [GFT-2]{} ($J=2$)[^3]. We generate ${\mathbf{T}}_j$ by a two-layer MLP from ${\mathbf{l}}_{\text{BoW}}$. After training, each of the eight methods is evaluated for 10k test sessions, where the models saved for the final three passes (each pass contains 5k minibatches) of each method are used to obtain an average result. The comparison setting described in this section applies to both <span style="font-variant:small-caps;">xworld2D</span> and <span style="font-variant:small-caps;">xworld3D</span>. More details of the comparison methods and our method are described in Appendix \[app:model\]. Optimization Details -------------------- The optimization details described in this section apply to all the methods in both environments. We adopt RMSprop [@Tieleman2012] with a learning rate of $10^{-5}$, a damping factor of $\epsilon=0.01$, and a gradient moving average decay of $\rho=0.95$. The gradient has a momentum of 0.9. The batch size is set to $128$. The total number of training batches is 2 million. The parameters of each method are initialized with four different random seeds, the results of which are averaged and reported. Results for <span style="font-variant:small-caps;">xworld2D</span> ------------------------------------------------------------------ [Environment and action.]{} We modified the <span style="font-variant:small-caps;">xworld2D</span> environment [@Yu2018] to host our navigation tasks. The original fully-observable setting now becomes a partially-observable egocentric setting in 2D. The original action set is augmented with `turn_left` and `turn_right`, of which the yaw changes are both $90^{\circ}$. To increase the visual variance, at each session we randomly rotate each object and scale it randomly within $[0.5,1.0]$. Suppose each map is $X\times Y$, then a session will end after $3XY$ time steps if a success or failure is not achieved. In the experiment, we set $X=Y=8$. Each $8\times 8$ map contains 4 objects and 16 obstacles. We use Prim’s algorithm [@Prim1957] to randomly generate a minimal spanning tree for placing the obstacles so that the map is always a valid maze. The objects and the agent are then randomly initialized while complying with the sampled navigation task $k$. The agent can only see a $5\times 5$ area in front of it, excluding any region occluded by obstacles. Generalization to larger maps will be investigated in Appendix \[app:generalization\]. [Rewards.]{} A success (failure) according to the teacher’s command gives the agent a $+1$ ($-1$) reward. A failure is triggered whenever the agent hits any object that is not required by the command. The time penalty of each step is $-0.01$. No other extrinsic or intrinsic rewards are used. [Objects and vocabulary.]{} We use a collection of 345 object instances released by @Yu2018, constituting 115 object classes in total. The vocabulary contains 115 object words, 8 spatial-relation words, and 40 grammatical words, for a total size of 163. In total, there are 1,187,850 distinct sentences that can be generated by the teacher’s predefined context-free grammar (CFG). The lengths of these sentences range from 1 to 15. [Results.]{} The training curves of success rates are shown in Figure \[fig:curves\] (a). We observe that [GFT-1]{} has some marginal improvement on the success rates of the best-performing comparison methods such as [FiLM]{} and [Gated]{}. As we perform a second feature transformation, [GFT-2]{} produces a performance jump. Our explanation is that the visual recognition challenge, with random object yaws and scales in each session, requires an expressive language grounding function that can be better modeled by multiple steps of Eq. \[eq:staft\]. Table \[tab:rates\] (2D) shows the test results split into five navigation types. [GFT-2]{} produces the best numbers in all the five columns. Unsurprisingly, `nav_avoid` has the highest rate because the agent only has to go to an arbitrary target which is not the specified one. `nav` has the lowest rate because the agent cannot exploit any object arrangement pattern like in `nav_bw`. See Figure \[fig:2d-examples\] Appendix \[app:examples\] for example navigation sessions. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Success rates (averaged over four random restarts) vs. number of training samples (time steps). The shaded area around each curve denotes the standard deviation. Although each method trains the same number of minibatches, for details explained in Appendix \[app:ga2c\], the total number of actions taken by the agent might vary slightly for different methods. []{data-label="fig:curves"}](xworld2d_rates "fig:"){width="48.00000%"} ![Success rates (averaged over four random restarts) vs. number of training samples (time steps). The shaded area around each curve denotes the standard deviation. Although each method trains the same number of minibatches, for details explained in Appendix \[app:ga2c\], the total number of actions taken by the agent might vary slightly for different methods. []{data-label="fig:curves"}](xworld3d_rates "fig:"){width="48.00000%"} \(a) <span style="font-variant:small-caps;">xworld2D</span> \(b) <span style="font-variant:small-caps;">xworld3D</span> -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Results for <span style="font-variant:small-caps;">xworld3D</span> ------------------------------------------------------------------ [Environment and action.]{} The environment layout and the agent’s action are both discrete in <span style="font-variant:small-caps;">xworld3D</span>. An $X\times Y$ map consists of $XY$ square grids, each grid as a unit containing an object, an obstacle, or nothing. An object always has a unit scale and a random yaw when initialized. An obstacle has one of four scales: $1.0$, $0.7$, $0.5$, and $0.3$, which is randomly sampled when the obstacle is initialized. The agent walks (`move_{forward,backward,left,right}`) roughly half of a grid per time step. The yaw change of the agent when it turns (`turn_{left,right}`) is $45^{\circ}$ per time step. A session will end after $10XY$ time steps if a success or failure is not achieved. In the experiment, we set $X=Y=8$. Each $8\times 8$ map contains 4 objects and 16 obstacles, and its initialization follows the same process in the 2D case. Generalization to larger maps will be investigated in Appendix \[app:generalization\]. [Rewards.]{} The reward function is the same with the 2D case. [Objects and vocabulary.]{} <span style="font-variant:small-caps;">xworld3D</span> contains 88 different objects[^4]. There are three types of obstacles: brick, crate, and cube. The vocabulary is the same with the 2D case, except that there are 88 different object words and the vocabulary size is now 136. In total, there are 709,383 distinct sentences that can be generated by the teacher according to the same set of syntax rules in the 2D case. The lengths of these sentences also range from 1 to 15. [Results.]{} The training curves of success rates are shown in Figure \[fig:curves\] (b). We observe that [GFT-1]{} already has a huge advantage over the best-performing comparison method [FiLM]{}. On top of this, [GFT-2]{} shows a faster performance increase during the training time. This suggests that the original feature space cannot easily comply with various language commands. A rotation of the feature space depending on the input command, an operation which [FiLM]{} lacks but [GFT-1]{} owns, is important for producing better grounding results. Table \[tab:rates\] (3D) shows the test results split into five navigation types. Unlike in the 2D case, now for [GFT-2]{}, `nav` has the second-best success rate while `nav_dir` has the lowest rate. Visualization of the `nav_dir` test cases reveals that due to severe perspective distortion in 3D, the agent has some difficulty of grounding the spatial-relation words, especially when multiple objects are located nearby. But still, GFT agents obtain much better `nav_dir` results than the comparison methods (over a 15% increase on average). Several example navigation sessions are shown in Figure \[fig:3d-examples\] Appendix \[app:examples\]. Limitations ----------- While GFT is theoretically more expressive and empirically better than some of the existing language grounding modules, there are certain limitations of it. First, because each ${\mathbf{T}}_j$ is generated from the command, it requires a large projection matrix. In our implementation, the projection matrix that converts a hidden sentence embedding of length 128 to ${\mathbf{T}}_j$ has a size of $128\times D(D+1)=128\times 4160$ assuming $D=64$. Thus usually GFT has more parameters to be learned compared with its simplified versions like FiLM. An alternative might be to explicitly constrain ${\mathbf{T}}_j$ to be sparse and possibly low-rank. Second, GFT performs several steps of transformations, each of which has a different transformation matrix. This further linearly increases the number of learnable parameters. One solution would be to set ${\mathbf{T}}_1=\ldots={\mathbf{T}}_j=\ldots={\mathbf{T}}_J$, [i.e.,]{} do a recurrent feature transformation. However, this method has been observed slightly worse than the current GFT in the performance. Third, depending on the actual value of $J$, GFT might be slower in computation than other gated networks which perform a single transformation. These three issues are left to our future work. Conclusions =========== We have presented GFT, a simple but general neural language grounding module for embodied agents. GFT provides a unifying view of some existing language grounding modules, and further generalizes on top of them. The evaluation results on two challenging navigation environments suggest that GFT can be easily adapted from one problem to another robustly. Although evaluated on navigation, we believe that GFT could potentially serve as a general-purpose language grounding module for embodied agents that need to follow language instructions in a variety of scenarios. Appendices {#appendices .unnumbered} ============== Agent Architecture {#app:ga2c} ================== The agent history ${\mathbf{h}}^{[t]}$ has two constituents: an action history ${\mathbf{h}}_a^{[t]}$ summarizing previous taken actions and a visual history ${\mathbf{h}}_m^{[t]}$ summarizing previous visual experience. We instantiate $f_{\theta}$ and $h_{\theta}$ as GRUs (Figure \[fig:overview\]): $$\label{eq:arch} \begin{array}{r@{\hskip 0.04in}l} {\mathbf{h}}_m^{[t]} &= GRU(m_{\theta}({\mathbf{o}}^{[t]},{\mathbf{l}}), {\mathbf{h}}_m^{[t-1]}),\\ {\mathbf{h}}_a^{[t]} &= GRU(a^{[t]}, {\mathbf{h}}_a^{[t-1]}),\\ {\mathbf{f}}^{[t]} &= GRU({\mathbf{h}}_m^{[t]}, {\mathbf{h}}_a^{[t-1]}, {\mathbf{f}}^{[t-1]}),\\ {\mathbf{h}}^{[t]} &= ({\mathbf{h}}_a^{[t]}, {\mathbf{h}}_m^{[t]}).\\ \end{array}$$ Our RL training design is synchronous advantage actor-critic (ParallelA2C) [@Clemente2017]. We run $N_{\text{agent}}$ agents in parallel with model parameters shared among them to encourage exploration and reduce variance in the policy gradient. Each backpropagation is done with a minibatch of $N_{\text{batch}}$ time steps collected from all the agents, each agent contributing $\frac{N_{\text{batch}}}{N_{\text{agent}}}$ time steps. In either environment, every agent gets blocked until the network parameters are updated with the current minibatch, after which it forwards $\frac{N_{\text{batch}}}{N_{\text{agent}}}$ time steps again with the updated model parameters. To speed up training, we adopt an $n$-step temporal difference (TD) when computing the advantage $A^{[t]}$ (Eq. \[eq:advantage\]), in a forward manner similar to @Mnih2016. Finally, we empirically set the discount $\gamma$ to $0.99$, the entropy weight $\kappa$ to $0.05$, and the value regression weight $\eta$ to $1.0$ throughout the experiments. Sometimes an agent might provide fewer than $\frac{N_{\text{batch}}}{N_{\text{agent}}}$ actions for each minibatch due to the end of an episode, because once an agent hits an episode end, it will be initialized in a new session for the next minibatch. Thus the actual size of the minibatch might be smaller than $N_{\text{batch}}$. In the experiments, we set $N_{\text{agent}}=32$, and $N_{\text{batch}}=128$. The agent is trained purely from reward signals, without: prior visual knowledge such as a pre-trained CNN, prior linguistic knowledge such as a parser, or any auxiliary task such as image reconstruction [@Das2018; @Hermann17], reward prediction [@Hermann17], or language prediction [@Hermann17; @Yu2018]. Method Details {#app:model} ============== [General.]{} The sentence embedding ${\mathbf{l}}_{\text{BoW}}$ is obtained by sum-pooling word embeddings. A word embedding has a length of $128$, except for the [Concept]{} method (see below). The agent perceives $84\times 84$ egocentric RGB images in <span style="font-variant:small-caps;">xworld3D</span> and $80 \times 80$ egocentric RGB images in <span style="font-variant:small-caps;">xworld2D</span>. Regardless of the image dimensions, the CNN has three convolutional layers for processing the image: $(8, 4, 32)$, $(4, 2, 64)$, and $(3, 1, 64)$, where $(a,b,c)$ represents a layer configuration of $c$ filters of size $a\times a$ at a stride of $b$. Each action is embedded as a vector of size $128$ before being input to $GRU_{\theta}^a$ to generate the action history ${\mathbf{h}}_a$ which also has $128$ units. The other two recurrent layers $GRU_{\theta}^m$ and $GRU_{\theta}^f$ both have $512$ units, and both have an extra hidden layer of size $512$ to preprocess their inputs. The policy network is a two-layer MLP where the first layer has $512$ units and the second layer is a softmax for outputting actions. The value network is a two-layer MLP where the first layer has $512$ units and the second layer outputs a scalar value without any activation. Unless otherwise stated, all the layer outputs are ReLU activated. [Concat.]{} Both ${\mathbf{l}}_{\text{BoW}}$ and ${\mathbf{C}}$ are projected to a latent space of $512$ dimensions. [Gated.]{} The MLP for generating ${\mathbf{l}}_{\text{gate}}$ from ${\mathbf{l}}_{\text{BoW}}$ has two layers: the first layer has $128$ units and the second layer has $D=64$ units which are sigmoid activated. [CGated.]{} The gate vector ${\mathbf{l}}_{\text{gate}}$ has $512$ units which are sigmoid activated. [FiLM.]{} The MLP for generating $\lambda_d$ and $b_d$ has two layers: the first layer has $128$ units and the second layer has $D+1=65$ units without any activation. [Concept.]{} Because the sentence embedding ${\mathbf{l}}_{\text{BoW}}$ is directly used as the $1\times 1$ filter, each word embedding has a length of $D=64$. The attention map and environment map are both ReLU activated. [EncDec.]{} This method has a slightly reorganized computational flow compared to the one in Eq. \[eq:arch\]. We refer the reader to the original paper [@Perez2018] for details. Except this, the configuration of each layer is the same with that of its counterpart that can be found in [Concat]{}. [GFT.]{} The MLP for generating the transformation matrix ${\mathbf{T}}_j$ from ${\mathbf{l}}_{\text{BoW}}$ has two layers: the first layer has $128$ units and the second layer has $D\times (D+1)=64\times 65=4160$ units. The second layer has no activation. For [GFT-2]{}, we share the parameters of the first layers between ${\mathbf{T}}_1$ and ${\mathbf{T}}_2$. We set the activation function $g$ in Eq. \[eq:staft\] as ReLU. Vocabulary ========== The following 8 spatial-relation words and 40 grammatical words are shared between the <span style="font-variant:small-caps;">xworld2D</span> and <span style="font-variant:small-caps;">xworld3D</span>. [Spatial-relation]{} (8) [Grammatical]{} (40) ------------------------------ ---------------------------------------- behind, !, ., ?, and, anything, avoid, besides, but, can, collect, could, destination, between, do, done, end, except, go, by, goal, grid, in, is, location, front, move, navigate, not, object, of, left, place, please, reach, target, that, near, the, time, to, up, well, right. will, wrong, you, your. The two environments have two different sets of object words: [@l\|l@]{}\ <span style="font-variant:small-caps;">xworld3D</span> (88) & <span style="font-variant:small-caps;">xworld2D</span> (115)\ apples, backpack, barbecue, barrel, basket, & apple, armadillo, artichoke, avocado, banana, bat,\ basketball, bathtub, bed, bench, boiler, & bathtub, beans, bear, bed, bee, beet,\ books, bookshelf, bottle, bread, brush, & beetle, bird, blueberry, bookshelf, broccoli, bull,\ bucket, burger, cake, calender, camera, & butterfly, cabbage, cactus, camel, carpet, carrot,\ candle, car, carpet, cat, chair, & cat, centipede, chair, cherry, clock, coconut,\ chessboard, clock, comb, cooker, crib, & corn, cow, crab, crocodile, cucumber, deer,\ cup, dart, dog, dog-house, drawers, & desk, dinosaur, dog, donkey, dragon, dragonfly,\ drums, fan, fence, firehydrant, flashlight, & duck, eggplant, elephant, fan, fig, fireplace,\ flowers, fountain, gift, guitar, hair-dryer, & fish, fox, frog, garlic, giraffe, glove, horse,\ headphones, horse, iron, lamp, laptop, & goat, grape, greenonion, greenpepper, hedgehog,\ mailbox, milk, oven, pan, phone, & kangaroo, knife, koala, ladybug, lemon, light,\ photo, piano, pillow, plant, pool-table, & lion, lizard, microwave, mirror, monitor, monkey,\ puzzle, rabbit, rooster, scale, scissors, & monster, mushroom, octopus, onion, orange, ostrich,\ screen, slippers, sofa, speaker, squeezer, & owl, panda, peacock, penguin, pepper, pig,\ staircase, stove, sunglasses, table, & pineapple, plunger, potato, pumpkin, rabbit, racoon,\ table-tennis, toilet, towel, train, & rat, rhinoceros, rooster, seahorse, seashell, seaurchin,\ trampoline, trashcan, treadmill, tricycle, & shrimp, snail, snake, sofa, spider, squirrel,\ umbrella, vacuum, vase, wardrobe, & stairs, strawberry, tiger, toilet, tomato, turtle,\ wheelchair. & vacuum, wardrobe, washingmachine, watermelon,\ & whale, wheat, zebra.\ [Level]{} [Map size ($X=Y$)]{} [number of goals per map]{} [number of obstacles per map]{} --------------- -------------------------- --------------------------------- ------------------------------------- 1 3 2 0 2 4 2 3 3 5 2 6 4 6 4 9 5 7 4 12 6 8 4 16 : The curriculum used for training the agents.[]{data-label="tab:curriculum"} Curriculum Learning =================== To help the agent learn, we adopt curriculum learning [@Bengio2009] to gradually increase the environment size and complexity, according to the curriculum in Table \[tab:curriculum\]. The training always starts from level 1. During training, the teacher maintains the average success rate of each task type (Table \[tab:tasks\]), for a total of 200 most recent sessions. If at some point, all the five average success rates are above a predefined threshold of $0.7$, then the teacher allows the agent to enter the next level and resets the maintained rates. The progress of this curriculum is computed separately for each of the 32 agents running in parallel. The above curriculum applies to all the methods in both environments. It should be noted that the training curves and test results in Section \[sec:exp\] are computed for the final level with the maximal difficulty, without being affected by the curriculum. Generalization to Larger Maps {#app:generalization} ============================= We evaluate the agent models, trained for $X=Y=8$ in Section \[sec:exp\], on three larger maps: $X=Y=9$, with 6 goals and 20 obstacles, $X=Y=10$, with 6 goals and 24 obstacles, and $X=Y=11$, with 8 goals and 28 obstacles. The evaluation results are shown in Table \[tab:generalization-9\], Table \[tab:generalization-10\], and Table \[tab:generalization-11\], respectively. Although the performance is not as good as on $8\times 8$ maps (except for `nav_avoid` whose chance performance tends to peak as there are more goals on the map before the map size becoming too large), the GFT agents achieve reasonable generalizations and still greatly outperform the comparison methods. This further demonstrates that our GFT agents are not trained to memorize environments in specific settings. Navigation Examples {#app:examples} =================== Below we show some navigation examples for the [GFT-2]{} agents trained in 2D and 3D. For each session, we present four key frames on the navigation path. More full-length navigation sessions are shown in a video demo at <https://www.youtube.com/watch?v=bOBb1uhuJxg>. Visualization of Transformation Matrices {#app:visual} ======================================== In this section, we visualize the transformation matrices of several example commands received by the [GFT-2**]{} agent trained in <span style="font-variant:small-caps;">xworld3D</span>. We are particularly interested in comparing the ${\mathbf{T}}_1$ and ${\mathbf{T}}_2$ of two different commands that have the same semantics or have a minimal semantic difference. Figure \[fig:mats\] shows the visualization results. Unsurprisingly, we observe that Two completely different commands with the same semantics (for our problem) will yield almost identical transformation matrices (Figure \[fig:mats\] row 1). Two commands with a minimal difference will yield matrices similar in general but differ in small places that capture the difference (Figure \[fig:mats\] rows 2-4). [^1]: Video demo at <https://www.youtube.com/watch?v=bOBb1uhuJxg> [^2]: In the remainder of this paper, a projection denotes a fully-connected (FC) layer followed by a nonlinear activation function. [^3]: According to our observation, $J=3$ appears to be a saturation point whose performance has almost no gain over $J=2$. [^4]: Downloaded from <http://www.sweethome3d.com/freeModels.jsp>
--- abstract: 'We show that biological networks with serial regulation (each node regulated by at most one other node) are constrained to [direct functionality]{}, in which the sign of the effect of an environmental input on a target species depends only on the direct path from the input to the target, even when there is a feedback loop allowing for multiple interaction pathways. Using a stochastic model for a set of small transcriptional regulatory networks that have been studied experimentally [@Guet:2002p38], we further find that all networks can achieve all functions permitted by this constraint under reasonable settings of biochemical parameters. This underscores the functional versatility of the networks.' author: - Andrew Mugler - Etay Ziv - Ilya Nemenman - 'Chris H. Wiggins' bibliography: - 'qbio\_paper4.bib' title: 'Serially-regulated biological networks fully realize a constrained set of functions' --- A driving question in systems biology in recent years has been the extent to which the topology of a biological network determines or constrains its function. Early works have suggested that the function follows the topology [@ShenOrr:2002p101; @Mangan:2003p41; @Guet:2002p38; @Kollmann:2005p103], and this continues as a prevailing view even though later analyses (at least in a small corner of biology) have questioned the paradigm [@Wall:2005p20; @Ziv:2007p88]. It remains unknown if a small biochemical or regulatory network can perform multiple functions, and whether the function set is limited by the network’s topological structure. To this extent, in this paper, we develop a mathematical description of the functionality of a certain type of biological network, and show that the answer to both questions is “yes”: the networks can perform many, but not all possible functions, and the set of attainable functions is constrained by the topology. We illustrate these results in the context of an experimentally realized system [@Guet:2002p38]. Following [@Guet:2002p38] and our earlier work [@Ziv:2007p88], we focus on the steady-state functionality of transcriptional regulatory networks. In this case, the input is the “chemical environment,” that is a binary vector of presence/absence of small molecules that affect the regulation abilities of the transcription factors; and the output is the steady-state expression of a particular gene, hereafter called the reporter. Different functions of the network correspond then to different ways to map the small molecule concentrations into the reporter expression. In our setup, the effect of introducing a small molecule S$_j$ specific to a transcription factor X$_j$ is to modify the affinity of X$_j$ to its binding site. Equivalently one can think of S$_j$ as modulating or renormalizing the transcription factor concentration $X_j$ by some factor $s_j$, making the effective concentration $\chi_j = \chi_j(X_j, s_j)$. A simple example of such a modulation function is $$\label{eq:chi0} \chi_j(X_j, s_j) \equiv X_j/s_j,$$ in which the presence of the small molecule reduces the effective concentration of transcription factor by the factor $s_j$. The function of the circuit will depend on how the steady-state expression $G^$ of the reporter gene G changes as the modulation factor $s_j$ is varied from some “off” value $s_j^-$ to some “on” value $s_j^+$: $$\label{eq:deltaG} \frac{\Delta G^}{\Delta s_j} = \frac{G^(s_j^+) - G^(s_j^-)}{\Delta s_j} = \frac{1}{\Delta s_j}\int_{s_j^-}^{s_j^+} \frac{dG^}{ds_j} \, ds_j,$$ where $\Delta s_j = s_j^+ - s_j^-$. For example, if $\chi_j = X_j/s_j$, then $s_j^- = 1$, indicating that the small molecule is absent, and $s_j^+ > 1$ is the factor by which effective concentration is reduced when the small molecule is present. If the sign of $dG^/ds_j$ does not change for $s_j\in [s_j^-, s_j^+]$, then the sign of $\Delta G^$ is fixed. For networks with only serial regulation, i.e. each gene is regulated by at most one other gene, we will show that the sign of $dG^/ds_j$ is unique and in accord with the direct path from S$_j$ to G, a property we term [direct functionality]{}. This constrains the possible responses and hence the functionality of serial networks. Importantly, we will then show that all admissible functions indeed can be attained by all the networks we studied operating at different parameter values. While throughout this work we focus on the setup pioneered experimentally by Guet et al. [@Guet:2002p38], we also show that the constraint to direct functionality holds for any network with serial regulation. Direct functionality in small networks {#sec:direct} ====================================== As in Guet et al. [@Guet:2002p38], we consider networks with $N$ = 4 genes (three transcription factors plus a reporter G), in which each gene is regulated by exactly one other gene. This admits three topologies and a total of 24 networks, as described in Fig.\[fig:topos\]. All three topologies consist of a cycle and a cascade that begins in the cycle and ends at the reporter gene G. Once outside the cycle, there is only one path to G, so it suffices to study a topology consisting of an $n$-gene cycle with a gene G immediately outside (Fig. \[fig:topos\]c is an example with $n = 3$), and extensions to topologies where the cycle is connected to the reporter by a linear cascade are trivial. ![Four-gene networks (3 transcription factors X$_i$ plus 1 reporter gene G) in which each gene is regulated by one other gene, as studied in [@Guet:2002p38; @Ziv:2007p88]. Transcription factor efficacies are influenced by small molecules S$_i$. Regulation functions $\tilde{\alpha}_i$ are assigned to the edges. The three edges $\tilde{\alpha}_1$, $\tilde{\alpha}_2$, and $\tilde{\alpha}_3$ can be up-regulating or down-regulating, giving $3 \times 2^3 = 24$ possibilities; the reporter gene is repressed in all cases.[]{data-label="fig:topos"}](topos_pic.pdf) In this section, we will perform the steady-state analysis of such single-cycle networks to lay the groundwork for understanding the effect of topology on allowed functionality. The process of protein expression has been modeled with remarkable success by combining transcription and translation into one step and directly coupling genes by a deterministic dynamics [@Elowitz:2000p37; @Gardner:2000p77; @Hasty:2001p84]. Accordingly we model mean expressions $\bar{X}_i$ (we later distinguish between entire probability distributions $P(X_i)$ and the means of these distributions $\bar{X}_i$; cf. Appendix) with the system of ordinary differential equations $$\begin{aligned} \label{eq:ode1} \frac{d\bar{X}_1}{dt} &=& \tilde{\alpha}_1(\bar{\chi}_n) - r_1 \bar{X}_1,\\ \label{eq:odei} \frac{d\bar{X}_i}{dt} &=& \tilde{\alpha}_i(\bar{\chi}_{i-1}) - r_i \bar{X}_i \quad (2 \leq i \leq n),\\ \label{eq:odeG} \frac{d\bar{G}}{dt} &=& \tilde{\alpha}_{n+1}(\bar{\chi}_n) - r_{n+1} \bar{G},\end{aligned}$$ where the $\tilde{\alpha}_i$ are creation rates for the species X$_i$ (and $X_{n+1} \equiv G$), each monotonically regulated by the effective concentration $\bar{\chi}_{\pi_i}$ of its parent $\pi_i$, and the $r_i$ are the decay rates. Note that we have set $$\begin{aligned} \pi_1 &=& n, \\ \pi_i &=& i-1 \quad (2 \leq i \leq n+1)\end{aligned}$$ to create the $n$-gene cycle with one gene immediately outside. The regulation functions $\tilde{\alpha}_i$ will be up- or down-regulating according to the network topology. A common example is the familiar Hill functions, $$\begin{aligned} \label{eq:hill1_0} \tilde{\alpha}(\bar{\chi}) &=& a_0 + a\frac{\bar{\chi}^h}{K^h + \bar{\chi}^h} \quad \textrm{(up-regulating)},\\ \label{eq:hill2_0} \tilde{\alpha}(\bar{\chi}) &=& a_0 + a\frac{K^h}{K^h + \bar{\chi}^h} \quad \textrm{(down-regulating)},\end{aligned}$$ with basal and maximal expression levels $a_0$ and $a_0 + a$ respectively, Michaelis-Menten constants $K$, and cooperativities $h$. Although we use the functional forms in Eqns. (\[eq:hill1\_0\]-\[eq:hill2\_0\]), as well as the functional form for the modulation function in Eqn. (\[eq:chi0\]), for our numerical experiment (cf. Numerical Results), the analytic result derived in this section will be valid for any monotonic functions $\tilde{\alpha}(\bar{\chi})$ and any function $\bar{\chi}(\bar{X}, s)$. Fixed points of the dynamical system in Eqns. (\[eq:ode1\]-\[eq:odeG\]) satisfy $$\begin{aligned} \label{eq:fp1} \bar{X}_1^* &=& \alpha_1(\bar{\chi}_n^),\\ \bar{X}_i^ &=& \alpha_i(\bar{\chi}_{i-1}^) \quad (2 \leq i \leq n),\\ \label{eq:fp3} \bar{G}^ &=& \alpha_{n+1}(\bar{\chi}_n^),\end{aligned}$$ where we define $$\alpha_i \equiv \tilde{\alpha}_i/r_i.$$ We may now, as in [@Kholodenko:1997p55; @Kholodenko:2002p43], use the chain rule to calculate the derivative of $\bar{G}^$ with respect to a particular input factor $s_j$. For illustration, we will do so first for the concrete example in Fig. \[fig:topos\]c, in which $n=3$. Let us consider the derivative of $\bar{G}^$ with respect to $s_1$: $$\label{eq:3chain1} \frac{d\bar{G}^}{ds_1} = \frac{\partial\alpha_4}{\partial \bar{X}_3} \frac{\partial\alpha_3}{\partial \bar{X}_2} \left[\frac{\partial\alpha_2}{\partial s_1}+\frac{\partial\alpha_2}{\partial \bar{X}_1}\frac{d\bar{X}_1^}{d s_1}\right],$$ where all derivatives are evaluated at the fixed point, and it is understood that $\alpha_i$ depends on either $\bar{X}_{\pi_i}$ or $s_{\pi_i}$ through $\bar{\chi}_{\pi_i}$, that is, that $$\label{eq:chiXs} \frac{\partial\alpha_i}{\partial \bar{X}_{\pi_i}} = \frac{\partial\alpha_i}{\partial \bar{\chi}_{\pi_i}} \frac{\partial\bar{\chi}_{\pi_i}}{\partial \bar{X}_{\pi_i}} \quad {\rm and} \quad \frac{\partial\alpha_i}{\partial s_{\pi_i}} = \frac{\partial\alpha_i}{\partial \bar{\chi}_{\pi_i}} \frac{\partial\bar{\chi}_{\pi_i}}{\partial s_{\pi_i}}.$$ If we introduce the notation $$\begin{aligned} \alpha_i' &\equiv& \partial\alpha_i/\partial \bar{X}_{\pi_i}, \\ \dot{\alpha_i} &\equiv& \partial \alpha_i/\partial s_{\pi_i},\end{aligned}$$ then Eqn. (\[eq:3chain1\]) becomes $$\label{eq:3chain2} \frac{d\bar{G}^}{ds_1} = \alpha'_4 \alpha'_3 \left[ \dot{\alpha}_2 + \alpha'_2 \frac{d\bar{X}_1^}{ds_1} \right].$$ The first term reflects the direct chain to G from S$_1$, and the second term incorporates further contributions around the cycle and will need to be evaluated self-consistently. For a cycle of arbitrary length $n$ and for an arbitrary input factor $s_j$ ($1 \le j \le n$), Eqn. (\[eq:3chain2\]) generalizes to $$\label{eq:dGds} \frac{d\bar{G}^}{ds_j} = \left[ \dot{\alpha}_{j+1} + \alpha'_{j+1} \frac{d\bar{X}_j^}{ds_j} \right] \prod_{k=j+2}^{n+1} \alpha'_k,$$ where we use the convention that $$\prod_{k=a}^b [\cdot] = 1 \quad {\rm if} \quad a>b.$$ We may also use the chain rule for $d\bar{X}_j^/ds_j$, $$\begin{aligned} \frac{d\bar{X}_j^}{ds_j} &=& \left[ \dot{\alpha}_{(j \, {\rm mod} \, n)+1} + \alpha'_{(j \, {\rm mod} \, n)+1} \frac{d\bar{X}_j^}{ds_j} \right] \nonumber \\ && \times \frac{\prod_{k=1}^n \alpha'_k}{\alpha'_{(j \, {\rm mod} \, n)+1}}\end{aligned}$$ and now we may solve for $d\bar{X}_j^/ds_j$ self consistently: $$\label{eq:dXjdsj} \frac{d\bar{X}_j^}{ds_j} = \frac{\dot{\alpha}_{(j \, {\rm mod} \, n)+1} }{\alpha'_{(j \, {\rm mod} \, n)+1}} \frac{\prod_{k=1}^n \alpha'_k}{ 1 - \prod_{l=1}^n \alpha'_l}.$$ For the special case of $j=n$, where $(j \, {\rm mod} \, n)+1 = 1$, substituting Eqn. (\[eq:dXjdsj\]) into Eqn. (\[eq:dGds\]) obtains $$\begin{aligned} \frac{d\bar{G}^}{ds_n} &=& \left[ \frac{1}{1 - \prod_{l=1}^n \alpha'_l} \right] \nonumber \\ && \times \left[ \dot{\alpha}_{n+1} + ( \dot{\alpha}_1 \alpha'_{n+1} - \dot{\alpha}_{n+1} \alpha'_1 ) \prod_{k=2}^n \alpha'_k \right]\\ %\frac{d\bar{G}^}{ds_n} &=& % \left[ \frac{1}{1 - \prod_{l=1}^n \alpha'_l} \right] % \left[ \dot{\alpha}_{n+1} + ( \dot{\alpha}_1 \alpha'_{n+1} - % \dot{\alpha}_{n+1} \alpha'_1 ) \prod_{k=2}^n \alpha'_k \right]\\ \label{eq:dGdsn} &=& \left[ \frac{1}{1 - \prod_{l=1}^n \alpha'_l} \right] \dot{\alpha}_{n+1},\end{aligned}$$ where the second step follows from $$\begin{aligned} \dot{\alpha}_1 \alpha'_{n+1} &=& \left( \frac{d\alpha_1}{d\bar{\chi}_n}\frac{\partial \bar{\chi}_n}{\partial s_n} \right) \left( \frac{d\alpha_{n+1}}{d\bar{\chi}_n}\frac{\partial \bar{\chi}_n}{\partial \bar{X}_n} \right)\\ &=& \left( \frac{d\alpha_1}{d\bar{\chi}_n}\frac{\partial \bar{\chi}_n}{\partial \bar{X}_n} \right) \left( \frac{d\alpha_{n+1}}{d\bar{\chi}_n}\frac{\partial \bar{\chi}_n}{\partial s_n} \right)\\ &=& \alpha'_1 \dot{\alpha}_{n+1},\end{aligned}$$ in which the first step recalls Eqn. (\[eq:chiXs\]). For $1 \leq j \leq n-1$, where $(j \, {\rm mod} \, n)+1 = j+1$, substituting Eqn. (\[eq:dXjdsj\]) into Eqn. (\[eq:dGds\]) obtains $$\label{eq:dGds_1} \frac{d\bar{G}^}{ds_j} = \left[ \frac{1}{1 - \prod_{l=1}^n \alpha'_l} \right] \dot{\alpha}_{j+1} \prod_{k=j+2}^{n+1} \alpha'_k,$$ which, upon inspection of Eqn. (\[eq:dGdsn\]), is valid for $j=n$ as well. Stability of the fixed point $\bar{X}^_j$ requires that the Jacobian of Eqns. (\[eq:ode1\]-\[eq:odei\]), $$\label{eq:jacobian} J = \left[ \begin{array}{cccccc} -r_1 &&&&& \tilde{\alpha}'_1 \\ \tilde{\alpha}'_2 & -r_2 &&&& \\ & \tilde{\alpha}'_3 & -r_3 &&& \\ && \ddots & \ddots && \\ &&& \tilde{\alpha}'_{n-1} & -r_{n-1} & \\ &&&& \tilde{\alpha}'_n & -r_n \\ \end{array} \right],$$ be negative definite or, since the determinant is the product of the eigenvalues, that $$\begin{aligned} 0 &<& (-1)^n\det(J)\\ &&= \prod_{k=1}^n r_k - \prod_{l=1}^{n} \tilde{\alpha}'_l\\ \label{eq:stability} &&= \prod_{k=1}^n r_k \left( 1 - \prod_{l=1}^{n} \alpha'_l \right).\end{aligned}$$ Since the decay rates $r_k$ are positive, Eqn. (\[eq:stability\]) says that the term inside the brackets in Eqn. (\[eq:dGds\_1\]) is positive for stable fixed points. For the networks in Fig. \[fig:topos\], where in the 1- and 2-cycles the reporter is attached by means of intermediates, the analog of Eqn. (\[eq:dGds\_1\]) is calculated similarly to be $$\label{eq:dGds24} \frac{d\bar{G}^}{ds_j} = \left[ \frac{1}{1 - \theta(n-j) \prod_{l=1}^n \alpha'_l} \right] \dot{\alpha}_{j+1} \prod_{k=j+2}^{N} \alpha'_k,$$ where $N=4$ is the number of genes, $1 \le j \le N-1$ for each of the 3 possible small molecule inputs, and $n$ is the length of the cycle ($1 \le n \le N-1$). Here $\theta$ is the Heaviside function, for which we use the convention $\theta(0) = 1$. Its presence reduces the bracketed term to 1 when the input S$_j$ is outside the cycle, leaving only the contribution corresponding to the cascade from S$_j$ to G, as must be the case. In Eqn. (\[eq:dGds24\]), the term outside the brackets represents the direct (i.e., the shortest) path from S$_j$ to G and fixes the sign of $d\bar{G}^/ds_j$ (since the term inside the brackets is positive at a stable fixed point). If the creation rates are monotonic (which is the usual model for transcriptional regulation, but may be violated in protein signaling due to competitive inhibition and other effects), this sign is unique and fixes the sign of $\Delta \bar{G}^/\Delta s_j$ via Eqn.(\[eq:deltaG\]). Importantly, this says that the feedback in each of the topologies in Fig. \[fig:topos\] is irrelevant in determining the sign of $\Delta \bar{G}^/\Delta s_j$ for a steady-state analysis. As an example, for the network in Fig. \[fig:func\]a (inset), $\bar{G}^$ changes with increasing $s_1$ according to $\dot{\alpha}_2\alpha'_3\alpha'_4$, which, since S$_1$ inhibits the activation, is negative $\times$ positive $\times$ negative $=$ positive, just as one would expect if the feedback was ignored. Direct functionality corresponds to specific orderings of output states ----------------------------------------------------------------------- Consider the case in which there are only two small molecule inputs, S$_1$ and S$_2$, as in Fig. \[fig:func\]a (inset). Since each input can be absent or present, $S_1, S_2 \in \{-, +\}$, there are four chemical input states $c = S_1S_2 \in \{--, -+, +-, ++\}$. Direct functionality admits only two orderings of the four output states $\bar{G}^_c$, and hence the functionality of the network is severely limited by its serial topology. To see this, note that for Fig. \[fig:func\]a (inset) we have $$\begin{aligned} \Delta \bar{G}^/\Delta s_1 \geq 0 &\Rightarrow& \bar{G}^_{+-} \geq \bar{G}^_{--} \textrm{ and} \nonumber \\ && \bar{G}^_{++} \geq \bar{G}^_{-+};\\ \Delta \bar{G}^/\Delta s_2 \geq 0 &\Rightarrow& \bar{G}^_{-+} \geq \bar{G}^_{--} \textrm{ and} \nonumber \\ && \bar{G}^_{++} \geq \bar{G}^_{+-}. %\Delta \bar{G}^/\Delta s_1 \geq 0 &\Rightarrow& \quad \bar{G}^_{+-} \geq \bar{G}^_{--} % \quad \textrm{and} \quad \bar{G}^_{++} \geq \bar{G}^_{-+};\\ %\Delta \bar{G}^/\Delta s_2 \geq 0 &\Rightarrow& \quad \bar{G}^_{-+} \geq \bar{G}^_{--} % \quad \textrm{and} \quad \bar{G}^_{++} \geq \bar{G}^_{+-}.\end{aligned}$$ These conditions permit only the following output orderings, irrespective of biochemical parameters: $$\begin{aligned} \bar{G}^_{--} &\leq& \bar{G}^_{-+} \leq \bar{G}^_{+-} \leq \bar{G}^_{++} \textrm{ or} \nonumber \\ \label{eq:2ords} \bar{G}^_{--} &\leq& \bar{G}^_{+-} \leq \bar{G}^_{-+} \leq \bar{G}^_{++}.\end{aligned}$$ These two orderings nevertheless allow the realization of a significant subset of all possible logical functions that one can build with two binary inputs, depending on the distinguishability of the four output states, as described in the next section. Quantifying the distinguishability demands careful treatment of the noise with a stochastic equivalent of our deterministic dynamical system, as described in the Appendix. Numerical results {#sec:results} ================= We numerically solved the system in Eqns.(\[eq:ode1\]-\[eq:odeG\]) \[with stochastic effects given by Eqn.(\[eq:lyap\])\] with many parameter settings for all 24 networks represented in Fig. \[fig:topos\]. In addition to verifying the restriction to direct functionality, we find that all networks can achieve all possible direct functions, suggesting that the networks are still quite versatile within the functional constraint. For all networks, we consider the case of two small molecule inputs S$_1$ and S$_2$, as in the experimental setup of Guet et al. [@Guet:2002p38], and as shown for an example network in Fig. \[fig:func\]a (inset). We take $s_j$ to be a multiplicative factor by which the transcription factor concentration $\bar{X}_j$ is effectively scaled, i.e.$$\label{eq:chi} \bar{\chi}_j(\bar{X}_j, s_j) \equiv \bar{X}_j/s_j.$$ Then $s_j^- \equiv 1$ for the “off” settings, and the $s_j^+ > 1$ are free parameters for the “on” settings. We model the regulation using the familiar Hill form (which is monotonic and thus satisfies the direct functionality conditions) $$\begin{aligned} \label{eq:hill1} \tilde{\alpha}(\bar{\chi}) &=& a_0 + a\frac{\bar{\chi}^h}{K^h + \bar{\chi}^h} \quad \textrm{(up-regulating)},\\ \label{eq:hill2} \tilde{\alpha}(\bar{\chi}) &=& a_0 + a\frac{K^h}{K^h + \bar{\chi}^h} \quad \textrm{(down-regulating)},\end{aligned}$$ with basal and maximal expression levels $a_0$ and $a_0 + a$ respectively, Michaelis-Menten constants $K$, and cooperativities $h$. For the 4-gene networks in Fig. \[fig:topos\], with only two small molecule inputs S$_1$ and S$_2$, this gives 22 parameters in total (cf. Table \[tab:params\]). Parameters Range ------------------------------------- --------------------- decay rates, $r_i$ $10^{-4} - 10^{-3}$ Michaelis-Menten constants, $K_i$ $10^0 - 10^3$ basal expression levels, $a_{0, i}$ $10^{-3} - 10^{-2}$ expression level ranges, $a_i$ $10^0 - 10^2$ cooperativities, $h_i$ $10^0 - 10^1$ “on” input factors, $s_j^+$ $10^2 - 10^3$ : Parameters and ranges from which each is randomly drawn, with $1 \leq i \leq 4$ for the four genes, and $1 \leq j \leq 2$ for the two small molecule inputs. Ranges are representative of typical cell conditions [@Elowitz:2000p37; @weiss].[]{data-label="tab:params"} For a given parameter set, we numerically solve Eqns.(\[eq:ode1\]-\[eq:odeG\]) (using Matlab’s `ode15s`) for each input state $c \in \{--, -+, +-, ++\}$ to find mean steady-state concentrations $\bar{G}^_c$. We then solve Eqn. (\[eq:lyap\]) to find fluctuations around these means, giving probability distributions $P(G^\|c)$ (cf. Appendix). The function is defined by the ranking of the conditional distributions $P(G^\|c)$. That is, if two distributions are distinguishable, then the one with the larger mean is ranked higher. We consider two distributions to be indistinguishable when their means are separated by less than the smaller of their standard deviations (alternative definitions do not change our results qualitatively), in which case they both take on the average of their two ranks. When there are only two distinguishable output states, this rank-based classification reduces to that defining the familiar binary logical functions [AND]{}, [OR]{}, [XOR]{} etc. (see, for example, Fig. \[fig:func\]b, (ii-v)). More generally, for one, two, three, and four distinguishable responses, there are 75 total rankings (as listed on the horizontal axis of Fig. \[fig:func\]a). However, only 12 of these satisfy the ordering constraints for each network analogous to those in Eqn.(\[eq:2ords\]) and therefore correspond to direct functions (for the newtork in Fig. \[fig:func\]a these 12 are shown in green on the horizontal axis). We ran 50,000 trials for each of the 24 networks, in which the parameters were randomly selected (using a distribution uniform in log-space) from the ranges in Table \[tab:params\]. We found the steady-state reporter expression distributions $P(G^\|c)$ and classified the responses by ranking. All 24 networks displayed only direct functions. However, every network was able to achieve all 12 of its direct functions with parameters selected via Table \[tab:params\], meaning that the networks fully realized all the functionality allowed by the constraint. This suggests that the networks studied are both constrained and versatile, and that a cell may still use a serial network to perform multiple logical functions by varying biochemical parameters, despite the restriction to direct functionality. Fig. \[fig:func\] shows a histogram of functions and an example of each type of direct function for a representative network. We note that Guet et al. experimentally observed both direct and indirect functions [@Guet:2002p38]. However, they explicitly call the indirect functions into question, citing several possible unanticipated effects including RNA polymerase read-through. We have not incorporated such effects into the current model. Multiple fixed points ===================== For the 12 networks in which the overall sign of the feedback cycle is positive, there are parameter settings that support multiple stable fixed points. In this section we evaluate the extent to which the presence of multiple fixed points affects the constraint to direct functionality, and we find that violation of the constraint is possible but unlikely. While the function of a network has been defined in terms of $P(G^\|c)$, the linear noise approximation (cf. Appendix) only gives us access to $P(G^\|c, \bar{\bf X}^_m)$, the distribution expanded around a particular fixed point $\bar{\bf X}^_m$. The two are related by a weighted sum, $$P(G^\|c) = \sum_m \pi_m P(G^\|c, \bar{\bf X}^_m),$$ where the probabilities $\pi_m$ of being near the $m$th fixed point will depend on the basins of attraction and curvatures near the fixed points. Numerical solution for $P(G^\|c)$ directly is possible in principle, although computationally difficult. Whether the statistical steady state distribution is calculated numerically or is approximated as in this manuscript, if we continue to define the function of the network by the ranking of the means of the $P(G^\|c)$, we have $$\begin{aligned} \frac{d\bar{G}^}{ds_j} &=& \frac{d}{ds_j} \int dG^* G^* P(G^\|c) \\ &=& \sum_m \frac{d}{ds_j} \pi_m \int dG^ G^* P(G^\|c, \bar{\bf X}^_m) \\ \label{eq:dGdsm} &=& \sum_m \left( \pi_m \frac{d\bar{G}^_m}{ds_j} + \frac{d\pi_m}{ds_j} \bar{G}^_m \right).\end{aligned}$$ The expressions for the individual $d\bar{G}^_m/ds_j$ are given by Eq. (\[eq:dGds24\]), so the first term in Eqn. (\[eq:dGdsm\]) exhibits direct functionality. If the weights $\pi_m$ do not depend appreciably on the $s_j$, the second term will be small, and the restriction to direct functionality will be maintained. If, on the other hand, the weights do change appreciably (an obvious case might be the presence of a bifurcation at a particular value of $s_j$), then the second term may overpower the first enough to change the sign of $\Delta \bar{G}^/\Delta s$ and violate the restriction to direct functionality. We investigate this effect in two ways. First, we show analytically that, in the case of a 1-cycle, crossing a bifurcation does not violate direct functionality. Second, we subject all positive-feedback networks to a numerical test to estimate the dependence of the weights $\pi_m$ on the $s_j$. The results of both techniques suggest that the likelihood of a violation of direct functionality due to the presence of multiple fixed points is low. Bifurcations do not violate direct functionality (1-D) ------------------------------------------------------ Consider the case of a positive 1-cycle with a gene G immediately outside, as shown in Fig. \[fig:1cyc\]a (inset). For $n=1$, Eqns. (\[eq:fp1\]-\[eq:fp3\]) become $$\begin{aligned} \bar{X}^* &=& \alpha_1(\bar{\chi}^),\\ \label{eq:fp2_1} \bar{G}^ &=& \alpha_2(\bar{\chi}^),\end{aligned}$$ where unnecessary subscripts are dropped and $\bar{\chi} = \bar{X}/s$ as in Eqn. (\[eq:chi\]). With $\alpha_1$ of the form in Eqn.(\[eq:hill1\]), there are at most two stable fixed points $\bar{X}_1^$ and $\bar{X}_2^$, with $\bar{X}_1^ < \bar{X}_2^$, as illustrated by an example in Fig. \[fig:1cyc\]a. As shown in Fig.\[fig:1cyc\]b, bifurcations occur at $s_1$ and $s_2$ such that only $\bar{X}_2^$ exists when $s < s_1$, only $\bar{X}_1^$ exists when $s > s_2$, and $\bar{X}_1^$ and $\bar{X}_2^$ are found with (unknown) probabilities $\tilde{\pi}_1(s)$ and $\tilde{\pi}_2(s) = 1 - \tilde{\pi}_1(s)$ respectively when $s_1 < s < s_2$. These statements can be combined such that $$\pi_1(s) = \theta(s-s_1)\theta(s_2-s)\tilde{\pi}_1(s) + \theta(s-s_2)$$ and $\pi_2(s) = 1 - \pi_1(s)$ define the probabilities of approaching $\bar{X}_1^$ and $\bar{X}_2^$ respectively for any $s$. Here $\theta$ is the Heaviside function. As we go from an “off” value $s^-$ to an “on” value $s^+$, let us assume that we hit both bifurcations, such that $s^- < s_1 < s_2 < s^+$. To test for direct functionality, we investigate the sign of $$\begin{aligned} \frac{\Delta \bar{G}^}{\Delta s} &=& \frac{1}{\Delta s}\int_{s^-}^{s^+} \sum_m \pi_m \frac{d\bar{G}^_m}{ds} \, ds \nonumber \\ && + \frac{1}{\Delta s}\int_{s^-}^{s^+} \sum_m \frac{d\pi_m}{ds} \bar{G}^_m \, ds \\ % \frac{1}{\Delta s}\int_{s^-}^{s^+} \sum_m \pi_m \frac{d\bar{G}^_m}{ds} \, ds % + \frac{1}{\Delta s}\int_{s^-}^{s^+} \sum_m \frac{d\pi_m}{ds} \bar{G}^_m \, ds \\ &\equiv& T_1 + T_2,\end{aligned}$$ obtained using Eqns. (\[eq:deltaG\]) and (\[eq:dGdsm\]). The first term $T_1$ depends on $$\frac{d\bar{G}^_m}{ds} = \frac{\dot{\alpha_2}}{1-\alpha'_1}$$ (from Eqn. (\[eq:dGds\_1\]); $\alpha' \equiv \partial \alpha/\partial \bar{X}$ and $\dot{\alpha} \equiv \partial \alpha/\partial s$ as before, both evaluated at the $m$th fixed point), which, as previously discussed, is always of the sign of $\dot{\alpha}_2$, consistent with direct functionality. The second term $T_2$ can be written $$\begin{aligned} T_2 &=& \frac{1}{\Delta s}\int_{s^-}^{s^+} \left( \frac{d\pi_1}{ds} \bar{G}^_1 + \frac{d\pi_2}{ds} \bar{G}^_2 \right) \, ds \\ &=& \frac{1}{\Delta s}\int_{s^-}^{s^+} -\frac{d\pi_1}{ds} (\bar{G}^_2 - \bar{G}^_1) \, ds,\end{aligned}$$ and since $$\begin{aligned} \frac{d\pi_1}{ds} &=& \theta(s_2-s)\tilde{\pi}_1(s)\delta(s-s_1) \nonumber \\ && + \left[1 - \theta(s-s_1)\tilde{\pi}_1(s)\right]\delta(s-s_2) \nonumber \\ && + \theta(s-s_1)\theta(s_2-s)\frac{d\tilde{\pi}_1}{ds},\end{aligned}$$ (where $\delta$ is the Dirac delta function) we have $$\begin{aligned} \label{eq:T2_1} T_2 &=& \frac{1}{\Delta s} \left\{ -\left[ \tilde{\pi}_1(\bar{G}^_2 - \bar{G}^_1)\right]_{s_1} - \left[ \tilde{\pi}_2(\bar{G}^_2 - \bar{G}^_1)\right]_{s_2} \right. \nonumber \\ && \left. - \int_{s_1}^{s_2} \frac{d\tilde{\pi}_1}{ds} (\bar{G}^_2 - \bar{G}^_1) \, ds \right\}.\end{aligned}$$ The first two terms in Eqn. (\[eq:T2\_1\]) represent the contributions from crossing the bifurcations at $s_1$ and $s_2$ respectively. Using Eqn. (\[eq:fp2\_1\]) we may write them as $$\begin{aligned} \label{eq:T2_2} T_2 &=& \frac{1}{\Delta s} \left\{\sum_{m=1}^2\left[ \tilde{\pi}_m\left(-\frac{\Delta\alpha_2}{\Delta\bar{\chi}^}\right) \Delta\bar{\chi}^\right]_{s_m} \right. \nonumber \\ && \left. - \int_{s_1}^{s_2} \frac{d\tilde{\pi}_1}{ds} (\bar{G}^_2 - \bar{G}^_1) \, ds \right\},\end{aligned}$$ where $\Delta \alpha_2 = \alpha_2(\bar{\chi}_2^) - \alpha_2(\bar{\chi}_1^)$ and $\Delta \bar{\chi}^ = \bar{\chi}_2^* - \bar{\chi}_1^* = (\bar{X}_2^* - \bar{X}_1^)/s > 0$. Since $\alpha_2$ is monotonic in $\bar{X}$, $-\Delta \alpha_2 / \Delta \bar{\chi}^$ at fixed $s$ is of the same sign as $-\alpha'_2$, which is of the same sign as $\dot{\alpha}_2$ since $s$ effectively reduces $X$ (Eqn.(\[eq:chi\])). Therefore the contributions to $\Delta \bar{G}^/\Delta s$ from crossing the bifurcations do not violate direct functionality. A violation, at least in the case of a 1-cycle, can only come from variations in the probabilities $\tilde{\pi}_m$ within the region $s_1 < s < s_2$, as described by the last term in Eqn. (\[eq:T2\_2\]). Next we describe a numerical test that suggests such violations are rare. Numerics suggest violations from multiple fixed points are rare --------------------------------------------------------------- For each of the 12 positive-feedback networks, we numerically found the steady state of the dynamical system with randomly sampled parameters as before (cf. Numerical Results). However now for each parameter set we solved the system many times with randomly selected initial conditions. When multiple fixed points were found, the fraction of trials approaching the $m$th fixed point was used for the weight $\pi_m$. This assumes the $\pi_m$ are determined only by the basins of attraction of each fixed point, and by the distribution of the initial conditions. However, different distributions of initial conditions do not result in qualitative different results. For each network, $2,000$ parameter sets were selected (uniform randomly in log-space), at which the system was solved $100$ times with initial protein counts selected uniform randomly from $0$ to $1,000$ proteins per cell. Over all positive-feedback networks, $37\%$ of the parameter sets supported multiple fixed points for at least one of the settings of S$_1$ and S$_2$. However only $0.46\%$ of parameter sets produced violations of direct functionality. Moreover this number is likely an overestimate, as no distinguishability criterion was imposed as was done in the single-fixed point case (cf. Numerical Results). It is likely that this fraction would remain low if the estimation of the $\pi_m$ was refined to incorporate the curvatures of the fixed points, or if alternative distributions were used for the sampling. All networks with serial regulation exhibit only direct functionality {#sec:general} ===================================================================== In this section, we extend our analytic constraint as derived in the context of the system studied experimentally by Guet et al.[@Guet:2002p38] to show that any network with only serial regulation—each node having 0 or 1 parent—exhibits only direct functionality, i.e. any target node X$_i$ changes with any input S$_j$ according to the direct path between them. We first consider a connected directed graph in which every node has in-degree 1, called a [contrafunctional graph]{} [@harary]. One can show that a contrafunctional graph has exactly one cycle, each of whose nodes is the root of a tree if the cycle edges are ignored [@harary]. Now consider changing one node’s in-degree to 0, or equivalently, removing an edge. If the edge is in the cycle, the graph remains connected and becomes a tree. If the edge is not in the cycle, the graph is cut into two components: a contrafunctional graph and a tree. A tree exhibits only direct functionality since there is at most one path from an input S$_j$ to a gene X$_i$, which is therefore the direct path. In a contrafunctional graph, we first consider the case where the target node X$_i$ is inside the cycle. Only inputs S$_j$ that are inside the cycle can affect X$_i$ because the rest of the graph consists of trees that all point away from the cycle. Since we can start labeling nodes at any point in the cycle, we may take $i \leq j$ without loss of generality. Then, using the chain rule, $$\begin{aligned} \frac{d\bar{X}_i^}{ds_j} &=& \left[ \dot{\alpha}_{(j \, {\rm mod} \, n)+1} + \alpha'_{(j \, {\rm mod} \, n)+1} \frac{d\bar{X}_j^}{ds_j} \right] \nonumber \\ && \times \frac{\prod_{k=1}^n \alpha'_k}{\prod_{l=i}^j \alpha'_{(l \, {\rm mod} \, n)+1}}\\ \label{eq:dXidsj} &=& \left[ \frac{1}{1 - \prod_{m=1}^n \alpha'_m} \right] \nonumber \\ && \times \dot{\alpha}_{(j \, {\rm mod} \, n)+1} \frac{\prod_{k=1}^n \alpha'_k}{\prod_{l=i}^j \alpha'_{(l \, {\rm mod} \, n)+1}}, %\frac{d\bar{X}_i^}{ds_j} &=& % \left[ \dot{\alpha}_{(j \, {\rm mod} \, n)+1} % + \alpha'_{(j \, {\rm mod} \, n)+1} \frac{d\bar{X}_j^}{ds_j} \right] % \frac{\prod_{k=1}^n \alpha'_k}{\prod_{l=i}^j \alpha'_{(l \, {\rm mod} \, n)+1}}\\ %\label{eq:dXidsj} %&=& \left[ \frac{1}{1 - \prod_{m=1}^n \alpha'_m} \right] % \dot{\alpha}_{(j \, {\rm mod} \, n)+1} % \frac{\prod_{k=1}^n \alpha'_k}{\prod_{l=i}^j \alpha'_{(l \, {\rm mod} \, n)+1}},\end{aligned}$$ where the second step follows from Eqn. (\[eq:dXjdsj\]). We next consider the case where the target node is outside the cycle. An input S$_j$ can only affect the node if it is either in the cycle or above the node in its tree. The portion of the path in the tree will exhibit direct functionality. Therefore in looking for possible indirect functionality we may, without loss of generality, take the node to be immediately outside the cycle, as we did for G in the previous section. $d\bar{G}^/ds_j$ is then given by Eqn.(\[eq:dGds\_1\]). In both Eqns. (\[eq:dXidsj\]) and (\[eq:dGds\_1\]), the term outside the brackets represents the direct path from S$_j$ to the target node, and the term inside the brackets is positive for stable fixed points. Therefore, a contrafunctional graph exhibits only direct functionality. Since each connected component of a network in which every node has in-degree 0 or 1 is either a contrafunctional graph or a tree, such networks exhibit only direct functionality. Thus, in general, the possible logical functions of topologies with at most one regulator per node are severely constrained. Appendix: The stochastic model {#sec:model} ============================== The dynamical system in Eqns. (\[eq:ode1\]-\[eq:odeG\]) provides a deterministic description of mean expression levels but fails to capture fluctuations around these means. A full stochastic description is given by the chemical master equation. For $N$ species participating in $R$ elementary reactions in a system with volume $\Omega$, the master equation reads $$\label{eq:master} \frac{dP({\bf n}, t)}{dt} = \Omega \sum_{j=1}^R \left( \prod_{i = 1}^N E^{-Z_{ij}} - 1\right) f_j({\bf X}, \Omega) P({\bf n}, t),$$ where $P({\bf n}, t)$ is the probability of having the copy number vector ${\bf n} = \Omega {\bf X} = \Omega (X_1, \dots, X_N )$ at time $t$, $Z_{ij}$ is the $N \times R$ stochiometric matrix, $E^{-Z_{ij}}$ is the step operator which acts by removing $Z_{ij}$ molecules from $n_i$, and $f_j$ is the rate for reaction $j$. The $f_j$ are the $\tilde{\alpha}_j$ and $r_j X_j$ of Eqns. (\[eq:ode1\]-\[eq:odeG\]) in the macroscopic limit. As in previous work [@Ziv:2007p88], we employ the much-used [ linear noise approximation]{} [@Elf:2003p86; @Paulsson:2004p85; @Elf:2003p87; @vanKampen] to make Eqn. (\[eq:master\]) tractable by expanding in orders of $\Omega^{-1/2}$. Introducing ${\bf \xi}$ such that $n_i = \Omega X_i + \Omega^{1/2} \xi_i$ and treating ${\bf \xi}$ as continuous, the first two terms in the expansion yield the macroscopic rate equations (e.g. Eqns. (\[eq:ode1\]-\[eq:odeG\]) in our case) and the linear Fokker-Plank equation, respectively: $$\begin{aligned} \sum_{i=1}^N \frac{\partial \bar{X}_i}{\partial t} \frac{\partial P({\bf \xi}, t)}{\partial \xi_i} = \sum_{i=1}^N \sum_{j=1}^R Z_{ij} f_j({\bf \bar{X}}) \frac{\partial P({\bf \xi}, t)}{\partial \xi_i}, \\ \label{eq:fokker} \frac{\partial P({\bf \xi}, t)}{\partial t} = - \sum_{i, k} J_{ik} \frac{\partial (\xi_k P)}{\partial \xi_i} + \frac{1}{2} \sum_{i, k} D_{ik} \frac{\partial^2 P}{\partial \xi_i \partial \xi_k},\end{aligned}$$ where $J_{ik} = \sum_{j=1}^R Z_{ij} (\partial f_j/\partial X_k)$ is the Jacobian matrix (e.g. Eqn. (\[eq:jacobian\])) and $D_{ik} = \sum_{j=1}^R Z_{ij} Z_{kj} f_j({\bf X})$ is a diffusion-like matrix. The steady-state solution to Eqn. (\[eq:fokker\]) is the multivariate Gaussian $$P({\bf \xi}) = \left[ (2\pi)^N \det \Xi \right]^{-1/2} \exp \left( - \frac{-\xi^T \Xi \xi}{2} \right),$$ where the covariance matrix $\Xi$ satisfies $$\label{eq:lyap} J \Xi + \Xi J^T + D = 0.$$ We solve for $\Xi$ using standard matrix Lyapunov equation solvers (e.g., Matlab’s `lyap`). Thus fluctuations are captured to leading order by Gaussian distributions with means $\bar{X}_i$ given by the macroscopic equation and variances given by the diagonal entries of $\Xi$. For example, Gaussian distributions $P(G^\|c)$ are shown in Fig. \[fig:func\]b for the steady-state concentration of the reporter gene G under chemical input states $c$. In [@Ziv:2007p88] we have compared the distributions $P(G^\|c)$ obtained using the linear noise approximation to those obtained via direct stochastic simulations [@Gillespie:1977p54] and found the results almost indistinguishable for molecular copy number above 10-20. We are grateful to the organizers and participants of The First q-bio Conference, where a preliminary version of this work was presented. This work was partially supported by NSF Grant No. ECS-0425850 to CW and IN. IN was further supported by LANL LDRD program under DOE Contract No. DE-AC52-06NA25396.
--- abstract: 'We investigate two different invariants for the Rubik’s Magic puzzle that can be used to prove the unreachability of many spatial configurations, one of these invariants, of topological type, is to our knowledge never been studied before and allows to significantly reduce the number of theoretically constructible shapes.' address: ' Dipartimento di Matematica, Università Cattolica “Sacro Cuore”, Brescia, Italy E-mail: paolini@dmf.unicatt.it' author: - Maurizio Paolini title: 'A new topological invariant for the “Rubik’s Magic” puzzle' --- Introduction {#sec:intro} ============ The Rubik’s Magic is another creation of Ernő Rubik, the brilliant hungarian inventor of the ubiquitous “cube” that is named after him. The Rubik’s Magic puzzle is much less known and not very widespread today, however it is a really surprising object that hides aspects that renders it quite an interesting subject for a mathematical analysis on more than one level. We investigate here two different invariants that can be used to prove the unreachability of many spatial configurations of the puzzle, one of these invariants, of topological type, is to our knowledge never been studied before and allows to significantly reduce the number of theoretically constructible shapes. However even in the special case of planar “face-up” configurations (see Section \[sec:piane\]) we don’t know whether the combination of the two invariants, together with basic constraints coming from the mechanics of the puzzle, is complete, [i.e.]{} characterize the set of constructible configurations. Indeed there are still a few planar face-up configurations having both vanishing invariants, but that we are not able to construct. In this sense this Rubik’s invention remains an interesting subject of mathematical analysis. In Section \[sec:rompicapo\] we describe the puzzle and discuss its mechanics, the local constraints are discussed in Section \[sec:vincolilocali\]. The addition of a ribbon (Section \[sec:nastro\]) allows to introduce the two invariants, the metric and the topological invariants are described respectively in Sections \[sec:invariantemetrico\] and \[sec:invariantetopologico\]. The special “face-up” planar configurations are defined in Section \[sec:piane\] and their invariant computed in Section \[sec:forsecostruibili\]. In Section \[sec:costruibili\] we list the planar face-up configuration that we were able to actually construct. We conclude the paper with a brief description of the software code used to help in the analysis of the planar configurations (Section \[sec:code\]). The puzzle {#sec:rompicapo} ========== ![[]{data-label="fig:black2x4"}](black_2x4.jpg "fig:"){width="7cm"} ![[]{data-label="fig:black2x4"}](orientedtiles "fig:"){width="5cm"} ![[]{data-label="fig:blacksolved"}](black_solved.jpg){width="6cm"} The Rubik’s Magic puzzle (see Figure \[fig:black2x4\] left) is composed by $8$ decorated square tiles positioned to form a $2\times 4$ rectangle. They are ingeniously connected to each other by means of nylon strings lying in grooves carved on the tiles and tilted at $45$ degrees [@basteleien:web]. The tiles are decorated in such a way that on one side of the $2\times 4$ original configuration we can see the picture of three unconnected rings, whereas on the back side there are non-matching drawings representing parts of rings with crossings among them. The declared aim is to manipulate the tiles in order to correctly place the decorations on the back, which can be done only by changing the global outline of the eight tiles. The solved puzzle is shown in Figure \[fig:blacksolved\] with the tiles positioned in a $3 \times 3$ square with a missing corner and overturned with respect to the original configuration of Figure \[fig:black2x4\]. Detailed instructions on how the puzzle can be solved and more generally on how to construct interesting shapes can be copiously found in the internet, we just point to the Wikipedia entry [@wikipedia:web] and to the web page [@basteleien:web]. The booklet [@Nou:86] contains a detailed description of the puzzle and illustrated instructions on how to obtain particular configurations of the tiles. The decorations can be used to distinguish a “front” and a “back” face of each tile and to orient them by suitably chosing an “up” direction. After dealing with the puzzle for some time it becomes apparent that a few local constraints are always satisfied. In particular the eight tiles remain always connected two by two in such a way to form a cyclic sequence. To fix ideas let us denote the eight tiles by $T_0$, $T_1$, ..., $T_7$, with $T_0$ the lower-left tile in Figure \[fig:black2x4\] and the others numbered in counterclockwise order. For example tile $T_3$ is the one with the Rubik’s signature in its lower-right corner (see Figure \[fig:black2x4\]). With this numbering tile $T_i$ is always connected through one of its sides to both tiles $T_{i+1}$ and $T_{i-1}$. Here and in the rest of this paper we shall always assume the index $i$ in $T_i$ to be defined “modulo $8$”, [i.e.]{} that for example $T_8$ is the same as $T_0$. We shall conventionally orient the tiles such that in the initial configuration of Figure \[fig:black2x4\] all tiles are “face up” (i.e. with their front face visible), the $4$ lower tiles ($T_0$ to $T_3$) are “straight” (not upside down), the $4$ upper tiles ($T_4$ to $T_7$) are “upside down” (as for a map with its north turned down), see Figure \[fig:black2x4\] right. At a more accurate examination it turns out that only half of the grooves are actually used (those having the nylon threads in them). These allow us to attach to a correctly oriented tile (face up and straight) a priviledged direction: direction ${\boxslash}$ (“slash”: North-East to South-West) and direction ${\boxbslash}$ (“backslash”: North-West to South-East). The used groves are shown in Figure \[fig:black2x4\] right. From now on we shall disregard completely the unused grooves. In the initial configuration tiles $T_i$ with even $i$ are all tiles of type ${\boxslash}$, whereas if $i$ is odd we have a tile of type ${\boxbslash}$. The direction of the used grooves in the back of a tile is opposite (read orthogonal) to the direction of the used grooves of the front face, but beware that when we reverse (turn over) a tile a ${\boxslash}$ groove becomes ${\boxbslash}$, so that the reversed tile remains of the same type (${\boxslash}$ or ${\boxbslash}$). From the point of view of a physical modeling we shall assume that the tiles are made of a rigid material and with infinitesimal thickness. This allows two or more tiles to be juxtaposed in space, however in such a case we still need to retain the information about their relative position (which is above of which). The nylon threads are assumed to be perfectly flexible but inextensible (and of infinitesimal thickness). Whether this is a suitable physical model for the puzzle is debatable, indeed there might exist configurations of the real puzzle that require a minimum of elasticity of the threads and thus cannot be obtained by the idealized model. On the contrary the real puzzle has non-infinitesimal tile thickness, which can lead to configurations that are allright for the physical model but that are difficult or impossible to achieve (because of the imposed stress on the nylon threads) with the real puzzle. Undecorated puzzle ------------------ We are here mainly interested in the study of the shapes in space that can be obtained, so we shall neglect the decorations on the tiles and only consider the direction of the grooves containing the nylon threads. In other words we only mark one diagonal on each face of the tiles, one connecting two opposite vertices on the front face and the other connecting the remaining two vertices on the back face. Now the tiles (marked with these two diagonals) are indistinguishable; distinction between ${\boxslash}$ and ${\boxbslash}$ is only possible after we have “oriented” a tile and in such a case rotation of $90$ degrees or a reflection will exchange ${\boxslash}$ with ${\boxbslash}$. \[def:orientation\] A tile can be oriented by drawing on one of the two faces an arrow parallel to a side. We have thus eight different possible orientations. We say that two adjacent tiles are compatibly oriented if their arrows perfectly fit together (parallel pointing at the same direction) when we ideally rotate one tile around the side on which they are hinged to make it juxtaposed to the other. There is exactly one possible orientation of a tile that is compatible with the orientation of an adjacent tile. A configuration of tiles is orientable if it is possible to orient all tiles such that they are pairwise compatibly oriented. For an orientable configuration we have eight different choices for a compatible orientation of the tiles. An example of compatible orientation of a configuration is shown in Figure \[fig:black2x4\] right, which makes the initial $2 \times 4$ configuration orientable. Once we have a compatible orientation for a configuration, we can classify each tile as ${\boxslash}$ or ${\boxbslash}$ according to the relation between the orienting arrow and the marked diagonal: a tile is of type ${\boxslash}$ if the arrow alignes with the diagonal after a clockwise rotation of $45$ degrees, it is of type ${\boxbslash}$ if the arrow alignes with the diagonal after a counterclockwise rotation of $45$ degrees. Two adjacent tiles are always of opposite type. \[def:chiral\] A spatial configuration of the puzzle that is not congruent (also considering the marked diagonals) after a rigid motion with its mirror image will be called chiral, otherwise it will be called achiral. Note that a configuration is achiral if and only if it is specularly symmetric with respect to some plane. The initial $2 \times 4$ configuration is achiral since it is specularly symmetric with respect to a plane orthogonal to the tiles. We say that an orientable spatial configuration of the puzzle (without decorations) is constructible if it can be obtained from the initial $2 \times 4$ configuration through a sequence of admissible moves of the puzzle. Once we have identified all the constructible spatial configurations, we also have all constructible configurations of the decorated puzzle. This is a consequence of the fact that all possible $2 \times 4$ configurations of the decorated puzzle are well understood (see for example [@basteleien:web] or [@Nou:86]). We note here that all $2 \times 4$ configurations of the undecorated puzzle are congruent, however the presence of the marked diagonal might require a reversal of the whole configuration (so that the faces previously visible become invisible) in order to obtain the congruence. For chiral configuratione (those that cannot be superimposed with their specular images) the following result is useful. \[teo:mirrorconstructible\] A spatial configuration of the undecorated puzzle is constructible if and only if its mirror image is constructible If a configuration is constructible we can reach it by a sequence of moves of the puzzle starting from the initial $2 \times 4$ configuration. However the initial $2 \times 4$ configuration is specularly symmetric, hence we can perform the specular version of that sequence of moves to reach the specular image of the configuration that we are considering. Local constraints {#sec:vincolilocali} ================= We now consider a version of the puzzle where in place of the usual decoration we draw arrows on the “front” face of the tiles as in Figure \[fig:black2x4\] right. The linking mechanism with the nylon threads is such that two consecutive tiles $T_i$ and $T_{i+1}$ are always “hinged” together through one of their sides. In particular, if we suitably orient $T_i$ with its “front” face visible and “straight”, i.e. with the arrow visible and pointing up) and we rotate tile $T_{i+1}$ such that its center is as far away as possible from the center of $T_i$ (like an open book), then also $T_{i+1}$ will have its arrow visible and - pointing up if the two tiles are hinged through a vertical side (the right or left side of $T_i$); - pointing down it the two tiles are hinged through a horizontal side (the top or bottom side of $T_i$). The surprising aspect of the puzzle is that when we “close the book”, i.e. we rotate $T_{i+1}$ so that it becomes superimposed with $T_i$, we than can “reopen the book” with respect to a different hinging side. The new hinging side is one of the two sides that are orthogonal to the original hinging side, which one depending on the type of the involved tiles (direction of the marked diagonals) and can be identified by the rule that the new side is not separated from the previous one by the “inner” marked diagonals. For example, if $T_i$ is of type ${\boxslash}$ (hence $T_{i+1}$ is of type ${\boxbslash}$) and they are hinged through the right side of $T_i$ (as $T_0$ and $T_1$ of Figure \[fig:black2x4\] right) then after closing the tiles by rotating $T_{i+1}$ up around its left side and placing it on top of (superimposed above) $T_i$, then we can reopen the tiles with respect to the bottom side. On the contrary, if we rotate down $T_{i+1}$, so that it becomes superimposed below $T_i$ (and the involved marked diagonal of $T_i$ is the one on the back face), the new hinging side will be the upper side. We remark that if a configuration does not contain superimposed consecutive tiles, then the hinging side of any pair of consecutive tiles is uniquely determined. If the tiles are (compatibly) oriented, than for each tile $T_i$ we have a unique side (say East, North, West or South, in short $E$, $N$, $W$ or $S$) about which it is hinged with the preceding tile $T_{i-1}$ and a unique side ($E$, $N$, $W$ or $S$) about which it is hinged with $T_{i+1}$. The two sides can be the same. \[def:localshape\] For a given spatial oriented configuration of the (undecorated) puzzle without superimposed consecutive tiles we say that a tile is straight : if the two hinging sides are opposite; curving : if the two hinging sides are adjacent (but not the same). In this case we can distinguish between tiles curving left and tiles curving right with the obvious meaning and taking into account the natural ordering of the tiles induced by the tile index; a flap : if it is hinged about the same side with both the previous and the following tile. Flaps {#sec:flaps} ----- Flap tiles (those that, following Definition \[def:localshape\], have a single hinging side with the two adjacent tiles) require a specific analysis. The term “flap” is the same used in [@Nou:86] and refers to the similarity with the flaps of a plane, that can rotate about a single side. Given an oriented configuration with a flap $T_i$, let us fix the attention to the three consecutive tiles $T_{i-1}$, $T_i$, $T_{i+1}$ and ignore all the others. Place the configuration so that $T_i$ is horizontal, with its front face up and the arrow pointing North, then rotate $T_{i-1}$ and $T_{i+1}$ at maximum distance from $T_i$ so that they become reciprocally superimposed. Now all three tiles have their front face up and we can distinguish between two situations: \[def:flaps\] Tile $T_i$ is an ascending flap if tile $T_{i-1}$ is below tile $T_{i+1}$; it is a descending flap in the opposite case. Tile $T_i$ is a horizontal ascending/descending flap if it is hinged at a vertical side (a side parallel to the arrow indicating the local orientation of the flap tile), it is a vertical ascending/descending flap otherwise. The ribbon trick {#sec:nastro} ================ In order to introduce the metric and the topological invariants we resort to a simple expedient: we insert a ribbon in between the tiles that more or less follows the path of the nylon threads. The ribbon is colored red on one side (front side) and blue in its back side and is oriented with longitudinal arrows printed along its length that allows to follow it in the positive or negative direction. ![[]{data-label="fig:nastro"}](nastro){height="4cm"} Let the tiles have side of length $1$, then the ribbon has width that does not exceed $\frac{\sqrt{2}}{4}$ (the distance between two nearby grooves), so that it will not interfere with the nylon threads. We insert the ribbon as shown in Figure \[fig:nastro\]. More precisely take the $2 \times 4$ initial configuration of the puzzle and start with tile $T_2$. Position the ribbon such that it travels diagonally along the front face of $T_2$ as shown in Figure \[fig:nastro\], then wrap the ribbon around the top side of $T_2$ and travel downwards along the back of $T_2$ to reach the right side. At this point we move from the back of $T_2$ to the front of $T_3$ (the ribbon now has its blue face up) and continue downward until we reach the bottom side of $T_3$, wrap the ribbon on the back and so on. In general, every time that the ribbon reaches a side of a tile that is not a hinge side with the following tile, we wrap it around the tile (from the front face to the back face or from the back face to the front face) as if it “bounces” against the side. Every time the ribbon reaches a hinging side of a tile with the following tile it moves to the next tile and crosses from the back (respectively front) side of one tile to the front (respectively back) side of the other and maintains its direction. In all cases the ribbon travels with sections of length $\delta = \frac{\sqrt{2}}{2}$ between two consecutive “touchings” of a side. It can stay adjacent to a given tile during one, two or three of such $\delta$ steps: one or three if the tile is a curving tile (Definition \[def:localshape\]), two if the tile is a straight tile. After having positioned the ribbon along all tiles, it will close on itself nicely (in a straight way and with the same orientation) on the starting tile $T_2$, and we tape it with itself. In this way the total length of the ribbon is $16 \delta$ with an average of $2\delta$ per tile, moreover if we remove the ribbon without cutting it (by making the tiles “disappear”), we discover that we can deform it in space into the lateral surface of a large and shallow cylinder with height equal to the ribbon thickness and circumference $16 \delta$. Direct inspection also shows that the inserted ribbon does not impact on the possible puzzle moves, whereas its presence allows us to define the two invariants of Sections \[sec:invariantemetrico\] and \[sec:invariantetopologico\]. We remark a few facts: 1. The ribbon is oriented: it has arrows on it pointing in the direction in which we have inserted it, and while traversing the puzzle along the ribbon the tiles are encountered in the order given by their index. 2. Each time the ribbon “bounces” at the side of a tile (moving from the front face to the back face or viceversa) its direction changes of $90$ degrees and simultaneously it turns over. This does not happen when the ribbon moves from one tile to the next, it does not change direction and it does not turn over. 3. Each $\delta$ section of the ribbon connects a horizontal side to a vertical side or viceversa; consequently the ribbon touches alternatively horizontal sides and vertical sides. 4. Each time the ribbon touches a lateral side it goes from one side of the tiles to the other (from the front to the back or from the back to the front). The above points 3 and 4 prove the following Following the orientation of the ribbon, when the ribbon touches/crosses a vertical side, it “emerges” from the back of the tiles to the front, whereas when it touches/crosses a horizontal side, it “submerges” from the front to the back. Here vertical or horizontal refers to the local orientation assigned to the tiles. Behaviour of the ribbon at a flap tile. {#sec:nastroflap} --------------------------------------- It is not obvious how the ribbon behaves at a flap tile (such tiles are not present in the initial $2 \times 4$ configuration). We can reconstruct the ribbon position by imagining a movement that transforms a configuration without flaps to another with one flap. It turns out that there are two different situations. In one case the ribbon completely avoids to touch the flap tile $T_i$ and directly goes from $T_{i-1}$ to $T_{i+1}$, this happens when in a neighbourhood of the side where the flap tile is hinged the ribbon is on the front face of the upper tile and on the bottom face of the lower tile (in the configuration where $T_{i-1}$ and $T_{i+1}$ are furthest away from $T_i$, hence superposed), this situation is illustrated in Figure \[fig:nastroflap\] left. In the other case the ribbon wraps around $T_i$ with four $\delta$ sections alternating between the front face and the back face, this situation is illustrated in Figure \[fig:nastroflap\] right. The first of the two cases arises at an ascending flap hinged at a vertical side (horizontal ascending flap) or at a descending flap hinged at a horizontal side (vertical descending flap); this is independent of the type ${\boxslash}$ or ${\boxbslash}$ of the flap tile. The second of the two cases arises at a vertical ascending flap or at a horizontal descending flap. ![[]{data-label="fig:nastroflap"}](nastroflap0 "fig:"){height="2cm"} ![[]{data-label="fig:nastroflap"}](nastroflap4 "fig:"){height="2cm"} Metric invariant {#sec:invariantemetrico} ================ Whatever we do to the puzzle (with the ribbon inserted) there is no way to change the length of the ribbon! This allows to regard the length of the ribbon associated to a given spatial configuration as an invariant, it cannot change under puzzle moves. The computation of the ribbon length can be carried out by following a few simple rules, they can also be found in [@Nou:86]. The best way to proceed is to compute for each tile $T_i$ how many $\delta$ sequences of the ribbon wrap it and subtract the mean value $2$. The resulting quantity will be called $\Delta_i$ and its value is: - $\Delta_i = 0$ if $T_i$ is a straight tile (Definition \[def:localshape\]); - $\Delta_i = -1$ if $T_i$ is of type ${\boxslash}$ and is “curving left”, or if it is of type ${\boxbslash}$ and is “curving right”; - $\Delta_i = +1$ if $T_i$ is of type ${\boxslash}$ and is curving right, or if it is of type ${\boxbslash}$ and is curving left; - $\Delta_i = -2$ if $T_i$ is a horizontal ascending flap (Definition \[def:flaps\]) or a vertical descending flap (see Figure \[fig:nastroflap\] left); - $\Delta_i = +2$ if $T_i$ is a horizontal descending flap or a vertical ascending flap (see Figure \[fig:nastroflap\] right). The last two cases ($\|\Delta_i\| = 2$) follow from the discussion in Section \[sec:nastroflap\]. We call $\Delta = \sum_{i=0}^7 \Delta_i$, the sum of all these quantities, then the total length of the ribbon will be $16 \delta + \Delta \delta$ and hence $\Delta$ is invariant under allowed movements of the puzzle. Since in the initial configuration we would have $\Delta = 0$ it follows that \[teo:invariantemetrico\] Any constructible configuration of the puzzle necessarily satisfies $\Delta = 0$. This invariant can also be found in [@Nou:86 page 19], though it is not actually justified. A few configurations (e.g. the $3 \times 3$ shape without the central square, called “window shape” in [@Nou:86]) can be ruled out as non-constructible by computing the $\Delta$ invariant. The “window shape” has a value $\Delta = \pm 4$, the sign depending on how we orient the tiles. It is non-constructible because $\Delta \neq 0$. Another interesting configuration that can be ruled out using this invariant is sequence , to be discussed in Section \[sec:noflaps\]. Topological invariant {#sec:invariantetopologico} ===================== Sticking to the ribbon idea (Section \[sec:nastro\]) we seek a way to know whether a given ribbon configuration (with the tiles and nylon threads removed) can be obtained by deformations in space starting from the configuration where the ribbon is the lateral surface of a cylinder. Topologically the ribbon is a surface with a boundary, its boundary consists of two closed strings. One thing that we may consider is the center line of the ribbon: it is a single closed string that can be continuously deformed in space and is not allowed to cross itself. Mathematically we call this a “knot”, a whole branch of Mathematics is dedicated to the study of knots, one of the tasks being finding ways to identify “unknots”, i.e. tangled closed strings that can be “unknotted” to a perfect circle. This is precisely our situation: the center line of the ribbon must be an unknot, otherwise the corresponding configuration of the puzzle cannot be constructed. However we are not aware of puzzle configurations that can be excluded for this reason. Another (and more useful) idea consists in considering the two strings forming the boundary of the ribbon. In Mathematics, a configuration consisting in possibly more than one closed string is called a “link”. Here we have a two-components link that in the starting configuration can be deformed into two unlinked perfect circles. There is a topological invariant that can be easily computed, the linking number between two closed strings, that does not change under continuous deformations of the link (again prohibiting selfintersections of the two strings or intersections of one string with the other). In the original configuration of the puzzle, the two strings bordering the ribbon have linking number zero: it then must be zero for any constructible configuration. Computing the linking number ---------------------------- In the field of knot theory a knot, or more generally a link, is often represented by its . It consists of a drawing on a plane corresponding to some orthogonal projection of the link taken such that the only possible selfintersections are transversal crossings where two distinct points of the link project onto the same point. We can always obtain such a generic projection possibly by changing a little bit the projection direction. We also need to add at all crossings the information of what strand of the link passes “over” the other. This is usually done by inserting a small gap in the drawing of the strand that goes below the other, see Figure \[fig:linkingnumber\]. ![[]{data-label="fig:linkingnumber"}](linkingnumber.png){height="2cm"} ![[]{data-label="fig:twistsign"}](twistplus "fig:"){height="3cm"} ![[]{data-label="fig:twistsign"}](twistminus "fig:"){height="3cm"} ![[]{data-label="fig:crosssign"}](crossplus "fig:"){height="3cm"} ![[]{data-label="fig:crosssign"}](crossminus "fig:"){height="3cm"} In order to define the linking number between two closed curves we need to select an orientation (a traveling direction) for the two curves. In our case the orientation of the ribbon induces an orientation for the two border strings by following the same direction. The linking number changes sign if we revert the orientation of one of the two curves, so that it becomes insensitive upon the choice of orientation of the ribbon. Once we have an orientation of the two curves, we can associate a signature to each crossing as shown in Figure \[fig:linkingnumber\] and a corresponding weight of value $\pm \frac{1}{2}$. Crossings of a component with itself are ignored in this computation. The linking number is given by the sum of all these contributions. Since the number of crossings in between the two curves in the diagram is necessarily even, it follows that the linking number is an integer and it can be proved that it does not change under continuous deformations of the link in space. Two far away rings have linking number zero, two linked rings have linking number $\pm 1$. In our case we shall investigate specifically the case where all tiles are horizontal and “face-up”, in which case we have two different situations that produce crossings between the two boundary strings. We shall then write the linking number as the sum of a “twist” part ($L_t$) and a “ribbon crossing” part ($L_c$) $$L = L_t + L_c$$ where we distinguish the two cases: 1. The ribbon wraps around one side of a tile (Figure \[fig:twistsign\]). This entail one crossing in the diagram, that we shall call “twist crossing” since it is actually produced by a twist of the ribbon. A curving tile (as of Definition \[def:localshape\]) can contain only zero or two of this type of crossings, and if there are two, they are necessarily of opposite sign. This means that curving tiles do not contribute to $L_t$. 2. The ribbon crosses itself (Figure \[fig:crosssign\]). Consequently there are four crossing of the two boundary strings, two of them are selfcrossings of one of the strings and do not count, the other two contribute with the same sign for a total contribution of $\pm 1$ to $L_c$. The presence of this type of crossings is generally a consequence of the spatial disposition of the sequence of tiles and in the specific case of face-up planar configurations (to be considered in Section \[sec:piane\]) there can be crossings of this type when we have superposed tiles, or in presence of flap tiles, however the computation of $L_c$ must be carried out case by case. Contribution of the straight tiles to Lt. {#sec:ltstraigth} ----------------------------------------- The ribbon “bounces” exactly once at each straight tile (Definition \[def:localshape\]), hence it contributes to $L_t$ with a value $\delta L_t = \pm \frac{1}{2}$. After analyzing the various possibilities we conclude for tile $T_i$ as follows: - $\delta L_t = +\frac{1}{2}$ if $T_i$ is a “vertical” tile (connected to the adjacent tiles through its horizontal sides) of type ${\boxslash}$, or if it is a horizontal tile of type ${\boxbslash}$; - $\delta L_t = -\frac{1}{2}$ if $T_i$ is a horizontal tile of type ${\boxslash}$ or a vertical tile of type ${\boxbslash}$. Contribution of the flap tiles to Lt. {#sec:ltflaps} ------------------------------------- A flap tile can be covered by the ribbon either with four sections (three “bounces”) of none at all. In this latter case there is still a “bounce” of the ribbon when it goes from the previous tile to the next (superposed) tile: the ribbon travels from below the lower tile to above the upper tile or viceversa. We need to keep track of this extra bounce. After analysing the possibilities we conclude for tile $T_i$ as follows: - $\delta L_t = +\frac{1}{2}$ if $T_i$ is a vertical flap of type ${\boxslash}$ (connected to the adjacent tile through a horizontal side), or if it is a horizontal flap of type ${\boxbslash}$; - $\delta L_t = -\frac{1}{2}$ if $T_i$ is a horizontal flap of type ${\boxslash}$ or a vertical flap of type ${\boxbslash}$. Linking number of constructible configurations. ----------------------------------------------- \[teo:invariantetopologico\] A constructible spatial configuration of the puzzle necessarily satisfies $L = 0$. The linking number $L$ does not change under legitimate moves of the puzzle, so that it is sufficient to compute it on the initial configuration of Figure \[fig:black2x4\]. There are no superposed tiles nor flaps, so that the ribbon does not cross itself, hence $L_c = 0$. The only contribution to $L_t$ comes from the four straight tiles, and using the analysis of Section \[sec:ltstraigth\] it turns out that their contribution cancel one another so that also $L_t = 0$ and we conclude the proof. Examples of configurations with nonzero linking number. ------------------------------------------------------- Due to Theorem \[teo:invariantetopologico\] such configurations of the puzzle cannot be constructed. ![[]{data-label="fig:lunga"}](lunga){width="11cm"} One such configuration is shown in Figure \[fig:lunga\] and would realize the maximal possible diameter for a configuration. The metric invariant of Section \[sec:invariantemetrico\] is $\Delta = 0$ so that it is not enough to exclude this configuration, however we shall show that in this case $L \neq 0$ and conclude that we have a nonconstructible configuration. It will be studied in Section \[sec:forsecostruibili\]. Another interesting configuration that can be excluded with the topological invariant and not with the metric one is a “figure eight” corresponding to the sequence of Section \[sec:forsecostruibili\]. Planar face-up configurations {#sec:piane} ============================= We shall apply the results of the previous sections to a particular choice of spatial configurations, we shall restrict to planar configurations (all tiles parallel to the horizontal plane) with non-overlapping consecutive tiles. Superposed nonconsecutive tiles are allowed. They can be obtained starting from strings of cardinal directions in the following way. The infinite string $s : {\mathbb Z}\to \{E,N,W,S\}$ is a typographical sequence with index taking values in the integers ${\mathbb Z}$ where the four symbols stand for the four cardinal directions East, North, West, South. On $s$ we require 1. Periodicity of period $8$: $s_{n+8} = s_n$ for any $n \in {\mathbb Z}$; 2. Zero mean value: in any subsequence of $8$ consecutive characters (for example in $\{s_0, \dots, s_7\}$) there is an equal number of characters $N$ as of characters $S$ and of characters $E$ as of characters $W$. An admissible sequence is one that satisfies the two above requirements. Periodicity allows us to describe an admissible sequence by listing $8$ consecutive symbols, for definiteness and simplicity we shall then describe an admissible sequence just by listing the symbols $s_1$ to $s_8$. The character $s_i$ of the string indicates the relative position between the two consecutive tiles $T_{i-1}$ and $T_i$, that are horizontal and face-up. The first tile $T_0$ can be of type ${\boxslash}$ or type ${\boxbslash}$, all the others $T_i$ are of the same type as $T_0$ if $i$ is even, of the opposite type if $i$ is odd. The local constraints allows to recover a spatial configuration of the puzzle from an admissible sequence with two caveats: 1. For at least one of the tiles, say $T_0$, it is necessary to specify if it is of type ${\boxslash}$ or ${\boxbslash}$. We can add this information by inserting the symbol ${\boxslash}$ or ${\boxbslash}$ between two consecutive symbols, usually before $s_1$; 2. In case of superposed tiles (same physical position) it is necessary to clarify their relative position (which is above which). We can add this information by inserting a positive natural number between two consecutive symbols that indicates the “height” of the corresponding tile. In the real puzzle the tiles are not of zero width, so that their height in space cannot be the same. In case of necessity we shall insert such numbers as an index of the symbol at the left. \[rem:assemblabili\] Given an admissible sequence it is possible to compute the number of superposed tiles at any given position. An assemblage of a sequence entails a choice of the height of each of the superposed tiles (if there is more than one). We do this by adding an index between two consecutive symbols. However for this assemblage to correspond to a possible puzzle configuration we need to require a condition. We hence say that an assemblage is admissible if whenever tile $T_i$ is superposed to tile $T_j$, $i \neq j$, and also $T_{i \pm 1}$ is superposed to $T_{j+1}$, then the relative position of the tiles in the two pairs cannot be exchanged. This means that if $T_i$ is at a higher height than $T_j$, then $T_{i \pm 1}$ cannot be at a lower height than $T_{j+1}$. It is possible that a given admissible sequence does not allow for any admissible assemblage or that it can allow for more than one admissible assemblage. Observe that the mirror image of an oriented spatial configuration of the undecorated puzzle entails a change of type, ${\boxslash}$ tiles become of type ${\boxbslash}$ and viceversa. If the mirror is horizontal the reflected image is a different assemblage of the same admissible sequence with all tiles of changed type and an inverted relative position of the superposed tiles. On the set of admissible sequences we introduce an equivalence relation defined by $s \equiv t$ if one of the following properties (or a combination of them) holds: 1. (cyclicity) The two sequences coincide up to a translation of the index: $s_n = t_{n+k}$ for all $n$ and some $k \in {\mathbb Z}$; 2. (order reversal) $s_n = t_{k-n}$ for all $n$ and some $k \in {\mathbb Z}$; 3. (rotation) $s$ can be obtained from $t$ after substituting $E \to N$, $N \to W$, $W \to S$, $S \to E$; 4. (reflection) $s$ can be obtained from $t$ after substituting $E \to W$ e $W \to E$. Let us denote by ${\mathcal S}$ the set of equivalence classes. We developed a software code capable of finding a canonical representative of each of these equivalence classes, they are $71$ (cardinality of ${\mathcal S}$). In Table \[tab:sequenze\] we summarize important properties of these canonical sequences, subdivided with respect to the number of flap tiles. It is worth noting that some of the $71$ sequences admit more than one nonequivalent admissible assemblages in space due to the arbitrariness in choosing the type of tile $T_0$ and the ordering of the superposed tiles. A few of the $71$ admissible sequences do not admin any admissible assemblage, one of these is the only sequence with $8$ flaps: $EWEWEWEW$. Since the constructability of a spatial configuration is invariant under specular reflection (which entails a change of type of all tiles) we can fix the type of tile $T_0$, possibly reverting the order of the superposed tiles. The canonical representative of an equivalence class in ${\mathcal S}$ is selected by introducing a lexicographic ordering in the finite sequence $s_1, \dots, s_8$ where the ordering of the four cardinal directions is fixed as $E < N < W < S$. Then the canonical representative is the smallest element of the class with respect to this ordering. The source of the software code can be downloaded from the web page [@mine:web]. ---------- ----------- ---------- -------------- ----------- ----- ------------ number $\Delta = 0$ non non of flaps sequences assembl. assembl. sequences classified none 7 6 4 2 5 - 1 7 14 7 3 4 - 2 22 44 20 3+2 13 7+1 3 10 50 15 3 6 2 4 18 38 11 16 2 5 2 12 1 - 2 - 6 4 4 1 - 4 - 8 1 0 - - 1 - total 71 168 59 10+2 11+1 ---------- ----------- ---------- -------------- ----------- ----- ------------ \[tab:sequenze\] In Table \[tab:2e4flap\] the sequences with two and four flaps are subdivided based on the distribution of the flaps in the sequence. \[teo:ltiszero\] All planar face-up configurations have zero “twist” contribution to the topological invariant: $L_t = 0$. Consequently we have $L = L_c$ and to compute the linking number it is sufficient to compute the contributions coming from the crossing of the ribbon with itself. Any planar face-up configuration with an odd number of selfintersectons of the ribbon with itself has $L \neq 0$. [^1] We denote with $k_1$, ..., $k_s$ the number of symbols in contiguous subsequence of $E$, $W$ (horizontal portions) or of $N$, $S$ (vertical portions). Each portion of $k_i$ symbols contains $k_i - 1$ straight tiles or flaps, all “horizontal” or “vertical”, hence each tile contributes to $L_t$ with alternating sign due to the fact that the tiles are alternatively of type ${\boxslash}$ and ${\boxbslash}$. If $k_i-1$ is even, then the contribution of this portion is zero, while if it is odd it will be equal to the contribution of the first straight or flap tile of the portion. It is not restrictive to assume that the first portion of $k_1$ symbols is horizontal and the last (of $k_s$ symbols) is vertical. In this way if $i$ is odd, then $k_i$ is the number of symbols in a horizontal portion whereas if $i$ is even, then $k_i$ is the number of symbols in a vertical portion. Up to a change of sign of $L_t$ we can also assume that the first tile is of type ${\boxslash}$. Finally we observe that $k_i > 0$ for all $i$. Twice the contribution to $L_t$ of the $i$-th portion is given by $$\label{eq:ltparziale} (-1)^{i-1} (-1)^{k_1 + k_2 + ... + k_{i-1}} (1 + (-1)^{k_i})$$ where the last factor in parentheses is zero if $k_i$ is odd and is $2$ if $k_i$ is even; the sign changes on vertical portions with respect to horizontal portions (factor $(-1)^{i-1}$) and changes when the type (${\boxslash}$ or ${\boxbslash}$) of the first straight or flap til of the portion changes (factor $(-1)^{k_1 + k_2 + ... + k_{i-1}}$). Summing up \[eq:ltparziale\] on $i$ and expanding we have $$\begin{aligned} 2 L_t &= & \displaystyle{\sum_{i=1}^s (-1)^{i-1} (-1)^{k_1 + k_2 + ... + k_{i-1}} + \sum_{i=1}^s (-1)^{i-1} (-1)^{k_1 + k_2 + ... + k_{i-1}} (-1)^{k_i}} \\ &= & \displaystyle{- \sum_{i=1}^s (-1)^i (-1)^{k_1 + k_2 + ... + k_{i-1}} + \sum_{i=1}^s (-1)^{i+1} (-1)^{k_1 + k_2 + ... + k_i}} \\ &= & \displaystyle{- \sum_{i=1}^s (-1)^i (-1)^{k_1 + k_2 + ... + k_{i-1}} + \sum_{i=2}^{s+1} (-1)^i (-1)^{k_1 + k_2 + ... + k_{i-1}}} \\ &= & 1 + (-1)^{s+1} (-1)^{k_1 + k_2 + ... + k_s} = 0\end{aligned}$$ because $s$ is even and $k_1 + \dots + k_s = 8$, even. Configurations with vanishing invariants {#sec:forsecostruibili} ======================================== We shall identify admissible assemblages whenever they correspond to equivalent puzzle configurations, where we also allow for specular images. In particular this allows us to assume the first tile to be of type ${\boxslash}$. Assemblages corresponding to non-equivalent sequences cannot be equivalent, on the contrary there can exist equivalent assemblages of the same sequence and this typically happens for symmetric sequences. The two invariants can change sign on equivalent sequences or equivalent assemblages, this is not a problem since we are interested in whether the invariants are zero or nonzero. In any case the computations are always performed on the canonical representative. The contribution $\Delta_c$ of $\Delta = \Delta_c + \Delta_f$ (coming from the curving tiles) can be computed on the sequence (it does not depend on the assemblage). On the contrary the contribution $\Delta_f$ coming from the flap tiles depends on the actual assemblage. With the aid of the software code we can partially analyze each canonical admissible sequence and each of the possible admissible assemblages of a sequence. In particular the software is able to compute the metric invariant of an assemblage, so that we are left with the analysis of the topological invariant, and we shall perform such analysis only on assemblages having $\Delta = 0$, since our aim is to identify as best as we can the set of constructible configurations. Sequences with no flaps {#sec:noflaps} ------------------------- There are seven such sequences, three of them do not have any superposed tiles, so that they cover a region of the plane corresponding to $8$ tiles (configurations of area $8$). For these three sequences we only have one possible assemblage (having fixed the type ${\boxslash}$ of tile $T_0$). The sequence $$\label{eq:seq2x4} EEENWWWS$$ corresponds to the initial configuration $2\times 4$ of the puzzle. The sequence $$\label{eq:seq3x3} EENNWWSS$$ corresponds to the “window shape”, a $3\times 3$ square without the central tile. The sequence $$\label{eq:seqelle} EENNWSWS$$ corresponds to the target configuration of the puzzle (Figure \[fig:blacksolved\]). Two sequences cover $7$ squares of the plane (area $7$), the sequence $$\label{eq:seqottoobliquo} EENWSSWN$$ and the sequence $$\label{eq:seqbirillo} ENENWSWS .$$ A sequence without flaps and area $6$ (two pairs of superposed tiles) is $$\label{eq:seqottoverticale} ENESWNWS .$$ The last possible sequence (with area $4$) would be $$\label{eq:seqspirale} ENWSENWS ,$$ this however cannot be assembled in space since it consists of a closed circuit of $4$ tiles traveled twice (see Remark \[rem:assemblabili\]). The metric invariant is nonzero (hence the corresponding assemblage is not constructible) for the two sequences and , the topological invariant $L$ further reduces the number of possibly constructible configuration by excluding also the two sequences e . The remaining two configurations, corresponding to sequences ed , are actually constructible (Figures \[fig:black2x4\] and \[fig:blacksolved\]). Sequences with one flap ------------------------- We find seven (nonequivalent) sequences with exactly one flap. Three of these have area $7$: $$\label{eq:seqf1a7a} EENNWSSW $$ $$\label{eq:seqf1a7b} EENWNSWS $$ $$\label{eq:seqf1a7c} EEENWWSW $$ and four have area $6$: $$\label{eq:seqf1a6a} EENWSWSN $$ $$\label{eq:seqf1a6b} EENWSWNS$$ $$\label{eq:seqf1a6c} EENWSNWS$$ $$\label{eq:seqf1a6d} EEENWSWW . $$ In all cases it turns out that there are two nonequivalent assemblages of each of these sequences according to the flap tile being ascending or descending, and they have necessarily a different value of $\Delta$, so that at most one (it turns out exactly one) has $\Delta = 0$. We shall restrict the analysis of the topological invariant to those having $\Delta = 0$. The two sequences and have $\Delta = 0$ if the (horizontal) flap tile is descending (Figure \[fig:nastroflap\] right). The linking number reduces to $L = L_c$ (Theorem \[teo:ltiszero\]). Since in both cases we have exactly one crossing of the ribbon with itself we conclude that $L \neq 0$ and the sequences are not constructible. To have $\Delta = 0$ the vertical flap of the sequence must be descending. Then there is one crossing of the ribbon with itself, so that $L \neq 0$ and the configuration is not constructible. Sequences and have $\Delta = 0$ provided their flap is ascending. We have now two crossings of the ribbon with itself and they turn out to have opposite sign in their contribution to $L_c$, so that $L = 0$ and the two sequences “might” be constructible. Sequences and have $\Delta = 0$ provided their flap is ascending. Sequence is then not constructible because there is exactly one selfcrossing of the ribbon so that $L = L_c \neq 0$. On the contrary, sequence exhibits two selfcrossings with opposite sign and $L = L_c = 0$. In conclusion of the $7$ different sequences with one flap, four are necessarily non constructible because the topological invariant is non-zero, the remaining three sequences: , , are actually constructible as we shall see in Section \[sec:costruibili\], Figures \[fig:flap1a6a\], \[fig:flap1a6b\], \[fig:flap1a6d\]. The two sequences with two adjacent flaps ------------------------------------------- Adjacency of the two flaps entails that both are ascending or both descending (Remark \[rem:assemblabili\]) and also they are both horizontal or both vertical since they are hinged to each other so that they contribute to the metric invariant $\Delta_f = \pm 4$ whereas $\Delta_c = 0$. Hence the metric invariant is nonzero and the two sequences are non-constructible. The five sequences with two flaps separated by one tile --------------------------------------------------------- ### Sequence $EENWSEWW$ $\Delta = 0$ implies that the two (horizontal) flaps are one ascending and one descending. There are two non-equivalent admissible assemblages satisfying $\Delta = 0$, computation of the topological invariant gives $L = L_c = \pm 4$ for one of the two assemblages whereas the other has $L = L_c = 0$ and might be constructible: $${\boxslash}E_3 E_2 N W S_2 E_1 W_1 W .$$ ### Sequence $ENEWSNWS$ $\Delta = 0$ implies that both flaps (one is horizontal and one vertical) are ascending or both descending. The two corresponding distinct admissible assemblages have both $L = L_c = 0$. The two assemblages are: $${\boxslash}E_2 N_3 E W_2 S_2 N_3 W S \quad \text{and} \quad {\boxslash}E_1 N_1 E W_2 S_2 N_3 W S .$$ Of these, the first is actually constructible (Figure \[fig:flap2a5a\]), the other one remains unclassified. ### Sequence $ENEWNSWS$ We can fix the first tile $T_0$ to be of type ${\boxslash}$, then $\Delta_c = 4$ and $\Delta = 0$ implies that the first flap (horizontal) is ascending and the second (vertical) is descending. Computation of the topological invariant gives $L = L_c = 0$ and we have another unclassified sequence: $${\boxslash}E N_1 E W_3 N S_2 W S .$$ The two lowest superposed tiles can be exchanged, however the resulting assemblage is equivalent due to the reflection symmetry of the sequence of symbols. ### Sequence $ENWESNWS$ Imposing $\Delta = 0$ the two flaps (one is horizontal and one is vertical) must be both ascending or both descending. In both cases we compute $L = L_c = 0$. Actually the two assemblages are equivalent by taking advantage of the symmetry of the sequence, one of these (unclassified) is $${\boxslash}E_1 N_1 W_1 E_2 S_2 N_3 W_2 S . $$ ### Sequence $EENWSSNW$ Imposing $\Delta = 0$ the two flaps (one horizontal and one vertical) must be both ascending or both descending. In both cases we compute $L = L_c = 0$. The two assemblages are equivalent as in the previous case, one of these (unclassified) is $${\boxslash}E_1 E N W S_3 S N_2 W .$$ The three sequences with two flaps separated by two tiles ----------------------------------------------------------- All three admissible sequences with two flaps at distance $3$ (separated by two tiles) have $\Delta_c = 0$. Two of these sequences have both horizontal or both vertical flaps, so that $\Delta = 0$ entails that one flap is ascending and one is descending. The third sequence has an horizontal flap and a vertical flap so that $\Delta = 0$ entails that both flaps are ascending or both descending. In all cases we have two selfcrossings of the ribbon with opposite sign, hence $L = L_c = 0$ and might be constructible. Each of the three sequences admit two distinct assemblages both with $\Delta = L = 0$: $$\begin{aligned} {\boxslash}E_2 E_2 E W_1 N W S_1 W \quad \text{,} \quad {\boxslash}E_1 E_1 E W_2 N W S_2 W \label{eq:seqf2a6bc} \\ {\boxslash}E_2 E N W_1 N S_2 S_1 W \quad \text{,} \quad {\boxslash}E_1 E N W_2 N S_1 S_2 W \nonumber \\ {\boxslash}E_2 E N W_2 W E_1 S_1 W \quad \text{,} \quad {\boxslash}E_1 E N W_1 W E_2 S_2 W \label{eq:seqf2a6a}\end{aligned}$$ The first two and the last two are actually constructible (Figures \[fig:flap2a6b\], \[fig:flap2a6c\], \[fig:flap2a6a\], \[fig:flap2a6aiso\], Section \[sec:costruibili\]). The twelve sequences with two flaps in antipodal position --------------------------------------------------------- ---------------- -------------- sequences distribution with $2$ flaps of flaps 2 ffxxxxxx 5 fxfxxxxx 3 fxxfxxxx 12 fxxxfxxx ---------------- -------------- sequences dist. of flaps ----------- ---------------- 1 ffffxxxx 5 fffxxfxx 1 ffxffxxx 4 ffxfxxfx 1 ffxxffxx 6 fxfxfxfx \[tab:2e4flap\] Of the $12$ sequences with two flaps in opposite (antipodal) position we first analyze those (they are $10$) in which the tiles follow the same path from one flap to the other and back. One of these is shown in Figure \[fig:lunga\]. All have $\Delta_c = 0$ so that the contribution of the two flaps must have opposite sign in order to have $\Delta = 0$. If one flap is horizontal and the other vertical, then they must be both ascending or both descending and we have no possible admissible assemblage (Remark \[rem:assemblabili\]). We are then left with those sequences having both horizontal or both vertical flaps, one ascending and one descending. In this situation we find that the ribbon has $3$ selfcrossings, so that necessarily $L = L_c \neq 0$ and these sequences are also not constructible. We remain with the two sequences $EENEWWSW$ and $EENNSWSW$ that both have a contribution $\Delta_c = -4$ (fixing $T_0$ of type ${\boxslash}$), so that the two flaps must contribute with a positive sign to the metric invariant. The first sequence has both horizontal flaps, and they must be both descending, this is now possible thanks to the different path between the two flaps. The second sequence has one horizontal and one vertical flap, so that the first must be ascending and the second descending. There are exactly two selfcrossings of the ribbon in both cases, however they have the same sign in the first case implying $L = L_c \neq 0$, hence non constructible. They have opposite sign in the second case and we have both zero invariants. In conclusion the only one of the $12$ sequences that might be constructible is $${\boxslash}E_1 E N_1 N S_2 W S_2 W .$$ Sequences with three flaps ---------------------------- Of the $10$ admissible sequences with three flaps there is only one with all adjacent flaps, having $\Delta_c = \pm 2$. The three flaps being consecutive are all horizontal or all vertical and all ascending or all descending, with a total of $\Delta_f = \pm 6$ and the metric invariant cannot be zero. Four sequences have two adjacent flaps, and in all cases $\Delta_c = \pm 2$. Imposing $\Delta = 0$ allows to identify a unique assemblage for each sequence (with one exception). In all cases a direct check allows to compute $L = L_c = 0$. These sequences are: $$\begin{aligned} {\boxslash}E_3 E_2 W_{1,2} E_1 N W S_{2,1} W \quad \text{,} \quad {\boxslash}E_3 E N W_1 S_1 N_2 S_2 W \label{eq:seqf3a5c} \\ {\boxslash}E_2 E N_2 W_2 E_1 W_1 S_1 W \quad \text{,} \quad {\boxslash}E_2 E_1 N_1 S_2 N_2 W S_1 W . \label{eq:seqf3a5a}\end{aligned}$$ The last two are actually constructible (Figures \[fig:flap3a5a\] and \[fig:flap3a5b\], Section \[sec:costruibili\]). One of the two assemblages of the left sequence in can be actually constructed (Figure \[fig:flap3a5c\]). If the puzzle has sufficiently deformable nylon threads and tiles we could conceivably deform the first assemblage into the second. We do not know at present if the right sequence in is constructible (unclassified). The five remaining sequences all have $\Delta_c = \pm 2$. Imposing $\Delta = 0$ leaves us with $10$ different assemblages: the sequence with all three horizontal flaps has three different assemblages with $\Delta = 0$, three of the remaining four sequences (with two flaps in one direction and the third in the other direction) have two assemblages each, the remaining sequence has only one assemblage with $\Delta = 0$. In all cases a direct check quantifies in $3$ or $5$ (in any case an odd value) the number of selfcrossings of the ribbon, so that $L = L_c \neq 0$. None of these sequences is then constructible. Sequences with four flaps --------------------------- There are $18$ such sequences. Six of these have a series of at least three consecutive flaps and a contribution $\Delta_c = 0$. They are not constructible because the consecutive flaps all contribute with the same sign to $\Delta_f$. The sequences $ENWEWSNS$ and $ENWEWSEW$ have $\Delta_c = 0$ and two pairs of adjacent flaps oriented in different directions in the first case and in the same direction in the second case. To have $\Delta = 0$ they must contribute with opposite sign and hence must be all ascending or all descending in the first case whereas in the second case one pair of flaps must be ascending and one descending. Thanks to the symmetry of the sequences the two possible assemblages of each are actually equivalent. The linking number turns out to be $L = 0$ and we have two possibly constructible configurations. There are four sequences with a single pair of adjacent flaps, the other two being isolated, all with $\Delta_c = 0$. The two isolated flaps must contribute to the metric invariant with the same sign, opposite to the contribution that comes from the two adjacent flaps. In three of the four cases the two isolated flaps have the same direction and hence both must be ascending or both descending. It turns out that there is no admissible assemblage with such characteristics. The pair of adjacent flaps of the remaining sequence ($EENSWEWW$) are horizontal. If they are ascending the remaining horizontal flap must be descending whereas the vertical flap must be ascending (to have $\Delta = 0$). This situation (or the one with a descending pair of adjacent flaps) is assemblable and we can compute the linking number, which turns out to be $L = L_c = \pm 2$. Even this configuration is not constructible. The remaining six sequences (flaps alternating with non-flap tiles) all have the non-flap tiles superposed to each other. An involved reasoning, or the use of the software code, allows to show that for two of this six sequences, having area $3$, namely $EEWWEEWW$ and $ENSWENSW$, there is no possible admissible assemblage with $\Delta = 0$. ### Sequence ENSEWNSW This sequence has area $4$, with two of the four flaps superposed to each other. Using the software code we find two different assemblages having $\Delta = 0$. Computation of the linking number leads in both cases to $L = L_c = \pm 2$, hence this sequence is not constructible. ### Sequence EEWNSEWW This sequence has also area $4$, with two of the four flaps superposed to each other. Using the software code we find only one assemblages having $\Delta = 0$. Computation of the linking number leads in to $L = L_c = \pm 2$, hence this sequence is also not constructible. ### Sequence EEWNSSNW This sequence has area $5$ with no superposed flaps and contribution $\Delta_c = 0$, so that to have $\Delta = 0$ two flaps contribute positively and two contribute negatively to the metric invariant. The software code gives three different assemblages with $\Delta = 0$. An accurate analysis of the selfcrossings of the ribbon due to the flaps shows that flaps that contribute positively to the metric invariant also contribute with an odd number of selfcrossings of the ribbon, besides there is one selfcrossing due to the crossing straight tiles. In conclusion we have an odd number of selfcrossings, hence the sequence is not constructible. ### Sequence ENSEWSNW This sequence has also area $5$ with no superposed flaps, but now the contribution of the curving tiles to the metric invariant is $\Delta_c = 4$, so that to have $\Delta = 0$ exactly one of the flaps has positive contribution to the metric invariant. This flap will also contribute with an odd number of selfcrossings of the ribbon. In this case there are no other selfcrossings of the ribbon because there are no straight tiles, so that we again conclude that the number of selfcrossings of the ribbon is odd and that $L = L_c \neq 0$. This sequence is also not constructible. Sequences with five flaps --------------------------- Both sequences have $\Delta_c = -2$. The software code quickly shows that the sequence $EEWNSNSW$ does not have any admissible assemblage with $\Delta = 0$. The other sequence is $EEWEWNSW$ and to have $\Delta = 0$ the three consecutive horizontal flaps must be ascending, and also the vertical flap must be ascending. There is one admissible assemblage satisfying these requirements, but the resulting number of selfcrossings of the ribbon is odd, and so also this configuration is not constructible. Sequences with six flaps -------------------------- None of the four sequences with six flaps is constructible. Indeed it turns out that all have $\Delta_c = 0$, so that in order to have $\Delta = 0$ they must have three flaps with positive contribution and three with negative contribution to $\Delta_f$. Two of the four sequences have four or more flaps that are consecutive and hence all contribute with the same sign to $\Delta_f$, so that $\Delta \neq 0$. The sequence $EEWEWWEW$ must have three ascending consecutive flaps and three consecutive descending flaps (all flaps are horizontal). Analyzing the ribbon configuration shows that there are an odd number of ribbon selfcrossings, hence $L = L_c \neq 0$. Finally, all the six flaps of sequence $ENSNSWEW$ must be ascending in order to have $\Delta = 0$, there is no admissible assemblage with this property. Sequences with seven flaps ---------------------------- There is none. Sequences with eight flaps ---------------------------- The only one is $EWEWEWEW$, but there is no admissible assemblage of this sequence. Constructible configurations {#sec:costruibili} ============================ Some of the sequences (or better, admissible assemblages of a sequence) for which both invariant vanish are actually constructible with the real puzzle, starting from the initial $2 \times 4$ configuration. We shall list these configurations in this Section together with a snapshot of the real puzzle taken using a clone of the puzzle that we used for the experimentation. We follow the same notation as in Section \[sec:forsecostruibili\] to describe a given face-up planar configuration, the sequence we list is always the canonical representative (see Section \[sec:piane\]), so that the first cardinal direction is always $E$ and the second is never $S$. The initial ${\boxslash}$ or ${\boxbslash}$ symbol indicates the type of the first tile $T_0$, we made an effort to normalize the configuration so that the first tile is of type ${\boxslash}$, however this sometimes entails going from a configuration to its specular image which is proved to be possible (Theorem \[teo:mirrorconstructible\]) but requires the application of moves towards the $2 \times 4$ configuration and then back using the specular moves in reverse, which cannot be done if we do not remember how we obtained the configuration. For consistency of notation, in one snapshot (Figures \[fig:flap3a5b\]) we used the trick of reflecting the image left-right (as if taking the snapshot through a mirror). ![[]{data-label="fig:magic27"}](magic27.png){height="6cm"} Intermediate 3D configurations ------------------------------ In order to reach the planar configurations in which we are interested it is necessary to walk through various 3D shapes. Figure \[fig:magic27\], taken from the web page [@basteleien:web], shows a collection of constructible 3D symmetric shapes such that all angles are multiples of $90$ degrees and there are no overlapping tiles (double walls). We shall refer to each shape of Figure \[fig:magic27\] by indicating the coordinates, row and column, in which they appear; for example shape (5,10) refers to the first green shape, the tenth of the fifth row. It is important to observe that in Figure \[fig:magic27\] there is no indication of the orientation of the tiles, which is however an essential information. For example, the shape with coordinates (6,10) can be obtained with two completely different configurations of the puzzles: one can be constructed following the instructions in section “cube” of the web page [@basteleien:web] during the process to obtain the cubic shape, the other can be constructed by following the instructions of [@mine:web] during the construction of the planar shape denoted by `seqf1a6a`. Similarly, in Figure \[fig:magic27\] there is no indication of the relative position of the tiles, [i.e.]{} how they are positioned in the 3D shape with respect to their circular ordering, although in most cases there is only one possible circular ordering (possibly up to symmetries of the puzzle, including reflection) of the tiles in the displayed shapes such that two consecutive tiles are hinged together. We shall sometimes refer to planar but not necessarily “face-up” configurations. These are configurations of the puzzle where all the tiles (supposed to be of infinitesimal thickness) lie in the same plane, but are allowed to be “stacked” onto each other. In a stack of tiles we can identify a “front” tile (the one nearest to the observer), a “back” tile and possibly some intermediate tiles. For definiteness planar configurations are always positioned vertically in the 3D space, so that we can locate a tile (or a stack of tiles) using adjectives like left/right or upper/lower. In this situation a “horizontal fold” always refers to a $180$ degrees rotation of one or more tiles around a common vertical axis, conversely for a “vertical fold”. In [@Nou:86] there are detailed instructions on how to obtain particular planar configurations that are not “face-up”. They consist of four adjacent stacks of two tiles, one face-up and one face-down, and turn out to be particularly useful as starting point for various constructions, they are grouped in three families: - I-1, I-2, I-3 with the four stacks aligned in a straight row to resemble a capital letter ‘I’; They are distinguised by how they are connected by the nylon strings. Shape I-1 for example can be directly obtained from the $2 \times 4$ configuration just by folding the four tiles of the top row onto the lower row of tiles; - S-1 to S-4 where the four stacks are positioned to form an ‘S’ shape; - L-1 to L-8 with the four stacks positioned in an ‘L’ shape. The variety of structurally distinct shapes that can be formed even in the very simple shapes of these three families is one of the marvels of the Rubik’s Magic. ![[]{data-label="fig:shapes"}](shape_1_9 "fig:"){width="2.25cm"} ![[]{data-label="fig:shapes"}](shape_6_9 "fig:"){width="4.25cm"} ![[]{data-label="fig:shapes"}](nsshape "fig:"){width="3.25cm"} The shape of Figure \[fig:shapes\] (ns) is not included in the table of Figure \[fig:magic27\] because it is not symmetric. It can be obtained from S-3 of [@Nou:86] as follows. Open S-3 to obtain one of the two different realizations of scheme (4,6) of Figure \[fig:magic27\]; close the left “wing” against one of the lateral faces of the “tube” (closing the other wing will produce the specular version of the final shape); this allows to free the corresponding tile of the lateral face and lift it into a horizontal position and then to push it further up “inside” the tube into a vertical position against one of the upper tiles; open that upper tile towards the outside and turn the whole structure upside-down to reach the position illustrated in Figure \[fig:shapes\] (ns). Constructible configurations with no flaps ------------------------------------------ These are well-known. The $2 \times 4$ rectangle of Figure \[fig:black2x4\] is just the (trivially obtaines) initial configuration, sequence $$E E E N W W W S.$$ The puzzle solution is shown in Figure \[fig:blacksolved\], sequence $$EENNWSWS.$$ One flap -------- The sequence is a $2 \times 3$ rectangle (Figure \[fig:flap1a6b\]): $${\boxslash}E_2 E N W S_1 W_1 N S_2.$$ This configuration can be obtained starting from configuration I-2 of [@Nou:86], to be folded as shown in Figure \[fig:chains1e2\] left (each segment of the sketch represents a tile seen from a side) and transformed with a couple of moves into the sketch in the right of Figure \[fig:chains1e2\]. Now we can open the lowest strip of three consecutive tiles to obtain a $2 \times 3$ rectangle. Finally we overturn the pair of adjacent tiles that are overlapped to two other. ![[]{data-label="fig:chains1e2"}](chain1 "fig:"){width="4cm"} ![[]{data-label="fig:chains1e2"}](chain2 "fig:"){width="6cm"} ![[]{data-label="fig:chains4e5"}](chain4 "fig:"){width="4cm"} ![[]{data-label="fig:chains4e5"}](chain5 "fig:"){width="6cm"} The configuration of Figure \[fig:flap1a6a\] $${\boxslash}E_2 E N W S_1 W_1 S N_2$$ is equivalent to the sequence . It can be obtained with a long sequence of moves starting from the configuration denoted by S-1 in [@Nou:86], from which we obtain the configuration of Figure \[fig:shapes\] (1,9) to be rotated in such a way that the two frontal tiles be of type ${\boxslash}$ the left one and ${\boxbslash}$ the right one. The sequence of moves from there is visually illustrated in the web page [@mine:web]. The configuration of Figure \[fig:flap1a6d\] $${\boxslash}E_2 E_2 E N W S_1 W_1 W$$ coincides with sequence \[eq:seqf1a6d\]. Two intermediate steps are schematically shown in Figure \[fig:chains4e5\]. The left configuration of Figure \[fig:chains4e5\] can be obtained starting from L-6 of [@Nou:86], to be modified by rotating down of one position the four tiles of the front stratum, then rotating two vertically adjacent tiles of the back stratum to the right and lifting a “flap” tile located on the right in the back stratum to obtain a $2 \times 3$ flat shape (by opening it we obtain the 3D shape in position $(5,10)$ of Figure \[fig:magic27\], the first green configuration). We now fold upwards the three lower tiles (two of them are stacked onto each other) with a “valley” fold into a $2 \times 2$ flat shape to be again folded horizontally with a vertical “valley” fold into a $2 \times 1$ flat shape that corresponds to the left sketch of Figure \[fig:chains4e5\]. Two flaps --------- A $2 \times 3$ rectangular shape corresponding to the sequence has both assemblages constructible (Figures \[fig:flap2a6b\] and \[fig:flap2a6c\]: $${\boxslash}E_1 E N W_1 W E_2 S_2 W \qquad \text{,} \qquad {\boxslash}E_2 E N W_2 W E_1 S_1 W .$$ The constructing procedure of the first can be found in [@Nou:86 page 54] during the transition from configuration L-3 to S-2. The second can be found in [@Nou:86 page 52] during the transition from configuration S-3 to L-8. A third constructible configuration (sequence ) with two flaps, together with a different assemblage (Figures \[fig:flap2a6a\] and \[fig:flap2a6aiso\]) are $${\boxslash}E_1 E_1 E W_2 N W S_2 W \quad \text{and} \quad {\boxslash}E_2 E_2 E W_1 N W S_1 W$$ The first assemblage can be obtained starting from L-1 of [@Nou:86] that can be opened to form the shape $(1,6)$ of Figure \[fig:magic27\] (in the configuration with the two front tiles of type of type ${\boxbslash}$, the left one, and ${\boxslash}$, the right one). By pushing all the way down the middle hinge of the top “roof” and making horizontal again the two adjacent tiles, now forming a kind of “floor” we can open the two most lateral tiles and reach the shape of Figure \[fig:shapes\] (6,9). The final shape can be obtained from here by grabbing the two flaps, translating them towards each other and then back again with a change of “hinging”. The second assemblage can be obtained from the shape of Figure \[fig:shapes\] (ns) by closing the upper “wing” clockwise against the “tube”; this allows to lower the corresponding tile of the tube in a horizontal position; finally we push further the same tile into the tube and open the corresponding tile into the required configuration. It turns out that these two configurations can both be opened into shape $(3,4)$ of Figure \[fig:magic27\]. A configuration with two flaps of area $5$ is shown in Figure \[fig:flap2a5a\], obtained starting from the chiral version of the configuration of Figure \[fig:flap1a6a\] as follows. First orient the configuration as in Figure \[fig:flap1a6a\] with front/back reflection. Turn the two rightmost tiles back with a “mountain” fold; flip up backwards one of the right tiles, the back one in a stack of three; flip down one of the right tiles, the front one in a stack of three; finally flip right two vertically adjacent tiles located on the right in the back of the corresponding stacks. Three flaps ----------- In Figures \[fig:flap3a5b\] and \[fig:flap3a5a\] we find the configurations $${\boxslash}E_2 E_1 N_1 S_2 N_2 W S W \quad \text{and} \quad {\boxslash}E_2 E N_2 W_2 E_1 W_1 S_1 W$$ corresponding the sequences . The configuration of Figure \[fig:flap3a5a\] can be obtained from the shape of Figure \[fig:shapes\] (ns) by first “closing” it and then lowering a flap and lifting another flap. Configuration \[fig:flap3a5b\] is just a few moves away from there. Note that these two configurations are connected by a few moves to the configuration of Figure \[fig:flap1a6a\], thus giving an alternative way to obtain it. The configuration ${\boxslash}E_3 E_2 W_1 E_1 N W S_2 W$ is shown in Figure \[fig:flap3a5c\]. It can be obtained starting from the configuration of Figure \[fig:flap1a6a\] as follows. Flip the two rightmost tiles with a “valley” fold; lift a single tile on the right, located on the right in front of a stack of three tiles; flip to the left a set of three tiles positioned on the right behind the two front-most tiles in the two stacks; finally flip up a back tile on the left and flip down a front tile on the left. By differently pocketing one of the tiles it is possible to also obtain the assemblage ${\boxslash}E_3 E_2 W_2 E_1 N W S_1 W$, however such operation requires a great deal of elasticity on the nylon threads and on the tiles, and we did not try it on the real puzzle. We don’t know if there is some other (less stressing) way to obtain it. ![[]{data-label="fig:flap1a6d"}](flap1a6d.jpg){height="5cm"} ![[]{data-label="fig:flap2a5a"}](flap2a5a.jpg){height="4cm"} ![[]{data-label="fig:flap3a5c"}](flap3a5c.jpg){height="4cm"} The software code {#sec:code} ================= The software code can be downloaded from [@mine:web] and should work on any computer with a `C` compiler. If run without arguments, it will search for all canonical representatives of the set ${\mathcal S}$ of equivalent classes of sequences. this is part of its output: $ \textit{./rubiksmagic} EEEEWWWW f=2 area=5 Dc=0 symcount=8 assemblages=1 deltaiszero=1 EEENWWWS f=0 area=8 Dc=0 symcount=4 assemblages=1 deltaiszero=1 EEENWWSW f=1 area=7 Dc=-2 symcount=1 assemblages=2 deltaiszero=1 [...] ENSWENSW f=4 area=3 Dc=0 symcount=8 assemblages=0 deltaiszero=0 EWEWEWEW f=8 area=2 Dc=0 symcount=32 assemblages=0 deltaiszero=0 Found 71 sequences $ It searches for all admissible sequences that are the canonical representative of their equivalent class in ${\mathcal S}$ (it finds $71$ equivalent classes), for each one it prints the sequence followed by some information (to be explained shortly). The software allows for puzzles with a different number of tiles, for example for the large version with $12$ tiles of the puzzle it finds $4855$ equivalence classes, with a command like $ \textit{./rubiksmagic -n 12} EEEEEEWWWWWW f=2 area=7 Dc=0 symcount=8 assemblages=1 deltaiszero=1 EEEEENWWWWWS f=0 area=12 Dc=0 symcount=4 assemblages=1 deltaiszero=1 EEEEENWWWWSW f=1 area=11 Dc=-2 symcount=1 assemblages=2 deltaiszero=1 [...] ENSNSWENSNSW f=8 area=3 Dc=0 symcount=4 assemblages=0 deltaiszero=0 ENSNSWENSWEW f=8 area=3 Dc=0 symcount=2 assemblages=0 deltaiszero=0 ENSWENSWENSW f=6 area=3 Dc=0 symcount=12 assemblages=0 deltaiszero=0 EWEWEWEWEWEW f=12 area=2 Dc=0 symcount=48 assemblages=0 deltaiszero=0 Found 4855 sequences $ however the computational complexity grows exponentially with the number of tiles. Another use of the code allows to ask for specific properties of a given sequence, we illustrate this with an example: $ \textit{./rubiksmagic -c EEWENWSW} EEWENWSW f=3 area=5 Dc=-2 symcount=1 assemblages=6 deltaiszero=2 Assemblage with delta = 0: sla E3 E2 W2 E1 N1 W1 S1 W1 Assemblage with delta = 0: sla E3 E2 W1 E1 N1 W1 S2 W1 $ The first line of output displays some information about the sequence given in the command line, specifically we find - the sequence itself; - the number of flaps ($3$ in this case); - the area of the plane covered ($5$); - the computed contribution $\Delta_c$ coming from the curving tiles; - the cardinality of the group of symmetries of the sequence, this particular sequence does not have any symmetry; - the number of admissible assemblages of the sequence, counting only those that start with $T_0$ of type ${\boxslash}$ and identifying assemblages that are equivalent under transformations in the group of symmetries of the sequence; - the number of admissible assemblages with vanishing metric invariant ($\Delta = 0$), we have two in this case. Then we have one line for each of the possible assemblages with $\Delta = 0$ with a printout of each assemblage, the numbers after each cardinal direction tells the level of the tile reached with that direction. It will be $1$ for tiles that are not superposed with other tiles, otherwise it is an integer between $1$ and the number of superposed tiles. The option ‘`-c`’ on the command line can be omitted in which case the software computes the canonical representative of the given sequence and prints all the informations for both the original sequence and the canonical one. Note that the sign of the invariants is sensitive to equivalence transformations. [99]{} Rubik’s Magic - Wikipedia, <https://en.wikipedia.org/wiki/Rubik's_Magic>, retrieved Jan 15, 2014. J. Köller, Rubik’s Magic, <http://www.mathematische-basteleien.de/magics.htm>, retrieved Jan 15, 2014. Jaap Scherphuis, Rubik’s Magic Main Page, <http://www.jaapsch.net/puzzles/magic.htm>, retrieved Jan 15, 2014. M. Paolini, Rubik’s Magic, <http://dmf.unicatt.it/~paolini/rubiksmagic/>, retrieved Jan 15, 2014. J.G. Nourse, Simple Solutions to Rubik’s Magic, New York, 1986. [^1]: This proof is due to Giovanni Paolini, Scuola Normale Superiore of Pisa.
--- abstract: 'In the field of quantum process tomography, the average fidelity quantifies how well the quantum channel preserves quantum information. In present work, we shall develop a protocol to estimate the average fidelity for the bipartite system. We show that the average fidelity should be known if the three measurable quantities, the average survive probability of the product state and the average survive probability of each subsystem, have been decided. Our protocol can be also applied to decide the selected element of the quantum process matrix.' address: 'College of Physical Science and Technology, Sichuan University, 610064, Chengdu, China.' author: - Long Huang - Xiaohua Wu title: A protocol to estimate the average fidelity of the bipartite system --- introduction ============ The characterization of the evolution of a quantum system is one of the main tasks to accomplish to achieve quantum information processing. A general class of methods, which have been developed in quantum information theory to accomplish this task is known as quantum process tomography (QPT)- for a review of quantum tomography, see Refs. \[$1-3$\]. The various protocols of getting the complete information about the quantum process can be divided into two classes: The standard quantum process tomography (SQPT) \[1,4,5\], with the central idea of preparing a set of linearly independent inputs and measuring the outputs via the quantum state tomography (QST), works without requiring any additionally quantum resources. Another is the so-called ancilla-assisted quantum process tomography (AAQPT) with the approach of encoding all information about the transformation into a single bipartite system-ancilla quantum state \[6-8\]. As a known fact, the complete characterization of a unknown quantum channel is a non-scalable task: For the $N$ $d$-level system, there are about $d^{4N}$ elements to be decided. Naturally, one may ask: Can the number of experiment be scalable if partial information about the quantum process is to be characterized ? The average fidelity is an important quantity in QPT. It quantifies how well the quantum map preserves quantum information \[9,10\]. Recently, it has been shown that it is possible to estimate the average fidelity via a technique known as twirling \[9-15\]. The Haar-twirl channel can be produced by the so-called Haar twirling procedure. It consists of applying a unitary, which is randomly chosen with the Haar measure, before the process to be characterized, followed by the inverse of the same unitary. The average fidelity can be estimated by preparing an arbitrary pure state for the input of the Haar-twirl channel and then measuring its survive probability ( the overlap between the input and output). Another twirling procedure, where the unitary is sampled uniformly from the Clifford group, has been proposed in \[12\]. The resulted Clifford-twirl channel was shown to be equivalent with the Haar-twirl one. Still, the exact measurement of the average fidelity with the Clifford twirling protocol, which involves finite but exponentially large resources, is a non-scalable task. However, to experimentally characterize the fidelity of a quantum process on $n$ qubits for a desired accuracy, an efficient protocol has been constructed with quantum circuits of size $O(n)$ without requiring any ancilla qubits \[12 \]. Besides the twirling protocol, another important ancilla-less way for deciding the average fidelity has also been developed. The average fidelity should be known if the survive probability of each state, which belongs to the state 2-design, has been decided \[16-18\]. Furthermore, it was found that the state 2-design protocol can be also adapted to estimate an arbitrary element of the quantum process matrix. For the $D$-dimensional system, the state 2-design usually has more than $D^2$ elements in it. Therefore, an efficient method to estimate a selected element, where the error scales as $\sqrt{1/M}$ with $M$ the number for the repetitions of the experiment, has also been proposed in \[16-18\]. The concept of the average fidelity can be generalized to the gate fidelity, a quantity characterizing how well the quantum map approximate a quantum gate \[9\]. It is demonstrated that twirling experiments previously used to characterize the average fidelity of quantum memories efficiently can be easily adapted to estimate the average fidelity of the experimental implementation of important quantum computation processes, such as untaries in the Clifford group, in a practical and efficient manner\[19\]. In present work, we shall develop a protocol to estimate the average fidelity of the quantum channel for a bipartite system. Our work is motivated by such an interesting case: In the Bell-type experiment, two spin-$s$ particles are initially prepared in an arbitrary state from a quantum source, then each particle is sent to Alice and Bob, the two users who are space separated, respectively. In general, we suppose that there exists a quantum map which relates the initial state (for the two particles in the source) to the final state (for the two particles in the users’ hand). Now, the average fidelity is still an important quantity to characterize the quantum process where the state should be kept unchanged. For such a case, the average fidelity is hard to be measured in a directly way: By its definition, one should measure the survive probability of an arbitrary state. However, the problem appears when the two particles are prepared in an entangled state. As a solution for it, we introduce three directly measurable quantities, the average survive probability of the product state and the average survive probability of each subsystem, for the bipartite system and give a formula to estimate the average fidelity with the introduced quantities. Furthermore, we show that our protocol can be also applied to decide an arbitrary selected element of the quantum process matrix. The rest content of present work can be divided into following parts. In Sec. II we shall give a brief review of the known ancilla-less methods used to estimate the average fidelity of the quantum channel. Especially, we introduce the convenient tool where a bounded matrix is related to a vector in the enlarged Hilbert space. As an application of it, we show that the average fidelity can be calculated as the expectation of the quantum process super operator with the separable Werner state. In Section III we shall firstly define the three quantities, the average survive probability of the product state and the average survive probability of each subsystem, for the bipartite system and then design several protocols to measure them. A formula, where the average fidelity is related to the three average survive probabilities, should be constructed there. In Sec. IV we define the quantum process matrix and its elements in an explicit way. By following the idea presented in \[18\], the protocol used to measure the average fidelity is adapted to decide an arbitrary element of the quantum process matrix. In Sec. V we shall develop an efficient protocol to measure the average fidelity. Finally, we end our work with a short discussion. Measuring the average fidelity with the protocol of twirling ============================================================== In this section, we shall firstly give a brief review of the known ancilla-less methods applied to measure the average fidelity of the quantum channel. Let $\{\vert i\rangle\}_{i=1}^D$ the basis of a $D-$diamensional Hilbert space ${\mathrm{H}}_{D}$. For the state vector $\vert\psi\rangle=\sum_{i=1}^Dc_i\vert i\rangle$, the conjugated state vector $\vert \psi ^\rangle $ is defined as $\vert\psi^\rangle=\sum_{i=1}^Dc_i^\vert i\rangle$. A corresponding capital letter, $\Psi$, is used to denote the projective operator, $\Psi=\vert\psi\rangle\langle\psi\vert$. With these denotations in hands, the average fidelity of the quantum map $\varepsilon$ can be defined as, $$f^{\mathrm{avg}}(\varepsilon)=\int d\mu_{\mathrm{H}}(\Psi)\mathrm{Tr}[\Psi\varepsilon(\Psi)],$$ with $d\mu_{\mathrm{H}}(\Psi)$ the Haar-measure of states in $\mathrm{H}_D$ and the process super operator $\varepsilon$ to be $\varepsilon(\rho)=\sum_{n}A_n\rho A_n^{\dagger}$. Usually, we suppose $\varepsilon$ is trace preserving, $\sum _{n} A_{n}^{\dagger}A_{n}=\mathrm{I}_{D}$. As it is depicted in FIG. 1a, we prepare an arbitrary state $\vert \psi\rangle$ for input of the quantum channel $ \varepsilon$, after the evolution, measure the survival probability $\mathrm{Tr}[\Psi\varepsilon(\Psi)]$. By sampling the state $\vert \psi\rangle$ with the Haar measure, the average fidelity of the channel $ \varepsilon$ should be decided. Let $\vert \psi_0\rangle$ to be a fixed state in ${\mathrm{H}}_{D}$, one may relate the arbitrary state $\vert \psi\rangle $ to a unitary transformation $U$ ($U\in \mathrm{U}(D)$), $\Psi=U\Psi_{0}U^{\dagger}$. Now, the average fidelity in (1) can be rewritten as $$f_b^{\mathrm{avg}}(\varepsilon)=\int d\mu_{\mathrm{H}}(U)\mathrm{Tr}[\Psi_{0}U^{\dagger}\varepsilon(U\Psi_0U^{\dagger})U].$$ In the derivation of it, we have applied the property of the trace operation, $\mathrm{Tr}[ABC]=\mathrm{Tr}[BCA]$. Here, we use a subscript $b$ to indicate that the average fidelity can be estimated with the Haar twirling protocol depicted in FIG. 1b: Apply a random unitary $U$ to the initial state $\vert \psi_0\rangle$, followed by the quantum operation $\varepsilon$, and then apply $U^{\dagger}$ to the output state. Then from (2), the average fidelity can be estimated by repeating the procedure with $U$ sampled randomly from the Haar measure in each experiment. ![\[fig:epsart\] (a)The average fidelity involves measuring the survive probability of an arbitrary state sampled with the Haar measure. It can be estimated with different protocols: (b) The Haar twirling, (c) the Clifford twirling, and (d) the sate 2-design.](fig1.eps) In general, one may view the Haar twirling procedure as to prepare a so-called Haar-twirl channel $\varepsilon^{\mathrm{HT}}$, $$\begin{aligned} \varepsilon^{\mathrm{HT}}(\rho)&=&\int d\mu_{\mathrm{H}}(U) \mathcal{U}^{\dagger}\circ\varepsilon\circ \mathcal{U}(\rho) \nonumber\\ &=&\int d\mu_{\mathrm{H}}(U) U^{\dagger}\varepsilon(U\rho U^{\dagger})U].\nonumber \end{aligned}$$ In this picture, the average fidelity in (2) can be interpreted as the survive probability of the fixed state $\Psi_0$ in the Haar-twirl channel $\varepsilon^{\mathrm{HT}}$, $f_b^{\mathrm{avg}}(\varepsilon)=\mathrm{Tr}[\Psi_0\varepsilon^{\mathrm{HT}}(\Psi_0)]$. The Haar twirling is hard to realize in experiment since that it involves preparing a continuous set of unitary operations. To alleviate it, the so-called Clifford-twirl channel $\varepsilon^{\mathrm{CT}}$ was introduced in \[12\], $$\begin{aligned} \varepsilon^{\mathrm{CT}}(\rho)&=&\frac{1}{K}\sum_{i=1}^{K}(\mathcal {C}^{\dagger}_{i}\circ \varepsilon\circ \mathcal{C}_{i})(\rho) \nonumber\\ &=&\frac{1}{K}\sum_{i=1}^{K}C_i^{\dagger}\varepsilon( C_i \rho C_{i}^{\dagger}) C_i,\nonumber\end{aligned}$$ where $C_i$ are the elements of the Clifford group of ${\mathrm{H}}_{D}$. To carry out the Clifford twirling in experiments, only a finite number ($K$) of operations should be prepared. It has been proven that the two different twirling procedures should result the same channel $\varepsilon^{\mathrm{CT}}=\varepsilon^{\mathrm{HT}}$. Certainly, as it is shown in FIG. 1c, the Clifford twirling protocol can be also applied to get the average fidelity. Formally, we express it as $$\begin{aligned} f_{c}^{\mathrm{avg}} (\varepsilon)=\frac{1}{K}\sum_{i=1}^{K}\mathrm{Tr}[\Psi_0C_{i}^{\dagger} \varepsilon(C_i\Psi_0 C_{i}^{\dagger}) C_i ].\end{aligned}$$ Recently, it has been found that a state 2-design can be also applied to measure the average fidelity. The state 2-design, $\{\Psi_x\}_{x=1}^{N}$, is set of states satisfying the constraint that $$\frac{1}{N}\sum_{x=1}^N \Psi_x\otimes \Psi_x=\frac{1}{D(D+1)}\sum_{i,j=1}^{D}\vert jk\rangle \langle jk\vert +\vert jk \rangle \langle kj\vert.$$ Straitly to say, this definition is suitable for the case where all the states $\Psi_x$ are equal weighted. For a more general definition of the state 2-design, please see \[20\]. For the case $N=D(D+1)$, the state 2-design is known to be a complete set of mutually unbiased bases (MUBs)\[21-22\]: That is a set of $D+1$ bases for ${\mathrm{H}}_{D}$ with a constant overlap of $1/D$ between elements of different bases, $$\vert \langle \psi_{m}^{j}\vert \psi^{j'}_{m'}\rangle\vert^2=\{\begin{array}{c} \delta_{mm'}~~j=j' \\ \frac{1}{D}~~~~~j\neq j'. \end{array}$$ If $N=D^2$, the state 2-design is unique: It’s just the symmetric information complete (SIC) set $\{\vert \psi^{\mathrm{SIC}}_x\rangle\}_{x=1}^{D^2}$ introduced in \[23\]. The normalized states $\vert \psi^{\mathrm{SIC}}_x\rangle$ have the property that $$\vert\langle \psi^{\mathrm{SIC}}_x\vert \psi^{\mathrm{SIC}}_y\rangle\vert^2=\frac{1+D\delta_{xy}}{1+D}.$$ The way of applying the state 2-design for estimate the average fidelity is shown in FIG. 1d: Preparing a state $\vert \psi_x\rangle$, which belongs to a given set of a state 2-design, for the quantum channel $\varepsilon$, after the evolution, one measure the expectation value of the projective operator $\Psi_x$ with the output $\varepsilon(\Psi_{x})$. The average fidelity should be known by repeating this process with a number of $N$ different inputs, $$f_{d}^{\mathrm{avg}}( \varepsilon)=\frac{1}{N}\sum_{x=1}^{N}\mathrm{Tr}[\Psi_{x}\varepsilon(\Psi_{x})].$$ In the above argument, we have assumed that all the quantities $f_k^{\mathrm{avg}}(\varepsilon)$, which are measured with the protocols depicted from FIG. 1a to FIG. 1d, should equal the average fidelity defined in (1), $$f^{\mathrm{avg}}_k ( \varepsilon)\equiv f^{\mathrm{avg}}(\varepsilon), ~~k=a,b,c,d$$ (Here, we suppose $f^{\mathrm{avg}}_a ( \varepsilon)\equiv f^{\mathrm{avg}}(\varepsilon)$.) Although this equivalence has been verified in previous works, for the convenience of reading, we would still like to give a self -contained proof for it. To complete this task, we shall at first introduce the convenient tool where a bounded matrix in $\mathrm{H}_{D}$ is related to a vector in the enlarged Hilbert space $\mathrm{H}_{D}^{\otimes 2}$. Let $A$ to be a bounded matrix in the $D-$ dimensional Hilbert space $\mathrm{H}_D$, with $A_{ij}=\langle i\vert A\vert j\rangle$ the matrix elements for it, an isomorphism between $A$ and a $D^2-$dimensional vector $\vert A\rangle\rangle$ is defined as $$\vert A\rangle\rangle =\sqrt{D} A\otimes \mathrm{I}_{D}\vert S_+\rangle=\sum_{i,j=1}^D A_{ij}\vert ij\rangle,$$ in which $\vert S_+\rangle$ is the maximally entangled state for $\mathrm{H}_{D}^{\otimes 2}$, $\vert S_+ \rangle =\frac{1}{\sqrt{D}}\sum_{k=1}^{D}\vert kk\rangle$ with $\vert ij\rangle=\vert i\rangle\otimes \vert j\rangle$. This isomorphism offers a one-to-one mapping between the matrix and its vector form. Suppose that $A$ , $B$, and $\rho$ are three arbitrary bounded matrices in $\mathrm{H}_{D}$, there should be $$\mathrm{Tr}[A^{\dagger}B]=\langle\langle A\vert B\rangle\rangle, \vert A\rho B\rangle\rangle =A\otimes B^{\mathrm{T}}\vert \rho \rangle\rangle,$$ with $B^{\mathrm{T}}$ denoting the transpose of B. Especially, if $A$ takes the form $A=\vert\psi\rangle\langle \phi\vert$, its corresponding vector should be $$\vert \vert\psi\rangle\langle \phi\vert\rangle\rangle= \vert \psi\rangle\otimes\vert \phi^\rangle.$$ With the isomorphism in (9) and its properties in (10-11), recalling $\varepsilon(\rho)=\sum_{n}A_n\rho A_n^{\dagger}$, we are able to express the average quantities measured in FIG. 1 in the way like $$f^{\mathrm{avg}}_k(\varepsilon )=\mathrm{Tr}[\hat{F}_k (\sum _{n}A_n\otimes A_n^)],~~ k=a, b, c, d,$$ where $\hat{F}_{k}$, the super operators in $\mathrm{H}_{D}^{\otimes 2}$, are defined as $$\begin{aligned} \hat{F}_a &=&\int d\mu_{\mathrm{H}}(\Psi)\vert \Psi\rangle\rangle\langle\langle \Psi\vert, \\ \hat{F}_b &=&\int d\mu_{\mathrm{H}}(U)U\otimes U^\vert \Psi_0\rangle\rangle\langle\langle \Psi_0\vert (U\otimes U^)^{\dagger}, \\ \hat{F}_c &=& \frac{1}{K}\sum_{j=1}^{K}C_j\otimes C^_j\vert \Psi_0\rangle\rangle\langle\langle \Psi_0\vert (C_j\otimes C_j^)^{\dagger},\\ \hat{F}_d &=&\frac{1}{N}\sum_{x=1}^{N}\vert \Psi_x\rangle\rangle\langle\langle \Psi_x\vert. \end{aligned}$$ Now, if a separable Werner state $\rho^{\mathrm{sep}}_{\mathrm{W}}$ for $\mathrm{H}_{D}^{\otimes 2}$ is introduced as $$\begin{aligned} \rho^{\mathrm{sep}}_{\mathrm{W}}&=&\frac{1}{D(D+1)}(\mathrm{I}_D\otimes \mathrm{I}_D +D\vert S_+\rangle\langle S_+\vert)\nonumber\\ &=&\frac{1}{D(D+1)}(\mathrm{I}_D\otimes \mathrm{I}_D +\vert\mathrm{I}_D\rangle\rangle\langle \langle\mathrm{I}_D\vert), \end{aligned}$$ one may conclude that all the super operators $\hat{F}_k$ are equivalent since that $$\hat{F}_k=\rho^{\mathrm{sep}}_{\mathrm{W}}.$$ A simple reasoning, where the above conclusion can be achieved at, is like this: At first, we take it for granted that $\hat{F}_a=\hat{F}_b$ since that the arbitrary state $\vert \psi\rangle$ in FIG. 1a is related to the arbitrary unitary transformation $U$ in FIG. 1b via the simple relation, $\vert \psi\rangle= U\vert \psi_0\rangle$. At the same time, the relation $\hat{F}_b=\hat{F}_c$ should also hold because that both the Haar twirling and the Clifford twirling will result the same channel, $\varepsilon^{\mathrm{CT}}=\varepsilon^{\mathrm{HT}}$. The proof for $\hat{F}_b=\rho^{\mathrm{sep}}_{\mathrm{W}}$ is given in Appendix. Recall that $\Psi_x$ is a Hermitian operator, $\Psi_x^=\Psi^{\mathrm{T}}_x$. By performing the partial transposition on both sides of (4), the definition of the state 2-design may have an equivalent version: $$\frac{1}{N}\sum_{x=1}^{N}\Psi_x\otimes \Psi_x^=\rho^{\mathrm{sep}}_{\mathrm{W}}.$$ Noting that $\Psi_x\otimes \Psi_x^$ are product states, this is the reason why we call the Wernner state in (17) the separable one. From (11), there should be $ \vert\Psi_x\rangle\rangle\langle\langle\Psi\vert =\Psi_x\otimes \Psi_x^$. Therefore, the relation, $\hat{F}_d=\rho^{\mathrm{sep}}_{\mathrm{W}}$, can be easily verified. estimating the average fidelity of the bipartite system ======================================================== In this section, we shall develop a protocol to estimate the fidelity of a bipartite system $\mathrm{H}=\mathrm{H}^A_D\otimes \mathrm{H}^B_D$. For this $D^2$-dimensional Hilbert system, we use $\Lambda$ to describe a trace preserving quantum map: $$\Lambda(\rho)=\sum_{m}E_m\rho(E_m)^{\dagger}, \sum_{m}(E_m)^{\dagger}E_m=\mathrm{I}_D^{\otimes 2}.$$ For an arbitrary $\Lambda$, we can introduce the following three average quantities, $$\begin{aligned} \bar{f}_{AB}(\Lambda)=\int\int d\mu_{\mathrm{H}}(\Psi) d\mu_{\mathrm{H}}(\Phi) \mathrm{Tr} [\Psi\otimes \Phi\Lambda(\Psi\otimes \Phi)], \\ \bar{f}_A(\Lambda)=\int\int d\mu_{\mathrm{H}}(\Psi) d\mu_{\mathrm{H}}(\Phi) \mathrm{Tr} [\Psi\otimes \mathrm{I}_D\Lambda(\Psi\otimes \Phi)],\\ \bar{f}_B(\Lambda)=\int\int d\mu_{\mathrm{H}}(\Psi) d\mu_{\mathrm{H}}(\Phi) \mathrm{Tr} [\mathrm{I}_D\otimes \Phi\Lambda(\Psi\otimes \Phi)],\end{aligned}$$ where $\Psi\otimes \Phi$ denotes an arbitrary product state in $\mathrm{H}^A_D\otimes \mathrm{H}^B_D$ while $d\mu_{\mathrm{H}}(\Psi)$ and $d\mu_{\mathrm{H}}(\Phi)$ are the Haar measures of the states on $\mathrm{H}_{D}$. In present work, $\bar{f}_{AB}(\Lambda)$ is referred to as the average survival probability of the product states (for channel $\Lambda$), $\bar{f}_A(\Lambda)$ and $\bar{f}_B(\Lambda)$ is the average survival probability for subsystem system $\mathrm{H}_D^A$ and $\mathrm{H}^{B}_D$, respectively. An experimental protocol of measuring the above three quantities is depicted in FIG. 2a: Preparing an arbitrary product state $\vert\psi\rangle\otimes \vert\phi\rangle$ as the input for the quantum channel $\Lambda$, after the evolution, one may simultaneously measure the three expectations with the output $\Lambda(\Psi\otimes \Phi)]$, $f_{AB}=\mathrm{Tr} [\Psi\otimes \Phi\Lambda(\Psi\otimes \Phi)]$, $f_{A}=\mathrm{Tr} [\Psi\otimes \mathrm{I}_D\Lambda(\Psi\otimes \Phi)]$, and $f_B= \mathrm{Tr} [\mathrm{I}_D\otimes \Phi\Lambda(\Psi\otimes \Phi)]$. By sampling $\Psi$ and $\Phi$ randomly with the Haar measure, the average survival probability in (19-21) should be known. The average survive probabilities are defined in an integral version. To carry out the integrations, we shall also apply the isomorphism in (9). To let it has a form suitable for the bounded operators in the $D^2$-dimensional Hilbert space case, we introduce the following definition: Letting $\vert \Omega\rangle$ be a maximally entangled states in $\mathrm{H}^{\otimes 4}$, $\vert \Omega\rangle=\frac{1}{D} \sum_{i,j=1}^D \vert ij ij\rangle$ with $\vert ijkl\rangle=\vert i\rangle\otimes \vert j\rangle\otimes\vert k\rangle\otimes \vert l\rangle$, a vector $\vert \Gamma)$ in $\mathrm{H}_{D}^{\otimes 4}$ is related to the bounded operator $\Gamma$ in $H_D^{\otimes 2}$, with its matrix elements to be $\Gamma_{ij;kl}\equiv \langle ij\vert \Gamma\vert kl\rangle$, via the isomorphism, $$\vert \Gamma)=D\cdot\Gamma\otimes \mathrm{I}_{D}^{\otimes 2}\vert\Omega\rangle=\sum_{i,j,k,l=1}^D \Gamma_{ij;kl}\vert ijkl\rangle.$$ Suppose that $\Gamma$, $\Delta$, and $\Sigma$ are three arbitrary bounded matrices in $\mathrm{H}_{D}^{\otimes 2}$, there should be $$\mathrm{Tr}[ \Gamma^{\dagger}\Delta]=(\Gamma\vert\Delta), \vert \Gamma\Sigma\Delta)=\Gamma\otimes \Delta^{\mathrm{T}} \vert \Sigma ).$$ If $\Gamma=\vert \Psi\rangle\rangle\langle\langle \Phi\vert $, there exists such a relation, $$\vert\vert \Psi\rangle\rangle\langle\langle \Phi\vert )=\vert \Psi\rangle\rangle\otimes \vert \Phi^\rangle\rangle.$$ ![\[fig:epsart\] (a) Circuit representation of measuring the average survive probabilities defined in (19-21). These quantities can be also measured in different protocols: (b)The product Haar twirling, (c) the product Clifford Twirling, and the product state 2-design.](fig2.eps) As an application of the above isomorphism, we find that the average survive probabilities in (19-21) can be rewritten as $$\begin{aligned} \bar{f}_{AB}(\Lambda)&=& \mathrm{Tr}[\hat{F}_{AB}\lambda],\\ \bar{f}_A(\Lambda)&=&\mathrm{Tr}[\hat{F}_A\lambda],\\ \bar{f}_B(\Lambda)&=&\mathrm{Tr}[\hat{F}_B\lambda],\end{aligned}$$ where the four super operators in $H_{D}^{\otimes 4}$ are defined as $$\begin{aligned} \hat{F}_{AB}&=&\int\int d\mu_{\mathrm{H}}(\Psi) d\mu_{\mathrm{H}}(\Phi)\vert \Psi\otimes\Phi)(\Psi\otimes\Phi\vert,\\ \hat{F}_{A}&=&\int\int d\mu_{\mathrm{H}}(\Psi) d\mu_{\mathrm{H}}(\Phi) \vert \Psi\otimes\Phi)(\Psi\otimes \mathrm{I}_D\vert,\\ \hat{F}_{B}&=&\int\int d\mu_{\mathrm{H}}(\Psi) d\mu_{\mathrm{H}}(\Phi)\vert \Psi\otimes\Phi)(\mathrm{I}_D\otimes\Phi\vert, \\ \lambda&=&\sum_m E_m\otimes E_m^.\end{aligned}$$ Here, it should emphasize that $\lambda$ is a physical meaningful super operator: Suppose $\rho$ to be an arbitrary input for the quantum channel $\Lambda$ and $\Lambda(\rho)=\sum_{m} E_m\rho E_m^{\dagger}$ to be the corresponding output. Applying the result in (23), we see that the two vectors, $\vert \Lambda(\rho))$ and $\vert \rho)$, are simply related by $\lambda$, $$\vert \Lambda(\rho))=\vert \sum_{m} E_m\rho E_m^{\dagger})\equiv\lambda\vert \rho).$$ To carry out the integrations above, we introduce a special unitary transformation $\beta$ in $\mathrm{H}_D^{\otimes 4}$, $$\beta=\sum_{i,j,k,l=1}^{D}\vert ijkl\rangle\langle ikjl\vert.$$ One may check that $\beta$ is also a Hermitian operator, $$\beta=\beta^{\dagger}=\beta^{-1}.$$ It has a nice property that $$\beta\vert \Psi\otimes \Phi)=\vert \Psi\rangle\rangle\otimes \vert \Phi\rangle\rangle.$$ With the above property of $\beta$, we can reexpress $\hat{F}_{AB}$ as $$\hat{F}_{AB}=\beta(\int d\mu_{\mathrm{H}}(\Psi)\vert \Psi\rangle\rangle\langle\langle\Psi\vert \otimes \int d\mu_{\mathrm{H}}(\Phi)\vert \Phi\rangle\rangle\langle\langle \Phi\vert)\beta.$$ Recalling our results in (13-18), we have $$\hat{F}_{AB}=\beta(\rho^{\mathrm{sep}}_{\mathrm{W}}\otimes\rho^{\mathrm{sep}}_{\mathrm{W}})\beta.$$ Different protocols of measuring the $\bar{f}_{AB}(\Lambda)$ are depicted in FIG.2. We call the one in FIG. 2b as the product Haar twirling procedure: Let $\vert \psi_0\rangle$ and $\vert \Phi_0\rangle$ the fixed state for the subsystem $H_D^A$ and $H_D^{B}$, respectively. The two arbitrary states in (19), $\Psi$ and $\Phi$, may be related to the arbitrary unitary operation $U$ and $V$ $$\Psi=U\Psi_0U^{\dagger}, \Phi=V\Phi_0V^{\dagger},~~~U,V\in \mathrm{U}(D),$$ respectively. The average survive probability of the product states may be expressed as $\bar{f}_{AB}^b(\Lambda)=\int d\mu_{H}(U)\int d\mu_{H}(V)\mathrm{Tr}[\Psi_0\otimes \Phi_0 (\mathcal{U\otimes V })^{\dagger}\circ\Lambda\circ\mathcal{U\otimes V}(\Psi_0\otimes\Phi_0)]$. In experiment, we first prepare $\vert \psi_0\rangle\otimes\vert \phi_0\rangle $ as the fixed input, then apply the operation $U\otimes V$ before the map $\Lambda$ and an operation $(U\otimes V)^{\dagger}$ after. Finally, we measure the survive probability of $\vert \psi_0\rangle\otimes\vert \phi_0\rangle $ with the output. By sampling $U$ and $V$ randomly with the Haar measure of $\mathrm{U}(D)$, we shall get the quantity $\bar{f}_{AB}^b(\Lambda)$ defined above. With the isomorphism in (22), one may express $\bar{f}_{AB}^b(\Lambda)$ as $$\bar{f}_{AB}^b(\Lambda)=\mathrm{Tr}[\hat{F}^b_{AB}\lambda],$$ where the super operator $\hat{F}^b_{AB}$ has the form $$\hat{F}^b_{AB}=\beta(\hat{F}_b\otimes\hat{F}_b)\beta$$ with $\hat{F}_b$ defined in (14). Applying the result in (18), we find $ \bar{f}_{AB}^b(\Lambda)$ equals the quantity $\bar{f}_{AB}(\Lambda)$. Therefore, the product Haar twirling procedure in FIG. 2b represents a possible way of getting $\bar{f}_{AB}(\Lambda)$. The way of applying the product Clifford twirling procedure to measure the average survive probability of the product states is shown in FIG. 2c. Its experimental data can be collected as $\bar{f}_{AB}^c(\Lambda)=\frac{1}{K^2}\sum_{i,j=1}^{K}\mathrm{Tr}[\Psi_0\otimes \Phi_0 (\mathcal{C}_{i}\otimes \mathcal{C}_j)^{\dagger}\circ\Lambda\circ\mathcal{C}_i\otimes \mathcal{C}_j(\Psi_0\otimes\Phi_0)]$, where $\{C_i\}_{i=1}^K$ is the Clifford group of $\mathrm{H}_D$. Via the similar argument above, we have $\bar{f}_{AB}^c(\Lambda)=\mathrm{Tr}[\hat{F}^c_{AB}\lambda]$ with the super operator $\hat{F}^c_{AB}$ to be $ \hat{F}^c_{AB}=\beta(\hat{F}_c\otimes\hat{F}_c)\beta .$ Obviously, $\bar{f}_{AB}^c(\Lambda)$ measured with the product Clifford twirling protocol equals $\bar{f}_{AB}(\Lambda)$ in (19) since that $\hat{F}^c_{AB}=\hat{F}_{AB}$. Let $\{\Psi_x\}_{x=1}^{N}$ and $\{\Phi_y\}_{y=1}^N$ to be the state 2-designs defined in (4), the quantity $\bar{f}_{AB}^d(\Lambda)$, $\bar{f}_{AB}^d(\Lambda)=\frac{1}{N^2}\sum_{x,y=1}^{N}\mathrm{Tr}[(\Psi_x\otimes \Phi_y)\Lambda(\Psi_x\otimes \Phi_y)]$, can be directly measured through the method in FIG. 2d. Jointing the result $\bar{f}_{AB}^d(\Lambda)=\mathrm{Tr}[\beta(\hat{F}_d\otimes \hat{F}_d)\beta \lambda]$ with (18) and (33), we conclude that $\bar{f}_{AB}^d(\Lambda)= \bar{f}_{AB}(\Lambda)$. After giving a detail discussion about how to measure $\bar{f}_{AB}(\Lambda)$ in experiment, we shall focus on $\bar{f}_{A}(\lambda)$ and $\bar{f}_{B}(\Lambda)$, the average survive probabilities for the subsystems, defined in (20) and (21), respectively. Let’s consider $\bar{f}_{A}(\Lambda)$ at first. Using Shur’s lemma, $$\int d\mu_{\mathrm{H}}(\Phi)\Phi=\frac{1}{D}\mathrm{I}_D,$$ we can simplify the expression of $\hat{F}_A$ in (29) as $$\hat{F}_A=\frac{1}{D}\int d\mu_{\mathrm{H}}(\Psi) \vert \Psi\otimes\mathrm{I}_D)(\Psi\otimes \mathrm{I}_D\vert.$$ With the unitary $\beta$ in (32), it has an equivalent form $$\hat{F}_A=\beta(\int d\mu_{\mathrm{H}}(\Psi)\vert \vert\Psi\rangle\rangle\langle \Psi\vert \otimes \frac{\vert\mathrm{I}_D\rangle\rangle\langle\langle \mathrm{I}_D\vert}{D})\beta.$$ Recalling our definition of the super operator $\hat{F}_a$ in (13) and the result in (18), we can get the formula $$\hat{F}_A=\beta(\rho_{\mathrm{W}}^{\mathrm{sep}}\otimes \frac{\vert \mathrm{I}_D\rangle\rangle\langle\langle \mathrm{I}_D\vert}{D})\beta.$$ Via a similar argument, there should be $$\hat{F}_B=\beta(\frac{\vert \mathrm{I}_D\rangle\rangle\langle\langle \mathrm{I}_D\vert}{D}\otimes\rho_{\mathrm{W}}^{\mathrm{sep}} )\beta.$$ With the result in (18), where the different ways to expand the separable Werner are given, we are able to prove that the average survive probability for a selected subsystem can be measured by the various protocols in FIG. 2. For example, when the survive probability of subsystem $H^{\mathrm{A}}_{D}$ is measured with the product state 2-design protocol depicted in FIG. 2d, the experimental data can be organized in the way like $$\bar{f}_A^d(\Lambda)=\frac{1}{N^2}\sum_{x,y=1}^N\mathrm{Tr}[(\Psi_x\otimes \mathrm{I}_D)\Lambda(\Psi_x\otimes\Phi_y)].$$ Formally, it can be transferred into $\bar{f}_A^d(\Lambda)=\mathrm{Tr}[\hat{F}_{A}^{d}\lambda]$ with the super operator $\hat{F}_{A}^{d}$ to be $$\hat{F}_{A}^{d}=\frac{1}{N^2}\sum_{x,y=1}^N\vert \Psi_x\otimes\Phi_y)(\Psi_x\otimes \mathrm{I}_D\vert.$$ As it is shown in \[20\], the state 2-design has the property that $$\frac{1}{N}\sum_{y=1}^N\Phi_y=\frac{1}{D}\mathrm{I}_D.$$ With this property in hand, we rewrite $\hat{F}_{A}^{d}$ as $$\hat{F}_{A}^{d}=\beta[ \frac{1}{N}\sum_{x=1}^{N}(\Psi_x\rangle\rangle\langle\langle \Psi_x\vert)\otimes \frac{\vert \mathrm{I}_D\rangle\rangle\langle\langle \mathrm{I}_D\vert }{D} ]\beta.$$ Jointing it with (16) and (18), we find $\hat{F}_{A}^{d}=\hat{F}_{A}$. Therefore, $\bar{f}_A(\Lambda)$ can be measured in the way presented in (36). With the separable Werner state defined in (17), we shall get a relation, $\mathrm{I}_D^{\otimes 2}= D(D+1)\rho_{\mathrm{W}}^{\mathrm{sep}}-\vert \mathrm{I}_D\rangle\rangle\langle\langle I_D\vert$. From it, we can expand the identity operator $\mathrm{I}_{D}^{\otimes 4}$ as $$\begin{aligned} \mathrm{I}_{D}^{\otimes 4}=D^2(D+1)^2\rho_{\mathrm{W}}^{\mathrm{sep}}\otimes \rho_{\mathrm{W}}^{\mathrm{sep}} +\vert \mathrm{I}_D\rangle\rangle\langle\langle I_D\vert\otimes\vert \mathrm{I}_D\rangle\rangle\langle\langle I_D\vert \nonumber\\ -D(D+1)[\rho_{\mathrm{W}}^{\mathrm{sep}}\otimes \vert \mathrm{I}_D\rangle\rangle\langle\langle \mathrm{I}_D\vert+ \vert\mathrm{I}_D\rangle\rangle\langle\langle \mathrm{I}_D\vert\otimes \rho_{\mathrm{W}}^{\mathrm{sep}}]\nonumber\end{aligned}$$ Note that $\mathrm{I}_{D}^{\otimes 4}$ is invariant under the transformation of $\beta$, $\mathrm{I}_{D}^{\otimes 4}=\beta\mathrm{I}_{D}^{\otimes 4}\beta$. Recalling our definitions of the super operators, $\hat{F}_{AB}$ in (33), $\hat{F}_{A}$ in (34), and $\hat{F}_{B}$ in (35), we shall find that the identity operator, $\mathrm{I}_{D}^{\otimes 4}$, can also be expanded as $$\mathrm{I}_{D}^{\otimes 4}=D^2(D+1)^2[\hat{F}_{AB}-\frac{\hat{F}_{A}+\hat{F}_{B}}{D+1}]+\vert \mathrm{I}_D^{\otimes 2})(\mathrm{I}_{D}^{\otimes 2}\vert.$$ For the quantum channel $\Lambda$, $\Lambda(\rho)=\sum_{m}E_m \rho E^{\dagger}_m$, of the joint system $\mathrm{H}=\mathrm{H}_{D}^A\otimes \mathrm{H}_{D}^B$, we introduce the entanglement fidelity $ f^{\mathrm{ent}}(\Lambda)$, which has been proposed to characterize the noise strength in $\Lambda$ \[24\], in the way like $$\begin{aligned} f^{\mathrm{ent}}(\Lambda) &=&\langle\Omega\vert( \mathrm{I}_D^{\otimes 2}\otimes \Lambda)(\vert\Omega\rangle\langle\Omega\vert)\vert \Omega\rangle \nonumber\\ &=&\frac{1}{D^4}\sum_{m}\vert \mathrm{Tr}[E_m]\vert^2,\end{aligned}$$ where $\vert \Omega\rangle$, as it has been introduced in (22), is the maximally entangled state of $\mathrm{H}\otimes \mathrm{H}^{\mathrm{anc}}$ with $\mathrm{H}^{\mathrm{anc}}$ to be a $D^2$-dimensional ancilla system. With the super operator $\lambda$ in (31), one may express the entanglement fidelity with a more compact form: $$f^{\mathrm{ent}}(\Lambda)=\frac{1}{D^4}\mathrm{Tr}[\lambda].$$ Jointing it with our expanding of the identity operator in (37) and the equations (25-27), we find that the entanglement fidelity should be known if the three average survive probabilities, $\bar{f}_{AB}(\Lambda)$, $\bar{f}_{A}(\Lambda)$ and $\bar{f}_{B}(\Lambda)$, have been decided, $$f^{\mathrm{ent}}(\Lambda)=\frac{1}{D^2}\{1+(D+1)^2\bar{f}_{AB}-(D+1)[\bar{f}_{A}+\bar{f}_{B}]\}.$$ (For simplicity, we have omitted the symbol $(\Lambda)$ in the expression for each average survive probability.) In the derivation of it, we have used the formula $$(\mathrm{I}_{D}^{\otimes 2}\vert \lambda =\sum_{m}(E_m^{\dagger}E_m\vert=(\mathrm{I}_{D}^{\otimes 2}\vert,$$ which can be viewed as a vector-form expression of trace-preserving condition of $\Lambda$. Now, we are able to show that the average fidelity can also be estimated with the three quantities, $\bar{f}_{AB}(\Lambda)$, $\bar{f}_{A}(\Lambda)$ and $\bar{f}_{B}(\Lambda)$. As a well-known fact \[9,10\], there exists a beautiful formula where the average fidelity is simply related to the entanglement fidelity. Instead of directly citing this formula, we shall give a simple reasoning to recover it. We use $\tilde{\rho}^{\mathrm{sep}}_{\mathrm{W}}$ as the generalization of the separable Werner state (for the $D^2$-dimensional system) in (17) for the $D^4$-dimensional system, $$\tilde{\rho}^{\mathrm{sep}}_{\mathrm{W}}=\frac{1}{D^2(D^2+1)}(\mathrm{I}_D^{\otimes 4}+\vert \mathrm{I}_D^{\otimes 2})(\mathrm{I}_D^{\otimes2} \vert).$$ Correspondingly, the results in (12) and (18) can also generalized into the following form $$f^{\mathrm{avg}}(\Lambda)=\mathrm{Tr}[\tilde{\rho}^{\mathrm{sep}}_{\mathrm{W}}\lambda].$$ By jointing it with (39) and (41), the well-known relation, for the case where the process super operator $\Lambda$ is defined for the $D^2$-dimensional system, is recovered here $$f^{\mathrm{avg}}(\Lambda)=\frac{D^2f^{\mathrm{ent}}(\Lambda)+1}{D^2+1}.$$ Putting (40) into it, we shall arrive at one of the main results of present work, $$f^{\mathrm{avg}}(\Lambda)=\frac{1}{D^2+1}\{2+(D+1)^2\bar{f}_{AB}-(D+1)[\bar{f}_{A}+\bar{f}_{B}]\},$$ where we show that the average fidelity $f^{\mathrm{avg}}(\Lambda)$ can be estimated with the average survival probabilities, $\bar{f}_{AB}(\Lambda)$, $\bar{f}_{A}(\Lambda)$ and $\bar{f}_{B}(\Lambda)$. measuring the selected element of the process matrix ===================================================== The proposal, which suggests that the protocol designed for measuring the average fidelity can be also applied to decide an arbitrary element of the quantum process matrix, originated from the work in \[16\]. In the first version of it, the off-diagonal elements should be measured by introducing ancilla system. Recently, an improved scheme, where all the elements can be estimated without introducing any ancilla system, has been presented in \[18\]. In the argument below, following the idea presented in \[18\], we shall develop a method where an arbitrary element of the process matrix can be decided with the average survive probabilities. At beginning, let’s introduce the definition of the process matrix. For a reason which will be clear later, we suppose $\bar{\Lambda}$ to be an arbitrary quantum map for the joint system $\mathrm{H}=\mathrm{H}_{D}^A\otimes \mathrm{H}_{D}^B$, $$\bar{\Lambda}(\rho)=\sum_{n}\bar{E}_{n}\rho \bar{E}_{n}, \sum_{n}\bar{E}_{n}^{\dagger}\bar{E}_n=\mathrm{I}_{D}^{\otimes 2}.$$ Letting $\{\Gamma_{\mu}\}_{\mu=1}^{D^4}$ to be an orthogonal operator basis for $\mathrm{H}$, $$\mathrm{Tr}[\Gamma^{\dagger}_{\mu}\Gamma_{\nu}]=D^2\delta_{\mu\nu},$$ one may rewrite (45) as $$\bar{\Lambda}(\rho)=\sum_{\mu,\nu=1}^{D^4} \Gamma_{\mu}\rho \Gamma_{\nu}^{\dagger}\chi_{\mu;\nu}(\bar{\Lambda}),$$ where the coefficients $\chi_{\mu;\nu}(\bar{\Lambda})$, $$\chi_{\mu;\nu}(\bar{\Lambda})=\frac{1}{D^4}\sum_{n}\mathrm{Tr}[\Gamma_{\mu}^{\dagger} \bar{E}_n](\mathrm{Tr}[\Gamma_{\nu}^{\dagger} \bar{E}_n])^,$$ are the entries of a $D^4\times D^4$ process matrix $\chi(\bar{\Lambda})$ which is Hermitian by definition. Here, it should be noted the factor $D^{-4}$ comes from the fact that the basis operators in (46) are not normalized. As it has been done in previous works, we suppose that $\Gamma_{\mu}$ are Hermitian and unitary operators, $$\Gamma_{\mu}=\Gamma_{\mu}^{\dagger}=\Gamma_{\mu}^{-1}.$$ Furthermore, we rewrite (48) with a convenient form $$\chi_{\mu;\nu}(\bar{\Lambda})=\frac{1}{D^4}\mathrm{Tr}[(\sum_{n}\bar{E}_n\otimes \bar{E}_n^)(\Gamma_{\mu}\otimes \Gamma_{\nu}^)].$$ From it, we see that the diagonal matrix elements should take the form, $$\chi_{\mu;\mu}(\bar{\Lambda})=\frac{1}{D^4}\mathrm{Tr}[(\sum_{n}\bar{E}_n\otimes \bar{E}_n^)(\Gamma_{\mu}\otimes \Gamma_{\mu}^)].$$ Now, let’s assume that the quantum map $\Lambda$ discussed in above section is related to $\bar{\Lambda}$ via the simple form, $$\Lambda=\bar{\Lambda} \circ {\Gamma}_{\mu},$$ where the process super operator $\bar{\Lambda} \circ {\Gamma}_{\mu}$ is defined as $\bar{\Lambda} \circ {\Gamma}_{\mu}(\rho)=\sum_n \bar{E}_n(\Gamma_{\mu}\rho\Gamma_{\mu}^{\dagger})\bar{E}_n^{\dagger}$. We call the so-defined $\Lambda$ the modified map (of $\bar{\Lambda}$). Based on this assumption, we find that the supper operator $\lambda$ in (31) should be $$\lambda=(\sum_{n}\bar{E}_n\otimes \bar{E}_n^)(\Gamma_{\mu}\otimes \Gamma_{\mu}^).$$ Recalling the formula for the calculation of the entanglement fidelity in (39) and the one for the diagonal matrix elements in (50), we shall find an interesting result, $$\chi_{\mu;\mu}(\bar{\Lambda})=f^{\mathrm{ent}}(\bar{\Lambda} \circ {\Gamma}_{\mu})\equiv f^{\mathrm{ent}}(\Lambda).$$ In other words, one may transfer the task of measuring the diagonal matrix element, which is defined for the quantum map $\bar{\Lambda}$, into the one of deciding the entanglement fidelity of the modified map $\Lambda=\bar{\Lambda} \circ {\Gamma}_{\mu}$. Based on this principle, the diagonal matrix element $\chi_{\mu;\mu}(\bar{\Lambda})$ can be estimated with $\bar{f}_{AB}(\bar{\Lambda}\circ {\Gamma}_{\mu})$, $\bar{f}_{A}(\bar{\Lambda}\circ {\Gamma}_{\mu})$ and $\bar{f}_{B}(\bar{\Lambda}\circ {\Gamma}_{\mu})$, which are the average survive probabilities of the modified quantum map $\bar{\Lambda}\circ {\Gamma}_{\mu}$, $$\begin{aligned} \chi_{\mu;\mu}(\bar{\Lambda})= \frac{1}{D^2}\{1+ (D+1)^2\bar{f}_{AB}(\bar{\Lambda}\circ {\Gamma}_{\mu}) \nonumber\\ - (D+1)[\bar{f}_{A}(\bar{\Lambda}\circ {\Gamma}_{\mu})+\bar{f}_{B}(\bar{\Lambda}\circ {\Gamma}_{\mu})]\}.\end{aligned}$$ ![\[fig:epsart\] Circuit representation of measuring the diagonal matrix element of the quantum process matrix.](fig3.eps) In FIG. 3, we give a protocol to measure the diagonal elements of the process matrix: Suppose that $\Psi_x$ and ${\Phi_y}$ are the elements of the state 2-design in (4). Preparing $\vert\psi_x\rangle \otimes\vert \phi_y\rangle$ as the input, perform an operation $\Gamma_{\mu}$ before the quantum $\bar{\Lambda}$, then measure the expectations ${f}_{AB}(\bar{\Lambda}\circ \Gamma_{\mu})=\mathrm{Tr}[\Psi_x\otimes \Phi_y \bar{\Lambda}\circ \Gamma_{\mu}(\Psi_x\otimes \Phi_y)] $, ${f}_{A}(\bar{\Lambda}\circ \Gamma_{\mu})=\mathrm{Tr}[\Psi_x\otimes \mathrm{I}_D \bar{\Lambda}\circ \Gamma_{\mu}(\Psi_x\otimes \Phi_y)] $ and ${f}_{B}(\bar{\Lambda}\circ \Gamma_{\mu})=\mathrm{Tr}[\mathrm{I}_D\otimes \Phi_y \bar{\Lambda}\circ \Gamma_{\mu}(\Psi_x\otimes \Phi_y)] $. By repeating this process with a number of $N^2$ different inputs, one may get the averaged survive probabilities which can be used to estimate the diagonal element of the process matrix according to (52). As it has been shown in \[18\], the way to decide the off-diagonal matrix elements is different form the one to get the diagonal matrix elements. Let’s organize the proposal developed there in the way like: For the two known operators, $\Gamma_{\mu}$ and $\Gamma_{\nu}$, ($\mu\neq\nu$), define the following four operators, $$\Gamma_{\pm}=\Gamma_{\mu}\pm\Gamma_{\nu}, \tilde{\Gamma}_{\pm}=\Gamma_{\mu}\pm i\Gamma_{\nu}.$$ From it, one may easily verify that $$\begin{aligned} \Gamma_{\mu}\otimes\Gamma^_{\nu}=\frac{1}{2}[\Gamma_+\otimes\Gamma_+^ -\Gamma_-\otimes\Gamma_-^\nonumber\\ -i(\tilde{\Gamma}_+\otimes\tilde{\Gamma}_+^- \tilde{\Gamma}_-\otimes\tilde{\Gamma}_-^* ], \nonumber\\ \Gamma_{\nu}\otimes \Gamma_{\mu}^=\frac{1}{2}[\Gamma_+\otimes\Gamma_+^ -\Gamma_-\otimes\Gamma_-^\nonumber\\ +i(\tilde{\Gamma}_+\otimes\tilde{\Gamma}_+^- \tilde{\Gamma}_-\otimes\tilde{\Gamma}_-^* ]. \nonumber\end{aligned}$$ Jointing this result with (49), we observe that the off-diagonal elements of the process matrix can be estimated by the way $$\begin{aligned} \chi_{\mu;\nu}(\bar{\Lambda})=\frac{1}{2}[\omega_+-\omega_--i(\tilde{\omega}_+-\tilde{\omega}_-)],\nonumber\\ \chi_{\nu;\mu}(\bar{\Lambda})=\frac{1}{2}[\omega_+-\omega_-+i(\tilde{\omega}_+-\tilde{\omega}_-)],\nonumber\end{aligned}$$ in which the four quantities, $\omega_{\pm}$ and $\tilde{\omega}_{\pm}$, are defined as $$\begin{aligned} \omega_{\pm}=\frac{1}{D^4}\mathrm{Tr}[(\sum_{n}\bar{E}_n\otimes \bar{E}_{n})(\Gamma_{\pm}\otimes \Gamma_{\pm}^)]\\ \tilde{\omega}_{\pm}=\frac{1}{D^4}\mathrm{Tr}[(\sum_{n}\bar{E}_n\otimes \bar{E}_{n})(\tilde{\Gamma}_{\pm}\otimes \tilde{\Gamma}_{\pm}^)]\end{aligned}$$ Let’s consider $\omega_+$ at first. Following the argument from (50) to (51), we can also interpret $\omega_+$ as the entanglement fidelity of the non-physical map $\bar{\Lambda}\circ \Gamma_+$ $$\omega_+=f^{\mathrm{ent}}(\bar{\Lambda}\circ \Gamma_+)$$ with the process super operator $\bar{\Lambda}\circ \Gamma_+ $ defined as $\bar{\Lambda} \circ {\Gamma}_{+}(\rho)=\sum_n \bar{E}_n(\Gamma_{+}\rho\Gamma_{+}^{\dagger})\bar{E}_n^{\dagger}$. With a simple calculation, which is similar with the one for deriving (40), we shall get $$\begin{aligned} \omega_+=&& \frac{1}{D^2}\{2+ (D+1)^2\bar{f}_{AB}(\bar{\Lambda}\circ {\Gamma}_{+}) \nonumber\\ &-& (D+1)[\bar{f}_{A}(\bar{\Lambda}\circ {\Gamma}_{+})+\bar{f}_{B}(\bar{\Lambda}\circ {\Gamma}_{+})]\}.\end{aligned}$$ In the derivation of it, we have used the relation, $\mathrm{Tr }[\vert \mathrm{I}^{\otimes 2}_D)( \mathrm{I}^{\otimes 2}_D\vert(\sum_{n}\bar{E}_n\otimes \bar{E}_{n})(\Gamma_{\pm}\otimes \Gamma_{\pm}^)]=2D^2.$ Now, we shall encounter the problem of measuring the average survive probabilities for a non-physical map $\bar{\Lambda}\circ \Gamma_+$. A solution for it, as it has been suggested in \[18\], is to prepare a set of states $\{\vert \psi_{xy}\rangle \}_{x,y=1}^{N}$, $$\vert \psi_{xy}\rangle =\Gamma_+(\vert \psi_x\rangle\otimes \vert \phi_y\rangle),$$ as the inputs for the quantum channel $\bar{\Lambda}$, then decide the averaged survived probabilities, which are needed in (56), through the way in below: $$\begin{aligned} \bar{f}_{AB}(\bar{\Lambda}\circ {\Gamma}_{+})&=&\frac{1}{N^2}\sum_{x,y=1}^{N}\mathrm{Tr}[(\Psi_x\otimes\Phi_y)\bar{\Lambda}(\Psi_{xy})], \nonumber\\ \bar{f}_{A}(\bar{\Lambda}\circ {\Gamma}_{+})&=&\frac{1}{N^2}\sum_{x,y=1}^{N}\mathrm{Tr}[(\Psi_x\otimes \mathrm{I}_D)\bar{\Lambda}(\Psi_{xy})],\nonumber\\ \bar{f}_{B}(\bar{\Lambda}\circ {\Gamma}_{+})&=&\frac{1}{N^2}\sum_{x,y=1}^{N}\mathrm{Tr}[(\mathrm{I}_D\otimes\Phi_y)\bar{\Lambda}(\Psi_{xy})].\nonumber\end{aligned}$$ The product state 2-design protocol designed to get the above average quantities is depicted in FIG. 4. The above method, which is developed for measuring $\omega_+$, can be easily generalized for the cases where the rest of the quantities defined in (54-55) should be measured. ![\[fig:epsart\] (a) Measuring the survive probabilities of a non-physical map $\bar{\Lambda}\circ \Gamma_+$ can be realized by the apparatus in (b) with the inputs $\vert \psi_{xy}\rangle=\Gamma_+\vert \psi_x\rangle\otimes\vert\phi_y\rangle$.](fig4.eps) approximated measurement of the average survive probabilities ============================================================= In section III, we have introduced three types average survive probabilities for a bipartite system. Several ways of measuring these quantities are depicted in FIG. 2. It should be noted that these measurements are non-scalable: Taking the product state 2-design protocol, which is given in FIG. 2d, for an example, there are about $N^2 (N\ge D^2)$ runs of experiment to be performed. A solution for this problem is to find an efficient approximated way where the error introduced by the approximation should depend on the number of operations irrespective of the dimension of the joint system. In this section, we shall develop such an efficient way to measure of the average survive probabilities. In what follows we restrict ourself to the case where a set of product states, $\{\Psi_{i}\otimes \Psi_j\}_{i,j=1}^{M}$ should be prepared as the inputs of the quantum channel $\Lambda$ while the average survive probabilities defined in (19-21) are approximated with the way like $$\begin{aligned} \bar{f}^{\mathrm{appr}}_{AB}=\frac{1}{M^2}\sum_{i,j,=1}^M\mathrm{Tr}[(\Psi_i\otimes \Psi_j)\Lambda(\Psi_i\otimes \Psi_j)],\nonumber\\ \bar{f}^{\mathrm{appr}}_{A}=\frac{1}{M^2}\sum_{i,j,=1}^M\mathrm{Tr}[(\Psi_i\otimes \mathrm{I}_{D})\Lambda(\Psi_i\otimes \Psi_j)],\nonumber\\ \bar{f}^{\mathrm{appr}}_{B}=\frac{1}{M^2}\sum_{i,j,=1}^M\mathrm{Tr}[(\mathrm{I}_D\otimes \Psi_j)\Lambda(\Psi_i\otimes \Psi_j)].\nonumber\end{aligned}$$ These average quantities can be also measured with the twirling procedures by letting $\Psi_j=U_j\Psi_0U_j^{\dagger}$ with $\Psi_0$ and $U_j$ to be a fixed state and the arbitrary unitary transformations in $\mathrm{H}_{D}$, respectively. By following the steps for getting the super operators in (33-35), the above equations can be also expressed $$\bar{f}^{\mathrm{appr}}_{\alpha}=\mathrm{Tr}[\hat{F}^{\mathrm{appr}}_{\alpha}\lambda], ~~\alpha=A, B, AB.$$ in which the super operators $\hat{F}^{\mathrm{appr}}_{\alpha}$ are defined as $$\begin{aligned} \hat{F}^{\mathrm{appr}}_{A}=\beta(\frac{1}{M^2}\sum_{i,j=1}^{M}\vert\Psi_i\rangle\rangle\langle\langle \Psi_i\vert\otimes \vert\Psi_j\rangle\rangle\langle\langle \mathrm{I}_D\vert)\beta,\\ \hat{F}^{\mathrm{appr}}_{B}= \beta(\frac{1}{M^2}\sum_{i,j=1}^{M}\vert\Psi_i\rangle\rangle\langle\langle \mathrm{I}_D\vert\otimes \vert\Psi_j\rangle\rangle\langle\langle \Psi_j\vert)\beta,\\ \hat{F}^{\mathrm{appr}}_{AB}=\beta(\frac{1}{M^2}\sum_{i,j=1}^{M}\vert\Psi_i\rangle\rangle\langle\langle \Psi_i\vert\otimes \vert\Psi_j\rangle\rangle\langle\langle \Psi_j\vert)\beta.\end{aligned}$$ In above sections, we have shown that the average fidelity can expressed as the expectation of the super operator $\lambda$, which is defined in (31), with the separable Werner state $\tilde{\rho}^{\mathrm{sep}}_{\mathrm{W}}$, $f^{\mathrm{avg}}(\Lambda)=\mathrm{Tr}[\tilde{\rho}^{\mathrm{sep}}_{\mathrm{W}}\lambda]$. With the expanding formula of the identity operation in (37), one may rewrite $\tilde{\rho}^{\mathrm{sep}}_{\mathrm{W}}$ as $$\tilde{\rho}^{\mathrm{sep}}_{\mathrm{W}}=\frac{2}{D^2(D^2+1)}\vert \mathrm{I}^{\otimes 2}_ D)(\mathrm{I}^{\otimes 2}_D\vert+\frac{1}{D^2+1}\Delta,$$ in which the operator $\Delta$ is defined as $$\begin{aligned} \Delta&=& \frac{1}{D^2}(\mathrm{I}^{\otimes 4}_D- \vert \mathrm{I}^{\otimes 2}_ D)(\mathrm{I}^{\otimes 2}_D\vert)\nonumber\\ &=& (D+1)^2\hat{F}_{AB}-(D+1)(\hat{F}_A+\hat{F}_B). \end{aligned}$$ Now, the average fidelity is approximated in the way like $$f_{\mathrm{appr}}^{\mathrm{avg}}(\Lambda)=\mathrm{Tr}[\tilde{\rho}_{\mathrm{appr}}\lambda]$$ with $\tilde{\rho}_{\mathrm{appr}}$ the approximated separable Werner state, $$\begin{aligned} \tilde{\rho}_{\mathrm{appr}}=\frac{2}{D^2(D^2+1)}\vert \mathrm{I}^{\otimes 2}_ D)(\mathrm{I}^{\otimes 2}_D\vert+\frac{1}{D^2+1}\Delta_{\mathrm{appr}},\\ \Delta_{\mathrm{appr}}=(D+1)^2\hat{F}^{\mathrm{appr}}_{AB}-(D+1)(\hat{F}^{\mathrm{appr}}_A+\hat{F}^{\mathrm{appr}}_B). \end{aligned}$$ Considering the fact that the super operator $\lambda$, which is decided by the quantum map $\Lambda$, is usually unknown, for simplicity, we suppose that the accuracy of the approximation should depend on the difference between the separable Werner state and its approximated form. As it has been done in \[25\], we introduce the Hilbert-Schmidt norm of their difference, $$\vert \vert \tilde{\rho}^{\mathrm{sep}}_{\mathrm{W}}-\tilde{\rho}_{\mathrm{appr}}\vert\vert:=\mathrm{Tr}[(\tilde{\rho}^{\mathrm{sep}}_{\mathrm{W}}- \tilde{\rho}_{\mathrm{appr}}) (\tilde{\rho}^{\mathrm{sep}}_{\mathrm{W}}-\tilde{\rho}_{\mathrm{appr}})^{\dagger}],$$ to characterize how well the separable Werner state is approximated. With the definitions from (62) to (64), we simplify the above equation into the form $$\vert \vert \tilde{\rho}^{\mathrm{sep}}_{\mathrm{W}}-\tilde{\rho}_{\mathrm{appr}}\vert\vert=\frac{1}{(D^2+1)^2}\vert\vert\Delta-\Delta_{\mathrm{appr}}\vert\vert.$$ In order to perform the above calculation in an easy way, we shall introduce the operators $R_{i}$, $$R_{i}=(D+1)\Psi_i-\mathrm{I}_{D},$$ and reexpress the operator $\Delta_{\mathrm{appr}}$ in (65) with a more compact form $$\Delta_{\mathrm{appr}}=\frac{1}{M^2}\sum_{i,j=1}^{M}\vert \Psi_i\otimes \Psi_j)(R_i\otimes R_j-\mathrm{I}^{\otimes 2}_D\vert.$$ With above formula, one may easily verify that $$\mathrm{Tr}[\Delta (\Delta)^{\dagger}]=\mathrm{Tr}[\Delta (\Delta)^{\dagger}_{\mathrm{appr}}]=\mathrm{Tr}[\Delta_{\mathrm{appr}} (\Delta)^{\dagger}]=\frac{2(D^2-1)}{D^2}.$$ Therefore, the Hilbert-Schmidt norm is $$\vert \vert \tilde{\rho}^{\mathrm{sep}}_{\mathrm{W}}-\tilde{\rho}_{\mathrm{appr}}\vert\vert=\frac{1}{(D^2+1)^2}(\vert\vert\Delta_{\mathrm{appr}} \vert\vert-\frac{2(D^2-1)}{D^2})$$ where the quantity $\vert\vert\Delta_{\mathrm{appr}}\vert\vert$ should be decided by our actual choice of the states $\Psi_i$. Now, we assume that the states $\Psi_i$ are chosen from the symmetric information complete set (SIC) introduced in (6), $\Psi_i\neq \Psi_j$ for $i\neq j$. Based on this assumption, we find that $$\begin{aligned} \vert\vert\Delta_{\mathrm{appr}} \vert\vert&=& \frac{D^4(D+2)^2+(D^2-2)D^2}{M^2(1+D)^2}\nonumber\\ && -\frac{4D}{M(1+D)}+\frac{D-1}{D+1}.\end{aligned}$$ Putting it back into (70), we can get the upper bound of the Hilbert-Schmidt norm, $$\vert \vert \tilde{\rho}^{\mathrm{sep}}_{\mathrm{W}}-\tilde{\rho}_{\mathrm{appr}}\vert\vert\{\begin{array}{c} =0,~~~~~~~~~~ M=D^2, \\ < \frac{1}{M^2}\frac{(1+D)^2}{D^2},~~1< M < D^2. \end{array}$$ For the case $D\ge 4$, there is $\frac{(1+D)^2}{D^2}<2$. Therefore, we conclude that the separable Werner state is approximated in an efficient way: The upper bound of the error introduced by the approximation process scales better than $\frac{2}{M^2}$ with the number of repetitions $M^2$ of the experiment. Discussion ========== In previous works, the state 2-design has also been defined in the way like: $$\frac{1}{N}\sum_{x=1}^{N}\mathrm{Tr}[\Psi_x A \Psi_x B]=\frac{\mathrm{Tr}[A]\mathrm{Tr}[B]+\mathrm{Tr}[AB]}{D(D+1)}.$$ One may check that the definition in (4) is consisted with it. Applying the isomorphism introduced in (9), we rewrite the left side of the above equation as $$\frac{1}{N}\sum_{x=1}^{N}\mathrm{Tr}[\Psi_x A \Psi_x B] =\mathrm{Tr}[(\frac{1}{N}\sum_{x=1}^{N} \Psi_x \otimes \Psi_x^)(A\otimes B^{\mathrm{T}})].$$ As it has been shown in section II, from the definition of the state 2-design in (4), we can get an extremely simple relation between the state 2-design and the separable Werner state, $\frac{1}{N}\sum_{x=1}^{N} \Psi_x \otimes \Psi_x^=\rho^{\mathrm{sep}}_{\mathrm{\mathrm{W}}}.$ With the results, $\mathrm{Tr}[B]=\mathrm{Tr}[B^{\mathrm{T}}]$ and $(\mathrm{I}^{\otimes 2}_D\vert A\otimes B^{\mathrm{T}}\vert \mathrm{I}^{\otimes 2}_D)=\mathrm{Tr}[AB]$, we have $$\mathrm{Tr}[\rho^{\mathrm{sep}}_{\mathrm{W}}(A\otimes B^)]=\frac{\mathrm{Tr}[A]\mathrm{Tr}[B]+\mathrm{Tr}[AB]}{D(D+1)}.$$ From it, one may conclude that the definition in (4) is consisted with the one in (73). Finally, let’s end our work with a short conclusion. For the bipartite system, we have introduced three directly measurable quantities, the average survive probability of the product states and the survive probability for each subsystem, and developed several protocols to measure them. These average quantities can be applied to estimate the average fidelity of the quantum channel and the selected element of the quantum process matrix. Appendix. In Section II, we have introduced a super operator $\hat{F}_b$, $\hat{F}_b =\int d\mu_{\mathrm{H}}(U)U\otimes U^\vert \Psi_0\rangle\rangle\langle\langle \Psi_0\vert (U\otimes U^)^{\dagger}$, and stated that it should equal with the separable Werner state defined in (17). Although this statement can be viewed as a known result in other works, (for example, the one in \[10\]), for the completeness of present work, we shall still give a compact proof for it. At first, applying the isomorphism in (22), we reexpress $\hat{F}_b$ in the vector form as $\vert \hat{F}_b )=\mathcal{T}\vert \Psi_0\rangle\rangle\otimes \vert \Psi^_0\rangle\rangle$ with the super operator $\mathcal{T}$ to be $$\mathcal{T}=\int d\mu_{\mathrm{H}}(U) U\otimes U^\otimes U^\otimes U.$$ In stead of directly carrying out the integration, we shall cite a result given by Scott in \[26\]: $$\begin{aligned} &&\int d\mu_{\mathrm{H}}(U)U\otimes U\otimes U^{\dagger}\otimes U^{\dagger}\nonumber\\ &&=\frac{1}{D^2-1}(P_{3412}+P_{4321})-\frac{1}{D(D^2-1)}(P_{4312}+P_{3421}),\nonumber\end{aligned}$$ with $P_{abcd}$ defined as the permutation operator (For its definition in detail, please see the original work in \[26\].) With the above calculation in hands, we find the expression of the $\mathcal{T}$ should be $$\begin{aligned} \mathcal{T}=&&\frac{1}{D^2-1}[\vert \mathrm{I}_{D}^{\otimes 2})(\mathrm{I}_{D}^{\otimes 2}\vert- \vert \mathrm{I}_{D}^{\otimes 2})(\hat{S}_+\vert \nonumber \\ &&+D^2\vert \hat{S}_+)(\hat{S}_+\vert-\vert \hat{S}_+)(\mathrm{I}_{D}^{\otimes 2}\vert ],\nonumber\end{aligned}$$ in which we use $\hat{S}_+$ to denote the projective operator, $\hat{S}_+=\vert S_+\rangle\langle S_+\vert$ with $\vert S_+\rangle$ the maximally entangled state defined in (9). Performing the operation $\mathcal{T}$ on an arbitrary product state $\vert \Psi_0\rangle \rangle\otimes \vert \Psi^_0\rangle\rangle$, we shall get $$\mathcal{T}(\vert \Psi_0\rangle\rangle\otimes \vert \Psi^_0\rangle\rangle)=\vert \rho^{\mathrm{sep}}_{\mathrm{W}}),$$ the vector form of the desired result $\hat{F}_b =\rho^{\mathrm{sep}}_{\mathrm{W}}$. M.A. Nielson, and I. L. 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--- abstract: 'In this paper, we investigate relativistic quantum dynamics of spin-$0$ massive charged particle subject to a homogeneous magnetic field in the Gödel-type space-time with potentials. We solve the Klein-Gordon equation subject to a homogeneous magnetic field in a topologically trivial flat class of Gödel-type space-time in the presence of a Cornell-type scalar and Coulomb-type vector potentials and analyze the effects on the energy eigenvalues and eigenfunctions.' --- =18 pt [Faizuddin Ahmed]{}[^1]\ [Ajmal College of Arts and Science, Dhubri-783324, Assam, India]{} [Keywords]{}: Gödel-type space-time, Klein-Gordon equation, electromagnetic field, energy spectrum, wave-functions [PACS Number:]{} 03.65.Pm, 03.65.Ge Introduction ============ The first solution to the Einstein’s field equations containing closed time-like curves is the cylindrical symmetry Gödel rotating Universe [@KG]. Reboucas [et al.]{} [@MJR; @MJR2; @MOG] investigated the Gödel-type solutions characterized by vorticity, which represents a generalization of the original Gödel metric with possible sources and analyzed the problem of causality. The line element of Gödel-type solution is given by $$ds^2=-\left (dt+A_{i}\,dx^{i} \right)^2+\delta_{ij}\,dx^{i}\,dx^{j}, \label{godel}$$ where the spatial coordinates of the space-time are represented by $x^{i}$ and $i,j=1,2,3$. The different classes of Gödel-type solutions have discussed in [@EPJC4]. Investigation of relativistic quantum dynamics of spin-zero and spin-half particles in Gödel Universe and Gödel-type space-times as well the Schwarzschild, the Kerr black holes solution has been addressed by several authors. The study of relativistic wave-equations, particularly, the Klein-Gordon and Dirac equations in the background of Gödel-type space-time was first studied in [@BDF]. The close relation between the relativistic energy levels of a scalar particle in the Som-Raychaudhuri space-time with the Landau levels was studied in [@ND]. The same problem in the Som-Raychaudhuri space-time was investigated and compared with the Landau levels [@SD]. The Klein-Gordon equation in the background of Gödel-type space-times with a cosmic string was studied in [@JC], and analyzed the similarity of the energy levels with Landau levels in flat space. The Klein-Gordon oscillator in the background of Gödel-type space-time under the influence of topological defects was studied in [@JC2]. The relativistic quantum dynamics of scalar particle in the presence of an external fields in the Som-Raychaudhuri space-time under the influence of topological defects was studied in [@ZW]. The relativistic quantum motion of spin-$0$ particles in a flat class of Gödel-type space-time was studied in [@EPJC2]. The study of spin-$0$ system of DKP equation in a flat class of Gödel-type space-time was studied in [@MPLA]. In all the above system, the influence of topological defects and vorticity parameter characterizing the space-time on the relativistic energy eigenvalues was analyzed. Linear confinement of a scalar particle in the Som-Raychaudhuri space-time with cosmic string was studied in [@RR1] (see also, [@Ahmed1]). The behavior of scalar particle with Yukawa-like confining potential in the Som-Raychaudhuri space-time in the presence of topological defects was investigated in [@ME]. Ground state of a bosonic massive charged particle in the presence of an external fields in a Gödel-type space-time was investigated in [@EOS] (see also, [@Ahmed2]). The relativistic quantum dynamics of spin-zero particles in 4D curved space-time with the cosmic string subject to a homogeneous magnetic field was studied in [@Ahmed3]. In addition, the relativistic wave-equations in (1+2)-dimensional rotational symmetry space-time background was investigated in [@AOP; @AOP2; @GERG; @MPLA2; @AOP3; @GERG2]. Furthermore, Dirac and Weyl fermions in the background of the Som-Raychaudhuri space-times in the presence of topological defects with torsion was studied in [@GQG]. Weyl fermions in the background of the Som-Raychaudhuri space-times in the presence of topological defects was studied in [@GQG2] (see, Refs. [@SGF; @AH]). The relativistic wave-equations for spin-half particles in the Melvin space-time, a space-time where the metric is determined by a magnetic field was studied in [@LCNS]. The fermi field and Dirac oscillator in the Som-Raychaudhuri space-time was studied in [@MM]. The Fermi field with scalar and vector potentials in the Som-Raychaudhuri space-time was investigated in [@PS]. The Dirac particles in a flat class of Gödel-type space-time was studied in [@EPJC4]. Dirac fermions in (1+2)-dimensional rotational symmetry space-time background was investigated in [@Ahmed4]. The relativistic quantum dynamics of a scalar particle subject to different confining potentials have been studied in several areas of physics by various authors. The relativistic quantum dynamics of scalar particles subject to Coulomb-type potential was investigated in [@QWC; @HM; @FY; @ALC]. It is worth mentioning studies that have dealt with Coulomb-type potential in the propagation of gravitational waves [@HA], quark models [@CLC], and relativistic quantum mechanics [@HWC; @VRK; @ERFM; @KB]. Linear confinement of scalar particles in a flat class of Gödel-type space-time, were studied in [@EPJC3]. The Klein-Gordon equation with vector and scalar potentials of Coulomb-type under the influence of non-inertial effect in cosmic string space-time was studied in [@LCNS2]. The Klein-Gordon oscillator in the presence of Coulomb-type potential in the background space-time generated by a cosmic string was studied in [@KB; @KB2]. Other works on the relativistic quantum dynamics are the Klein-Gordon scalar field subject to a Cornell-type potential [@MH], and survey on the Klein-Gordon equation in a Gödel-type space-time [@MS]. Our aim in this paper is to investigate the quantum effects on bosonic massive charged particle by solving the Klein-Gordon equation subject to a homogeneous magnetic field in the presence of a Cornell-type scalar and Coulomb-type vector potentials in Gödel-type space-time. We see that the presence of magnetic field as well as various potential modifies the energy spectrum. Bosonic charged particle : The KG-Equation ============================================== The relativistic quantum dynamics of a charged particle of modifying mass $ m \rightarrow m+S$, where $S$ is the scalar potential is described by the following equation [@ERFM] $$\left [\frac{1}{\sqrt{-g}}\,D_{\mu} (\sqrt{-g}\,g^{\mu\nu}\,D_{\nu})-(m+S)^2-\xi\,R \right]\,\Psi=0, \label{1}$$ where $g$ is the determinant of metric tensor with $g^{\mu\nu}$ its inverse, $D_{\mu}=\partial_{\mu}-i\,e\,A_{\mu}$ is the minimal substitution, $e$ is the electric charge and $A_{\mu}$ is the electromagnetic four-vector potential, and $\xi$ is the non-minimal coupling constant with the background curvature. We choose the electromagnetic four-vector potential $A_{\mu}=(-V,\vec{A})$ with [@GBA] $$A_{y}=-x\,B_0\quad ,\quad \vec{A}=(0,A_y,0) \label{2}$$ such that the constant magnetic field is along the axis $\vec{B}=\vec{\nabla}\times \vec{A}=-B_0\,\hat{z}$. Consider the following stationary space-time [@CTP] (see, [@EPJC4; @EPJC2; @MPLA; @EPJC3; @CTP2]) in the Cartesian coordinates $(x^0=t, x^1=x, x^2=y, x^3=z)$ is given by $$ds^2=-\left (dt+\alpha_0\,x\,dy \right)^2+\delta_{ij}\,dx^i\,dx^j, \label{3}$$ where $\alpha_0 > 0$ is a real positive constant. In Ref. [@EPJC4], we have discussed different classes of Gödel-type space-time. For the space-time geometry (\[3\]), it belongs to a linear or flat class of Gödel-type metrics. The parameter $\alpha_0=2\,\Omega$ where, $\Omega$ characterize the vorticity parameter of the space-time. For $\Omega \rightarrow 0$, the study space-time reduces to four-dimensional flat Minkowski metric. The determinant of the corresponding metric tensor $g_{\mu\nu}$ is $$det\;g=-1. \label{4}$$ The scalar curvature of the metric is $$R=\frac{\alpha_{0}^2}{2}=2\,\Omega^2. \label{5}$$ For the space-time geometry (\[3\]), the equation (\[1\]) becomes $$\begin{aligned} &&[-(\frac{\partial}{\partial t}+i\,e\,V)^2+\left\{(\frac{\partial}{\partial y}-i\,e\,A_y)-2\,\Omega\,x\,(\frac{\partial}{\partial t}+i\,e\,V)\right \}^2+\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial z^2}\nonumber\\ &&-(m+S)^2-2\,\xi\,\Omega^2]\,\Psi=0. \label{6}\end{aligned}$$ Since the metric is independent of $t, y, z$. One can choose the following ansatz for the function $\Psi$ $$\Psi (t, x, y, z)=e^{i\,(-E\,t+l\,y+k\,z)}\,\psi(x), \label{7}$$ where $E$ is the total energy of the particle, $l=0,\pm\,1,\pm\,2,...$ are the eigenvalues of the $y$-component operator, and $-\infty < k < \infty $ is the eigenvalues of the $z$-component operator. Substituting the ansatz Eq. (\[7\]) into the Eq. (\[6\]), we obtain the following differential equation for $\psi (x)$ : $$\begin{aligned} &&[\frac{d^2}{dx^2}+(E-e\,V)^2-k^2-\{(l+e\,B_0\,x)+2\,\Omega\,x\,(E-e\,V)\}^2\nonumber\\ &&-(m+S)^2-2\,\xi\,\Omega^2]\,\psi(x)=0. \label{8}\end{aligned}$$ Interaction with Cornell-type and Coulomb-type potentials --------------------------------------------------------- Here we study a spin-$0$ massive charged particle by solving the Klein-Gordon equation in the presence of an external fields in a flat class of Gödel-type space-time subject to a Cornell-type scalar and Coulomb-type vector potentials. We obtain the energy eigenvalues and eigenfunctions and analyze the effects due to various physical parameters. The Cornell-type potential contains a confining (linear) term besides the Coulomb interaction and has been successfully accounted for the particle physics data [@DHP]. This type of potential is a particular case of the quark-antiquark interaction, which has one more harmonic-type term [@MKB]. The Coulomb potential is responsible by the interaction at small distances and the linear potential leads to the confinement. The quark-antiquark interaction potential has been studied in the ground state of three quarks [@CA], and systems of bound heavy quarks [@JDG; @EE; @EE2]. This type of interaction has been studied by several authors ([@ZW; @MPLA2; @ERFM; @MSC; @RLLV6; @RLLV2; @RLLV3; @RLLV4; @RLLV5; @LFD]). We consider the scalar $S$ to be Cornell-type [@ERFM] $$S=\frac{\eta_c}{x}+\eta_{L}\,x, \label{9}$$ where $\eta_{c},\eta_{L}$ are the Coulombic and confining potential constants, respectively. Another potential that we are interest here is the Coulomb-type potential which we discussed in the introduction. Therefore, the Coulomb-type vector potential is given by $$V=\frac{\xi_c}{x}, \label{vector}$$ where $\xi_{c}$ is the Coulombic potential constants. Substituting the potentials (\[9\]) and (\[vector\]) into the Eq. (\[8\]), we obatin the following equation: $$\left [\frac{d^2}{dx^2} + \lambda-\omega^2\,x^2-\frac{j^2}{x^2}-\frac{a}{x}-b\,x \right]\,\psi(x)=0, \label{10}$$ where we have defined $$\begin{aligned} &&\lambda=E^2-m^2-k^2-2\,\eta_c\,\eta_L-2\,\xi\,\Omega^2-(l-2\,e\,\Omega\,\xi_c)^2,\nonumber\\ &&\omega=\sqrt{4\,(\Omega\,E+m\,\omega_c)^2+\eta^2_{L}},\nonumber\\ &&j=\sqrt{\eta^2_{c}-e^2\,\xi^2_{c}},\nonumber\\ &&a=2\,(e\,\xi_c\,E+m\,\eta_c),\nonumber\\ &&b=2\,[m\,\eta_L+\omega\,(l-2\,e\,\Omega\,\xi_c)],\nonumber\\ &&\omega_c=\frac{e\,B_0}{2\,m} \label{11}\end{aligned}$$ is called the cyclotron frequency of the particle moving in the magnetic field. Let us define a new variable $r=\sqrt{\omega}\,x$, Eq. (\[10\]) becomes $$\left [\frac{d^2}{dr^2} + \beta-r^2-\frac{j^2}{r^2}-\frac{\eta}{r}-\theta\,r \right]\,\psi(r)=0, \label{12}$$ where $$\beta=\frac{\lambda}{\omega}\quad,\quad \eta=\frac{a}{\sqrt{\omega}}\quad,\quad \theta=\frac{b}{\omega^{\frac{3}{2}}}. \label{13}$$ We now use the appropriate boundary conditions to investigate the bound states solution in this problem. It is known in relativistic quantum mechanics that the radial wave-functions must be regular both at $r \rightarrow 0$ and $r \rightarrow \infty $. Then we proceed with the analysis of the asymptotic behavior of the radial eigenfunctions at origin and in the infinite. These conditions are necessary since the wave-functions must be well-behaved in these limit, and thus, the bound states of energy eigenvalues for this system can be obtained. Suppose the possible solution to the Eq. (\[12\]) is $$\psi (r)=r^{j}\,e^{-\frac{1}{2}\,(\theta+r)\,r}\,H(r). \label{14}$$ Substituting the solution Eq. (\[14\]) into the Eq. (\[12\]), we obtain $$\frac{d^2 H}{dr^2}+\left [\frac{\gamma}{r}-\theta-2\,r \right]\,\frac{d H}{dr}+\left [-\frac{\chi}{r}+\Theta \right]\,H (r)=0, \label{17}$$ where $$\begin{aligned} &&\gamma=1+2\,j,\nonumber\\ &&\Theta=\beta+\frac{\theta^2}{4}-2\,(1+j),\nonumber\\ &&\chi=\eta+\frac{\theta}{2}\,(1+2\,j). \label{18}\end{aligned}$$ Equation (\[17\]) is the biconfluent Heun’s differential equation [@ERFM; @KB; @KB2; @RLLV6; @RLLV2; @RLLV3; @RLLV4; @RLLV5; @AR; @SYS] with $H(r)$ is the Heun polynomials function. Writing the function $H (r)$ as a power series expansion around the origin [@GBA]: $$H(r)=\sum^{\infty}_{i=0} c_{i}\,r^{i}. \label{19}$$ Substituting the series solution into the Eq. (\[17\]), we obtain the following recurrence relation: $$c_{n+2}=\frac{1}{(n+2)(n+1+\gamma)}\,[\{\chi+\theta\,(n+1)\}\,c_{n+1}-(\Theta-2\,n)\,c_n]. \label{20}$$ Few coefficients of the series solution are $$\begin{aligned} &&c_{1}=\left(\frac{\eta}{\gamma}+\frac{\theta}{2} \right)\,c_0,\nonumber\\ &&c_{2}=\frac{1}{2\,(1+\gamma)}\,[(\chi+\theta)\,c_1-\Theta\,c_0]. \label{21}\end{aligned}$$ As the function $H (r)$ has a power series expansion around the origin in Eq. (\[19\]), then, the relativistic bound states solution can be achieved by imposing that the power series expansion becomes a polynomial of degree $n$ and we obtain a finite degree polynomial for the biconfluent Heun series. Furthermore, the wave-function $\psi$ must vanish at $r \rightarrow \infty$ for this finite degree polynomial of power series otherwise the function diverge for large values of $r$. Therefore, we must truncate the power series expansion $H (r)$ a polynomial of degree $n$ by imposing the following two conditions [@ZW; @RR1; @ERFM; @KB; @EPJC3; @KB2; @RLLV6; @RLLV2; @RLLV3; @RLLV4; @RLLV5; @AV; @JM]: $$\begin{aligned} \Theta&=&2\,n,\quad (n=1,2,...)\nonumber\\ c_{n+1}&=&0. \label{22}\end{aligned}$$ By analyzing the condition $\Theta=2\,n$, we have the second degree eigenvalues equation $$\begin{aligned} &&E^{2}_{n,l}=m^2+k^2+2\,\eta_c\,\eta_L+2\,\xi\,\Omega^2+2\,\omega\,(n+1+\sqrt{\eta^2_{c}-e^2\,\xi^2_{c}})\nonumber\\ &&-\frac{m^2\,\eta^2_{L}}{\omega^2}-\frac{2\,m\,\eta_L\,(l-2\,e\,\xi_c\,\Omega)}{\omega}. \label{23}\end{aligned}$$ The corresponding eigenfunctions is given by $$\psi_{n,l} (r)=r^{\sqrt{\eta^2_{c}-e^2\,\xi^2_{c}}}\,e^{-\frac{1}{2}\,(r+\theta)\,r}\,H (r). \label{24}$$ Note that Eq. (\[23\]) does not represent the general expression of the eigenvalue problem. One can obtain the individual energy eigenvalues one by one, that is, $E_1$, $E_2$, $E_3$,.. by imposing the additional recurrence condition $c_{n+1}=0$ on the eigenvalues. The solution with Heun’s equation makes it possible to obtain the individual eigenvalues one by one as done in [@ZW; @RR1; @ERFM; @KB; @EPJC3; @KB2; @RLLV6; @RLLV2; @RLLV3; @RLLV4; @RLLV5; @AV; @JM]. In order to analyze the above conditions, we must assign values to $n$. In this case, consider $n=1$, that means we want to construct a first degree polynomial to $H (r)$. With $n=1$, we have $\Theta=2$ and $c_2=0$ which implies from Eq. (\[21\]) $$\begin{aligned} &&(\chi+\theta)\,c_1=2\,c_0\Rightarrow \frac{\eta}{\gamma}+\frac{\theta}{2}=\frac{2}{\chi+\theta},\nonumber\\ &&(\frac{a_{1,l}}{1+2\,j}+\frac{b_{1,l}}{2\,\omega_{1,l}})(a_{1,l}+\frac{b_{1,l}}{\omega_{1,l}}(j+\frac{3}{2}))=2\,\omega_{1,l}\nonumber\\ &&\omega^{3}_{1,l}-\frac{a^2}{2\,(1+2\,j)}\,\omega^2_{1,l}-a\,b\,(\frac{1+j}{1+2\,j})\,\omega_{1,l}-\frac{b^2}{8}\,(3+2\,j)=0, \label{25}\end{aligned}$$ a constraint on the physical parameter $\omega_{1,l}$. The relation given in Eq. (\[25\]) gives the possible values of the parameter $\omega_{1,l}$ that permit us to construct first degree polynomial to $ H (r)$ for $n=1$ [@ERFM; @KB; @KB2; @RLLV6]. Note that its values changes for each quantum number $n$ and $l$, so we have labeled $\omega \rightarrow \omega_{n,l}$. In this way, we obtain the following energy eigenvalue $E_{1,l}$: $$\begin{aligned} E_{1,l}&=&\pm\,\{m^2+k^2+2\,\eta_c\,\eta_L+2\,\xi\,\Omega^2+2\,(2+\sqrt{\eta^2_{c}-e^2\,\xi^2_{c}})\,\omega_{1,l}\nonumber\\ &&-\frac{m^2\,\eta^2_{L}}{\omega^2_{1,l}}-\frac{2\,m\,\eta_L\,(l-2\,e\,\xi_c\,\Omega)}{\omega_{1,l}}\}^{\frac{1}{2}}. \label{26}\end{aligned}$$ Then, by substituting the real solution $\omega_{1,l}$ from Eq. (\[25\]) into the Eq. (\[26\]) it is possible to obtain the allowed values of the relativistic energy levels for the radial mode $n=1$ of a position- dependent mass system. We can see that the lowest energy state is defined by the real solution of the algebraic equation Eq. (\[25\]) plus the expression given in Eq. (\[26\]) for the radial mode $n=1$, instead of $n=0$. This effect arises due to the presence of Cornell-type potential in the system. Note that, it is necessary physically that the lowest energy state is $n=1$ and not $n=0$, otherwise the opposite would imply that $c_1=0$ which is not possible. The corresponding radial wave-function for $n=1$ is given by $$\psi_{1,l} (r)=r^{\sqrt{\eta^2_{c}-e^2\,\xi^2_{c}}}\,e^{-\frac{1}{2}\,(r+\frac{b}{\omega^{\frac{3}{2}}_{1,l}})\,r}\,(c_0+c_1\,r), \label{27}$$ where $$c_1=\frac{1}{\sqrt{\omega_{1,l}}}\,[\frac{a}{1+2\,\sqrt{\eta^2_{c}-e^2\,\xi^2_{c}}}+\frac{b}{2\,\omega_{1,l}}]\,c_0. \label{28}$$ Interaction without potential ----------------------------- Here we study a spin-$0$ massive charged particle by solving the Klein-Gordon equation in the presence of an external fields in a Gödel-type space-time without potential and obtain the relativistic energy eigenvalue. We choose here zero scalar and vector potentials, $S=0=V$. In that case, Eq (\[8\]) becomes $$\left [\frac{d^2}{dx^2} + E^2-m^2-k^2-l^2-2\,\xi\,\Omega^2-\omega^2\,x^2-2\,\omega\,l\,x \right]\,\psi(x)=0, \label{bb1}$$ The above equation can be expressed as $$\left [\frac{d^2}{dx^2} + E^2-m^2-k^2-2\,\xi\,\Omega^2-\omega^2\,(x+\frac{l}{\omega})^2 \right]\,\psi(x)=0, \label{bb2}$$ Let us define a new variable $r=(x+\frac{l}{\omega})$, Eq. (\[bb2\]) becomes $$\psi''(r)+(\delta-\omega^2\,r^2)\,\psi(r)=0, \label{bb3}$$ where $$\delta=E^2-m^2-k^2-2\,\xi\,\Omega^2. \label{bb4}$$ Again introducing a new variable $\rho=\sqrt{\omega}\,r$ into the Eq. (\[bb3\]), we obtain $$\psi''(\rho)+(\frac{\delta}{\omega}-\rho^2)\,\psi (\rho)=0 \label{bb5}$$ which is similar a harmonic-type oscillator equation. Therefore, the energy eigenvalues equation is $$\begin{aligned} &&\frac{\delta}{\omega}=2\,n+1\Rightarrow \delta=(2\,n+1)\,\omega\nonumber\\\Rightarrow &&E^2_{n}-2\,\Omega\,(2\,n+1)\,E_{n}-m^2-k^2-2\,\xi\,\Omega^2-2\,m\,\omega_c\,(2\,n+1)=0.\quad \label{bb6}\end{aligned}$$ The energy eigenvalues associated with $n^{th}$ modes is $$\begin{aligned} E_{n}&=&(2\,n+1)\,\Omega+\sqrt{(2\,n+1)^2\,\Omega^2+m^2+k^2+2\,\xi\,\Omega^2+2\,m\,\omega_c\,(2\,n+1)}\nonumber\\ &=&(2\,n+1)\,\Omega+\sqrt{(2\,n+1)^2\,\Omega^2+m^2+k^2+2\,\xi\,\Omega^2+\|e\,B_0\|\,(2\,n+1)}, \label{bb7}\end{aligned}$$ where $n=0,1,2,.$. We can see that the energy eigenvalues (\[bb7\]) depend on the parameter $\Omega$ characterizing the vorticity parameter of the space-time geometry, and the external magnetic field $B_0$ as well the non-minimal coupling constant $\xi$ with the background curvature. In absence of external magnetic fields, $B_0 \rightarrow 0$, and without non-minimal coupling constant, $\xi \rightarrow 0$, the eigenvalue (\[bb7\]) becomes $$E_{n}=(2\,n+1)\,\Omega+\sqrt{(2\,n+1)^2\,\Omega^2+m^2+k^2}. \label{bb8}$$ Equation (\[bb8\]) is the energy eigenvalues of spin-$0$ particle in the background of a flat class of Gödel-type space-time and consistent with the result in [@EPJC2]. Thus we can see that the energy eigenvalues (\[bb7\]) in comparison to the result in [@EPJC2] get modify due to the presence of external fields and the non-minimal coupling constant with the background curvature. Therefore, the individual energy levels for $n=0,1$ using (\[bb7\]) are follows: $$\begin{aligned} &&n=0\quad:\quad E_0=\Omega+\sqrt{\Omega^2+2\,\xi\,\Omega^2+2\,m\,\omega_c+m^2+k^2},\nonumber\\ &&n=1\quad:\quad E_1=3\,\Omega+\sqrt{9\,\Omega^2+2\,\xi\,\Omega^2+6\,m\,\omega_c+m^2+k^2}. \label{bb9}\end{aligned}$$ A special case corresponds to $m=0=k$, the energy eigenvalues (\[bb7\]) reduces to $$E_{n}=(2\,n+1)\,\Omega+\sqrt{(2\,n+1)^2\,\Omega^2+2\,\xi\,\Omega^2+\|e\,B_0\|\,(2\,n+1)}. \label{bb10}$$ The individual energy levels for $n=0,1$ in that case are follows: $$\begin{aligned} &&n=0\quad:\quad E_0=\Omega+\sqrt{\Omega^2+2\,\xi\,\Omega^2+\|e\,B_0\|},\nonumber\\ &&n=1\quad:\quad E_1=3\,\Omega+\sqrt{9\,\Omega^2+2\,\xi\,\Omega^2+3\,\|e\,B_0\|}. \label{bb11}\end{aligned}$$ And others are in the same way. We can see that the presence of external magnetic field $B_0$ as well as the non-minimal coupling constant $\xi$ causes asymmetry in the energy levels and hence, the energy levels are not equally spaced. The eigenfunctions is given by $$\psi_{n} (\rho)=\|N\|\,H_{n} (\rho)\,e^{-\frac{\rho^2}{2}}, \label{bb12}$$ where $\|N\|=\sqrt{\frac{1}{2^{n}\,{n!}\,\sqrt{\pi}}}$ is the normalization constant and $H_{n} (\rho)$ are the Hermite polynomials and define as $$H_{n} (\rho)=(-1)^{n}\,e^{\rho^2}\,\frac{d^{n}}{d{\rho^n}}\,(e^{-\rho^2}),\quad \int^{\infty}_{-\infty} e^{-\rho^2}\,H_{n} (\rho)\,H_{m} (\rho)\,d\rho=\sqrt{\pi}\,2^{n}\,{n!}\,\delta_{nm}. \label{bb13}$$ The Klein-Gordon Oscillator =============================== Here we study a spin-$0$ massive charged particle by solving the Klein-Gordon equation of the Klein-Gordon oscillator in the presence of an external fields in a Gödel-type space-time subject to a Cornell-type scalar and Coulomb-type vector potentials. We analyze the effects on the relativistic energy eigenvalue and corresponding eigenfunctions due to various physical parameters. To couple Klein-Gordon field with oscillator [@Bru; @Dvo], the generalization of Mirza [et al.]{} prescription [@Mirza], in which the following change in the momentum operator is taken: $$p_{\mu}\rightarrow p_{\mu}+i\,m\,\omega_0\,X_{\mu}, \label{cc1}$$ where $m$ is the particle mass at rest, $\omega_0$ is the frequency of the oscillator and $X_{\mu}=(0,x,0,0)$, with $x$ being the distance of the particle. In this way, the Klein-Gordon oscillator equation becomes $$\frac{1}{\sqrt{-g}}\,(D_{\mu}+m\,\omega_0\,X_{\mu})\sqrt{-g}\,g^{\mu\nu}\,(D_{\nu}-m\,\omega_0\,X_{\nu})\,\Psi=(m+S)^2\,\Psi. \label{cc2}$$ Using the space-time (\[1\]), we obtain the following equation $$\begin{aligned} &&[-(\frac{\partial}{\partial t}+i\,e\,V)^2+\{(\frac{\partial}{\partial y}-i\,e\,A_y)-\alpha\,x\,(\frac{\partial}{\partial t}+i\,e\,V)\}^2+\frac{\partial}{\partial z^2}\nonumber\\ &&+(\frac{\partial}{\partial x}+m\,\omega_0\,x)\,(\frac{\partial}{\partial x}-m\,\omega_0\,x)-(m+S)^2]\,\Psi=0. \label{cc3}\end{aligned}$$ Using the ansatz (\[7\]) into the above Eq. (\[cc3\]), we arrive at the following equation $$\begin{aligned} &&\frac{d^2\,\psi}{dx^2}+[(E-e\,V)^2-\{ \alpha_0\,x\,(E-e\,V)+(l-e\,A_y) \}^2-k^2-m\,\omega_0\nonumber\\ &&-m^2\,\omega^2_{0}\,x^2-(m+S)^2]\,\psi=0. \label{cc4}\end{aligned}$$ Substituting the potentials Eq. (\[2\]), (\[9\]) and (\[vector\]) into the equation (\[cc4\]), we obtain the following equation: $$\psi''(x)+[\tilde{\lambda}-\tilde{\omega}^2\,x^2-\frac{j^2}{x^2}-\frac{a}{x}-\tilde{b}\,x]\,\psi(x)=0, \label{dd1}$$ where we have defined $$\begin{aligned} &&\tilde{\lambda}=E^2-m^2-k^2-2\,\eta_c\,\eta_L-2\,\xi\,\Omega^2-(l-2\,e\,\Omega\,\xi_c)^2-m\,\omega_0,\nonumber\\ &&\tilde{\omega}=\sqrt{4\,(\Omega\,E+m\,\omega_c)^2+m^2\,\omega^2_{0}+\eta^2_{L}},\nonumber\\ &&j=\sqrt{\eta^2_{c}-e^2\,\xi^2_{c}},\nonumber\\ &&a=2\,(e\,\xi_c\,E+m\,\eta_c),\nonumber\\ &&\tilde{b}=2\,[m\,\eta_L+\tilde{\omega}\,(l-2\,e\,\Omega\,\xi_c)]. \label{dd2}\end{aligned}$$ Let us define a new variable $r=\sqrt{\tilde{\omega}}\,x$, Eq. (\[dd1\]) becomes $$\left [\frac{d^2}{dr^2} +\tilde{\beta}-r^2-\frac{j^2}{r^2}-\frac{\tilde{\eta}}{r}-\tilde{\theta}\,r \right]\,\psi(r)=0, \label{dd7}$$ where $$\tilde{\beta}=\frac{\lambda}{\tilde{\omega}}\quad,\quad \tilde{\eta}=\frac{a}{\sqrt{\tilde{\omega}}}\quad,\quad \tilde{\theta}=\frac{\tilde{b}}{\tilde{\omega}^{\frac{3}{2}}}. \label{dd8}$$ Suppose the possible solution to the Eq. (\[dd7\]) is $$\psi (r)=r^{j}\,e^{-\frac{1}{2}\,(\tilde{\theta}+r)\,r}\,H(r). \label{dd9}$$ Substituting the solution Eq. (\[dd9\]) into the Eq. (\[dd7\]), we obtain $$H'' (r)+[\frac{\gamma}{r}-\tilde{\theta}-2\,r]\,H' (r)+[-\frac{\tilde{\chi}}{r}+\tilde{\Theta}]\,H (r)=0, \label{dd3}$$ where $\gamma$ is given earlier and $$\begin{aligned} &&\tilde{\Theta}=\tilde{\beta}+\frac{\tilde{\theta}^2}{4}-2\,(1+j),\nonumber\\ &&\tilde{\chi}=\tilde{\eta}+\frac{\tilde{\theta}}{2}\,(1+2\,j) \label{dd4}\end{aligned}$$ Equation (\[dd3\]) is the biconfluent Heun’s differential equation [@ERFM; @KB; @KB2; @RLLV6; @RLLV2; @RLLV3; @RLLV4; @RLLV5; @AR; @SYS]. Substituting the series solution Eq. (\[19\]) into the Eq. (\[dd3\]), we obtain the following recurrence relation: $$c_{n+2}=\frac{1}{(n+2)(n+1+\gamma)}\,[\{\tilde{\chi}+\tilde{\theta}\,(n+1)\}\,c_{n+1}-(\tilde{\Theta}-2\,n)\,c_n]. \label{dd10}$$ Few coefficients of the series solution are $$\begin{aligned} &&c_{1}=\left(\frac{\tilde{\eta}}{\gamma}+\frac{\tilde{\theta}}{2} \right)\,c_0,\nonumber\\ &&c_{2}=\frac{1}{2\,(1+\gamma)}\,[(\tilde{\chi}+\tilde{\theta})\,c_1-\tilde{\Theta}\,c_0]. \label{dd11}\end{aligned}$$ The power series expansion becomes a polynomial of degree $n$ by imposing following two conditions [@ZW; @RR1; @ERFM; @KB; @EPJC3; @KB2; @RLLV6; @RLLV2; @RLLV3; @RLLV4; @RLLV5; @AV; @JM]: $$\begin{aligned} \tilde{\Theta}&=&2\,n \quad (n=1,2,..)\nonumber\\ c_{n+1}&=&0 \label{dd12}\end{aligned}$$ Using the first condition we obtain the following energy eigenvalues : $$\begin{aligned} &&E^{2}_{n,l}=m^2+k^2+2\,\eta_c\,\eta_L+2\,\xi\,\Omega^2+m\,\omega_0+2\,(n+1+\sqrt{\eta^2_{c}-e^2\,\xi^2_{c}})\,\tilde{\omega}\nonumber\\ &&-\frac{m^2\,\eta^2_{L}}{\tilde{\omega}^2}-\frac{2\,m\,\eta_L\,(l-2\,e\,\xi_c\,\Omega)}{\tilde{\omega}}. \label{dd5}\end{aligned}$$ The corresponding eigenfunctions is given by $$\psi_{n,l}(r)=r^{\sqrt{\eta^2_{c}-e^2\,\xi^2_{c}}}\,e^{-\frac{1}{2}\,(r+\tilde{\theta})\,r}\,H (r). \label{dd6}$$ As done earlier, we obtain the individual energy levels by imposing the recurrence condition $c_{n+1}=0$. For $n=1$, we have $c_2=0$ which implies from Eq. (\[dd11\]) $$\begin{aligned} \tilde{\omega}^{3}_{1,l}-\frac{a^2}{2\,(1+2\,j)}\,\tilde{\omega}^2_{1,l}-a\,\tilde{b}\,(\frac{1+j}{1+2\,j})\,\tilde{\omega}_{1,l}-\frac{\tilde{b}^2}{8}\,(3+2\,j)=0, \label{dd13}\end{aligned}$$ a constraint on the physical parameter $\tilde{\omega}_{1,l}$. The relation given in Eq. (\[dd13\]) gives the possible values of the parameter $\tilde{\omega}_{1,l}$ that permit us to construct first degree polynomial to $ H (r)$ for $n=1$ [@ERFM; @RLLV6; @KB; @KB2]. In this way, we obtain the following second degree algebraic equation for $E_{1,l}$: $$\begin{aligned} &&E_{1,l}=\pm\,\{m^2+k^2+2\,\eta_{c}\,\eta_{L}+2\,\xi\,\Omega^2+2\,(2+\sqrt{\eta^2_{c}-e^2\,\xi^2_{c}})\,\tilde{\omega}_{1,l}\nonumber\\ &&-\frac{m^2\,\eta^2_{L}}{\tilde{\omega}^2_{1,l}}-\frac{2\,m\,\eta_{L}\,(l-2\,e\,\xi_{c}\,\Omega)}{\tilde{\omega}_{1,l}}\}^{\frac{1}{2}}. \label{dd14}\end{aligned}$$ Then, by substituting the real solution $\tilde{\omega}_{1,l}$ from Eq. (\[dd13\]) into the Eq. (\[dd14\]) it is possible to obtain the allowed values of the relativistic energy levels for the radial mode $n=1$ of a position- dependent mass system. We can see that the lowest energy state is defined by the real solution of algebraic equation Eq. (\[dd13\]) plus the expression given in Eq. (\[dd14\]) for the radial mode $n=1$, instead of $n=0$. This effect arises due to the presence of Cornell-type potential in the system. Note that, it is necessary physically that the lowest energy state is $n=1$ and not $n=0$, otherwise the opposite would imply that $c_1=0$ which is not possible. The corresponding radial wave-function for $n=1$ is given by $$\psi_{1,l} (r)=r^{\sqrt{\eta^2_{c}-e^2\,\xi^2_{c}}}\,e^{-\frac{1}{2}\,(r+\frac{b}{\tilde{\omega}^{\frac{3}{2}}_{1,l}})\,r}\,(c_0+c_1\,r), \label{dd15}$$ where $$c_1=\frac{1}{\sqrt{\tilde{\omega}_{1,l}}}\,[\frac{2\,(e\,\xi_c\,E_{1,l}+m\,\eta_c)}{1+2\,\sqrt{\eta^2_{c}-e^2\,\xi^2_{c}}}+\frac{2\,\left(m\,\eta_L+\omega_{1,l}\,(l-2\,e\,\Omega\,\xi_c) \right)}{2\,\tilde{\omega}_{1,l}}]\,c_0. \label{dd16}$$ Conclusions =========== The relativistic quantum system of scalar and spin-half particles in Gödel-type space-times was investigated by several authors ([e. g.]{}, [@ND; @SD; @JC; @ZW; @EPJC2; @MPLA; @RR1; @EPJC3; @RLLV6]). They demonstrated that the energy eigenvalues of the relativistic quantum system get modified and depend on the global parameters characterizing the space-times. In this work, we have investigated the influence of vorticity parameter on the relativistic energy eigenvalues of a relativistic scalar particle in a Gödel-type space-time subject to a homogeneous magnetic field with potentials. We have derived the radial wave-equation of the Klein-Gordon equation in a class of flat Gödel-type space-time in the presence of an external fields with(-out) potentials by choosing a suitable ansatz of the wave-function. In [sub-section 2.1]{}, we have introduced a Cornell-type scalar and Coulomb-type vector potentials into the considered relativistic system and obtained the energy eigenvalue Eq. (\[23\]) and corresponding eigenfunctions Eq. (\[24\]). We have seen that the presence of a uniform magnetic field and potential parameters modifies the energy spectrum in comparison those result obtained in [@EPJC2]. By imposing the additional recurrence condition $c_{n+1}=0$, we have obtained the ground state energy levels Eq. (\[26\]) and wave-functions Eq. (\[27\])–(\[28\]) for $n=1$. In [sub-section 2.2]{}, we have considered zero potential into the relativistic system and solved the radial wave-equation of the Klein-Gordon equation in the presence of an external field. We obtained the energy eigenvalues Eq. (\[bb7\]) and compared with the results obtained in [@EPJC2]. We have seen that the relativistic energy eigenvalues Eq. (\[bb7\]) get modify in comparison to those in [@EPJC2] due to the presence of a homogeneous magnetic field. In [section 3]{}, we have solved the Klein-Gordon equation of the Klein-Gordon oscillator in a Gödel-type space-time subjected to a homogeneous magnetic field in the presence of a Cornell-type scalar and Coulomb-type vector potentials. We have obtained the energy eigenvalue Eq. (\[dd5\]) and corresponding eigenfunctions Eq. (\[dd6\]). We have seen that the presence of a uniform magnetic field and potential parameters modifies the energy spectrum in comparison to those in [@EPJC2]. By imposing the additional recurrence condition $c_{n+1}=0$, we have obtained the ground state energy levels Eq. (\[dd14\]) and wave-function Eq. (\[dd15\]) for $n=1$ and others are in the same way. So, in this paper we have some results which are in addition to the previous results obtained in [@ND; @SD; @JC; @ZW; @EPJC2; @MPLA; @EPJC3; @CTP2; @RLLV6] present may interesting effects. 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--- author: - Atsushi Hariki - Mathias Winder - Jan Kuneš bibliography: - 'ccfo.bib' title: 'Supplementary material of “Metal-insulator transition in CaCu$_3$Fe$_4$O$_{12}$”' --- Supplementary information for the LDA+$U$ calculation ----------------------------------------------------- Following previous DFT+$U$ studies for CaCu$_3$Fe$_4$O$_{12}$ [@Hao09; @Ueda13], we used $U=$ 4.0 eV and 7.0 eV for Fe and Cu sites, respectively. To check the robustness of our result to the choice of the $U$ values, we performed calculations with different sets of $U$ values and obtained the energy-well structure with its minimum around the experimental $Q_{\rm R}$ and $Q_{\rm B}$ amplitude, see Figs. \[fig:ldau1\](a)(b). The energy minima at finite $Q_{\rm B}$ become shallow when the $U$ value at Fe site gets close to the one at Cu site. For $U=$ 4.0 eV and 7.0 eV at Fe and Cu sites, we performed the calculation with the generalized gradient approximation (GGA) scheme as the exchange correlation potential [@Perdew96], Fig. \[fig:ldau1\](c), that yields an energy profile in consistent with that obtained with the LDA scheme. The low-energy bands for the minority spin, complementary to Fig. 3 in the main text, are summarized in Fig. \[fig:ldau2\]. The minority-spin bands show a larger gap (compared to the one in the majority-spin channel) for the distortion amplitudes near the experimental value. Fig. \[fig:ldau3\] shows the LDA+$U$ results for a wide range of the $Q_{\rm R}$ and $Q_{\rm B}$ amplitudes. Supplementary information for the LDA+DMFT calculation ------------------------------------------------------ Fig. \[fig:dmft1\] shows the phase diagram covering a wide range of the double-counting corrections $\mu_{\rm dc}$ at the Fe and Cu sites. The ferrimagnetic insulating phase, observed in the experiment, is realized in large range of the $\mu_{\rm dc}$ values. With unrealistically small or large $\mu_{\rm dc}$ at the Cu site, the metallic solution realizes in which the spectral weights at the Fermi energy is dominated by the Cu 3$d$ states in the minority spin channel. There, the Cu valency departs from the divalent ($d^{9}$), which is inconsistent with the experimental report by Cu $L$-edge x-ray absorption spectroscopy method [@Mizumaki11]. Fig. \[fig:dmft2\] shows the total and Cu $d$-projected spectra, complementary to Fig. 5 in the main text. The LDA+DMFT calculation with a paramagnetic constraint on the Cu site yields a ferrimagnetic insulating solution with almost unchanged Fe bands, see Fig. \[fig:dmft3\]. Supplementary information for the distortion-mode analysis ---------------------------------------------------------- Starting with the low-temperature crystal structure ($Pn\overline{3}$, lattice constant $a=7.2612 \AA$), reported in Ref. [@Yamada08], the displacement of the oxygen atoms with respect to the ideal perovskite structure ($Im\overline{3}$) is quantified by a symmetry-adoped mode analysis using AMPLIMODE [@Perez10]. Since the Cu (Ca) atoms sitting on 3/4 (1/4) of the A sites show no displacements from the cubic cage, undistorted CaFeO$_3$ ($Pm\overline{3}m$) is used as a reference system in the mode analysis. We obtained three modes: breathing mode, $M^{+}_2$, and $M^{+}_1$, visualized in Fig. 1 in the main text, which have the amplitudes ($\AA$) of 0.20, 3.22, 0.33, respectively. Note that $M^{+}_1$ and $M^{+}_2$ modes exist in both high-temperature metallic and low-temperature insulating phases, while the breathing mode is obtained only in the low-temperature insulating phase. Toy model --------- In order to demonstrate the effect of spin polarization we calculate the spectral weights of toy model $$G^{-1}(k,\omega)= \begin{pmatrix} \omega+\mu - m - \frac{d^2}{\omega+m}& \cos k \\ \cos k & \omega + \mu -\Delta -m \\ \end{pmatrix},$$ with $\mu=-0.35$, $d=0.6$, $\Delta=0.2$, $-0.6<m<0.6$ (in eV). The parameters were chosen to mimic the quarter-filled Fe $e_g$ band of CCFO. Parameter $m$, which represents the exchange splitting (Hartree term of the self-energy), varies from 0.0 (PM state) to 0.6 (FM state). Positive and negative $m$ represents the majority and minority spin channel, respectively. The long-bond (LB) and short-bond (SB) sites correspond to $G_{11}$ and $G_{22}$, respectively. The spectral weight is represented by thickness of the symbols. In the attached movie the LB and SB spectral weights are in the left and right column, respectively. The majority and minority spin projections are in the top and bottom rows, respectively.
--- abstract: 'We report the development and application of a new method for carrying out computational investigations of the effects of mass and force-constant (FC) disorder on phonon spectra. The method is based on the recently developed typical medium dynamical cluster approach (TMDCA), which is a Green’s function approach. Excellent quantitative agreement with previous exact diagonalization results establishes the veracity of the method. Application of the method to a model system of binary mass and FC-disordered system leads to several findings. A narrow resonance, significantly below the van Hove singularity, that has been termed as the boson peak, is seen to emerge for low soft particle concentrations. We show, using the typical phonon spectrum, that the states constituting the boson peak cross over from being completely localized to being extended as a function of increasing soft particle concentration. In general, an interplay of mass and FC disorder is found to be cooperative in nature, enhancing phonon localization over all frequencies. However, for certain range of frequencies, and depending on material parameters, FC disorder can delocalize the states that were localized by mass disorder, and vice-versa. Modeling vacancies as weakly bonded sites with vanishing mass, we find that vacancies, even at very low concentrations, are extremely effective in localizing phonons. Thus, inducing vacancies is proposed as a promising route for efficient thermoelectrics. Finally, we use model parameters corresponding to the alloy system, Ni$_{1-x}$Pt$_x$, and show that mass disorder alone is insufficient to explain the pseudogap in the phonon spectrum; the concomitant presence of FC disorder is necessary.' author: - Wasim Raja Mondal - 'T. Berlijn' - 'M. Jarrell' - 'N. S. Vidhyadhiraja' bibliography: - 'ref.bib' title: 'Phonon localization in binary alloys with diagonal and off-diagonal disorder: A cluster Green’s function approach' --- Introduction ============ Most forms of disorder in a crystal structure have two essential consequences, namely a randomness in mass, and a concomitant change in the bond strengths. The phonon spectrum is, naturally, affected strongly by the presence of mass and bond disorder, and can exhibit Anderson localization (AL) [@PhysRev.109.1492] depending on the nature and strength of disorder, dimensionality and other factors. The possibility of phonon localization due to disorder has evoked great interest over the past several decades, and several theoretical and experimental investigations have been carried out. In recent years, there has been a resurgence of interest in the field due to several direct experimental observations of phonon localization [@PhysRevLett.118.145701; @PhysRevLett.113.175501]. The role of AL in the formation of polar nanoregions in ferroelectrics has also generated a lively debate [@ferrolocalization]. Theoretical investigations of spectral dynamics in mass and force constant (FC)[^1] disordered systems received a big impetus with the development of mean-field based approaches [@PhysRevB.9.1783; @PhysRevB.14.3462; @PhysRevB.24.1872; @0305-4608-6-6-014]. However, single-site theories are, by construction, incapable of incorporating the full non-local nature of force constants. Additionally, some of the Green’s function based attempts failed to maintain Herglotz analytic properties [@PhysRevB.18.5291; @0022-3719-6-10-003], which are essential to produce physically acceptable results. Various other extensions of single-site theories[@0022-3719-17-6-010; @0022-3719-18-22-010; @0305-4608-18-10-008; @PhysRevB.66.214206] have been attempted. Nevertheless, these perturbative methods are plagued by uncontrollable approximations, and hence are unable to treat AL properly. As a non-perturbative route to understand AL, exact diagonalization (ED) [@PhysRev.175.1201] is the most heavily employed method. Although the method does not suffer from approximations, and yields the disorder averaged spectrum, it does have quite a few disadvantages. The state space increases exponentially with system size implying a severe difficulty in simulating large system sizes. The restriction on system sizes, in turn, leads to difficulties in obtaining information of AL[@PhysRev.175.1201]. The consideration of three independent force constants namely $\phi_{AA}, \phi_{AB}$ and $\phi_{BB}$ is necessary for a minimal description of FC disorder in a binary alloy system. Such a consideration compounds the computational expense involved in ED calculations. Another important exact method for studying AL of phonons is the transfer matrix method (TMM)[@0295-5075-97-1-16007]. Disorder effects in masses and force constants are intertwined with each other in most of the binary alloys, but the state-of-the-art TMM calculations [@0295-5075-97-1-16007] have, thus far, treated the ionic masses and force constants as uncorrelated variables, which is quite unrealistic. In addition, such an approximation can lead to a violation of sum rules. Thus, despite extensive attempts, a satisfactory, reliable method for studying phonon localization in a strong mass and FC disordered binary alloy is still lacking, which calls for further theoretical development. In a recent work[@PhysRevB.96.014203], we described the development and application of a typical medium dynamical cluster approximation (TMDCA) for investigating the effect of mass disorder on the AL of phonons. In this work, we incorporate the effects of FC disorder into the existing framework, thus taking a step closer to realistic disorder. The present study has two objectives, namely (i) the development of a formalism for mass and FC disordered systems, that is non-perturbative, systematically convergent, causal, computationally feasible, and is well-benchmarked; and (ii) the application of this formalism to address several open questions (see below). Our first objective has two parts: to develop (a) the DCA to obtain phonon spectra and (b) the TMDCA to investigate phonon localization, in a mass and FC disordered binary alloy. The DCA yields the average density of states (ADOS), which is experimentally observable and is crucial for a basic understanding of disordered lattice vibrations, at a dramatically less computational cost compared to other methods like ED. Concomitantly, the most salient feature of the TMDCA compared to other theories of localization is its ability in predicting localization based on a single-particle order parameter, namely the typical density of states (TDOS). The main development in this work is a cluster adaptation of the Blackman, Esterling and Berk (BEB) formalism [@PhysRevB.4.2412], which was originally proposed for bond-disordered electronic systems. In this study, we adopt a scalar binary alloy model, that consists of a single branch and a single basis atom within the harmonic approximation. In order to assess the validity of the method, we carry out quantitative benchmarks, and find excellent agreement. Subsequently, through an application of the method, we attempt to address the following questions/issues related to the effects of pure FC disorder and the interplay of mass and FC disorder: (1) Structurally disordered glasses are known to exhibit a low frequency anomaly, known as the Boson peak, in the density of states. Such an anomaly has also been observed in disordered lattice models, though the origin of the Boson peak in the two families of systems might be quite different. It has been argued that a purely FC disordered system can also exhibit such behaviour. We carry out a comprehensive analysis of the deviations from low frequency Debye behaviour, and ask- What are the reasons and conditions for the emergence of a Boson peak in a binary mass and spring disordered system? (2) In a related context, are the modes associated with the Boson peak localized or delocalized? (3) As shown in our previous work[@PhysRevB.96.014203], lighter isotopic impurities lead to strongly localized, short-wavelength phonons in impurity bands. In this work, we ask whether the effect of FC disorder is to reinforce or negate the localization induced by mass disorder? (4) And finally, as relevant for improving the figure of merit in thermoelectrics, we explore the efficacy of vacancies in realizing strong phonon localization over a broad range of frequencies. The manuscript is organized as follows. In Section 2, we present our model containing both, the mass and FC disorder, and give a detailed description of the DCA and TMDCA formalism as well as their numerical implementation. In Section 3, we present our results and discussion. We summarize our analysis with some future perspectives in Section 4. Method ====== The Hamiltonian for lattice vibrations involving a single basis atom is: $$H=\sum_{l\alpha}\frac{{p^2_\alpha (l)}}{2M (l)} + \frac {1}{2} \sum_{ll^\prime,\alpha\beta} \Phi^{\alpha\beta}(l,l^\prime)u_\alpha(l)u_\beta(l^\prime)\,, \label{eq:ham}$$ where $u_\alpha(l)$ is the displacement of an ion in the $\alpha^{\rm th}$ direction having mass $M(l)$ in the $l^{th}$ unit cell coupled by the force constant $\Phi_{\alpha\beta}(l,l^\prime)$ tensor to $u_\beta(l^\prime)$. We note that the mass $M(l)$ in Eq. can vary randomly from site to site. Since we are considering the Hamiltonian in Eq. for a binary alloy, the site $l$ can be occupied by either an A-type atom or a B-type atom, [i.e]{} $M(l) \in \{M_A, M_B\}$ with certain probabilities depending on the relative concentrations of the A or B-type atoms. To this end, it is convenient to introduce occupation indices $(x,y)$ for host (A-type atoms) and guest (B-type atoms) as (following Blackman, Ester and Berk (BEB) [@PhysRevB.4.2412]) $$\begin{aligned} \begin{split} x_l=1, y_l=0, & \;\text{if $l \in \rm A $}\\ x_l=0, y_l=1, & \;\text{if $l \in \rm B$} \end{split}\end{aligned}$$ These occupation indices must obey the following properties: $$\begin{aligned} \begin{split} x_{l}y_{l}=0, x_{l}^2=x_l \\ \langle x_{l}\rangle =c_{\rm A}, \langle y_{l}\rangle =c_{\rm B} \end{split} \label{eq:occuprop}\end{aligned}$$ Note that double occupancy of a a given site is prohibited in this formalism. With this assumption, we are ready to express the randomness in the masses as $$M(l)= \begin{cases} x_l M(l) x_l=M_{\rm A}\\ y_l M(l) y_l=M_{\rm B} \\ x_l M(l) y_l=M_{\rm AB}=0\\ y_l M(l) x_l=M_{\rm BA}=0\\ \end{cases} \label{eq:diffmass}$$ We incorporate such randomness in our formalism by defining a local disorder potential matrix ${\hat{V}}$, as $$\left({\hat{V}}\right)_{ll^\prime} = \left(1 - M(l)/M_0 \right)\delta_{l,l^\prime}\,, \label{eq:dispot}$$ where $M_0$ is a reference (host) mass, and as a convention, we have chosen A to be the host, hence $M_0=M_A$. The corresponding probability distribution for binary disorder reads as $$P(V_l)=c_{\rm A} \delta(V_l-V_{\rm A})+c_{\rm B}\delta(V_l-V_{\rm B})\,,$$ where $c_{\rm A}$ and $c_{\rm B}=1-c_{\rm A}$ are the concentrations of A and B type of atoms respectively. Mass disorder can be isotopic or non-isotopic. In general, mass disorder will be accompanied by a [corresponding]{} randomness in the force constants as $$\Phi(l,l^\prime)= \begin{cases} x_l \Phi^{\alpha\beta}(l,l^\prime)x_{l^\prime}=\Phi^{\alpha\beta,\rm AA}(l,l^\prime)\\ y_l \Phi^{\alpha\beta}(l,l^\prime)y_{l^\prime}=\Phi^{\alpha\beta,\rm BB}(l,l^\prime)\\ x_l \Phi^{\alpha\beta,}(l,l^\prime)y_{l^\prime}=\Phi^{\alpha\beta,\rm AB}(l,l^\prime)\\ y_l \Phi^{\alpha\beta}(l,l^\prime)x_{l^\prime}=\Phi^{\alpha\beta,\rm BA}(l,l^\prime)\\ \end{cases} \label{eq:spring}$$ Note that $\Phi(l,l^\prime)$ can be decomposed into diagonal $\Phi(l,l)$ and off-diagonal parts $$\Phi^{\alpha\beta}(l,l^\prime) = \delta_{\alpha\beta}(\Phi_D\delta_{l,l^\prime} + \Phi_{nn}\delta_{\bR_{l^\prime},\bR_l+\vec{\delta}})\,, \label{eq:phiform}$$ where $\Phi_D$ and $\Phi_{nn}$ are the diagonal, and the off-diagonal component of the tensor, respectively, and $\vec{\delta}$ is defined as a vector from a site to its nearest neighbors. Force constant tensor must obey a sum rule, namely $\sum_{l^\prime} \Phi^{\alpha\beta}(l,l^\prime)=0$. For satisfying the sum rule, the formalism must incorporate multi-site correlations, because the spring constant tensor is off-diagonal in nature. We are satisfying this property by systematically by increasing $N_c$. Now, we are in a position to apply the equation of motion method to the Hamiltonian (Eq.) and obtain the Dyson equation as $$M(l) \omega^2 D(l,l^\prime,\omega) = \delta_{ll^\prime} +\sum_{l^{\prime\prime}l^\prime} {\Phi (l, l^{\prime\prime})} D(l^{\prime\prime}, l^\prime,\omega)\, \label{eq:eom}$$ Next, we premultiply and postmultiply the above Eq. by $x_l$ and $y_l$, which would generate the four possible configurations of the binary alloy. Combining this with Eqs., and yields four self-consistent equations for the Green’s functions as given below: $$\begin{aligned} D_{\rm AA}(l,l^\prime) &= \delta_{ll^\prime} + d_{\rm A}(l) \sum_{l^{\prime\prime}\neq l} \Phi^{\rm AA} (l,l^{\prime\prime})D_{\rm AA}(l^{\prime\prime}, l^\prime) \nnu \\ &+ d_{\rm A}(l) \sum_{l^{\prime\prime} \neq l} \Phi^{\rm AB} (l,l^{\prime\prime})D_{\rm BA}(l^{\prime\prime}, l^\prime)\, \label{eq:eomAA}\\ D_{\rm AB}(l,l^\prime) &= d_{\rm A}(l) \sum_{l^{\prime\prime}\neq l} \Phi^{\rm AA} (l,l^{\prime\prime})D_{\rm AB}(l^{\prime\prime}, l^\prime) \nnu \\ &+ d_{\rm A}(l) \sum_{l^{\prime\prime} \neq l} \Phi^{\rm AB} (l,l^{\prime\prime})D_{\rm BB}(l^{\prime\prime}, l^\prime)\, \label{eq:eomAB}\\ D_{\rm BA}(l,l^\prime) &= d_{\rm B}(l) \sum_{l^{\prime\prime}\neq l} \Phi^{\rm BA} (l,l^{\prime\prime})D_{\rm AA}(l^{\prime\prime}, l^\prime) \nnu \\ &+ d_{B}(l) \sum_{l^{\prime\prime} \neq l} \Phi^{\rm BB} (l,l^{\prime\prime})D_{\rm BA}(l^{\prime\prime}, l^\prime)\, \label{eq:eomBA}\\ D_{\rm BB}(l,l^\prime) &= \delta_{ll^\prime} + d_{\rm B}(l) \sum_{l^{\prime\prime}\neq l} \Phi^{\rm BA} (l,l^{\prime\prime})D_{\rm AB}(l^{\prime\prime}, l^\prime) \nnu \\ &+ d_{\rm B}(l) \sum_{l^{\prime\prime} \neq l} \Phi^{\rm BB} (l,l^{\prime\prime})D_{\rm BB}(l^{\prime\prime}, l^\prime)\, \label{eq:eomBB}\end{aligned}$$ where the four configuration dependent Green’s function are defined as $$\begin{aligned} & x_l D(l,l^{\prime})x_{l^{\prime}}=D_{\rm AA}(l,l^\prime) \nnu \\ & x_l D(l,l^{\prime})y_{l^{\prime}}=D_{\rm AB}(l,l^\prime) \nnu \\ & y_l D(l,l^{\prime})x_{l^{\prime}}=D_{\rm BA}(l,l^\prime) \nnu \\ & y_l D(l,l^{\prime})y_{l^{\prime}}=D_{\rm BB}(l,l^\prime) \\end{aligned}$$ and the bare locators, $d_{\rm A}$ and $d_{\rm B}$ are given by $$\begin{aligned} x_l d(l) x_l &=d_{\rm A}(l) =\frac{1}{ \left[M_{\rm A} \omega^2-\Phi^{\rm AA}(l,l) \right]}\\ y_l d(l) y_l &= d_{\rm B}(l) =\frac{1}{\left[ M_{\rm B} \omega^2-\Phi^{\rm BB}(l,l) \right]}\end{aligned}$$ The four self-consistent equations, Eqs.-Eqs. may be combined in a convenient $2\times 2$ matrix form as $${ \begin{pmatrix} D_{\rm AA} & D_{\rm AB} \\ D_{\rm BA} & D_{\rm BB} \end{pmatrix} }_{ll^\prime} = { \begin{pmatrix} d_{\rm A} & 0 \\ 0 & d_{\rm B} \end{pmatrix} }_{l}\delta_{ll^\prime}\\ +\\$$ $${ \begin{pmatrix} d_{\rm A} & 0 \\ 0 & d_{\rm B} \end{pmatrix} }_{l}\\ \sum_{l^{\prime\prime}\neq l}{ \begin{pmatrix} \Phi^{\rm AA} & \Phi^{\rm AB} \\ \Phi^{\rm BA} & \Phi^{\rm BB} \end{pmatrix} }_{ll^{\prime\prime}} \\ { \begin{pmatrix} D_{\rm AA} & D_{\rm AB} \\ D_{\rm BA} & D_{\rm BB} \end{pmatrix} }_{l^{\prime\prime} l^{\prime}}$$ And finally, even this matrix equation can be compactified to get $$\begin{aligned} \underline{D}=\underline{d}+ \underline{d}\times \underline{\Phi} \times \underline{D} \end{aligned}$$ Here $\underline{D}$ and $\underline{d}$ are matrices of size $2N\times 2N$, where $N$ is the system size. Thus, we have obtained an equation which has a structure similar to the one obtained in the BEB formalism [@PhysRevB.4.2412] for the electronic problem. It is interesting to note that there is no randomness associated with $\Phi$ matrices. All the randomness is absorbed in the $d$ matrices, and the origin of this randomness lies in the mass term. The $\Phi$ matrices will take the values depending on the random values associated with the mass term. Hence, diagonal mass disorder and off-diagonal spring disorder are dependent on each other. The other point to note is that we can consider three different spring constants, which has been a computational limitation for some theoretical approaches [@1742-6596-286-1-012025; @PhysRevB.87.134203]. In order to solve these equations, we have adopted the dynamical cluster approximation. The formalism is similar to the one presented in our previous work on mass disorder, but there are certain steps that are unique to the spring disorder case. In the next section, we provide the details of the formalism. Dynamical Cluster Approximation (DCA) ------------------------------------- The advent of dynamical mean field theory (DMFT) led to a sort of revolution in the understanding of quantum many body lattice systems. However, since DMFT ignores non-local dynamical correlations, several phenomena such as d-wave superconductivity, Anderson localization, and low dimensional physics are out of scope of this framework. Hence, quantum cluster approaches, that go beyond DMFT, have assumed great importance. One such approach, that is based on momentum space clusters, is the dynamical cluster approximation (DCA). The DCA may be viewed as an approximation to the wave-vector sums that occur in Feynman-Dyson Perturbation Theory. Here, the first Brillouin zone containing $N$ wavenumbers $\bk$ is broken into $N_c$ non-overlapping coarse graining cells. We then approximate the integrals associated with each diagram by its sum of average/coarse-grained estimates of the integrand within each cell. There is considerable freedom in how this is done. For example, if the integrand is composed of the product of two functions of the integration variable, do we take the product of the two averages, or the average of the products to define the approximate value in the cell? Since these two approximations have the same error provided that the number of such cells is large, we can use this freedom to simplify the approximation. To do this, we define the many-to-few mapping $M(\bk) = \bK$, where $\bK$ labels the cells including $\bk$, so that $\bk=\bK+\bkt$, where $\bkt$ labels the wave-numbers within each cell. The corresponding transformation of the Lie algebra is $ {\bar{c}}_\bk = \sum_\bkt c_{\bK+\bkt}\,. $ It is easy to see that this transformation preserves the Lie algebra. So it maps bosons onto bosons and fermions onto fermions. The mapping is not canonical though since information is lost in the process. Nevertheless, this mapping ensures that FDPT may be used to analyze lattice plus the quantum impurity problem. Under this transformation, all points within each cell are considered to be equivalent, and are mapped to a single point $\bK$. So, each Green’s function within the cell may be replaced by its average value within the cell. Equivalently, each $G(\bk)$ in a Feynman graph may be replaced by its coarse grained analog. More significantly, each sum over $\bk$ is replaced by a sum over $\bK$ thereby dramatically reducing the complexity of the problem of order $N$ to order $N_c$. The associated FDPT is the same as a small self-consistently embedded periodic cluster problem. Once the cluster problem is solved, we calculate the corresponding irreducible self energy and vertex functions. We use them in the Dyson and Bethe-Salpeter equation to calculate the single-particle spectra and the two-particle susceptibilities. For example, the DMFT framework may be represented as a mapping of the entire first Brillouin zone to just one momentum at the centre for the zone. The main simplification in the DMFT framework is the absence of momentum conservation at the vertices in the Feynman diagrams, thus leading to a local self-energy. The DCA targets this lacuna of DMFT and replaces the Dirac delta function that represents true momentum conservation at the vertices by a Laue function that conserves momentum, but only for the cluster momenta. This brings back the momentum dependence in the Green’s functions and self-energy, lost at the DMFT level. Thus, as the number of clusters increases, the Brillouin zone is sampled more densely, and hence the thermodynamic limit is systematically approached. We refer the reader to review articles [@RevModPhys.77.1027; @TMDCAreview] for details of the DCA, and its applications to a variety of problems. The DCA algorithm that we have implemented is derived using the formalism described in the previous subsection and is described below: 1\. We start with an initial guess of the hybridization function as $$\underline{\Delta (\mathbf K,\omega)}= \begin{pmatrix} \Delta^{\rm AA}(\mathbf K,\omega) & \Delta^{\rm AB}(\mathbf K,\omega)\\ \Delta^{\rm BA}(\mathbf K,\omega) & \Delta^{\rm BB}(\mathbf K,\omega) \end{pmatrix}$$ This guess may be obtained through a coarse graining of the non-disordered Green’s function, or from a previously converged calculation. 2\. As a first step for solving the cluster problem, we generate random configurations of the disorder potential V. The disorder potentials $V_{A}$ and $V_{B}$ are assigned depending on whether the site is occupied by A or B type of atom. We generate some random number and if it is less than a given impurity concentration $c_A$, we assign a given site as A-type, else it is assigned as B-type. 3\. We define $\Phi^{\prime}$ as configuration dependent force-constants which can be obtained by configuration dependent Fourier transform as shown below $$\Phi^{\prime}(l,l^\prime)= \begin{cases} \sum_{\mathbf K} {\big(\underline{\omega^2_{\bK}}\big)^{\rm AA}} e^{i\mathbf K \cdot(\vec R_l-\vec R_{l^\prime})},&\text{if $l \in \rm A$, $l^{\prime}\in \rm A$ } \\ \sum_{\mathbf K} {\big(\underline{\omega^2_{\bK}}\big)^{\rm AB}} e^{i\mathbf K \cdot(\vec R_l-\vec R_{l^\prime})},&\text{if $l \in \rm A$, $l^{\prime}\in \rm B$ }. \\ \sum_{\mathbf K} {\big(\underline{\omega^2_{\bK}}\big)^{\rm BA}} e^{i\mathbf K \cdot(\vec R_l-\vec R_{l^\prime})},&\text{if $l \in \rm B$, $l^{\prime}\in \rm A$ }. \\ \sum_{\mathbf K} {\big(\underline{\omega^2_{\bK}}\big)^{\rm BB}} e^{i\mathbf K \cdot(\vec R_l-\vec R_{l^\prime})},&\text{if $l \in \rm B$, $l^{\prime}\in \rm B$ }. \end{cases}$$ where the 2X2 dispersion matrix is given by $${\underline {\omega^2_{\mathbf K}}}= \begin{pmatrix} \Phi^{\rm AA}& \Phi^{\rm AB}\\ \Phi^{\rm BA}& \Phi^{\rm BB} \end{pmatrix} \bar \omega_{\mathbf K}^2$$ and the coarse-grained dispersion is given by $$\begin{gathered} \bar \omega_{\mathbf K}^2= \frac{N_c}{N} \sum_{\tilde k}\sin^2 \bigg ( \frac{(K_x +\tilde k_x)a}{2}\bigg)\\ + \sin^2 \bigg (\frac{(K_y+\tilde k_y)a}{2}\bigg) + \sin^2 \bigg (\frac{(K_z+\tilde k_z)a}{2}\bigg)\\\end{gathered}$$ The real space hybridization function $\Delta^{\prime}$ is obtained from the configuration dependent Fourier transform as below: $$\Delta^{\prime}(l,l^\prime)= \begin{cases} \sum_{\mathbf K}\lbrack \Delta^{\rm AA}(\mathbf K,\omega)\rbrack e^{i\mathbf K.(R_l-R_{l^\prime})},&\text{if $l \in \rm A$, $l^{\prime}\in \rm A$ } \\ \sum_{\mathbf K}\lbrack \Delta^{\rm AB}(\mathbf K,\omega)\rbrack e^{i\mathbf K.(R_l-R_{l^\prime})},&\text{if $l \in \rm A$, $l^{\prime}\in \rm B$ }. \\ \sum_{\mathbf K}\lbrack \Delta^{\rm BA}(\mathbf K,\omega)\rbrack e^{i\mathbf K.(R_l-R_{l^\prime})},&\text{if $l \in \rm B$, $l^{\prime}\in \rm A$ }. \\ \sum_{\mathbf K}\lbrack \Delta^{\rm BB}(\mathbf K,\omega)\rbrack e^{i\mathbf K.(R_l-R_{l^\prime})},&\text{if $l \in \rm B$, $l^{\prime}\in \rm B$ }. \end{cases}$$ After constructing $\Phi^{\prime}$, $\Delta^{\prime}$ and $V$, we compute the corresponding cluster Green function through the mass-weighted Dyson’s equation [@PhysRevB.96.014203] as $$\begin{aligned} & D^c(l,l^\prime,\omega,V)= \nnu \\ & \sqrt{1-(\hat V)_l} (\omega^2 I-\underline{\Phi^{\prime}} -\underline{\Delta^{\prime}} -\underline{\hat V})_{ll^{\prime}}^{-1} \sqrt{1-(\hat V)_{l^\prime}}\,. \label{eq:mwdysonspring}\end{aligned}$$ 4. The next step is disorder averaging over disorder configurations denoted by $\langle (...)\rangle$. These disorder averaged Green’s functions correspond to a translationally invariant system, and are denoted by the DCA subscript: $$\begin{aligned} \big (D^c_{\scriptscriptstyle{\rm DCA}}\big)_{\rm AA} = \bigg\langle &D^c(l,l^\prime,\omega) \bigg\rangle, &\text{if $l \in \rm A$, $l\in \rm A$ }\nnu \\ \big (D^c_{\scriptscriptstyle{\rm DCA}}\big)_{\rm AB} = \bigg\langle &D^c(l,l^\prime,\omega) \bigg\rangle, &\text{if $l \in \rm A$, $l\in \rm B$ }\nnu \\ \big (D^c_{\scriptscriptstyle{\rm DCA}}\big)_{\rm BA} = \bigg\langle &D^c(l,l^\prime,\omega) \bigg\rangle, &\text{if $l \in \rm B$, $l\in \rm A$ }\nnu \\ \big (D^c_{\scriptscriptstyle{\rm DCA}}\big)_{\rm BB} = \bigg\langle &D^c(l,l^\prime,\omega) \bigg\rangle, &\text{if $l \in \rm B$, $l\in \rm B$ }\nnu \\ \label{eq:dcaave}\end{aligned}$$ Next, we construct a matrix of the cluster Green function by re-expanding the Green function to a $2N_c \times 2N_c$ matrix. It can be represented as $$\underline{{D^c_{DCA}}}= \begin{pmatrix} \big (D^c_{\scriptscriptstyle{\rm DCA}}\big)_{\rm AA} & \big (D^c_{\scriptscriptstyle{\rm DCA}}\big)_{\rm AB} \\ \\ \big (D^c_{\scriptscriptstyle{\rm DCA}}\big)_{\rm BA} & \big (D^c_{\scriptscriptstyle{\rm DCA}}\big)_{\rm BB}\\ \end{pmatrix}$$ 5. As mentioned above, after disorder averaging, the translation symmetry is restored and we can perform Fourier transform for each component to get disorder averaged $\mathbf K$ dependent cluster Green function as $$\underline{D^c(\mathbf K, \omega)}= \begin{pmatrix} {D^c_{\rm AA}(\mathbf K,\omega) } & { D^c_{\rm AB}(\mathbf K, \omega) } \\ {D^c_{\rm BA}(\mathbf K, \omega)} & { D^c_{\rm BB}(\mathbf K, \omega)} \end{pmatrix}$$ 6. Once the cluster problem is solved, we calculate the coarse-grained lattice Green function as $$\begin{aligned} & \underline{D^{\rm CG}(\mathbf K, \omega)} \nnu \\ & =\frac{N_c}{N}\sum_{\tilde k}\big \lbrack \underline{D^c(\mathbf K,\omega)}^{-1}+\underline{\Delta(\mathbf K,\omega)}-\underline{\omega^2_k} +\underline{\bar \omega^2 (\mathbf K)}\big \rbrack \label{coarsegroff}\end{aligned}$$ which in explicit matrix form is given by, $$\underline{D^{\rm CG}(\mathbf K, \omega)} = \begin{pmatrix} {D^{\rm CG}_{\rm AA}(\mathbf K,\omega)} & {D^{\rm CG}_{\rm AB}(\mathbf K, \omega)}\\ \\ {D^{\rm CG}_{\rm BA}(\mathbf K, \omega) }& {D^{\rm CG}_{\rm BB}(\mathbf K, \omega)} \end{pmatrix}$$ 7. The DCA self consistency condition requires that the disorder averaged cluster Green function equal the coarse-grained lattice Green’s function $$\underline{D^c(\mathbf K,\omega)}=\underline{D^{\rm CG}(\mathbf K,\omega)}$$ 8\. The self consistency condition is used for updating the hybridization function for each component $$\begin{aligned} &{\Delta_{n}^{\rm AA}}(\mathbf K,\omega)= {\Delta_{o}^{\rm AA}}(\mathbf K,\omega)\nnu \\ &+\xi \left[ {\big (D^c_{\rm AA}(\mathbf K,\omega)\big)}^{-1} - {\big (D^{CG}_{\rm AA}(\mathbf K,\omega)\big)}^{-1}\right]\\ &{\Delta_{n}^{\rm BB}}(\mathbf K,\omega)= {\Delta_{o}^{\rm BB}}(\mathbf K,\omega)\nnu \\ &+\xi \left[ {\big(D^c_{\rm BB}(\mathbf K,\omega) \big)}^{-1}- {\big (D^{CG}_{\rm BB}(\mathbf K,\omega)\big)}^{-1}\right]\\ &{\Delta_{n}^{\rm AB}}(\mathbf K,\omega)= {\Delta_{o}^{\rm AB}}(\mathbf K,\omega)\nnu \\ &+\xi \left[ {\big (D^c_{\rm AB}(\mathbf K,\omega)\big)}^{-1}- {\big (D^{CG}_{\rm AB}(\mathbf K,\omega)\big)}^{-1}\right]\\ &{\Delta_{n}^{\rm BA}}(\mathbf K,\omega)= {\Delta_{o}^{\rm BA}}(\mathbf K,\omega)\nnu \\ &+\xi \left[ {\big (D^c_{\rm BA}(\mathbf K,\omega)\big)}^{-1}- {\big (D^{CG}_{\rm BA}(\mathbf K,\omega)\big)}^{-1}\right] \label{eq:selfoff}\end{aligned}$$ In the above equations, the $\xi$ is a mixing parameter that determines the fraction of the updated hybridization that should be mixed with the existing one, thus ensuring smooth convergence of the DCA iterations. As is well known, the Anderson localization of phonons requires us to go beyond DCA. The arithmetic averaging procedure needs to be modified, and a typical averaging ansatz needs to be evolved. Such an ansatz has been worked out in the electronic case, and has been benchmarked against known results. [@PhysRevB.90.094208]. We have adopted the same ansatz in the phonon case, and have found that it yields the same level of benchmarks as the electronic case. The formalism that employs this typical averaging ansatz is called the typical medium DCA or the TMDCA, and is detailed in the next section. Typical Medium Dynamical Cluster Approximation (TMDCA) ------------------------------------------------------ As mentioned in our previous discussion[@PhysRevB.96.014203], the effective medium is constructed via algebraic averaging in the DCA, while the TMDCA utilizes geometric averaging to construct the effective medium. We employ the same ansatz for evaluating the typical density of states as in the electronic case [@PhysRevB.90.094208], as: $$\begin{gathered} \underline{\rho_{typ}^c(\bK, \omega)}= \exp \bigg( \frac{1}{N_c} \sum_{i=1}^{N_c} \langle \rm ln \rho_{ii} (\omega) \rangle\bigg) \times \\ \begin{pmatrix} \left\langle \frac{-\frac{1}{\pi} \rm Im D^c_{\rm AA}(\bK,\omega)}{\frac{1}{N_c}\sum_{i=1}^{N_c}\big (-\frac{1}{\pi} \rm Im D^c_{ii} (\omega)\big )}\right\rangle & \left\langle \frac{-\frac{1}{\pi} \rm Im D^c_{\rm AB}(\bK,\omega)}{\frac{1}{N_c}\sum_{i=1}^{N_c}\big (-\frac{1}{\pi} \rm Im D^c_{ii} (\omega)\big )}\right\rangle\\ \\ \left\langle \frac{-\frac{1}{\pi} \rm Im D^c_{\rm BA}(\bK,\omega)}{\frac{1}{N_c}\sum_{i=1}^{N_c}\big (-\frac{1}{\pi} \rm Im D^c_{ii}(\omega)\big )}\right\rangle & \left\langle \frac{-\frac{1}{\pi} \rm Im D^c_{\rm BB}(\bK,\omega)}{\frac{1}{N_c}\sum_{i=1}^{N_c}\big (-\frac{1}{\pi} \rm Im D^c_{ii}(\omega)\big )}\right\rangle \end{pmatrix}\,, \label{eq:anstzoff}\end{gathered}$$ where $D^c_{ii}$ is defined as $$\begin{aligned} &D^c_{ii}(\omega)=\sum_{\bf K} \bigg( D^c_{\rm AA}(\mathbf{K},\omega)+D^c_{\rm BB}(\bf K, \omega) \nnu \\ &+D^c_{\rm AB}(\mathbf{K},\omega)+D^c_{\rm BA}(\bf K,\omega) \bigg)\\ &{\text{ and the local spectral function is given by}} \nnu \\ &\rho_{ii}(\omega)=-\frac{1}{\pi} \left[ D^c_{ii}(\omega) \right]\,.\end{aligned}$$ Next, we calculate the cluster-averaged typical Green function which is also a $2\times 2$ matrix $$\underline{{D^c_{\rm typ}}}= \begin{pmatrix} \big (D^c_{\scriptscriptstyle{\rm typ}}\big)_{\rm AA} & \big (D^c_{\scriptscriptstyle{\rm typ}}\big)_{\rm AB} \\ \\ \big (D^c_{\scriptscriptstyle{\rm typ}}\big)_{\rm BA} & \big (D^c_{\scriptscriptstyle{\rm typ}}\big)_{\rm BB}\\ \end{pmatrix}$$ We compute each component of the cluster-averaged typical Green’s function from the corresponding component of the typical density of states using the Hilbert transform, $$\begin{aligned} \big (D^c_{\scriptscriptstyle{\rm typ}}\big)_{\rm AA} = \mathcal{P}\int d\omega^{\prime} \frac{\rho_{typ}^{\rm AA}(\bK,\omega^\prime)}{\omega^2-\omega^{\prime^2}}-i \frac{\pi}{2\omega} \rho_{typ}^{\rm AA}\nnu \\ \big (D^c_{\scriptscriptstyle{\rm typ}}\big)_{\rm AB} = \mathcal{P}\int d\omega^{\prime} \frac{\rho_{typ}^{\rm AB}(\bK,\omega^\prime)}{\omega^2-\omega^{\prime^2}} -i \frac{\pi}{2\omega} \rho_{typ}^{\rm AB}\nnu \\ \big (D^c_{\scriptscriptstyle{\rm typ}}\big)_{\rm BA} = \mathcal{P}\int d\omega^{\prime} \frac{\rho_{typ}^{\rm BA}(\bK,\omega^\prime)}{\omega^2-\omega^{\prime^2}} -i \frac{\pi}{2\omega} \rho_{typ}^{\rm BA}\nnu \\ \big (D^c_{\scriptscriptstyle{\rm typ}}\big)_{\rm BB} = \mathcal{P}\int d\omega^{\prime} \frac{\rho_{typ}^{\rm BB}(\bK,\omega^\prime)}{\omega^2-\omega^{\prime^2}}-i \frac{\pi}{2\omega} \rho_{typ}^{\rm BB} \label{eq:hilbertoff}\end{aligned}$$ Once the disorder-averaged cluster Green function is calculated using , the self-consistency follows the same steps as in the DCA presented in previous section. The coarse-grained lattice Green’s function is then calculated using , which is utilized to update the hybridization function in . Using DCA and TMDCA, we can calculate the arithmetically averaged density of states (ADOS) and typical density of states (TDOS) respectively as follows: $$\begin{aligned} {\rm ADOS}(\omega^2)=-\frac{2\omega}{N_c\pi}{\rm Im} \sum_{{\mathbf K},\sigma\sigma^{\prime}} \big(D^{c}_{\scriptscriptstyle{\rm DCA}}({\mathbf K},\omega)\big)_{\sigma\sigma^\prime} \\ {\rm TDOS}(\omega^2)=-\frac{2\omega}{N_c\pi}{\rm Im} \sum_{{\mathbf K},\sigma\sigma^\prime} \big(D^{c}_{\scriptscriptstyle{\rm typ}}(\mathbf K,\omega)\big)_{\sigma\sigma^\prime}\,,\end{aligned}$$ where $\sigma, \sigma^\prime = A/B$. The formalism described above has been implemented, and we present results and discussion in the following section. Results and discussions ======================= We begin our discussion with a validation of the method. To that end we compute the ADOS via exact diagonalization (ED) of a large number of phonon models of large sized disordered supercells. Within the supercells the impurities are randomly distributed and beyond the supercell boundaries the impurity distributions periodically repeat. Specifically for each set of model parameters we derive the force constant matrices of 100 supercells each with 60 impurities and roughly 400 sites on average. The dynamical matrix of each supercell is then evaluated and diagonalized on a $10\times10\times10$ supercell momentum space grid. We note that not only the impurity distributions but also the shapes of the supercells are randomized under the following constraints. The supercell volumes vary within 375 and 430 sites and the angles between the vectors that span the supercells vary between 75 and 105 degrees. The left panels of figure \[fig:fig1\] show DCA results (red solid lines) and ED results (black solid lines) for a binary alloy ($c=0.15; V=0.67$) with three different spring constant combinations. A good agreement is seen over all scales thus validating the formalism. We also note that the DCA can access detailed information of the ADOS with relatively small cluster sizes, i.e $N_c=64$ which shows that the DCA is dramatically less expensive compared to ED while being numerically exact. To check the sensitivity of the DCA results on the choice of cluster size $N_c$, we present the ADOS for different $N_c$ in the right panel of Fig.\[fig:fig1\]. We find that the ADOS for $N_c=64$ and $N_c=125$ are almost identical which implies a rapid convergence of our calculations with respect to increasing $N_c$. In contrast to the results for $N_c=64$ and $N_c=125$, the single-site ($N_c=1$) calculations (solid blue lines) are unable to capture non-local fluctuations and disagree significantly with the converged spectra. The result presented above, namely a comparison of DCA with ED, lends strong credence to results from DCA. Hence, we employ DCA, and subsequently TMDCA to investigate the interplay of spring and mass disorder on phonon spectra and on AL of phonons. We begin with an investigation of the effect of pure spring disorder on phonon spectra. The upper panel in figure \[fig:BP\_pure\_spring\] shows the average DOS for pure spring disorder (i.e. $V\rightarrow 0$) with spring constant values $\phi_{AA}=1.0, \;\phi_{BB}=0.1$ and $\phi_{AB}=0.3$ for various impurity concentrations ($c_B$) ranging from 0.95 to 0.05. The parameters have been chosen to mimic the values obtained in a recent experiment [@PhysRevE.87.052301] on crystals of binary hard-soft microgel particles with three distinct interparticle potentials. The spring constant values imply that A are hard particles, while B are soft. Hence, $c_B=0.95$ corresponds to B-particle concentration of 95% which implies hard sphere concentration of 5%. As expected, the spectrum for a higher concentration of hard particles (stiffer springs, $c_B=0.05$, $c_A=0.95, \phi_{AA}/\phi_{BB}=10$) has almost entire spectral weight at higher frequencies, and as $c_B$ varies from $0.05$ to $0.95$, spectral weight is transferred to lower frequencies. The DOS at 20% to 40% soft particles ($c_B=0.2-0.4$) shows a clear excess density of states around a frequency, that occurs far below the van Hove singularities of the pure hard particle system. Such behavior is strongly reminiscent of disordered systems, where the origin of such an excess of DOS, termed a Boson peak, has generated a lot of debate. We briefly review a few theoretical and experimental results relevant to this issue, and place our results in perspective. It has been shown theoretically [@PhysRevLett.81.136], that a strongly disordered three-dimensional system of coupled harmonic oscillators with a [continuous]{} force constant distribution exhibits an excess low-frequency DOS (boson peak) as a generic feature. Specifically, if the system is proximal to the borderline of stability, a low-frequency peak (i.e the Boson peak) appears in the quantity $g(\omega)/\omega^2$ as a precursor of the instability. Our results have been obtained for a binary alloy with three values of spring constants, and we see that a Boson peak appears in a regime of lower soft particle concentration. Experimental measurements [@science] of normal modes and the DOS in a disordered colloidal crystal showed Debye-like behavior at low energies and an excess of modes, or Boson peak, at higher energies. The normal modes took the form of plane waves, that hybridized with localized short wavelength features in the Debye regime but lost both longitudinal and transverse plane-wave character at a common energy near the Boson peak. More recently, experiments[@PhysRevE.87.052301] on deformable microgel colloidal particles with random stiffness appear to contradict the theoretical results of Ref[@PhysRevLett.81.136]. The authors create crystals of binary hard-soft microgel particles with three distinct interparticle potentials distributed randomly on a two-dimensional triangular lattice. The nearest-neighbor bonds are either very stiff ($\phi_{AA}$), very soft ($\phi_{BB}$), or of intermediate stiffness ($\phi_{AB}$). Subsequently, they obtain, experimentally, the phonon modes in crystals with bond strength disorder as a function of increasing dopant concentration. The interesting feature of the microgel crystal is that although the bonds are randomly distributed, the masses are nearly identical, hence the disorder is purely off-diagonal. The experimental results[@PhysRevE.87.052301] show the absence of a boson peak, although an excess in the density of states as compared to conventional Debye behaviour was observed. In the lower panel of figure \[fig:BP\_pure\_spring\], we present integrated DOS as a function of frequency. The results indicate conventional Debye behaviour ($\sim \omega^d$) at lowest frequencies followed by a deviation, and finally a convergence to the normalization value of one at the highest frequencies. In the experiment, the hard particle concentration has been varied from 0 to about 21%. A comparison to figure 3 of Yodh [et. al]{}[@PhysRevE.87.052301] shows that our results concur well with the experiments. An absence of Boson peak, as concluded in the experiments, is natural since a clear Boson peak occurs only in the opposite limit of lower soft particle concentration. To summarize, within the framework of DCA, we find (see figure \[fig:BP\_pure\_spring\]), for a binary alloy, that a transfer of spectral weight to lower frequencies results in the Boson peak, which emerges as a crossover feature between a pure host system and a pure guest system. A question that has been much debated in the literature is about the nature of states within the boson peak - Are they localized or delocalized? This question can be effectively answered through the evaluation of the typical DOS, since the typical spectral weight is a measure of the proximity to Anderson localization. A subtle issue about the interpretation of the typical density of states must be mentioned here. A non-zero typical DOS signifies the presence of extended states. According to Mott, a degeneracy of localized and extended states should lead to their hybridization, and hence an eventual delocalization of the localized states. The average DOS and the typical DOS, being different at a given energy, is thus immaterial regarding the identification of the states being extended or localized. If the typical DOS is non-zero, the states at that energy should be interpreted as being extended. Concomitantly, a large difference between the average and typical DOS does indicate a proximity to the Anderson localization transition (ALT). In figure \[fig:BP\_TDOS\], we show, for the same parameters as figure \[fig:BP\_pure\_spring\], a series of average (black solid lines) and the corresponding typical spectra (red shaded part). \[t!\] The boson peak is seen to have numerically negligible typical spectral weight (TSW) until $c_B\sim 0.4$, beyond which the low frequency peak acquires finite and significant TSW. Thus, the states in the BP exhibit a kind of crossover from being almost localized (proximal to ALT) to being delocalized (relatively smaller difference between average and typical DOS) with increasing $c_B$. The overall typical spectral weight, shown in the bottom left panel, is non-monotonic, and shows a minimum at $c_B\sim 0.5$, showing that the overall system is closest to the ALT when the ratio of the concentrations of the two species is roughly equal to one. In order to understand the interplay of mass and spring disorder, we consider two protocols. In the first, we keep the mass ratio parameter, $V=0.67$ and the impurity concentration $c=0.5$ as constants, and increase the spring disorder systematically by varying $\phi_{AA}/\phi_{BB}$ (with $\phi_{BB}=1$, and $\phi_{AB}=\left(\phi_{AA}+\phi_{BB}\right)/2$) from $0.2$ to $2.0$, representing a change of host spring constants from very soft to very stiff. The resulting spectra (ADOS as solid black lines and TDOS as red shaded part) are shown in figure \[fig:fig5\], while the inset shows the integrated typical spectral weight (TSW, solid blue circles) as a function of $\phi_{AA}$. For soft A-springs, the characteristic frequencies of the system must be lower than a pure B-type system, and as the $\phi_{AA}$ is increased, spectral weight in the second, high frequency peak increases, as also the bandwidth of the system. So, nominally, it appears that the system is getting delocalized, as the host springs are made stiffer for a fixed mass disorder. However, the inset shows a decrease in TSW with increasing stiffness of $\phi_{AA}$, which implies that the order parameter for AL is decreasing, and hence the system is moving closer to localization. If we focus on a fixed frequency, say $\omega^2=5.0$, then we see that for $\phi_{AA}=0.2$, the ADOS and TDOS are zero, while for $\phi_{AA}=2.0$, both the average and typical DOS are non-zero, suggesting the interpretation that spring disorder is of a delocalizing nature and counters the localization produced by mass disorder alone. However, the order parameter for AL, namely the TSW, decreases sharply. Thus the interplay of mass and spring disorder is quite subtle, and an interpretation of the results need to be done carefully. It must be emphasized here, that the subtlety of this interplay has been uncovered through the application of TMDCA, which is able to produce, simultaneously, the typical and the average DOS. The second protocol is to vary the mass ratio parameter, $V=1-M_{imp}/M_{host}$, keeping the relative concentrations as well the spring constants fixed. The ADOS and TDOS are shown in figure \[fig:fig6\] for $V=0.01$ to $0.9$, implying a systematic decrease in the B-site ionic mass. Again, lighter impurities imply transfer of spectral weight to higher frequencies, and the B-site band appears as a separate feature, which blue shifts significantly with increasing $V$. In parallel with the results of the first protocol, this result lends itself to an interpretation of delocalization of modes at higher frequencies, but the vanishing of the typical density of states shows that the high frequency modes for $V\rightarrow 1$ are almost localized. The inset shows that the TSW (solid blue circles) decreases sharply with increasing $V$, and this also implies that increasing mass disorder in the presence of fixed spring disorder pushes the system closer to the AL transition. The insight we gain from the study of the interplay of mass and spring disorder is that an inference of localization/delocalization of specific modes cannot be made on the basis of ADOS alone, and the TDOS must be concomitantly examined. Vacancies, even at low concentrations, can lead to strong localization of phonons. Within the present theoretical framework, we model vacancies as weakly bonded sites with vanishing mass. So, we choose $M_{imp}=M_{host}/20$, which is equivalent to $V=0.95$, and spring constants as $\phi_{AA}=1, \phi_{BB}=0.15, \phi_{AB}=0.15$. For these parameters, in figure \[fig:fig7\], we show the average DOS (upper panel) and typical DOS (lower panel) for three different guest concentrations, namely $c=0.1$(solid black), $c=0.2$(dashed red), and $c=0.3$(dotted blue). The upper panel shows that the average DOS hardly changes with increasing concentration, while the corresponding typical DOS (lower panel) undergoes significant changes. The inset in the lower panel shows the rapid decrease of the integrated typical spectral weight (blue solid circles) with increasing concentration, $c$. Modeling real vacancies is quite challenging, but the present analysis with a very crude model for vacancies is already indicative of their strong localization effects. The figure of merit for thermoelectrics is inversely proportional to the thermal conductivity, and directly proportional to electrical conductivity. So, in order to maximize the figure of merit, the vacancy concentration, $c$ should be tuned to an optimal value such that it is less than, but not too close to the percolation limit, implying that the electrical conductivity is not too significantly affected, but the thermal conductivity due to phonons gets drastically reduced due to the strong localization of acoustic phonons in a large part of the spectrum. Finally, we attempt a qualitative comparison with experiments. It has been argued for a Ni$_x$Pt$_{1-x}$ alloy, that $x=0.65$ constitutes weak force-constant disorder, while $x=0.5$ constitutes strong mass and force-constant disorder. Ghosh [et. al.]{}[@PhysRevB.66.214206] demonstrate that for $x=0.5$, a CPA-level consideration of inter-atomic force-constants leads to a split-band spectrum. The authors show that a proper treatment of force-constant disorder using itinerant CPA leads to a closure of the gap. Our calculations are in full qualitative agreement with these conclusions as argued below. Figure \[fig:DCAalloy\] shows ADOS for a binary alloy with M$_{\rm imp}$=M$_{\rm host}$/3 as appropriate for Ni impurities in Pt host. The left panel is for $N_c=1$, equivalent to a CPA calculation, while the right panel is for $N_c=64$, which is equivalent to the thermodynamic limit. The impurity concentration used is $c=0.5$, which implies strong mass disorder; and the spectra corresponding to three distinct force-constant combinations are shown. The black solid line corresponds to pure mass-disorder, which at the CPA-level shows a split-band (left panel), while at the DCA-level (right panel), the spectrum has a two peak structure with a soft-gap between the peaks. The red and green lines correspond to weak and strong force-constant disorder respectively. Again, the CPA results are hardly affected by an increase in disorder, while the DCA results for $N_c=64$ show that increasing force-constant disorder leads to significant spectral weight transfer, especially a filling-up of the soft-gap. Since the force constant combination represented by the green line most closely corresponds to the Ni$_{0.5}$Pt$_{0.5}$ alloy, we conclude that our results agree qualitatively with the ICPA results as well as experimental neutron scattering data for Ni$_{0.5}$Pt$_{0.5}$ [@PhysRevB.19.2876]. Conclusions =========== The incorporation of the BEB formalism for off-diagonal disorder into the TMDCA yields a reliable, and computationally feasible approach for investigating binary mass and spring-disordered alloys. Such a conclusion is supported by the benchmarking studies discussed in the initial part of the results section. For a fixed mass ratio, and fixed spring constants ($\phi_{AA}, \phi_{BB}$ and $\phi_{AB}$, increasing the soft particle concentration leads to an excess density of states below the first van Hove singularity of the host hard particle system. In the present context, it may be identified as the Boson peak, commonly observed in structurally disordered glasses as well as disordered lattice systems, albeit with different origins. We conclude that the origin of the Boson peak in the disordered binary alloy system is a transfer of spectral weight from the guest to the host system, with a necessary condition being the presence of off-diagonal disorder. We emphasize that with pure mass disorder, even though spectral weight transfer does occur, such a BP does not emerge. Additionally, we find that at very low soft particle concentrations, the states in the BP are completely localized, but some of the states crossover to being extended as their concentration is increased. The BP eventually ceases to be an anomaly, as the soft particle system becomes the host, and the hard particles assume the role of impurities. The interplay of mass and spring disorder is found to be quite subtle. The overall typical spectral weight decreases upon increasing either of the types of disorder, which indicates that there is a co-operative interplay. However, an added clause is that the interpretation of a co-operative or competitive interplay is also frequency selective, since different parts of the spectrum can transform from being localized to delocalized or vice-versa, depending on the protocol. With a crude modeling of vacancies, we suggest that tuning the concentration of vacancies to an optimal level, which is below but not close to the percolation limit, should be an optimal route to maximizing the figure of merit of thermoelectric materials. All the above results finally culminate in an attempt to understand experiments on Ni$_x$Pt$_{1-x}$ alloys, where, in agreement with ICPA results, we show that $x=0.65$ constitutes weak force-constant disorder, while $x=0.5$ represents a system with strong mass and force-constant disorder. Of course, such a conclusion is qualitative at best, because the real system has a non-trivial structure with multiple branches and spring-constant disorder, while the present implementation is restricted to a scalar approximation and force-constant disorder. The extension of the present framework to incorporate multiple branches, which will allow us to treat disordered phonons in real materials, is in progress. A portion of this research (T.B.) was conducted at the Center for Nanophase Materials Sciences, which is a Department of Energy (DOE) Office of Science User Facility. This material is also based upon work supported by the National Science Foundation under the NSF EPSCoR Cooperative Agreement No. EPS-1003897 with additional support from the Louisiana Board of Regents (M.J., W.R.M.). N.S.V and W.R.M acknowledge funding from JNCASR, India. M.J acknowledges support from DOE grant DE-SC0017861. References {#references .unnumbered} ========== [^1]: We use the term force-constant disorder to imply that the force constants have been randomly chosen and are fixed i.e. no attempt has been made to re-compute force-constants based on structural reorganization.
--- author: - 'Nicolas Martinet, Douglas Clowe, Florence Durret, Christophe Adami, Ana Acebrón, Lorena Hernandez-García, Isabel Márquez, Loic Guennou, Florian Sarron, Mel Ulmer' bibliography: - 'wl.bib' title: 'Weak lensing study of 16 DAFT/FADA clusters: substructures and filaments. [^1]' --- Introduction {#sec:intro} ============ In Cold Dark Matter (CDM) theories, our Universe can be represented as an ensemble of Large Scale Structures (LSS) made of voids and galaxy clusters that are connected through filamentary structures [@Bond+96]. In this scenario, matter collapses into halos that then grow through accretion and merging with other halos. Galaxy clusters are the highest density structures resulting from this hierarchical formation. N-body simulations [e.g. Millennium: @Springel+05] and low redshift observations [e.g. SDSS: @Tegmark+04] have confirmed this evolutionary scheme. In this framework, galaxy clusters can be used to constrain cosmological models. Indeed, the distribution of clusters with mass and redshift contains information on the mentioned hierarchical formation scenario [e.g. @Allen+11]. The main challenge is to calibrate the so-called observable-mass relation, that links true cluster masses to the mass proxy used in the survey. With its ability of being insensitive to the matter dynamical state, Weak Lensing (WL) appears as a major tool in determining the masses of galaxy clusters with sufficient precision to derive cosmological constraints. However, this technique requires a large amount of clusters, and therefore more and more WL surveys with increasing numbers of clusters are conducted [e.g. @Dahle+02; @Cypriano+04; @Clowe+06; @Gavazzi+07; @Hoekstra07; @Okabe+10; @vonderLinden+14; @Hoekstra+15]. In a similar idea, @Martinet+15b recently showed that counting shear peaks can constrain cosmological parameters almost as well as counting galaxy clusters, without requiring any knowledge of the observable-mass relation, but needing a large number of cosmological simulations. As it directly traces the matter density, WL also allows to study the LSSs of our Universe. However, the low density of filaments compared to clusters makes their detection difficult. Several studies pioneered in using WL to detect such structures in the vicinity of clusters either by reporting low significance detection or questioning previous claims of detection [e.g. @Clowe+98; @Kaiser+98; @Gray+02; @Gavazzi+04; @Dietrich+05; @Heymans+08; @Dietrich+12]. Note that @Massey+07 found evidence for a cosmic network of filaments in the COSMOS field galaxy survey. @Mead+10 used the Millennium Simulation [@Springel+05] to test the ability of various WL techniques to detect nearby cluster filaments, and concluded that background galaxy density is key to filament detection. Future space-based missions are likely to detect many filaments, but today, the narrow field of view of the Advanced Camera for Surveys (ACS) on the Hubble Space Telescope (HST) does not allow such detection in a simple way. In this context, deep ground-based imaging can be very efficient as it often has a much wider field of view, and offers the possibility to cover clusters and their vicinity in a single image with Subaru/Suprime-Cam or CFHT/Megacam. Recently, @Jauzac+12 reported the first WL detection of a $z=0.54$ cluster with a filament, MACSJ0717.5+3745 based on a mosaic of HST/ACS images. This detection was latter confirmed by @Medezinski+13 from a Subaru/Suprime-Cam WL analysis. In this paper, we present the WL analysis of 16 clusters from the Dark energy American French Team (DAFT, in French FADA) survey. All are medium-high redshift ($0.4 \leq z \leq 0.9$) massive (M$\geq 2\times 10^{14}$ M$_\odot$) clusters of galaxies selected through their X-ray luminosities. This sample is comparable to other X-ray selected cluster studies such as [LOCUSS]{} at $0.15 \leq z \leq 0.3$ [@Okabe+10], Weighting the Giants at $0.15 \leq z \leq 0.7$ [@vonderLinden+14], and [CCCP]{} at $0.15 \leq z \leq 0.55$ [@Hoekstra07; @Hoekstra+15], with a slightly higher redshift, but with fewer clusters than the mentioned surveys which respectively contain 30, 51, and 50 galaxy clusters. Apart from estimating cluster masses, we take advantage of the large field of view of our images (8 CFHT/Megacam images with 1 deg$^2$ f.o.v. and 7 Subaru/Suprime-Cam images with $34\times27$ arcmin$^2$ f.o.v. - one of the Subaru images contains two clusters) to investigate galaxy cluster environments. In particular, we report the WL detection of several [elongated structures that might correspond to filaments]{}. This paper is structured as follows. Sect. \[sec:data\] describes our data set, Sect. \[sec:wl\] presents in detail the shear measurement we apply, and Sect. \[sec:mass\] the mass reconstruction process. In Sect.\[sec:clusters\], we estimate the cluster masses and in Sect. \[sec:environment\] we focus on the environment of clusters: substructures, mergers, and filaments. We conclude in Sect. \[sec:ccl\]. Throughout the paper, we use a fiducial flat $\Lambda$CDM cosmology with $\Omega_M=0.3$, $\Omega_\Lambda=0.7$, and $H_0=70$ km Mpc$^{-1}$ s$^{-1}$. [All displayed distances are comoving.]{} Data {#sec:data} ==== DAFT/FADA {#subsec:daft} --------- DAFT/FADA is a survey of $\sim 90$ medium-high-redshift ($0.4 \leq z \leq 0.9$) massive (M$\geq 2\times 10^{14}$ M$_\odot$) clusters of galaxies selected through their X-ray luminosities. All of the clusters have Hubble Space Telescope (HST) imaging available with either WFPC2 or ACS cameras. We also gathered multi-band optical and near infrared ground based imaging, using 4m class telescopes for most of the sample. This data set allows to accurately measure the ellipticity of galaxies from space and their photometric redshifts (hereafter [photo-$z$]{}) from the ground. The main goals of the survey are to form a comprehensive database to study galaxy clusters and their evolution, and to test cosmological constraints geometrically by means of weak lensing tomography. Several steps have been made towards the achievement of these two goals, and the current status of the survey, with a list of refereed publications, can be found at <http://cesam.lam.fr/DAFT/project.php>. Among other papers, @Murphy+14 performed a WL analysis of HST/ACS mosaic imaging data of ten massive, high-redshift ($z > 0.5$) DAFT/FADA galaxy clusters. Using the [photo-$z$s ]{}calculated by @Guennou+10, they explored their use for background galaxy discrimination. Our team is currently increasing this small sample of HST/ACS shear measurements to a larger number of clusters and also aims at combining ground-based and space-based shear catalogs to build a shear analysis which is both deep in the cluster central region and extended on larger scales. This will serve as the reference catalog to perform Weak Lensing Tomography with Clusters (WLTC) as described in @Jain+03. This study {#subsec:study} ---------- Cluster RA DEC z Instrument Filters Seeing G14 M15 ------------------- ------------- -------------- -------- -------------------- --------- -------- ----- ----- XDCScmJ032903 03 29 02.81 +02 56 25.18 0.4122 CFHT/Megacam r+(v,i) 0.73” Y Y MACSJ0454.1-0300 04 54 10.92 -03 01 07.14 0.5377 CFHT/Megacam r+(v,z) 0.76” Y Y ABELL0851 09 42 56.64 +46 59 21.91 0.4069 CFHT/Megacam i+(v,z) 0.80” Y Y LCDCS0829 13 47 31.99 -11 45 42.01 0.4510 CFHT/Megacam r+(v,i) 0.83” Y Y MS1621.5+2640 16 23 35.50 +26 34 13.00 0.4260 CFHT/Megacam r+(v,i) 0.65” Y N OC02J1701+6412 17 01 22.60 +64 14 09.00 0.4530 CFHT/Megacam r+(i,v) 0.73” N N NEP0200 17 57 19.39 +66 31 31.00 0.6909 CFHT/Megacam i+(v,r) 0.97” N N RXJ2328.8+1453 23 28 49.90 +14 53 12.01 0.4970 CFHT/Megacam r+(v,i) 0.70” Y N CLJ0152.7-1357 01 52 40.99 -13 57 45.00 0.8310 Subaru/Suprime-Cam r+(v,z) 0.70” Y Y MACSJ0717.5+3745 07 17 33.79 +37 45 20.01 0.5458 Subaru/Suprime-Cam r+(v,z) 0.69” N N BMW-HRIJ122657 12 26 58.00 +33 32 54.09 0.8900 Subaru/Suprime-Cam r+(i,z) 0.80” Y Y MACSJ1423.8+2404 14 23 48.29 +24 04 46.99 0.5450 Subaru/Suprime-Cam i+(v,r) 0.88” Y Y MACSJ1621.4+3810 16 21 23.99 +38 10 01.99 0.4650 Subaru/Suprime-Cam i+(v,r) 0.62” N Y RXJ1716.4+6708 17 16 49.60 +67 08 30.01 0.8130 Subaru/Suprime-Cam r+(v,z) 0.63” Y/N N CXOSEXSIJ205617\* 20 56 17.16 -04 41 55.10 0.6002 Subaru/Suprime-Cam r+(v,i) 0.61” Y N MS2053.7-0449\* 20 56 22.37 -04 37 43.42 0.5830 Subaru/Suprime-Cam r+(v,i) 0.61” Y N \CXOSEXSI\_J205617 and MS\_2053.7-0449 are on the same image. \[tab:data\] In this study, we focus on 16 galaxy clusters for which we have Subaru/Suprime-Cam or CFHT/Megacam wide field images for at least three optical bands among the v, r, i, and z bands. Having three bands is mandatory to be able to perform a color-color cut to remove foreground galaxies that dilute the lensing signal. The shear measurements are performed in the r or i bands depending on the image seeing. This choice is made to maximize the number of source galaxies as these bands are the deepest optical bands. The use of Suprime-Cam (34$\times$27 arcmin$^2$ field) and Megacam (1$\times$1 deg$^2$ field) imaging allows to study clusters within their virial radius and also to see how they interplay with the surrounding LSS at the selected redshifts ($0.4 \leq z \leq 0.9$). These fields of view are much wider than what can be achieved from current space telescopes, as the HST/ACS field of view is only 3.4$\times$3.4 arcmin$^{2}$. Besides, the Megacam and Suprime-Cam cameras present rather stable Point Spread Functions (PSFs) and contain a large number of stars within each pointing allowing to accurately estimate the PSF distortion due to the instrument and atmospheric biases. A list of the data for each cluster can be found in Table \[tab:data\]. Some of the clusters from the present study have been analyzed in previous DAFT/FADA papers. @Guennou+14 derived X-ray luminosities and temperatures for 12 out of these 16 clusters. A comparison of WL and X-ray total masses will be performed in Sect. \[subsec:xraymass\]. @Guennou+14 also searched for substructures using both X-ray data and optical galaxy spectroscopy. @Martinet+15a studied the optical emission of galaxy clusters and measured the Galaxy Luminosity Functions (GLFs) for 7 out of these 16 clusters. We indicate in Table \[tab:data\] for each cluster in which study it was included. With the present DM study, we will have a full understanding of the matter content of a sample of galaxy clusters: the DM halo, the X-ray Intra Cluster Medium (ICM), and the stars contained in galaxies. Even if we do not include all the clusters in each analysis, we will have a general knowledge of cluster behaviors as observed through WL, X-rays, and optical. Image reduction {#subsec:source} --------------- The Subaru and CFHT data presented here are archive data, either from previous studies, or from the early phases of DAFT/FADA. The CFHT/Megacam data have been reduced by the [TERAPIX]{} team at the Institut d’Astrophysique de Paris, using the astromatic softwares (<http://www.astromatic.net/>). Sources are detected with [SExtractor]{} [@Bertin+96] and an astrometric solution is found using [SCAMP]{} [@Bertin06]. The stacking of the dithered exposures is then performed using [SWarp]{} [@Bertin+02]. We measure the seeing by fitting a Gaussian surface brightness profile to the bright stars of the image with [ PSFEx]{} [@Bertin11]. The images obtained with the Subaru telescope and Suprime-Cam were retrieved in raw form from the SMOKA archive (<http://smoka.nao.ac.jp/>), together with calibration files (bias and sky flat field exposures), except the images of MACSJ0717, that were taken from @Medezinski+13. They were reduced in the usual way, by subtracting an average bias and dividing by the normalized flat field in each filter exactly in the same way as the images we observed ourselves. The reduced images were then calibrated astrometrically using the [SCAMP]{} and [SWarp]{} tools, and combined for each filter. The photometric calibration was made in priority with SDSS catalogs when available in the field and in the corresponding band. If not available, we used the observed standard stars. Shear measurement {#sec:wl} ================= The main idea of lensing is to reconstruct the mass distribution of a foreground object, designated as the lens, through the deflection it induces on the background object light, namely galaxy sources. In the WL regime, the deflection is smaller than the typical intrinsic ellipticity of a galaxy (of the order of the percent), so that we must take the mean of many shear measurements from individual galaxies to reach a high signal-to-noise (S/N) detection of the shear. For a complete description of this phenomenon, check e.g., the review by @BS01. The main difficulty of the method is to take into account all the galaxy shape distortions that are not due to the shear signal, such as atmospheric variations and instrumental biases. To correct for these biases, we apply a KSB+ method, initially proposed by @KSB95 and later refined by @Luppino+97 [@Hoekstra+98]. The KSB method suits well shear measurements in cluster fields as assessed by the various large surveys choosing this technique [@Okabe+10; @vonderLinden+14; @Hoekstra+15]. In addition, it has been accurately tested on simulated images such as, e.g. the STEP2 simulations by @Massey+07b. Most of the WL reduction presented here is similar to the technique applied in @Clowe+12. We first detect objects using [SExtractor]{} and clean the catalog from spurious detections (Sect. \[subsec:detection\]). We separate stars from galaxies and measure the instrument Point Spread Function (PSF) variation on stars (Sect. \[subsec:PSFm\]) using the [IMCAT]{} software (@Kaiser11: <http://www.ifa.hawaii.edu/~kaiser/imcat/>) with some additional developments. We correct galaxy shapes for the PSF anisotropies to obtain an individual object shear catalog (Sect. \[subsec:PSFc\]). We then smooth the shear measurement noise (Sect. \[subsec:noise\]) and correct for the methodology biases by testing our reduction on the STEP2 [@Massey+07b] shear simulations (Sect. \[subsec:bias\]). Source detection {#subsec:detection} ---------------- We use [SExtractor]{} to detect objects and measure their photometry in our images. In most cases, the precise alignments of the three bands are sufficient to allow a detection in double image mode. We then perform the initial detection in the band used for shape measurements and detect objects in the same apertures and positions in the two other images. For some Subaru images, we did not manage to align precisely the images from all three bands. The detection is then performed separately in each band and measurements are associated to those in the band on which the ellipticity measurement is done. This cross correlation is done through a minimization of matched object distances with a 2 arcsec limit. We detect all objects which lie on at least three pixels above 1.5 times the sky background after convolving the surface brightness profile with a Gaussian kernel of $7\times7$ pixel size and 3 pixel FWHM. We use 32 deblending sub-thresholds with a deblending contrast close to zero in order to remove most of the possible blended objects that would have a modified shape. Object magnitudes are measured with the [MAG\_AUTO]{} keyword. We then compute the signal-to-noise ratio of each object using the [getsig]{} [IMCAT]{} tool. This command convolves the object surface brightness profile with a Gaussian filter of increasing smoothing radius $r_g$ and selects the value of $r_g$ that maximizes the signal-to-noise. We obtain at the same time the best signal-to-noise ratio for the object and an estimate of its size with the $r_g$ parameter. The local background is computed by fitting a mean sky level and a 2-d linear slope of the sky brightness in an annulus centered on the object, ignoring all the pixels within $3r_g$ of any object to avoid contamination. Once this accurate signal-to-noise is computed, we remove all objects with signal-to-noise lower than 10. We measure the 1st to 4th order of the surface brightness profile of each object in a circular aperture of size $3r_g$ using a Gaussian weighting with $\sigma=r_g$, through the [getshapes]{} [ IMCAT]{} command. We reject objects for which the first moment of the surface brightness profile does not coincide within one pixel, with the object peak position as detected by [SExtractor]{}. We adjust the position of the remaining objects to the first moment of the surface brightness profile which represents a sub-pixel estimate of the object peak position and re-measure the object shape centered on this new position. We then apply a series of cuts to remove likely spurious detections. We first remove all objects that have a smaller size than the instrument PSF, i.e. having a radius $r_g$ smaller than the minimum radius of stars, selected in a magnitude versus $r_g$ diagram. We also remove all objects located at less than 20 pixels from the image edges to avoid measuring truncated objects. Finally, we remove bad pixel detection and only keep objects that do not have any neighbor within 10 pixels of their center. This catalog is then separated between stars and galaxies in a half light radius $r_h$ versus magnitude plot, as shown in Fig. \[fig:starngal\]. Stars are selected as objects lying on the constant radius sequence and with appropriate magnitudes. This magnitude range is set by hand to avoid saturated stars and too faint objects. Galaxies are selected as all objects larger than the star sequence at the same magnitude excluding the saturated objects that can be seen in the bright part of the diagram. ![Half light radius $r_h$ (in pixels) versus i-band magnitude diagram for MACSJ1621. Red dots are catalog objects. The star selection is represented by the black polygon. The sequence of saturated stars on the left part is removed and all remaining objects above the star sequence are considered as galaxies. []{data-label="fig:starngal"}](plots/starngal_macsj1621-eps-converted-to.pdf){width="9.cm"} PSF measurement {#subsec:PSFm} --------------- The PSF of a given image represents the response of the instrument to a point like source in the conditions of observation. Its variations across the image are due to the instrument characteristics and to the weather conditions. CFHT/Megacam and Subaru/Suprime-Cam have rather stable PSFs suitable for WL. Having a good seeing also diminishes the PSF correction that we need to apply. As stars are point like sources, they are suitable for measuring the PSF of an image. The large field of view of our images enables us to have enough stars in a single frame to correct for the PSF anisotropies, on the contrary of smaller field of view cameras that often require to use stars across several images. A general image distortion can be expressed by the two following quantities: the smear polarizability tensor $\vec{P^{\rm sm}}$ that describes the object response to the PSF anisotropy, and the shear polarizability tensor $\vec{P^{\rm sh}}$ that describes its response to the shear. These two tensors are measured from the 0th, 2nd and 4th order moments of an object surface brightness distribution. We refer the reader to @KSB95 and @Hoekstra+98 for the expression of these tensors. The ellipticity $\vec{e}$, is estimated from the 2nd order moments of this distribution. In the next subsection we will use the following quantities, as measured on stars, to infer the true shape of galaxies: $\vec{P}^{\rm sm}_{\rm star}$, $\vec{P}^{\rm sh}_{\rm star}$, and $\vec{e}_{\rm star}$. Before measuring those quantities, we refine the star catalog to the cleanest objects. We first remove all objects that are closer than 40 pixels to any other object. We then fit the star ellipticities with a two dimension polynomial of the 6th order and generate modeled ellipticities at each object position using this polynomial. Objects that have a measured ellipticity differing by more than 0.05 from their modeled ellipticity are rejected. This step is repeated three times, and permits to remove galaxies that might have been considered as stars. We chose an ellipticity cut at 0.05 as we found that it removes objects that are mainly out of the whole sample ellipticity distribution. Finally, a visual inspection is carried out to remove all remaining objects that could still suffer from blending issues or being close to saturated stars. The final catalogs contain $\sim 1000$ and $\sim 3000$ stars in average for Subaru and Megacam images respectively, leading to an average star density of 1.0 arcmin$^{-2}$ and 0.8 arcmin$^{-2}$ respectively for Subaru and Megacam. Star shapes are measured using the [getshapes]{} [IMCAT]{} tool. As $\vec{P^{\rm sm}}$ and $\vec{P^{\rm sh}}$ depend on object sizes, we have to measure them for various sets of weighting radii. Hence, we compute a series of tensors for each $r_g$ between 1 and 10 pixels with a step of 0.5 pixels, so that we can use the tensors corresponding to the galaxy radius when correcting for the PSF. Final quantities are fitted by 6th-order 2D polynomials as a function of position in order to have continuous functions defined at every point of the image. Here we chose to measure the PSF over the entire image, using a high order polynomial fit. However, in the case of large field-of-view images, one could also divide the frame into several small patches, and fit the PSF in each tile with a lower order polynomial. While the second approach is used in various studies [@Okabe+10; @Umetsu+11], @vonderLinden+14 applied and validated the first approach in the case of Subaru/Suprime-Cam images. For the CFHT/Megacam data, while fitting the PSF on each chip, @Hoekstra07 found negligible discontinuities in the PSF anisotropy between chips. Following e.g. @Massey+05, we compute the auto-correlation function of star ellipticities before and after the PSF correction and the cross correlation function between galaxy shear and star ellipticities in Appendix \[appendix:psf\], validating our PSF correction. PSF correction {#subsec:PSFc} -------------- In the absence of noise the shear of a background galaxy ($\vec{g}_{\rm gal}$) can be computed from the following equation: $$\label{eq:gamma} \vec{g}_{\rm gal} = \left(\vec{P}^{\rm g}_{\rm gal}\right)^{-1}\vec{\delta e}_{\rm gal},$$ where $\vec{P}^{\rm g}_{\rm gal}$ is the shear susceptibility tensor defined in eq. \[eq:polar\], and $\vec{\delta e}_{\rm gal}$ the apparent change in ellipticity, described in eq. \[eq:ell\]. Note that in this equation we neglect the intrinsic ellipticity that should be subtracted to the apparent ellipticity change ($\vec{\delta e}_{\rm gal}$). This is true if a sufficient number of galaxies is taken into account: the galaxies being randomly oriented, the intrinsic ellipticity is null in average. The shear susceptibility tensor represents the PSF corrected distortion, i.e. only due to the shear. We define it as in @Luppino+97: $$\label{eq:polar} \vec{P}^{\rm g}_{\rm gal} = \vec{P}^{\rm sh}_{\rm gal} - \vec{P}^{\rm sh}_{\rm star}\left(\vec{P}^{\rm sm}_{\rm star}\right)^{-1} \vec{P}^{\rm sm}_{\rm gal},$$ where the $_{\rm gal}$ index is for tensors measured on galaxies, and $_{\rm star}$ for tensors measured on stars. The apparent change in ellipticity is: $$\label{eq:ell} \vec{\delta e}_{\rm gal} = \vec{e}_{\rm gal} - \vec{P}^{\rm sm}_{\rm gal} \left(\vec{P}^{\rm sm}_{\rm star}\right)^{-1} \vec{e}_{\rm star},$$ where represents the object ellipticity. In order to compute a galaxy shear, we then need to measure its ellipticity vector, and its smear polarizability and shear polarizability tensors. This is again done with the [getshapes]{} tool. We also generate the star quantities corresponding to each galaxy radius $r_g$ using the polynomials computed in the last section. Prior to measuring the shape of galaxies, we reject QSOs and cosmic rays by removing objects that lie away from the principal sequence in a maximum flux versus magnitude diagram. We also remove objects in regions where the sky level is too bright to avoid star diffraction halos. We restrict our catalogs to objects larger than 1.5 times the PSF size, defined as the minimum star radius $r_g$, deleting objects on which the PSF deconvolution could be too noisy. Finally, we visually inspect the images to remove any object close to saturated stars or reduction artifacts that could have survived our previous cleaning. Noise smoothing and co-addition {#subsec:noise} ------------------------------- The individual shear values are noisy due to the sky noise in the measurements of the higher order moments of the light distribution of objects. As these moments are subtracted one to each other when computing the shear polarizibility tensor, the final signal value is reduced while the noise increases. We then have to smooth the noise in the shear polarizibility tensor measurement to avoid it dominating the shear measurement, using its distribution across the image. We fit each component of the shear polarizability tensor $\vec{P}^{\rm g}_{\rm gal}$ as a function of one component of the ellipticity and of the object size $r_g$ by a 4th order two dimension polynomial. We chose a 4th order polynomial after testing several orders, as we found that it was minimizing the noise. Also, we find that the shear polarizability tensor weakly depends on the ellipticity but is more sensitive to the object size. We then use this modeled tensor to re-generate the shear values of each object following eq. \[eq:gamma\]. We note that this step removes the noise that would cause negative values of the shear polarizability tensor. We verify that after this fitting procedure, we do not have $\vec{P}^{\rm g}_{\rm gal}$ values lower than 0.1. Finally, we weight the individual shear values according to their significance compared to their neighbors in the ($r_g$,$S/N$) plane. In practice, this weight factor is set to the inverse of the root mean square of the shear of the 50 nearest neighbors for a region around each galaxy size and significance. Generally, the small, faint galaxies are given a low weight and larger, bright galaxies are given a high weight, due to the larger galaxies being affected only by the intrinsic shape noise while the smaller, fainter galaxies also have a significant noise component coming from sky noise in their shear measurements. In addition, sub-areas presenting a large shear dispersion will contribute less than sub-areas with a low shear dispersion. Bias calibration {#subsec:bias} ---------------- We measure the bias of our method on the STEP2 simulations [@Massey+07b] that provide images computed with various PSFs, and with an added constant shear across each image. We use the sets of images characterized by a Subaru PSF with a seeing of 0.8 arcsec (PSF C). This PSF suits well our data as about half of our images are from Subaru and our image seeing lies between $0.6<\epsilon<1.0~{\rm arcsec}$. However, note that the STEP2 images are $7\times7$ arcmin$^2$ size, while our images are of the order of $34\times27$ arcmin$^2$ for Suprime-Cam and $60\times60$ arcmin$^2$ for MegaCam. Hence, the PSF should be better sampled in the true images. Applying our reduction pipeline, we calculate the average shear of each of the 64 simulated galaxy fields and fit the difference between our shear estimate and the true shear as a function of the true shear, according to the notation of eq. \[eq:gammastep2\] from @Massey+07b: $$\label{eq:gammastep2} \gamma_i - \gamma_i^{\rm true} = m_i\times\gamma_i^{\rm true}+c_i,$$ where i is the index for both shear components. The values we have found for the multiplicative biases $m_1$ and $m_2$ and the additive biases $c_1$ and $c_2$ are shown in Table \[tab:stepresults\]. m c ------------ -------------------- ------------------- $\gamma_1$ -0.053 $\pm$ 0.021 0.004 $\pm$ 0.001 $\gamma_2$ -0.021 $\pm$ 0.030 0.001 $\pm$ 0.001 : Multiplicative (m) and additive (c) shear biases derived from applying our WL reduction pipeline to the STEP2 simulations with a Subaru PSF and a seeing of 0.8” (PSF C). See eq. \[eq:gammastep2\] and text for details. \[tab:stepresults\] Our results compare well with the ones from other methods as described in the STEP2 challenge [@Massey+07b]. As expected, the additive bias is rather negligible and the shear is slightly underestimated with the KSB method. The multiplicative bias can be seen as an evaluation of the quality of the shear measurement. Our results hence show that we can measure the galaxy shear with an accuracy better than $\sim5\%$. We correct each component of the shear for the multiplicative bias, and thus obtain our final shear catalog. Note that we do not correct for the additive bias which is strongly PSF dependent, and rather prefer to leave it as a potential systematic bias, small compared to the other sources of errors. Mass reconstruction {#sec:mass} =================== We then translate the measured shear signal to a mass estimate. We first apply the standard @Seitz+95 inversion technique based on the @KS93 algorithm to calculate a convergence density map (Sect. \[subsec:massmap\]). This technique allows to draw significance contour levels on the cluster image to search for structures but does not allow to recover the true masses of objects. Indeed, the integration of the shear over a finite space introduces a constant called the mass sheet degeneracy that cannot be properly taken into account without a magnification study. To avoid this problem, we fit NFW shear profiles on clusters to infer their 3D mass distribution in Sect. \[subsec:3Dmass\]. In any case, we first have to select galaxies that lie behind the structures we aim to detect, to avoid diluting the shear signal. This is done in Sect. \[subsec:back\], where we also estimate the mean background galaxy redshift, as this quantity is required to convert the shear and the convergence into mass. Background galaxies {#subsec:back} ------------------- ### Color cuts Foreground and cluster galaxies are not lensed by the cluster. Hence, they will appear as noise in the co-adding of individual shear measurements, and have to be deleted. The most accurate way to select background galaxies is to use spectroscopic redshifts, but it requires too much observational time. Photometric redshifts are more promising, as less time-consuming, and are starting to give accurate redshift estimations. However, we do not have spectroscopic or photometric redshifts for all galaxies and therefore we must consider galaxy colors. Galaxy colors are linked to the galaxy formation history and can be used as a crude approximation of the galaxy redshift. We select background galaxies in a color-color diagram, comparing our galaxy colors to those from galaxy templates computed at various redshifts. We generate templates for early and late type galaxies using [EzGal]{} [@Mancone+12] with @BC03 models, assuming a @Chabrier03 Initial Mass Function (IMF), a formation redshift of $z_{\rm form}=4$, and a solar metallicity. The red early type galaxies are modeled with a single starburst model and the blue late type galaxies by an exponentially decaying star formation model. We remove all galaxies that correspond to the color-color area covered by template galaxies at redshift $z<z_{\rm clus}+0.2$. For example, we show the color-color diagram of [RXJ1716]{} with the removed area in Fig. \[fig:cccp\]. Note that the colors we use vary from one cluster to another according to the available optical bands (see Table \[tab:data\]). We also cut all the remaining galaxies with magnitudes brighter than $i=22$ or $r=22.5$ (depending on the image on which the shear measurement is performed), as they are very likely foreground galaxies given the high redshift of our clusters. In the same manner, galaxies fainter than $i=25$ or $r=25.5$ are removed as they are fainter than the depth of our images, and therefore not reliable. ![(v-r) versus (r-i) color-color diagram for [RXJ1716]{}. Black dots represent galaxies from our catalog. Circles are late type galaxy templates and squares early types. [Green]{} is for templates at $\pm0.2$ around the cluster redshift, blue for lower-redshift galaxy templates and red for higher-redshift galaxy templates. The black polygon circling [green]{} and blue points correspond to the color area we remove from our catalog. See text for details on used galaxy templates.[]{data-label="fig:cccp"}](plots/good_rxj1716-eps-converted-to.pdf){width="9.cm"} ### Boost factor {#subsec:boost} To check that the color-color cuts removed cluster dwarf galaxies, we computed the number density of galaxies in our lensing catalog as a function of radius from the brightest cluster galaxy, correcting for loss of sky area due to the presence of bright galaxies and stars in each radial bin. Due to the magnification depletion effect [@Smail+95], the number density of background galaxies should either be flat or decrease with decreasing cluster centric distance, with the exact effect depending on the slope of the change in number counts with increasing magnitude for galaxies in and slightly fainter than the lensing catalog. In contrast, dwarf galaxies number density should increase with decreasing cluster centric radius, and thus any increase seen in the number density of the lensing catalog towards the cluster center is indicative that not all cluster galaxies were removed by the color cuts. The ratio of the number density of galaxies in the lensing catalog a given annular bin compared to the number density at large cluster radius can then be used as an estimate of the contamination fraction of cluster galaxies. Under the assumption that the cluster galaxies’ shapes are uncorrelated and should average to zero shear, this correction factor can then be used to boost the measured shear in the inner regions of the clusters to correct for the presence of cluster galaxies in the lensing catalog [@Clowe+01]. It should be noted that this is a conservative estimate of the fraction of cluster galaxies as we are assuming the underlying density of background galaxies is flat and not depleted towards the cluster center, however as the cosmic variance of the slope of the background galaxy number density with magnitude relation on arc minute sized patches can be quite large, estimates of the magnification depletion effect for individual clusters are too noisy to provide better constraints [@Schneider+00]. [We fit the radial profile of the normalized galaxy density with an exponential function of the form:]{} $$\label{eq:boost} 1+f(r) = 1+ A\times \exp(-r/r_0),$$ [where $A$ and $r_{0}$ are constrained by the fit. We then apply this function to boost shear values in the cluster vicinity. The weights are also modified according to the error on the fit to the density profile. We show in Fig. \[fig:boost\] the stacked normalized galaxy density along with the best fit for our boost factor. The error bars are computed from the dispersion over all clusters and show that the boost factor varies from one cluster to another, requiring individual fits. The galaxy density profiles are computed using the WL peak as the center. As a sanity check, we also computed the density profiles centered on the BCG and found no significant variation in the mass estimates of our clusters. Note that we neglected the effect of magnification when estimating the radial galaxy density profile, but @Okabe+15 showed that doing so only decreases the amplitude of the shear profile by $\sim10$% on scales lower than one tenth of the virial radius. Applying the corrections above as a function of radius from the cluster center results in the increase in the measured cluster masses. To be exact, the boost factor affects the concentration, and then the mass as we fixed the concentration parameter to break the mass concentration degeneracy (see Sect. \[subsec:3Dmass\]). The largest increase in mass is 30% (MACJ0717), while the mean increase is 9% and the median 6%.]{}\ ![[Stacked normalized galaxy density profile for all clusters. Error bars are the dispersion of values in the stack. Radius is in comoving distance and in [kpc]{} units. Individual profiles are centered on the WL peak. The red curve is the best exponential fit (see eq. \[eq:boost\]) to the data.]{}[]{data-label="fig:boost"}](plots/stacked_etprof2-eps-converted-to.pdf){width="9.cm"} ### Distance measurements Another issue is to measure the distances of the lens and of the background galaxies. These observables are required to estimate the mass of the lens, which depends on the ratio of the source to observer distance over the source to lens distance: $D_{s}/D_{ls}$. We estimate the lens distance through the spectroscopic redshift of the cluster. The classical way of estimating the mean background galaxy distance is to average the distance ratio $D_{s}/D_{ls}$ over all source galaxies. See also @Applegate+14 for a method that uses all galaxy background photometric redshifts in a Bayesian formalism. As we do not have photometric redshifts for background galaxies, we consider an external redshift distribution. We use the [COSMOS]{} data [@Ilbert+09] as our redshift distribution. These data are well suitable as they cover a large area of about 1.7 deg$^2$ after masking, down to a magnitude of $i=25$, and are adapted to our redshift range. Furthermore, the photometric redshifts of [ COSMOS]{} are computed with a high precision, using 30 bands from near-UV to mid-IR. We first apply the same magnitude and color cuts than those applied to our shear catalog. We then remove all galaxies that have a photometric redshift smaller than that of the cluster and calculate the mean of the ratio of the source to lens versus source distances $D_{ls}/D_{s}$, applying an appropriate weight. The weighting function is generated on the COSMOS galaxy sub-sample from a 2D polynomial fitted on the shear weighting function in our data in a half-light radius versus magnitude plane. We use the magnitude instead of the S/N ratio as the second coordinate because the S/N in COSMOS and in our data can vary significantly. Finally, the weights generated on COSMOS are re-normalized to 1. The mean redshift of background galaxies is then set to the one that allows to find the measured mean distance ratio $D_{ls}/D_{s}$. These redshifts can be found in Table \[tab:resclus\]. 2D mass map {#subsec:massmap} ----------- We reconstruct the projected convergence field by inverting the shear in Fourier-space, following @Seitz+95. This technique is an iterative application of the @KS93 algorithm to correct for the fact that we measure the reduced shear, which is equal to the shear $\gamma$ divided by $1-\kappa$, and not the shear. We reconstruct the first convergence map assuming $\kappa=0$ in the shear, and then generate a map from the shear where the convergence is set to the previous map in the loop until the process converges. We find that the convergence map remains constant within 0.01% after three realizations. This technique allows to better estimate the mass map around high masses and is therefore particularly suitable for our cluster mass reconstruction. The convergence field is smoothed with a Gaussian filter of width $\theta_s=1$ arcmin at each step of the algorithm, before reading off which convergence to use to correct for a given galaxy. The noise level in the final convergence map can be estimated as eq. \[eq:sigkappa\] [@vanWaerbeke00]: $$\label{eq:sigkappa} \sigma_{\kappa}=\frac{\sigma_{\epsilon}}{\sqrt{4 \pi n_{\rm bg} \theta_s^2}},$$ where $n_{\rm bg}$ is the density of background galaxies and $\sigma_{\epsilon}$ the dispersion of the ellipticities of the background galaxies. $n_{\rm bg}$ and $\sigma_{\epsilon}$ are estimated independently for each image, taking into account the weight function of the shear. $\sigma_{\epsilon}$ ranges from 0.27 to 0.32 across our data, while $n_{\rm bg}$ can be found in Table \[tab:resclus\] for each cluster. One can then convert the convergence map into a surface mass density map using the definition of the convergence (eq. \[eq:kappa\]): $$\label{eq:kappa} \kappa=\frac{\Sigma}{\Sigma_{\rm crit}},$$ where $\Sigma$ is the surface mass density and $\Sigma_{\rm crit}$ the critical surface mass density defined in eq. \[eq:sig\]: $$\label{eq:sig} \Sigma_{\rm crit}=\frac{c^2}{4\pi G}\frac{D_{\rm s}}{D_{\rm l}D_{\rm ls}}.$$ c is the speed of light, G the gravitational constant, and $D_{\rm s}$, $D_{\rm l}$, and $D_{\rm ls}$ are respectively the distance to the source, the distance to the lens, and the distance between the source and the lens. This conversion hence only requires the knowledge of the lens and source redshifts, calculated in Sect. \[subsec:back\]. As we cannot properly account for the mass sheet degeneracy in our reconstruction, we did not try to estimate the mass of clusters through the convergence map. These mass maps are thus only used to detect clusters and their surrounding structures, while the cluster masses are estimated in the next section fitting an NFW profile to the shear. ![image](plots/macsj1621_gros_rot-eps-converted-to.pdf){width="9.cm"} ![image](plots/macsj1621_petit_rot-eps-converted-to.pdf){width="9.cm"} The significance of the detection is computed from a noise re-sampling technique, adding a random ellipticity to every galaxy for each realization. To preserve the shape noise properties of the sample, we draw the added ellipticities from the image galaxy catalog. Doing so, we neglect the additional shear signal as it is very unlikely that it correlates with the detected structures given the large number of galaxies in our catalogs. The shape noise used in eq. \[eq:sigkappa\] is increased by a factor of $\sqrt{2}$ as the ellipticity of galaxies now corresponds to the sum of two Gaussian distributions with a null mean and a width $\sigma_{\epsilon}$. We perform a hundred realizations for each catalog, computing the detection level of every structure at each step. The mean and dispersion of these detection levels give a strong estimate of the significance of the detection. We also measure the number of realizations in which the structure is detected at more than 3$\sigma$ above the map noise. For example we can be very confident in a structure detected at more than 3$\sigma$ in 95% of the realizations. In addition, this noise re-sampling allows to refine the measure of the position of each structure, computing the mean and dispersion of the local maximum position over all noise realizations. These quantities respectively correspond to an estimate of the structure center and to the error on its position. For example, we show in Fig. \[fig:massmapex\] the 3-band-color image with the convergence contours overlaid for MACSJ1621. The contours are spaced in units of the map noise computed from eq. \[eq:sigkappa\], starting at 3$\sigma$. We display the same figure for every cluster with X-ray emissivity and galaxy light density contours when available in Sect. \[sec:environment\]. As a sanity check, we computed the mass map with shear rotated by 45 degrees (white contours) and found that the signal due to the cluster presence disappears in this map, validating our convergence map reconstruction method. The position of the WL peaks are noted by white crosses with a 1 for the cluster and a 2 for the main secondary structure. The cluster is detected at $(6.8\pm1.4)\sigma_{\kappa}$ in the center region and an elongated structure aligned with the cluster major axis can be seen at a $(5.9\pm1.7)\sigma_{\kappa}$ confidence level computed from the mean and dispersion of a hundred realizations of the noise. These two structures are detected in respectively 97 and 96 % of the realizations. The nature of the secondary peak is discussed in Sect. \[sec:environment\] comparing the WL with other probes (X-ray and optics). The center positions are estimated with a precision of about 200 kpc. Also, we note an offset between the Brightest Cluster Galaxy (BCG) marked by a yellow cross and the WL peak. [This offset is discussed in Sect. \[subsec:bcg\], where we estimate cluster masses at both positions.]{} In spite of all our care to build accurate mass maps, some peaks will arise from the noise. One must evaluate the number of these fake peaks in order to discuss the detection of structures in the mass maps. As the number of fake peaks depends both on the density of background galaxies and on the redshifts of the lens and sources, we compute the fake peak probability for each cluster field. To do so, we assign a random position to each galaxy in the frame, to make sure that no structure from the original positions would be left in the simulation. We then use this new ellipticity catalog as an input to our mass map pipeline. The resulting convergence map should be representative of the noise. However, the presence of the cluster also modifies the distribution of fake peaks. To take this into account, we add to the ellipticity of each galaxy, shear values based on the fitted NFW profile of the corresponding cluster (see Sect. \[subsec:3Dmass\]). We find slightly fewer peaks when adding the cluster. This is due to the fact that some noise peaks can be aligned with the cluster, and also because the presence of the cluster is compensated by negative convergence values in the mass map as the mean convergence in the reconstruction is set to zero. We do a hundred realizations to capture the statistical properties of the fake peaks. For MACSJ0717, we also performed 10,000 realizations to check that our 100 realizations are sufficient. We find little difference between the two cases. Quantitatively, we find 11.1 peaks above 3$\sigma_\kappa$ and 1.3 above 4$\sigma_\kappa$ in the entire Suprime-Cam field for 100 realizations, and 10.9 and 1.2 above 3$\sigma_\kappa$ and 4$\sigma_\kappa$ for 10,000 realizations. In any case we find less than 0.1 fake peaks above 5$\sigma_\kappa$. When discussing the detection of structures in Sect. \[sec:environment\], we give the expected number of fake peaks in the displayed area for each cluster. We note that in Fig. \[fig:massmapex\], the white contours corresponding to the reconstruction of the orthonormal shear component, are in good agreement with the expected number of fake peaks for the displayed field (2.9 above 3$\sigma_\kappa$ and 0.4 above 4$\sigma_\kappa$ in the left-hand field). Cluster mass fit {#subsec:3Dmass} ---------------- To infer the cluster mass distribution, we choose to fit the shear profile centered on the cluster. This avoids having to measure the shear in the cluster core, and partially breaks the mass sheet degeneracy by imposing a given mass profile on the data. We note that using this radial technique on N-body simulated clusters, @Becker+11 [@Bahe+12] found a systematic underestimate of cluster masses of roughly 5%, which we do not correct for as the exact correction factor is likely to be a function of the chosen cosmologcial paramaters (and is small compared to the uncertainties for all of our clusters). The NFW density profile [@NFW] defined in eq. \[eq:nfw\] is among the best available profiles to fit observed galaxy clusters [e.g. @Umetsu+11]. $$\label{eq:nfw} \rho_{\rm NFW}(r)=\frac{\rho_{\rm s}}{\frac{r}{r_{\rm s}}(1+\frac{r}{r_{\rm s}})^2}$$ where $r_{\rm s}$ is the scale radius and $\rho_{\rm s}$ a density expressed as $\rho_{\rm crit} \delta_{\rm c}$. $\rho_{\rm crit} = 3H^2/8\pi G$ is the critical density of the Universe at the cluster redshift, and $\delta_{\rm c}$ is a dimensionless density that depends on the DM halo, and that can be expressed as a function of the concentration parameter: $$\delta_{\rm c} = \frac{\Delta}{3}\frac{c_\Delta^3}{\ln(1+c_\Delta)-\frac{c_\Delta}{1+c_\Delta}},$$ where $\Delta$ is the overdensity compared to the critical density, $c_\Delta = r_\Delta / r_s$ is the concentration parameter. By integration of the density under spherical symmetry, the mass $M_{\rm NFW,\Delta}$ in a given radius $r_\Delta$, can be estimated as a function of $r_\Delta$ and $c_\Delta$ only: $$\label{eq:nfwmass} M_{\rm NFW,\Delta}=\frac{4\pi\rho_{\rm s}r_{\rm \Delta}^3}{c_{\rm \Delta}^3}[\ln(1+c_{\rm \Delta})-\frac{c_{\rm \Delta}}{1+c_{\rm \Delta}}].$$ The radial shear profile has an analytic formula derived in e.g. @Wright+00, that we fit to the measured shear to obtain $r_\Delta$ and $c_\Delta$ which are converted into a cluster mass according to eq. \[eq:nfwmass\]. [There is a known degeneracy between the concentration $c_\Delta$ and the mass $M_\Delta$ [e.g. @Diemer+14; @Meneghetti+14], or equivalently $r_\Delta$ in our case. We show in Fig. \[fig:dege\] the degeneracy between both parameters of our NFW fit for two clusters representative of the large (MACSJ0717) and low (NEP200) significance detections. These plots highlight the need to break the degeneracy between the two parameters especially in the low significance case.]{} [This can be achieved using predictions of the typical concentration of clusters from cosmological N-body simulations, and one can either choose a mean concentration for all clusters in the sample [e.g. @Applegate+14] or use a mass-redshift-concentration relation [e.g. @Hoekstra+15]]{}. To break the degeneracy between $r_\Delta$ and $c_\Delta$, we fix the concentration parameter to $c_{200}=3.5$, since @Gao+08 demonstrated that very massive clusters have concentration parameters between 3 and 4 at the studied redshifts. This choice of a fixed concentration parameter imposes a systematic error on each individual cluster mass although the average should be correct. We quantify the error on the mass measurement due to the intrinsic scatter of 1.34 on the concentration parameter estimate in @Gao+08 by fixing the concentration parameter to 2.16 and 4.84, which represent the scatter around our chosen value of $c_{200}=3.5$. We find a variation of the mass of about $\pm 25\%$. This error is not added to the error budget of Table \[tab:resclus\]. [As a result of our choice of breaking the mass-concentration degeneracy by fixing the concentration parameter, any concentration effect, such as the boost factor (see Sect. \[subsec:boost\]) or the off-centering effect (see Sect. \[subsec:bcg\]), directly affects the mass estimate.]{} ![image](plots/macsj0717_c200r200-eps-converted-to.pdf){width="7.cm"} ![image](plots/nep200c200r200-eps-converted-to.pdf){width="7.cm"} The fit is done in an annulus where the inner radius is iteratively set to a value larger than the Einstein radius, to remove the area affected by strong lensing. We also require to have a minimum number of objects in every bin, which can push the inner radius to large physical values in the case of high redshift clusters. The outer radius is set to the value at which the output $r_\Delta$ does not significantly change (less than 1%) if we probe a larger area. We also ensure that the outer radius is at least larger than the output $r_\Delta$. The fit is performed on the tangential shear computed to the cluster center, which is defined as the highest peak close to the cluster position in the convergence map reconstruction. [We discuss the possibility of using the BCG instead of the WL peak as the center in Sect. \[subsec:bcg\], but we shall mainly discuss masses centered on the WL peak in the following. Cluster masses are shown in Table \[tab:resclus\].]{} An estimate of the significance of the fit is obtained by computing the $\Delta \chi^2$ between the best fit NFW model and a zero mass model. The tangential shear profiles for every cluster can be found in Appendix \[appendix:shearprof\], where the error bars correspond to the orthonormal shear that should be equal to zero in the absence of noise. We measure $r_{200}$ from the best NFW fit and then compute $M_{200}$, and $M_{500}$. We note that for clusters where the NFW fit has a low significance value ($\sigma<3$), the tangential shear profile presents error bars consistent with no signal. We then do not compute a mass for these clusters, as their shear profile is not reliable. The errors are computed using the same noise re-sampling method than for the mass maps (see Sect. \[subsec:massmap\]). A random ellipticity is drawn from our catalog and added to each galaxy. Then, the best NFW fit gives a new value for $r_{200}$ and $M_{200}$. The mean and the dispersion over a hundred noise realizations are used as the true value and its error. The $r_{200}$ and various mass values are given in Table \[tab:resclus\] of Sect. \[sec:clusters\]. Galaxy clusters {#sec:clusters} =============== In this section we present the results concerning the 16 galaxy clusters that we have studied. The discussion is based on the masses obtained from the NFW fits presented in Sect. \[subsec:3Dmass\] and given in Table \[tab:resclus\]. After discussing the WL masses (Sect. \[subsec:wlmass\]), we compare them to the X-ray values from the literature (Sect. \[subsec:xraymass\]), [and then analyze the effect of using the BCG as the cluster center instead of the WL center (Sect. \[subsec:bcg\]).]{} The comparison of individual cluster masses with other studies is done jointly with the environment discussion in the next section (Sect. \[sec:environment\]). ---------------- -------- -------------------- ----------------------- --------------------- ------------------------------ -------------------------------- -------------------------------- -------------------------------- Cluster z $\bar{z}_{\rm bg}$ $n_{\rm bg}$ $r^{NFW}_{200}$ $\sigma_{NFW}$/$\sigma_{2D}$ $M^{NFW}_{200}$ $M^{NFW}_{500}$ $M^{X}_{500}$ $({\rm arcmin}^{-2})$ (kpc.h$_{70}^{-1}$) $(10^{14}M_\odot.h_{70}^{-1})$ $(10^{14}M_\odot.h_{70}^{-1})$ $(10^{14}M_\odot.h_{70}^{-1})$ XDCS0329 0.4122 0.90 10.20 - 1.2/2.8 - - 2.9 $\pm$ 0.6 MACSJ0454 0.5377 0.99 9.96 - 1.9/5.1 - - 13.9 $\pm$ 3.0 ABELL0851 0.4069 0.92 8.30 1542 $\pm$ 160 3.9/7.6 6.6 $\pm$ 2.0 4.4 $\pm$ 1.4 5.5 $\pm$ 1.2 LCDCS0829 0.4510 0.93 8.79 1638 $\pm$ 218 3.8/5.5 8.5 $\pm$ 3.2 5.7 $\pm$ 2.1 16.9 $\pm$ 3.6 MS1621 0.4260 0.93 14.13 1718 $\pm$ 140 6.4/8.3 9.2 $\pm$ 2.2 6.2 $\pm$ 1.5 4.5$\pm$0.5$^{M12}$ OC02 0.4530 0.96 13.15 1202 $\pm$ 187 3.1/4.7 3.4 $\pm$ 1.5 2.3 $\pm$ 1.0 - NEP200 0.6909 1.02 5.80 1929 $\pm$ 306 3.3/5.1 18.9 $\pm$ 8.2 12.7 $\pm$ 5.5 - RXJ2328 0.4970 0.95 11.46 1393 $\pm$ 159 3.2/5.5 5.5 $\pm$ 1.9 3.7 $\pm$ 1.2 2.2 $\pm$ 0.5 CLJ0152 0.8310 1.19 14.94 1670 $\pm$ 194 3.8/8.3 14.0 $\pm$ 4.6 9.4 $\pm$ 3.1 8.8 $\pm$ 1.9 MACSJ0717 0.5458 0.98 13.16 2236 $\pm$ 206 5.2/10.9 23.6 $\pm$ 6.4 15.9 $\pm$ 4.3 17.8$\pm$1.7$^{M12}$ BMW1226 0.8900 1.43 10.12 - 0.2/- - - 12.1 $\pm$ 0.4 MACSJ1423 0.5450 0.93 8.98 1594 $\pm$ 214 3.4/5.0 8.8 $\pm$ 3.3 5.9 $\pm$ 2.2 5.7 $\pm$ 1.2 MACSJ1621 0.4650 0.94 16.39 1379 $\pm$ 185 4.2/6.8 5.2 $\pm$ 1.9 3.5 $\pm$ 1.3 4.3$\pm$0.4$^{M12}$ RXJ1716 0.8130 1.17 7.49 1685 $\pm$ 194 3.9/7.3 14.1 $\pm$ 4.7 9.5 $\pm$ 3.2 2.8$\pm$0.5$^{M12}$ MS2053\ 0.5830 0.98 14.44 1620 $\pm$ 195 4.6/8.7 9.5 $\pm$ 3.3 6.4 $\pm$ 2.2 4.9 $\pm$ 1.1 CXOSEXSI2056\* 0.6002 0.98 14.44 - 0.7/4.4 - - 3.6 $\pm$ 0.8 ---------------- -------- -------------------- ----------------------- --------------------- ------------------------------ -------------------------------- -------------------------------- -------------------------------- \* CXOSEXSI\_J205617 and MS\_2053.7-0449 are on the same image. \[tab:resclus\] WL Masses {#subsec:wlmass} --------- The results of the best NFW fit are given only when its significance is higher than $3\sigma$, because otherwise such masses would not be reliable. This means that we were not able to constrain the masses of all clusters (see Table \[tab:resclus\] and shear profiles in Appendix \[appendix:shearprof\]). The fact that some of our fits do not converge can have several explanations depending on each case. One obvious limitation is the background galaxy density: as the noise is proportional to the inverse square root of the background density, the deeper the observations, the higher the signal-to-noise of the shear. The data obtained with Subaru, which is an 8m class telescope, are less affected than those obtained with the CFHT, which is only a 4m class telescope. The masses of the clusters and the noise in the images are also important factors. A high mass cluster will tend to be detected even with a low background galaxy density. Finally, we note that the redshift of the cluster also plays a role. For example, BMW-HRI J122657 is a rather massive cluster, but at a redshift of $z=0.89$. As the lensing effect is measured on the galaxies behind the cluster, the higher the redshift, the more difficult it is to detect the cluster. A redshift of $z\sim0.9$ is close to the accessible limit, as lensing is most sensitive to structures at redshifts around $z\sim0.3-0.4$. [We present the individual shear profiles in Appendix \[appendix:shearprof\]. In Fig. \[fig:stackedetprof\] we show a stacked shear profile including all 12 clusters for which it was possible to compute a mass. [The black dots correpond to the stacking of all individual cluster shear profiles, the error bars being the dispersion of each shear bin values. In addition we also coadd the shear catalogs recentered to the WL peak and compute a global shear profile, using the mean redshift of the clusters to convert into comoving distance (blue points). In this case the error bars correspond to the rotated shear as for the individual profiles. Both methods agree very well. In the second case the error bars are smaller because we get more galaxies per radial bin, but it does not take into account the dispersion in the shears. In our study, we have enough signal-to-noise in each cluster to also do the stacking of the individual shear profiles.]{} Though the error bars are still large given that we have only a small number of clusters, most of the noisy or asymmetrical irregularities have been washed out, and the stacked shear profile is well represented by an NFW spherical profile.]{} ![[Stacked shear profile for the 12 clusters for which we were able to safely measure the mass. Black points correspond to the stacked profiles and blue points to the profile of the stacked shear catalogs (see text for details). In the first case error bars are the dispersion of values in the stack, and in the second the rotated shear. Radius is in comoving distance and in [kpc]{} units. Individual profiles are centered on the WL peak. The red curve is the best NFW fit to the stacked profile.]{}[]{data-label="fig:stackedetprof"}](plots/stacked_et_prof-eps-converted-to.pdf){width="9.cm"} For the clusters for which we were able to compute masses, we find error bars typical of WL studies. We note however, that using the noise re-sampling method to determine the mass increases our errors over using only the significance of the best NFW fit. We choose to show the former errors because they are more robust and more conservative. We do not statistically compare our masses with other WL studies because we have only few clusters in common. Three of our clusters are studied in the @Mahdavi+13 sample, three in the CCCP sample [@Hoekstra+15], three in the Weighting the Giants sample [@Applegate+14], two in the @Foex+12 sample, one is studied in @Jauzac+12 and @Medezinski+13, and one in @Israel+14. Nonetheless, a comparison of the WL masses, and also with the X-ray and strong lensing estimates, is done for each cluster in Sect. \[subsec:sbs\]. In the next subsection, we compare our WL masses with those derived from X-rays to evaluate potential biases in both measurements. X-ray and WL masses {#subsec:xraymass} ------------------- The X-ray masses come from two different samples. Most of them have XMM–Newton data and are taken from @Guennou+14. We add four clusters that have Chandra data and belong to the @Maughan+12 sample. MACSJ1423 has Chandra data but is also part of @Guennou+14. The masses from @Guennou+14 are obtained by applying the @Kravtsov+06 scaling relation to the X-ray derived temperature of the clusters. The error bars have been recomputed taking the scatter of this scaling relation into account, since they were too optimistic in @Guennou+14. The masses from Chandra observations have been computed in @Lagana+13 using both the temperatures and surface brightness profiles (see eq. 5 of the mentioned paper). We compare in Fig. \[fig:xvswl\] the cluster masses inferred from X-ray data and from WL, all computed in $r_{500}$, for the ten clusters that have both data. We see that the points are fairly distributed around the line of equality. Computing the [lognormal]{} mean ratio of the WL to X-ray masses, we find that WL masses are [ 8%]{} higher than the X-ray masses in the mean. Finding an offset is quite normal, as the X-ray masses rely on the assumption that clusters are relaxed, which is generally not the case. Weak lensing, on the other part, does not need such an assumption, and WL masses are usually more reliable. An underestimate of about 10 to 40% in the X-ray derived total cluster masses is commonly observed [@Rasia+06; @Nagai+07; @Battaglia+13]. We also note a departure from this relation for LCDCS0829, for which we cannot reproduce the high X-ray mass, and for RXJ1716 which has a very low mass in X-rays compared to its WL mass. In the first case we note that LCDCS0829 is highly asymmetrical as seen from its mass map in Fig. \[fig:massmaplcdcs0829\] (Sect. \[sec:environment\]). Hence, the hypothesis of spherical symmetry that we made for our NFW fit might explain why we find a low mass for this cluster. In general one can expect WL masses to be very accurate for individual clusters, but only for a large sample of clusters. ![X-ray versus WL masses. The red dashed line is the first bisector and represents the sequence on which X-ray and WL masses would be equal. All values can be found in Table \[tab:resclus\].[]{data-label="fig:xvswl"}](plots/XvsWL-eps-converted-to.pdf){width="9.cm"} BCG and WL offset {#subsec:bcg} ----------------- [In this section, we discuss the difference in the mass estimate when centering on the BCG instead of the WL peak. Note that we chose the latter center and apart from this section our WL masses discussed in this paper are computed centered on the WL peak.]{} [First, using our simulated clusters (see Sect. \[subsec:massmap\]) for different realizations of the noise, we measure the offset between the true input center and the highest WL peak. We find a mean offset of 0.32 arcmin with a scatter of 0.20 arcmin. We use angular distances here because the noise comes from the background galaxies. We can then say that using the BCG as the center of mass of the cluster is a good approximation only if the offset of the BCG and the WL peak is lower than 0.52 arcmin (one sigma above the mean offset due to the noise). For each realization, we also compute the mass centered on the true input center and on the highest WL peak. We find that centering on the WL peak systematically overestimates masses by about 8% in the mean with a scatter of 9%.]{} [For clusters which have a well identified BCG, we then compute the WL masses centered on the BCG in our data. The resulting masses are shown in Table \[tab:resclusbcg\]. We also plot one mass estimate against the other in Fig. \[fig:bcgvswl\]. Table \[tab:resclusbcg\] displays the offset between the BCG and the WL peak which can be high for some clusters. The mean angular distance between the WL and BCG centers is 0.67 arcmin, and ranges from 0.29 arcmin to 1.20 arcmin. Note that we also display the BCG offset in comoving distance in Table \[tab:resclusbcg\] to allow a comparison with the shear profiles that are computed within comoving radii. The mean offset between the BCG and DM centers in comoving distance is [246 h$_{70}^{-1}$.kpc, which is about 100 kpc higher than what is observed at lower redshift [e.g. @Oguri+10]]{}, and highlights the fact that our clusters are mostly not relaxed and have probably suffered from a complex merging history. According to our simulations, we can distinguish between two populations of clusters. Those with a BCG offset lower than 0.52 arcmin and those with a larger offset. For the first category, the BCG offset is compatible with the noise offset. Thus the BCG center assumption is valid and the masses centered on the BCG and the WL peak should agree. In the second case, the BCG is likely not the center of mass of the cluster, and masses centered on the BCG and on the WL peak will significantly disagree. [Additionally, we verify that clusters with small BCG offsets are indeed not ongoing mergers, looking at their convergence map. Only NEP200 presents signs of an ongoing merger with two peaks in the WL reconstruction, and is then counted in the merger category. We also note that the BCG offset for this cluster is very close to the acceptable limit.]{} These expectations are well met in Fig. \[fig:bcgvswl\], where we isolated the two types of clusters. When identifying clusters that have their BCG and WL peaks closer than 0.52 arcmin (red dots) we find that masses with the different centers agree well within the error bars. Note however that the error bars are not independent for the two measurements as the shear at large radius will be largely the same. The WL masses are still slightly higher when centered on the WL peak because centering on the WL peak maximizes the positive contribution of noise to the mass. Hence, choosing the center the way we did tends to overestimate the mass in relaxed clusters compared to centering on the BCG. The masses are lower by about 20 small BCG offsets, which is within the error bars of our simulations. However, the mass difference is significantly larger for unrelaxed clusters (black dots) and can be up to 60 the case of significant mergers (A851).]{} [For about half of our sample, the BCG centering assumption would then be correct here. However many of the clusters in this sample have significant merging activity and therefore the BCG is likely not the center of mass of the cluster currently. In addition, there are several clusters for which it is not possible to identify the BCG, and using a different center definition for these clusters would bias the mass estimate in our sample. Therefore we believe that our mass measurements are systematically high, but centering on the BCG would create masses that are systematically low, and that would not be reliable in the case of mergers, which a large fraction of our clusters are. A possibility would be to use the BCG center when this assumption is valid and the WL peak in the case of mergers, but we prefer to use the same center (WL peak) for the whole sample to be able to compare masses computed in the same way.]{} ----------- -------------------------- --------------------- -------------------------------- -------------------------------- Cluster $d^{\rm com}_{\|WL-BCG\|}$ $\theta_{\|WL-BCG\|}$ $M^{NFW}_{200}$ $M^{NFW,BCG}_{200}$ (kpc.h$_{70}^{-1}$) (arcmin) $(10^{14}M_\odot.h_{70}^{-1})$ $(10^{14}M_\odot.h_{70}^{-1})$ ABELL0851 384 1.18 6.6 $\pm$ 2.0 2.5$\pm$1.6 LCDCS0829 178 0.51 8.5 $\pm$ 3.2 6.5$\pm$2.9 MS1621 338 1.01 9.2 $\pm$ 2.2 5.6$\pm$1.8 OC02 74 0.21 3.4 $\pm$ 1.5 2.8$\pm$1.4 NEP200 209 0.49 18.9 $\pm$ 8.2 8.4$\pm$5.2 RXJ2328 343 0.94 5.5 $\pm$ 1.9 3.2$\pm$1.8 CLJ0152 339 0.74 14.0 $\pm$ 4.6 9.5$\pm$4.3 MACSJ1423 146 0.38 8.8 $\pm$ 3.3 7.2$\pm$3.3 MACSJ1621 422 1.20 5.2 $\pm$ 1.9 1.7$\pm$1.1 RXJ1716 190 0.42 14.1 $\pm$ 4.7 12.9$\pm$4.5 MS2053 87 0.29 9.5 $\pm$ 3.3 8.6$\pm$2.9 ----------- -------------------------- --------------------- -------------------------------- -------------------------------- : [Comparison of masses centered on the WL peak and on the BCG for 11 clusters. The first six clusters are observed with CFHT/Megacam and the last five with Subaru/Suprime-cam. The different columns correspond to \#1: cluster ID, \#2: $d^{\rm com}_{\|WL-BCG\|}$ comoving distance between the WL peak and the BCG in kpc, \#3: $\theta_{\|WL-BCG\|}$ angular distance between the WL peak and the BCG in arcmin, \#4: $M^{NFW}_{200}$ from the best NFW fit centered on the WL peak, \#5: $M^{NFW,BCG}_{200}$ from the best NFW fit centered on the BCG.]{} \[tab:resclusbcg\] ![[WL masses centered on the BCG versus WL masses centered on the WL peak. Red dots correspond to clusters for which the WL peak is closer than 0.52 arcmin from the BCG, and black dots for those with higher position offsets. The red dashed line is the first bisector and represents the sequence on which both masses would be equal. The different values can be found in Table \[tab:resclusbcg\]. [Note that NEP200 lies in the large offset category, even if its offset is slightly lower than 0.52 arcmin, because of its mass map reconstruction (see text for details).]{}]{}[]{data-label="fig:bcgvswl"}](plots/BCGvsWL-eps-converted-to.pdf){width="9.cm"} Environment {#sec:environment} =========== In this section, we use the 2D mass maps computed in Sect. \[subsec:massmap\] to discuss the structures detected in the vicinity of clusters. To have a full understanding of the different mass components we overplot on the images the WL contours at a 3$\sigma$ significance as well as the X-ray contours and the galaxy light distribution contours. To secure the WL detection of each structure we compute its significance level with respect to the map noise for a hundred realizations of the noise. We also count the percentage of simulations in which the structure is detected at more than 3$\sigma$ above the background. The last two quantities contain similar information, and are given in Table \[tab:resenv\]. The significance levels in this table are computed from the hundred realizations of the noise and can slightly differ from the contour levels shown in Figs. \[fig:massmapxdcs\] to \[fig:massmapms2053\] which correspond to the original mass maps. We also compute the number and significance of peaks expected to be due to the noise in the map reconstruction. This enables us to discuss the presence of WL peaks which do not show any optical or X-ray counterpart. We also note that in the case of the optical contours, we tried to select only cluster member galaxies, while the WL is sensitive to any line-of-sight structure, with a higher efficiency for structures at redshift around $z\sim0.3-0.4$. As a result, it is not surprising to find some peaks in the convergence map with no optical counterpart. The X-ray contours are plotted from XMM–Newton EPIC MOS1 or MOS2 images. The XMM images suit well our study, as XMM has a larger field of view than Chandra. However, when no XMM data are available, we show contours from Chandra images. Even with XMM, the field of view is limited to about 30 arcmin in diameter, and in some cases, several structures detected through weak lensing have no X-ray counterparts because only the cluster vicinity is in the X-ray field. The X-ray images have been binned in squares of 64 pixels and then smoothed with a Gaussian filter of 20 pixel width. The significance of the X-ray maps are computed from the dispersion of the values of the respective map avoiding the cluster region, and start at 2$\sigma$. We chose a 2$\sigma$ value to show better how our WL detections are embedded in the baryonic components, and because the X-ray maps are only used for qualitative description. The light density maps are built with the galaxies selected to have a high probability of being at the same redshift as the cluster. For this, we first extract all the objects from the images in two bands. We separate stars from galaxies and draw color-magnitude diagrams. For each cluster, we superimpose on the color-magnitude diagram the positions of the galaxies with spectroscopic redshifts coinciding with the cluster redshift range. This allows to define the red sequence drawn by the early type galaxies belonging to the cluster and to fit it with a linear function of fixed slope $-0.0436$, as in @Durret+11. We then select all the galaxies within $\pm 0.3$ magnitude of this sequence as probable cluster members and compute the density map of this galaxy catalog, using the same Gaussian kernel than that of the WL analysis. The pixel size chosen to compute these maps is 0.001 deg, and the number of bootstraps is 100. To derive the significance level of our detections, it is necessary to estimate the mean background of each image and its dispersion. For this, we draw for each density map the histogram of the pixel intensities. We apply a 2.5$\sigma$ clipping to eliminate the pixels of the image that have high values and correspond to objects in the image. We then redraw the histogram of the pixel intensities after clipping and fit this distribution with a Gaussian. For each cluster, the mean value and the width of the Gaussian will respectively give the mean background level and the dispersion, that we will call $\sigma$. We then compute the values of the contours corresponding to 3$\sigma$ detections as the background plus 3$\sigma$. In all the figures of the following subsection, we show contours starting at 3$\sigma$ and increasing by 1$\sigma$. We first discuss individually the mass map of every cluster in Sect. \[subsec:sbs\], and then make general considerations in Sect. \[subsec:fil\]. Individual clusters {#subsec:sbs} ------------------- In addition to discussing the reconstructed convergence maps, in this subsection, we also compare the WL masses computed from the NFW best fit (see Sect. \[subsec:3Dmass\]) to other masses from the literature. However we would like to warn the reader that WL masses from different studies can significantly vary. The reason for that lies in the estimate of the redshift distribution of the background galaxies. In the ideal case where every study selects the same background galaxies and agrees on their redshift distribution, they should get the same masses within errors coming just from the shear measurement. However, in most cases the selection of galaxies and the estimate of their redshift distribution significantly vary from one study to another, introducing large differences on cluster masses. In addition, cluster masses can present a bias, for example introduced by the choice of a given value or range of value for the concentration parameter, in order to break the mass-concentration degeneracy. For large WL cluster surveys, masses thus differ systematically by 20-30% in comparing the masses of each cluster across the survey. However the different teams generally agree with each other regarding which cluster are more massive.\ [XDCS0329, Fig. \[fig:massmapxdcs\]:]{} XDCS0329 is barely detected, with a significance of only 2.8$\sigma_{\kappa}$. It possesses a weak X-ray and optical counterpart. A larger structure is detected at the south with WL (3: 3.9$\sigma_{\kappa}$) and could correspond to a structure at a different redshift from that of the cluster or to a fake peak but with a weak probability given its signal-to-noise. The most massive structure in this field lies north west of the cluster (2: 5.6$\sigma_{\kappa}$), and does not present any X-ray or optical detection. In addition there are no known structure referenced at this position in NED, and its high significance detection cannot be reproduced by noise in the mass map reconstruction. A spectroscopic survey of the area would help determine the nature and redshift of this massive object. Finally, we note that XDCS0329 is a small cluster given its hydrodynamical mass of $M_{500}^{X}=(2.9\pm0.6)\times10^{14}M_\odot.h_{70}^{-1}$ found in @Guennou+14. It is even sometimes considered as a group rather than a cluster [e.g., @Mulchaey+06].\ ![Convergence density map for XDCS0329 overlaid on the 3-color CFHT/MegaCam image. Contour levels (cyan) are in signal-to-noise from 3$\sigma_{\kappa}$ with steps of 1$\sigma_{\kappa}$. Each weak lensing peak is noted as a white cross. The yellow cross indicates the position of the BCG. The X-ray contours starting at 2$\sigma_X$ are in magenta and the light density contours starting at 3$\sigma$ are in green. We expect 1.3 fake peaks above 3$\sigma_{\kappa}$ and 0.2 above 4$\sigma_{\kappa}$ in the displayed field (see Sect. \[subsec:massmap\] for details). [The scale is given in comoving distance.]{}[]{data-label="fig:massmapxdcs"}](plots/contours/new_xdcs-eps-converted-to.pdf){width="9.cm"} [MACSJ0454, Fig. \[fig:massmapmacsj0454\]:]{} MACSJ0454 has two substructures detected in WL: a first peak at 5.1$\sigma_{\kappa}$, and a second at 4.2$\sigma_{\kappa}$ defining [an elongated]{} structure, as already reported from the optical study of @Kartaltepe+08. [We note that these substructures are not detected in the WL reconstruction of @Soucail+15, probably because they use a larger smoothing kernel ($\theta=150"$ against $\theta=60"$ in our case). However, they found a clear elongation that matches those substructures.]{} The X-ray and optical contours are centered between these two substructures, and elongated in their direction. The fact that this cluster is highly substructured can explain why the NFW fit fails. In addition, this cluster is probably of low mass as @Zitrin+11 found a central mass of $M_{500}^{SL}=(0.41\pm0.03)\times10^{14}M_\odot.h_{70}^{-1}$ in their strong lensing analysis. We also detect several faint peaks. They are detected at levels of 4.4, 3.8, 4.2, and 4.0$\sigma_{\kappa}$ for structures 4, 5, 6, and 7 respectively. While structures 5 and 6 might have an optical counterparts, structure 4 and 7 very likely correspond to fake peaks, or to a small group at a different redshift for structure 4. [Structures 4 and 6 are also detected in @Soucail+15]{}. A larger structure is found at the south west (8 at 5.5$\sigma_{\kappa}$), which is not at the cluster redshift, given that it is not detected through the galaxy density contours, but could also be due to a contamination from stars in its vicinity.\ ![Same as Fig \[fig:massmapxdcs\] for MACSJ0454 on the 3-color CFHT/MegaCam image. We expect 3.0 fake peaks above 3$\sigma_{\kappa}$ and 0.6 above 4$\sigma_{\kappa}$ in the displayed field (see Sect. \[subsec:massmap\] for details).[]{data-label="fig:massmapmacsj0454"}](plots/contours/new_macsj0454-eps-converted-to.pdf){width="9.cm"} [ABELL 851, Fig. \[fig:massmapa851\]:]{} A851 is a massive cluster, detected at a high significance level ($7.6\sigma_{\kappa}$). It is highly sub-structured as already found in @Guennou+14, and confirmed here by the presence of three spatially separated components: the dark matter, the X-ray gas, and the galaxies. [No substructures are detected in the mass reconstruction of @Soucail+15, but they used a smoothing kernel more than twice larger than ours.]{} The most important substructures are those noted 2 and 3, the first to the south with a $5\sigma_{\kappa}$ significance and the second to the north-east with a $4.3\sigma_{\kappa}$ significance. These structures are also detected on the galaxy density map and perhaps also in X-rays, the contours of which are extended towards the substructure directions. Finally, we note a fourth and a fifth structures, north-east and south-west of the cluster. These are quite far from the cluster, and while 5 has an optical counterpart, 4 does not, and could either be a fake peak or a group at a different redshift. The 5th structure should lie at the same redshift as the cluster. We note that other studies reported a higher mass than the one we derived for this cluster. We find [$M_{500}^{NFW}=(4.4\pm1.4)\times10^{14}M_\odot.h_{70}^{-1}$]{} while @Mahdavi+13 found $M_{500}=(10.5\pm2.5)\times10^{14}M_\odot.h_{70}^{-1}$ and @Hoekstra+15 found $M_{500}^{NFW}=(12.5\pm3.0)\times10^{14}M_\odot.h_{70}^{-1}$. Finally, we note that the hydrodynamical masses from X-ray studies are lower: $M_{500}^{\rm X}=(7.4\pm2.3)\times10^{14}M_\odot.h_{70}^{-1}$ from @Mahdavi+13 and $M_{500}^{X}=(5.5\pm1.2)\times10^{14}M_\odot.h_{70}^{-1}$ in the present study.\ ![Same as Fig \[fig:massmapxdcs\] for A851 on the 3-color CFHT/MegaCam image. We expect 3.8 fake peaks above 3$\sigma_{\kappa}$ and 0.6 above 4$\sigma_{\kappa}$ in the displayed field (see Sect. \[subsec:massmap\] for details).[]{data-label="fig:massmapa851"}](plots/contours/new_a851-eps-converted-to.pdf){width="9.cm"} [LCDCS0829, Fig. \[fig:massmaplcdcs0829\]:]{} LCDCS0829 is at first view an isolated cluster, with an elongation to the north-west. [An elongation is also detected in the WL reconstruction of @Soucail+15.]{} It is detected with our three probes. However, at a larger scale there is another structure (3: $4.7\sigma_{\kappa}$) about 1.5-2 Mpc south-west from the cluster, that could be in interaction, and is detected both with WL and galaxy density. Farther away but still at the same redshift according to our galaxy density map lies a $4.5\sigma_{\kappa}$ structure (2) that could be a group connecting to the main cluster through a filamentary structure passing by 3, that remains to be detected. For this cluster we find a mass of [$M_{500}^{NFW}=(5.7\pm2.1)\times10^{14}M_\odot.h_{70}^{-1}$]{}, which agrees within the error bars with the WL study of @Mahdavi+13 ($M_{500}=(9.3\pm2.9)\times10^{14}M_\odot.h_{70}^{-1}$), but is low compared to that of @Foex+12 ($M_{500}^{NFW}=(17.7\pm2.2)\times10^{14}M_\odot.h_{70}^{-1}$).\ ![Same as Fig \[fig:massmapxdcs\] for LCDCS0829 on a 3-color CFHT/MegaCam image. We expect 4.6 fake peaks above 3$\sigma_{\kappa}$ and 0.8 above 4$\sigma_{\kappa}$ in the displayed field (see Sect. \[subsec:massmap\] for details).[]{data-label="fig:massmaplcdcs0829"}](plots/contours/new_lcdcs0829-eps-converted-to.pdf){width="9.cm"} [MS1621, Fig. \[fig:massmapms1621\]:]{} This cluster is massive, and highly substructured at large scales. The main cluster is detected at 8.3$\sigma_{\kappa}$, and is also seen on the X-ray and galaxy density maps. It is elongated towards structures 2 and 3 detected at 4.3 and 3.5$\sigma_{\kappa}$, with also an elongation in the X-ray and galaxy density contours for structure 2, while 3 might just be a fake peak. Finally, the galaxy density contours show a structure south-east of substructure 3 that could be a close group. [Structures 1 and 2 are detected as a single structure in @Soucail+15, because of the larger smoothing scale they apply to the mass map. Their reconstruction is clearly elongated in the direction of these substructures.]{} We note that @Foex+12 found a mass of $M_{500}^{WL}=(8.5\pm1.5)\times10^{14}M_\odot.h_{70}^{-1}$, slightly higher than our value of [$M_{500}^{NFW}=(6.2\pm1.5)\times10^{14}M_\odot.h_{70}^{-1}$]{}, but in worse agreement with the hydrodynamical mass inferred from X-rays: $M_{500}^{X}=(4.5\pm0.5)\times10^{14}M_\odot.h_{70}^{-1}$.\ ![Same as Fig \[fig:massmapxdcs\] for MS1621 on the 3-color CFHT/MegaCam image. We expect 2.0 fake peaks above 3$\sigma_{\kappa}$ and 0.4 above 4$\sigma_{\kappa}$ in the displayed field (see Sect. \[subsec:massmap\] for details).[]{data-label="fig:massmapms1621"}](plots/contours/new_ms1621-eps-converted-to.pdf){width="9.cm"} [OC02, Fig. \[fig:massmapoc02\]:]{} OCO2 is detected with the three probes, with a 4.7$\sigma_{\kappa}$ from WL. It seems to be merging with a smaller group on the south, detected at 4.2$\sigma_{\kappa}$ (3). Finally, we note a massive structure detected at 5.8$\sigma_{\kappa}$, with an X-ray counterpart and only a faint optical counterpart. This means it is a group or cluster, at a different redshift from OC02. By checking on NED, we find that structure 2 corresponds in fact to Abell 2246, a foreground cluster at $z=0.225$. Finally, OC02, also known as CL1701+6414 is a low mass cluster. We find a mass of [$M_{500}^{NFW}=(2.3\pm1.0)\times10^{14}M_\odot.h_{70}^{-1}$]{}, slightly higher than @Israel+14, who found a WL mass of $M_{500}^{WL}=0.33\times10^{14}M_\odot.h_{70}^{-1}$ or $M_{500}^{WL}=1.41\times10^{14}M_\odot.h_{70}^{-1}$ depending on the chosen concentration parameter. We also investigate the bias in the mass estimate from OC02’s shear profile due to the presence of the foreground cluster A2246. To do this, we first compute the expected shear profile for the foreground cluster, using an X-ray derived total mass from @Wang+14: $M_{200}^{X}=(3.3\pm0.6)\times10^{14}M_\odot.h_{70}^{-1}$, and assuming a concentration parameter of $c_{200}=3.5$. We note that X-ray derived masses should not be biased by the proximity of both clusters as they are derived in a much smaller region than the WL. We then subtract this expected shear contribution to every galaxy in the field and compute again the mass of OC02 by fitting an NFW profile to its new shear profile. We find a new mass which is 7% lower than the value from Table \[tab:resclus\]. We conclude that the presence of the foreground cluster only weakly affects the cluster mass estimate in this case, and do not correct for it as it is low compared to the other sources of error, and to avoid biasing our sample in applying a different method to one of our cluster.\ ![Same as Fig \[fig:massmapxdcs\] for OC02 on the 3-color CFHT/MegaCam image. We expect 1.0 fake peaks above 3$\sigma_{\kappa}$ and 0.2 above 4$\sigma_{\kappa}$ in the displayed field (see Sect. \[subsec:massmap\] for details).[]{data-label="fig:massmapoc02"}](plots/contours/new_oc02-eps-converted-to.pdf){width="9.cm"} [NEP200, Fig. \[fig:massmapnep200\]:]{} NEP200 is detected in X-rays, optical, and WL, with a detection significance of 5.1$\sigma_{\kappa}$. It seems to be merging with a companion on the west (2: 4.6$\sigma_{\kappa}$), while it is probably a projection effect given that it is not detected in the optical contours. Spectroscopic redshifts would be needed to confirm this hypothesis. We also note several peaks at $\sim3\sigma_{\kappa}$ which could correspond to fake peaks or faint structures at different redshifts. As this cluster has not been widely studied yet, we derive a first WL mass of [$M_{500}^{NFW}=(12.7\pm5.5)\times10^{14}M_\odot.h_{70}^{-1}$]{} for NEP200.\ ![Same as Fig \[fig:massmapxdcs\] for NEP200 on the 3-color CFHT/MegaCam image. We expect 1.6 fake peaks above 3$\sigma_{\kappa}$ and 0.3 above 4$\sigma_{\kappa}$ in the displayed field (see Sect. \[subsec:massmap\] for details).[]{data-label="fig:massmapnep200"}](plots/contours/new_nep200-eps-converted-to.pdf){width="9.cm"} [RXJ2328, Fig. \[fig:massmaprxj2328\]:]{} This cluster is detected at 5.5$\sigma_{\kappa}$ from WL, and also has X-ray and optical counterparts. From the WL contours, it seems to be merging with an infalling group detected at 3.9$\sigma_{\kappa}$ in the south. However, this structure is not detected in X-rays or in the galaxy density map, suggesting that it is at a different redshift, and therefore not in interaction with RXJ2328. Note the presence of the Pegasus dwarf galaxy in the south that has been masked in our analysis, but could still bias our measurements. We find a WL mass of [$M_{500}^{NFW}=(3.7\pm1.2)\times10^{14}M_\odot.h_{70}^{-1}$]{}.\ ![Same as Fig \[fig:massmapxdcs\] for RXJ2328 on the 3-color CFHT/MegaCam image. We expect 1.8 fake peaks above 3$\sigma_{\kappa}$ and 0.3 above 4$\sigma_{\kappa}$ in the displayed field (see Sect. \[subsec:massmap\] for details).[]{data-label="fig:massmaprxj2328"}](plots/contours/new_rxj2328-eps-converted-to.pdf){width="9.cm"} [CLJ0152, Fig. \[fig:massmapclj0152\]:]{} This cluster is highly sub-structured and has several neighboring groups nearby, implying a complex recent merging history [e.g., @Massardi+10]. The cluster is massive ([$M_{500}^{NFW}=(9.4\pm3.1)\times10^{14}M_\odot.h_{70}^{-1}$]{}) and rather elongated in a north-south direction (see structure 2 detected at $6.4\sigma_{\kappa}$) and in a lesser extent in the east-west direction. Several structures are also detected in the south, and are aligned horizontally: 3 ($4.8\sigma_{\kappa}$), 4 ($6.6\sigma_{\kappa}$), 5 ($4.7\sigma_{\kappa}$), and 6. Structures 3 and 4, and also maybe 5, are detected in X-rays, while 4 and 6 have optical counterparts. Structures detected in WL and X-rays have a high probability to be groups, while those detected through the galaxy density maps should be around the same redshift as CLJ0152. Given the extension of the galaxy density map compared to that of the main cluster, structure 3 is probably a foreground group. One possible explanation is that the cluster recently underwent a merging event with the group 4 that passed through CLJ0152 from the north-west to the south-east. Structure 2 would be a remnant of this merging, while 3 should not have taken part in that scenario. Also structure 6 could have been created in the same event or being now interacting with structure 4. An X-ray temperature map would be valuable to check the direction of the past merger events.\ ![Same as Fig \[fig:massmapxdcs\] for CLJ0152 on the r band Subaru/Suprime-Cam image. We expect 4.4 fake peaks above 3$\sigma_{\kappa}$ and 1.1 above 4$\sigma_{\kappa}$ in the displayed field (see Sect. \[subsec:massmap\] for details).[]{data-label="fig:massmapclj0152"}](plots/contours/new_clj0152-eps-converted-to.pdf){width="9.cm"} [MACSJ0717, Fig. \[fig:massmapmacsj0717\]:]{} MACSJ0717 is famous for being one of the most massive clusters, as can be seen from its WL contours, which reach a significance of $10.9\sigma_{\kappa}$. We note also that it is strongly elongated towards a south east structure noted 2 with a $8.2\sigma_{\kappa}$ significance. Both structures are also detected from the optical density map (as in @Kartaltepe+08), suggesting that they are at the same redshift, but only the main cluster is strongly emitting in X-rays. Structure 2 is thus poor in hot gas, which makes us think that it corresponds to a filament rather than a group which would have produced more hot gas in its formation. The absence of a BCG agrees with this idea. Structure 3 could also be a continuation of this filament. Note that this filament has first been studied by @Jauzac+12 from composite HST data, and later by @Medezinski+13. We compared our WL contours with those from @Jauzac+12, and found good agreement. Concerning the mass of the cluster, [@Zitrin+11] and @Limousin+12 found strong lensing masses of respectively $M_{r<350kpc.h_{70}^{-1}}^{SL}=(7.4\pm0.5)\times10^{14}M_\odot.h_{70}^{-1}$ and $M_{r<960kpc.h_{70}^{-1}}^{SL}=(21.1\pm2.3)\times10^{14}M_\odot.h_{70}^{-1}$. From WL, various masses have been calculated in different radii. In $r_{500}$, we have a mass of [$M_{500}^{WL}=(15.9\pm4.3)\times10^{14}M_\odot.h_{70}^{-1}$]{} to be compared to @Mahdavi+13 and @Hoekstra+15 who respectively found $M_{500}^{WL}=(16.6\pm3.4)\times10^{14}M_\odot.h_{70}^{-1}$ and $M_{500}^{WL}=(22.3\pm5.2)\times10^{14}M_\odot.h_{70}^{-1}$. The first estimate is closed to ours, but the second is larger and agrees only within the error bars. In a radius of 0.5 Mpc, we have [$M_{r<0.5Mpc.h_{70}^{-1}}^{WL}=(4.4\pm1.2)\times10^{14}M_\odot.h_{70}^{-1}$]{}, somewhat lower than @Jauzac+12 who found a mass of $M_{r<0.53Mpc.h_{70}^{-1}}^{WL}=(11.0\pm0.8)\times10^{14}M_\odot.h_{70}^{-1}$. However we find a good agreement with masses from the [CLASH]{} collaboration WL follow up [@Medezinski+13] who found $M_{r<0.5Mpc.h_{70}^{-1}}^{WL}=(5.4\pm1.2)\times10^{14}M_\odot.h_{70}^{-1}$. @Applegate+14 also found higher masses within 1.5 Mpc, with $M_{r<1.5Mpc.h_{70}^{-1}}^{WL}=(25.3\pm4.2)\times10^{14}M_\odot.h_{70}^{-1}$ or $M_{r<1.5Mpc.h_{70}^{-1}}^{WL}=(23.1\pm3.8)\times10^{14}M_\odot.h_{70}^{-1}$, in the first case using the full distribution of photometric redshifts of the background galaxies and in the second the standard color-color cut, while we have [$M_{r<1.5Mpc.h_{70}^{-1}}^{WL}=(16.1\pm4.5)\times10^{14}M_\odot.h_{70}^{-1}$]{}. We see that the mass estimates vary strongly for this cluster; we tend to find a lower value, but in any study (including ours) MACSJ0717 appears to be one of the most massive cluster.\ ![Same as Fig \[fig:massmapxdcs\] for MACSJ0717 on the 3-color Subaru/Suprime-Cam image. We expect 1.6 fake peaks above 3$\sigma_{\kappa}$ and 0.3 above 4$\sigma_{\kappa}$ in the displayed field (see Sect. \[subsec:massmap\] for details).[]{data-label="fig:massmapmacsj0717"}](plots/contours/new_macsj0717-eps-converted-to.pdf){width="9.cm"} [BMW1226, Fig. \[fig:massmapbmw1226\]:]{} This cluster is not detected through WL, probably due to its high redshift: $z=0.89$ which decreases the number of background galaxies usable for the WL reconstruction. However a large elongated structure (1) is detected, and could be a filament linked to BMW1226. It is detected at 5.6$\sigma_{\kappa}$ and has an optical counterpart, such that is should not be too far from the cluster redshift. The small structure (2) west of the cluster is not very significant (2.9$\sigma_{\kappa}$) and is probably due to the noise in the convergence map reconstruction. This cluster has been studied by @Jee+09 under its other name: CLJ1226+3332. Using deep HST data, they manage to have a sufficient number of background galaxies to reconstruct the WL map around the cluster. However, the small field of view of the ACS camera does not allow them to study the filamentary structure that we see east of the cluster.\ ![Same as Fig \[fig:massmapxdcs\] for BMW1226 on the r-band Subaru/Suprime-Cam image. We expect 1.0 fake peaks above 3$\sigma_{\kappa}$ and 0.1 above 4$\sigma_{\kappa}$ in the displayed field (see Sect. \[subsec:massmap\] for details).[]{data-label="fig:massmapbmw1226"}](plots/contours/new_bmw1226-eps-converted-to.pdf){width="9.cm"} [MACSJ1423, Fig. \[fig:massmapmacsj1423\]:]{} MACSJ1423 looks rather isolated on small scales, with a good alignment between the WL, X-ray, and optical centers. @Kartaltepe+08 also classified it as a relaxed cluster according to its optical contours. A small structure is detected north-east from WL but not from the optical data and should correspond to a group at a different redshift. The X-ray data come from Chandra in this case, so structure 2 has no X-ray imaging. This cluster has been studied in strong lensing by @Zitrin+11 and also by @Limousin+10 who found a single central mass component, which agrees with our smooth contours. @Applegate+14 also computed WL masses for this cluster finding values of $M_{r<1.5Mpc.h_{70}^{-1}}^{WL}=(3.7\pm2.8)\times10^{14}M_\odot.h_{70}^{-1}$ or $M_{r<1.5Mpc.h_{70}^{-1}}^{WL}=(8.8\pm3.6)\times10^{14}M_\odot.h_{70}^{-1}$, in the first case using the full distribution of photometric redshifts of the background galaxies and in the second the standard color-color cut. We note that our value of [$M_{r<1.5Mpc.h_{70}^{-1}}^{NFW}=(7.9\pm3.1)\times10^{14}M_\odot.h_{70}^{-1}$]{} is in good agreement with the one obtained with the color-color cut method (the one which we used).\ ![Same as Fig \[fig:massmapxdcs\] for MACSJ1423 on the i band Subaru/Suprime-Cam image. We expect 2.1 fake peaks above 3$\sigma_{\kappa}$ and 0.3 above 4$\sigma_{\kappa}$ in the displayed field (see Sect. \[subsec:massmap\] for details).[]{data-label="fig:massmapmacsj1423"}](plots/contours/new_macsj1423-eps-converted-to.pdf){width="9.cm"} [MACSJ1621, Fig. \[fig:massmapmacsj1621\]:]{} MACSJ1621 presents a large substructure (2: $5.9\sigma_{\kappa}$ significance) that could be an infalling group. Another structure (3) is detected south-east at more than $5\sigma_{\kappa}$, and could be embedded in a filament linking it to the cluster, as suggested by the galaxy light density map. Note that structure 3 is also detected by @vonderLinden+14. An X-ray counterpart is detected only for the cluster and not for structure 2, that has then good chance of being part of the filament rather than being an infalling group. The WL mass that we measure for this cluster agrees with the value of @Applegate+14 within the error bars: we find [$M_{r<1.5Mpc.h_{70}^{-1}}^{NFW}=(5.3\pm1.9)\times10^{14}M_\odot.h_{70}^{-1}$]{} and they have $M_{r<1.5Mpc.h_{70}^{-1}}^{WL}=(8.5\pm2.3)\times10^{14}M_\odot.h_{70}^{-1}$ or $M_{r<1.5Mpc.h_{70}^{-1}}^{WL}=(8.8\pm2.2)\times10^{14}M_\odot.h_{70}^{-1}$ in the first case using the full distribution of photometric redshifts of the background galaxies and in the second the standard color-color cut. However, we do not reproduce the high mass found in @Hoekstra+15: $M_{500}^{NFW}=(11.2\pm2.5)\times10^{14}M_{\odot}.h_{70}^{-1}$.\ ![Same as Fig \[fig:massmapxdcs\] for MACSJ1621 on the 3-color Subaru/Suprime-Cam image. We expect 1.2 fake peaks above 3$\sigma_{\kappa}$ and 0.2 above 4$\sigma_{\kappa}$ in the displayed field (see Sect. \[subsec:massmap\] for details).[]{data-label="fig:massmapmacsj1621"}](plots/contours/new_macsj1621-eps-converted-to.pdf){width="9.cm"} [RXJ1716, Fig. \[fig:massmaprxj1716\]:]{} RXJ1716 (1: 7.3$\sigma_{\kappa}$) shows a very elongated profile pointing towards two groups: 2 and 3 detected at respectively 4.9 and 5.4$\sigma_{\kappa}$. However those structures are not detected in the galaxy density map and must then lie at a different redshift. The main cluster is also detected with the X-ray and galaxy density contours. The [elongated structure]{} to the north east of the cluster is also seen in the WL reconstruction of @Clowe+98. This is a massive cluster with [$M_{500}^{NFW}=(9.5\pm3.2)\times10^{14}M_{\odot}.h_{70}^{-1}$]{}.\ ![Same as Fig \[fig:massmapxdcs\] for RXJ1716 on the r-band Subaru/Suprime-Cam image. We expect 2.1 fake peaks above 3$\sigma_{\kappa}$ and 0.5 above 4$\sigma_{\kappa}$ in the displayed field (see Sect. \[subsec:massmap\] for details).[]{data-label="fig:massmaprxj1716"}](plots/contours/new_rxj1716-eps-converted-to.pdf){width="9.cm"} [MS2053/CXOSEXSI2056, Fig. \[fig:massmapms2053\]:]{} MS2053 is detected with a high level of significance: 8.7$\sigma_{\kappa}$ and with a mass of [$M_{500}^{NFW}=(6.4\pm2.2)\times10^{14}M_{\odot}.h_{70}^{-1}$]{}. It is also detected in the X-ray and galaxy density contours. CXOSEXSI2056 is a smaller cluster detected at a 4.4$\sigma_{\kappa}$ significance, and also presents an X-ray counterpart. It seems to be merging with a wide structure (3: 4.5$\sigma_{\kappa}$) on the east and might also be linked to the small structure 4 but the significance of the latter structure remains low (3.2$\sigma_{\kappa}$) and it is more likely a fake peak due to noise. For this field we did not try to estimate the masses of each cluster by removing the contribution from the other, as we did for OC02, because the significance of their detections are too different. CXOSEXSI has little chance to significantly affect the shear profile of MS2053, and on the contrary, removing such a big cluster as MS2053 would introduce an other large bias in the mass estimate of CXOSEXSI. In addition we did not compute any mass for this latter cluster.\ ![Same as Fig \[fig:massmapxdcs\] for MS2053 and CXOSEXSI2056 on the r-band Subaru/Suprime-Cam image. We expect 1.3 fake peaks above 3$\sigma_{\kappa}$ and 0.2 above 4$\sigma_{\kappa}$ in the displayed field (see Sect. \[subsec:massmap\] for details).[]{data-label="fig:massmapms2053"}](plots/contours/new_sexi-eps-converted-to.pdf){width="9.cm"} General discussion {#subsec:fil} ------------------ We summarize the structure detection in Table \[tab:resenv\], where we show the average significance of the WL detection obtained from 100 realizations of the noise along with the percentage of realizations in which the structures are detected at more than 3$\sigma$ above the map noise defined in eq. \[eq:sigkappa\]. We also indicate for each structure if it has X-ray and optical counterparts, and conclude on the current status of the cluster and the possible presence of filaments. Cluster z structure $\sigma_{2D}$ detection percentage X-ray galaxies Cluster status ---------------- -------- ----------- --------------- ---------------------- -------- ---------- ----------------------------------------------------------- XDCS0329 0.4122 1 2.8 44% Y Y - 2 5.6 96% N N 3 3.9 74% N $\sim$ MACSJ0454 0.5377 1 5.1 91% Y Y recent or present merger (3) / [elongation or filament]{} 2 4.2 76% Y Y 3 3.7 67% N $\sim$ 4 4.4 88% N N 5 3.8 70% N $\sim$ 6 4.2 82% N $\sim$ 7 4.0 78% N N 8 5.5 95% N N ABELL851 0.4069 1 7.6 100% Y Y recent or present merger (3) / [elongation or filament]{} 2 5.0 89% $\sim$ Y 3 4.3 81% $\sim$ Y 4 4.4 86% N N LCDCS0829 0.4510 1 5.5 98% Y Y past merger (2) 2 4.5 86% - Y 3 4.7 92% - Y MS1621 0.4260 1 8.3 100% Y Y recent or present merger (3) / [elongation or filament]{} 2 4.3 77% $\sim$ Y 3 3.5 67% N N 4 3.1 53% N N OC02 0.4530 1 4.7 88% Y Y recent or present merger (3) 2 5.8 98% Y $\sim$ foreground cluster (A2246) 3 4.2 78% N $\sim$ NEP200 0.6909 1 5.1 93% Y Y recent or present merger (3) 2 4.6 84% N N RXJ2328 0.4970 1 5.5 93% Y Y recent or present merger (3) 2 3.9 73% N N CLJ0152 0.8310 1 8.3 100% Y Y recent or present merger (3) / [elongation or filament]{} 2 6.4 98% Y N 3 4.8 85% Y N 4 6.6 98% Y Y 5 4.7 89% $\sim$ N MACSJ0717 0.5458 1 10.9 100% Y Y recent or present merger (3) / [elongation or filament]{} 2 8.2 100% $\sim$ Y 3 5.7 98% N Y BMW1226 0.8900 0 - - Y Y - / [elongation or filament]{} 1 5.6 97% N Y 2 2.9 46% N N MACSJ1423 0.5450 1 5.0 91% Y Y Relaxed (1) MACSJ1621 0.4650 1 6.8 97% Y Y recent or present merger (3) / [elongation or filament]{} 2 5.9 96% N Y RXJ1716 0.8130 1 7.3 98% Y Y past merger (2) 2 4.9 85% N N 3 5.4 86% N N MS2053\* 0.5830 1 8.7 100% Y Y past merger (2) CXOSEXSI2056\* 0.6002 2 4.4 84% Y N recent or present merger (3) 3 4.5 84% N N 4 3.2 55% N N \CXOSEXSI\_J205617 and MS\_2053.7-0449 are on the same image. \[tab:resenv\] The first conclusion from the study of this sample is that all the clusters appear very different, especially when considering their close environment. Several hypotheses made for the mass calculation are then questionable. Most of these clusters are not spherical, and present either a preferential direction, or several substructures. The NFW profile used in Sect. \[subsec:3Dmass\] seems simplistic compared to these results, and it appears very difficult to find a mass profile that fits every cluster, when extending to radii higher than the cluster core. Despite these very different behaviors, we try to classify our sample according to the smoothness of their WL contours and the presence of substructures or infalling groups: \(1) The only relaxed cluster of our sample is MACSJ1423. On small scales we see smooth symmetrical contours and no substructures. However, even for this cluster, we find that it might be embedded in at least one filamentary structure at larger scales. \(2) The second category gathers clusters which are highly asymmetrical but do not present any clear substructure or infalling group: LCDCS0829, RXJ1716, and MS2053. These clusters are probably recovering from old merger events, the direction of interaction of which only remains visible. \(3) The last category encompasses clusters with high levels of substructuring or apparent merging events. These clusters are recovering from a recent merging event or even are presently merging. Such behaviors are observed for MACSJ0454, A851, MS1621, OC02, NEP200, RXJ2328, CLJ0152, MACSJ0717, MACSJ1621, and CXOSEXSIJ2056. Six clusters among this last list seem to be part of particularly intense [extended structures]{}: MACSJ0454, A851, MS1621, CLJ0152, MACSJ0717, and MACSJ1621. In addition, BMW1226 shows a large filament despite the fact that the cluster is not detected itself. However, fainter [elongated]{} structures linking the different mass peaks can be seen in many cases [as a result of smoothing scale $\theta_S=1~\mathrm{arcmin}$]{}, suggesting that every cluster lies in a large scale structure. These LSSs are often not clearly detected, as they are too diffuse compared to the mass peaks corresponding to either infalling groups, or small merger events. Finally, we note that most of our clusters are either past mergers ($\sim21.5\%$) or recent or present mergers ($\sim71.5\%$). This supports the standard hierarchical scenario in which clusters grow through the merging of smaller structures. In addition, it means that most massive clusters at $0.4<z<0.9$ are still evolving through this merging process. XDCS0329 is not discussed as it is only weakly detected. This classification is summarized in Table \[tab:resenv\]. Conclusion {#sec:ccl} ========== We accurately measured galaxy shears for eight CFHT/MegaCam and seven Subaru/Suprime-Cam images. We successfully estimated the mass of twelve clusters out of sixteen, by fitting their shear profiles with an NFW profile. Comparing with masses from X-ray data (XMM–Newton and Chandra* observations), we found that our masses are generally higher than those from X-rays by about [8%]{}, an expected result given that the X-ray masses rely on the hypothesis of hydrostatical equilibrium. However, our sample is small and we need higher statistics to compare both masses, and also to better compare to the WL literature. We inverted the shear to obtain convergence maps, and overlaid the WL contours on images. We estimated the significance of each detected structure with a hundred realizations with a random ellipticity added to each galaxy. Comparing with X-ray contours and galaxy light density contours, we studied the environment of every cluster. We found that clusters are very different on large scales and doubt they can all be fitted with a simple NFW profile. We separated our sample between isolated relaxed clusters, asymmetrical clusters with no substructures and clusters which have a more complex environment. The second category corresponds to past mergers and the third one to recent or present mergers. Most of the sampled clusters are in the last two categories, providing strong observational support to the hierarchical growth scenario, and implying that clusters are still evolving through this process at $0.4<z<0.9$. Temperature maps from deep X-ray imaging could help characterize the different merging phases that we observe [see e.g. @Durret+11 and references therein]. Even in the isolated case, we found that clusters are embedded in complex large scale structures, often connecting to another group on megaparsec scales. We report possible filament detections in CLJ0152, MACSJ0454, MACSJ0717, A851, BMW1226, MS1621, and MACSJ1621, the first one also experiencing recent complex merger events. Finally, it is important to note that the distinction between a filament and an infalling group or small cluster is almost a semantic problem. However, groups and small clusters should contain more X-ray gas than filaments, and are more likely to possess a BCG, at least in the case of clusters. A more detailed study of each cluster with separate simulations is required to help distinguish between the two possibilities. We intend to study the galaxy populations of the proposed filaments in the framework of the DAFT/FADA survey, a work that will also help discriminating the nature of these structures. We thank Raphael Gavazzi and Hugo Capelato for useful discussion, and Mathilde Jauzac for sharing her WL contours of MACSJ0717 with us. We are grateful to the anonymous referee for his/her careful reading and comments that improved the quality of the manuscript. We thank Nick Kaiser for authorizing us to use the [IMCAT]{} software, Emmanuel Bertin for making his [ astromatic]{} softwares publicly available, and Martin Kilbinger for his [ATHENA]{} software. DC acknowledges support from the National Science Foundation under Grant No. 1109576. FD acknowledges long-term support from CNES. IM acknowledges financial support from the Spanish Ministry of Economy and Competitiveness through the grant AYA2012-42227P. Validating the PSF correction {#appendix:psf} ============================= In order to validate our PSF correction, we compute the auto correlation functions of star ellipticities before and after correcting for the PSF (Fig. \[fig:staracf\]), and we also compare the auto correlation function of the shear to the cross correlation function between the galaxy corrected shears and the stellar ellipticities before correction (Fig. \[fig:ssccf\]). Results are shown for MACSJ0717 and RXJ2328 which respectively correspond to the Subaru/Suprime-Cam and CFHT/Megacam data. The correlation functions are computed using the [ATHENA]{} software [@Kilbinger+14; @Schneider+02], from a 1 arcmin separation angle to 30 arcmin. $C_1$ and $C_2$ correspond to the rotated 1 and 2 components, i.e., when taking the correlation between a given pair, $C_1$ is comparing the shear that is tangential to the line connecting the pairs and $C_2$ is the 45 degree component. ![image](plots/star_autocorrel_MACSJ0717_1-eps-converted-to.pdf){width="48.00000%"} ![image](plots/star_autocorrel_RXJ2328_1-eps-converted-to.pdf){width="48.00000%"}\ ![image](plots/star_autocorrel_MACSJ0717_2-eps-converted-to.pdf){width="48.00000%"} ![image](plots/star_autocorrel_RXJ2328_2-eps-converted-to.pdf){width="48.00000%"} ![image](plots/gal_autocorrel_MACSJ0717_1-eps-converted-to.pdf){width="48.00000%"} ![image](plots/gal_autocorrel_RXJ2328_1-eps-converted-to.pdf){width="48.00000%"}\ ![image](plots/gal_autocorrel_MACSJ0717_2-eps-converted-to.pdf){width="48.00000%"} ![image](plots/gal_autocorrel_RXJ2328_2-eps-converted-to.pdf){width="48.00000%"} Fig. \[fig:staracf\] shows that the PSF correction has reduced the star ellipticity auto correlation function by about three orders of magnitude both for the Suprime-cam and Megacam data. In addition, we see in Fig. \[fig:ssccf\] that the correlation between shear and stars is consistent with zero and thus that the residual bias from the PSF correction does not significantly affect the shear, which shows classical auto correlation functions. Shear profiles {#appendix:shearprof} ============== In this section we present the shear profiles for every cluster. See Sect. \[subsec:3Dmass\] for details about how shear profiles are computed and how the NFW fit is performed. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- ![image](plots/shearprof/xdcs_etprof-eps-converted-to.pdf){width="24.00000%"} ![image](plots/shearprof/macsj0454_etprof-eps-converted-to.pdf){width="24.00000%"} ![image](plots/shearprof/a851_etprof-eps-converted-to.pdf){width="24.00000%"} ![image](plots/shearprof/lcdcs0829_etprof-eps-converted-to.pdf){width="24.00000%"} ![image](plots/shearprof/ms1621_etprof-eps-converted-to.pdf){width="24.00000%"} ![image](plots/shearprof/oc02_etprof-eps-converted-to.pdf){width="24.00000%"} ![image](plots/shearprof/nep200_etprof-eps-converted-to.pdf){width="24.00000%"} ![image](plots/shearprof/rxj2328_etprof-eps-converted-to.pdf){width="24.00000%"} ![image](plots/shearprof/clj0152_etprof-eps-converted-to.pdf){width="24.00000%"} ![image](plots/shearprof/macsj0717_etprof-eps-converted-to.pdf){width="24.00000%"} ![image](plots/shearprof/bmw1226_etprof-eps-converted-to.pdf){width="24.00000%"} ![image](plots/shearprof/macsj1423_etprof-eps-converted-to.pdf){width="24.00000%"} ![image](plots/shearprof/macsj1621_etprof-eps-converted-to.pdf){width="24.00000%"} ![image](plots/shearprof/rxj1716_etprof-eps-converted-to.pdf){width="24.00000%"} ![image](plots/shearprof/ms2053_etprof-eps-converted-to.pdf){width="24.00000%"} -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- ------------------------------------------------------------------------------- -- -- ![image](plots/shearprof/sexi_etprof-eps-converted-to.pdf){width="24.00000%"} ------------------------------------------------------------------------------- -- -- [^1]: Based on observations obtained with MegaCam, a joint project of CFHT and CEA/IRFU, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l’Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. Also based on archive data collected at the Subaru Telescope, which is operated by the National Astronomical Observatory of Japan. This research made use of data obtained from the Chandra Data Archive provided by the Chandra X-ray Center (CXC), and data obtained from the XMM–Newton Data Archive provided by the XMM–Newton Science Archive (XSA).
--- abstract: 'The maximal number of cells in a polyomino with no four cells equally spaced on a straight line is determined to be 142. This is based on several partial results, each of which can be verified with computer assistance.' author: - \| <span style="font-variant:small-caps;">Jan Kristian Haugland</span>\ `admin@neutreeko.net` title: Largest polyomino with no four cells equally spaced on a straight line --- Definitions =========== A polyomino is an object in the plane formed by joining one or more unit squares (called cells) edge to edge. It can be viewed as a graph with the cells as vertices and an edge joining two vertices if the corresponding cells are adjacent. A polyomino is called admissible if no four cells (i.e., their centres) are equally spaced on a straight line. A path is a polyomino that either consists of a single cell, or contains two cells of degree 1 (called endpoints) while all the remaining cells have degree 2. Suppose a subset of the cells of a polyomino $P$ form a path $Q$ with endpoints $A$ and $B$. If the graph distance between $A$ and $B$ in $P$ is equal to the graph diameter $d$ of $P$, and $Q$ consists of $d+1$ cells, then $Q$ is said to be a maximal path in $P$. The radius of $P$ with respect to $Q$ is then the minimal value of $r$ such that for any cell $C$ in $P$, there exists a cell $D$ in $Q$ for which the graph distance between $C$ and $D$ is at most $r$. A loop is a polyomino for which all the cells have degree 2. Introduction ============ The objective of this paper is to outline a verification of the following result: An admissible polyomino may have at most 142 cells. It is straightforward to generate all admissible paths with a computer program, and this will serve as the basis of the verification. Suppose, temporarily, that we only considered a special type of paths in which we could traverse the cells from one endpoint to the other by always moving either East or North, say, from one cell to the next. Instead of the polyomino itself, we could view the path as a sequence on the symbols E, N (East, North). The requirement that no four cells are equally spaced on a straight line would then be equivalent to requiring that no three consecutive “blocks” of symbols are permutations of each other. Dekking [@Dekking:1979] has shown that such a sequence could be infinitely long if no four consecutive blocks are permutations of each other, and mentions that the case with three blocks is easily checked to only have finite solutions. But in our more general case, the paths can have as many as 120 cells. Analysis ======== The basic idea in verifying Theorem 1 is to go through the admissible paths (or a subset of them, as we shall see later), and for each one, either find the largest admissible polyominoes that contain it as a maximal path, or find an upper bound for their size. As a polyomino is built one cell at a time from a maximal path, it can be useful to keep track of the graph diameter of the intermediate polyominoes. Therefore, we start with a result on the existence of loops in large admissible polyominoes. An admissible polyomino with at least 67 cells can not contain any loop of length greater than 4. Table 1 shows all admissible loops (up to isometry) of length greater than or equal to 8, together with the maximal number of cells that may be added (so that the resulting polyomino remains admissible), and the maximal total number of cells. Each loop is given by a set of directions for moving from one cell to the next around the loop, with E, N, W and S representing East, North, West and South respectively. ---------------------------------------------------------- -------- ------------- -------------- Max. no. of Max. total Loop Length extra cells no. of cells EENNWWSS 8 58 66 EENENNWWSWSS 12 14 26 EENEENNWNWWSSWSS 16 7 23 EENENNWNWWSWSSES 16 4 20 EENEENNWNWWSWWSSES 18 6 24 EENEENNWWNWWSWSSES 18 4 22 EENEENNWNNWWSWWSSESS 20 1 21 EENEENNWNNEENEENNWNNWWSWSWWNWWSWSSESESSWSSES 44 8 52 EENEENNWNNEENEENNWNWWSWWNWNWWSSWSSESESSWSSES 44 10 54 EENEENNWNNEENEENNWNNWWSWSWWNNWWSSWSSESESSWSSES 46 6 52 EENEENNWNNEENEENNWNNWWSSWWNWNWWSSWSSESESSWSSES 46 6 52 EENEESESEENENNWNNENENNWNWWSWWNWNWWSWSSESSWSWSSES 48 16 64 EENEESESEENENNWNWNNENNWNWWSWWNWNWWSWSSESESSWSSES 48 4 52 EENEENNWNWNNENNWNWWSWWNWNWWSWWSSESESSWSSESEENEESES 50 8 58 EENEENNWWNNENENNWNWWSWWNWNWWSWSSESSWSWSSESEENEESES 50 14 64 EENEENNWWNNENENNWNNWWSSWWNWNWWSWSSESSWSWSSESEENEESES 52 12 64 EENEENNWWNNENENNWNWWSWWNWNWWSWWSSEESSWSWSSESEENEESES 52 12 64 EENEENNWNNEENEENNWNWWSWWNWNWWSWWSSESSWWSWWSSESEENEESES 54 8 62 EENEENNWWNNENENNWNNWWSSWWNWNWWSWWSSEESSWSWSSESEENEESES 54 10 64 EENEENNWWNNENENNWNNWWSSWWNWNWWSWWSSEESSWSWSSESSEENNEESES 56 8 64 ---------------------------------------------------------- -------- ------------- -------------- Unlike larger loops, a loop of length 4 does not have its graph diameter increased if a cell is removed. This leads to the following result. We can build any admissible polyomino $P$ with a least 67 cells by adding one by one cell to a maximal path in $P$, without altering the graph diameter at any point. If $P$ is an admissible polyomino that does not contain any loop of length greater than 4, and $Q$ is a maximal path in $P$, then the radius of $P$ with respect to $Q$ is at most 5. There is only one admissible polyomino (up to isometry) that is the union of three paths of graph diameter 6 that only overlap in one common endpoint, shown here. ![image](fork.png){width="0.25\linewidth"} However, it contains a loop of length 12, and is not an actual counterexample. If $Q$ is an admissible path, suppose its set of cells is partitioned into one or more disjoint subsets $$Q=\bigcup\limits_{i=1, ..., k} Q_i$$ It seems natural to restrict our attention to the cases in which each $Q_i$ is connected, although this is not strictly required. Let $f(i)$ denote the maximal number of cells that can be added to $Q$ by the following iterative steps, assuming that a cell can only be added to an admissible polyomino if the resulting polyomino is also admissible, and if the graph diameter is not altered.\ Step 1: Add only cells that are adjacent to at least one cell in $Q_i$ Step $j \in \{ 2, 3, ... \}$: Add only cells that are adjacent to at least one cell that was added in step $j-1$\ It follows from Lemma 2 that five iterative steps is sufficient if the resulting polyomino does not contain any loop of length greater than 4. Ideally, we want to use $k=1$ and find $f(1)$, the exact number of cells that can be added, but this can be time consuming. For higher values of $k$, an upper bound for the number of cells that can be added is given by $$f(1) + f(2) + ... + f(k)$$ which is often good enough if we are only interested in the global maximum. A combination of the two approaches is also possible: Using upper bounds, we can determine which paths are good candidates for creating large admissible polyominoes, and then we can run a full analysis on those. Results ======= We do not need to go through all paths. For example, it can be verified that if $Q$ contains 48 cells, and we take the first 16, the middle 16 and the last 16 cells as the $k=3$ subsets, then we have $f(1) \leq 9$, $f(2) \leq 8$ and $f(3) \leq 9$. By Lemma 2, this gives us valid upper bounds also for other paths partitioned into segments of 16 cells in a similar way (i.e., 9 at the ends and 8 everywhere in between), and we can cover all diameters from 47 to 90. It has been verified that for any graph diameter less than 106, we have an upper bound for the total number of cells that is less than 142. For each value of the graph diameter from 106 and upwards, the exact maximal number of cells has been determined. ---------- ------------------------ ---------------------- Graph No. of admissible Maximal size of an diameter paths (up to isometry) admissible polyomino 119 6 138 118 30 138 117 55 140 116 75 141 115 117 142 114 144 142 113 187 142 112 221 141 111 266 140 110 332 138 109 478 136 108 679 134 107 963 133 106 1308 132 ---------- ------------------------ ---------------------- All the admissible polyominoes with 142 cells can be generated by including exactly one cell of each colour other than black in the figure (provided, of course, that it remains connected), and admissible polyominoes with maximal size for graph diameters 108 through 112 can be obtained by “pruning” them. ![image](142omino.png){width="0.86\linewidth"} Likewise, the largest admissible polyominoes having the maximal graph diameter of 119 can be generated by including exactly one cell of each colour other than black in the next figure. ![image](138omino.png){width="0.9\linewidth"} [99]{} Dekking, F. M. (1979). Strongly non-repetitive sequences and progression-free sets. , 27:181–185.
--- abstract: 'Continuous Galerkin Petrov time discretization scheme is tested on some Hamiltonian systems including simple harmonic oscillator, Kepler’s problem with different eccentricities and molecular dynamics problem. In particular, we implement the fourth order Continuous Galerkin Petrov time discretization scheme and analyze numerically, the efficiency and conservation of Hamiltonian. A numerical comparison with some symplectic methods including Gauss implicit Runge-Kutta method and general linear method of same order is given for these systems. It is shown that the above mentioned scheme, not only preserves Hamiltonian but also uses the least CPU time compared with upto-date and optimized methods.' author: - 'M. A. Qureshi' - 'S. Hussain' - Ghulam Shabbir title: Conservation of Hamiltonian using Continuous Galerkin Petrov time discretization scheme --- [Mathematics Subject Classification:]{} Hamiltonian systems, Continuous Galerkin Petrov time discretization, G-symplectic general linear methods, Runge-Kutta Mathod, Simple harmonic oscillator, Kepler’s problem and Molecular dynamics problem Introduction {#intro} ============ Non-dissipative phenomena arising in the fields of classical mechanics, molecular dynamics, accelerator physics, chemistry and other sciences are modeled by Hamiltonian systems. Hamiltonian systems define equations of motion based on generalised co-ordinates $q_{i}=(q_{1},q_{2},\cdots,q_{n})$ and generalised momenta $p_{i}=(p_{1},p_{2},\cdots,p_{n})$ and are given as, $$\label{hamiltonian} \frac{dp_{i}}{dt}=-\frac{\partial{H}}{\partial{q_{i}}},\hspace{0.5in}\frac{dq_{i}}{dt}=\frac{\partial{H}}{\partial{p_{i}}},\hspace{1in} i=1,\cdots,n,$$ having $n$ degrees of freedom. $H:\mathbb{R}^{2n}\times \mathbb{R}^{2n}\to \mathbb{R}$ is the total energy of the Hamiltonian system. A separable Hamiltonian has the structure $$H(p,q)=T(p)+V(q)$$ in mechanics, $T=\frac{1}{2}P^T M^{-1} P$ represents the kinetic energy and $V$ being the potential energy. The Hamiltonian system in partitioned form takes the form $$\label{hamiltonian_partitioned} \frac{dp_{i}}{dt}=-\triangledown_q V,\hspace{0.5in}\frac{dq_{i}}{dt}=\triangledown_p T = M^{-1} p.$$ The first observation is that, for autonomous Hamiltonian systems, $H$ is an invariant, thus by differentiating $H(p,q)$ with respect to time we have, $$\label{energy_conservation} \frac{dH}{dt} = \displaystyle\sum_{i=1}^{n}\Big{(}\frac{\partial{H}}{\partial{p_{i}}}\frac{dp_{i}}{dt} + \frac{\partial{H}}{\partial{q_{i}}}\frac{dq_{i}}{dt}\Big{)} = 0 .$$ We can write $ y = (p,q)$, then can be written as, $$\label{hamiltonian-compact} y' = J^{-1} \nabla H,$$ where $'$ represents the derivative with respect to time, $\nabla$ is a gradient operator and $J$ is a skew symmetric matrix consisting of zero matrix $0$ and $n \times n$ identity matrix $I$, $$\label{J_matrix} J = \left[ \begin{array}{cc} 0 & I \\ -I & 0 \end{array} \right].$$ Another property of Hamiltonian systems is that its flow is symplectic, i.e. for a linear transformation $ \Psi:\mathbb{R}^{2n} \mapsto \mathbb{R}^{2n}$, the jacobian matrix $\Psi'(y)$ satisfies $$\begin{aligned} \Psi'^{T}(y) \mathbf{\mathrm{J}} \Psi'(y) =\mathbf{\mathrm{J}}.\end{aligned}$$ Conservation laws for Hamiltonian systems are generally lost while integrating these system. It is generally desirable to preserve the underlying qualitative property of solutions of Hamiltonian systems. This is achieved by using symplectic integrators from the class of one step, multistep and general linear methods. A lot of attention has been paid on the construction and implementation of such integrators, for details see [@hairer_gni], [@rf1], [@rf2] and [@sanzserna]. The continuous Galerkin Petrov time discretization scheme (cGP) was investigated in [@Schieweck2010] for the system of ordinary differential equations (ODEs). In [@Hussain2011], this scheme was studied for the heat equation. In particular, the cGP(2) scheme has found to be 4th order accurate in the discrete time point and is A-stable method. The objective of this paper is to provide analysis of cGP(2) scheme [@Schieweck2010; @Hussain2011; @Aziz1989; @Vidar2006] on some Hamiltonian systems and comparing it with other symplectic methods of order four including Gauss implicit Runge-Kutta method represented as irk4 [@butcher-book] and a g-symplectic general linear method represented by glm4 of same order developed in [@butcher_yousaf] and [@yousafthesis]. In section two a brief introduction about the methods is given. The tested problems of Hamiltonian systems along with numerical experiments of these methods on Hamiltonian systems are described in third section. Conclusion based on numerical comparison of third section is given in fourth section. The Methods =========== [Continuous]{} Galerkin-Petrov method (cGP) \[cGP\_method\] -------------------------------------------------------------- As a model problem we consider the ODE system given in : [Find $u:[0,T_m]\to W$ such that]{} $$\label{cont_eq} \begin{array}{rcll} d_t u(t) &=& F(t, u(t)) \quad \text{for} \quad t\in(0,T_m) , \\ u(0) &=& 0 \end{array}$$ The [weak formulation]{} of problem reads: Find $u\in X$ such that $u(0)=u_0$ and $$\label{weak_eq} \int_0^{T_m} {\left< d_t u(t), v(t) \right>} dt = \int_0^{T_m} {\left< F(t, u(t)), v(t) \right>} dt \qquad\forall\; v\in Y ,$$ where $X$ denotes the [solution space]{} and Y the test space. To describe the time discretization of problem let us introduce the following notation. We denote by $I = [0, T_m]$ the time interval with some positive final time $T_m$. We start by decomposing the time interval $I$ into $N$ subintervals $I_n := (t_{n-1},t_n)$, where $n\in\{1, \dots, N\}$ and $$0=t_0 < t_1 < \dots < t_{N-1} < t_N = T_m.$$ In our time discretization, we approximate the [continuous solution]{} $u(t)$ of problem on each time interval $I_n$ by a polynomial function: $$\label{poly_ansatz} u(t) \approx u_{h}(t) := \sum_{j=0}^{k} U_n^j \phi_{n,j}(t) \qquad\forall \; t\in I_n ,$$ where the ”coefficients” $U_n^j$ are elements of the Hilbert space $W$ and the basis functions $\phi_{n,j} \in{{\mathbb P}}_{k}(I_n)$ are linearly independent elements of the standard space of polynomials on the interval $I_n$ with a degree not larger than a given order $k$. For a given time interval $J\subset{{\mathbb R}}$ and a Banach space $B$, we introduce the linear space of $B$-valued time polynomials with degree of at most $k$ as $${{\mathbb P}}_k(J,B) := \left\{ u: J\to B \,:\; u(t) = \sum_{j=0}^k U^j t^j\,,\;\forall\, t\in J,\; U^j\in B, \;\forall\, j \right\}.$$ Now, the [discrete solution space]{} for the global approximation $u_h:I\to W$ is the space $X_h^{k}\subset X$ defined as $$\label{Xt_def} X_h^{k} := \{ u\in C(I,W) :\; u\big\|_{\bar{I}_n} \in {{\mathbb P}}_k(\bar{I}_n, W) \quad\forall\; n=1,\dots,N \}$$ and the [discrete test space]{} is the space $Y_h^{k}\subset Y$ given by $$\label{Yt_def} Y_h^{k} := \{ u\in L^2(I,W) :\; u\big\|_{I_n} \in {{\mathbb P}}_{k-1}(I_n, W) \quad\forall\; n=1,\dots,N \}.$$ The symbol $h$ denotes the [discretization parameter]{} which acts in the error estimates as the maximum time step size $\; h := \max_{1\le n\le N} h_n$, where $\;h_n := t_n - t_{n-1}$ is the length of the $n$-th time interval $I_n$. Let us denote by $X_{h,0}^{k} := X_h^{k}\cap X_0$ the subspace of $X_h^{k}$ with zero initial condition. Then, it is easy to see that the dimensions of the spaces $X_{h,0}^{k}$ and $Y_h^{k}$ coincide such that it makes sense to consider the following [discontinuous Galerkin-Petrov discretization]{} of [order $k$]{} for the weak problem : Find $u_h\in u_0 + X_{h,0}^{k}$ such that $$\label{cGP_form} \int_0^{T_m} {\left< d_t u_h(t), v_h(t) \right>} dt = \int_0^{T_m} {\left< F(t, u_h(t)), v_h(t) \right>} dt \qquad\forall\; v_h\in Y_h^{k}.$$ We will denote this discretization as the ”[exact]{} cGP(k)-[method]{}”. Since the discrete test space $Y_h^{k}$ is discontinuous, problem can be solved in a time marching process. Therefore, we choose test functions $v_h(t) = v \psi_{n,i}(t)$ with an arbitrary $v\in W$ and a scalar function $\psi_{n,i} : I\to{{\mathbb R}}$ which is zero on $I\setminus \bar{I}_n$ and a polynomial $\psi_{n,i}\in{{\mathbb P}}_{k-1}(\bar{I}_n)$ on the time interval $\bar{I}_n = [t_{n-1},t_{n}]$. Then, we obtain for each $i = 0, \dots, k-1$ $$\label{lin_evol_In} \int_{I_n} {\left< d_t u_h(t), v \right>} \psi_{n,i}(t) dt = \int_{I_n} {\left< F(t, u_h(t)), v \right>} \psi_{n,i}(t) dt \qquad\forall\; v\in W .$$ By the definition of the weak time derivative we get for $u_h$ represented by the equation $$\label{dtu_eq} \int_{I_n} {\left< d_t u_h(t), v \right>} \psi_{n,i}(t) dt = \int_{I_n}\sum_{j=0}^{k} {\left( U_n^j, v \right)_{H}} \phi_{n,j}'(t) \psi_{n,i}(t)\, dt \qquad\forall\; v\in W .$$ We define the basis functions $\phi_{n,j}\in{{\mathbb P}}_{k}(\bar{I}_n)$ of via the reference transformation where ${\hat{I}}:= [-1,1]$ and $$\label{ref_tra} t = {\upomega}_n({\hat{t}}) := \frac{t_{n-1}+t_n}{2} + \frac{h_n}{2} {\hat{t}}\in \bar{I}_n \qquad\forall\; {\hat{t}}\in{\hat{I}}, \; n=1,\ldots, N.$$ Let ${\hat{\phi}}_j\in{{\mathbb P}}_{k}({\hat{I}})$, $j=0,\ldots, k$, be suitable basis functions satisfying the conditions $$\label{phi_lr} {\hat{\phi}}_j(-1) = \delta_{0,j}, \qquad {\hat{\phi}}_j(1) = \delta_{k,j},$$ where $\delta_{k,j}$ denotes the usual Kronecker symbol. Then, we define the basis functions on the original time interval $\bar{I}_n$ by $$\label{phi_ref} \phi_{n,j}(t) := {\hat{\phi}}_j({\hat{t}}) \qquad\text{with}\qquad {\hat{t}}:= {\upomega}_n^{-1}(t) = \frac{2}{h_n} \left( t - \frac{t_n - t_{n-1}}{2} \right) \in {\hat{I}}.$$ Similarly, we define the test basis functions $\psi_{n,i}$ by suitable reference basis functions ${\hat{\psi}}_i\in{{\mathbb P}}_{k-1}({\hat{I}})$, i.e., $$\label{psi_ref} \psi_{n,i}(t) := {\hat{\psi}}_i( {\upomega}_n^{-1}(t) ) \qquad\forall\; t\in \bar{I}_n ,\; i = 0, \ldots , k-1 .$$ By the property , the initial condition and the continuity (with respect to time) of the discrete solution $u_h:I\to W$ is equivalent to the conditions: $$\label{U-left} U_1^{0} = u_0 \qquad\text{and}\qquad U_n^{0} = U_{n-1}^{k} \quad\forall\; n>2 .$$ We transform the integrals in to the reference interval ${\hat{I}}$ and obtain the following system of equations for the ”coefficients” $U_n^j\in W$, $j=1,\ldots,k$, in the ansatz : $$\label{lin_evol_sys} \sum_{j=0}^{k} \alpha_{i,j} {\left( U_n^j,v \right)_{H}} = \frac{h_n}{2} \int_{{\hat{I}}} {\left< F\left( {\upomega}_n({\hat{t}}), \sum_{j=0}^{k} U^j_n {\hat{\phi}}_j({\hat{t}}) \right), v \right>} {\hat{\psi}}_i({\hat{t}}) \,d{\hat{t}}\qquad\forall\; v\in W$$ where $i=0,\ldots,k-1$, $$\alpha_{i,j} := \int_{{\hat{I}}} {d_{{\hat{t}}} {\hat{\phi}}_{j}}({\hat{t}}) {\hat{\psi}}_i({\hat{t}}) \,d{\hat{t}},$$ and the ”coefficient” $U_n^0\in W$ is known. We approximate the integral on the right hand side of by the ($k+1$)-point [Gau[ß]{}-Lobatto quadrature formula]{}: $$\label{f_GL_approx} \int_{{\hat{I}}} {\left< F\left( {\upomega}_n({\hat{t}}), \sum_{j=0}^{k} U^j_n {\hat{\phi}}_j({\hat{t}}) \right), v \right>} {\hat{\psi}}_i({\hat{t}}) \,d{\hat{t}}\;\approx \; \sum_{\mu=0}^{k} {\hat{w}}_{\mu} {\left< F\left( {\upomega}_n({\hat{t}}_{\mu}), \sum_{j=0}^{k} U^j_n {\hat{\phi}}_j({\hat{t}}_{\mu}) \right), v \right>} {\hat{\psi}}_i({\hat{t}}_{\mu}),$$ where ${\hat{w}}_{\mu}$ are the weights and ${\hat{t}}_{\mu}\in[-1,1]$ are the integration points with ${\hat{t}}_0=-1$ and ${\hat{t}}_k=1$. Let us define the mapped Gau[ß]{}-Lobatto points $t_{n,\mu}\in \bar{I}_n$ and the coefficients $\beta_{i,\mu}$, $\gamma_{j,\mu}$ by $$t_{n,\mu}:={\upomega}_n({\hat{t}}_{\mu}) , \qquad \beta_{i,\mu} := {\hat{w}}_{\mu} {\hat{\psi}}_i({\hat{t}}_{\mu}) , \qquad \gamma_{j,\mu} := {\hat{\phi}}_j({\hat{t}}_{\mu}) .$$ Then, the system is equivalent to the following system of equations for the $k$ unknown ”coefficients” $U_n^j\in W$, $j=1,\ldots,k$, $$\label{lin_sys} \sum_{j=0}^{k} \alpha_{i,j} {\left( U_n^j, v \right)_{H}} = \frac{h_n}{2} \sum_{\mu=0}^{k} \beta_{i,\mu} {\left< F\left( t_{n,\mu}, \sum_{j=0}^{k} \gamma_{j,\mu} U^j_n \right), v \right>} \qquad\forall\; v\in W.$$ with the $k$ ”equations” $i=0,\ldots,k-1$ where $U^0_n = U^k_{n-1}$ for $n>1$ and $U^0_1= u_0$. Once we have solved this system we enter the next time interval and set the initial value of the new time interval $I_{n+1}$ to $U_{n+1}^0 := U_n^{k}$. If the Gau[ß]{}-Lobatto formula would be exact for the right hand side of this time marching process would solve the global time discretization exactly. Since in general there is an integration error we call the time marching process corresponding to simply the ”cGP(k)-[method]{}”. In principle, we have to solve a coupled system for the $U_n^j\in W$ which could be very expensive. However, by a clever choice of the functions ${\hat{\phi}}_j$ and ${\hat{\psi}}_i$ it is possible to uncouple the system to a large extend. In the following, we will discuss this issue for the special methods cGP(1), cGP(2) and for the general method cGP($k$), $k\ge 3$. In all cases, we choose the basis functions ${\hat{\phi}}_j\in{{\mathbb P}}_k({\hat{I}})$ as the Lagrange basis functions with respect to the Gau[ß]{}-Lobatto points ${\hat{t}}_{\mu}$, i.e., $$\label{phih_choice} {\hat{\phi}}_j({\hat{t}}_{\mu}) = \delta_{j,\mu} \qquad\forall\; j, \mu\in \{0, \ldots, k \}.$$ Then, the method reduces to $$\label{lin_sys_1} \sum_{j=0}^{k} \alpha_{i,j} {\left( U_n^j, v \right)_{H}} = \frac{h_n}{2} \sum_{j=0}^{k} \beta_{i,j} {\left< F\left( t_{n,j}, U^{j}_n \right), v \right>} \qquad\forall\; v\in W,\; i = 0, \ldots, k-1,$$ and by the choice of the test basis functions ${\hat{\psi}}_i\in{{\mathbb P}}_{k-1}({\hat{I}})$ we try to get suitable values for the coefficients $\alpha_{i,j}$ and $\beta_{i,j}$. In the following, we will use the following abbreviation and assumption: $$\label{F_j} F_n^j(U_n^j) := F(t_{n,j}, U_n^j) \in H' \qquad\forall\; j = 0, \ldots, k, \; n = 1, \ldots, N.$$ ### The cGP(1) method\[cGP1\_method\] We use the 2-point Gau[ß]{}-Lobatto formula (trapezoidal rule) with ${\hat{w}}_0 = {\hat{w}}_1 = 1$ and ${\hat{t}}_0=-1$, ${\hat{t}}_1=1$. The only test function ${\hat{\psi}}_0$ is chosen as ${\hat{\psi}}_0({\hat{t}})=1$. Then, we obtain $$\alpha_{0,0}=-1, \quad \alpha_{0,1}=1, \quad \beta_{0,0} = \beta_{0,1} = 1 .$$ Using the notation $U^{n-1} := u_h(t_{n-1}) = U_n^0$ and $U^{n} := u_h(t_{n}) = U_n^1$, we obtain the following equation for the ”unknown” $U^n\in W$ : $$\label{cGP1_eq} {\left( U^n,v \right)_{H}} - {\left( U^{n-1},v \right)_{H}} = \frac{h_n}{2} \left\{ {\left< F(t_{n-1}, U^{n-1}) + F(t_{n}, U^n), v \right>} \right\}$$ for all $v\in W$ which is the well-known [Crank-Nicolson method]{}. In operator notation it can be written in the equivalent form: $$\label{cGP1_M_form} U^n = U^{n-1} + \frac{h_n}{2} {M}^{-1}\left\{ F(t_{n-1}, U^{n-1}) + F(t_{n}, U^n) \right\}.$$ ### The cGP(2) method\[cGP2\_method\] We use the 3-point Gau[ß]{}-Lobatto formula (Simpson rule) with ${\hat{w}}_0 = {\hat{w}}_2 = 1/3$, ${\hat{w}}_1 = 4/3$ and ${\hat{t}}_0=-1$, ${\hat{t}}_1=0$, ${\hat{t}}_2=1$. For the test functions ${\hat{\psi}}_i\in{{\mathbb P}}_1({\hat{I}})$, we choose $${\hat{\psi}}_0({\hat{t}}) = - \frac{3}{4} {\hat{t}}, \quad {\hat{\psi}}_1({\hat{t}}) = 1 .$$ Then, we get $$(\alpha_{i,j}) = \begin{pmatrix} -1/2 & 1 & -1/2 \\ -1 & 0 & 1 \end{pmatrix}, \quad (\beta_{i,j}) = \begin{pmatrix} 1/4 & 0 & -1/4 \\ 1/3 & 4/3 & 1/3 \end{pmatrix} \quad$$ and the assumption , the system to compute the ”unknowns” $U_n^1, U_n^2 \in W$ from the known $U_n^0 = U_{n-1}^2$ reads: $$\begin{aligned} \label{cGP2_1} U_n^1 &=& \frac{1}{2} U_n^0 + \frac{1}{2} U_n^2 + \frac{h_n}{8} M^{-1} \left\{ F_n^0(U_n^0) - F_n^2(U_n^2) \right\} \\[1mm] \label{cGP2_2} U_n^2 &=& U_n^0 + \frac{h_n}{6} M^{-1} \left\{ F_n^0(U_n^0) + 4 F_n^1(U_n^1) + F_n^2(U_n^2) \right\} . \end{aligned}$$ Let us denote the value for $U_n^1$ computed from and depending on $U_n^2$ by $U_n^1 = G_n^1( U_n^2 )$ where $G_n^1: W \to W$ in general is a nonlinear operator. We substitute this in the equation and get, for the unknown $U_n^2\in W$, the following fixed point equation : $$\label{cGP2_fixpoint} U_n^2 = G_n^2 (U_n^2) := U_n^0 + \frac{h_n}{6} {M}^{-1}\left\{ F_n^0(U_n^0) + 4 F_n^1( G_n^1(U_n^2) ) + F_n^2(U_n^2) \right\}$$ The mapping $G_n^2 : W \to W$ is a contraction if the time step size ${{\uptau}_{n}}$ is sufficiently small. Gauss implicit Runge-Kutta methods ---------------------------------- For the general autonomous first order differential equations $$\label{aut_de} y'(t)=f(y(t)),$$ where for system , we choose $ y = \left( \begin{array}{cc} p \\ q \end{array} \right)$ and $f(y) = \left( \begin{array}{cc} -\triangledown_q V(q) \\ -\triangledown_p T(q) \end{array} \right).$ Runge-Kutta methods are defined as $$y_{n+1}=y_{n}+h\sum _{i=1}^{s} b_{i}f(Y_{i})$$ and $$Y_{i}=y_{n}+h\sum _{i=1}^{s} a_{ij}f(Y_{j})$$ where the coefficients $a_{ij}$, $b_i$ and stage $s$ determine the method. The Gauss methods have the highest possible order $r=2s$ and are symplectic and symmetric. We exclusively consider $s=2$, fourth order method for a fair comparison. General linear methods ---------------------- General linear methods provide numerical solutions of initial value problems of the form A general linear method is of the form, $$\begin{aligned} Y &= h(A\otimes I)f(Y)+ (U\otimes I)y^{[n-1]},\\ y^{[n]}&= h(B\otimes I)f(Y)+ (V\otimes I)y^{[n-1]}.\end{aligned}$$ where $A\otimes I$ is the Kronecker product of the matrix $A$ and the identity matrix $I$ and $h$ represents the step size. The $s-$component vector $Y$ are the stages and $f(Y)$ are the stage derivatives. The vector $y^{[n-1]}$ with $r-$components is an input at the beginning of a step and results in output approximation $y^{[n]}$. With a slight abuse of notation, we can write, $$\begin{aligned} \label{general_lin} Y &= hAf(Y)+ Uy^{[n-1]},\nonumber \\ y^{[n]} &= hBf(Y)+ Vy^{[n-1]}.\end{aligned}$$ The matrices $A$, $U$, $V$ and $B$ represent a particular general linear method and are generally displayed as, $$\label{matrix_glm} \left[ {\begin{array}{c\|c} A & U \\ \hline B & V \end{array} } \right].$$ A fourth order symmetric $G$-symplectic general linear method is constructed with four stages $(s=4)$ and three input values $(r=3)$. The coefficeints of the method are given in [@butcher_yousaf]. Numerical Experiments ===================== We performed numerical comparisons of the continuous Galerkin Petrov time discretization scheme , general linear method and implicit Gauss R-K method all having the same order four, for some Hamiltonian systems including simple harmonic oscillator, Kepler’s problem with different eccentricities and molecular dynamical problems. Throughout the comparison, continuous Galerkin Petrov scheme is denoted by acronym cGP(2), while general linear method and implicit Gauss R-K method are represented by the acronym glm4 and irk4 respectively. The emphasis in our comparison is on the accuracy of solution, including the phase information, enrgy conservation and CPU time using above discussed methods. For each method and problem, we used different stepsizes and several intervals of integration. Stepsizes were chosen as a compromise between having small truncation error and performing efficient integration on each step. The accuracy of the solution was measured by the $L_2$ norm of the absolute global error in the position and velocity coordinates and is denoted by $E_g(t)$. The relative error in Hamiltonian is defined as $$E_e(t)=\frac{E(t)-E(0)}{E(0)}.$$ Growth of global error is measured for first two problems as their exact solution exists, while relative error in Hamiltonian $E_e(t)$ is calcultaed for all problems. We also measured computational effort using the CPU time. All the comparisons are done on the same machine and are optimized using MATLAB. Simple Harmonic Oscillator {#simple-harmonic-oscillator .unnumbered} -------------------------- As an example of simple harmonic oscillator a mass spring system having kinetic energy ${p^{2}}/({2m})$, where $p=mv$ is the momentum of the system and potential energy $\frac{1}{2}kq^{2}$. Where $q$ is distance from the equilibrium, $m$ is the mass of the body which is attached to spring and $k$ is constant of proportionality often called as spring constant. Here the Hamiltonian is the total energy of the system and has one degree of freedom $$H(q,p)=\frac{1}{2}kq^{2} + \frac{p^{2}}{2m}.$$ The equations of motion from the Hamiltonian are $$q'=\frac{\partial H(q,p)}{\partial p}=p,\;\;\;\;\;\;\;\;\:\:\:\:p'=-\frac{\partial H(q,p)}{\partial q}=-q.$$ We compared the problem using different stepsizes of $h=0.005, 0.01, 0.025,$ and $0.05$. Figure \[fig:sho\_ge\] gives the log-log graph for time versus global error $E_g(t)$ and relative error in Hamiltonian $E_e(t)$ using stepsize $h=0.005$ for the time interval \[0, 1000\]. We found almost the same behavior of error growth for position and Hamiltonian using the rest of stepsizes. In Figure \[fig:sho\_ge\], the top plot gives the growth of global error and is approximately same for all tested methods, irk4 and cGP(2) having the least error while glm4 with slightly bigger error. In bottom plot of figure \[fig:sho\_ge\], the error in Hamiltonian is conserved by the methods. We also calculated the error growth according to Brouwer’s law [@bl], our calculation shows that the exponent of time is 1 and 0.6 for $E_g(t)$ and $E_e(t)$ respectively, closed to its expected value. Table \[sho\_table\] gives the cost of integration for simple harmonic oscillator using all stepsizes. The table lists the stepsizes, maximum of global error, maximum of Hamiltonian error and CPU time. We observe from the Table \[sho\_table\] that cGP(2) used the least CPU time and also having the least value for maximum of global error except for $h=0.005$, where irk4 having the least end point global error, may be because of entering in a dip also depicted in Figure \[fig:sho\_ge\]. The methods irk4 and glm4 are using eight and sixteen times more CPU time than cGP(2) giving similar accuracy for $h=0.005$. [l\|cccc]{} & & & &\ & & & &\ \ cGP(2) & $0.05$ & $8.67\times10^{-6}$ & $4.08\times10^{-14}$& $2.8$\ cGP(2) & $0.025$ & $5.42\times10^{-7}$ & $1.07\times10^{-13}$& $5.3$\ cGP(2) & $0.01$ & $1.38\times10^{-8}$ & $1.12\times10^{-13}$& $12.1$\ cGP(2) & $0.005$ & $8.68\times10^{-10}$ & $1.31\times10^{-13}$& $23.9$\ \ glm4 & $0.05$ & $2.51\times10^{-5}$ & $4.67\times10^{-11}$& $3.2$\ glm4 & $0.025$ & $1.57\times10^{-6}$ & $7.57\times10^{-13}$& $11.6$\ glm4 & $0.01$ & $4.09\times10^{-8}$ & $3.1\times10^{-15}$& $78.6$\ glm4 & $0.005$ & $3.34\times10^{-9}$ & $1.14\times10^{-15}$& $395.3$\ \ irk4 & $0.05$ & $8.67\times10^{-6}$ & $6.43\times10^{-15}$& $5.7$\ irk4 & $0.025$ & $5.41\times10^{-7}$ & $2.51\times10^{-14}$& $11.5$\ irk4 & $0.01$ & $1.31\times10^{-8}$ & $1.97\times10^{-14}$& $45.0$\ irk4 & $0.005$ & $6.98\times10^{-11}$ & $3.68\times10^{-14}$& $191.8$\ ![The growth of global error and relative error in Hamiltonian for Simple harmonic oscillator using stepsize $h=0.005$.[]{data-label="fig:sho_ge"}](comp_both_1000_0.005_sho.eps){width="5in" height="2in"} ### Kepler’s Problem {#keplers-problem .unnumbered} Kepler’s problem is two body orbital problem in which the bodies are moving under their mutual gravitational forces. We can assume that one body is fixed at the origin and the second body is located in the plane with coordinates $(q_{1},q_{2}).$ The solution of this problem is used in many important applications which includes the determination of orbits for new asteroids and the measurement of orbits for the two primary bodies in a restricted three body problem. The Hamiltonian of the system can be written in separable form as [@hairer_gni] $$H(q,p)=\frac{1}{2} (p_{1}^{2}+p_{2}^{2}) - \frac{1}{\sqrt{q_{1}^{2}+q_{2}^{2}}}$$ This can be written as $H = T + V$, where $T = (p_{1}^{2}+p_{2}^{2})/{2}$ and $V = -1/{\sqrt{q_{1}^{2}+q_{2}^{2}}}$ are kinetic and potential energy of the system respectively. As like the previous problem, this system is also autonomous so the Hamiltonian $H$ is a conserved quantity. The equations of motion are $$q'_{1}=p_{1},\:\:\:\:\:\:\;\;\;\:\;\:q'_{2}=p_{2}$$ $$\label{kepler} p'_{1}=q''_{1}=-\frac{q_{1}}{(q_{1}^2+q_{2}^2)^\frac{3}{2}}$$ $$p'_{2}=q''_{2}=-\frac{q_{2}}{(q_{1}^2+q_{2}^2)^\frac{3}{2}}$$ with the initial conditions $$q_{1}(0)=1-e,\;\;\;q_{2}(0)=0,\;\;\;q'_{1}(0)=0,\;\;\;q_{2}(0)=\sqrt{\frac{1+e}{1-e}}$$ where $e$ is eccentricity $0 \leq e < 1$. The exact solution of the above equations (\[kepler\]) is $$y_{1} = \cos(E) - e,\;\;\;\;\;y_{2} = \sqrt{1-e^{2}}\sin(E),$$ and $$y_{1}' = -\sin(E)(1-e\cos(E))^{-1},\;\;\;\;\;y_{2}' = \sqrt{(1-e^{2})}\cos(E)(1-e\cos(E))^{-1},$$ where the eccentric anomaly $E$ satisfies Kepler’s equation $t=E-e\sin(E)$. Since Kepler’s equation is implicit in $E$, the equation is usually solved using a non-linear equation solver, although useful analytical approximations can be found for smaller eccentricity. The integrations are performed for Kepler’s problem with different eccentricities $e=0, 0.5~ \mathtt{and}~ 0.9$. The integration is done for $1000$ periods for $e=0$ and 100 periods for $e=0.5 ~ \mathtt{and}~ 0.9$. For each method, we measured $E_g(t)$ and $E_e(t)$ throughout the interval of integration. A variety of different stepsizes are used to analyze the behaviour of error growth. We used the stepsizes of $h=\frac{2\pi}{400}$, $h=\frac{2\pi}{800}$ $h=\frac{2\pi}{1600}$, $h=\frac{2\pi}{3200}$ and $h=\frac{2\pi}{6400}$ for eccentricities $e=0,0.5 ~\mathtt{and}~ 0.9$. A log-log plot of time against error is given for Kepler’s problem in Figures \[fig:kep0\_ge\] and \[fig:kep0p9\_ge\] using eccentricities 0 and 0.9 respectively. Growth of errors in both quantities behave in the same manner as for e=0.5. It is seen that the global error growth is approximately linear for cGP(2), irk4 and glm4, i. e., growing as $t^{0.9}$ (see figures \[fig:kep0\_ge\] and \[fig:kep0p9\_ge\]). The error in Hamiltonian remains conserved for cGP(2), irk4 and glm4 for the intervals of integration. Our calculation shows that for $E_e(t)$ grows as $t^{0.6}$, showing a good agreement to its expected value. The cGP(2) exhibits a smaller error even the problem becomes more eccentricitic (see Figures \[fig:kep0\_ge\] and \[fig:kep0p9\_ge\]). We also measured the cost of integration for Kepler’s problem using all stepsizes for all three eccentricities. Tables \[kepe0\_table\], \[kep0p5\_table\] and \[kep0p9\_table\] lists the stepsizes, maximum of global error, maximum of Hamiltonian error and CPU time for $e=0, e=0.5 ~ \mathtt{and}~ 0.9$ respectively. We observe from the information depicted in tables, that cGP(2) used the least CPU time and also having the least value for maximum of global error for all the stepsizes. For e=0, using the least stepsize i.e $h=\frac{2\pi}{6400}$, irk4 and glm4 used 506 and 26 times more CPU time than cGP(2). While for e=0.5 and 0.9, irk4 and glm4 used nearly 55 and 24 times more CPU time than cGP(2). [l\|cccc]{} & & & &\ & & & &\ \ cGP(2) & $2\pi/400$ & $4.88\times10^{-6}$ & $4.07\times10^{-13}$& $93.4$\ cGP(2) & $2\pi/800$ & $2.54\times10^{-7}$ & $7.54\times10^{-12}$& $192.7$\ cGP(2) & $2\pi/1600$ & $6.56\times10^{-8}$ & $1.28\times10^{-11}$& $378$\ cGP(2) & $2\pi/3200$ & $1.38\times10^{-8}$ & $2.28\times10^{-12}$& $760.5$\ cGP(2) & $2\pi/6400$ & $4.54\times10^{-9}$ & $1.49\times10^{-12}$& $1487$\ \ glm4 & $2\pi/400$ & $2.36\times10^{-5}$ & $2.35\times10^{-14}$& $3464$\ glm4 & $2\pi/800$ & $1.56\times10^{-6}$ & $3.06\times10^{-14}$& $12172$\ glm4 & $2\pi/1600$ & $1.66\times10^{-7}$ & $9.39\times10^{-14}$& $47724$\ glm4 & $2\pi/3200$ & $8.59\times10^{-8}$ & $5.32\times10^{-13}$& $180716$\ glm4 & $2\pi/6400$ & $1.05\times10^{-8}$ & $9.98\times10^{-12}$& $752864$\ \ irk4 & $2\pi/400$ & $1.04\times10^{-5}$ & $4.72\times10^{-14}$& $1598$\ irk4 & $2\pi/800$ & $5.68\times10^{-7}$ & $2.79\times10^{-14}$& $6348$\ irk4 & $2\pi/1600$ & $2.98\times10^{-7}$ & $3.28\times10^{-14}$& $23609$\ irk4 & $2\pi/3200$ & $6.58\times10^{-8}$ & $5.17\times10^{-13}$& $93215$\ irk4 & $2\pi/6400$ & $1.27\times10^{-8}$ & $7.18\times10^{-12}$& $39460$\ [l\|cccc]{} & & & &\ & & & &\ \ cGP(2) & $2\pi/400$ & $1.06\times10^{-4}$ & $2.83\times10^{-8}$& $10.2$\ cGP(2) & $2\pi/800$ & $6.63\times10^{-6}$ & $1.77\times10^{-9}$& $19.9$\ cGP(2) & $2\pi/1600$ & $4.15\times10^{-7}$ & $1.11\times10^{-10}$& $39.8$\ cGP(2) & $2\pi/3200$ & $2.54\times10^{-8}$ & $7.01\times10^{-12}$& $79.2$\ cGP(2) & $2\pi/6400$ & $2.9\times10^{-9}$ & $1.02\times10^{-12}$& $157.5$\ \ glm4 & $2\pi/400$ & $2.1\times10^{-4}$ & $5.97\times10^{-8}$& $43.8$\ glm4 & $2\pi/800$ & $1.31\times10^{-5}$ & $3.74\times10^{-9}$& $182.3$\ glm4 & $2\pi/1600$ & $8.2\times10^{-7}$ & $2.34\times10^{-10}$& $688.2$\ glm4 & $2\pi/3200$ & $3.31\times10^{-8}$ & $1.46\times10^{-11}$& $2366$\ glm4 & $2\pi/6400$ & $2.16\times10^{-8}$ & $1.38\times10^{-12}$& $8789$\ \ irk4 & $2\pi/400$ & $8.84\times10^{-5}$ & $1.89\times10^{-8}$& $22.1$\ irk4 & $2\pi/800$ & $5.53\times10^{-6}$ & $1.18\times10^{-9}$& $68.5$\ irk4 & $2\pi/1600$ & $3.43\times10^{-7}$ & $4.71\times10^{-11}$& $244$\ irk4 & $2\pi/3200$ & $1.04\times10^{-8}$ & $4.66\times10^{-12}$& $955$\ irk4 & $2\pi/6400$ & $2.51\times10^{-8}$ & $3.13\times10^{-13}$& $3830$\ [l\|cccc]{} & & & &\ & & & &\ \ cGP(2) & $2\pi/400$ & $4.87$ & $5.23\times10^{-3}$& $10.4$\ cGP(2) & $2\pi/800$ & $1.68$ & $2.43\times10^{-4}$& $20.2$\ cGP(2) & $2\pi/1600$ & $1.92\times10^{-1}$ & $1.42\times10^{-5}$& $41.1$\ cGP(2) & $2\pi/3200$ & $1.3\times10^{-2}$ & $8.74\times10^{-7}$& $80.6$\ cGP(2) & $2\pi/6400$ & $8.41\times10^{-6}$ & $5.43\times10^{-8}$& $162.1$\ \ glm4 & $2\pi/400$ & $111.8$ & $6.5\times10^{-4}$& $44.1$\ glm4 & $2\pi/800$ & $4.54$ & $2.23\times10^{-4}$& $172$\ glm4 & $2\pi/1600$ & $1.46$ & $1.62\times10^{-5}$& $676$\ glm4 & $2\pi/3200$ & $9.77\times10^{-2}$ & $1.04\times10^{-6}$& $2482$\ glm4 & $2\pi/6400$ & $6.14\times10^{-3}$ & $6.53\times10^{-8}$& $8402$\ \ irk4 & $2\pi/400$ & $4.54$ & $1.73\times10^{-5}$& $22.4$\ irk4 & $2\pi/800$ & $1.27$ & $1.36\times10^{-5}$& $67.8$\ irk4 & $2\pi/1600$ & $1.32\times10^{-1}$ & $1.38\times10^{-6}$& $257.5$\ irk4 & $2\pi/3200$ & $9.01\times10^{-3}$ & $9.45\times10^{-8}$& $1086$\ irk4 & $2\pi/6400$ & $5.73\times10^{-4}$ & $6.03\times10^{-9}$& $3786$\ ![The growth of global error and relative error in Hamiltonian for kepler’s problem with $e=0$ using stepsize $2\pi/6400$ for $10^3$ periods.[]{data-label="fig:kep0_ge"}](comp_ke0_both_1000p_2pi1600.eps){width="5in" height="2in"} ![The growth of global error and relative error in Hamiltonian for kepler’s problem with $e=0.9$ using stepsize $2\pi/1600$ for $10^2$ periods.[]{data-label="fig:kep0p9_ge"}](comp_ke0p9_both_100p_2pi6400.eps){width="5in" height="2in"} Molecular Dynamical Problem {#molecular-dynamical-problem .unnumbered} --------------------------- We consider the interaction of seven Argon atoms in two dimension, where one of the atom is centered by six atoms which are symmetrically arranged [@bies_skeel]. The Hamiltonian for the molecular dynamics is written as [@hairer_gni] $$H(q,p)=\frac{1}{2} \sum_{i=1}^{7} \frac{1}{m_{i}}p_{i}^{T}p_{i} + \sum_{i=2}^{7}\sum_{j=1}^{i-1} V_{ij}\Vert q_{i}-q_{j}\Vert$$ where $V_{ij}(r)$ are potential functions. Here $q_{i}$ and $p_{i}$ are positions and generelized momenta for the atoms. And $m_{i}$ denotes the atomic mass of the $i$th atom. $$V_{ij}(r)=4\varepsilon_{ij}\left( \left( \frac{\sigma_{ij}}{r}\right)^{12} - \left( \frac{\sigma_{ij}}{r}\right)^6 \right).$$ The equations of motion for the frozen Argon crystals are given as $$q''_{i}(t)=\frac{24\varepsilon\sigma^6}{m_{i}} \sum_{j=1,j\neq i}^{7}\left[ \frac{(q_{j}-q_{i})}{\Vert q_{j}-q_{i}\Vert _{2}^{8}}-2\sigma^6\frac{(q_{j}-q_{i})}{\Vert q_{j}-q_{i}\Vert _{2}^{14}}\right ],\hspace{0.2in} i=1,...,7,$$ where $r=\sigma_{ij} \sqrt[6]{2}$, $m_{i}=66.34\times 10^{-27}\mathtt{[kg]}$, $\sigma_{ij}=\sigma=0.341\mathtt{[nm]}$ and $\varepsilon=1.654028284\times10^{-21}\mathtt{[J]}$. Initial positions and initial velocities are taken in \[nm\] and \[nm/sec\] respectively [@hairer_gni]. In molecular dynamics, since much ineterst is emphasized on macroscopic quantities like Hamiltonian. So we also discussed only the energy conservation of atoms over an interval of length $2 \times 10^5$ \[fsec\] ($1\mathtt{fsec}=10^{-6}$). The experiments are done using the stepsizes of 0.5 fsec, 1 fsec, 2 fsec and 4 fsec. The graphical results are only shown for $h=0.5\times 10^{-6}\mathtt{[fsec]}$ as the error growth using other stepsizes was approximately same. Figure \[fig:mole\_reha\] shows that the tested methods conserve the value of Hamiltonian $H$ even though the conservation is of highly oscillatory, while the error in Hamiltonian for cGP(2) grows as $t^{0.7}$. On the other hand, for irk4 and glm4 the exponent of time is 0.59 and 0.61 respectively. Table \[mole\_table\] gives the cost of integration for molecular dynamical problem using all stepsizes. The table lists the stepsizes, maximum of Hamiltonian error and CPU time. It is observed from the Table \[mole\_table\] that cGP(2) used the least CPU time for all the stepsizes used but exhibiting slightly big maximum of Hamiltonian error. The methods irk4 and glm4 having almost the same error growth for the integrated interval. [l\|ccc]{} & & &\ & & &\ \ cGP(2) & $4$ & $1.24\times10^{-9}$ & $658$\ cGP(2) & $2$ & $6.15\times10^{-11}$ & $1323$\ cGP(2) & $1$ & $7.05\times10^{-12}$ & $2680$\ cGP(2) & $0.5$ & $3.73\times10^{-13}$ & $5821$\ \ glm4 & $4$ & $9.24\times10^{-11}$ & $1772$\ glm4 & $2$ & $5.58\times10^{-12}$ & $4300$\ glm4 & $1$ & $3.53\times10^{-13}$ & $11367$\ glm4 & $0.5$ & $9.41\times10^{-14}$ & $34591$\ \ irk4 & $4$ & $3.72\times10^{-11}$ & $2060$\ irk4 & $2$ & $2.32\times10^{-12}$ & $4272$\ irk4 & $1$ & $1.59\times10^{-13}$ & $10020$\ irk4 & $0.5$ & $2.34\times10^{-14}$ & $23596$\ ![The growth of relative error in Hamiltonian using $h=0.5\times 10^{-6}\mathtt{[fsec]}$ for molecular dynamical problem over an interval of $2 \times 10^5$ \[fsec\].[]{data-label="fig:mole_reha"}](comp_rehe_0p5em6.eps){width="5in" height="2in"} Summary ======= We implemented and analyzed the cGP(2) for Hamiltonian systems such as harmonic oscillator, Kepler’s problem and molecular dynamical problem. The obtained results are also compared with symplectic methods irk4 and glm4. It is shown that the cGP(2) method conserves the hamiltonian as other tested symplectic methods do. Moreover, giving the efficiency approximately same as other methods yield, cGP(2) uses marginally less CPU time than compared methods. [99]{} Hairer, E., Lubich, C., Wanner, G. Geometric Numerical Integration, Springer-Verlag, Berlin, Heidelberg Germany, 2006. Eirola, T. and Sanz-Serna, J.M. Conservation of integrals and symplectic structure in the integration of differential equations by multistep methods, Numer. Math., 61:281-290, 1992. Hairer, E. Conjugate-symplecticity of linear multistep methods, J. Comput. Math., 26(5):657–659, 2008. Sanz-Serna, J. M., Calvo, M. P. Numerical Hamiltonian Problems, Chapman and Hall, Great Britain, 1994. Schieweck, F. A-stable discontinuous Galerkin-Petrov time discretization of higher order. J. Numer. Math., 18(1):25 – 57, 2010. Hussain, S., Schieweck, F. and Turek, S. Higher order Galerkin time discretizations and fast multigrid solvers for the heat equation. J. Numer.Math., 19(1):41–61, 2011. Aziz, A. K. and Monk, P. Continuous finite elements in space and time for the heat equation. Math. Comp., 52(186):255-274, 1989. Thom´ee, V. Galerkin finite element methods for parabolic problems, volume 25 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, second edition, 2006. Butcher, J. C. Numerical Methods for Ordinary Differential Equations, John Wiley and Sons, Ltd, 2008. Butcher, J. C., Habib, Y., Hill, A. T. and Norton, T. J. T. The control of parasitism in G-symplectic methods, SIAM J. Numer. Anal., 52(5):2440–2465, 2014. Habib, Y. Long-Term Behaviour of G-symplectic Methods, PhD Thesis, The University of Auckland, 2010. https://researchspace.auckland.ac.nz/handle/2292/6641 Brouwer D. On the accumulation of errors in numerical integration. Astron. J. , 46:149-153, 1937 Biesiadecki, J. J. and Skeel, R. D. 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--- abstract: 'Let $G$ be a reductive affine algebraic group defined over $\mathbb C$, and let $\nabla_0$ be a meromorphic $G$-connection on a holomorphic $G$-bundle $E_0$, over a smooth complex curve $X_0$, with polar locus $P_0 \subset X_0$. We assume that $\nabla_0$ is irreducible in the sense that it does not factor through some proper parabolic subgroup of $G$. We consider the universal isomonodromic deformation $(E_t\to X_t, \nabla_t, P_t)_{t\in \mathcal{T}}$ of $(E_0\to X_0, \nabla_0, P_0)$, where $\mathcal{T}$ is a certain quotient of a certain framed Teichmüller space we describe. We show that if the genus $g$ of $X_0$ satisfies $g\geq 2$, then for a general parameter $t\in \mathcal{T}$, the $G$-bundle $E_t\to X_t$ is stable. For $g\geq 1$, we are able to show that for a general parameter $t\in \mathcal{T}$, the $G$-bundle $E_t\to X_t$ is semistable.' address: - 'School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India' - 'Institut de Recherche Mathématique Avancée, 7 rue René-Descartes, 67084 Strasbourg Cedex, France' - 'Department of Mathematics, McGill University, Burnside Hall, 805 Sherbrooke St. W., Montreal, Que. H3A 0B9, Canada' author: - Indranil Biswas - Viktoria Heu - Jacques Hurtubise title: Isomonodromic deformations of irregular connections and stability of bundles --- Introduction ============ The natural correspondence between a flat connection on a principal bundle defined over a variety and its monodromy representation is a recurrent theme in mathematics, with a long history, as evidenced by the name, Riemann–Hilbert problem, given to one of the core questions of the subject. This basic problem consists in asking when a representation of the fundamental group of a punctured Riemann sphere can be realized by a flat connection on a holomorphically trivial bundle, with simple poles at the punctures; the answer, which is most of the time, but not always ([@Plemelj], [@Dekkers], [@Bolibruch1], [@Bolibruch2], [@Kostov]), is in itself an interesting chapter of the history of mathematics. If one relaxes the condition of triviality, and asks whether the representation can be realized on a principal bundle, then the answer is always yes, and indeed the correspondence is quite natural. The question then becomes that of whether the bundle can be made trivial, either by some twists at the punctures (Schlesinger transformations) or by deforming the location of the punctures (isomonodromic deformations). The deformation theoretic version of the Riemann–Hilbert problem becomes: Given a logarithmic connection $(E_0\, ,\nabla_0)$ on $\mathbb{P}^1_{\mathbb C}$ with polar divisor $D_0$ of degree $n$, is there a point $t$ of the Teichmüller space $\mathrm{Teich}_{0,n}$ such that the underlying holomorphic vector bundle $E_t\,=\,\mathcal{E}\vert_{\mathbb{P}^1_{\mathbb C} \times\{t\}}$ in the universal isomonodromic deformation $(\mathcal{E}\, , \nabla)$ for $(E_0\, ,\nabla_0)$ is trivial? A partial answer to this question is given, in the case of vector bundles of rank two, by the following theorem of Bolibruch: \[thmBolibruch\] Let $(E_0\, ,\nabla_0)$ be an irreducible trace–free logarithmic rank two connection with $n\,\geq\, 4$ poles on $\mathbb{P}^1_{\mathbb C}$ such that each singularity is resonant. There is a proper closed complex analytic subset $\mathcal{Y}\,\subset\, \mathrm{Teich}_{0,n}$ such that for all $t\,\in\, \mathrm{Teich}_{0,n}\setminus \mathcal{Y}$, the holomorphic vector bundle $E_t\,=\,\mathcal{E}\vert_{\mathbb{P}^1_{ \mathbb C}\times\{t\}}$ underlying the universal isomonodromic deformation $(\mathcal{E}\, , \nabla)$ of $(E_0,\nabla_0)$ is trivial. In [@Viktoria1], it is shown that the resonance condition in Theorem \[thmBolibruch\] is redundant. This gives an indication for the Riemann sphere; one can actually consider a similar problem for an arbitrary Riemann surface. Indeed, triviality of a vector bundle over the Riemann sphere is equivalent to being semi-stable of degree zero. On a general Riemann surface, the question of whether one can realize a representation by a semi-stable vector bundle of degree zero was considered in [@Helene1; @Helene2]. The deformation version, whether a logarithmic connection on a bundle over an arbitrary Riemann surface admits an isomonodromic deformation to a logarithmic connection on a stable or semi-stable bundle, was treated in [@BHH]; see also [@Viktoria1] for rank two. We recall from [@BHH], [@Viktoria1]: \[R-O\] Let $X$ be a compact connected Riemann surface of genus $g$, and let $D_0\, \subset\, X $ be an ordered subset of it of cardinality $n$. Let $G$ be a reductive affine algebraic group defined over $\mathbb C$. Let $E_G$ be a holomorphic principal $G$–bundle on $X$ and $\nabla$ a logarithmic connection on $E_G$ with polar divisor $D_0$. Let $(\mathcal{E}_G\, , \nabla)$ be the universal isomonodromic deformation of $(E_G\, ,\nabla_0)$ over the universal Teichmüller curve $\tau\,:\, (\mathcal{X}\, , \mathcal{D})\,\longrightarrow\, \mathrm{Teich}_{g,n}$. For any point $t\, \in\, \mathrm{Teich}_{g,n}$, the restriction $\mathcal{E}_G\vert_{\tau^{-1}(t)}\,\longrightarrow\, {\mathcal X}_t\,:=\, \tau^{-1}(t)$ will be denoted by $\mathcal{E}^t_G$. 1. Assume that $g\, \geq\, 2$ and $n\,=\, 0$. Then there is a closed complex analytic subset $\mathcal{Y}\, \subset\, \mathrm{Teich}_{g,n}$ of codimension at least $g$ such that for any $t\,\in\, \mathrm{Teich}_{g,n} \setminus \mathcal{Y}$, the holomorphic principal $G$–bundle $\mathcal{E}^t_G\,\longrightarrow\, {\mathcal X}_t$ is semistable. 2. Assume that $g\,\geq\, 1$, and if $g\,=\, 1$, then $n\, >\, 0$. Also, assume that the initial monodromy representation for $\nabla $ at $t=0$ does not factor through some proper parabolic subgroup of $G$. Then there is a closed complex analytic subset $\mathcal{Y}'\, \subset\, \mathrm{Teich}_{g,n}$ of codimension at least $g$ such that for any $t\,\in\, \mathrm{Teich}_{g,n} \setminus \mathcal{Y}'$, the holomorphic principal $G$–bundle $\mathcal{E}^t_G$ is semistable. 3. Assume that $g\,\geq\, 2$. Assume that the monodromy representation for $\nabla $ at $t=0$ does not factor through some proper parabolic subgroup of $G$. Then there is a closed complex analytic subset $\mathcal{Y}''\, \subset\,\mathrm{Teich}_{g,n}$ of codimension at least $g-1$ such that for any $t\,\in\, \mathrm{Teich}_{g,n}\setminus \mathcal{Y}''$, the holomorphic principal $G$–bundle $\mathcal{E}^t_G$ is stable. Our aim here is to extend this result to connections with irregular singularities, that is connections with higher order poles. Let us consider a triple $$(E_G \longrightarrow X\, , D\, ,\nabla),$$ where $E_G$ is a holomorphic principal $G$–bundle over a compact connected Riemann surface $X$, and $\nabla$ is an integrable holomorphic connection on $E_G$, with possibly irregular singularities bounded by a divisor $D$ on $X$; that is, if $\nabla$ has poles of order $n_i$ at points $p_i$ of $X$, we set $D\,=\, \sum_{i=1}^m n_ip_i$, and let $D_0\,= \,\sum_{i=1}^m p_i$ denote the reduced divisor. We will suppose that the leading order term (i.e., coefficient of $z^{-n_i}$) of the connection at $p_i$ is conjugate to a regular semisimple element $h_{i,-n_i}$ of a fixed Cartan subalgebra $\mathfrak h$ of the Lie algebra $\mathfrak g$ of $G$. By a gauge transformation at the poles, the polar part of the connection can be conjugated to $$h_i(z) dz\,=\, (h_{i,-n_i}z^{-n_i} + h_{i,-n_i+1}z^{-n_i+1}+\cdots +h_{i,-1}z^{-1}) dz\, .$$ If one allows a formal gauge transformation, then the connection itself can be put in this form at the puncture; the power series that effects this transformation though does not typically converge. Instead, there is additional monodromy data, given by Stokes matrices [@JMU]. A good introduction to the theory can be found in [@Sa], and the more advanced results we need have been established in [@Boa1; @Boa2]. We now give a brief outline of the basic ideas. For each irregular singular point, one chooses disks $\Delta_i$ centered at $p_i$, $1\,\leq\, i\,\leq\, m$. On $\Delta_i$, as noted, one has a formal solution $$H_i(z) \,=\, \exp(\int h_i (z) dz)$$ with a monodromy $\mu_i \,=\, \exp( 2\pi\sqrt{-1} h_{i,-1} )$; one partitions the disk into $2n_i-2$ angular sectors ${\mathcal{S}}_{i,j}$ determined by the $ h_{i,-n_i}$. Associated to the intersections ${\mathcal{S}}_{i,2j}\cap {\mathcal{S}}_{i,2j+1}$, there is a fixed (independent of $j$) unipotent radical $U_{+,i}$ of a Borel subgroup associated to $\mathfrak h$; to the intersections ${\mathcal{S}}_{i,2j+1}\cap {\mathcal{S}}_{i,2j+2}$ one has associated the opposite unipotent radical $U_{-,i}$. We choose a base point $q_i$ in ${\mathcal{S}}_{i,1}$. One can then consider on each sector actual integrals $g_{i,j}(z) \,\in\, G$ of the connection asymptotic to the $H_i$, and passing from the sector ${\mathcal{S}}_{i,2j}$ to ${\mathcal{S}}_{i,2j+1}$, the two solutions are related by Stokes factors $u_{+,i,j}$ lying in $U_{+,i}$. In passing from ${\mathcal{S}}_{i,2j+1}$ to ${\mathcal{S}}_{i,2j+2}$, the two solutions are related by Stokes factors $u_{-,i,j}$ lying in $U_{-,i}$. The monodromy of the connection around the singularity is the product $$\rho_i \,= \,\mu_i u_{-,i, n-1}\cdots u_{+,i,2} u_{-,i,1}u_{+,i,1}\, .$$ This monodromy and its decomposition into Stokes factors is defined up to the action of a torus. For the deformations, one has a generalized Teichmüller space $\mathrm{Teich}_{\mathfrak h, g,m}$, which combines the standard $\mathrm{Teich}_{g,m}$ with the irregular polar parts. Note that this combines parameters on the curves with parameters associated to the group; in addition to the standard Teichmüller parameters of the punctured curve, one considers the extra parameters of the “irregular type”, realized as the formal singularity $H_i(z) \,=\, \exp(\int h_i (z) dz)$. More details can be found below. Lying above this deformation on the base, there is a theory of isomonodromic deformations of such connections, generalizing the one we have for the logarithmic case. Over the base parameters, in particular $H_i(z) \,=\,\exp(\int h_i (z) dz)$, which becomes an Abelian transition function at the puncture, one fixes the Stokes factors $u_{\pm,i, j}$ at the irregular singularities, and the representation $\pi_1(X\setminus D_0)\,\longrightarrow \,\text{GL}(n,{\mathbb{C}})$ of the fundamental group. Fixing such isomonodromy data gives a lift of the Teichmüller deformations to a deformation of singular connections. By Malgrange’s theorem, such isomonodromic deformations exist, and determine the connection up to gauge transformations [@Ma] [@Viktoria2]. Our aim will be to show: \[Result\] Assume that the monodromy representation for $\nabla_0$ is irreducible in the sense that it does not factor through some proper parabolic subgroup of $G$. 1. If $g\,\geq\, 1$, then there is a closed complex analytic subset $\mathcal{Y}\, \subset\, \mathrm{Teich}_{\mathfrak h, g,m}$ of codimension at least $g$ such that for any $t\,\in\, \mathrm{Teich}_{\mathfrak h, g,m} \setminus \mathcal{Y}$, the holomorphic principal $G$–bundle $\mathcal{E}^t_G$ is semistable. 2. If $g\,\geq\, 2$, then there is a closed complex analytic subset $\mathcal{Y}'\, \subset\,\mathrm{Teich}_{\mathfrak h, g,m}$ of codimension at least $g-1$ such that for any $t\,\in\, \mathrm{Teich}_{\mathfrak h, g,m}\setminus \mathcal{Y}'$, the holomorphic principal $G$–bundle $\mathcal{E}^t_G$ is stable. The base space ============== We will describe the space $\mathrm{Teich}_{\mathfrak h, g,m}$. The Teichmüller space $\mathrm{Teich}_{ g,m}$ for genus $g$ curves with $m$ marked points is a contractible complex manifold of complex dimension $3g-3+m$, assuming that $3g-3+m\, >\, 0$. We first build a framed Teichmüller space. If the singularity divisor is $D\,=\, \sum_{i=1}^m n_ip_i$, we can enrich the Teichmüller space $\mathrm{Teich}_{ g,m}$ by adding to each point $(\Sigma,\, \sum_{i=1}^m p_i)$ of $\mathrm{Teich}_{g,m}$, the additional data of a coordinate $z_i$ centered at $p_i$, defined to order $n_i-1$ inclusively, for all $n_i\,>\,1$. We note that this additional data at $p_i$ is the choice of an isomorphism of the algebra $m_{p_i}/m^{n_i}_{p_i}$ with $z{\mathbb C}[z]/ z^{n_i}{\mathbb C}[z]$, where $m_{p_i}$ is the ring of holomorphic functions defined around $p_i$ that vanish at $p_i$. The Teichmüller space $\mathrm{Teich}_{ g,m}$ together with the above data produce a framed Teichmüller space $\mathrm{FTeich}_{g,m,n_1,\cdots ,n_m}$. Now consider the extra data of the parameters $h_{i,j}$ of the polar parts of the connection. We set our framed Teichmüller space for deformations of the irregular part of the connections plus punctured curves to simply be a product: $$\mathrm{FTeich}_{{\mathfrak h}, g,m,n_1,\cdots ,n_m} \,=\, \mathrm{FTeich}_{g,m,n_1, \cdots ,n_m}\times \prod_{i=1}^m ({\mathfrak h_0} \times {\mathfrak h }^{n_i-1})\, .$$ Our desired space of deformations $\mathrm{Teich}_{\mathfrak h, g,m}$ will be the quotient of this space by the groups of germs of diffeomorphisms of neighborhoods of $p_i$ which fix $p_i$, acting diagonally on the factors. As the action at each puncture is on truncated power series, one need only act by groups of jets $$J_{p_i, n_i} = \left\{ z\mapsto a_1z+ a_2z^2+\ldots +a_{n_i}z^{n_i}~\middle\|~a_j\in \mathbb{C}, ~ a_1\neq 0\right\};$$ one has $$\mathrm{Teich}_{\mathfrak h, g,m} = \mathrm{FTeich}_{{\mathfrak h}, g,m,n_1,\cdots ,n_m}/_{\prod_iJ_{p_i,n_i}} .$$ (In fact, one would want to go to a universal cover, but for our purposes, this is not necessary as we are just considering the local deformations.) Let us now see what this gives us for infinitesimal deformations. The tangent space of $\mathrm{T}_X(-D_0)$ at any element $(X\, , D_0)\, \in\, \mathrm{T}_X(-D_0)$ is $$\mathrm{H}^1(X,\, \mathrm{T}_X(-D_0))\, .$$ We note that a $1$- cocycle $v$ can be thought of as giving an infinitesimal deformation of the coordinate changes from one patch to another; the $1$-coboundaries must be taken with values in the vector fields vanishing at $D_0$. Such a coboundary, however, affects the form of the irregular polar parts at $p_i$. Indeed, these are not well defined in themselves, as they are acted on by diffeomorphisms of the curve fixing $p_i$. This must be taken into account in the deformation theory. Consider an infinitesimal local diffeomorphism of the curve given at the puncture $z\,=\,0$ by a vector field $v(z) \partial/\partial z$. As we are considering punctured curves, we want $v(0) \,=\, 0$. The changes in the function $H_i(z)$ caused by a change in parametrization, infinitesimally a vector field, should be considered as trivial: in other words, $$H_i(z+{\epsilon}v(z)) \,=\, H_i(z) (1+ {\epsilon}H_i^{-1}(z)H_i'(z) v(z)) \,=\, H_i(z) (1+ {\epsilon}h_i(z)v(z))\, .$$ Thus, for our deformations, we will be interested in the complex $${\mathcal{C}}\,:\, \mathrm{T}_X(-D_0)\, \buildrel{F}\over{ \longrightarrow}\,{{\mathcal{P}}{\mathcal{P}}}\, = \, {\mathcal{O}}_{D-D_0} \otimes \mathfrak h \,=\,\bigoplus_i\mathfrak h^{\oplus n_i-1}\, ;$$ the second sheaf is a sum of skyscraper sheaves supported at the points of $D_0$; the homomorphism $F$ sends a vector field $v$ around $p_i$ to the irregular polar part (${{\mathcal{P}}{\mathcal{P}}}$) of the contraction of $v$ with the connection matrix $h_i$: $$F\,:\, v(z) \,\longmapsto\, {{\mathcal{P}}{\mathcal{P}}}((h_i(z) v(z))) .$$ The first order deformations of the marked curve are given by the cohomology group $\mathrm{H}^1(X,\, \mathrm{T}_X(-D_0) )$; adding in the deformations of the irregular polar parts gives us the global hypercohomology group $${\mathbb{H}}^1(X,\, {\mathcal{C}}) \, = \, \mathrm{T}_{(X,D,H)}\mathrm{Teich}_{\mathfrak h, g,m} \, .$$ We note that ${\mathbb{H}}^1(X,\, {\mathcal{C}})$ coincides with the space of admissible deformations in [@Boa3] of the irregular curve defined by the triple $(X,\, D_0,\, \bigoplus_i\mathfrak h^{\oplus n_i-1})$. In [@Boa3], the space of objects, consisting of a Riemann surface, some marked points on it and irregular types at the marked points, are defined more intrinsically. We have an exact sequence $$\bigoplus_i \mathfrak h^{\oplus n_i-1} \, \longrightarrow \, {\mathbb{H}}^1(X,\,{\mathcal{C}}) \, \longrightarrow \, \mathrm{H}^1(X,\, \mathrm{T}_X(-D_0) ) \, .$$ The elements $\beta$ of $\mathrm{H}^1(X,\, \mathrm{T}_X(-D_0))$ encode extensions $$0 \, \longrightarrow \, \mathrm{T}_X(-D_0) \, \longrightarrow \, {\mathcal{T}}\,\longrightarrow \, {\mathcal{O}}_X \, \longrightarrow \, 0 \, .$$ This can be viewed as the tangent bundle to the infinitesimal one-parameter family of bundles represented by the element $\beta$, with the structure sheaf ${\mathcal{O}}_X$ representing a trivial normal bundle. An element $\widehat\beta$ of ${\mathbb{H}}^1(X,\,{\mathcal{C}})$ mapping to $\beta$ encodes a bit more, namely a diagram $$\begin{xy}\xymatrix{\, \,0\, \, \ar[r]&\, \,\mathrm{T}_X(-D_0)\, \, \ar[r]\ar[d]_{F}&\, \, {\mathcal{T}}\, \,\ar[r]\ar[d]& \, \,{\mathcal{O}}_X\, \,\ar[r]& 0 \\ &\, \,{{\mathcal{P}}{\mathcal{P}}}\, \,\ar@{=}[r]&\, \, {{\mathcal{P}}{\mathcal{P}}}\, .}\end{xy}$$ Deforming the bundle ==================== The Lie algebra of $G$ will be denoted by $\mathfrak g$. Let $\text{ad}(E_G)\,=\, E_G\times^G{\mathfrak g}$ be the adjoint bundle for $E_G$ over $X$. Let $\text{At}(E_G)$ denote the Atiyah bundle for $E_G$; it fits in the Atiyah exact sequence over $X$ $$0\, \longrightarrow\, \text{ad}(E_G) \, \longrightarrow\,\text{At}(E_G) \, \longrightarrow\, \mathrm{T}_X \, \longrightarrow\, 0$$ [@At]. The Atiyah bundle for $E_G$ represents over the base the $G$–invariant vector fields on the principal $G$–bundle $E_G$; the subbundle of invariant vector fields tangent to the fibers is ${\rm ad}(E_G)$. The Atiyah exact sequence produces a short exact sequence $$0\, \longrightarrow\, \text{ad}(E_G) \, \longrightarrow\,\text{At}_{D_0}\,:=\, \text{At}_{D_0}(E_G)\, \longrightarrow\,\mathrm{T}_X(-D_0) \, \longrightarrow\, 0\, ,$$ where $D_0$ is the reduced singular locus of the connection. In [@BHH] it is shown that the deformations of the logarithmic connection, over a curve $X$ that is also being deformed, are parametrized by $\mathrm{H}^1(X,\, \text{At}_{D_0})$. To deal with the higher order poles, we need to consider the sheaf $\mathrm{At}_{D_0}( D-D_0)$ of meromorphic sections of $\mathrm{At}_{D_0}$ with poles living only in the ${\rm ad}(E_G)$ factor, bounded by $D-D_0$: $$0\, \longrightarrow \, \mathrm{ad}(E_G)(D-D_0)\, \longrightarrow \, \mathrm{At}_{D_0}( D-D_0) \, \longrightarrow \, \mathrm{T}_X(-D_0) \, \longrightarrow \, 0\, .$$ Now let us consider deformations of these. We cover our Riemann surface away from the punctures with Stokes sectors $S_{i,j}$, as well as other contractible open sets $V_\nu$; choose flat trivializations on these sets, with the ones on Stokes sectors being compatible with the formal asymptotics. The transition functions on the bundle for these trivializations are then constants, with those between the Stokes sectors being the Stokes matrices. For the puncture, we have the transition functions $H_i(z)$. Re-label the Stokes sectors as being in the set of $V_\nu$; we then have constant transition functions $\Theta_{\nu_1,\nu_2}$ away from the puncture, and $H_i(z)$ at the puncture. Now take a variation $H_i(z)(1 + {\epsilon}\int k_i(z))$ and a cocycle $v_{\nu_1,\nu_2}$ for $\mathrm{T}_X(-D_0)$ , which corresponds to infinitesimal displacements of the coordinate patches with respect to each other; these together arise from a class $\hat \beta$ in ${\mathbb{H}}^1(X,\,{\mathcal{C}})$. We are, in our isomonodromic deformations, deforming the bundle above the curve by keeping the same $\Theta_{\nu_1,\nu_2}$, and modifying the transition function at the puncture by $$H_i(z)(1 + {\epsilon}\int k_i(z))\, .$$ As a deformation of the Atiyah bundle, the former consists of considering the mapping $$\nabla\,: \,\mathrm{T}_X(-D_0)\, \longrightarrow \, \mathrm{At}_{D_0}(D-D_0)\, ,$$ and taking the induced map on the cocycles, i.e. taking $\nabla(v_{\nu_1,\nu_2})$ as a cocycle for $$\mathrm{At}_{D_0}(D-D_0)\, ,$$ which, as we are away from the punctures, we can take to be a cocycle in for $\mathrm{At}_{D_0}$. To this, we add the element $ k_i(z)$ as a cocycle for the deformation of the transition function from the disk around the puncture to the Stokes sectors, for the subbundle $\mathrm{ad}(E_G)$ of $\mathrm{At}_{D_0}$; the sum of the cocycles gives a class $\gamma$ in $\mathrm{H}^1(X,\, \mathrm{At}_{D_0})$. As for the deformation of the curves, an element $\gamma$ of $\mathrm{H}^1(X,\, \mathrm{At}_{D_0} )$ defines an extension $$0\,\longrightarrow \, \mathrm{At}_{D_0}\, \longrightarrow \,{\mathcal{A}}\, \longrightarrow\,{\mathcal{O}}_X\,\longrightarrow\, 0\,,$$ mapping to corresponding extensions of $\mathrm{T}_X(-D_0)$, and so gives a diagram $$\begin{xy}\xymatrix{ &\, \,\mathrm{ad}(E_G)\, \,\ar@{=}[r]\ar[d] &\, \,\mathrm{ad}(E_G)\, \,\ar[d] \\ 0\ar[r]& \, \,\mathrm{At}_{D_0}\, \,\ar[d] \ar[r]& \, \,{\mathcal{A}}\, \,\ar[r]\ar[d] &\, \, {\mathcal{O}}_X\, \,\ar[r]\ar@{=}[d]& \, \, 0 \, \, \\ \, \,0\, \,\ar[r]&\, \,\mathrm{T}_X ( -D_0)\, \,\ar[r]& \, \,{\mathcal{T}}\, \, \ar[r] & \, \,{\mathcal{O}}_X\, \,\ar[r]& \, \, 0 \, .}\end{xy}$$ This represents over ${\epsilon}\,=\, 0$ the $G$–invariant vector fields on our first order extension, the ${\mathcal{O}}_X$–quotient being the normal bundle. Deformations of reductions ========================== Extending a reduction --------------------- The stability of $G$-bundles concerns reductions to a parabolic subgroup: the bundle $E_G$ is stable (respectively, semistable) if for all its reductions $E_P$ to a parabolic subgroup $P$, the associated bundle $$\text{ad}(E_G)/\text{ad}(E_P)\,=\, E_P({\mathfrak g}/{\mathfrak p})$$ has positive (respectively, non-negative) degree, where ${\mathfrak g}$ and ${\mathfrak p}$ are the Lie algebras of $G$ and $P$ respectively. If we want to ensure that the set of non stable bundles is somehow small along the isomonodromic deformation, we must see how reductions to a parabolic extend along a deformation, and in particular try to understand the space of first order obstructions to such an extension. Given a reduction $E_P$, we now have two Atiyah bundles $\mathrm{At}_{D_0}^G$ and $\mathrm{At}_{D_0}^P$ over the surface associated to $E_G$ and $E_P$ respectively. These fit into a diagram: $$\label{e6} \begin{xy}\xymatrix{ & \, \,0\, \,\ar[d] & \, \,0\, \,\ar[d] \\ \, \,0\, \,\ar[r]&\, \, \text{ad}(E_P) \, \,\ar[r]\ar[d] & \, \,\mathrm{At}_{D_0}^P\, \, \ar[r]^{\beta}\ar[d]^{\xi}&\, \, \mathrm{T}X (-D_0)\, \, \ar[r]\ar@{=}[d] & \, \,0\, \,\\ \, \,0\, \,\ar[r]& \, \,\text{ad}(E_G)\, \, \ar[r] \ar[d]^{\omega_1}& \, \,\mathrm{At}_{D_0}^G\, \,\ar[r]^{\sigma}\ar[d]^{\omega} & \, \,\mathrm{T}X (-D_0) \ar[r]\, \, &\, \, \, 0\, .\\ & \, \,0\, \, & \quad \, \, 0\quad \,} \end{xy}$$ Now assume that the reduction to $P$ extends to first order along a first order deformation of the $G$-bundle over the curve. One then has extensions $$\label{deform-reductions} \begin{xy}\xymatrix{ \, \,0\, \,\ar[r] &\, \,\mathrm{At}_{D_0}^P \, \,\ar[r]\ar[d]& \, \,{\mathcal{A}}^P \, \,\ar[r]\ar[d] & \, \,{\mathcal{O}}_X\, \,\ar[r] \ar@{=}[d]&\, \,0\, \,\\ \, \,0\, \,\ar[r] &\, \,\mathrm{At}_{D_0}^G \, \,\ar[r]& \, \,{\mathcal{A}}^G \, \,\ar[r] & \, \,{\mathcal{O}}_X\, \,\ar[r]& \, \,0\, . } \end{xy}$$ given by extension classes $\gamma^P\,\in\, \mathrm{H}^1(X ,\, \mathrm{At}_{D_0}^P)$ and $\gamma^G\,\in\, \mathrm{H}^1(X ,\, \mathrm{At}_{D_0}^G)$. One has the lemma \[obstruction\] The above extension classes $\gamma^P, \gamma^G$ are related by $$\gamma^G \,=\, \xi(\gamma^P)\, .$$ Consequently, there is an obstruction to extending the reductions for deformations $\gamma^G$ given by $\omega(\gamma^G)\,\in\, \mathrm{H}^1(X,\, E_P({\mathfrak g}/{\mathfrak p}))$. A second fundamental form ------------------------- Assume now that there is a connection $\nabla$ on the bundle $E_G$. It does not of course, necessarily preserve the reduction to $E_P$. The failure to preserve $ E_P$ is measured by a second fundamental form: one takes the composition $$\label{sff} S(\nabla) \, = \, \omega\circ \nabla \, : \, \mathrm{T}_X(-D_0) \, \longrightarrow \, \mathrm{At}_{D_0} ( D-D_0) \, \longrightarrow \, E_P({\mathfrak g}/{\mathfrak p})(D-D_0) \, .$$ The connection preserves the reduction to $P$ if and only if $S(\nabla)\,=\,0$. Assume that $E_P$ satisfies the condition that $ S(\nabla)\,\not=\, 0\, . $ We define some line bundles. Let $${\mathcal M}({D-D_0})\, \subset\, \mathrm{At}_{D_0}(D-D_0)$$ be the holomorphic line subbundle generated by the image $ \nabla(\mathrm{T}_X(-D_0))$ in , and let $${\mathcal L}({D-D_0})\, \subset\, E_P({\mathfrak g}/{\mathfrak p})(D-D_0)$$ be the holomorphic line subbundle generated by the image $ \omega(\nabla(\mathrm{T}_X(-D_0)))$ in . More precisely, ${\mathcal M}_{D-D_0}$ (respectively, ${\mathcal L}_{D-D_0}$) is the inverse image in $\mathrm{At}_{D_0}(D-D_0)$ (respectively, $E_P({\mathfrak g}/{\mathfrak p})(D-D_0)$) of the torsion part of the quotient $\mathrm{At}_{D_0}(D-D_0)/\nabla(\mathrm{T}_X(-D_0)$ (respectively, $E_P({\mathfrak g}/{\mathfrak p})/(\omega\circ\nabla)(\mathrm{T}X(-D_0)))$. Set $$\label{l} {\mathcal M}\,=\, {\mathcal M}_{D-D_0}\cap \mathrm{At}_{D_0}\, , \quad \quad {\mathcal L} \,=\, {\mathcal L}_{D-D_0}\cap E_P({\mathfrak g}/{\mathfrak p})\, .$$ We then have the diagram of homomorphisms of line bundles, with the columns being exact: $$\begin{xy}\xymatrix{ \, \, \mathrm{T}_X(-D)\, \,\ar[r]\ar[d]& \, \,{\mathcal M}\, \,\ar[r]\ar[d] &\, \, {\mathcal L}\, \,\ar[d]\\ \, \,\mathrm{T}_X(-D_0)\, \,\ar[r]\ar[d]& \, \,{\mathcal M}(D-D_0)\, \,\ar[r]\ar[d] & \, \,{\mathcal L}(D-D_0)\, \,\ar[d]\\ \, \,Q_1\, \,\ar[d]&\, \,Q_2\, \,\ar[d]&\, \,Q_3\, \,\ar[d]\\ \, \,0\, \,&\, \,0\, \,&\,\, \,0\, . }\end{xy}$$ Note that $Q_1$, $Q_2$ and $Q_3$ are isomorphic torsion sheaves supported on $D-D_0$. \[surj1\] The horizontal homomorphisms in this diagram induce surjective maps on the level of first cohomology. The proof consists in noting that the cokernels of each of these homomorphisms are torsion sheaves. If one considers the homomorphism $\mathrm{T}_X(-D )\,\longrightarrow\, {\mathcal M} \,\subset\, \mathrm{At}_{D_0}$ given by the connection, one has that ${\mathcal M}$ lies in the “Cartan component" of the bundle to order $n_i-1$ at $p_i$, as it is a multiple of $h_i(z)$. For the sheaf ${{\mathcal{P}}{\mathcal{P}}}$ of polar parts of the connection, let us consider the subsheaf ${{\mathcal{P}}{\mathcal{P}}}_\parallel$ whose sections are multiples of $h_i(z)$; likewise, in our deformation space ${\mathbb{H}}^1(X,\,{\mathcal{C}})$, let us consider the subspace ${\mathbb{H}}^1_\parallel(X,\,{\mathcal{C}})$ of classes where the principal part is parallel to (i.e. a multiple of) $h_i(z)$. \[surj2\] We have a diagram $$\begin{xy}\xymatrix{ \, \,{{\mathcal{P}}{\mathcal{P}}}_\parallel\, \,\ar[r]\ar[d]&\, \, \mathrm{H}^0(Q_2)\, \,\ar[d]\\ \, \,{\mathbb{H}}^1_\parallel(X,\, {\mathcal{C}})\, \,\ar[r]\ar[d]& \, \,\mathrm{H}^1(X,\, {\mathcal M})\, \,\ar[d]\\ \, \,\mathrm{H}^1(X, \, \mathrm{T}_X(-D_0))\, \,\ar[r]&\, \, \,\mathrm{H}^1(X, \, {\mathcal M}(D-D_0))\, . }\end{xy}$$ The top horizontal homomorphism is an isomorphism, and the other two horizontal homomorphisms are surjective. On the top row, the components of ${{\mathcal{P}}{\mathcal{P}}}_\parallel$ are exactly those of the torsion sheaf $Q$. On the bottom row, one has elements $\beta$ of $\mathrm{H}^1(X,\, \mathrm{T}_X(-D_0))$ mapped by $\nabla$ to $\mathrm{H}^1(X,\,{\mathcal M}(D-D_0))$. As argued in Lemma \[surj1\], this map is surjective. One now wants to see that the top and bottom fit together correctly in the middle term. Let $\widehat\beta\,\in\, {\mathbb{H}}^1_\parallel(X,\,{\mathcal{C}})$ be represented by elements $k_i$ of ${{\mathcal{P}}{\mathcal{P}}}_\parallel$ at each puncture, and a representative cocycle $\beta$ of $\mathrm{H}^1(X, \, \mathrm{T}_X(-D_0))$. If one turns $k_i$ in the natural way into a cocycle supported on a punctured disk $\Delta_i$ at $p_i$, it gives precisely the element of $\mathrm{H}^1(X,\,{\mathcal M})$ which is the coboundary of $k_i$ thought of as an element of $Q_2$. In turn, the cocycle $\beta$ is simply mapped to $\mathrm{H}^1(X,\,{\mathcal M})$ by the sheaf map; the total map from ${\mathbb{H}}^1_\parallel(X,\,{\mathcal{C}})$ is given by the sum of these two contributions, as in the definition of ${\mathcal{A}}$ above. Since the top map is an isomorphism, and the bottom one is surjective, the middle map is also surjective. We now have a surjective map ${\mathbb{H}}^1_\parallel(X,\, {\mathcal{C}})\, \longrightarrow\, \mathrm{H}^1(X,\, {\mathcal M})$, which, when mapped on to $\mathrm{H}^1(X,\, \mathrm{At}_{X_0})$, defines the extension ${\mathcal{A}}$. We saw in turn that the map $$\omega\,:\, \mathrm{H}^1(X,\, \mathrm{At}_{X_0})\,\longrightarrow \, \mathrm{H}^1(X,\, E_P({\mathfrak g}/{\mathfrak p}))$$ gave an obstruction to extending to first order a reduction to $P$. We have a diagram: $$\begin{xy}\xymatrix{ \, \,{\mathbb{H}}^1_\parallel(X,\, {\mathcal{C}})\, \,\ar[r]& \, \,\mathrm{H}^1(X,\, {\mathcal M})\, \,\ar[r]^{\omega_{\mathcal L}}\ar[d]&\, \, \mathrm{H}^1(X,\, {\mathcal L})\, \,\ar[d]\\ &\, \,\mathrm{H}^1(X,\, \mathrm{At}_{X_0})\ar[r]^{\omega}\, \, &\, \,\, \mathrm{H}^1(X,\, E_P({\mathfrak g}/{\mathfrak p}))\, . }\end{xy}$$ We have, as in Proposition 4.3 of [@BHH]: \[prop1\] Let $\widehat\beta \in {\mathbb{H}}^1_\parallel(X,\, {\mathcal{C}})$ represent an isomonodromy deformation class, of a connection with non-vanishing second fundamental form, yielding a class $$\gamma_{\mathcal{M}}\, \in \, \mathrm{H}^1(X,\, {\mathcal M})\, ,$$ and a class $$\gamma \,\in\, \mathrm{H}^1(X,\, \mathrm{At}_{X_0})$$ representing the extension ${\mathcal{A}}$. The obstruction $\omega (\gamma)$ to extending a reduction to $P$ factors through ${\mathcal{L}}$, as $\omega_{\mathcal{L}}(\gamma_{\mathcal{M}})\,\in\, \mathrm{H}^1(X,\, {\mathcal{L}})$, and if the bundle reduces to $P$ then $\omega_{\mathcal{L}}(\gamma_{\mathcal{M}})\,\in\, \mathrm{H}^1(X,\, {\mathcal{L}})$ is also zero. The proof in essence works by taking the restriction of the Atiyah bundle and its extension which live above ${\mathcal{L}}$. We will want to estimate the dimension of the space spanned by the obstructions $\omega_L(\gamma_{\mathcal{M}})$, as this will give a bound on the codimensions of the stable locus, as explained in the next section. Harder-Narasimhan filtrations ============================= Let as before $ \mathrm{Teich}_{\mathfrak h, g,m}$ be our Teichmüller space; over it we have, locally at least, a universal family $(\mathcal{C},\, \mathcal{D},\, \mathcal{H})$ whose fiber at $q$ is a curve $\mathcal{C}(q)$, a divisor $$\mathcal{D}(q)\,=\, \sum_in_ip_i(q)$$ and a collection of formal solutions $H_i(q)$. Over this in turn the isomonodromy process, described in section 3, gives a $G-$bundle ${\mathcal{E}}_G\,\longrightarrow\,{\mathcal{C}}$, equipped with a flat connection, with the appropriate polar behavior at ${\mathcal{D}}$. For ${\mathcal{E}}_G$, one has a Harder-Narasimhan filtration for families of $G$-bundles, as propounded in [@Nitsure] (see also [@Sh]); the filtration is trivial if and only if the bundle is semi-stable. \[lemNitsure\] Let $\mathcal{E}_G\,\longrightarrow\, \mathcal{C}\,\longrightarrow\,\mathcal{T}_{\mathfrak h,g,m}$ be as above. For each Harder–Narasimhan type $\kappa$, the set $$\mathcal{Y}_\kappa \,:=\, \{t\,\in\, \mathcal{T}_{\mathfrak h,g,m} ~\mid ~ \mathcal{E}_G\vert_{{\mathcal C}_t}\ \text{ is\ of\ type }\ \kappa\}$$ is a (possibly empty) locally closed complex analytic subspace of $\mathcal{T}_{\mathfrak h,g,m}$. More precisely, for each Harder–Narasimhan type $\kappa$, the union $\mathcal{Y}_{\leq \kappa}\,:=\,\bigcup_{\kappa'\leq \kappa}\mathcal{Y}_{\kappa'}$ is a closed complex analytic subset of $\mathcal{T}_{\mathfrak h,g,m}$. Moreover, the principal $G$–bundle $$\mathcal{E}_G\vert_{\tau^{-1}(\mathcal{Y}_\kappa)}\,\longrightarrow\, \tau^{-1}(\mathcal{Y}_\kappa)$$ possesses a canonical holomorphic reduction of structure group inducing the Harder–Narasimhan reduction of $\mathcal{E}_G\vert_{{\mathcal C}_t}$ for every $t\,\in \,\mathcal{Y}_\kappa$. It is our aim to show that all of these strata except that corresponding to the trivial filtration (and hence to semi-stable bundles) are of codimension $g$, by showing that there is a $g$-dimensional family of directions for which the reduction does not extend. \[prop2\] Assume that the monodromy representation for $\nabla_0$ is irreducible in the sense that it does not factor through some proper parabolic subgroup of $G$. 1. If $g\,\geq\, 1$, then there is a closed complex analytic subset $\mathcal{Y}\, \subset\, \mathrm{Teich}_{\mathfrak h, g,m}$ of codimension at least $g$ such that for any $t\,\in\, \mathrm{Teich}_{\mathfrak h, g,m} \setminus \mathcal{Y}$, the holomorphic principal $G$–bundle $\mathcal{E}^t_G$ is semi-stable. 2. If $g\,\geq\, 2$, then there is a closed complex analytic subset $\mathcal{Y}'\, \subset\,\mathrm{Teich}_{\mathfrak h, g,m}$ of codimension at least $g-1$ such that for any $t\,\in\, \mathrm{Teich}_{\mathfrak h, g,m}\setminus \mathcal{Y}'$, the holomorphic principal $G$–bundle $\mathcal{E}^t_G$ is stable. Let $g\, >\, 1$. Let ${\mathcal Y}\, \subset\, {\mathcal T}_{{\mathfrak h}, g,m}$ denote the (finite) union of all Harder-Narasimhan strata ${\mathcal Y}_\kappa$ as in Lemma \[lemNitsure\] with non-trivial Harder-Narasimhan type $\kappa$. From Lemma \[lemNitsure\] we know that ${\mathcal Y}$ is a closed complex analytic subset of ${\mathcal T}_{{\mathfrak h}, g,m}$. Take any $t\, \in\, \mathcal{Y}_\kappa\, \subset\, {\mathcal Y}$. Let $E_G\,=\, \mathcal{E}_G\vert_{{\mathcal C}_t}$ be the holomorphic principal $G$–bundle on $$X\,:=\, {\mathcal C}_t\, .$$ The holomorphic connection on $E_G$ obtained by restricting the universal isomonodromy connection will be denoted by $\nabla$. Since $E_G$ is not semistable, there is a proper parabolic subgroup $P\, \subsetneq\, G$ and a holomorphic reduction of structure group $E_P\, \subset\, E_G$ to $P$, such that $E_P$ is the Harder–Narasimhan reduction [@Be], [@AAB]; the type of this Harder–Narasimhan reduction is $\kappa$. From Lemma \[lemNitsure\] we know that $E_P$ extends along its stratum to a holomorphic reduction of structure group of the principal $G$–bundle $\mathcal{E}_G\vert_{\tau^{-1}(\mathcal{Y}_\kappa)}$ to the subgroup $P$. Let $\mu_{\rm max}$ be the maximal slope (degree/rank) of the terms of the Harder Narasimhan-filtration. We have $$\label{deg2} \mu_{\rm max}(E_P({\mathfrak g}/{\mathfrak p}))\, <\, 0$$ [@AAB p. 705]. In particular $$\label{deg3} \text{degree}(E_P({\mathfrak g}/{\mathfrak p})) \, <\, 0\, .$$ Form the irreducibility of the connection, we know that the second fundamental form $S(\nabla)$ does not vanish, and so we can build the line bundle ${\mathcal{L}}$ as above in . $${\mathcal L}\, \subset\, E_P({\mathfrak g}/{\mathfrak p}).$$ From we have $$\label{deg} \text{degree}({\mathcal L}) \, <\, 0\, .$$ Therefore, $\mathrm{h}^0(X,\,{\mathcal{L}}) =0$, and $\mathrm{h}^1(X,\,{\mathcal{L}})\geq g$. On the other hand, Lemma \[surj1\] and Lemma \[surj2\] gave a surjective map from our deformation space ${\mathbb{H}}^1_\parallel(X,\, {\mathcal{C}})$ to our obstruction space $\mathrm{H}^1(X,\,{\mathcal{L}})$. The space ${\mathcal Y}$ is thus of codimension at least $g$. For the second case, when stability fails, the degree of ${\mathcal{L}}$ is only less than or equal to zero, and so $\mathrm{h}^1(X,\,{\mathcal{L}})\geq g-1$, giving the announced codimension. [ZZZZ]{} B. Anchouche, H. 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--- abstract: 'We discuss ground-state projector simulations of a modified two-dimensional $S=1/2$ Heisenberg model in the valence bonds basis. Tuning matrix elements corresponding to the diagonal and off-diagonal terms in the quantum dimer model, we show that there is a quantum phase transition from the antiferromagnet into a columnar valence-bond-solid (VBS). There are no signs of discontinuities, suggesting a continuous or very weakly first-order transition. The Z$_4$-symmetric VBS order parameter exhibits an emergent U$(1)$ symmetry as the phase transition is approached. We extract the associated length-scale governing the U$(1)$–Z$_4$ cross-over inside the VBS phase.' author: - Jie Lou - 'Anders W. Sandvik' title: \| Z$_{\rm 4}$–U(1) crossover of the order parameter symmetry\ in a two-dimensional valence-bond-solid --- A valence-bond-solid (VBS) is a magnetically disordered state of a quantum spin system in which translational symmetry is spontaneously broken due to the formation of a pattern of strong and weak bond correlations $\langle {\bf S}_i \cdot {\bf S}_j\rangle$ (where $i,j$ are nearest-neighbor sites). Using an SU$(N)$ generalization of the Heisenberg model, Read and Sachdev showed that a four-fold degenerate columnar VBS ground state can be expected on the square lattice [@read]. Numerical studies have found evidence for such VBS states in frustrated SU$(2)$ symmetric systems [@j1j2], but because of technical limitations, in particular the sign problem in quantum Monte Carlo (QMC) simulations [@henelius], the nature of the strongly-frustrated ground state remains controversial [@isaev]. Another challenging issue is how the ground state evolves from an antiferromagnet (AF) into a VBS. According to the “Landau rules”, one would expect a direct transition between these states to be first-order [@sirker], because unrelated symmetries are broken. There could also be an intervening disordered (spin liquid) phase [@anderson] or a coexistence region. Senthil [et al.]{} recently suggested an alternative scenario for a generic continuous transition based on a “deconfined” quantum critical point (DQCP) associated with spinon deconfinement [@senthil; @nogueira]. This proposal has generated significant interest, as well as controversy. An extended “J-Q” Heisenberg model has been introduced [@sandvik1] which is not frustrated, in the standard sense, but includes a four-spin interaction which destroys the AF order and leads to a VBS ground state. This model is amenable to large scale QMC studies, which show scaling behavior consistent with a DQCP [@sandvik1; @melko]. Other studies dispute these findings, however [@jiang]. Numerical studies of the proposed field theory describing the deconfined quantum critical point are also subject to conflicting interpretations [@motrunich; @kuklov]. Further studies of AF–VBS transitions is thus called for. In this Letter we address an important aspect of the VBS state and the AF–VBS transition, namely, the nature of the quantum fluctuations of the VBS order parameter. In the DQCP theory, the Z$_4$ symmetric lattice-imposed structure of the VBS is a dangerously irrelevant, and, as a consequence, U$(1)$ symmetry emerges close to the DQCP [@senthil]. An U$(1)$ symmetric VBS order parameter was indeed confirmed in the studies of the J-Q model [@sandvik1; @melko; @jiang], and also in simulations of the SU$(N)$ Heisenberg model with $N > 4$ [@kawashima]. However, the expected cross-over into a Z$_4$ symmetric distribution inside the VBS phase was not observed. This can be interpreted as the lattice sizes studied so far being smaller than the spinon confinement length-scale $\Lambda$, which governs the U$(1)$–Z$_4$ cross-over [@lou1]. $\Lambda$ should diverge as $\xi_d^a$, where $\xi_d$ is the dimer (VBS) correlation length and $a>1$ [@senthil], and for a finite lattice with $L \ll \Lambda$ the distribution should be U$(1)$ symmetric. The models studied so far have a rather weak VBS order, and hence $\xi_d$ is large, which likely makes it difficult to satisfy $L \gg \xi_d^a$. The exponent $a$ is not known. Here we introduce a way to generate much more robust VBS states, with which we can study the U$(1)$–Z$_4$ cross-over already on small lattices. Our approach is based on a ground-state projector QMC method operating in the valence bond (VB) basis [@sandvik2; @sandvik3; @liang2]. Starting from some trial state $\|\Psi\rangle$, the ground state of a hamiltonian $H$ can be obtained by applying a high power of $H$; $\|\Psi_0\rangle \sim H^m\|\Psi\rangle$. Consider the $S=\frac{1}{2}$ Heisenberg model written as a sum of singlet projection operators $H_{ij}$, $$H = -J\sum_{\langle i,j\rangle} H_{ij},~~~~H_{ij}=\hbox{$\frac{1}{4}$}-{\bf S}_i \cdot {\bf S}_j,$$ where $\langle i,j\rangle$ denotes nearest neighbors on a square lattice of $N=L^2$ sites. In the VB basis the trial state $\|\Psi\rangle$ is a superposition of singlet products $\|(a_1,b_1)\cdots(a_{N/2},b_{N/2})\rangle$, where $(a,b)=(\uparrow_{a}\downarrow_{b}-\downarrow_{a}\uparrow_{b})/\sqrt{2}$ with $a$ and $b$ sites on different sublattices. We here use the amplitude-product state of Liang [et al.]{} [@liang1; @lou2]. A singlet projector can have two different effects upon acting on a VB state; $$\begin{aligned} && H_{ab}\|\cdots (a,b)(c,d)\cdots \rangle = 1\|\cdots (a,b)(c,d)\cdots \rangle \label{vbdia} \\ && H_{ad}\|\cdots (a,b)(c,d)\cdots \rangle = \hbox{$\frac{1}{2}$}\|\cdots (a,d)(c,b)\cdots \rangle. \label{vboff}~~ ~~~~~\end{aligned}$$ These rules form the basis of the VB projector method [@sandvik3; @liang2], where $H^m$ is expanded in its strings of $m$ singlet projectors. Expectation values of operators $O$ are obtained by importance-sampling the VBs and operator strings produced when expanding $$\langle O \rangle = \frac{\langle \Psi\|H^m OH^m\|\Psi\rangle}{\langle \Psi\|H^mH^m\|\Psi\rangle}.$$ For details of these procedures we refer to Refs. [@sandvik2; @sandvik3]. ![(Color on-line) Diagonal (a) and off-diagonal (b) singlet projection operations (indicated by the arches) on VB pairs on a plaquette. The matrix elements (\[vbdia\]) and (\[vboff\]) corresponding to these operations are multiplied by $Q_v$ and $Q_k$, respectively. In (a), the factor is $2Q_v$ if there is a VB also on to the left side of the operator. For all other bond configurations the matrix elements remain those in (\[vbdia\]) and (\[vboff\]).[]{data-label="fig1"}](fig1.eps){width="4.5cm"} -3mm In Ref. [@sandvik1], the J-Q model, which includes a four-spin coupling consisting of terms $-QH_{ij}H_{kl}$, with $ij$ and $kl$ site pairs on two opposite edges of a plauqtte, was studied using the VB projector method. The $Q$ term naturally favors singlet formation on plaquettes, and this was shown to lead to a VBS state when $J/Q<(J/Q)_c$, $(J/Q)_c \approx 0.04$. Here we introduce another mechanism leading to VBS formation. We define an effective hamiltonian based on the Heisenberg model in the VB basis, by changing the diagonal matrix element $1$ in Eq. (\[vbdia\]) and the off-diagonal matrix element $\frac{1}{2}$ in Eq. (\[vboff\]) to $Q_v$ and $\frac{1}{2}Q_k$, respectively, for $H_{ab}$ acting on VBs on opposite edges of the same plaquette. These operations, illustrated in Fig. \[fig1\], correspond to the kinetic- and potential-energy terms of the quantum dimer model [@rokhsar]. There, however, the Hilbert space consists of only dimers connecting nearest-neighbor sites, whereas we here keep the full space of VBs connecting any pair of sites on different sublattices. In the quantum dimer model, the dimer configurations are also considered as orthogonal states, whereas we here keep the singlet nature of the VBs, whence the states are non-orthogonal. The non-orthogonality may at first sight seem problematic, because when $Q_v,Q_k \not =1$ the hamiltonian is non-hermitean. We therefore refer to it as an [pseudo hamiltonian]{} in the VB basis. However, in spite of this, the states generated by the projection procedure (with the sampling weights modified by the presence of the factors $Q_v$ and $Q_k$, and taking the power $m$ large enough for convergence to the $m=\infty$ limit) are completely well-defined SU$(2)$ invariant quantum states. We can thus think of the modified projection technique as a means of generating a family of states parametrized by $Q_v$ and $Q_k$. Moreover, there must be some corresponding hamiltonians, defined in terms of the standard spin operators ${\bf S}_i$, which have these states as their ground states. Although we are not able to write down these hamiltonians (which likely contain multi-spin interactions, possibly long-ranged), it is still useful to study the evolution of the states as a function of $Q_v$ and $Q_k$. Here we will consider two cases; $Q_v \ge 1, Q_k=1$ and $Q_k \ge 1, Q_v=1$, which we refer to as the $Q_v$ and $Q_k$ models, respectively. Both these models indeed undergo AF–VBS transitions. ![(Color on-line) Finite-size scaling of the spin and dimer correlation lengths. The inset shows $\xi/L$ versus the coupling, with crossing points tending toward $Q_v^c \approx 1.40$.[]{data-label="xi"}](fig2.eps){width="7.25cm"} -3mm We calculate the square of the staggered magnetization, $M^2=\langle {\bf M} \cdot {\bf M}\rangle$, where $${\bf M}=\frac{1}{N}\sum_{x,y}(-1)^{x+y} {\bf S}_{x,y}$$ is the operator for the AF order parameter. The columnar VBS operator for $x$-oriented bonds is $$\begin{aligned} D_x=\frac{1}{N}\sum_{x,y}(-1)^{x} {\bf S}_{x,y} \cdot {\bf S}_{x+1,y},\end{aligned}$$ and $D_y$ is defined analogously. We calculate the squared order parameter, $D^2=\langle D_x^2+D_y^2\rangle$, and the distribution $P(D_x,D_y)$ as in [@sandvik1]. Results for these quantities and the corresponding spin and dimer correlation lengths $\xi_s$ (spin) and $\xi_d$ (defined through the momentum-space second moments of the spin and dimer correlation functions) indicate coinciding critical points for the AF and VBS order parameters. In the following we first discuss the finite size scaling behavior of the $Q_v$ model. ![(Color on-line) Finite size scaling of the squared AF (top panel) and VBS (bottom panel) order parameters of the $Q_v$ model. The insets show the unscaled data.[]{data-label="MD"}](fig3.eps){width="6.5cm"} -3mm We define a reduced coupling $q=Q_v-Q^c_v$. Then, if there indeed is a single critical point, there is AF order for $q<0$ and VBS order for $q>0$, and in the thermodynamic limit the squared spin and dimer order parameters should scale as $M^2 \sim (-q)^{2\beta_s}$ and $D^2 \sim q^{2\beta_d}$ inside the respective phases. To extract $Q_v^c$ and the exponents, we use standard finite-size scaling forms, $$\begin{aligned} M^2 &=& L^{\sigma_s}(1+aL^{- \omega})F_s(qL^{1/\nu}), \\ D^2 &=& L^{\sigma_d}(1+aL^{- \omega})F_d(qL^{1/\nu}), \\ \xi_{s,d} &=& L(1+aL^{- \omega}) G_{s,d}(qL^{1/\nu}).\end{aligned}$$ where $\sigma_s=2\beta_s/\nu$, $\sigma_d=2\beta_d/\nu$ and the correlation length exponent $\nu$ is the same for all the quantities (as required in the DQCP theory). The scaling functions $F_{s,d}$ and $G_{s,d}$ are extracted in the standard way by adjusting the critical point and exponents to collapse finite-size data onto common curves. Since our lattices are not very large, $L \le 24$, a subleading correction helps significantly to scale the data. In all cases we find that $\omega=1$ works well (the prefactor $a$ is quantity-dependent, however). Results are shown in Fig. \[xi\] and \[MD\]. All the data can be scaled with $Q^c_v=1.400(5)$, $\nu=0.78(3)$, $\beta_s=0.27(2)$ and $\beta_d=0.68(3)$. Here $\nu$ and $\beta_d$ are approximately the same, within error bars, as in the J-Q model [@sandvik1], while $\beta_s$ is very different—for the J-Q model $\beta_s \approx \beta_d=0.63(2)$ was found. The range of system sizes is quite small and we cannot, of course, exclude drifts in the exponents for larger lattices, nor a very weakly first-order transition. Turning to the $Q_k$ model, it is more demanding computationally, because the critical point is rather large, $Q_k= 2.5(1)$, leading to a lower acceptance rate in simulations close to the transition than for the $Q_v$ model. We can therefore not reach the same level of precision for the exponents. The results are nevertheless consistent with a continuous transition and exponents similar to those of the $Q_v$ model. ![(Color on-line) VBS order=parameter distributions. The left column is for $Q_k=1,Q_v=1.44,1.48,1.54$ (from top) on $12 \times 12$ lattices. The right column is for $Q_v=1,Q_k=2.5,3.5,5.0$ (from top) on $16\times 16$ lattices.[]{data-label="histo"}](fig4.eps){width="5cm"} -3mm Unfortunately, we cannot easily calculate the dynamic exponent $z$ with the present approach, because it requires access to the triplet sector, e.g., to extract the spin gap $\Delta \sim L^{-z}$. Our model is explicitly defined only in the singlet sector. While one can extend the VB basis and the projection scheme to triplets [@sandvik2; @sandvik3], the extension of the $Q_v$ and $Q_k$ models to this sector is not unique, and $z$ may depend on how that is accomplished. We could in principle calculate gaps in the singlet sector, but this is much more complicated. Our main interest in studying the $Q_v$ and $Q_k$ models is in the distribution $P(D_x,D_y)$ of the columnar dimer order parameter. While this is a basis dependent quantity, it still provides direct information on the order parameter symmetry. In a columnar symmetry-broken VBS state, we expect a distribution with a single peak located on the $x$ or $y$ axis, while in a plaquette state the peak should be on one of the $45^\circ$ rotated axes. In simulations that do not break the symmetry, we expect four-fold symmetric distributions, with peak locations corresponding to the type of VBS as above. In previous studies of VBS states, only ring-shaped distributions were observed [@sandvik1; @melko; @jiang; @kawashima], however, which can be taken as a confirmation of the predicted [@senthil] emergent $U(1)$ symmetry close to a DQCP. One would then expect the four-fold symmetry to appear for large systems, $L \gg \Lambda$, inside the VBS phase, as has been observed explicitly in a classical XY model including dangerously irrelevant $Z_q$ ($q\ge 4$) perturbations [@lou1]. With the $Q_v$ and $Q_k$ models, we can reach further inside the VBS phases than in the previously studied quantum spin systems, and, as seen in Fig. \[histo\], we can indeed follow the evolution from $U(1)$ to $Z_4$ symmetric distributions as a function of the coupling constants even for modest system sizes. The peak locations correspond to columnar VBS states for both models, although the shapes of the distributions are qualitative different in other respects. The previously observed ring-shaped distributions [@sandvik1; @kawashima] are more reminicant of those for the $Q_k$ model. ![(Color on-line) Finite-size scaling of anisotropy order parameter. The inset shows the unscaled data.[]{data-label="z4"}](fig5.eps){width="7.25cm"} -3mm To study the length scale $\Lambda$ which governs the $Z_4$–$U(1)$ crossover (and is related to the scaling dimension of a dangerously irrelevant field [@lou1]) we define an order parameter $D_4$ which is sensitive to the $Z_4$ anisotropy $$\begin{aligned} D_4&=&\int_{-1}^{1}dD_x\int_{-1}^{1}dD_yP(D_x,D_y) r_{xy}\cos(4\theta) \nonumber \\ &=&\int_{0}^{1}dr\int_0^{2\pi}d\theta r^2P(r,\theta)\cos(4\theta),\end{aligned}$$ where $r_{xy}=(D_x^2+D_y^2)^{1/2}$. This order parameter should obey the finite-size scaling form [@oshikawa; @lou1]; $$D_4 = L^{\sigma_d/a_4}F_4(qL^{1/a_4\nu}),$$ with $a_4>1$. The data can be scaled with $a=1.30(5)$, as shown in Fig. \[z4\] (where we use the same $Q^c_v,\sigma_d$, and $\nu$ as in Figs. \[MD\] and \[xi\]). Here the error bars on the raw data, as seen in the inset, are much larger than for $D^2$, reflecting that slow angular fluctuations of the VBS order parameter in the simulations (which do not affect the rotationally-invariant $D^2$). In the classical XY model with $Z_4$ perturbation $a_4 \approx 1.1$ [@carmona; @lou1], and, thus, the $Q_v$ model has more prominent angular fluctuations. For the $Q_k$ model, we find an even larger $a_4 \approx 1.5$, although the error bars are rather large and we cannot say for sure that it is different from the $Q_v$ model. To summarize, by tuning specific matrix elements in valence-bond QMC simulations, we are able to study a family of SU$(2)$ symmetric states undergoing AF–VBS phase transitions. Unlike previous studies of quantum spin models with VBS states, we are able to observe both the $Z_4$ symmetry of the order-parameter distribution (which arises from the nature of the VBS on the square lattice) deep inside the VBS phase and the cross-over into an emergent $U(1)$ symmetry upon approaching the transition point. We extracted the length scale $\Lambda$, which is associated with spinon deconfinement in the DQCP theory. While the correlation length exponent $\nu$ and the VBS order-parameter exponent $\beta_d$ are roughly consistent with those found previously for the J-Q model [@sandvik1; @melko] (which is the best candidate so far for a DQCP), the AF exponent $\beta_s$ is significantly smaller. It thus appears that these transitions are in different universality classes. 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--- abstract: \| We define a positive operator valued measure $E$ on $ [0,2\pi] \times\mathbb R$ describing the measurement of randomly sampled quadratures in quantum homodyne tomography, and we study its probabilistic properties. Moreover, we give a mathematical analysis of the relation between the description of a state in terms of $E$ and the description provided by its Wigner transform. address: - \| Dipartimento di Fisica, Università di Genova, Via Dodecaneso 33,\ Genova, 16146, Italy\ and I.N.F.N., Sezione di Genova, Via Dodecaneso 33,\ Genova, 16146, Italy\ albini@fisica.unige.it - \| Dipartimento di Scienze per l’Architettura, Università di Genova, Stradone S. Agostino 37\ Genova, 16123, Italy,\ and I.N.F.N., Sezione di Genova, Via Dodecaneso 33\ Genova, 16146, Italy\ devito@dima.unige.it - \| Dipartimento di Informatica, Università di Genova, Via Dodecaneso 35\ Genova, 16146, Italy,\ and I.N.F.N., Sezione di Genova, Via Dodecaneso 33\ Genova, 16146, Italy\ toigo@ge.infn.it author: - Paolo Albini - Ernesto De Vito - Alessandro Toigo title: QUANTUM HOMODYNE TOMOGRAPHY AS AN INFORMATIONALLY COMPLETE POSITIVE OPERATOR VALUED MEASURE --- Introduction ============ Quantum homodyne tomography [@Art; @DAr; @Leo; @VR] allows to determine the state of a single mode radiation field by repeated measurements of the quadrature observables $X_\theta$, the phases $\theta$ being chosen randomly in ${\mathbb{T}}=[0,2\pi]$. This can be seen as a consequence of the fact [@Cas] that, for a large class of observables $O$, there exists an associated function $f_O:{\mathbb{T}}\times{\mathbb{R}}\to{\mathbb{R}}$ such that $$\label{} {{\rm tr} \left[ O\rho \right]} = \int_0^{2\pi} \left[\int_{{\mathbb{R}}} f_O (\theta,x) {{\rm d}}\nu_\theta^\rho (x) \right] \frac{{{\rm d}}\theta}{2\pi} ,$$ where $\nu_\theta^\rho$ is the probability distribution of outcomes obtained for the quadrature $X_\theta$ measured on the state $\rho$ and ${{\rm d}}\theta / 2\pi$ is the uniform probability distribution on ${\mathbb{T}}$. Actual reconstrunction schemes are strictly related to a statistical interpetation of formulas of this kind. Indeed, quantum tomography experiments output a $n$-uple $\{(\Theta_i, X_i)\}_{i = 1}^n$ of pairs in ${\mathbb{T}}\times {\mathbb{R}}$, each one of which represents the outcome $X_i$ of a measurement of the quadrature observable corresponding to the randomly picked phase $\Theta_i$. If one assumes such pairs to be samples from a random variable on ${\mathbb{T}}\times {\mathbb{R}}$ distributed accordingly to a probability measure $\mu^\rho$ such that $$\label{spezz} {{\rm d}}\mu^\rho(\theta, x)={{\rm d}}\nu_\theta^\rho (x) \frac{{{\rm d}}\theta}{2\pi},$$ one can use the experimental outcomes to estimate integrals such as (see Ref. [@DAr] and references therein), for example by replacing ${{\rm d}}\mu^\rho$ with its empirical estimate $\frac{1}{n} \sum_{i=1}^n \delta_{(\Theta_i,X_i)}$. The above reconstruction formula, although very popular, is not the only scheme used for tomographical state estimation: other ones are known which don’t rely on it [@DAr]. The hypotesis that experimental results are distributed accordingly to lies however under both every proposed reconstruction algorithm and its statistical analysis [@Art; @DAr; @Leo; @Butu; @Guta]. Actually, although given for granted in the cited literature, well-definiteness of a joint probability distribution such as $\mu^\rho$ in is [a priori]{} not trivial. In the first part of our paper we prove well defineteness of $\mu^\rho$ by showing there exists a positive operator valued measure (POVM) $E$ on ${\mathbb{T}}\times{\mathbb{R}}$ such that $$\mu^\rho(Z)=tr{[E(Z)\rho]}\qquad\text{for any Borel subset $Z$ of }{\mathbb{T}}\times{\mathbb{R}}.$$ According to the physical meaning of $\mu^\rho$, the POVM $E$ is the [generalized observable]{} associated with the quantum homodyne tomography experimental setup. In particular, we show that $\mu^\rho$ has density $p^\rho(\theta,x)$ with respect to the Lebesgue measure on ${\mathbb{T}}\times {\mathbb{R}}$, its support is always an unbounded set, and the mapping $\rho\mapsto \mu^\rho$ is injective (i. e. $E$ is [ informationally complete]{}). The intertwining property $ X_\theta= e^{i\theta N}X e^{-i\theta N}$, where $N$ is the number operator and $X$ is the position operator, turns out to be crucial for the definition of $E$ (or, equivalently, for the definition of $\mu^\rho$). We remark that the introduction of a POVM for the homodyne tomography measurement process is already present in physical literature (see section 2.3.2 in Ref. [@DAr]), but it is grounded on a formal construction. We provide here an alternative, rigorous formulation. In their seminal paper on quantum homodyne tomography [@VR], Vogel and Risken argued that the the Radon transform of $W(\rho)$, where $W(\rho)$ is the Wigner function associated to $\rho$, is precisely the probability density function $p^\rho$ generated by the homodyne tomography measurement, so that the following commutative diagram holds: $$\xymatrix{ & \text{States on }\mathcal H \ar_W[dl]\ar^{p^{\cdot}}[dr] & \\ \text{Wigner functions on }{\mathbb{R}}^2 \ar@<0.5ex>[rr]^{\text{Radon transform}} & & \text{probability densities on }{\mathbb{T}}\times{\mathbb{R}}}$$ The suggested estimation procedure, applied also in the first homodyne tomography experiments, is then based on the inversion of the Radon transform by means of classical techniques in medical tomography. However, the derivation of this fact is once again rather formal, and never given a rigorous basis in the literature on the subject: in the second part of the paper, we will thus address the problems that arise in looking at such formulation of quantum tomography from a rigorous point of view. First of all, we will recall that in order for the Radon transform to be well-defined, we need the Wigner function $W(\rho)$ to be integrable on ${\mathbb{R}}^2$. Then we will show that the support of $W(\rho)$ can [never]{} be bounded. This is a potential problem, since the estimation techniques used in classical tomography are explicitly devised for compactly supported objects. One can however by-pass the problem and still give an inverse for the Radon transform if he assumes that the Wigner function under observation is a Schwartz function on ${\mathbb{R}}^2$. This is precisely what happens in most homodyne tomography experiments, where the states under observation are linear combinations of coherent or number states. In Section \[secRadon\] we show that this assumption on the Wigner function is equivalent to suppose that $\rho$ has a kernel which is a Schwartz function on ${\mathbb{R}}^2$ (since $\rho$ is an Hilbert-Schmidt operator, $\rho$ is an integral operator whose kernel is a function on ${\mathbb{R}}^2$). Under this assumption on $\rho$ we prove that the Radon transform of $W(\rho)$ is $p^\rho$ and the inversion formula holds true. Preliminaries and notations {#preliminari} =========================== In this section, we will introduce the notations and give a very brief description of the mathematical structure of quantum homodyne tomography. Notations --------- Let ${\mathcal{H}}$ be a complex, separable Hilbert space with norm ${\left\\|\cdot\right\\|}$ and scalar product ${\left\langle \cdot \, , \, \cdot \right\rangle}$ linear in the second entry. Denote by ${\mathcal{L}(\mathcal{H})}$ the Banach space of the bounded operators on ${\mathcal{H}}$ with uniform norm ${\left\\|\cdot\right\\|}_{{\mathcal{L}}}$. Let ${\mathcal{I}_1 (\mathcal{H})}$ be the Banach space of the trace class operators on ${\mathcal{H}}$ with trace class norm ${\left\\|\cdot\right\\|}_1$, and let ${\mathcal{S(H)}}$ be the convex subset of positive trace one elements in ${\mathcal{I}_1 (\mathcal{H})}$. Finally, let ${\mathcal{I}_2(\mathcal{H})}$ be the Hilbert-Schmidt operators on ${\mathcal{H}}$, with norm ${\left\\|A\right\\|}_2 = \left[{{\rm tr} \left[ A^\ast A\right]}\right]^{1/2}$. We recall that the elements of ${\mathcal{S(H)}}$ are the [states]{} of the quantum system whose associated Hilbert space is ${\mathcal{H}}$. Suppose $\Omega$ is a Hausdorff locally compact second countable topological space. Let ${{\mathcal{B}}(\Omega)}$ be the Borel $\sigma$-algebra of $\Omega$. We recall the following definition of positive operator valued measure. \[defPOVM1\] A [positive operator valued measure]{} (POVM) on $\Omega$ with values in ${\mathcal{H}}$ is a map $E: {{\mathcal{B}}(\Omega)} {\longrightarrow}{\mathcal{L}(\mathcal{H})}$ such that - $E(A) \geq 0$ for all $A \in{{\mathcal{B}}(\Omega)}$; - $E(\Omega) = I$; - if $\{ A_i \}_{i\in I}$ is a denumerable sequence of pairwise disjoint sets in ${{\mathcal{B}}(\Omega)}$, then $$E (\cup_i A_i) = \sum\nolimits_i E(A_i) ,$$ where the sum converges in the weak (or, equivalently, ultraweak or strong) topology of ${\mathcal{L}(\mathcal{H})}$. $E$ is a [projection valued measure]{} (PVM) if $E(A)^2 = E(A)$ for all $A\in{{\mathcal{B}}(\Omega)}$. If $E$ is a POVM and $T\in{\mathcal{I}_1 (\mathcal{H})}$, we define $$\mu^T_E (A) = {{\rm tr} \left[ E(A) T\right]} \quad \forall A\in{{\mathcal{B}}(\Omega)} .$$ Then, $\mu^T_E$ is a bounded complex measure on $\Omega$. If $\rho\in{\mathcal{S(H)}}$, $\mu^\rho_E$ is actually a probability measure on $\Omega$, and $\mu^\rho_E (A)$ is the probability of obtaining a result in $A$ when performing a measurement of $E$ on the state $\rho$. The mathematics of quantum homodyne tomography ---------------------------------------------- The physical system of quantum homodyne tomography is a single radiation mode of the electromagnetic field. The associated Hilbert space is ${\mathcal{H}}= {L^2 \left( {\mathbb{R}}\right)}$. Let $${\mathcal{A}}= \left\{ p(x) \, e^{-\frac{x^2}{2}} \mid p \textrm{ is a polinomial}, \right\}$$ which is a dense subspace of ${\mathcal{H}}$. As usual, we denote by $X$ and $P$ the position and momentum operators, respectively. Their action on ${\mathcal{A}}$ is explicitly given by $$(Xf)(x)=x f(x) \qquad\text{and}\qquad (Pf)(x)=-i\frac{{{\rm d}}f}{{{\rm d}}x}(x).$$ Letting ${\mathbb{T}}=[0,2\pi]$, for any $\theta\in{\mathbb{T}}$ the corresponding quadrature is the self-adjoint operator $X_\theta$ on ${L^2 \left( {\mathbb{R}}\right)}$, whose action on ${\mathcal{A}}$ is $$X_\theta =\cos{\theta} X + \sin{\theta} P.$$ If $x,y\in{\mathbb{R}}$, and $x = r\cos \theta$, $y = r\sin\theta$, we have $$\left[ e^{ir X_\theta} f \right] (z) = \left[ e^{i (xX + yP)} f \right] (z) = e^{i\left( \frac{xy}{2} + xz \right)} f(z+y)$$ for all $f\in {L^2 \left( {\mathbb{R}}\right)}$. We denote by $\Pi_\theta$ the PVM on ${\mathbb{R}}$ associated to $X_\theta$ by spectral theorem. In particular, $\Pi (A) := \Pi_0 (A)$ is just multiplication in ${L^2 \left( {\mathbb{R}}\right)}$ by the characteristic function $1_A$ of $A$, while $\Pi_{\frac{\pi}{2}} (A) = {\mathcal{F}}^\ast \Pi (A) {\mathcal{F}}$, where ${\mathcal{F}}$ is the Fourier transform $$\label{FT} {\mathcal{F}}f = \frac{1}{\sqrt{2\pi}} \int_{{\mathbb{R}}} e^{-ixy} f(y) {{\rm d}}y \quad f\in L^1\cap L^2 ({\mathbb{R}}) .$$ The number operator is the essentially self-adjoint operator $N$ whose action on ${\mathcal{A}}$ is $$N = \frac{1}{2} {\left(}X^2 + P^2 - 1 {\right)}.$$ For all $\theta\in{\mathbb{T}}$, we let $V(\theta)=e^{i\theta N}$. Since the spectrum of $N$ is ${\mathbb N}$, the map $\theta\to V(\theta)$ is a unitary continuous representation of ${\mathbb{T}}$ acting on ${L^2 \left( {\mathbb{R}}\right)}$, where we regard ${\mathbb{T}}$ as a topological abelian group with addition modulo $2\pi$. The number representation $V$ intertwines the quadratures $X_\theta$, in the sense that $$X_\theta = V(\theta) X V(\theta)^\ast$$ for all $\theta\in{\mathbb{T}}$, and $$\Pi_\theta (A) = V(\theta) \Pi (A) V(\theta)^\ast$$ for all $\theta\in{\mathbb{T}}$ and $A\in{{\mathcal{B}}({\mathbb{R}})}$. Finally, given $\rho\in{\mathcal{S(H)}}$ and $\theta\in{\mathbb{T}}$, we denote by $\nu^\rho_\theta$ the probability distribution on ${\mathbb{R}}$ of the outcomes of the quadrature $X_\theta$ measured on the state $\rho$, namely $$\label{nurt} \nu^\rho_\theta (A) = {{\rm tr} \left[ \rho \Pi_\theta (A)\right]} = {{\rm tr} \left[ \rho V(\theta) \Pi (A) V(\theta)^\ast\right]} \quad \forall A \in {{\mathcal{B}}({\mathbb{R}})} .$$ Main results {#mainresults} ============ In this section, we will describe explicitly the POVM which intervenes in homodyne tomography and the associated probability distributions on states. The first result studies some properties of the family of probability measures $\nu^\rho_\theta$ defined by . In its proof and in the statement of some of the following results, we will make use of the concept of section through some $\theta \in {\mathbb{T}}$ of a Borel set $B \in {{\mathcal{B}}({\mathbb{T}}\times {\mathbb{R}})}$, defined as follows: $$B^\theta = \{x \in {\mathbb{R}}\vert (\theta, x) \in B\}.$$ Given $\rho\in{\mathcal{S(H)}}$ and $\theta\in{\mathbb{T}}$ - the probability measure $\nu^\rho_\theta$ has density $p_\theta^\rho \in {L^1 \left( {\mathbb{R}}\right)}$ with respect to the Lebesgue measure on ${\mathbb{R}}$; - the map $\theta \mapsto \nu^\rho_\theta (B^\theta)$ is measurable for any $B\in{{\mathcal{B}}({\mathbb{T}}\times {\mathbb{R}})}$. If $A\in {{\mathcal{B}}({\mathbb{R}})}$ has zero Lebesgue measure, then $\Pi (A) f = 1_A f = 0$ for all $f\in {L^2 \left( {\mathbb{R}}\right)}$. Therefore, $\nu^\rho_\theta (A) = {{\rm tr} \left[ \rho V(\theta) \Pi (A) V(\theta)^\ast\right]} = 0$. Thus, the first claim follows. If $\{ e_n \}_{n\in{\mathbb N}}$ is a Hilbert basis of ${\mathcal{H}}$, then $$\begin{aligned} {{\rm tr} \left[ \rho V(\theta) \Pi (B^\theta) V(\theta)^\ast\right]} & = & \sum_n {\left\langle e_n \, , \, V(\theta)^\ast \rho V(\theta) \Pi (B^\theta) e_n \right\rangle} \\ & = & \sum_n \sum_m {\left\langle e_n \, , \, V(\theta)^\ast \rho V(\theta) e_m \right\rangle} {\left\langle e_m \, , \, \Pi (B^\theta) e_n \right\rangle} . \end{aligned}$$ Since the map $\theta \mapsto {\left\langle e_n \, , \, V(\theta)^\ast \rho V(\theta) e_m \right\rangle}$ is continuous and the map $\theta \mapsto {\left\langle e_m \, , \, \Pi (B^\theta) e_n \right\rangle} = \int 1_B (\theta,x) e_n (x) \overline{e_m (x)} {{\rm d}}x$ is measurable by Fubini theorem, measurability of $\theta \mapsto {{\rm tr} \left[ \rho V(\theta) \Pi (B^\theta) V(\theta)^\ast\right]}$ follows. Next theorem shows the existence of a POVM associated to quantum homodyne tomography. This theorem should be compared with the formal derivation of $E$ given in Ref. [@DAr] (see eq. (2.34) therein). \[PropPOVM\] There exists a unique positive operator valued measure $E$ on ${\mathbb{T}}\times {\mathbb{R}}$ acting in ${L^2 \left( {\mathbb{R}}\right)}$ such that $$\label{rabbia} {{\rm tr} \left[ \rho E(B)\right]}= \int_{{\mathbb{T}}} \nu_\theta^\rho (B^\theta) \frac{{{\rm d}}\theta}{2\pi}$$ for all $\rho\in{\mathcal{S(H)}}$ and $B\in{{\mathcal{B}}({\mathbb{T}}\times {\mathbb{R}})}$. Eq. suggests to define the POVM as $$E(B) = \int_{{\mathbb{T}}} V(\theta) \Pi (B^\theta) V(\theta)^\ast \frac{{{\rm d}}\theta}{2\pi} .$$ To prove that the above definition is correct, we first show that the map $\theta \mapsto V(\theta) \Pi (B^\theta) V(\theta)^\ast$ is $\frac{{{\rm d}}\theta}{2\pi}$-ultraweakly integrable for all Borel subsets $B$ of ${\mathbb{T}}\times{\mathbb{R}}$, and then we prove that $B\mapsto E(B)$ is a POVM. Now, given $B\in{{\mathcal{B}}({\mathbb{T}}\times {\mathbb{R}})}$ and $\rho\in{\mathcal{S(H)}}$, the map $\theta \mapsto {{\rm tr} \left[ \rho V(\theta) \Pi (B^\theta) V(\theta)^\ast\right]}$ is measurable by the previous proposition, and $$\left\| {{\rm tr} \left[ \rho V(\theta) \Pi (B^\theta) V(\theta)^\ast\right]} \right\| \leq {\left\\|\rho\right\\|}_1 {\left\\|\Pi (B^\theta)\right\\|}_{{\mathcal{L}}} \leq 1 \quad \forall \theta \in {\mathbb{T}}.$$ Therefore, it is $\frac{{{\rm d}}\theta}{2\pi}$-integrable. This shows that $\theta \mapsto V(\theta) \Pi (B^\theta) V(\theta)^\ast$ is $\frac{{{\rm d}}\theta}{2\pi}$-ultraweakly integrable. Suppose $T\in{\mathcal{I}_1 (\mathcal{H})}$. Then $T = \sum_{k=0}^3 i^k T_k$, with $T_k \geq 0$ and ${\left\\|T_0\right\\|}_1 + {\left\\|T_2\right\\|}_1 = {\left\\|T_1\right\\|}_1 + {\left\\|T_3\right\\|}_1 \leq {\left\\|T\right\\|}_1$. Setting $\rho_k = T_k / {\left\\|T_k\right\\|}_1$ (with $0/0 = 0$), we see that $$\begin{aligned} \left\| \int_{{\mathbb{T}}} {{\rm tr} \left[ T V(\theta) \Pi (B^\theta) V(\theta)^\ast\right]} \frac{{{\rm d}}\theta}{2\pi} \right\| & \leq & \sum_{k=0}^3 {\left\\|T_k\right\\|}_1 \int_{{\mathbb{T}}} \left\| {{\rm tr} \left[ \rho_k V(\theta) \Pi (B^\theta) V(\theta)^\ast\right]} \right\| \frac{{{\rm d}}\theta}{2\pi} \\ & \leq & \sum_{k=0}^3 {\left\\|T_k\right\\|}_1 \leq 2 {\left\\|T\right\\|}_1 . \end{aligned}$$ This shows the existence of $E(B)\in{\mathcal{L}(\mathcal{H})}$. Clearly, $E(B) \geq 0$, and $E({\mathbb{T}}\times {\mathbb{R}}) = I$. If $\{ B_n \}_{n\in{\mathbb N}}$ is a monotone increasing family of elements in ${{\mathcal{B}}({\mathbb{T}}\times {\mathbb{R}})}$, with $B_n \uparrow B$, then, for all $\theta$, $${{\rm tr} \left[ \rho V(\theta) \Pi (B_n^\theta) V(\theta)^\ast\right]} = \nu^\rho_\theta (B_n^\theta) \uparrow \nu^\rho_\theta (B^\theta) = {{\rm tr} \left[ \rho V(\theta) \Pi (B^\theta) V(\theta)^\ast\right]} .$$ By dominated convergence theorem $$\int_{{\mathbb{T}}} {{\rm tr} \left[ \rho V(\theta) \Pi (B_n^\theta) V(\theta)^\ast\right]} \frac{{{\rm d}}\theta}{2\pi} \uparrow \int_{{\mathbb{T}}} {{\rm tr} \left[ \rho V(\theta) \Pi (B^\theta) V(\theta)^\ast\right]} \frac{{{\rm d}}\theta}{2\pi} ,$$ and ultraweak $\sigma$-additivity of $E$ follows. We let $\mu^\rho = {{\rm tr} \left[ E(\cdot) \rho\right]}$ be the probability distribution on ${\mathbb{T}}\times {\mathbb{R}}$ associated to a measurement of $E$ performed on the state $\rho$. By definition it follows that $$\label{murho} \mu^\rho (B) = \int_{{\mathbb{T}}} \nu_\theta^\rho (B^\theta) \frac{{{\rm d}}\theta}{2\pi}$$ as wanted. The following theorem gives some properties of $\mu^\rho$. \[TeoMis\] Let $\rho\in{\mathcal{S(H)}}$. - The measure $\mu^\rho$ has density with respect to $\frac{{{\rm d}}\theta}{2\pi} \, {{\rm d}}x$. We denote such density by $p^\rho$. - For $\frac{{{\rm d}}\theta}{2\pi}$-almost all $\theta$, $p^\rho (\theta,x) = p^\rho_\theta (x)$ for ${{\rm d}}x$-almost all $x$. - The marginal probability distribution induced by $\mu^\rho$ on ${\mathbb{T}}$ is the Haar measure $\frac{{{\rm d}}\theta}{2\pi}$, and the conditional probability distribution induced by $\mu^\rho$ on ${\mathbb{R}}$ is $\nu^\rho_\theta$ for $\frac{{{\rm d}}\theta}{2\pi}$-almost all $\theta$. <!-- --> - If $B\in{{\mathcal{B}}({\mathbb{T}}\times {\mathbb{R}})}$ is a $\frac{{{\rm d}}\theta}{2\pi} \, {{\rm d}}x$-null set, then $B^\theta$ is ${{\rm d}}x$-null for $\frac{{{\rm d}}\theta}{2\pi}$-almost all $\theta$ by Fubini theorem, so, for such $\theta$’s, $\nu^\rho_\theta (B^\theta) = 0$. Therefore, $\mu^\rho (B) = 0$ by , thus showing that $\mu^\rho$ has density with respect to $\frac{{{\rm d}}\theta}{2\pi} \, {{\rm d}}x$. - If $Z\in{{\mathcal{B}}({\mathbb{T}})}$, $A\in{{\mathcal{B}}({\mathbb{R}})}$, we have $$\int_Z \frac{{{\rm d}}\theta}{2\pi} \int_A p^\rho (\theta,x) {{\rm d}}x = \mu^\rho (Z\times A) = \int_Z \nu^\rho_\theta (A) \frac{{{\rm d}}\theta}{2\pi} .$$ This holds for all $Z$, implying that there exists a $\frac{{{\rm d}}\theta}{2\pi}$-null set $N_A \in {{\mathcal{B}}({\mathbb{T}})}$ such that $p^\rho (\theta,\cdot)$ is ${{\rm d}}x$-integrable with $$\int_A p^\rho (\theta,x) {{\rm d}}x = \nu^\rho_\theta (A)$$ for all $\theta\notin N_A$. Let $\{ A_n \}_{n\in{\mathbb N}}$ be a sequence in ${{\mathcal{B}}({\mathbb{R}})}$ with the following property: if $\mu_1 , \, \mu_2$ are positive measures on ${\mathbb{R}}$ such that $\mu_1 (A_n) = \mu_2 (A_n)$ for all $n$, then $\mu_1 = \mu_2$ (such sequence exists since ${\mathbb{R}}$ is second countable by Theorems C §5 and A §13 in Ref. [@Hal]). Let $N = \cup_n N_{A_n}$. Then $N$ is $\frac{{{\rm d}}\theta}{2\pi}$-null, and, if $\theta\notin N$, $p^\rho (\theta,\cdot)$ is integrable with $$\int_{A_n} p^\rho (\theta,x) {{\rm d}}x = \int_{A_n} p^\rho_\theta (x) {{\rm d}}x \quad \forall n .$$ This implies that, if $\theta\notin N$, $p^\rho (\theta,x) = p^\rho_\theta (x)$ for ${{\rm d}}x$-almost all $x$. - This is just . As a consequence of item ([iii]{}) in the above proposition, a well known result on conditional probability distribution ensures that, if $\phi$ is a $\mu^\rho$-integrable function, then $\phi(\theta,\cdot)$ is $\nu^\rho_\theta$-integrable for $\frac{{{\rm d}}\theta}{2\pi}$-almost all $\theta$, the map $\theta\mapsto \int_{\mathbb{R}}\phi(\theta,x) {{\rm d}}\nu^\rho_\theta (x)$ is $\frac{{{\rm d}}\theta}{2\pi}$-integrable, and $$\int_{{\mathbb{T}}\times {\mathbb{R}}} \phi(\theta,x) {{\rm d}}\mu^\rho (\theta,x) = \int_{\mathbb{T}}\left[ \int_{\mathbb{R}}\phi(\theta,x) {{\rm d}}\nu^\rho_\theta (x) \right] \frac{{{\rm d}}\theta}{2\pi}.$$ By Theorem \[TeoMis\], $E$ is the POVM associated to the measurement of a quadrature $X_\theta$ chosen randomly from ${\mathbb{T}}$ with uniform probability $\frac{{{\rm d}}\theta}{2\pi}$. The next corollary shows that the probability distribution $\mu^\rho$ can not have compact support for any $\rho\in{\mathcal{S(H)}}$. \[suppprho\] For all $R>0$, we have $$\int_{{\mathbb{T}}} \int_{\|x\| > R} p^\rho (\theta,x) {{\rm d}}x \frac{{{\rm d}}\theta}{2\pi} > 0 .$$ With $A_R = \{ x\in{\mathbb{R}}\mid \|x\|>R \}$, we have $$\begin{aligned} \int_{{\mathbb{T}}} \int_{\|x\| > R} p^\rho (\theta,x) {{\rm d}}x \frac{{{\rm d}}\theta}{2\pi} & = & \mu^\rho ({\mathbb{T}}\times A_R) = \int_{{\mathbb{T}}} {{\rm tr} \left[ \rho V(\theta) \Pi (A_R) V(\theta)^\ast\right]} \frac{{{\rm d}}\theta}{2\pi} \\ & = & {{\rm tr} \left[ \rho^\prime \Pi (A_R)\right]} ,\end{aligned}$$ with $\rho^\prime = \int_{{\mathbb{T}}} V(\theta)^\ast \rho V(\theta) \frac{{{\rm d}}\theta}{2\pi}$. $\rho^\prime$ is a trace one positive operator. Since it commutes with the representation $V$ of ${\mathbb{T}}$, it is diagonal in the number basis $\{e_n\}_{n\in{\mathbb N}}$ of ${L^2 \left( {\mathbb{R}}\right)}$. Since ${\left\langle e_n \, , \, \Pi (A_R) e_n \right\rangle} > 0$ for all $n$, the claim follows. As a consequence, the map $\rho\mapsto p^\rho$ from ${\mathcal{S(H)}}$ to the set $P({\mathbb{T}}\times {\mathbb{R}})$ of probability densities in ${L^1 \left( {\mathbb{T}}\times {\mathbb{R}}\right)}$ is not surjective. The next corollary shows that it is actually injective, i. e. the POVM $E$ is [ informationally complete]{} [@OQP]. If $\rho, \sigma\in{\mathcal{S(H)}}$ and $\rho \neq \sigma$, then $\mu^\rho \neq \mu^\sigma$. If $\rho, \sigma\in{\mathcal{S(H)}}$, then $\mu^\rho = \mu^\sigma$ if and only if $p^\rho = p^\sigma$ (in ${L^1 \left( {\mathbb{T}}\times {\mathbb{R}}\right)}$), which amounts to say that $p^\rho_\theta = p^\sigma_\theta$ (in ${L^1 \left( {\mathbb{R}}\right)}$) for $\frac{{{\rm d}}\theta}{2\pi}$-almost all $\theta$. This is in turn equivalent to $\nu^\rho_\theta = \nu^\sigma_\theta$ for $\frac{{{\rm d}}\theta}{2\pi}$-almost all $\theta$. For $r\in{\mathbb{R}}$ and $\theta \in{\mathbb{T}}$, we have by spectral theorem $$\int_{{\mathbb{R}}} e^{irx} {{\rm d}}\nu^\rho_\theta (x) = \int_{{\mathbb{R}}} e^{irx} {{\rm tr} \left[ \rho \Pi_\theta ({{\rm d}}x)\right]} = {{\rm tr} \left[ \rho e^{ir X_\theta}\right]} = \sqrt{2\pi} \, [V(\rho)](r\cos \theta , r\sin\theta) ,$$ where $$[V(\rho)] (x,y) = \frac{1}{\sqrt{2\pi}} {{\rm tr} \left[ \rho e^{i(xX + yP)}\right]} .$$ Since the map $V:{\mathcal{I}_1 (\mathcal{H})}{\longrightarrow}C({\mathbb{R}}^2)$ is injective (see for example Ref. [@Fol]), injectivity of the map $\rho\mapsto \mu^\rho$ follows. The Radon transform of the Wigner function and Radon reconstruction formula {#secRadon} =========================================================================== In the previous section, by means of the POVM $E$ defined in Theorem \[PropPOVM\] we estabilished a convex injective correspondence $\rho\mapsto p^\rho$ between states and the set of probability densities on ${\mathbb{T}}\times{\mathbb{R}}$. However, no explicit formula relating $\rho$ to the function $p^\rho$ was given, due to the fact that, if $\rho$ does not have a simple expression in terms of the number basis, the expression ${{\rm tr} \left[ V(\theta)^\ast \rho V(\theta) \Pi (A)\right]}$ can not be explicitly computed. In this section, we will show that, if the state $\rho$ is sufficiently regular, $p^\rho$ can indeed be evaluated, being in fact the Radon transform of the Wigner function $W(\rho)$ of $\rho$. This is a very well known fact in quantum tomography, going back to the seminal paper of Vogel and Risken [@VR]. However, no attention has never been paid in the literature to the fact that performing the Radon transform of $W(\rho)$ makes sense only for a restricted class of states, namely for those $\rho\in{\mathcal{S(H)}}$ such that $W(\rho)\in{L^1 \left( {\mathbb{R}}^2 \right)}$. This constraint becomes even more stringent when one considers the inverse formula reconstructing $\rho$ (or, better, $W(\rho)$) from its associated probability density $p^\rho$. We will see that, in order to derive mathematically consistent formulas both for $p^\rho$ and the reconstruction of $W(\rho)$, one needs to assume that the state belongs to the set of Schwartz functions on ${\mathbb{R}}^2$. This seems a rather strong limitation, as the very natural attempt to extend the Radon transform and Radon reconstruction to the whole set ${\mathcal{S(H)}}$ by means of distribution theory fails in the quantum context (Remark \[RemDistr\]). Our main reference to the results below is Ref. [@Hel]. If $T\in {\mathcal{I}_1 (\mathcal{H})}$, we introduce the bounded continuous function $V(T)$ on ${\mathbb{R}}^2$, given by $$\label{WW} [V(T)](x,y) = \frac{1}{\sqrt{2\pi}} \, {{\rm tr} \left[ T e^{i(xX+yP)}\right]} .$$ It is well known (see for example Ref. [@Fol]) that $V(T) \in {L^2 \left( {\mathbb{R}}^2 \right)}$, and $V$ uniquely extends to a unitary operator $V : {\mathcal{I}_2(\mathcal{H})}{\longrightarrow}{L^2 \left( {\mathbb{R}}^2 \right)}$. The [Wigner transform]{} of $A\in {\mathcal{I}_2(\mathcal{H})}$ is just (up to a constant) the Fourier transform of $V(A)$, i. e. $$\label{W} W(A) = \frac{1}{\sqrt{2\pi}} \, {\mathcal{F}}_2 V(A) ,$$ where ${\mathcal{F}}_2 = {\mathcal{F}}\otimes {\mathcal{F}}$ on ${L^2 \left( {\mathbb{R}}^2 \right)} = {L^2 \left( {\mathbb{R}}\right)} \otimes {L^2 \left( {\mathbb{R}}\right)}$, with ${\mathcal{F}}$ defined in . If $f\in {L^1 \left( {\mathbb{R}}^2 \right)}$, the [Radon transform]{} of $f$ is the complex function $Rf\in {L^1 \left( {\mathbb{T}}\times {\mathbb{R}}\right)}$ given by $$\label{TdR} Rf (\theta , r) = \int_{-\infty}^{+\infty} f (r\cos \theta - t\sin \theta , r\sin \theta + t\cos \theta) {{\rm d}}t$$ $\frac{{{\rm d}}\theta}{2\pi} \, {{\rm d}}r$-almost everywhere. We have the following fact. \[PropTdRdW\] If $W(\rho)\in{L^1 \left( {\mathbb{R}}^2 \right)}$, then $$\label{TdRdW} [RW(\rho)] (\theta,r) = p^\rho (\theta,r)$$ for $\frac{{{\rm d}}\theta}{2\pi} {{\rm d}}r$-almost all $(\theta,r)$. Let $\gamma : {\mathbb{T}}\times {\mathbb{R}}{\longrightarrow}{\mathbb{R}}^2$ be the map $$\gamma (\theta,r) = {\left(}r\cos\theta , r\sin\theta {\right)}.$$ We have $$[V(\rho)\circ \gamma] (\theta,r) = \frac{1}{\sqrt{2\pi}} {{\rm tr} \left[ \rho e^{ir {X}_\theta}\right]} = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} e^{irt} p^\rho_\theta (t) {{\rm d}}t = \left[ {\mathcal{F}}^{-1} p^\rho_\theta \right] (r)$$ by spectral theorem. On the other hand, $$\begin{aligned} && [{\mathcal{F}}_2^{-1} W(\rho)\circ \gamma] (\theta,r) = \frac{1}{2\pi} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} e^{i (xr\cos\theta + yr\sin\theta)} [W(\rho)] (x,y) {{\rm d}}x {{\rm d}}y \\ && \qquad \qquad = \frac{1}{2\pi} \int_0^{\pi} \int_{-\infty}^{+\infty} e^{i t r (\cos\phi\cos\theta + \sin\phi\sin\theta)} [W(\rho)] (t\cos\phi , t\sin\phi) \|t\| {{\rm d}}t \frac{{{\rm d}}\phi}{2\pi} \\ && \qquad \qquad = \frac{1}{2\pi} \int_0^{\pi} \int_{-\infty}^{+\infty} e^{i tr \cos (\phi-\theta)} [W(\rho)] (t\cos\phi , t\sin\phi) \|t\| {{\rm d}}t \frac{{{\rm d}}\phi}{2\pi} \\ && \qquad \qquad = \frac{1}{2\pi} \int_0^{\pi} \int_{-\infty}^{+\infty} e^{i tr \cos\phi} [W(\rho)] (t\cos(\phi + \theta) , t\sin(\phi + \theta)) \|t\| {{\rm d}}t \frac{{{\rm d}}\phi}{2\pi} \\ && \qquad \qquad = \frac{1}{2\pi} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} e^{i r x} [W(\rho)] (x\cos\theta - y\sin\theta , y\cos\theta + x\sin\theta) {{\rm d}}x {{\rm d}}y \\ && \qquad \qquad = \frac{1}{2\pi} \int_{-\infty}^{+\infty} e^{i r x} [RW(\rho)](\theta , x) {{\rm d}}x \\ && \qquad \qquad = \frac{1}{\sqrt{2\pi}} \left[ {\mathcal{F}}^{-1} [RW(\rho)](\theta , \cdot) \right] (r) .\end{aligned}$$ By injectivity of Fourier transform, the claim then follows by comparison. \[CorSuppW\] The support of $W(\rho)$ is an unbounded subset of ${\mathbb{R}}^2$ for all $\rho\in{\mathcal{S(H)}}$. Suppose by contradiction that $W(\rho) = 0$ almost everywhere outside the disk $D_R$ of radius $R$ in ${\mathbb{R}}^2$. Then $W(\rho) \in {L^1 \left( {\mathbb{R}}^2 \right)}$, and so $[RW(\rho)] (\theta,r) = p^\rho (\theta,r)$ by the above proposition. We have $$\begin{aligned} \int_{0}^{2\pi} \int_{\|r\| > R} \|RW(\rho) (\theta,r)\| {{\rm d}}r \frac{{{\rm d}}\theta}{2\pi} & \leq & \int_{0}^{2\pi} \iint_{{\mathbb{R}}^2 \setminus D_R} \|W(\rho) (r\cos \theta - t\sin \theta , r\sin \theta + t\cos \theta)\| {{\rm d}}r {{\rm d}}t \frac{{{\rm d}}\theta}{2\pi} \\ & = & \int_{0}^{2\pi} \iint_{{\mathbb{R}}^2 \setminus D_R} \|W(\rho) (r,t)\| {{\rm d}}r {{\rm d}}t \frac{{{\rm d}}\theta}{2\pi} = 0 ,\end{aligned}$$ which contradicts Corollary \[suppprho\]. The first formal derivation of is contained in Ref. [@VR], without the assumption $W(\rho)\in {L^1 \left( {\mathbb{R}}^2 \right)}$. We stress that if $W(\rho)\notin {L^1 \left( {\mathbb{R}}^2 \right)}$, then does not make sense, and the only possible definition of $p^\rho$ is by means of item 1 in Theorem \[TeoMis\]. If we denote by ${\mathcal{S}^1 (\mathcal{H})}$ the subset of states $\rho\in{\mathcal{S(H)}}$ such that $W(\rho)\in{L^1 \left( {\mathbb{R}}^2 \right)}$, then we have estabilished the following diagram $$\xymatrix{ & {\mathcal{S}^1 (\mathcal{H})}\ar_W[dl]\ar^{p^{\cdot}}[dr] & \\ {L^1 \left( {\mathbb{R}}^2 \right)} \ar@<0.5ex>[rr]^R & & P({\mathbb{T}}\times{\mathbb{R}})}$$ Now we turn to the problem of reconstructing $W(\rho)$ given $p^\rho$. If $W(\rho)\in S({\mathbb{R}}^2)$, the space of Schwartz functions on ${\mathbb{R}}^2$, Radon inversion formula is applicable, and we can obtain $W(\rho)$ from $p^\rho$ in a rather explicit way. Before stating Radon inversion theorem, according to Ref. [@Hel] we need to introduce the set $S_H ({\mathbb{P}}^2)$ of functions $\phi : {\mathbb{T}}\times {\mathbb{R}}{\longrightarrow}{\mathbb C}$ such that - $\phi\in C^\infty ({\mathbb{T}}\times {\mathbb{R}})$; - $\sup_{\theta, \, r} \left\| {\left(}1+\|r\|^k {\right)}\frac{\partial^l}{\partial r^l} \frac{\partial^m}{\partial \theta^m} \phi(\theta,r) \right\| < \infty$; - $\phi(\theta,r) = \phi(2\pi -\theta,-r)$ for all $\theta,r$; - for each $k\in{\mathbb N}$, $\int_{-\infty}^{+\infty} \phi(\theta,r) r^k {{\rm d}}r$ is a homogeneous polynomial in $\sin\theta$, $\cos\theta$ of degree $k$. It is shown in Ref. [@Hel] that $Rf\in S_H ({\mathbb{P}}^2)$ if $f\in S({\mathbb{R}}^2)$, and the map $R: S({\mathbb{R}}^2) {\longrightarrow}S_H ({\mathbb{P}}^2)$ is one-to-one and onto. Thus, in our case $W(\rho) \in S({\mathbb{R}}^2)$ is equivalent $p^\rho\in S_H ({\mathbb{P}}^2)$ by Proposition \[PropTdRdW\]. The next theorem is a restatement of Theorem 3.6 in Ref. [@Hel] (see also Ref. [@VR] for a formal derivation of ). We stress that the hypothesis $W(\rho)\in S({\mathbb{R}}^2)$ (or, equivalently, $p^\rho\in S_H ({\mathbb{P}}^2)$) is needed in order to give meaning to and to define the integral in . Suppose $W(\rho)\in S({\mathbb{R}}^2)$. Then $$\label{ATprho} W(\rho) = \frac{1}{4\pi^2} R^\# [\Lambda p^\rho]$$ where $$\label{DefLambda} \Lambda p^\rho (\theta,r) = \sqrt{\frac{\pi}{2}} \left[ {\mathcal{F}}_t [\|t\|] \ast p^\rho (\theta,\cdot) \right] (r) = {\rm PV} \left[ \int_{-\infty}^{+\infty} \frac{1}{r-t} \frac{\partial p^\rho (\theta,t)}{\partial t} {{\rm d}}t \right]$$ and $$\label{backpr} R^\# f (x,y) = \int_0^{2\pi} f(\theta , x\cos\theta + y\sin\theta) \frac{{{\rm d}}\theta}{2\pi} \quad \forall f\in C^\infty ({\mathbb{T}}\times {\mathbb{R}})$$ (in , the Fourier transform of $\|t\|$ and the convolution are interpreted in the sense of tempered distributions, and ${\rm PV}$ is the Cauchy principal value of the integral). We devote the rest of this section to find the subset of states $\rho\in{\mathcal{S(H)}}$ such that $W(\rho)\in S({\mathbb{R}}^2)$, i. e. to which both Radon transform and Radon reconstruction formula are applicable. Each $T\in{\mathcal{I}_1 (\mathcal{H})}$, being a Hilbert-Schmidt operator on ${L^2 \left( {\mathbb{R}}\right)}$, is an integral operator, whose kernel $K_T$ is in ${L^2 \left( {\mathbb{R}}^2 \right)}$. We have the following fact. Suppose $K\in S({\mathbb{R}}^2)$. Then the integral operator $L_K$ with kernel $K$ is in ${\mathcal{I}_1 (\mathcal{H})}$, and its trace is $$\label{trLK} {{\rm tr} \left[ L_K\right]} = \int_{-\infty}^{+\infty} K(x,x) {{\rm d}}x .$$ Moreover, $L_K \in {\mathcal{S(H)}}$ if and only if $K$ is positive semidefinite[^1] and $\int_{-\infty}^{+\infty} K(x,x) {{\rm d}}x = 1$. Let $I = (-\pi , \pi)$, and let $\Phi : {L^2 \left( {\mathbb{R}}\right)} {\longrightarrow}{L^2 \left( I \right)}$ be the following unitary operator $$\Phi f (y) = (1+\tan^2 y)^{1/2} f (\tan y) .$$ $\Phi$ intertwines $L_K$ with the integral operator $L_{\tilde{K}}$ on ${L^2 \left( I \right)}$ with kernel $$\tilde{K} (y_1 , y_2) = (1+\tan^2 y_1)^{1/2} K (\tan y_1 , \tan y_2) (1+\tan^2 y_2)^{1/2} \quad y_1, y_2 \in (-\pi, \pi) .$$ Since $\tilde{K}$ extends to a $C^\infty$-function on $\overline{I\times I}$ by setting $\tilde{K} = 0$ in the frontier of $\overline{I\times I}$, by Lemma 10.11 in Ref. [@Knapp] $L_{\tilde{K}}$ is a trace class operator on ${L^2 \left( I \right)}$, whose trace is given by $${{\rm tr} \left[ L_{\tilde{K}}\right]} = \int_{-\pi}^{\pi} \tilde{K} (y,y) {{\rm d}}y = \int_{-\infty}^{+\infty} K (x,x) {{\rm d}}x .$$ Since $L_K = \Phi^{-1} L_{\tilde{K}} \Phi$, eq. follows. It is easy to check that, if $K$ is positive semidefinite, then the integral operator $L_K$ is positive. Conversely, suppose $L_K$ is a positive operator. Fix a Dirac sequence $\{ f_n \}_{n\in{\mathbb N}}$, and let $g_n = \sum_{i=1}^N c_i f_n^{x_i}$, where $f_n^{x_i} (x) = f_n (x-x_i)$. We have $$\begin{aligned} 0 & \leq & {\left\langle g_n \, , \, L_K g_n \right\rangle} = \sum_{i,j=1}^N c_i \overline{c_j} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \overline{f_n (x-x_j)} K(x,y) f_n (y-x_i) {{\rm d}}x {{\rm d}}y \\ && \mathop{{\longrightarrow}}_{n\to\infty} \, \sum_{i,j=1}^N c_i \overline{c_j} K(x_j,x_i) ,\end{aligned}$$ from which positive definiteness of $K$ follows. The last claim in the statement is thus clear. We introduce the following linear subspace of ${\mathcal{I}_1 (\mathcal{H})}$ $${\mathcal{I}_1^S (\mathcal{H})}= \left\{ T\in{\mathcal{I}_1 (\mathcal{H})}\mid K_T \in S({\mathbb{R}}^2) \right\} ,$$ and define $${\mathcal{S}^S (\mathcal{H})}= {\mathcal{S(H)}}\cap {\mathcal{I}_1^S (\mathcal{H})}.$$ If $T\in{\mathcal{I}_1^S (\mathcal{H})}$, we can explicitly evaluate the trace in and the Fourier transform in defining $V(T)$ and $W(T)$ respectively. We find $$\begin{gathered} [V(T)](x,y) = {\mathcal{F}}_t^{-1}\left[ K_T ( t + y/2 , t - y/2) \right] (x) \\ [W(T)](x,y) = \frac{1}{\sqrt{2\pi}} {\mathcal{F}}_t\left[ K_T ( x + t/2 , x - t/2) \right] (y) ,\end{gathered}$$ where we denoted by ${\mathcal{F}}_t$ the Fourier transform with respect to the variable $t$. The second formula proves the next proposition. $W : {\mathcal{I}_1^S (\mathcal{H})}{\longrightarrow}S({\mathbb{R}}^2)$ is a bijection. Restricting to states in ${\mathcal{S}^S (\mathcal{H})}$, we have thus arrived at the following diagram. $$\xymatrix{ & {\mathcal{S}^S (\mathcal{H})}\ar_W[dl]\ar^{p^{\cdot}}[dr] & \\ S({\mathbb{R}}^2) \ar@<0.5ex>[rr]^R & & S_H ({\mathbb{P}}^2) \ar@<0.5ex>[ll]^{\frac{1}{4\pi^2} R^\# \Lambda}}$$ \[RemDistr\] Unfortunately, one can not use the definition of Radon transform of distributions to extend to whole ${L^2 \left( {\mathbb{R}}^2 \right)}$, or reconstruction formula to a larger set than ${\mathcal{S}^S (\mathcal{H})}$. In fact, as explained in §5 of Ref. [@Hel], the distributional Radon transform can be defined only as a map $R : {\mathcal{E}}^\prime ({\mathbb{T}}\times {\mathbb{R}}) {\longrightarrow}{\mathcal{E}}^\prime ({\mathbb{T}}\times {\mathbb{R}})$, ${\mathcal{E}}^\prime ({\mathbb{T}}\times {\mathbb{R}})$ being the set of compactly supported distributions on ${\mathbb{T}}\times {\mathbb{R}}$. Corollary \[CorSuppW\] then prevents us from giving any distributional sense to . Similarly, eq. has no distributional analogue, as the reconstruction formula $T = \frac{1}{4\pi^2} R^\# [\Lambda RT]$ (Theorem 5.5 in Ref. [@Hel]) again holds only for compactly supported distributions $T$. Being able to exhibit an explicit inversion formula of the Radon transform only for Wigner functions which are Schwartz class does not* imply a failure of quantum tomographical methods in reconstructing states with weaker regularity properties, as the associated POVM remains informationally complete on the whole of ${\mathcal{S(H)}}$, as we have shown in the first part of this paper. In fact, mainly in order to address issues of numerical stability, actual reconstruction methods usually do not involve $\frac{1}{4\pi^2} R^\# \Lambda$ directly, but some approximated technique involving regularizations; proofs of consistency are available [@Art] for some of these regularized estimators which holds on the whole of quantum state space. [99]{} L. M. Artiles, R. D. Gill and M. I. Guta J. Roy. Stat. Soc. (B) 67 109-134 (2005). G. M. D’Ariano, M. G. A. Paris and M. F. Sacchi Advances in Imaging and Electron Physics, 128 205-308 (2003). U. Leonhardt Phys. Rev. Lett. 74 4101-4105 (1995). K. Vogel and H. Risken Phys. Rev. A 40 2847-2849 (1989). G. Cassinelli, G. M. D’Ariano, E. De Vito and A. Levrero J. Math. Phys. 41 7940-7951 (2000). M. I. Guta, L. Artiles and C. Butucea Ann. Statis. 35 465-494 (2007) M. I. Guta, L. M. Artiles Math. Methods Statist. 16 1-15 P. R. Halmos . Van Nostrand, New York, 1950. P. Busch, M. Grabowski, and P.J. Lahti. . (Springer-Verlag, Berlin, 1997). second corrected printing. G. B. Folland , Annals of Math. Studies. Princeton University Press, Princeton, New York, 1989. S. Helgason , Progress in Mathematics. Birkhäuser, Boston, 1999. A. W. Knapp . Princeton University Press, Princeton, New Jersey, 1986. [^1]: We recall that a function $K:{\mathbb{R}}^2 {\longrightarrow}{\mathbb C}$ is [positive semidefinite]{} if $\sum_{i,j = 1}^N c_i \overline{c_j} K(x_j , x_i) \geq 0$ for all $N\in{\mathbb N}$, ${c_1 , c_2 \ldots c_N} \subset {\mathbb C}$ and ${x_1 , x_2 \ldots x_N} \subset {\mathbb{R}}$.
--- abstract: 'The liquid-gas phase transition in finite nuclei is studied in a heated liquid-drop model where the drop is assumed to be in thermodynamic equilibrium with the vapour emanated from it. Changing pressure along the liquid-gas coexistence line of the systems, symmetric or asymmetric, suggests that the phase transition is a continuous one. This is further corroborated from the study of the thermal evolution of the entropy at constant pressure.' --- =8.1 true in =5.9 true in \ 1.0cm Tapas Sil$^1$, S. K. Samaddar$^1$, J. N. De$^2$ and S. Shlomo$^3$\ $^1$Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700064, India\ $^2$Variable Energy Cyclotron Centre, 1/AF Bidhannagar, Kolkata 700064, India\ $^3$The Cyclotron Institute, Texas A$\&$M University, College Station,\ Texas 77845,USA 1.0cm PACS Number(s): 25.70.-z, 21.65.+f, 24.10.Pa The study of liquid-gas phase transition in finite nuclear systems is of considerable contemporary interest [@poc; @hau; @de1; @nat1; @lee; @ell1; @gul; @das; @paw]. Experimental analyses of the accumulated data on multifragmentation and caloric curves show compelling evidence of such a transition. Phase transitions are normally signalled by peaks in the specific heat at constant volume $C_V$ with rise in temperature. Theoretical models of different genres, such as the microcanonical [@gro] or the canonical [@bon] description of multifragmentation, the lattice-gas model [@gul; @das; @sam] or even the microscopic treatment in a relativistic [@sil] or a nonrelativistic [@de1] Thomas-Fermi framework support such a structure in the heat capacity. A clear idea about the subtle details of the liquid-gas phase transition in finite nuclei, however, has not emerged yet. Confusion remains about whether the system has evolved dynamically through the critical point [@ell1; @nat2; @rad]; a coherent picture about the order of the phase transition is also missing. Analyses of the EOS group [@ell2] and the ISiS group [@kle] in the scaling model give strong circumstantial evidence for a continuous (second order) phase transition. Calculations performed in a mean field model [@lee] also lead to a similar conclusion. Predictions from the lattice-gas models [@gul; @pan] are, however, compatible with a first order transition. Symmetric infinite nuclear matter is effectively a one-component system; with heating, it undergoes a first order phase transition. On the other hand, in case of two-component asymmetric nuclear matter, as was shown in a comprehensive analysis by M$\ddot u$ller and Serot [@mul] in a relativistic mean-field framework, the separate conservation of the neutron and proton number densities leads the system to a continuous phase transition over a finite temperature interval. This is also supported in the nonrelativistic calculations by Kolomietz [et al]{} [@kol]. Unlike symmetric nuclear matter, even a finite symmetric nucleus ($N=Z$) behaves like a two-component system as the Coulomb interaction lifts the isospin degeneracy. A finite nucleus is then expected to undergo a continuous liquid-gas phase transition, if at all. The conflicting predictions from the previous model analyses do not have realistic inputs of nuclear physics as relevant for a quantum system of interacting fermions with a short-range nuclear force as also with the long-range Coulomb interaction. In this communication, we focus on the nature of the phase transition once specific features concerning an atomic nucleus are properly taken into account. For our study, we choose a representative system, namely, Rhenium with $A$= 186 and $Z$= 75. Such a nucleus is likely to be formed from the reaction $^{124}Sn+^{124}Sn$ at energies around 50 MeV per nucleon [@xu] after some nucleons have left out from the reaction zone as preequilibrium particles. We also investigate $^{150}Re$ to explore the isospin asymmetry effects in phase transition. The nucleus is viewed as a spherical liquid drop with asymmetry $X_0$ defined as $(N_0 - Z_0)/A_0$ where $A_0= N_0+Z_0$, is the mass number of the total system. At a finite temperature, we assume the depleted nucleus to be enveloped by its own vapour and the system to be in complete thermodynamic equilibrium conserving the total number of neutrons and protons. The nucleon distributions in the liquid and gas are assumed to be uniform in each phase. This definition allows to explore the liquid-gas coexistence region for a finite system in close analogy with bulk nuclear matter. In absence of a well-defined way to write the energy density functional of a finite nucleus in terms of volume, surface, symmetry and Coulomb terms, we write the free energy of the nuclear system at temperature $T$ in the single phase as $$F=A_0f_{nm}(\rho,X_0)+F_C+F_{surf}$$ where $f_{nm}(\rho,X_0)$ is the free energy per particle of infinite nuclear matter at density $\rho$ with asymmetry $X_0$ at the same temperature $T$, $F_C$ the Coulomb free energy and $F_{surf}$ the temperature and asymmetry dependent surface free energy. In the liquid-gas coexistence region, the free energy is given by $$F_{co}=F^l+F^g,$$ where the liquid free energy is, $$F^l= A^l f_{nm}(\rho^l,X^l)+F_C^l+ F_{surf}^l,$$ and the free energy of the emanated gas is $$F^g= A^g f_{nm}(\rho^g,X^g)+F_C^g.$$ Here $A^l$ and $A^g$ are the number of nucleons in the liquid and the gas phase, $\rho^l$, $\rho^g$ and $X^l$, $X^g$ are the corresponding density and asymmetry. The free energy of infinite nuclear matter is evaluated in the finite temperature Thomas-Fermi framework with a modified Seyler-Blanchard interaction [@de2]. The Coulomb free energies for the liquid and the gas, $F_C^l$ and $F_C^g$ are calculated corresponding to a uniform charged sphere and a spherical-shell, respectively. For simplicity, their mutual interaction is neglected. The surface free energy of the liquid part $F_{surf}^l$ is taken as [@lev] $$F_{surf}^l=\sigma(X^l,T)(A^l)^{2/3},$$ where the surface energy coefficient is $$\sigma(X,T)=[\sigma(X=0)-a_sX^2][1+1.5 T/T_c][1-T/T_c]^{3/2} .$$ Here, $\sigma(X=0)=18$ MeV, $a_s=28.66$ MeV and $T_c$, the critical temperature of the symmetric nuclear matter is 15 MeV. The surface energy coefficient decreases with density; we neglect its density dependence for the liquid part. Since the gas density is very low, its surface energy is neglected. The total surface energy of the liquid is $$E_{surf}^l=F_{surf}^l+TS_{surf}^l$$ where the total surface entropy is given by $$S_{surf}^l=-\left(\frac{\partial F_{surf}^l}{\partial T}\right)_V .$$ The total entropy $S_0$ of the system is then calculated as $$S_0=A^ls_{nm}^l(\rho^l,X^l)+A^gs_{nm}^g(\rho^g,X^g)+S_{surf}^l .$$ The per particle entropy $s_{nm}$ of homogeneous nuclear matter is obtained from the standard mean-field prescription. The chemical potentials $\mu_n^l$ and $\mu_z^l$ for neutron and proton in the liquid are given by $$\begin{aligned} \mu_n^l&=&\frac{\partial F^l}{\partial N^l},\nonumber\\ \mu_z^l&=&\frac{\partial F^l}{\partial Z^l}.\end{aligned}$$ The liquid pressure is obtained from $$P=\rho^2\frac{\partial (F/A)}{\partial \rho}.$$ Similar equations follow for the gas phase. For thermodynamic equlibrium between the liquid and the gas, the two chemical potentials and the pressures in both phases must be the same, i.e, $\mu_n^l=\mu_n^g,\; \mu_z^l=\mu_z^g$ and $P^l=P^g$. Along the coexistence line, the mass of the liquid drop changes. For a chosen $A^l$, the quantities $\rho^l$, $\rho^g$, $X^l$, $X^g$ and $A^g$ are determined by exploiting the three thermodynamic equlibrium conditions and the constraints of baryon number and the total isospin conservation: $$\begin{aligned} \rho_{n,z}&=&\lambda\rho_{n,z}^l + (1-\lambda)\rho_{n,z}^g\nonumber,\\ \rho X_0&=&\lambda\rho^lX^l+(1-\lambda)\rho^gX^g. \label{iso}\end{aligned}$$ Here $\rho$ is the average nucleon density and $\lambda$ is the liquid volume fraction. The isotherms for the nucleus $^{186}Re$ at $T=7,8,9$ and 10 MeV are displayed in Fig.1. For comparison, the isotherm for nuclear matter with asymmetry same as that of $^{186}Re$ is also shown at $T=10$ MeV. The difference between the isotherms for the infinite and the finite system is not insignificant. Though the Coulomb and surface have opposing effects on the pressure, the former wins over the latter at this temperature. The liquid-gas coexistence lines for $^{186}Re$ for the four temperatures mentioned are shown by the dotted lines. It is seen that the pressure changes along the coexistence line, as seen earlier in the case of asymmetric nuclear matter [@mul]; the slope of the coexistence lines also increases with temperature. It is further noted that at a given temperature and asymmetry, because of the Coulomb effect, the slope of the coexistence line of a finite nucleus is more compared to that of asymmetric nuclear matter. The variation of pressure along the coexistence line is a pointer to a continuous phase transition. Unlike nuclear matter, the coexistence lines do not extend from the pure gas phase to the pure liquid phase. At relatively lower temperatures, it is found that as the system expands, the size of the liquid drop depletes and reaches a minimum mass beyond which no thermodynamic equilibrium is possible; at higher temperatures, with compression the liquid drop attains a limiting mass which decreases with increasing temperature as is evident from the figure. As an example, the minimum liquid-drop mass at $T=7$ MeV is $A^l=24$ (marked as A in the figure); at $T=10$ MeV, the limiting liquid-drop mass is $A^l=130$ (marked as B). This implies that at lower temperatures, a gas of a finite number of nucleons when compressed start nucleating with a minimum mass for the seed in order to remain in thermodynamic equilibrium. Similarly, at higher temperatures, for the coexisting finite system, the evaporated gas should contain a minimum number of nucleons. The isospin fractionation along the coexistence line for the nucleus $^{186}Re$ at $T=8 $ MeV is shown in Fig.2. The system has proton fraction $Y_0=0.403$ (defined as $Y=Z/A$). As the system prepared in the gaseous phase is compressed, the two-phase region is encountered at the point A with the emergence of a minimum liquid mass ($A^l=20$) at the point B with a proton fraction $Y_B$ larger than $Y_0$. With further compression, the gas phase depletes from A to C while the liquid phase grows from B to D attaining the total mass $A_0$ and proton fraction $Y_0$. During compression, the proton fractions in both phases decrease, but the total proton fraction $Y_0$ remains fixed, as dictated by the conservation of the total isospin given by Eq.(\[iso\]). It is evident from the figure that the gas phase is more neutron-rich compared to the total system while the liquid phase is comparatively neutron-deficient as also observed experimentally [@xu]. This feature becomes more prominent with increasing liquid mass. In order to explore the asymmetry effect on isospin fractionation in phase transition, the calculated results for the symmetric system $^{150}Re$ are also displayed in the figure. Contrary to the relatively neutron-rich nucleus $^{186}Re$ , here it is found that the gas phase is proton-rich. In symmetric nuclei, since $\mu_z$ is greater than $\mu_n$ because of Coulomb interaction, the separation of the gas phase behaves more like that of proton-rich nuclear matter. The occurrence of liquid-gas phase transition in symmetric medium-heavy nuclei should then lead to preponderance of proton-rich isotopes in energetic heavy-ion collisions and this can be tested in experiments. The heat capacity per particle $C_V$ for $^{186}Re$ at constant volume (defined as $\left(\frac{d(E^/A_0)}{dT}\right)_V$, where $E^$ is the total excitation energy of the system) is displayed in Fig.3 at a representative volume $V=10V_0$ which can be interpreted as a [freeze-out]{} volume. Here $V_0$ is the normal volume of the nucleus calculated with the radius parameter $r_0=1.16$ fm. A very broad bump in $C_V$ with a maximum at $T\sim 10 $ MeV is seen. The system then corresponds to a liquid part with $A^l$ around 80, the rest of the nucleons being in the gas phase. This is contrary to the results in the microscopic mean-field calculations obtained earlier [@de1; @sil] at around the same freeze-out volume, where a much sharper peak was observed when the system just vaporises completely. At very high temperature, the heat capacity saturates at 1.5 corresponding to a pure classical gas. The thermal evolution of the entropy per particle for the nucleus $^{186}Re$ at a constant pressure $P=0.06$ MeV fm$^{-3}$ is shown by the solid line in figure 4. In contrast to a first order phase transition where the entropy at constant pressure exhibits a discontinuity at a particular temperature (the phase transition temperature), here the entropy change is continuous. The noticeable rise in entropy in the temperature range 8.4 to 10.3 MeV is the manifestation of a liquid-gas phase transition in this temperature domain for the chosen pressure. The dashed line in the figure corresponds to the entropy evolution for the symmetric nucleus $^{150}Re$. This is quite similar to that of the asymmetric nucleus considered, but markedly different from that of symmetric nuclear matter [@mul] which shows a sharp discontinuity at the transition temperature. The striking similarity between the two results shows that the finite size and Coulomb effects are the dominant factors in determining the nature of phase transition in finite nuclei. The continuous change of entropy at a constant pressure plead in favour of the characterisation of the liquid-gas phase transition in atomic nuclei, symmetric or asymmetric, as a continuous one. The liquid-gas phase transition in finite nuclei with explicit conservation of the baryon number and the total isospin has been investigated in this communication in a heated liquid drop model. The peaked structure in the heat capacity, though broad, signals the occurrence of a liquid-gas phase transition. From the evolution of entropy at constant pressure, one sees that the transition occurs over a range of temperatures; this strongly suggests that the liquid-gas phase transition in a finite nuclear system is continuous. The simplified assumptions in the model may affect the results somewhat quantitatively, but the general qualitative features are expected to remain unaltered. The thermodynamic concepts may not be very meaningful when the number of particles in one of the phases is very small, still this model serves as a window to understand the basic features of liquid-gas phase transition in finite systems. [99]{} J. Pochodzalla [et al]{}, Phys. Rev. Letts. [75]{}, 1040 (1995). J. 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The corresponding coexistence line is shown by the filled circles. - Evolution of proton fraction ($Y$) along the liquid-gas coexistence line at $T=$ 8 MeV for the systems as shown. - The specific heat capacity $C_V$ as a function of temperature for $^{186}Re$ in a freeze-out volume 10$V_0$. - Entropy per particle as a function of temperature at a constant pressure for the systems $^{186}Re$ and $^{150}Re$.
--- abstract: 'We use thirty-eight high-resolution simulations of galaxy formation between redshift 10 and 5 to study the impact of a 3 keV warm dark matter (WDM) candidate on the high-redshift Universe. We focus our attention on the stellar mass function and the global star formation rate and consider the consequences for reionization, namely the neutral hydrogen fraction evolution and the electron scattering optical depth. We find that three different effects contribute to differentiate warm and cold dark matter (CDM) predictions: WDM suppresses the number of haloes with mass less than few $10^9$ M$_{\odot}$; at a fixed halo mass, WDM produces fewer stars than CDM; and finally at halo masses below $10^9$ M$_{\odot}$, WDM has a larger fraction of dark haloes than CDM post-reionization. These three effects combine to produce a lower stellar mass function in WDM for galaxies with stellar masses at and below $~10^7$ M$_{\odot}$. For $z > 7$, the global star formation density is lower by a factor of two in the WDM scenario, and for a fixed escape fraction, the fraction of neutral hydrogen is higher by 0.3 at $z \sim 6$. This latter quantity can be partially reconciled with CDM and observations only by increasing the escape fraction from 23 per cent to 34 per cent. Overall, our study shows that galaxy formation simulations at high redshift are a key tool to differentiate between dark matter candidates given a model for baryonic physics.' author: - 'Boyan K. Stoychev$^{1}$, Keri L. Dixon$^{1}$[^1], Andrea V. Macciò$^{1,2}$, Marvin Blank$^{1,3}$,' - \| Aaron A. Dutton$^1$\ \ $^{1}$New York University Abu Dhabi, PO Box 129188, Saadiyat Island, Abu Dhabi, United Arab Emirates\ $^{2}$Max Planck Institute für Astronomie, Königstuhl 17, D-69117 Heidelberg, Germany\ $^{3}$Institut für Theoretische Physik und Astrophysik, Christian-Albrechts-Universität zu Kiel, Leibnizstr. 15, D-24118 Kiel, Germany bibliography: - 'bibliography.bib' title: Clues to the nature of dark matter from first galaxies --- \[firstpage\] cosmology: theory – dark matter – galaxies: formation – galaxies: high-redshift – galaxies: kinematics and dynamics – methods: numerical Introduction ============ In the next several years, we expect a large amount of data on the high-redshift ($z>6$) Universe to become available to the scientific community. Facilities like Atacama Large Millimetre/sub-millimetre Array (ALMA) and James Webb Space Telescope (JWST) will open a completly new observational window on the first billion years of the life of our Cosmos. These data will help us understand the early phases of galaxy formation, but they might also guide us towards a better understanding of the dark side of our Universe, as for example the nature of dark matter. The current leading model for dark matter is based on a Cold candidate (CDM), initially motivated by the possible existence of Weakly Interactive Massive Particles (WIMPs) predicted by some extensions of the Standard Particle Models [@Bertone2005]. Such candidates should be in the mass range of several GeV [@Bergstrom2012] and have been extensively searched for in underground labs, but with very little success to date [see @Roszkowski2017 for a recent review]. The lack of direct detection of a possible CDM particles raises the question if the dark matter particle might have a much lower mass than initially thought, and possibly lie in the keV range, being what is usually dubbed Warm Dark Matter (WDM). WDM was initially put forward as a possible solution of the so called ‘small-scale crisis’ of CDM [e.g @Moore1994; @Klypin1999; @Moore1999; @Colin2000]. More recently, there has been a mounting evidence that the solution of these problems most likely lies in the baryonic sector. Recent simulations and semi-analytical models have been able to reconcile a CDM universe with observations both on the number of galactic satellites [@Bullock2000; @Maccio2010; @Sawala2016; @Buck2018] and on the dark matter distribution at small scales by simply providing a more accurate treatment of galaxy formation and baryonic physics within a CDM model [@Governato2010; @Zolotov2012; @DiCintio2014a; @Onorbe2015; @Frings2017]. Moreover, current limits from the Lyman-$\alpha$ (Ly$~\alpha$) forest on the matter power spectrum [@Irsic2017; @Yeche2017] have set very stringent limits on the mass of a possible (thermal) WDM candidate, which is now constrained to $m_{\nu} >3.0$ keV. With such a mass, a WDM candidate will be practically indistinguishable from a cold one regarding the dark matter distribution on small scales [@Colin2008; @Maccio2012b; @Shao2013; @Herpich2014] or the number of satellites [@Maccio2010b; @Lovell2016]. The similarity of currently allowed WDM models to CDM on small scales calls for a new venue on where to look for more clues on the actual mass of the dark matter candidate, and this venue can lie in the high-redshift Universe. In WDM, due to the lack of power on small scales, structure formation is delayed with respect to a cold scenario. At present time, the effect of such an initial delay has been erased by the nonlinearity of galaxy formation, but it can still present itself in the very early stages of the Universe, when the first galaxies were formed. This signature can manifest itself in the formation of first stars [e.g. @Gao2007], or first galaxies [e.g. @Dayal2015]. Recently, [@Corasaniti2017] have used measurements of the galaxy luminosity function at $z =$ 6, 7, and 8 to derive constraints on WDM. They have combined very high resolution $N$-body (pure gravity) simulations with an empirical approach based on halo abundance matching to derive predictions for the luminosity function in different dark matter models. They have obtained a lower limit on the WDM thermal relic particle mass of 1.5 keV at a 2$\sigma$ level. This study relies on dark matter only simulations that might miss some of the nonlinear effects between the nature of dark matter and galaxy formation, which is a possible limitation. The first galaxies are also thought to be the main source of the ionizing photons responsible for cosmic reionization, which may then provide another constraint to the nature of dark matter. Many recent studies have studied the connection between WDM and reionization [e.g. @Lapi2015; @Tan2016; @Lopez2017; @Das2018]. In particular, @Carucci2019 extend the work of [@Corasaniti2017] to derive the requisite escape fractions of ionizing photons to match recent observations, but given the uncertain nature of this quantity, the results largely independent of underlying dark matter scenario. Two studies employing semi-analytic methods to study WDM in the context of high redshift and reionization also find that with a dark matter scenario consistent with observational constraints the ionizing population is shifted to lower halo masses, but that distinguishing CDM and WDM with current observations would be difficult [@Bose2016; @Dayal2017]. Furthermore, [@Villanueva2018] have used hydrodynamical simulations in a box of 20 $h^{-1}$ Mpc to derive constraints on WDM from galaxy luminosity functions, the ionization history, and the Gunn-Peterson effect. They have concluded that while some effects are indeed present, until the modelling of baryonic effects (star formation and feedback mainly) are constrained better, any conclusions on the nature of dark matter derived from reionization observables remain model-dependent. One possible way to anchor the parameters describing the baryonic physics (besides having better constraints from observations) is to use observations at one redshift (usually $z=0$) as a constraint and then check predictions at a much higher redshift. In this study, we attempt to connect the high- and low-redshift Universe via hydrodynamical simulations of galaxy formation. We want to extend to the high-redshift Universe ($z>6$) the baryonic physics used in the NIHAO project [@Wang2015], which has proven very successful in reproducing key observations in the low-redshift Universe ($0<z<2$), as for example the stellar mass - halo mass relation [@Wang2015], the local velocity function [@Maccio2016], and galaxies scaling relations [@Dutton2017]. We use 19 high-resolution, zoom-in simulations in each cosmology (for a total of 38) to accurately sample the stellar mass function in the redshift range $5<z<10$, and we study the impact of a 3 keV dark matter candidate on several current and future observables, like the stellar mass function, the global star formation rate, the hydrogen neutral fraction, and the Cosmic Microwave Background (CMB) optical depth. The paper is organized as follows. In Section \[sec:setup\], we present the set up of our galaxy formation simulations, in Section \[sec:results\], we present our results and forecasts for current and future observations. Finally in Section \[sec:concl\], we discuss the implications or our results and present our conclusions. Simulations {#sec:setup} =========== All simulations herein were ran in a flat $\rm \Lambda CDM$/$\rm \Lambda WDM$ cosmology, with parameters from the Planck Collaboration et al. [@Planck2014]: Hubble Parameter $H_{0} = 67.1$ km s$^{-1}$ Mpc$^{-1}$, matter density $\Omega_{\rm m} = 0.3175$, radiation density $\Omega_{\rm r} = 0.00008$, dark energy density $\Omega_{\Lambda} = 1 - \Omega_{\rm m} - \Omega_{\rm r} = 0.6824$, power spectrum normalization $\sigma_{\rm s} = 0.8344$, and power spectrum slope $n = 0.9624$. Initial Conditions {#ssec:IC} ------------------ Initial conditions for both the $N$-body and hydrodynamical, zoom-in simulations of galaxy formation were generated using a modified version of the [grafic2]{} package [@Bertschinger2001; @Penzo2014]. For WDM, the process is identical to CDM except for a modification to the power spectrum. The same random seed was used to generate all pairs of CDM/WDM 3 keV initial conditions. WDM power spectra were computed with the use of a relative transfer function, as in [@Bode2001]: $$P_{\rm WDM}(k) = [T_{\rm WDM}(k)]^2 P_{\rm CDM}(k),$$ where $$T_{\rm WDM}(k) = \left[1+(\alpha k)^{2\nu}\right]^{-5/\nu}.$$ Here, $\nu = 1.12$ [@Viel2005] and $\alpha$ is the length scale of the break in the WDM power spectrum: $$\alpha = 0.049\left(\frac{m_{\rm x}}{1 {\rm keV}}\right)^{-1.11}\left(\frac{\Omega_{\rm x}}{0.25}\right)^{0.11} \left(\frac{h}{0.7}\right)^{1.22} h^{-1} {\rm Mpc},$$ where $m_{\rm x}$ is the mass of the WDM particle and $\Omega_{\rm x}$ the density. $N$-body Simulations -------------------- Box Size N m$_{\rm dm}$ $\epsilon_{\rm dm}$ ----- ------- --------- -------------------- --------------------- Mpc M$_{\rm \odot}$ kpc 1 $5$ $250^3$ $1.00 \times 10^6$ $0.50$ 2 $15$ $250^3$ $2.70 \times 10^7$ $1.50$ 3 $20$ $500^3$ $8.00 \times 10^6$ $1.00$ 4 $40$ $600^3$ $3.70 \times 10^7$ $1.67$ 5 $100$ $500^3$ $1.00 \times 10^9$ $5.00$ 6 $200$ $500^3$ $8.00 \times 10^9$ $10.0$ : Size and resolution of the 6 $N$-body simulation boxes.[]{data-label="tab:box"} First, we run a series of $N$-body simulations that only include dark matter, detailed in Table \[tab:box\]. We used six boxes that range from 5 to 200 Mpc length on a side with varying particle number. Softening was set to 1/40 of the intra-particle distance on the initial conditions grid. From this set of simulations, we are then able to sample a wide range of halo masses at $z= 5 - 10$. Hydrodynamical Simulations -------------------------- From the dark matter-only simulations, 19 haloes were chosen to be re-simulated at higher resolution with baryons included. Table \[tab:res\] shows the five resolution levels used in our simulations. These levels were chosen to have approximately one million particles inside the virial radius of the galaxy across the whole range of halo masses. Level $m_{\rm DM}$ $m_{\rm gas}$ $\epsilon_{\rm DM}$ $\epsilon_{\rm gas}$ ------- -------------------- -------------------- --------------------- ---------------------- M$_{\rm \odot}$ M$_{\rm \odot}$ pc pc 1 $1.95 \times 10^3$ $3.56 \times 10^2$ $63$ $27$ 2 $1.56 \times 10^4$ $2.85 \times 10^3$ $125$ $53$ 3 $1.25 \times 10^5$ $2.28 \times 10^4$ $250$ $107$ 4 $4.21 \times 10^5$ $7.70 \times 10^4$ $375$ $160$ 5 $5.78 \times 10^5$ $1.06 \times 10^5$ $417$ $178$ : Resolution levels used in our zoom-in simulations.[]{data-label="tab:res"} All simulations were performed with the SPH code [gasoline]{} [@Wadsley2004; @Wadsley2017]. The code was setup in the framework of the NIHAO project [@Wang2015], including metal cooling, chemical enrichment, star formation, and feedback from massive stars and supernovae (SN). Stars are formed from gas cooler than $T$ = 15,000 K and denser than $n_{\rm th}=10{\rm ~cm}^{-3}$. The star formation efficiency used was $c_\star$ = 0.1. Cooling via hydrogen, helium, and various metal-lines is included as in @Shen2010, including photoionization and heating from ultraviolet (UV) background in @Haardt2012. SN feedback is implemented using the blastwave approach as described in @Stinson2013, which relies on delaying the cooling of nearby particles to a SN event. We also include what we dubbed Early Stellar Feedback: we inject 13 per cent of the UV luminosity of the stars as thermal energy before any SN events take place without disabling the cooling [see @Stinson2013 for more details]. NIHAO galaxies have been shown to be consistent with a wide range of low-redshift galaxy properties, such as the stellar mass - halo mass relation and star formation rates [@Wang2015]; stellar disk kinematics [@Obreja2016]; cold and hot gas content [@Stinson2015; @Wang2016; @Gutcke2016]; and resolve the too-big-to-fail problem [@Dutton2016]. We are therefore confident that the simulations presented herein can be used as plausible tools to study the effect of a WDM cosmology on the high-redshift Universe. Haloes in all simulations were identified using the MPI+OpenMP hybrid Amiga Halo Finder [^2] [<span style="font-variant:small-caps;">AHF</span>; @ahf]. The virial masses of the haloes are defined as the mass within a sphere containing $\Delta = 200$ times the cosmic critical matter density. The virial (total) mass is denoted as $M_{200}$, and $M_{\star}$ indicates the total stellar mass within the virial radius. Beyond the 19 target galaxies, we include any galaxy within our zoom simulations that has greater than 10,000 particles. We also only consider central haloes, i.e. not satellites, and require no pollution from larger low-resolution dark matter particles, i.e. any particle larger than the lowest level in Table \[tab:res\]. The result is of order 100 galaxies at $z = 5$, and tens of galaxies at $z = 10$. Simulation analysis was done using the Python package [pynbody]{} [@pynbody]. All parameter fits were performed using Markov Chain Monte Carlo (MCMC) via the Python package [emcee]{} [@emcee]. Results {#sec:results} ======= In the following, we will present the results of our $N$-body and galaxy simulations in CDM and WDM cosmologies at $z = 5-10$. In all plots, CDM results are shown in blue with circles, while 3 keV WDM results are shown in red with squares. Section \[sec:hmf\] shows the halo mass functions obtained from the $N$-body simulations. In Section \[sec:smhm\], we find the stellar mass - halo mass relation obtained from our simulations. Section \[sec:frac\] shows our results for the fraction of galaxies that form stars as a function of halo mass. Section \[sec:smf\] shows the stellar mass functions obtained by convolving all prior results. Finally in Section \[sec:reion\], we compare our results for the reionization history and CMB optical depth to observational constraints. Halo Mass Function {#sec:hmf} ------------------ The $N$-body boxes were used to construct and fit halo mass functions. We used a five-parameter, modified Schechter function [@Schecter1976]: $$\label{eq:hmf} \log\left(\frac{dN}{d\log M}\right) = A-B \log M - C{\rm e}^{\left(M_{0}-\log M\right)^{\alpha}},$$ where $A$, $B$, $C$, $\alpha$, and $M_0$ are fitting parameters that are given in Appendix \[app:hmf\] and $M$ represents $M_{200}/\Msun$. The original Schechter function struggled to fit the WDM results, so we adopted this modified version for both cosmologies. The uncertainty for each halo mass bin is considered Poissonian. ![Halo mass function at $z = 6$ (upper panel) and 10 (lower panel). The blue circles and red squares correspond to CDM and WDM results as obtained from the $N$-body simulations, respectively. The error bars represent Poisson errors, reflecting the smaller number of massive haloes at high redshift. The familiar turnover of the halo mass function for WDM at low halo mass is evident in both panels.[]{data-label="fig:hmf"}](mf_z6.pdf "fig:"){width="80mm"} ![Halo mass function at $z = 6$ (upper panel) and 10 (lower panel). The blue circles and red squares correspond to CDM and WDM results as obtained from the $N$-body simulations, respectively. The error bars represent Poisson errors, reflecting the smaller number of massive haloes at high redshift. The familiar turnover of the halo mass function for WDM at low halo mass is evident in both panels.[]{data-label="fig:hmf"}](mf_z10.pdf "fig:"){width="80mm"} The halo mass function represents the abundance of haloes at a given mass. As shown in Fig. \[fig:hmf\], the halo mass functions exhibit a suppression of the number density of low-mass haloes in a WDM cosmology, a direct consequence of the lack of power on small scales in the WDM power spectrum. The lines represent the MCMC parameter fit, the values of which are given in Appendix \[app:hmf\]. The relations for the two cosmologies follow the same trend at $z = 5-10$. Importantly, the difference between CDM and WDM is larger at higher redshift. At $z = 0$, the halo mass functions would be identical for the two scenarios. Stellar Mass - Halo Mass Relation {#sec:smhm} --------------------------------- ![Stellar mass - halo mass relation at $z = 6$ (upper panel) and 10 (lower panel). Each blue circle and red square corresponds to a hydrodynamical zoom-in simulation in CDM and WDM, respectively. The blue and red lines show a simple power-law fit. The dashed line shows the well-known Moster relation with the grey area showing the $1\sigma$ scatter [@Moster2013]. Importantly, this relation is extrapolated to higher redshift and lower masses than the original fit.[]{data-label="fig:smhm"}](moster_z6.pdf "fig:"){width="80mm"} ![Stellar mass - halo mass relation at $z = 6$ (upper panel) and 10 (lower panel). Each blue circle and red square corresponds to a hydrodynamical zoom-in simulation in CDM and WDM, respectively. The blue and red lines show a simple power-law fit. The dashed line shows the well-known Moster relation with the grey area showing the $1\sigma$ scatter [@Moster2013]. Importantly, this relation is extrapolated to higher redshift and lower masses than the original fit.[]{data-label="fig:smhm"}](moster_z10.pdf "fig:"){width="80mm"} Fig. \[fig:smhm\] shows the stellar mass - halo mass relation obtained from the set of zoom-in simulations run in CDM and WDM cosmologies.[^3] Based on this relation, CDM and WDM galaxies cannot be distinguished from each other. While our simulations closely follow the abundance matching relation(s) at lower redshift [see @Wang2015], our predicted relation at these high redshifts consistently lies above the extrapolation from [@Moster2013].[^4] This offset (minor in our simulations) suggests that high-redshift extrapolations of relations calibrated at $z < 4$ should be used with caution. Many studies at high redshift exhibit similar offsets, see e.g. [@Rosdahl2018], but also see [@Ma2018]. This figure also indicates that a larger fraction of WDM galaxies remain dark at high redshift compared to CDM, as seen by the smaller number of red points relative to blue. This effect is further explored in the next section. The solid lines represent a linear fit to the logarithmic data, which is done independently for each cosmology and redshift. The median (1$\sigma$) parameters derived from the MCMC sample generate the solid lines (shaded regions), the values of which are located in Appendix \[app:hmf\] These (small) uncertainties are propagated to the stellar mass function results. As shown in Fig. \[fig:smhm\], CDM and WDM behave similarly. We find weak evidence for redshift evolution of the normalization in that lower redshift has a higher value for the intercept of the linear fit, which is consistent with @Ceverino2017. However, [@Ma2018] find no redshift evolution. Dark Fraction {#sec:frac} ------------- Star formation becomes more inefficient at low masses, to the point where no gas is able to collapse to the center and halo and form stars due to the UV background [@Gnedin2000]. Fig. \[fig:smhm\] seems to suggest a larger fraction of ‘dark’ galaxies in WDM w.r.t. CDM, given that there are fewer red squares than blue circles. To quantify this effect, we want to compute the fraction of objects able to form stars as a function of their virial mass. A halo is considered dark if it contains no stellar particles. Results are presented in Fig. \[fig:frac\] for $z = 6$ and $z = 10$. The fitting curve is a hyperbolic tangent two-parameter function $$f_{\star} = \frac{1+\tanh\left[\, \beta ({\rm log}M-M_{1})\right]}{2}, \label{eq:fstar}$$ where $\beta$ and $M_1$ are parameters and $M$ once again represents $M_{200}/\Msun$. The points represent the histogram of the $f_{\star} = 1 - N_{\rm dark}/N_{\rm bin}$, where $N_{\rm dark}$ are the number of dark haloes and $N_{\rm bin}$ are the total number of haloes in the bin. When fitting this curve, the uncertainties on $f_{\star}$ are derived from the number of haloes in each bin. The resultant median (1$\sigma$) uncertainties of the fitted parameters are generated from the MCMC sample and are represented by the solid line (shaded region), all values are found in Appendix \[app:hmf\]. We find weak evidence that WDM exhibits a steeper dark fraction fraction transition, which is consistent with [@Maccio2019]. In these simulations, reionization occurs $z \gtrsim 9$, and the shift of minimum halo mass for star formation is evident. At earlier times (higher redshift), the $M_{200}$ at which stars form is lower due to a lower UV background. At $z > 8$, WDM has a lower minimum mass for star formation than CDM, and vice versa for lower redshift. Another interesting result is that only approaching $M_{200} > 5\times10^{10} \Msun$ guarantees that stars will form in galaxies at this redshift. The main caveat for these observations is that the uncertainties for $f_{\star}$ are large given our small halo numbers, and distinguishing with confidence between the two cosmologies is beyond our statistical reach. ![Fraction of galaxies containing stars as a function of their halo mass. The points represent the binned values for CDM (squares) and WDM (circles). The solid lines are the best-fitting from the MCMC sample, and the shaded regions represent the 1$\sigma$ uncertainties on the parameters derived for the sample.[]{data-label="fig:frac"}](fstar_z6.pdf "fig:"){width="85mm"} ![Fraction of galaxies containing stars as a function of their halo mass. The points represent the binned values for CDM (squares) and WDM (circles). The solid lines are the best-fitting from the MCMC sample, and the shaded regions represent the 1$\sigma$ uncertainties on the parameters derived for the sample.[]{data-label="fig:frac"}](fstar_z10.pdf "fig:"){width="85mm"} Stellar Mass Function {#sec:smf} --------------------- The next step is to convolve the results from previous sections in order to obtain the stellar mass function. We started from the fitted stellar mass - halo mass relation, including the uncertainty in the parameters, for our set of galaxy simulations (Fig. \[fig:smhm\]) to convert the halo masses in Fig. \[fig:hmf\] into stellar masses. We then multiplied by the fraction, including the uncertainty of the fit, of galaxies that actually formed stars (Fig. \[fig:frac\]) to obtain the differential number density of galaxies as a function of their stellar mass, which is shown in Fig. \[fig:smf\]. The solid line represents the best-fitting (and median) parameters and the shaded region is the propagated uncertainties, where the $M_{\star}$-$M_{200}$ relation dominates at high stellar masses (few high-mass haloes) and $f_{\star}$ dominates at low stellar masses (suppression of star formation becomes important). The $M_{\star}$-$M_{200}$ uncertainty is reflected the stellar massCeverino2017 uncertainty ($x$ axis), and $f_{\star}$ drives the uncertainty in the number density ($y$ axis). ![image](smf){width="85mm"} ![image](smf_err_z8.pdf){width="85mm"} As expected, the stellar mass functions substantially differ only at low masses, while the small differences at high masses are due to cosmic variance and low number statistics. WDM predicts fewer low-mass galaxies than CDM, with a larger difference at high redshift. For our choice of warm candidate mass (3 keV), the relative difference between CDM and WDM is only a factor of a few stellar masses around $10^6$ [see also @Villanueva2018], below what is currently observable [@Bouwens2015; @Bouwens2017; @Ceverino2017]. On the other hand, our results are in agreement with [@Corasaniti2017], who found a lower limit of 2 keV from analysis of the luminosity function. Future observations and facilities might improve this limit (e.g. JWST), but then a very careful understanding of the baryonic physics implemented in the simulations will be needed, as discussed in [@Villanueva2018]. Star Formation Rates {#sec:sfr} -------------------- The star formation rate (SFR) of a galaxy is not only of interest on its own, the reionization history of the Universe is crucially dependent on this quantity. First, we compute the SFR-$M_{\star}$ relation for our galaxies as a function of redshift, where the SFR includes star formation occurring within the past 100 Myr at the redshift in question. As shown in Fig. \[fig:smsfr\], CDM and WDM follow the same relation at $z = 7$, and the same is true for lower and higher redshifts. With this in mind, we fit this relation with a single power law for both models at each redshift (shown by the dashed line). The usual MCMC procedure was performed with each SFR value weighted by the number of stellar particles that formed in the past 100 Myr. The best-fitting slope and intercept values[^5] are found in Appendix \[app:hmf\] and are similar to those found in @Ma2018. Contrary to @Ma2018, we find evidence of redshift evolution in the intercept or normalization, though no evolution in the slope. Therefore, at lower redshift, a lower SFR is expected for the same stellar mass. ![Star formation rate - stellar mass relation in CDM (blue circles) and WDM (red squares) at $z = 7$. Here, the SFR is averaged over the previous 100 Myr. The dashed line represents the linear (in log space) fit to the combined data set of CDM and WDM, as there is no significant difference in the relation between the two cosmologies.[]{data-label="fig:smsfr"}](sfr_100_z7){width="80mm"} We then combine our stellar mass functions with the fits of SFR versus $M_{\star}$ to obtain an integrated global SFR density, $\rho_{\rm SFR}$, as a function of redshift (Fig. \[fig:sfr\]). Stellar mass functions were integrated between $10^3 - 10^{12}~\Msun$, and the results are insensitive to order-of-magnitude changes in these limits. The values of $\rho_{\rm SFR}$ obtained for $z = 5-10$ were fitted following: $$\label{eq:sfrdens} \log\left(\frac{\rho_{\rm SFR}(z)}{\rm M_{\odot} \, Mpc^{-3} \, yr^{-1}}\right) = \kappa - \lambda z - \mu e^{z_{0} - z},$$ where $\kappa$, $\lambda$, $\mu$, and $z_0$ are parameters, the best-fitting values derived from the usual MCMC procedure for cosmologies are found in Table \[tab:sfr\]. Note that the errorbars represent the propagated uncertainties from the previous relations and decrease at lower redshift as we have more statistical power. At $z = 10$, WDM is a factor of two lower than the CDM case, and this factor decreases as redshift decreases. By $z = 7$, the two cosmologies are very similar and essentially indistinguishable at even lower redshift. Both scenarios lie within the range of values between the uncorrected and dust-corrected results from [@Bouwens2015] as indicated by the shaded region in the figure. Note that we do not correct for dust or consider any effects beyond our intrinsic star formation. Cosmology $\kappa$ $\lambda$ $\mu$ $z_{0}$ ----------- ---------- ----------- ------- --------- CDM 0.00131 0.254 6.03 0.0937 WDM 0.676 0.377 6.16 0.381 : Best-fitting parameter values for global SFR density, given by equation (\[eq:sfrdens\]).[]{data-label="tab:sfr"} ![Global star formation rate density as a function of redshift. The shaded region encompasses the data with and without dust correction. The WDM scenario has lower $\rho_{\rm SFR}$ than CDM at higher redshift, but by $z = 6$, the two cosmologies roughly reach parity. []{data-label="fig:sfr"}](globalSFR.pdf){width="85mm"} Reionization History {#sec:reion} -------------------- In this section, we use the predicted galaxy populations in the CDM and WDM models to explore the effects of the WDM cosmology on cosmic reionization, using a commonly used, simple ODE model. We emphasize that this model is a rough estimate of the reionization history, but is a sufficient first step given the uncertainties in reionization modelling. In particular, we treat $f_{\rm esc}$, the fraction of ionizing photons that escape from galaxies into the IGM, as a free parameter that is tuned to match current observational measurements. We follow the approach described in [@Kuhlen2012], starting with this differential equation for the ionized fraction: $$\label{eq:ionfrac} \dot{Q}_{\ion{H}{ii}} = \frac{\dot{n}_{\rm ion}}{\langle n_{\rm H} \rangle} - \frac{Q_{\ion{H}{ii}}}{t_{\rm rec}}.$$ Here, $\langle n_{\rm H} \rangle = X_{\rm p} \Omega_{\rm b} \rho_{\rm c}$ is the comoving hydrogen density, in terms of the hydrogen mass fraction $X_{\rm p} = 0.75$, the baryon density $\Omega_{\rm b} = 0.049$, and the critical density $\rho_{\rm c}$. The IGM recombination time is: $$\begin{aligned} t_{\rm rec} & = \left[C_{\ion{H}{ii}}\alpha_{\rm B}(T)\left(1+\frac{Y_{\rm p}}{4X_{\rm p}}\right)\langle n_{\rm H} \rangle (1+z)^3\right]^{-1} \\ & \approx 0.97\,{\rm Gyr} \left(\frac{C_{\ion{H}{ii}}}{3}\right)^{-1} \left(\frac{T}{2 \times 10^{4}\,{\rm K}} \right)^{0.7} \left(\frac{1+z}{7} \right)^{-3}, \end{aligned}$$ where $\alpha_{\rm B}(T) $ is the case-B recombination coefficient at an IGM temperature $T = 2 \times 10^4$ K, with the value of $1.6 \times 10^{-13}$ cm$^3$ s$^{-1}$ [@Storey1995], and $Y_{\rm p} = 0.25$ being the helium mass fraction. $C_{\ion{H}{ii}}$ is the effective clumping factor of ionized gas in the IGM; we adopt a constant value of 3 in this work. Of course, a lower value would decrease the resultant $f_{\rm esc}$ and vice versa. For example, $C_{\ion{H}{ii}}$ = 1 requires an $f_{\rm esc}$ that is approximately 10 per cent lower. Finally, $\dot{n}_{\rm ion}$ is the global production rate of ionizing photons: $$\dot{n}_{\rm ion} = f_{\rm esc}\ \xi_{\rm ion}\ \rho_{\rm SFR},$$ where $f_{\rm esc}$ is the effective fraction of photons that escape from galaxies to reionize the IGM, and $ \xi_{\rm ion}$ is the ionizing photon production efficiency for a typical stellar population per unit time per unit SFR: $$\rm log \left(\frac{\xi_{ion}}{photons\ s^{-1}\ M_{\odot}\ yr} \right) = 53.14,$$ where the right-hand value is chosen to be consistent with recent works, e.g. [@Lovell2018]. In fact, some observational work points to a higher value [e.g. @Topping2015]. The exact value is unimportant as $\xi_{\rm ion}$ is completely degenerate with $f_{\rm esc}$, and a higher $\xi_{\rm ion}$ merely indicates a lower $f_{\rm esc}$ and vice versa. The differential equation was evolved backwards in time, starting with an ionized fraction of $0.1$ at $z = 10$, a value consistent with results from simulations, e.g. [@Dixon2016], and observations of the CMB optical depth measurements [@Planck2016]. This simplification was necessary due to our small sample of sufficiently massive haloes at $z > 10$, limiting the accuracy of predicted stellar mass functions (see Fig. \[fig:hmf\]). The $f_{\rm esc}$ was treated as a free parameter and tuned separately for the CDM and WDM scenarios with the goal of matching observations and reaching $Q_{\ion{H}{ii}}(z=6) = 1$. We found that in CDM $f_{\rm esc} = 0.23$ matched the observed data well, while WDM can be forced to match CDM with an increased $f_{\rm esc} = 0.34.$ Fig. \[fig:ionfrac\] summarizes our ionization history results. The initial ionization fraction assumption does not significantly impact our results. For example, doubling the initial ionized fraction decreases the resultant $f_{\rm esc}$ at the percent level. The physical value of $f_{\rm esc}$, including the potential redshift and halo mass dependence, is a hotly debated topic. Measurements of nearby galaxies have yielded very small values of a few per cent [e.g. @Rutkowski2017; @Steidel2018]. [@Faisst2016] have observed a somewhat larger $f_{\rm esc}$ and evidence of increasing values with redshift. Simulations have not reached a consensus, but $f_{\rm esc}$ appears to vary significantly from halo to halo [e.g. @Paardekooper2015; @Trebitsch2017; @Rosdahl2018]. Although our values are generally larger, the high redshift Universe remains uncertain, and $f_{\rm esc}$ and $\xi_{\rm ion}$ are degenerate, meaning a larger $\xi_{\rm ion}$ could bring our results more in line with observations. ![Neutral fraction of hydrogen as a function of redshift. The blue solid line shows the CDM ionization with a tuned $f_{\rm esc} = 0.23$. The red solid line shows the ionization in WDM with the same escape fraction. The red dashed line shows the WDM ionization forced to match CDM results with $ f_{\rm esc} = 0.34$. Our results are compared to observational inferences from Ly $\alpha$ damping wings (squares; @Greig2017 [@Davies2018; @Greig2019]), dark Ly $\rm \alpha$ forest pixels (triangles; @McGreer2011 [@McGreer2015]), GRB damping wing absorption (diamonds; @McQuinn2008 [@Chornock2013]), decline in Ly $\rm \alpha$ emitters (hexagons; @Ota2008 [@Ouchi2010]), and Ly $\rm \alpha$ clustering (pentagons; @Ouchi2010)[]{data-label="fig:ionfrac"}](ionfrac.pdf){width="85mm"} Finally, we used the resultant ionized fractions to compute the CMB optical depth: $$\tau(z) = c \langle n_{\rm H} \rangle \sigma_{\rm T} \int_{0}^{z} f_{\rm e} Q_{\ion{H}{ii}}(z') H^{-1}(z')(1+z')^2 dz',$$ where $c$ is the speed of light; $\sigma_{\rm T}$ is the Thompson cross section; $\rm H(z)$ is the Hubble parameter. $f_{\rm e} = 1\ + \ \eta\ Y_{\rm p}/4X_{\rm p}$ is the number of free electrons per hydrogen nucleus. We consider helium to be singly ionized $(\eta = 1)$ at the same rate as hydrogen for $z > 3$ and doubly ionized $(\eta = 2)$ at $z\leq 3$, which is consistent with recent observations [@Worseck2018]. Fig. \[fig:optdepth\] shows our optical depth results. ![The CMB optical depth as a function of redshift. The blue solid line shows the CDM ionization with a tuned $f_{\rm esc} = 0.23$. The red solid line shows the ionization in WDM with the same escape fraction. The red dashed line shows the WDM ionization forced to match CDM results with $ f_{\rm esc} = 0.34$. The shaded area shows the 1$\sigma$ confidence interval from [@Planck2018] with the thin line representing the best fit from all combined data. []{data-label="fig:optdepth"}](optdepth.pdf){width="85mm"} The WDM 3 keV model struggles to reach the $1\sigma$ confidence level of the Planck measurement for optical depth with the CDM-inspired $f_{\rm esc} = 0.23$. This result is a direct consequence of the delay in reionization shown in Fig. \[fig:ionfrac\]. The low optical depth even for our CDM results can be partially explained by the lack of simulation data at $z > 10$. Furthermore, reionization modelling that includes inhomogeneity and more physical processes will generically produce slower reionization histories and therefore a larger $\tau$. Conclusion {#sec:concl} ========== With upcoming facilities such as ALMA and [JWST]{}, we will have access to unprecedented data for the high-redshift Universe and the formation and properties of the first galaxies. The data are sensitive to the early phase of structure formation and hence might help shed light on the nature of dark matter. In this paper, we present a set a large set of high-resolution cosmological hydrodynamical simulations of galaxy formation performed in two different cosmological models, Cold Dark Matter (CDM) and Warm Dark Matter (WDM). For the WDM simulations, we have chosen a particle mass of 3 keV, in agreement with current constraints [@Irsic2017]. Our simulations cover the redshift range $5 < z < 10$ and span three orders of magnitude in halo mass, from $10^8$ to $10^{11}$ (at $z=6$) and more than four in stellar mass, for a total of 38 individual zoom-in simulations among the two cosmologies. In addition to the main zoom-in galaxy, we include all haloes with more than 10,000 particles, resulting in over 100 galaxies at $z = 5$. Simulations in the two different cosmologies have been performed with the same implementation of the baryonic physics as in the NIHAO project [@Wang2015], which has proven to be extremely successful in reproducing several observed properties of galaxies at low and intermediate redshift [@Wang2015; @Dutton2017] and for a very extended mass range [e.g. @Maccio2017; @Buck2018]. WDM simulations have a systematically lower stellar mass function at low masses. This deficit is due to a combination of three effects: a suppression of the number of haloes with mass less than $\approx 10^9~\Msun$, a slightly lower star formation efficiency at a fixed halo mass, and a larger fraction of dark haloes for masses below $\approx 10^9~\Msun$. At all redshifts, the two mass functions start to depart at a stellar mass of $10^7~\Msun$. The difference peaks at about an order of magnitude at $M_{\star} \approx 10^{5.5}~\Msun$ and is more pronounced at higher redshift. Unfortunately, for a realistic candidate of 3 keV, these differences are on the mass scales not yet accessible by current facilities, such as Hubble Space Telescope. The delayed structure formation in WDM models also manifests itself in the global star formation density rate (star formation rate per unit volume) as a function of redshift, which is about a factor of a few lower in WDM w.r.t. CDM at high redshift. By $z = 6$, this quantity is very similar in these two models. Finally, the different star formation histories in the two cosmologies also leave an imprint in the reionization history of the Universe. For a fixed photon escape fraction ($f_{\rm esc}=0.23$), the fraction of neutral hydrogen is higher by 0.3 in WDM for a redshift range between $5 < z < 7$; a larger escape fraction ($f_{\rm esc}=0.34$) is needed in order to reconcile WDM with current data and have a completely ionized Universe by $z \approx 6$. The same effect is visible in the CMB optical depth, where WDM is inconsistent with Planck constraints for $f_{\rm esc} = 0.23$, and a larger value than even 0.34 is needed to match the data. Our results suggest that high redshift is one of the best places to look for the signature of a possible warm candidate on structure formation, in agreement with recent studies from [@Corasaniti2017]. Even for a quite warm-ish mass of 3 keV, the delayed structure formation of WDM leaves a distinct imprint on observable quantities, such as the stellar mass function, and on the evolution of cosmic reionization. On the other hand, as recently shown by [@Villanueva2018], a firm comparison with current data is at the moment prevented by our lack on knowledge on how to parametrize the ‘baryonic’ side of structure formation, as for example the UV photon escape fraction. Our results, combined with previous works, strengthen the need to combine galaxy formation and cosmology if we want to shed light on the nature of the dark components of our Universe. Acknowledgments {#acknowledgments .unnumbered} =============== The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for funding this project by providing computing time on the GCS Supercomputer SuperMUC at the Leibniz Supercomputing Centre (www.lrz.de) and the High Performance Computing resources at New York University Abu Dhabi. Parameter fitting values {#app:hmf} ======================== In this appendix, we present the parameters derived from the MCMC fitting procedure for the majority of relations presented in the text. Table \[tab:hmf\] shows our best-fittingting parameters for the halo mass function at $z = 5-10$ in CDM and 3 keV WDM cosmologies, according to equation (\[eq:hmf\]). Table \[tab:moster\] includes the median and 1$\sigma$ parameter values for the $M_\star$-$M_{200}$ relation, which in linear in log-log space. Table \[tab:fstar\] gives the same quantities for the fraction of dark galaxies, as described by equation (\[eq:fstar\]). $z$ Cosmology $A$ $B$ $C$ $M_{0}$ $\alpha$ ----- ----------- ------- -------- ------- --------- ---------- CDM 52.93 3.641 15.31 11.05 0.1315 WDM 48.99 3.863 16.06 8.574 0.2371 CDM 53.23 3.851 17.12 9.568 0.1462 WDM 53.60 4.331 14.72 9.092 0.2569 CDM 47.86 3.819 14.86 8.572 0.1862 WDM 62.80 5.0117 17.48 9.201 0.2411 CDM 54.53 4.338 16.04 8.975 0.1837 WDM 60.06 4.987 14.13 9.495 0.2628 CDM 51.62 4.274 15.18 8.557 0.1963 WDM 58.73 4.960 15.83 8.878 0.2616 CDM 54.11 4.652 11.09 9.837 0.2199 WDM 65.11 5.537 21.02 8.175 0.2643 : Best-fitting parameter values for halo mass functions.[]{data-label="tab:hmf"} $z$ Cosmology slope intercept ----- ----------- --------------------------- ------------------------- CDM $1.536_{-0.051}^{+0.049}$ $-8.24_{-0.47}^{+0.50}$ WDM $1.683_{-0.062}^{+0.058}$ $-9.73_{-0.58}^{+0.61}$ CDM $1.490_{-0.052}^{+0.050}$ $-7.67_{-0.47}^{+0.50}$ WDM $1.621_{-0.070}^{+0.065}$ $-9.01_{-0.63}^{+0.67}$ CDM $1.448_{-0.060}^{+0.056}$ $-7.25_{-0.51}^{+0.56}$ WDM $1.483_{-0.082}^{+0.078}$ $-7.61_{-0.74}^{+0.78}$ CDM $1.358_{-0.069}^{+0.064}$ $-6.37_{-0.58}^{+0.62}$ WDM $1.444_{-0.100}^{+0.094}$ $-7.23_{-0.87}^{+0.93}$ CDM $1.354_{-0.083}^{+0.078}$ $-6.25_{-0.96}^{+0.73}$ WDM $1.350_{-0.13}^{+0.13}$ $-6.28_{-1.16}^{+1.21}$ CDM $1.43_{-0.12}^{+0.11}$ $-6.96_{-0.99}^{+1.06}$ WDM $1.41_{-0.20}^{+0.19}$ $-6.89_{-1.71}^{+1.81}$ : The median parameter values, including the 1$\sigma$ uncertainties, derived from the MCMC sample for the $M_\star$-$M_{200}$ relation in log-log space. []{data-label="tab:moster"} $z$ Cosmology $\beta$ $M_1$ ----- ----------- ------------------------ ------------------------- CDM $1.45_{-0.32}^{+0.63}$ $-8.73_{-0.17}^{+0.16}$ WDM $1.64_{-0.47}^{+1.04}$ $-8.96_{-0.17}^{+0.19}$ CDM $1.82_{-0.42}^{+0.88}$ $-8.46_{-0.12}^{+0.15}$ WDM $1.82_{-0.50}^{+1.05}$ $-8.62_{-0.16}^{+0.19}$ CDM $2.02_{-0.48}^{+0.96}$ $-8.35_{-0.12}^{+0.14}$ WDM $2.08_{-0.62}^{+1.22}$ $-8.59_{-0.16}^{+0.22}$ CDM $2.43_{-0.63}^{+1.15}$ $-8.15_{-0.13}^{+0.13}$ WDM $2.50_{-0.76}^{+1.31}$ $-8.20_{-0.22}^{+0.27}$ CDM $2.36_{-0.58}^{+1.11}$ $-7.99_{-0.14}^{+0.10}$ WDM $2.74_{-0.88}^{+1.30}$ $-7.99_{-0.30}^{+0.30}$ CDM $2.55_{-0.69}^{+1.19}$ $-8.03_{-0.15}^{+0.12}$ WDM $2.78_{-0.94}^{+1.29}$ $-7.97_{-0.38}^{+0.37}$ : The median parameter values, including the 1$\sigma$ uncertainties, resulting from the MCMC procedure for $f_{\star}$, which follows the functional form of equation (\[eq:fstar\]). []{data-label="tab:fstar"} $z$ slope intercept ----- -------- ----------- 5 0.999 $-8.947$ 6 1.019 $-8.493$ 7 1.070 $-8.802$ 8 1.030 $-8.408$ 9 0.9930 $-8.053$ 10 0.9994 $-8.026$ : The best-fitting parameters derived from the MCMC procedure for the SFR-$M_{\star}$ relation, which is linear in log-log space. []{data-label="tab:sfr"} [^1]: k.dixon@nyu.edu [^2]: http://popia.ft.uam.es/AMIGA [^3]: To be included in Fig. \[fig:smhm\] and the fit used in Section \[sec:reion\], we require that the haloes have at least 10 stellar particles. [^4]: The original NIHAO simulations of [@Wang2015] use the stellar mass within 20 percent of the virial radius, which would bring our results even closer to the relation. At high redshift, this cut is too conservative. [^5]: As a check, the $1\sigma$ uncertainties were calculated as detailed in previous relation fittings, but the resultant values were very small and would be entirely subdominant to the uncertainties in $M_{\star}$-$M_{200}$ and $f_{\star}$. As such, the uncertainty in this relation is not presented nor propagated to the stellar mass functions.
--- abstract: 'The quantization of the giant magnon away from the infinite size limit is discussed. We argue that this quantization inevitably leads to string theory on a $Z_M$-orbifold of $S^5$. This is shown explicitly and examined in detail in the near plane-wave limit.' author: - \| Bojan Ramadanovic and Gordon W. Semenoff\ \ [Department of Physics and Astronomy, University of British Columbia]{}\ [Vancouver, British Columbia, Canada V6T 1Z1]{} title: Finite size giant magnon --- A significant amount of work on the AdS/CFT correspondence [@Maldacena:1997re]-[@Witten:1998qj] has been inspired the idea that the planar limit of ${\cal N}=4$ Yang-Mills theory and its string dual might be integrable models which would be completely solvable using a Bethe Ansätz [@Minahan:2002ve]-[@Beisert:2003yb]. Computation of the conformal dimensions of composite operators in ${\cal N}=4$ Yang-Mills theory can be mapped onto the problem of solving an $SU(2,2\|4)$ spin chain. It is known that the spin chain simplifies considerably in the limit of infinite length where dynamics are encoded in the scattering of magnons and integrability would imply a factorized S-matrix [@Arutyunov:2004vx]. Beginning with this limit, a strategy advocated by Staudacher [@Staudacher:2004tk], Beisert showed that a residual $SU(2\|2)^2$ supersymmetry and integrability determine the ${\cal N}=4~$ S-matrix up to a phase [@Beisert:2005tm],[@Beisert:2006qh]. More recent work constrains [@Janik:2006dc] and essentially computes this phase [@Hernandez:2006tk]-[@Benna:2006nd]. An important problem that the integrability program would eventually have to address is that of finite size corrections. In fact, recent four-loop computations of short operators [@Fiamberti:2007rj],[@Keeler:2008ce] suggest that the most advanced form of the integrability Ansätz, due to Beisert, Eden and Staudacher [@Beisert:2006ez], is likely valid only in the infinite size limit and it is spoiled by finite size effects. In the gauge theory, these effects are thought to stem from wrapping interactions [@Ambjorn:2005wa]-[@Janik:2007wt]. A place where some progress has been made in studying finite size effects is the spectrum of a single magnon. The Bethe Ansätz implies that the energy spectrum of a single magnon (at least in infinite volume) has the form $$\label{spectrum} \Delta-J = \sqrt{1+\frac{\lambda}{\pi^2}\sin^2\frac{p_{\rm mag}}{2}}$$ Here, $\Delta$ is the conformal dimension and $J$ is a $U(1)\in SU(4)$ R-charge which also dictates the length of the spin chain. $p_{\rm mag}$ is the magnon momentum. The string theory dual of the magnon of the infinite size system, the “giant magnon”, was identified by Hofman and Maldacena [@Hofman:2006xt] who showed that it had energy spectrum the expected leading large $\lambda$ limit of (\[spectrum\]). Then it was noted that the giant magnon solution could also be found in finite volume [@Arutyunov:2006gs]-[@Klose:2008rx] where an asymptotic expansion of its spectrum is $$\label{spectrum2} \Delta-J = \frac{\sqrt{\lambda}}{\pi}\left\|\sin\frac{p_{\rm mag}}{2}\right\| -4\frac{\sqrt{\lambda}}{\pi}\left\|\sin\frac{p_{\rm mag}}{2}\right\|^3 e^{-2-J\pi/\sqrt{\lambda}\|\sin{\tiny\frac{p_{\rm mag}}{2}}\|}+\ldots$$ The finite size corrections are exponentially small with large $J$. This was first found in the beautiful paper by Arutyunov, Frolov and Zamaklar [@Arutyunov:2006gs] where, in their approach, it was noted that the spectrum depended on a light-cone gauge fixing parameter. This was not a problem in the strict infinite volume limit, which turned out to be gauge invariant, but it afflicted the exponentially small corrections. Arutyunov et.al. attributed this gauge variance to the fact that a single magnon with non-zero momentum $p_{\rm mag}$ is not a physical state of either the string theory or its dual, ${\cal N}=4$ Yang-Mills theory. In the gauge theory, a single magnon would be an excitation of the exactly known ferromagnetic ground state of the spin chain, the ${\tiny\frac{1}{2}}$-BPS chiral primary operator ${\rm Tr} Z^J$, which has exact conformal dimension the classical value, $\Delta=J$, protected by supersymmetry. We shall denote the three complex scalar fields of ${\cal N}=4$ Yang-Mills theory as $(Z,\Psi,\Phi)$. The sixteen states of the Magnon multiplet are obtained by a spin flip – a single insertion of $D_\mu Z$ or another scalar or a fermion into the trace. They form a short multiplet of the $SU(2\|2)\times SU(2\|2)$ subalgebra of $SU(2,2\|4)$ which commutes with ${\rm Tr }Z^J$. Because of cyclicity of the trace, all positions where one could flip one spin in the ground state of the spin chain are equivalent and $\sum_n e^{inp}{\rm Tr}Z^n\Psi Z^{J-n}\sim \delta(p)$, the magnon momentum must vanish[^1]. Single magnon states with finite magnon momentum do not exist.[^2] The Hofman-Maldacena giant magnon [@Hofman:2006xt] is a soliton solution of the bosonic part of the IIB sigma model propagating on an $R^1\times S^2$ subspace of $AdS_5\times S^5$. They showed that a magnon corresponds to a closed string with an open boundary condition, where the azimuth angle spanned by the two ends of the string corresponds to $p_{\rm mag}$. Ref. [@Arutyunov:2006gs] argued that the open boundary condition led to a modification of the level-matching condition and gauge parameter dependence of the spectrum was a result. In Ref. [@Astolfi:2007uz] it was suggested that the single magnon is well-defined as the twisted state of a closed string on an orbifold – where the orbifold group acts in such a way that it identifies the ends of the string, resulting in a legitimate state of closed string theory. This was advocated as a way to study the spectrum of a single magnon in a setting where it is a physical state and there are no issues with gauge invariance. The giant magnon spectrum was computed there and an asymptotic expansion in the size of the system yields (\[spectrum2\]) (all results in this picture are identical to what Ref. [@Arutyunov:2006gs] obtain if their gauge parameter $a$ is set to zero). In the following we shall develop this idea further. [Our main observation will be that if we consider the single magnon state in the IIB string theory with the boundary condition that the string is open in the direction of magnon motion, we are inevitably led to an orbifold.]{} To get the gist of our argument, consider the following (drastically oversimplified) example of the closed bosonic string on flat Minkowski spacetime where we legislate that one of the string coordinates is not periodic, but obeys the “magnon” boundary condition $ X^1(\tau,\sigma=2\pi) = X^1(\tau,0)+p_{\rm mag}$ and all other variables, including $ \partial_\sigma X^1(\tau,\sigma)$ are periodic. Then, a solution of the worldsheet equation of motion $\left(\partial_\tau^2-\partial_\sigma^2\right)X^1=0$ with the appropriate boundary condition is [@Polchinski:1998rq] $ X^1= x^1+ \alpha' p^1\tau + \frac{\sigma}{2\pi} p_{\rm mag} + {\rm ~oscillators} $. One of the Virasoro constraints is the level matching condition $ L_0 - \tilde L_0 = 0$ which takes the form $$N-\tilde N +p^1\frac{p_{\rm mag}}{2\pi} =0\label{level}$$ where $ N=\sum_{n=1}^\infty \alpha_{-n}\cdot\alpha_n$ and $\tilde N=\sum_{n=1}^\infty \tilde \alpha_{-n}\cdot\tilde \alpha_n $. Since the spectra of the operators $N$ and $\tilde N$ are integers, there is no solution of the level-matching condition unless $p^1p_{\rm mag} =2\pi\cdot{\rm integer}$, i.e. the momentum $p^1$ is quantized in units of integer$\cdot 2\pi/p_{\rm mag}$. This is identical to (and indistinguishable from) the situation where the dimension $X^1$ is compactified with radius $R= \frac{p_{\rm mag}}{\rm integer}$ and where we consider a wrapped string with fixed momentum which is then quantized in units of $\frac{2\pi}{R}$. We see that the magnon boundary condition leads us to string theory on a simple orbifold, a periodic identification of the direction in which the magnon boundary condition was taken. We shall observe a similar fact for the more complicated case of a single magnon on $AdS_5\times S^5$ background. The bosonic part of the IIB sigma model on $AdS_5\times S^5$ and in the conformal gauge is $$\begin{aligned} {\cal L}=-\frac{\sqrt{\lambda}}{4\pi} \left\{ -\left( \frac{ 1+\frac{Z^2}{4}}{1-\frac{Z^2}{4}}\right)^2\partial_aT\partial^aT +\left( \frac{ 1}{1-\frac{Z^2}{4}}\right)^2\partial_aZ\cdot\partial^aZ \right. \nonumber \\ \left. +\left( \frac{ 1-\frac{Y^2}{4}}{1+\frac{Y^2}{4}}\right)^2 \partial_a\chi\partial^a\chi+ \left( \frac{ 1}{1+\frac{Y^2}{4}}\right)^2 \partial_aY\cdot\partial^aY\right\} \label{sigmamodelaction}\end{aligned}$$ supplemented by Virasoro constraints. The eight fields $\vec Z$ and $\vec Y$ transform as 4-vectors under $SO(4)\times SO(4)\sim SU(2)^4$. We will impose the magnon boundary condition on the angle coordinate $$\label{chiboundarydoncition} \chi(\tau,\sigma=2\pi)=\chi(\tau,\sigma=0)+p_{\rm mag}$$ If $\chi(\tau,\sigma)=\tilde\chi(\tau,\sigma)+p_{\rm mag}\sigma/2\pi$ with $\tilde\chi$ periodic, $$\begin{aligned} \label{shiftedlagrangian} {\cal L}[T,\vec Z,\chi,\vec Y]={\cal L}[T,\vec Z,\tilde\chi,\vec Y] -\frac{\sqrt{\lambda}}{4\pi } \left(\left(\frac{p_{\rm mag}}{2\pi}\right)^2+ \frac{ p_{\rm mag}}{\pi}\tilde\chi'\right)\left( \frac{ 1-\frac{Y^2}{4}}{1+\frac{Y^2}{4}}\right)^2\end{aligned}$$ The effect of the magnon boundary condition is to add terms to the action. These, as well as similar terms which appear in the Virasoro constraints, will break some of the (super-)symmetries of the background. The last terms in (\[shiftedlagrangian\]) has the symmetries $SU(2)^2\times SU(2)^2\times R^2$ where the $R^2$ are translations of $T$ and $\tilde\chi$. The bosonic part of the level-matching condition is $$\begin{aligned} 0= \int_0^{2\pi}d\sigma\left\{\Pi_T T' +\Pi_Z Z' + \Pi_{\tilde\chi}\tilde\chi' + \Pi_YY'\right\} +\frac{p_{\rm mag}}{2\pi} J \label{levelmatching}\end{aligned}$$ where $\Pi_\mu\equiv\partial{\cal L}/\partial\dot X^\mu$ are the canonical momenta conjugate to coordinates $X^\mu$ and the charge $J $ is the generator of translations of $\tilde \chi$, $\chi\to\chi+{\rm const.}$ $$\label{J} J=\int_0^{2\pi}d\sigma\Pi_{\tilde\chi} =\frac{\sqrt{\lambda}}{2\pi} \int_0^{2\pi} d\sigma \left( \frac{ 1-\frac{Y^2}{4}}{1+\frac{Y^2}{4}}\right)^2 \dot{\tilde\chi}$$ Since $\chi\sim\chi+2\pi$, the eigenvalues of $J$ must be integers.[^3] Furthermore, being generators of translations of the worldsheet $\sigma$-argument of the fields, and the fields involved being periodic in $\sigma$, the first four terms in (\[levelmatching\]) must be integers plus a possible constant.[^4] Since the theory has a symmetry under $\sigma\to 2\pi-\sigma$, the constant must be either zero or one-half. Thus, the spectrum of the first terms in (\[levelmatching\]) is either integers or integers$+{\tiny\frac{1}{2}}$. To eliminate the second possibility, we shall see that, in the plane wave limit, we can solve for the spectrum explicitly and there we find that it is integers. Then, since the spectrum should not change discontinuously as the plane wave limit is taken, we conclude that it should always be integers. Since $J$ comes in units of integers, and the first four terms in (\[levelmatching\]) are integers, (\[levelmatching\]) will only have a solution if $\frac{p_{\rm mag}}{2\pi}$ is a rational number, $\frac{m}{M}$. Then, $J$ is quantized in units of $M$. This is identical to what should occur for a m-times wrapped string on a $Z_M$ orbifold of $AdS_5\times S^5$ where the orbifold group $Z_M$ makes the identification $\chi\to\chi+2\pi\frac{m}{M}$. To get the superstring, we must include the fermions. For this, we must decide what their boundary conditions will be. It is clear that, at large $J$, we will obtain the correct magnon supermultiplet if we add them in such a way that, in the modification of the Virasoro constraint (\[levelmatching\]), $J$ also contains the appropriate fermionic contribution $J\to\tilde J=\int (\Pi_{\tilde\chi}\tilde\chi'+\Pi_\psi\Sigma\psi')$. This gives the magnon boundary condition for the fermions $$\label{fermionboundarycondition} \psi(\tau,\sigma=2\pi)=e^{ip_{\rm mag}\tilde J}\psi(\tau,\sigma=0)e^{-ip_{\rm mag}\tilde J} =e^{ip_{\rm mag}\Sigma}\psi(\tau,\sigma=0)$$ where $\Sigma= {\rm diag}\left(\frac{1}{2}.-\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right)$ and the orbifold identification is $$\label{orbifold1} \left(\chi,\psi\right)\sim\left(\chi+p_{\rm mag}, e^{ip_{\rm mag}\Sigma}\psi\right)$$ All of the fermions have a twist in their boundary condition. With this identification, all supercharges transform non-trivially under the orbifold group and all of the supersymmetries will be broken (in fact, the supercharges are set to zero by the obtifold projection). This twist in the fermion boundary condition and concomitant breaking of supersymmetry is well known from orbifold constructions in string theory [@Douglas:1996sw] and was outlined in detail in a context similar to ours in Ref. [@Alday:2005ww]. Some supersymmetry can be saved if we impose a slightly more elaborate identification: $$\label{orbifold2} \left(\chi,Y_1+iY_2,\psi\right)\sim\left(\chi+p_{\rm mag}, e^{-ip_{\rm mag}}(Y_1+iY_2), e^{ip_{\rm mag}\tilde\Sigma}\psi\right)$$ where, now $\tilde\Sigma={\rm diag}\left( 0,0,1,-1\right)$. This contains the previous identification of the angle $\chi$ as well as a simultaneous rotation of the transverse $Y$-coordinates. Half of the fermions are un-twisted and this identification preserves half of the supersymmetries. The giant magnon can still be considered a wrapped state of this orbifold where the identified $Y$-coordinates are not excited. The gauge theory duals of both of these models are well-known orbifold projections of ${\cal N}=4$ theory [@Douglas:1996sw]. They are obtained by beginning with the parent theory, ${\cal N}=4$ super Yang-Mills with gauge group $SU(MN)$ and coupling constant $g_{YM}$. Then, we consider a simultaneous $R$-symmetry transformation by a generator of the $Z_M$ orbifold group and a gauge transform by a constant $SU(MN)$ matrix $\gamma={\rm diag}(1,\omega,\omega^2,...,\omega^{M-1})$ where $\omega$ is the $M$-th root of unity. Each diagonal element of the $MN\times MN$-matrix $\gamma$ is multiplied by the $N\times N$ unit matrix. The projection throws away all fields which are not invariant under the simultaneous transformation. This reduces a typical field which was an $MN\times MN$ matrix in the parent theory to $M~$ $N\times N$ blocks embedded in that matrix in the orbifold theory. For example, consider a field $Z$ of the parent theory which is charged under the orbifold group and transforms as $Z\to\omega Z$. The orbifold projection reduces it to a matrix which obeys $$\label{zorbifold} Z\gamma = \omega \gamma Z$$ By similar reasoning, a field $\Phi$ which was neutral in the parent theory commutes with $\gamma$ once the orbifold projection is imposed, $$\label{phiorbifold}\Phi\gamma = \gamma\Phi$$ Given any single-trace operator of the parent ${\cal N}=4$ theory, for example, a single magnon state such as ${\rm Tr} Z^J\Phi$, there are a family of $M$ states of the orbifold theory ${\rm Tr}\gamma^{m}Z^J\Phi$ with $m=0,1,...,M-1$. The operator must be neutral under the orbifold group transformation in the parent theory. To see this: we could insert $1=\gamma^{M-1}\gamma$ into the trace and use the commutators such as (\[zorbifold\]) and (\[phiorbifold\]) and cyclicity of the trace to show that the trace of any operator which is not a singlet under the orbifold group must vanish. In our example, if $\Phi$ is neutral, this requires quantization of $J$ in units of $M$, $J=kM$, in the state ${\rm Tr}\gamma^{m}Z^J\Phi$. This is the gauge dual of the quantization of the momentum $J$ in units of $M\cdot$integers, rather than integers after the orbifold projection is imposed in the sigma model, discussed above after Eq. (\[J\]). In addition, the single-trace operator of the parent theory descends to a family of $M$ operators which are distinguished an additional quantum number, $m$. It is easy to see that moving the position where $\Phi$ was inserted into ${\rm Tr}\gamma^{m}Z^J\Phi$ changes the operator by an overall factor of $\omega^m$. This implies that this trace is already an eigenstate of magnon momentum, $p_{\rm mag}=2\pi\frac{m}{M}$. The integer $m$ is the gauge theory dual of the wrapping number of the string state on the orbifold cycle. There is a theorem to the effect that, in the planar limit of the orbifold gauge theory, un-twisted operators (with $m=0$ in the above examples) have the same correlation functions with each other as those in the planar parent ${\cal N}=4$ gauge theory – with the only difference being a re-scaling of the coupling constant by the order of the orbifold group [@Bershadsky:1998cb]. For this reason, in the planar limit, the gauge theory resulting from either of the orbifold projections (\[orbifold1\]) or (\[orbifold2\]) is a conformal field theory. In the non-supersymmetric case (\[orbifold1\]) non-planar corrections would give a beta-function, whereas in the ${\cal N}=2$ supersymmetric case (\[orbifold2\]) the beta function would vanish in the full theory. On the orbifold, the spectrum of states in the ${\cal N}=4$ magnon super-multiplet are expected to be split according to the residual symmetries. In the two cases we considered, the first (\[orbifold1\]) has no supersymmetry but has $SU(2)^4\times R^2$ bosonic symmetry. We would expect that the fermionic states gain different energies than the bosonic states and that the $SU(2)$ multiplets within the bosonic states also split. In the other case (\[orbifold2\]), there remains ${\cal N}=2$ supersymmetry and the spectrum should represent the super-algebra $SU(2\|1)^2\times R^2$. The ${\cal N}=4$ magnon supermultiplet becomes $$\begin{aligned} {\rm Tr}\gamma^m D_{\mu}ZZ^{kM-1} \label{ads}\\ \label{s5} {\rm Tr}\gamma^m \Phi Z^{kM} ~~,~~ {\rm Tr}\gamma^m \bar\Phi Z^{kM} ~~,~~ {\rm Tr}\gamma^m\bar \Psi Z^{kM+1} ~~,~~ {\rm Tr}\gamma^m \Psi Z^{kM-1}\\ {\rm Tr}\gamma^m \chi_{1\alpha}Z^{kM}~,~{\rm Tr}\gamma^m \chi_{3\alpha}Z^{kM-1} ~,~{\rm Tr}\gamma^m \bar\chi^2_{\dot\alpha} Z^{kM}~,~{\rm Tr}\gamma^m \bar\chi^4_{\dot\alpha} Z^{kM+1} \label{ferms}\end{aligned}$$ Here $m$ gives the number of units of magnon momentum $p_{\rm mag}=\frac{2\pi}{M}m$ and $k$ is the number of units of space-time momentum $J=kM$. There are two limits where the operators in the set (\[ads\])-(\[ferms\]) are degenerate and have energies $\Delta-J=1$: One is when we turn off the ’tHooft coupling $\lambda=g_{YM}^2MN\to 0$ so that the operators have their classical conformal dimension. The other is when magnon momentum vanishes, $m=0$. In the latter, the “untwisted operator” with $m=0$ is known to have identical correlation functions with the operators in the parent ${\cal N}=4$ theory and therefore have exact conformal dimension $\Delta=J+1$. The spectrum away from these limits will depend on both $\lambda$ and $m$. It would be interesting to check the splitting of the supermultiplet in perturbative gauge theory, a task which we reserve for a later publication. In particular, it would be interesting to study the orbifold Bethe Ansätz [@Beisert:2005he]-[@Solovyov:2007pw]. To conclude, we examine the plane-wave limit of $AdS_5\times S^5$ where the string theory sigma model is exactly solvable [@Metsaev:2001bj]. We re-define the string coordinates as: $T=X^+$, $\chi=\frac{1}{\sqrt{\lambda}}X^--X^+$. This has been chosen so that $\Delta-J = \frac{1}{i}\left(\frac{\partial}{\partial T} - \frac{\partial}{\partial\chi}\right)= \frac{1}{i}\frac{\partial}{\partial X^+}$. In addition we re-scale the transverse coordinates $\vec Y\to \vec Y/\lambda^{\frac{1}{4}}$, $\vec Z\to\vec X/\lambda^{\frac{1}{4}}$. The appropriate plane-wave limit [@Berenstein:2002jq] then takes $\lambda\to \infty$ simultaneously with $\Delta\to\infty$ and $J\to\infty$ with $\Delta-J$ and $\frac{J}{\sqrt{\lambda}}$ finite. From (\[levelmatching\]) we see that the limit should be taken so that $p_{\rm mag}J$ is finite. This implies that $$\label{scalingofpmag} p_{\rm mag}\sim\frac{1}{\sqrt{\lambda}}$$ The magnon boundary condition (\[chiboundarydoncition\]) implies $$\label{identificationofxminus} X^-(\sigma=\pi)=X^-(\sigma=0) + p_{\rm mag} \sqrt{\lambda}$$ The scaling (\[scalingofpmag\]) then gives a finite radius for $X^-$. We have already argued that $J=\frac{1}{i}\frac{\partial}{\partial\chi}=\sqrt{\lambda}\frac{1}{i}\frac{\partial}{\partial X^-}$ should be quantized in integral units. In fact, in the magnon sector, we have argued that the level-matching condition (\[levelmatching\]) has a solution only when $p_{\rm mag}=2\pi\frac{m}{M}$ where $m$ and $M$ are integers and $J$ is quantized in units of $M$, $J=kM$ with $k$ an integer. To get the correct scaling of $p_{\rm mag}$ we must therefore take the plane wave limit by taking $M$ to be large so that $\frac{M}{\sqrt{\lambda}}$ is held finite. What is effectively the same limit was discussed in Ref. [@Mukhi:2002ck] where it was shown to result in a plane-wave background with a periodically identified null direction, $X^-\sim X^-+2\pi R^-$ where $R^- = \frac{\sqrt{\lambda}}{M}$. (To be consistent with (\[identificationofxminus\])), the integer $m$ which appears in $p_{\rm mag}$ is interpreted is a wrapping number.) The resulting discrete light-cone quantization of the string on the plane wave background is a simple generalization of Metsaev’s original solution [@Metsaev:2001bj]. Here, we are interested in a wrapped sector where $X^-(\sigma=2\pi)=X^-(\sigma=0)+2\pi R^-m$. In Ref. [@Mukhi:2002ck] the spectrum of the IIB string theory in this plane wave limit was matched with the appropriate generalization of the BMN limit of the ${\cal N}=2$ Yang-Mills theory which is obtained from ${\cal N}=4$ by the orbifold projection corresponding to (\[orbifold2\]). It was also used to study non-planar corrections [@De; @Risi:2004bc] and finite-size corrections at weak coupling [@Astolfi:2006is]. Together with the limit, we take the light-cone gauge, $X^+=p^+\tau$. Periodicity of $X^-$ quantizes $p^+= k/R^-$. We obtain the sigma model as a free massive worldsheet field theory $$\begin{aligned} {\cal L}=-\frac{1}{4\pi} \left\{ \partial_a \vec Y\cdot\partial^a\vec Y + \partial_a\vec Z\cdot\partial^a\vec Z + (p^+)^2(Y^2+Z^2)\right\} \nonumber \\ -\frac{ip^+}{2\pi} \left(\bar\psi\partial_-\bar\psi+\psi\partial_-\psi + 2ip^+\bar\psi\Pi\psi\right)\end{aligned}$$ with $\Pi={\rm diag}(1,1,1,1,-1,-1,-1,-1)$. In this limit, the magnon parameter $p_{\rm mag}$ does not appear in the Lagrangian or the mass-shell condition which determines the light-cone Hamiltonian: $$\begin{aligned} \label{ppham} p^-=\frac{1}{p^+} \sum_{n=-\infty}^\infty \sqrt{n^2+(p^+)^2} \left( \alpha_{n}^{\alpha_1\dot\alpha_1\dagger}\alpha_{n\alpha_1\dot\alpha_1}+ \alpha_{n}^{\alpha_2\dot\alpha_2\dagger}\alpha_{n\alpha_2\dot\alpha_2} \right. \nonumber \\ \left. + \beta_{n}^{\alpha_1\dot\alpha_2\dagger}\beta_{n\alpha_1\dot\alpha_2}+ \beta_{n}^{\alpha_2\dot\alpha_1\dagger}\beta_{n\alpha_2\dot\alpha_1}\right)\end{aligned}$$ Its only vestige is in the level-matching condition. $$\label{pplevel} km= \sum_{n=-\infty}^\infty n \left( \alpha_{n}^{\alpha_1\dot\alpha_1\dagger}\alpha_{n\alpha_1\dot\alpha_1}+ \alpha_{n}^{\alpha_2\dot\alpha_2\dagger}\alpha_{n\alpha_2\dot\alpha_2}+ \beta_{n}^{\alpha_1\dot\alpha_2\dagger}\beta_{n\alpha_1\dot\alpha_2}+ \beta_{n}^{\alpha_2\dot\alpha_1\dagger}\beta_{n\alpha_2\dot\alpha_1}\right)$$ where $k$ are the number of units of $J=kM$ and $m$ is the wrapping number. The bosonic $\alpha_{n..}$ and fermionic $\beta_{n..}$ oscillators have the non-vanishing brackets $$\begin{aligned} [\alpha_{m\alpha_1\dot\alpha_1},\alpha^{\beta_1\dot\beta_1\dagger}_{n} ]= \delta_{mn} \delta^{\alpha_1}_{\beta_1}\delta^{\dot\alpha_1}_{\dot\beta_1} &,& \{\beta_{m\alpha_1\dot\alpha_2},\beta^{\beta_1\dot\beta_2\dagger}_{n} \}= \delta_{mn} \delta^{\alpha_1}_{\beta_1}\delta^{\dot\alpha_2}_{\dot\beta_2} \\ \left[\alpha_{m\alpha_2\dot\alpha_2},\alpha^{\beta_2 \dot\beta_2\dagger}_{n}\right] =\delta_{mn} \delta^{\alpha_2}_{\beta_2}\delta^{\dot\alpha_2}_{\dot\beta_2}&,& \{\beta_{m\alpha_2\dot\alpha_1},\beta^{\beta_2\dot\beta_1\dagger}_{n} \}= \delta_{mn} \delta^{\alpha_2}_{\beta_2}\delta^{\dot\alpha_1}_{\dot\beta_1}\end{aligned}$$ and bi-spinors of $SO(4)\times SO(4)\sim SU(2)^4$.[^5] We confirm in (\[pplevel\]), which is the plane wave limit of (\[levelmatching\]), there is solution of the level matching constraint unless $\frac{p_{\rm mag}}{2\pi}J=$integer. Here, we can think of the null identification as the vestige of the orbifold identification. The level-matching condition (\[levelmatching\]) allows 1-oscillator states and the magnon supermultiplet is the sixteen states $$\label{planewavesupermultiplet} \alpha^{\dagger}_{km\alpha_1\dot\alpha_1}\|p^+> ~,~ \alpha^{\dagger}_{km\alpha_2\dot\alpha_2}\|p^+>~,~ \beta^{\dagger}_{km\alpha_1\dot\alpha_2}\|p^+> ~,~ \beta^{\dagger}_{km\alpha_2\dot\alpha_1}\|p^+>$$ These states are degenerate with spectrum given by $$p^-=\frac{1}{p^+} \sqrt{ (km)^2+(p^+)^2} = \sqrt{ 1+(R^-)^2 m^2}=\sqrt{ 1+\frac{\lambda'}{M^2} m^2}$$ has the form expected from the plane-wave limit of (\[spectrum\]) when $p_{\rm mag}=2\pi\frac{m}{M}$. Note that in this plane-wave limit the finite size corrections that occur in (\[spectrum2\]) vanish due to the limit of small $p_{\rm mag}$. The degeneracy of the states in (\[planewavesupermultiplet\]) can be attributed to an enhancement of the supersymmetry which is well known to occur in the Penrose limit. One would expect, and we shall confirm, that the supersymmetry is broken when corrections to the Penrose limit are taken into account. Before that, we recall that in Refs. [@Beisert:2005tm],[@Beisert:2006qh] Beisert argued magnon states form a sixteen dimensional short multiplet of an extended super-algebra $SU(2\|2)\times SU(2\|2)\times (R^1)^3$ where the spectrum (\[spectrum\]) is the shortening condition. The superalgebra $SU(2\|2)$ has generators ${\cal R}^{\alpha_1}_{~\beta_1}$ and ${\cal L}^{\dot\alpha_2}_{~\dot\beta_2}$ of $SU(2)\times SU(2)$, supercharges ${\cal Q}^{\dot\alpha_2}_{~\alpha_1}$ and ${\cal S}_{~\dot\alpha_2}^{\alpha_1}$ and the algebra $$\begin{aligned} \left[ {\cal R}^{\alpha_1}_{~\beta_1},{\cal J}^{\gamma_1}\right]=\delta^{\gamma_1}_{\beta_1} {\cal J}^{\alpha_1}-\frac{1}{2}\delta^{\alpha_1}_{\beta_1} {\cal J}^{\gamma_1} &,& \left[ {\cal L}^{\dot\alpha_2}_{~\dot\beta_2},{\cal J}^{\dot\gamma_2}\right]=\delta^{\dot\gamma_2}_{\dot\beta_2} {\cal J}^{\dot\alpha_2}-\frac{1}{2}\delta^{\dot\alpha_2}_{\dot\beta_2} {\cal J}^{\dot\gamma_2} \nonumber \\ \left\{{\cal Q}^{\dot\alpha_2}_{~\alpha_1},{\cal S}^{\beta_1}_{~\dot\beta_2}\right\}&=&\delta^{\beta_1}_{\alpha_1}{\cal L}^{\dot\alpha_2}_{~\dot\beta_2}+\delta^{\dot\alpha_2}_{\dot\beta_2} {\cal R}^{\beta_1}_{~\alpha_1}+\delta^{\beta_1}_{\alpha_1}\delta^{\dot\alpha_2}_{\dot\beta_2} {\cal C} \nonumber \\ \left\{ {\cal Q}^{\dot\alpha_2}_{~\alpha_1},{\cal Q}^{\dot\beta_2}_{~\beta_1}\right\}=\epsilon^{\dot\alpha_2\dot\beta_2} \epsilon_{\alpha_1\beta_1}{\cal P} ~~&,&~~ \left\{ {\cal S}_{~\dot\alpha_2}^{\alpha_1},{\cal S}_{~\dot\beta_2}^{\beta_1}\right\}=\epsilon_{\dot\alpha_2\dot\beta_2} \epsilon^{\alpha_1\beta_1}{\cal K} \nonumber \end{aligned}$$ ${\cal J}^{...}$ represents any generator with the appropriate index, ${\cal K}$, ${\cal P}$ and ${\cal C}$ are central charges. In our application, ${\cal C}=\Delta-J = p^-$ and $$\begin{aligned} {\cal R}^{\alpha_1}_{~\beta_1} &=&\sum_n\left\{\alpha^{\dagger\alpha_1\dot\gamma}_n\alpha_{n\beta_1\dot\gamma_1} +\beta_n^{\dagger\alpha_1\gamma_2}\beta_{\beta_1\gamma_2}\right\} - \frac{1}{2} \delta^{\alpha_1}_{~\beta_1}\sum_n \left\{\alpha^{\dagger\gamma_1\dot\gamma_1}_n\alpha_{n\gamma_1\dot\gamma_1} +\beta_n^{\dagger\gamma_1\gamma_2}\beta_{\gamma_1\gamma_2}\right\} \nonumber \\ {\cal L}^{\dot\alpha_2}_{~\dot\beta_2}&=& \sum_n\left\{\alpha^{\dagger\gamma_2\dot\alpha_2}_n\alpha_{n\gamma_2\dot\beta_2} +\beta_n^{\dagger\dot\alpha_2\dot\gamma_1}\beta_{\dot\gamma_1\dot\beta_2}\right\} - \frac{1}{2} \delta^{\dot\alpha_2}_{~\dot\beta_2}\sum_n \left\{\alpha^{\dagger\gamma_2\dot\gamma_2}_n\alpha_{n\gamma_2\dot\gamma_2} +\beta_n^{\dagger\dot\gamma_1\dot\gamma_2}\beta_{\dot\gamma_1\dot\gamma_2}\right\} \nonumber \\ {\cal Q}_{~\alpha_1}^{\dot\beta_2}&=&\frac{\bar\eta}{\sqrt{8 p^+}}\sum_n\left\{-e(n)\sqrt{\omega_n+p^+} \alpha^{\dagger}_{n\alpha_1\dot\gamma_1}\beta_{n}^{\dot\gamma_1\dot\beta_2} +i\sqrt{\omega_n-p^+}\alpha_{n\alpha_1\dot\gamma_1}\beta_{n}^{\dagger\dot\gamma_1\dot\beta_2} \right. \nonumber \\ &~&\left. ~~~~~~~~~~~~-i\sqrt{\omega_n-p^+}\beta^{\dagger}_{n\alpha_1\gamma_2}\alpha_{n}^{\gamma_2\dot\beta_2} + e(n)\sqrt{\omega_n+p^+} \beta_{n\alpha_1\gamma_2}\alpha^{\dagger\gamma_2\dot\beta_2}_n \right\}\nonumber\\ {\cal S}^{\alpha_1}_{~\dot\beta_2}&=&\frac{\bar\eta}{\sqrt{8 p^+}} \sum_n\left\{ \sqrt{\omega_n-p^+} \alpha^{\dagger\alpha_1\dot\gamma_1}_n\beta_{n\dot\gamma_1\dot\beta_2} -i e(n) \sqrt{\omega_n+p^+}\alpha^{\alpha_1\dot\gamma_1}_n\beta^{\dagger}_{n\dot\gamma_1\dot\beta_2} + \right. \nonumber \\ &~&\left. ~~~~~~~~~~~~ +i e(n) \sqrt{\omega_n+p^+}\beta^{\dagger\alpha_1\gamma_2}_{n}\alpha_{n\gamma_2\dot\beta_2} - \sqrt{\omega_n-p^+}\beta^{\alpha_1\gamma_2}_{n}\alpha^{\dagger}_{n\gamma_2\dot\beta_2} \right\}\nonumber \\\end{aligned}$$ where $\omega_n=\sqrt{(p^+)^2+n^2}$ and $e(n)=\frac{n}{\|n\|}$. We have used Metsaev’s [@Metsaev:2001bj] conventions for the supercharges (those called $Q^-$ and $\bar Q^-$) and notation for oscillators as summarized, for example, in Ref. [@Young:2007pk]. Computing their algebra, we find that the plane wave background supercharges indeed satisfy Beisert’s extended superalgebra with the central extensions set to the plane-wave limits of those found by Beisert [@Beisert:2005tm] $${\cal P}=-i\frac{\sqrt{\lambda}p_{\rm mag}}{4\pi}\leftarrow \frac{\sqrt{\lambda}}{4\pi}\left(e^{-ip_{\rm mag}}-1\right) ~,~ {\cal K}=i\frac{\sqrt{\lambda}p_{\rm mag}}{4\pi}\leftarrow \frac{\sqrt{\lambda}}{4\pi}\left(e^{ip_{\rm mag}}-1\right) \label{extension}$$ The existence of the central extension follows directly from the fact that the unextended algebra closes up to the level matching condition and the level-matching condition (\[levelmatching\]) contains the term with $km= \frac{1}{2\pi}2\pi\frac{m}{M}\cdot kM=\frac{1}{2\pi}p_{\rm mag}J$. A derivation of Beisert’s superalgebra in the context of the $AdS_3\times S^5$ sigma model was first given in Ref. [@Arutyunov:2006ak] and developed in Ref. [@Arutyunov:2006yd]. They worked with the un-orbifolded theory by “relaxing” the level-matching condition. Then, there is a central charge in the superalgebra which depends on the level miss-match. The idea is that, once the resulting algebraic structure is used to study magnon and multi-magnon states, the level-matching condition should be re-imposed so as to get a physical state of the string theory. They work in the “magnon limit”, where $J\to\infty$, but magnon momentum is not necessarily small (in our case it relaxes the plane-wave limit by taking $M$ not necessarily large). They obtain the full central extension, rather than the form linearized in $p_{\rm mag}$ that we have found in (\[extension\]). In their work, they use a generalized light-cone gauge $x^+=\tau=(1-a)T+a\chi$, $x_-=\chi-T$ with $a$ a parameter. They also use the identification, $x_-(\tau,\sigma=2\pi)-x_-(\tau,\sigma=0) =p_{\rm ws}$ with $p_{\rm ws}$ an eigenvalue of the level operator and $x_+=\tau$ trivially periodic in $\sigma$. For the variables in (\[sigmamodelaction\]), this amounts to using the boundary condition $\chi(\tau,\sigma=2\pi)-\chi(\tau,\sigma=0)= -(1-a)p_{\rm ws}$ and $T(\tau,\sigma=2\pi)-T(\tau,0)=ap_{\rm ws}$ which is different from the one which we use when $a\neq0$ (they primarily use $a=\frac{1}{2}$) - where $T(\tau,\sigma=2\pi)=T(\tau,\sigma=0)$ and $\chi(\tau,\sigma=2\pi)-\chi(\tau,\sigma=0)=p_{\rm mag}$. This makes no difference at infinite $J$ where the effect of $a$ is diluted by scaling. However, it matters at finite size. In fact, the same gauge fixing was used in Ref. [@Arutyunov:2006gs] and the $a$-dependence of the one-magnon spectrum found there (away from the infinite $J$ limit) can be attributed to this $a$-dependence of boundary conditions, rather than the gauge variance which is claimed there. To see how the spectrum will be split in the near plane-wave limit, we must include corrections to the Lagrangian and the Virasoro constraints that are of order $\frac{1}{\sqrt{\lambda}}$. A systematic scheme for including these corrections in the usual $p_{\rm mag}=0$ sector are outlined in the series of papers [@Callan:2004ev]-[@Callan:2003xr] and nicely summarized in Ref. [@McLoughlin:2005dh]. There they find that the corrections terms to the Hamiltonian add normal ordered terms which are quartic in oscillators. They also adjust the gauge by adjusting the worldsheet metric in such a way that the level-matching condition remains unmodified. We have shown, and will present elsewhere, that the modification of at procedure in the magnon sector are minimal. The corrections to the free field theory light-cone Hamiltonian are of two types, quartic normal ordered pieces from near-plane-wave limit corrections to the sigma model identical in form to those found in Refs. [@Callan:2004ev]-[@McLoughlin:2005dh] and terms such as the last one in Eq. (\[shiftedlagrangian\]) which arise from the orbifolding. To leading order in perturbation theory, the normal ordered quartic interaction Hamiltonian cannot shift the spectrum of 1-oscillator states. Furthermore, none of the extra terms displayed in Eq. (\[shiftedlagrangian\]) contribute in the leading order in $1/\sqrt{\lambda}$. However, recall that, to preserve some supersymmetry, the orbifold identification (\[orbifold2\]) that we have been discussing also acts on the transverse direction and this action must also be taken into account. This generates simple correction terms in the Hamiltonian to order $\frac{1}{\sqrt{\lambda}}$. The relevant part of the interaction Hamiltonian is $$H_{\rm int}= i\frac{p_{\rm mag}}{2\pi}\frac{1}{2\pi}\int_0^2\pi d\sigma\left( Y_{1_1\dot 2_1}Y_{2_1\dot 1_1}' +ip^+\left( \psi\tilde\Sigma\psi+\bar\psi\tilde\Sigma\bar\psi\right)\right)$$ With this orbifold identification exactly half of the supersymmetries are preserved in the near plane-wave limit. Specifically, out of the 16 supersymmetries $ {\cal Q}_{\a_1}^{\dot \a_2}, {\cal S}_{\dot \a_2}^{\a_1}$ only ${\cal S}^{\a_1}_{\dot 1_2} / {\cal Q}_{\a_1}^{\dot 1_2}$ and ${\cal S}^{2_2}_{\dot \a_1} / {\cal Q}_{2_2}^{\dot \a_1} $ survive. This leads to a splitting of the energies of the single impurity states. The original multiplet had 16 states (8 bosons - $\a^\dagger_{\a_1 \dot \a_1} \|0>, \a^\dagger_{\a_2 \dot \a_2} \|0>$ and 8 fermions - $\b^\dagger_{\a_1 \a_2} \|0>, \b^\dagger_{\dot \a_1 \dot \a_2} \|0>$). In the near plane-wave, it breaks up into 4 super-multiplets of the residual superalgebra: one with 9 elements (5 bosons and 4 fermions) and two with 3 elements (2 fermions and a boson in each) and one boson singlet. The following table illustrates the breaking of the original super-multiplet: \[multiplets\] Here, columns and rows with dashes represent the surviving supersymmetry transformations: ${\cal S}^{\a_1}_{\dot 1_2} / {\cal Q}_{\a_1}^{\dot 1_2}$ and ${\cal S}^{2_2}_{\dot \a_1} / {\cal Q}_{2_2}^{\dot \a_1} $. Columns and rows without dashes represent the broken supersymmetries: ${\cal S}^{\a_1}_{\dot 2_2} / {\cal Q}_{\a_1}^{\dot 2_2}$ and ${\cal S}^{1_2}_{\dot \a_1} / {\cal Q}_{1_2}^{\dot \a_1} $. The energy degeneracy of the original multiplet is likewise broken by the interaction Hamiltonian in the near plane-wave limit. One of the triplets gets positive energy shift, its energy becoming: $$\sqrt{ 1+\lambda\frac{m^2}{M^2} }+\frac{1}{2\sqrt{\lambda}}~ \frac{\lambda\frac{m^2}{M^2} }{ \sqrt{ 1+\lambda\frac{m^2}{M^2} }}$$ Other triplet gets equal but negative energy shift: $$\sqrt{ 1+\lambda\frac{m^2}{M^2} }-\frac{1}{2\sqrt{\lambda}}~ \frac{ \lambda\frac{m^2}{M^2} }{ \sqrt{ 1+\lambda\frac{m^2}{M^2} }}$$ Singlet and a 9-multiplet are annihilated by the interaction Hamiltonian and thus retain the energy of the original multiplet: $$\sqrt{1+\lambda\frac{m^2}{M^2} }$$ In conclusion, we have made an number of observations about the giant magnon solution of string theory. We observed that the previously noted resemblance of the magnon to a wrapped string on a $Z_M$ orbifold of $AdS_5\times S^5$ seems to be the only solution of the Virasoro constraints in the string sigma-model. 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[^2]: We also note that these states can be obtained by commutators of symmetry generators and ${\rm Tr}Z^J$ so they are all in the same ${\tiny\frac{1}{2}}$-BPS multiplet of the full $SU(2,2\|4)$ algebra and have exact conformal dimensions $\Delta-J=1$ which agrees with (\[spectrum\]) and (\[spectrum2\]) when $p=0$. [^3]: When fermions are included, they could be half-integers. [^4]: Consider the operator $\xi$ which has the property $\left[ \xi, \varphi(\sigma)\right]=i\frac{d}{d\sigma}\varphi(\sigma)$. Consider eigenstates $\|\alpha>$ and $\|\alpha'>$ where $\xi \|\alpha>=\alpha\|\alpha>$. If $<\alpha'\|\varphi(\sigma)\|\alpha>=<\alpha'\|e^{- i\sigma \xi}\varphi(0)e^{ i \sigma\xi}\|\alpha>=e^{i(\alpha-\alpha')\sigma}<\alpha'\| \varphi(0) \|\alpha>$, the matrix element obeys $<\alpha'\|\varphi(\sigma)\|\alpha>=<\alpha'\|\varphi(\sigma+2\pi)\|\alpha>$ only when $\alpha-\alpha`={\rm integers}$. The eigenvalues are equal to integers plus a constant which is common to all eigenvalues. If, there is a reflection symmetry $\sigma\to2\pi-\sigma$ under which $\xi\to-\xi$, the constant must be either an integer or half-integer. [^5]: Indices are raised and lowered with $\epsilon^{\alpha_i\beta_i}$ and $-\epsilon_{\alpha_i\beta_i}$, respectively, always operating from the left.
--- abstract: 'Let ${\mathbb{F}}$ be a finite field, an algebraically closed field, or the field of real numbers. Consider the vector space $V={\mathbb{F}}^3 \otimes {\mathbb{F}}^3$ of $3 \times 3$ matrices over ${\mathbb{F}}$, and let $G \leq \text{PGL}(V)$ be the setwise stabiliser of the corresponding Segre variety $S_{3,3}({\mathbb{F}})$ in the projective space ${\mathrm{PG}}(V)$. The $G$-orbits of lines in $\text{PG}(V)$ were determined by the first author and Sheekey as part of their classification of tensors in ${\mathbb{F}}^2 \otimes V$ in the article “Canonical forms of $2 \times 3 \times 3$ tensors over the real field, algebraically closed fields, and finite fields”, [Linear Algebra Appl.]{} [476]{} (2015) 133–147. Here we consider the related problem of classifying those line orbits that may be represented by [symmetric]{} matrices, or equivalently, of classifying the line orbits in the ${\mathbb{F}}$-span of the Veronese variety $\mathcal{V}_3({\mathbb{F}}) \subset S_{3,3}({\mathbb{F}})$ under the natural action of $K={\mathrm{PGL}}(3,{\mathbb{F}})$. Interestingly, several of the $G$-orbits that have symmetric representatives split under the action of $K$, and in many cases this splitting depends on the characteristic of ${\mathbb{F}}$. The corresponding orbit sizes and stabiliser subgroups of $K$ are also determined in the case where ${\mathbb{F}}$ is a finite field, and connections are drawn with old work of Jordan, Dickson and Campbell on the classification of pencils of conics in ${\mathrm{PG}}(2,{\mathbb{F}})$, or equivalently, of pairs of ternary quadratic forms over ${\mathbb{F}}$.' address: ' Michel Lavrauw, University of Padua, Italy and Sabanc[i]{} University, Turkey; Email: `mlavrauw@sabanciuniv.edu` Tomasz Popiel, Queen Mary University of London and The University of Western Australia Email: `tomasz.popiel@uwa.edu.au`' author: - 'Michel Lavrauw, Tomasz Popiel' title: 'The symmetric representation of lines in ${\mathrm{PG}}({\mathbb{F}}^3\otimes {\mathbb{F}}^3)$' --- [^1] Introduction {#intro} ============ Consider the vector space $V={\mathbb{F}}^3\otimes {\mathbb{F}}^3$ of $3\times 3$ matrices over a field ${\mathbb{F}}$, and recall that the corresponding [Segre variety]{} $S_{3,3}({\mathbb{F}})$ in the projective space ${\mathrm{PG}}(V) \cong {\mathrm{PG}}(8,{\mathbb{F}})$ is the image of the map taking $(\langle v \rangle,\langle w \rangle) \in {\mathrm{PG}}({\mathbb{F}}^3) \times {\mathrm{PG}}({\mathbb{F}}^3)$ to $\langle v \otimes w \rangle$. Let $G$ denote the setwise stabiliser of $S_{3,3}({\mathbb{F}})$ inside the group ${\mathrm{PGL}}(V)$. The classification of $G$-orbits of lines in ${\mathrm{PG}}(V)$ was obtained by the first author and Sheekey [@LaSh2015] as a consequence of their classification of tensors in ${\mathbb{F}}^2 \otimes V$ for ${\mathbb{F}}$ an algebraically closed field, ${\mathbb{F}}$ a finite field, and ${\mathbb{F}}=\mathbb R$. This lead to the classification in [@LaSh2017] of all subspaces of ${\mathrm{PG}}({\mathbb{F}}^2\otimes {\mathbb{F}}^3)$, and of the tensor orbits in ${\mathbb{F}}^2\otimes{\mathbb{F}}^3\otimes {\mathbb{F}}^r$ for every $r\geq 1$. Here we study the [symmetric representation]{} of the line orbits in ${\mathrm{PG}}(V)$, by which we mean the following. Let $\mathcal{O}$ be a $G$-orbit of lines in ${\mathrm{PG}}(V)$, and consider the subspace $V_\text{s} {\leqslant}V$ of symmetric $3 \times 3$ matrices over ${\mathbb{F}}$. If $\mathcal{O}$ happens to contain a line $L$ in ${\mathrm{PG}}(V_\text{s})$, then $L$ is called a [symmetric representative]{} of $\mathcal{O}$. If $\mathcal{O}$ has two symmetric representatives that are not in the same orbit under the natural action of $K={\mathrm{PGL}}(3,{\mathbb{F}})$, whereby a symmetric matrix $M$ is mapped by $D \in K$ to $DMD^\top$, then we say that the $G$-orbit $\mathcal{O}$ [splits]{} (under this action of $K$). Here we address the following natural problems concerning the $G$-orbits of lines in ${\mathrm{PG}}(V)$: - We determine which $G$-orbits of lines in ${\mathrm{PG}}(V)$ have a symmetric representative. - We classify those orbits that have symmetric representatives, under the action of $K$. - In the case where ${\mathbb{F}}$ is a finite field, we determine for each $K$-orbit the corresponding stabiliser subgroup of $K$ and the orbit size. Note that problem (ii) is equivalent to the classification of $K$-orbits of lines in the ${\mathbb{F}}$-span $\langle {\mathcal{V}}_3({\mathbb{F}}) \rangle$ of the [Veronese variety]{}, or [quadric Veronesean]{}, ${\mathcal{V}}_3({\mathbb{F}}) \subset S_{3,3}({\mathbb{F}})$, namely the image of the [Veronese map]{} $\nu_3 : {\mathrm{PG}}(2,{\mathbb{F}}) \rightarrow {\mathrm{PG}}(5,{\mathbb{F}})$ induced by the mapping taking $u \in {\mathbb{F}}^3$ to $u \otimes u$. Problems (i) and (ii) are addressed for the case of a finite field ${\mathbb{F}}$ in Section \[sec:orbits\]; the results are summarised in Table \[table:main\]. There are 14 orbits of lines in ${\mathrm{PG}}(V)$ under $G$, arising from the tensor orbits $o_4,\ldots,o_{17}$ in ${\mathbb{F}}^2 \otimes V$, in the notation of [@LaSh2015], which we adopt here for consistency. Of these 14 orbits, only three do not have symmetric representatives, namely those arising from the tensor orbits $o_4$, $o_7$ and $o_{11}$. Moreover, the line orbits (corresponding to the tensor orbits) $o_5$, $o_6$, $o_9$, $o_{10}$ and $o_{17}$ do not split for any value of the characteristic $\operatorname{char}({\mathbb{F}})$ of ${\mathbb{F}}$, while $o_{14}$ and $o_{15}$ split for odd characteristic but not for even characteristic, $o_{12}$ and $o_{16}$ split for even characteristic but not for odd characteristic, and $o_8$ and $o_{13}$ split for all values of $\operatorname{char}({\mathbb{F}})$. We note that no $G$-line orbit splits into more than two $K$-line orbits. For algebraically closed fields and the case ${\mathbb{F}}=\mathbb R$, problems (i) and (ii) are handled in Section \[sec:otherF\]. The situation is overall somewhat simpler than in the finite case, but we note in particular that the results for an algebraically closed field ${\mathbb{F}}$ do depend on whether or not $\operatorname{char}({\mathbb{F}})=2$. Problem (iii) is addressed in Section \[sec:stabs\], with the results summarised in Table \[table:stabs\]. As noted above, the splitting of a $G$-orbit under $K$ sometimes depends on whether $\operatorname{char}({\mathbb{F}})=2$ or not. It turns out that the structures of the corresponding line stabilisers inside $K$ can also depend on $\operatorname{char}({\mathbb{F}})$, and that, moreover, if ${\mathbb{F}}={\mathbb{F}}_q$ for an odd prime power $q$ then the structure of a line stabiliser can depend on whether $q \equiv 1$ or $3 \pmod 4$. It seems remarkable, therefore, that in the end the number of symmetric representatives of any given $G$-orbit turns out to be independent of the characteristic of ${\mathbb{F}}={\mathbb{F}}_q$. Because our results imply the classification of lines in $\langle {\mathcal{V}}_3({\mathbb{F}}) \rangle$ under ${\mathrm{PGL}}(3,{\mathbb{F}})$, they also imply the classification of pencils of conics in ${\mathrm{PG}}(2,{\mathbb{F}})$, or equivalently, of pairs of ternary quadratic forms over ${\mathbb{F}}$. This classification problem goes back to work of Jordan [@Jordan1906; @Jordan1907], Dickson [@Dickson1908], and Campbell [@Campbell1927], and we have included details in Section \[sec:pencils\]. Preliminaries {#prelims} ============= Here we collect some preliminary information for background and later reference. Orbits of tensors in $\mathbb{F}^2 \otimes \mathbb{F}^3 \otimes \mathbb{F}^3$ ----------------------------------------------------------------------------- Write $$V_1 = {\mathbb{F}}^2, \quad V={\mathbb{F}}^3 \otimes {\mathbb{F}}^3 \quad \text{and} \quad \overline{V} = V_1 \otimes V,$$ and let $G$ be the setwise stabiliser in $\text{GL}(\overline{V})$ of the set of fundamental tensors in $\overline{V}$, namely the tensors of the form $v_1 \otimes v_2 \otimes v_3$ with $v_1 \in {\mathbb{F}}^2$ and $v_2,v_3 \in {\mathbb{F}}^3$. The $G$-orbits of tensors in $\overline{V}$ were classified in [@LaSh2015 Main Theorem]. In particular, in the case where ${\mathbb{F}}$ is a finite field ${\mathbb{F}}_q$, there are precisely $18$ orbits, with representatives given in terms of a basis $\{e_1,e_2,e_3\}$ for $\mathbb{F}_q^3$ in the table on [@LaSh2015 p. 146]. For convenience, the information in this table is included here in Table \[pointTable\]. Orbit Representative Condition Rank dist. ---------- ------------------------------------------------------------------------------------------------------------------- ----------- ------------- $o_0$ $0$ $[0,0,0]$ $o_1$ $e_1 \otimes e_1 \otimes e_1 $ $[1,0,0]$ $o_2$ $e_1 \otimes (e_1 \otimes e_1+ e_2\otimes e_2)$ $[0,1,0]$ $o_3$ $e_1 \otimes e$ $[0,0,1]$ $o_4$ $e_1 \otimes e_1 \otimes e_1 + e_2\otimes e_1 \otimes e_2$ $[q+1,0,0]$ $o_5$ $e_1 \otimes e_1 \otimes e_1 + e_2\otimes e_2 \otimes e_2$ $[2,q-1,0]$ $o_6$ $e_1 \otimes e_1 \otimes e_1 + e_2\otimes (e_1 \otimes e_2 + e_2 \otimes e_1)$ $[1,q,0]$ $o_7$ $e_1 \otimes e_1 \otimes e_3 + e_2\otimes (e_1 \otimes e_1 + e_2 \otimes e_2)$ $[1,q,0]$ $o_8$ $e_1 \otimes e_1 \otimes e_1 + e_2\otimes (e_2 \otimes e_2 + e_3 \otimes e_3)$ $[1,1,q-1]$ $o_9$ $e_1 \otimes e_3 \otimes e_1 + e_2\otimes e$ $[1,0,q]$ $o_{10}$ $e_1\otimes (e_1\otimes e_1+ e_2\otimes e_2+u e_1\otimes e_2) + e_2\otimes (e_1\otimes e_2+v e_2\otimes e_1)$ ($$) $[0,q+1,0]$ $o_{11}$ $e_1\otimes (e_1 \otimes e_1 + e_2 \otimes e_2) + e_2\otimes (e_1 \otimes e_2 + e_2 \otimes e_3)$ $[0,q+1,0]$ $o_{12}$ $e_1\otimes (e_1 \otimes e_1 + e_2 \otimes e_2) + e_2\otimes (e_1 \otimes e_3 + e_3 \otimes e_2)$ $[0,q+1,0]$ $o_{13}$ $e_1\otimes (e_1 \otimes e_1 + e_2 \otimes e_2) + e_2\otimes (e_1 \otimes e_2 + e_3 \otimes e_3)$ $[0,2,q-1]$ $o_{14}$ $e_1\otimes (e_1 \otimes e_1 + e_2 \otimes e_2) + e_2\otimes (e_2 \otimes e_2 + e_3 \otimes e_3)$ $[0,3,q-2]$ $o_{15}$ $e_1\otimes (e+u e_1\otimes e_2) + e_2\otimes (e_1\otimes e_2+v e_2\otimes e_1)$ ($$) $[0,1,q]$ $o_{16}$ $e_1\otimes e + e_2\otimes (e_1 \otimes e_2 + e_2 \otimes e_3)$ $[0,1,q]$ $o_{17}$ $e_1\otimes e + e_2\otimes (e_1\otimes e_2 + e_2\otimes e_3 + e_3\otimes (\alpha e_1 + \beta e_2 + \gamma e_3))$ ($*$) $[0,0,q+1]$ : Orbits of tensors in $\overline{V} = \mathbb{F}_q^2 \otimes \mathbb{F}_q^3 \otimes \mathbb{F}_q^3$ under the setwise stabiliser in $\text{GL}(\overline{V})$ of the set of fundamental tensors in $\overline{V}$, as per [@LaSh2015 p. 146]. Representatives are given in terms of a basis $\{e_1,e_2,e_3\}$ of $\mathbb{F}_q^3$, with $e := \sum_{i=1}^3 e_i \otimes e_i$. The final column shows the rank distribution of the first contraction space of each representative. Condition ($$) is: $v\lambda^2+uv\lambda - 1 \neq 0$ for all $\lambda \in {\mathbb{F}}_q$. Condition ($*$) is: $\lambda^3+\gamma \lambda^2- \beta \lambda+ \alpha \neq 0$ for all $\lambda \in {\mathbb{F}}_q$.[]{data-label="pointTable"} In this paper, we are interested in symmetric representatives of line orbits in the projective space ${\mathrm{PG}}(V)$. The line orbits themselves can be obtained by considering the [first contraction spaces]{} of tensors in $\overline{V}$. As per [@LaSh2015 p. 136], the [first contraction space]{} of a tensor $A \in \overline{V}$ is the subspace $$A_1 = \langle w_1^\vee(A) : w_1^\vee \in V_1^\vee \rangle$$ of $V$. Here $V_1^\vee$ is the dual of $V_1$, and $w_1^\vee(A)$ is defined by its action on fundamental tensors via $w_1^\vee(v_1 \otimes v_2 \otimes v_3) = w_1^\vee(v_1)v_2 \otimes v_3$. Recall also that the [rank]{} of a point in ${\mathrm{PG}}(V)$ is the rank of any ($3 \times 3$) matrix representing that point (and that this does not depend on the choice of bases for the factors of the tensor product). In geometric terms, a point has rank 1 if it is contained in the Segre variety $S = S_{3,3}({\mathbb{F}}) \subset {\mathrm{PG}}(V)$, rank 2 if is not contained in $S$ but is contained in the secant variety of $S$, and rank 3 if it is not contained in the secant variety of $S$. Note that the last column of Table \[pointTable\] shows, for each orbit, the [rank distribution]{} of the first contraction space of a representative $A$, namely a list $[a_1,a_2,a_3]$ where $a_i$ is the number of points of rank $i$ in $\text{PG}(A_1)$. The first contraction spaces of the tensors in orbits $o_4,\ldots,o_{17}$ are lines of ${\mathrm{PG}}(V)$; in particular, for their corresponding rank distributions we have $a_1+a_2+a_3=q+1$. Properties of the quadric Veronesean {#V3props} ------------------------------------ Here we collect some facts about the quadric Veronesean ${\mathcal{V}}_3({\mathbb{F}})$ that are used throughout the paper. For proofs and/or further details, the reader may consult a standard reference such as or [@Harris p. 23] or [@HandT Chapter 4]. As noted in Section \[intro\], ${\mathcal{V}}_3({\mathbb{F}})$ is the image of the map $\nu_3 : {\mathrm{PG}}(2,{\mathbb{F}}) \rightarrow {\mathrm{PG}}(5,{\mathbb{F}})$ induced by the map that takes $u \in {\mathbb{F}}^3$ to $u \otimes u$, and the setwise stabiliser $K$ of ${\mathcal{V}}_3({\mathbb{F}})$ inside ${\mathrm{PGL}}(6,{\mathbb{F}})$ is isomorphic to ${\mathrm{PGL}}(3,{\mathbb{F}})$, with $D \in K$ mapping a symmetric matrix $M$ to $M^D := DMD^\top$. We also record the following facts, which are readily obtained from the relevant definitions: - The image of a line of ${\mathrm{PG}}(2,{\mathbb{F}})$ under $\nu_3$ is a conic. A plane of ${\mathrm{PG}}(5,{\mathbb{F}})$ intersecting ${\mathcal{V}}_3({\mathbb{F}})$ in a conic is called a [conic plane]{}, and each conic plane is equal to $\langle \nu_3(\ell)\rangle$ for some line $\ell$ of ${\mathrm{PG}}(2,{\mathbb{F}})$. - Each two points $P,Q$ of ${\mathcal{V}}_3({\mathbb{F}})$ lie on a unique conic ${\mathcal{C}}(P,Q)$ in ${\mathcal{V}}_3({\mathbb{F}})$, given by ${\mathcal{C}}(P,Q)=\nu_3(\langle \nu_3^{-1}(P),\nu_3^{-1}(Q)\rangle)$. - Each rank-2 point $R$ in $\langle {\mathcal{V}}_3({\mathbb{F}}) \rangle$ determines a unique conic ${\mathcal{C}}(R)$ in ${\mathcal{V}}_3({\mathbb{F}})$. The point $R$ is called an [exterior point]{} if it lies on a tangent to ${\mathcal{C}}(R)$, and an [interior point]{} otherwise. - The quadrics of ${\mathrm{PG}}(2,{\mathbb{F}})$ are mapped by $\nu_3$ onto the hyperplane sections of ${\mathcal{V}}_3({\mathbb{F}})$. A conic consisting of just one point (two distinct lines over the quadratic extension) corresponds to a hyperplane intersecting ${\mathcal{V}}_3({\mathbb{F}})$ in one point. A repeated line of ${\mathrm{PG}}(2,{\mathbb{F}})$ corresponds to a hyperplane meeting ${\mathcal{V}}_3({\mathbb{F}})$ in a conic; two distinct lines correspond to a hyperplane meeting ${\mathcal{V}}_3({\mathbb{F}})$ in two conics; and a non-degenerate conic corresponds to a hyperplane meeting ${\mathcal{V}}_3({\mathbb{F}})$ in a normal rational curve. - If $\operatorname{char}({\mathbb{F}})=2$, then the nuclei of all of the conics contained in ${\mathcal{V}}_3({\mathbb{F}})$ form a plane, called the [nucleus plane]{} of ${\mathcal{V}}_3({\mathbb{F}})$ (recall that a [nucleus]{} of a conic is a point contained in all of the tangents to the conic). In the representation of the points of ${\mathcal{V}}_3({\mathbb{F}})$ as symmetric $3\times 3$ matrices of rank 1, the nucleus plane comprises the matrices with zeroes on the main diagonal (and the other three variables free). The $K$-orbits of [points]{} in $\langle {\mathcal{V}}_3({\mathbb{F}})\rangle$ are well understood. For convenience, we note some facts about these point orbits in the case where ${\mathbb{F}}$ is a finite field ${\mathbb{F}}_q$: - There is one $K$-orbit of points of rank 1: $K$ acts transitively on the set of 4-tuples of points of ${\mathcal{V}}_3({\mathbb{F}}_q)$, no three of which are on a conic. This orbit has size $q^2+q+1$ (the number of points in ${\mathrm{PG}}(2,{\mathbb{F}})$). - There are two $K$-orbits of points of rank 2. In odd characteristic, one of these orbits consists of all the exterior points, and the other consists of all the interior points. Denoting these orbits by ${\mathcal{P}}_{2,\text{e}}$ and ${\mathcal{P}}_{2,\text{i}}$ respectively, we have $$\|{\mathcal{P}}_{2,\text{e}}\|=\frac{1}{2}q(q+1)(q^2+q+1) \quad \text{and} \quad \|{\mathcal{P}}_{2,\text{i}}\|=\frac{1}{2}q(q-1)(q^2+q+1).$$ In even characteristic, one orbit consists of all the points that lie on the nucleus plane of ${\mathcal{V}}_3({\mathbb{F}}_q)$, and the other orbit consists of all the other points of rank 2. Denoting these orbits by ${\mathcal{P}}_{2,\text{n}}$ and ${\mathcal{P}}_{2,\text{s}}$ respectively, we have $$\|{\mathcal{P}}_{2,\text{n}}\|=q^2+q+1 \quad \text{and} \quad \|{\mathcal{P}}_{2,\text{s}}\|=(q^2-1)(q^2+q+1).$$ Hence, regardless of the value of $q$, the total number of points of rank 2 is $q^2(q^2+q+1)$. - Finally, there is a unique $K$-orbit of rank 3 points, of size $$\frac{q^6-1}{q-1}-(q^2+1)(q^2+q+1)=q^5-q^2.$$ Line orbits in $\langle \mathcal{V}_3({\mathbb{F}}) \rangle$ for ${\mathbb{F}}$ a finite field {#sec:orbits} ============================================================================================== We now address problems (i) and (ii) of Section \[intro\] in the case where ${\mathbb{F}}$ is a finite field ${\mathbb{F}}_q$. Our strategy is as follows. We consider each of the orbits $o_0,\ldots,o_{17}$ of tensors in ${\mathbb{F}}_q^2 \otimes {\mathbb{F}}_q^3 \otimes {\mathbb{F}}_q^3$, which are shown in Table \[pointTable\]. Given a representative of an orbit $o_i$ from the second column of Table \[pointTable\], we consider the corresponding first contraction space $M_i$, which is a subspace of $V={\mathbb{F}}_q^3 \otimes {\mathbb{F}}_q^3$. When $i {\geqslant}4$ in Table \[pointTable\], $\text{PG}(M_i)$ is a line of ${\mathrm{PG}}(V)$, comprising $q+1$ points which we represent by $3 \times 3$ matrices. For each $i {\geqslant}4$, we first address problem (i) by checking whether $\text{PG}(M_i)$ can be mapped into $\langle \mathcal{V}_3({\mathbb{F}}_q) \rangle$ by the action of $\text{PGL}(3,{\mathbb{F}}_q) \times \text{PGL}(3,{\mathbb{F}}_q)$ induced by the action of ${\mathrm{GL}}(3,{\mathbb{F}}_q) \times {\mathrm{GL}}(3,{\mathbb{F}}_q)$ taking a $3 \times 3$ matrix $M$ to $M^{(B,C)} := BMC$ (where $B,C, \in {\mathrm{GL}}(3,{\mathbb{F}}_q)$). In other words, we check whether $M_i^{(B,C)}$ can be a subspace of [symmetric]{} $3 \times 3$ matrices. If it cannot, then the $G$-line orbit arising from the tensor orbit $o_i$ is not represented in $\langle \mathcal{V}_3({\mathbb{F}}_q) \rangle$, that is, it does not have a symmetric representative in the sense defined in Section \[intro\]. If it can, then we address problem (ii) by determining the orbits of the group $K = \text{PGL}(3,{\mathbb{F}}_q)$ in the action $M^D = DMD^\top$ for $M$ a symmetric matrix and $D \in K$. Note also that when considering problem (i) as described above, we may take $C$ to be the identity matrix, because the line ${\mathrm{PG}}(BM_iC)$ is equivalent under the action of $K$ to the projective space obtained from the vector subspace $$(BM_iC)^{(C^{-1})^\top} = (C^{-1})^\top(BM_iC)C^{-1} = (C^{-1})^\top B M_i.$$ This simplifies the proof of the fact that certain tensor orbits, namely $o_4$, $o_7$ and $o_{11}$, do [not]{} yield lines with symmetric representatives. The remaining tensor orbits $o_i$ (with $i {\geqslant}4$) do yield lines with symmetric representatives, and the representatives of the corresponding $K$-orbits are listed in Table \[table:main\]. It turns out that in each case there are at most two $K$-orbits of lines. The $K$-orbit in the second column of the table arises for all values of $q$, and sometimes there is another $K$-orbit, with representative shown in the third column if $q$ is odd and in the fourth column if $q$ is even. The following notation is used for brevity in Table \[table:main\] (and in the proofs): \[def:notation\] [The matrices in Table \[table:main\] represent subspaces of symmetric matrices over the finite field $\mathbb{F}_q$. The subscript “$x,y$” indicates that the pair $(x,y)$ ranges over all values in $\mathbb{F}_q^2$, and the symbol $\cdot$ denotes $0$. For example, in the first line of the table, $$\left[ \begin{matrix} x & \cdot & \cdot \\ \cdot & y & \cdot \\ \cdot & \cdot & \cdot \end{matrix} \right]_{x,y} = \left\{ \left[ \begin{matrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & 0 \end{matrix} \right] : (x,y) \in \mathbb{F}_q^2 \right\},$$ and the line orbit representative in $\langle {\mathcal{V}}_3({\mathbb{F}}_q) \rangle$ is the corresponding projective space. Moreover, the symbol $\Box$ denotes the set of squares in ${\mathbb{F}}_q$.]{.nodecor} ---------- ---------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------- Tensor Conditions orbit Common orbit (all $q$) $q$ odd $q$ even $o_5$ $\left[ \begin{matrix} x & \cdot & \cdot \\ \cdot & y & \cdot \\ \cdot & \cdot & \cdot \end{matrix} \right]_{x,y}$ $o_6$ $\left[ \begin{matrix} x & y & \cdot \\ y & \cdot & \cdot \\ \cdot & \cdot & \cdot \end{matrix} \right]_{x,y}$ $o_8$ $\left[ \begin{matrix} x & \cdot & \cdot \\ \cdot & y & \cdot \\ \cdot & \cdot & y \end{matrix} \right]_{x,y}$ $\left[ \begin{matrix} x & \cdot & \cdot \\ \cdot & y & \cdot \\ \cdot & \cdot & \gamma y \end{matrix} \right]_{x,y}$ $\left[ \begin{matrix} x & \cdot & \cdot \\ \cdot & \cdot & y \\ \cdot & y & \cdot \end{matrix} \right]_{x,y}$ $\gamma \not \in \Box$ $o_9$ $\left[ \begin{matrix} x & \cdot & y \\ \cdot & y & \cdot \\ y & \cdot & \cdot \end{matrix} \right]_{x,y}$ $o_{10}$ $\left[ \begin{matrix} vx & y & \cdot \\ y & x+uy & \cdot \\ \cdot & \cdot & \cdot \end{matrix} \right]_{x,y}$ ($$) $o_{12}$ $\left[ \begin{matrix} \cdot & x & \cdot \\ x & \cdot & y \\ \cdot & y & \cdot \end{matrix} \right]_{x,y}$ $\left[ \begin{matrix} \cdot & x & \cdot \\ x & x+y & y \\ \cdot & y & \cdot \end{matrix} \right]_{x,y}$ $o_{13}$ $\left[ \begin{matrix} \cdot & x & \cdot \\ x & y & \cdot \\ \cdot & \cdot & y \end{matrix} \right]_{x,y}$ $\left[ \begin{matrix} \cdot & x & \cdot \\ x & y & \cdot \\ \cdot & \cdot & \gamma y \end{matrix} \right]_{x,y}$ $\left[ \begin{matrix} \cdot & x & \cdot \\ x & x+y & \cdot \\ \cdot & \cdot & y \end{matrix} \right]_{x,y}$ $\gamma \not \in \Box$ $o_{14}$ $\left[ \begin{matrix} x & \cdot & \cdot \\ \cdot & x+y & \cdot \\ \cdot & \cdot & y \end{matrix} \right]_{x,y}$ $\left[ \begin{matrix} x & \cdot & \cdot \\ \cdot & \gamma(x+y) & \cdot \\ \cdot & \cdot & y \end{matrix} \right]_{x,y}$ $\gamma \not \in \Box$ $o_{15}$ $\left[ \begin{matrix} v_1y & x & \cdot \\ x & ux+y & \cdot \\ \cdot & \cdot & x \end{matrix} \right]_{x,y}$ $\left[ \begin{matrix} v_2y & x & \cdot \\ x & ux+y & \cdot \\ \cdot & \cdot & x \end{matrix} \right]_{x,y}$ ($$), $\begin{array}{ll}-v_1 \in \Box \\ -v_2 \not \in \Box \end{array}$ $o_{16}$ $\left[ \begin{matrix} \cdot & \cdot & x \\ \cdot & x & y \\ x & y & \cdot \end{matrix} \right]_{x,y}$ $\left[ \begin{matrix} \cdot & \cdot & x \\ \cdot & x & y \\ x & y & y \end{matrix} \right]_{x,y}$ $o_{17}$ $\left[ \begin{matrix} \alpha^{-1}x & y & \cdot \\ y & \beta y - \gamma x & x \\ \cdot & x & y \end{matrix} \right]_{x,y}$ ($$) ---------- ---------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------- : Representatives of line orbits in $\langle \mathcal{V}_3({\mathbb{F}}_q) \rangle$ under the action of $K={\mathrm{PGL}}(3,\mathbb{F}_q)$ on subspaces of ${\mathrm{PG}}({\mathbb{F}}_q^3 \otimes {\mathbb{F}}_q^3)$ induced by the action of ${\mathrm{GL}}(3,{\mathbb{F}}_q)$ on $3 \times 3$ matrices $M$ given by $M^D = DMD^\top$ (where $D \in {\mathrm{GL}}(3,{\mathbb{F}}_q)$). Notation is as in Definition \[def:notation\]. For brevity, the corresponding [vector]{} subspaces $M$ of ${\mathbb{F}}_q^3 \otimes {\mathbb{F}}_q^3$ are shown, so that the $K$-orbit representatives themselves are given by ${\mathrm{PG}}(M)$. Condition ($$) is: $v\lambda^2+uv\lambda - 1 \neq 0$ for all $\lambda \in {\mathbb{F}}_q$, where $v \in \{v_1,v_2\}$ in the case $o_{15}$. Condition ($*$) is: $\lambda^3+\gamma \lambda^2- \beta \lambda+ \alpha \neq 0$ for all $\lambda \in {\mathbb{F}}_q$.[]{data-label="table:main"} Let us now proceed with the case-by-case proof. We remind the reader that several of the facts recorded in Section \[prelims\] are used throughout. Tensor orbit $o_4$ {#tensor-orbit-o_4 .unnumbered} ------------------ The tensor orbit representative from Table \[pointTable\] is $e_1 \otimes e_1 \otimes e_1 + e_2 \otimes e_1 \otimes e_2$. Its first contraction space is $M_4 = \langle e_1 \otimes e_1 , e_1 \otimes e_2 \rangle$, and has rank distribution $[q+1,0,0]$. Let $B \in \text{GL}(3,\mathbb{F}_q)$ and suppose that the line $\text{PG}(BM_4)$ is contained in $\langle \mathcal{V}_3(\mathbb{F}_q) \rangle$. Then, in particular, $\text{PG}(BM_4)$ is contained in the Veronese variety ${\mathcal{V}}_3({\mathbb{F}})$, a contradiction. Therefore, the tensor orbit $o_4$ does not give rise to any line with a symmetric representative. Tensor orbit $o_5$ {#tensor-orbit-o_5 .unnumbered} ------------------ Here we have tensor orbit representative $e_1 \otimes e_1 \otimes e_1 + e_2 \otimes e_2 \otimes e_2$. The first contraction space is $M_5 = \langle e_1 \otimes e_1 , e_2 \otimes e_2 \rangle$, with rank distribution $[2,q-1,0]$. Note that ${\mathrm{PG}}(M_5)$ is contained in $\langle \mathcal{V}_3(\mathbb{F}_q) \rangle$: it is the $K$-line orbit representative given in Table \[table:main\] (in the second column). Now, if $\text{PG}(BM_5)$ is contained in $\langle \mathcal{V}_3(\mathbb{F}_q) \rangle$ for some $B \in \text{GL}(3,\mathbb{F}_q)$, then both of the matrices $B^1 \otimes e_1$ and $B^2 \otimes e_2$ must be symmetric and of rank $1$. This forces $B^1 = \alpha e_1$ and $B^2 = \beta e_2$ for some $\alpha,\beta \in \mathbb{F}_q^{\times}= \mathbb{F}_q \setminus \{0\}$, so $$B=\left[ \begin{matrix} \alpha & \cdot & \\ \cdot & \beta & * \\ \cdot & \cdot & * \end{matrix} \right],$$ where $$ denotes an unspecified element of ${\mathbb{F}}_q$ and $\cdot$ denotes $0$ as per Definition \[def:notation\]. Therefore, $$BM_5 = \left[ \begin{matrix} \alpha & \cdot & \\ \cdot & \beta & * \\ \cdot & \cdot & * \end{matrix} \right] \left[ \begin{matrix} x & \cdot & \cdot \\ \cdot & y & \cdot \\ \cdot & \cdot & \cdot \end{matrix} \right]_{x,y} = \left[ \begin{matrix} \alpha x & \cdot & \cdot \\ \cdot & \beta y & \cdot \\ \cdot & \cdot & \cdot \end{matrix} \right]_{x,y}.$$ Since $\alpha x$ and $\beta y$ range over all values in $\mathbb{F}_q$ as $x$ and $y$ do, we may relabel $\alpha x$ as $x$ and $\beta y$ as $y$ to see that $BM_5 = M_5$. That is, ${\mathrm{PG}}(BM_5)$ is contained in $\langle \mathcal{V}_3(\mathbb{F}_q) \rangle$ if and only if ${\mathrm{PG}}(BM_5)={\mathrm{PG}}(M_5)$, and so the orbit containing ${\mathrm{PG}}(M_5)$ is the [only]{} $K$-line orbit in $\langle \mathcal{V}_3(\mathbb{F}) \rangle$ arising from the tensor orbit $o_5$. Tensor orbit $o_6$ {#tensor-orbit-o_6 .unnumbered} ------------------ This tensor orbit has representative $e_1 \otimes e_1 \otimes e_1 + e_2 \otimes ( e_1 \otimes e_2 + e_2 \otimes e_1 )$. The first contraction space is $M_6 = \langle e_1 \otimes e_1 , e_1 \otimes e_2 + e_2 \otimes e_1 \rangle$, with rank distribution $[1,q,0]$. As in the previous case, we note that ${\mathrm{PG}}(M_6)$ is contained in $\langle \mathcal{V}_3(\mathbb{F}_q) \rangle$, and is the representative given in Table \[table:main\]. Now suppose that ${\mathrm{PG}}(BM_6)$ is contained in $\langle \mathcal{V}_3(\mathbb{F}_q) \rangle$ for some $B \in \text{GL}(3,\mathbb{F}_q)$. Then $B^1 \otimes e_1$ must be symmetric and of rank $1$, so $B^1 = \alpha e_1$ for some $\alpha \in \mathbb{F}_q^{\times}$. Moreover, $B^1 \otimes e_2 + B^2 \otimes e_1 = \alpha e_1 \otimes e_2 + B^2 \otimes e_1$ must be symmetric and of rank $2$, so $B^2 = \beta e_1 + \alpha e_2$ for some $\beta \in \mathbb{F}_q$. Hence, $$BM_6 = \left[ \begin{matrix} \alpha & \beta & * \\ \cdot & \alpha & * \\ \cdot & \cdot & * \end{matrix} \right] \left[ \begin{matrix} x & y & \cdot \\ y & \cdot & \cdot \\ \cdot & \cdot & \cdot \end{matrix} \right]_{x,y} = \left[ \begin{matrix} \alpha x + \beta y & \alpha y & \cdot \\ \alpha y & \cdot & \cdot \\ \cdot & \cdot & \cdot \end{matrix} \right]_{x,y}.$$ Again, we may relabel $\alpha x$ as $x$ and $\beta y$ as $y$ to deduce that $BM_6=M_6$. Hence, the orbit containing ${\mathrm{PG}}(M_6)$ is the only $K$-line orbit in $\langle \mathcal{V}_3(\mathbb{F}) \rangle$ arising from the tensor orbit $o_6$. Tensor orbit $o_7$ {#tensor-orbit-o_7 .unnumbered} ------------------ This tensor orbit has representative $e_1 \otimes e_1 \otimes e_3 + e_2 \otimes ( e_1 \otimes e_1 + e_2 \otimes e_2 )$. The first contraction space is $M_7 = \langle e_1 \otimes e_3 , e_1 \otimes e_1 + e_2 \otimes e_2 \rangle$, with rank distribution $[1,q,0]$. We claim that ${\mathrm{PG}}(BM_7)$ is not contained in $\langle \mathcal{V}_3(\mathbb{F}_q) \rangle$ for any $B \in \text{GL}(3,\mathbb{F}_q)$. If it were, then $B^1 \otimes e_3$ would have to be symmetric and of rank $1$, forcing $B^1 = \alpha e_3$ for some $\alpha \in \mathbb{F}_q^{\times}$. However, then $B^1 \otimes e_1 + B^2 \otimes e_2 = \alpha e_3 \otimes e_1 + B^2 \otimes e_2$ would not be symmetric, a contradiction. Hence, the tensor orbit $o_7$ does not give rise to any line with a symmetric representative. Tensor orbit $o_8$ {#tensor-orbit-o_8 .unnumbered} ------------------ Here the tensor orbit representative is $e_1 \otimes e_1 \otimes e_1 + e_2 \otimes ( e_2 \otimes e_2 + e_3 \otimes e_3 )$. The first contraction space is $M_8 = \langle e_1 \otimes e_1 , e_2 \otimes e_2 + e_3 \otimes e_3 \rangle$, with rank distribution $[1,1,q-1]$. We claim that this yields two $K$-orbits of lines in $\langle \mathcal{V}_3(\mathbb{F}_q) \rangle$. Suppose that ${\mathrm{PG}}(BM_8)$ is contained in $\langle \mathcal{V}_3(\mathbb{F}_q) \rangle$ for some $B \in \text{GL}(3,\mathbb{F}_q)$. Then $B(e_1 \otimes e_1) = B^1 \otimes e_1$ must be symmetric and of rank $1$, forcing $B^1 = \alpha e_1$ for some $\alpha \in \mathbb{F}_q^{\times}$. Moreover, $B^2 \otimes e_2 + B^3 \otimes e_3$ must be symmetric and of rank $2$, so $B^2 = \beta_2 e_2 + \gamma_2 e_3$ and $B^3 = \beta_3 e_2 + \gamma_3 e_3$, with $\gamma_2=\beta_3$ and $\beta_2\gamma_3 - \beta_3^2 = \beta_2 \gamma_3 - \gamma_2 \beta_3 \neq 0$. Therefore, $$BM_8 = \left[ \begin{matrix} \alpha & \cdot & \cdot \\ \cdot & \beta_2 & \beta_3 \\ \cdot & \beta_3 & \gamma_3 \end{matrix} \right] \left[ \begin{matrix} x & \cdot & \cdot \\ \cdot & y& \cdot \\ \cdot & \cdot & y \end{matrix} \right]_{x,y} = \left[ \begin{matrix} \alpha x &\cdot&\cdot \\ \cdot& \beta_2 y & \beta_3 y \\ \cdot& \beta_3 y & \gamma_3 y \end{matrix} \right]_{x,y}.$$ We now claim that ${\mathrm{PG}}(BM_8)$ is $K$-equivalent to either $$\label{o8-1} {\mathrm{PG}}\left( \left[ \begin{matrix} \alpha x&\cdot&\cdot \\ \cdot&\cdot&\beta y \\ \cdot&\beta y&\cdot \end{matrix} \right]_{x,y} \right) \quad \text{or} \quad {\mathrm{PG}}\left( \left[ \begin{matrix} \alpha x&\cdot&\cdot \\ \cdot&\beta y&\cdot \\ \cdot&\cdot&\gamma y \end{matrix} \right]_{x,y} \right) \quad \text{for some} \quad \beta,\gamma \in \mathbb{F}_q^{\times},$$ according to whether $\beta_2=\gamma_3 = 0$ or not. This is clear in the case where $\beta_2=\gamma_3 = 0$, as we simply relabel $\beta_3$ as $\beta$. On the other hand, if $\gamma_3 \neq 0$ then $$D \left[ \begin{matrix} \alpha x &\cdot&\cdot \\ \cdot& \beta_2 y & \beta_3 y \\ \cdot& \beta_3 y & \gamma_3 y \end{matrix} \right]_{x,y} D^\top = \left[ \begin{matrix} \alpha x&\cdot&\cdot \\ \cdot & (\beta_2-\beta_3^2\gamma_3^{-1})y & \cdot \\ \cdot&\cdot&\gamma_3 y \end{matrix} \right]_{x,y}, \quad \text{where} \quad D = \left[ \begin{matrix} 1 & \cdot & \cdot \\ \cdot & 1 & -\beta_3\gamma_3^{-1} \\ \cdot & \cdot & 1 \end{matrix} \right];$$ and if $\beta_2 \neq 0$ then $$D \left[ \begin{matrix} \alpha x &\cdot&\cdot \\ \cdot& \beta_2 y & \beta_3 y \\ \cdot& \beta_3 y & \gamma_3 y \end{matrix} \right]_{x,y} D^\top = \left[ \begin{matrix} \alpha x&\cdot&\cdot \\ \cdot & \beta_2 y & \cdot \\ \cdot&\cdot& (\gamma_3-\beta_3^2\beta_2^{-1})y \end{matrix} \right]_{x,y}, \quad \text{where} \quad D = \left[ \begin{matrix} 1 & \cdot & \cdot \\ \cdot & 1 & \cdot \\ \cdot & -\beta_3\beta_2^{-1} & 1 \end{matrix} \right].$$ The claim follows upon appropriately relabelling the variables $x$ and $y$. By relabelling the constants in , we then see that ${\mathrm{PG}}(BM_8)$ is $K$-equivalent to one of the lines $$L = {\mathrm{PG}}\left( \left[ \begin{matrix} x&\cdot&\cdot \\ \cdot&\cdot& y \\ \cdot& y&\cdot \end{matrix} \right]_{x,y} \right) \quad \text{or} \quad L_\gamma = {\mathrm{PG}}\left( \left[ \begin{matrix} x&\cdot&\cdot \\ \cdot& y&\cdot \\ \cdot&\cdot&\gamma y \end{matrix} \right]_{x,y} \right) \text{ with } \gamma \in \mathbb{F}^{\times}. $$ We now show that $L$ and $L_\gamma$ represent the same $K$-orbit if and only if $q$ is odd and $-\gamma \in \Box$ (that is, if $-\gamma$ is a square). Disregarding the first row and column, we have $$\left[ \begin{matrix} a&b \\ c&d \end{matrix} \right] \left[ \begin{matrix} 0&1 \\ 1&0 \end{matrix} \right] \left[ \begin{matrix} a&c \\ b&d \end{matrix} \right]= \left[ \begin{matrix} a&b \\ c&d \end{matrix} \right]\left[ \begin{matrix} b&d \\ a&c \end{matrix} \right] = \left[ \begin{matrix} 2ab&ad+bc \\ ad+bc&2cd \end{matrix} \right]=\left[ \begin{matrix} 1&0 \\ 0&\gamma \end{matrix} \right]$$ if and only if $2ab=1$, $ad+bc=0$ and $2cd=\gamma$, a contradiction if $q$ is even. If $q$ is odd then $b=(2a)^{-1}$, $d=\gamma (2c)^{-1}$, and $-\gamma = (ca^{-1})^2$ for $a\neq 0 \neq c$, so $L$ and $L_\gamma$ represent the same $K$-orbit if and only if $-\gamma \in \Box$. Finally, if $q$ is odd and $-1\in \Box$ then $L$ represents the same $K$-orbit as $L_1$, whereas if $-1\notin \Box$ then $L$ represents the same $K$-orbit as $L_\gamma$ with $\gamma\notin \Box$. Hence, we have the following two cases, as per Table \[table:main\]: if $q$ is even then there are two $K$-orbits, represented by $L$ and $L_1$; and if $q$ is odd then the two $K$-orbits are represented by $L_1$ and $L_\gamma$ with $\gamma\notin \Box$. Tensor orbit $o_9$ {#tensor-orbit-o_9 .unnumbered} ------------------ This tensor orbit has representative $$e_1 \otimes e_3 \otimes e_1 + e_2 \otimes ( e_1 \otimes e_1 + e_2 \otimes e_2 + e_3 \otimes e_3 ).$$ The first contraction space is $M_9 = \langle e_3 \otimes e_1 , e_1 \otimes e_1 + e_2 \otimes e_2 + e_3 \otimes e_3 \rangle$, with rank distribution $[1,0,q]$. If ${\mathrm{PG}}(BM_9)$ is contained in $\langle \mathcal{V}_3(\mathbb{F}_q) \rangle$, then $B^3 \otimes e_1$ must be symmetric and of rank $1$, forcing $B^3 = \alpha e_1$ for some $\alpha \in \mathbb{F}_q^{\times}$. Hence, $B^1 \otimes e_1 + B^2 \otimes e_2 + B^3 \otimes e_3 = B^1 \otimes e_1 + B^2 \otimes e_2 + \alpha e_1 \otimes e_3$ must be symmetric and of rank $3$. Writing $B^1 = \alpha_1 e_1 + \beta_1 e_2 + \gamma_1 e_3$ and $B^2 = \alpha_2 e_1 + \beta_2 e_2 + \gamma_2 e_3$, it follows that we must have $\alpha_1=\gamma_2=0$, $\gamma_1=\alpha$, $\alpha_2=\beta_1$ and $\beta_2 \neq 0$. Therefore, $$BM_9 = \left[ \begin{matrix} \cdot & \beta_1 & \alpha \\ \beta_1 & \beta_2 & \cdot \\ \alpha & \cdot & \cdot \end{matrix} \right] \left[ \begin{matrix} y & \cdot & \cdot \\ \cdot & y & \cdot \\ x & \cdot & y \end{matrix} \right]_{x,y} = \left[ \begin{matrix} \alpha x & \beta_1 y & \alpha y \\ \beta_1 y & \beta_2 y & \cdot \\ \alpha y & \cdot & \cdot \end{matrix} \right]_{x,y}.$$ Since $\beta_2 \neq 0$, we may set $\beta_2=1$. Relabelling also $\beta_1$ as $\beta$ and $x$ as $\alpha^{-1} x$, we see that $$BM_9 = \left[ \begin{matrix} x & \beta y & \alpha y \\ \beta y & y & \cdot \\ \alpha y & \cdot & \cdot \end{matrix} \right]_{x,y}.$$ We now see that ${\mathrm{PG}}(BM_9)$ is $K$-equivalent to the representative shown in Table \[table:main\], because $$D \left[ \begin{matrix} x & \beta y & \alpha y \\ \beta y & y & \cdot \\ \alpha y & \cdot & \cdot \end{matrix} \right]_{x,y} D^\top = \left[ \begin{matrix} x & \cdot & y \\ \cdot & y & \cdot \\ y & \cdot & \cdot \end{matrix} \right]_{x,y}, \quad \text{where} \quad D = \left[ \begin{matrix} 1 & \cdot & \cdot \\ \cdot & 1 & -\beta\alpha^{-1} \\ \cdot & \cdot & \alpha^{-1} \end{matrix} \right].$$ Tensor orbit $o_{10}$ {#tensor-orbit-o_10 .unnumbered} --------------------- This tensor orbit has representative $$e_1 \otimes (e_1 \otimes e_1 + e_2 \otimes e_2 + ue_1 \otimes e_2) + e_2 \otimes (e_1 \otimes e_2 + ve_2 \otimes e_1),$$ where $v\lambda^2 + uv\lambda - 1 \neq 0$ for all $\lambda \in \mathbb{F}$, namely condition ($$) in Tables \[pointTable\] and \[table:main\]. The first contraction space is $M_{10} = \langle e_1 \otimes e_1 + e_2 \otimes e_2 + ue_1 \otimes e_2 , e_1 \otimes e_2 + ve_2 \otimes e_1 \rangle$, with rank distribution $[0,q+1,0]$. If we take $$B = \left[ \begin{matrix} 1 & \cdot & \cdot \\ u & v^{-1} & \cdot \\ \cdot & \cdot & \end{matrix} \right],$$ then $$BM_{10} = \left[ \begin{matrix} 1 & 0 & \cdot \\ u & v^{-1} & \cdot \\ \cdot & \cdot & * \end{matrix} \right] \left[ \begin{matrix} x & ux+y & \cdot \\ vy & x & \cdot \\ \cdot & \cdot & \cdot \end{matrix} \right]_{x,y} = \left[ \begin{matrix} x & ux+y & \cdot \\ ux+y & u^2x+uy+v^{-1}x & \cdot \\ \cdot & \cdot & \cdot \end{matrix} \right]_{x,y},$$ so ${\mathrm{PG}}(BM_{10})$ lies in $\langle \mathcal{V}_3(\mathbb{F}_q) \rangle$. By relabelling $ux+y$ as $y$ and $v^{-1}x$ as $x$, we see that ${\mathrm{PG}}(BM_{10})$ is the $K$-line orbit representative given in Table \[table:main\]. Now, this line is a constant rank-$2$ line of $2 \times 2$ matrices, so is an external line to a conic. Since the group of a conic acts transitively on the set of external lines to the conic, there is only one $K$-line orbit in $\langle \mathcal{V}_3(\mathbb{F}_q) \rangle$ arising from the tensor orbit $o_{10}$. Tensor orbit $o_{11}$ {#tensor-orbit-o_11 .unnumbered} --------------------- This tensor orbit has representative $$e_1 \otimes ( e_1 \otimes e_1 + e_2 \otimes e_2 ) + e_2 \otimes ( e_1 \otimes e_2 + e_2 \otimes e_3 ).$$ The first contraction space is $M_{11} = \langle e_1 \otimes e_1 + e_2 \otimes e_2 , e_1 \otimes e_2 + e_2 \otimes e_3 \rangle$, with rank distribution $[0,q+1,0]$. We claim that ${\mathrm{PG}}(BM_{11})$ is not contained in $\langle \mathcal{V}_3(\mathbb{F}_q) \rangle$ for any $B \in \text{GL}(3,\mathbb{F}_q)$. If it were, then $B^1 \otimes e_1 + B^2 \otimes e_2$ would need to be symmetric, so in particular $B^1$ and $B^2$ would lie in the span of $e_1$ and $e_2$. However, then $B^1 \otimes e_2 + B^2 \otimes e_3$ would not be symmetric, a contradiction. Hence, the tensor orbit $o_{11}$ does not give rise to any line with a symmetric representative. Tensor orbit $o_{12}$ {#tensor-orbit-o_12 .unnumbered} --------------------- Here we have tensor orbit representative $$e_1 \otimes (e_1 \otimes e_1 + e_2 \otimes e_2) + e_2 \otimes (e_1 \otimes e_3 + e_3 \otimes e_2).$$ The first contraction space is $M_{12} = \langle e_1 \otimes e_1 + e_2 \otimes e_2 , e_1 \otimes e_3 + e_3 \otimes e_2 \rangle$, with rank distribution $[0,q+1,0]$. If ${\mathrm{PG}}(BM_{12})$ is contained in $\langle \mathcal{V}_3(\mathbb{F}_q) \rangle$ for some $B \in \text{GL}(3,\mathbb{F}_q)$, then $B^1 \otimes e_1 + B^2 \otimes e_2$ must be symmetric and of rank $2$, forcing $B^1 = \alpha_1 e_1 + \beta_1 e_2$ and $B^2 = \alpha_2 e_1 + \beta_2 e_2$, with $\alpha_2 = \beta_1$ and $\alpha_1 \beta_2 - \beta_1^2 = \alpha_1 \beta_2 - \alpha_2 \beta_1 \neq 0$. Writing $B^3 = \alpha_3 e_1 + \beta_3 e_2 + \gamma_3 e_3$, we then have $$B^1 \otimes e_3 + B^3 \otimes e_2 = \alpha_1 e_1 \otimes e_3 + \beta_1 e_2 \otimes e_3 + \alpha_3 e_1 \otimes e_2 + \beta_3 e_2 \otimes e_2 + \gamma_3 e_3 \otimes e_2,$$ which must also be symmetric and of rank $2$, forcing $\alpha_1 = 0$, $\alpha_3 = 0$ and $\gamma_3 = \beta_1$. Hence, $$BM_{12} = \left[ \begin{matrix} \cdot & \beta_1 & \cdot \\ \beta_1 & \beta_2 & \beta_3 \\ \cdot & \cdot & \beta_1 \end{matrix} \right] \left[ \begin{matrix} x & \cdot & y \\ \cdot & x & \cdot \\ \cdot & y & \cdot \end{matrix} \right]_{x,y} = \left[ \begin{matrix} \cdot & \beta_1 x & \cdot \\ \beta_1 x & \beta_2 x + \beta_3 y & \beta_1 y \\ \cdot & \beta_1 y & \cdot \end{matrix} \right]_{x,y}.$$ Since $\beta_1 \neq 0$, we can relabel this as $$BM_{12} = \left[ \begin{matrix} \cdot & x & \cdot \\ x & \alpha x + \beta y & y \\ \cdot & y & \cdot \end{matrix} \right]_{x,y} \quad \text{for some} \quad \alpha,\beta \in \mathbb{F}_q.$$ If $q$ is odd then ${\mathrm{PG}}(BM_{12})$ is $K$-equivalent to the line $$L = {\mathrm{PG}}\left( \left[ \begin{matrix} \cdot & x & \cdot \\ x & \cdot & y \\ \cdot & y & \cdot \end{matrix} \right]_{x,y} \right),$$ because $$\left[ \begin{matrix} \cdot & x & \cdot \\ x & \cdot & y \\ \cdot & y & \cdot \end{matrix} \right]_{x,y} = D \left[ \begin{matrix} \cdot & x & \cdot \\ x & \alpha x + \beta y & y \\ \cdot & y & \cdot \end{matrix} \right]_{x,y} D^\top, \quad \text{where} \quad D = \left[ \begin{matrix} 1 & \cdot & \cdot \\ -\tfrac{\alpha}{2} & 1 & -\tfrac{\beta}{2} \\ \cdot & \cdot & 1 \end{matrix} \right].$$ Therefore, there is a single $K$-line orbit, with representative $L$, as per Table \[table:main\]. Now suppose that $q$ is even. In this case we claim that ${\mathrm{PG}}(BM_{12})$ is $K$-equivalent either to $L$ or to the line $$L' = {\mathrm{PG}}\left( \left[ \begin{matrix} \cdot & x & \cdot \\ x & x+y & y \\ \cdot & y & \cdot \end{matrix} \right]_{x,y} \right),$$ according to whether $\alpha=\beta=0$ or not. These two lines lie in different $K$-orbits, characterised by the intersection of the line with the nucleus plane (see fact (F5) in Section \[V3props\]): $L$ is contained in the nucleus plane, and $L'$ intersects the nucleus plane in a point. It remains to prove the claim. If $\alpha = \beta = 0$ then ${\mathrm{PG}}(BM_{12})=L$. If $\alpha \neq 0 \neq \beta$ then ${\mathrm{PG}}(BM_{12})$ is $K$-equivalent to ${\mathrm{PG}}(M)$ for $$M=\left[ \begin{matrix} \cdot & \alpha x & \cdot \\ \alpha x & \alpha x + \beta y & \beta y \\ \cdot & \beta y & \cdot \end{matrix} \right]_{x,y} = D \left[ \begin{matrix} \cdot & x & \cdot \\ x & \alpha x + \beta y & y \\ \cdot & y & \cdot \end{matrix} \right]_{x,y} D^\top, \quad \text{where} \quad D = \left[ \begin{matrix} \alpha & \cdot & \cdot \\ \cdot & 1 & \cdot \\ \cdot & \cdot & \beta \end{matrix} \right];$$ and ${\mathrm{PG}}(M)$ is in turn $K$-equivalent to $L$, by relabelling $\alpha x$ as $x$ and $\beta y$ as $y$. If $\beta = 0$ and $\alpha \neq 0$ then ${\mathrm{PG}}(BM_{12})$ is $K$-equivalent to ${\mathrm{PG}}(M')$ for $$M'=\left[ \begin{matrix} \cdot & \alpha (x+y) & \cdot \\ \alpha (x+y) & \alpha x & \alpha y \\ \cdot & \alpha y & \cdot \end{matrix} \right]_{x,y} = D \left[ \begin{matrix} \cdot & x & \cdot \\ x & \alpha x & y \\ \cdot & y & \cdot \end{matrix} \right]_{x,y} D^\top, \quad \text{where} \quad D = \left[ \begin{matrix} \alpha & \cdot & \alpha \\ \cdot & 1 & \cdot \\ \cdot & \cdot & \alpha \end{matrix} \right];$$ and ${\mathrm{PG}}(M')$ is also $K$-equivalent to $L$, by relabelling $\alpha(x+y)$ as $x$ and $\alpha y$ as $y$. The case where $\alpha=0$ and $\beta \neq 0$ is analogous, and so the proof of the claim is complete. Tensor orbit $o_{13}$ {#tensor-orbit-o_13 .unnumbered} --------------------- This tensor orbit has representative $$e_1 \otimes (e_1 \otimes e_1 + e_2 \otimes e_2) + e_2 \otimes (e_1 \otimes e_2 + e_3 \otimes e_3).$$ The first contraction space is $M_{13} = \langle e_1 \otimes e_1 + e_2 \otimes e_2 , e_1 \otimes e_2 + e_3 \otimes e_3 \rangle$, with rank distribution $[0,2,q-1]$. Suppose that ${\mathrm{PG}}(BM_{13})$ is in $\langle \mathcal{V}_3(\mathbb{F}_q) \rangle$ for some $B \in \text{GL}(3,\mathbb{F}_q)$. Then $B^1 \otimes e_1 + B^2 \otimes e_2$ must be symmetric and of rank $2$, so $B^1 = \alpha_1 e_1 + \beta_1 e_2$ and $B^2 = \alpha_2 e_1 + \beta_2 e_2$, with $\alpha_2 = \beta_1$ and $\alpha_1 \beta_2 - \beta_1^2 = \alpha_1 \beta_2 - \alpha_2 \beta_1 \neq 0$. Writing $B^3 = \alpha_3 e_1 + \beta_3 e_2 + \gamma_3 e_3$, we then have $$B^1 \otimes e_2 + B^3 \otimes e_3 = \alpha_1 e_1 \otimes e_2 + \beta_1 e_2 \otimes e_2 + \alpha_3 e_1 \otimes e_3 + \beta_3 e_2 \otimes e_3 + \gamma_3 e_3 \otimes e_3,$$ which must also be symmetric and of rank $2$, forcing $\alpha_1 = \alpha_3 = \beta_3 = 0$. Hence, $$BM_{13} = \left[ \begin{matrix} \cdot & \beta_1 & \cdot \\ \beta_1 & \beta_2 & \cdot \\ \cdot & \cdot & \gamma_3 \end{matrix} \right] \left[ \begin{matrix} x & y & \cdot \\ \cdot & x & \cdot \\ \cdot & \cdot & y \end{matrix} \right]_{x,y} = \left[ \begin{matrix} \cdot & \beta_1 x & \cdot \\ \beta_1 x & \beta_2 x + \beta_1 y & \cdot \\ \cdot & \cdot & \gamma_3 y \end{matrix} \right]_{x,y},$$ or equivalently $$BM_{13} = \left[ \begin{matrix} \cdot & x & \cdot \\ x & \alpha x + y & \cdot \\ \cdot & \cdot & \gamma y \end{matrix} \right]_{x,y} \quad \text{for some} \quad \alpha \in \mathbb{F}_q, \gamma \in \mathbb{F}_q^{\times}.$$ First suppose that $q$ is odd. Then ${\mathrm{PG}}(BM_{13})$ is $K$-equivalent to $$L_\gamma = {\mathrm{PG}}\left( \left[ \begin{matrix} \cdot & x & \cdot \\ x & y & \cdot \\ \cdot & \cdot & \gamma y \end{matrix} \right]_{x,y} \right),$$ because $$\left[ \begin{matrix} \cdot & x & \cdot \\ x & y & \cdot \\ \cdot & \cdot & \gamma y \end{matrix} \right]_{x,y} = D \left[ \begin{matrix} \cdot & x & \cdot \\ x & \alpha x + y & \cdot \\ \cdot & \cdot & \gamma y \end{matrix} \right]_{x,y} D^\top, \quad \text{where} \quad D = \left[ \begin{matrix} 1 & \cdot & \cdot \\ -\frac{\alpha}{2} & 1 & \cdot \\ \cdot & \cdot & 1 \end{matrix} \right].$$ We claim that the lines $L_\gamma$ comprise two $K$-orbits, characterised by whether $\gamma \in \Box$ or not, as indicated in Table \[table:main\]. If $\gamma \in \Box$ then $L_\gamma$ is $K$-equivalent to $L_1$ because $L_1 = D L_\gamma D^\top$ for $D = \text{diag}(1,1,\delta^{-1})$ with $\delta^2=\gamma$. If $\gamma, \gamma' \not \in \Box$ then we may write $\gamma' = \gamma \mu^2$ for some $\mu \in \mathbb{F}_q^{\times}$, and so $L_{\gamma'} = D L_\gamma D^\top$ for $D = \text{diag}(1,1,\mu)$. It remains to show that if $\gamma \not \in \Box$ then $L_\gamma$ is not $K$-equivalent to $L_1$. To see this, first consider the line $L_\delta$ for an arbitrary $\delta \in \mathbb{F}_q^{\times}$, and let $P_\delta$ denote the rank-$2$ point on $L_\delta$ obtained by taking $x=0$. If $-\delta \in \Box$ then $P_\delta$ is an exterior point (see fact (F3) of Section \[V3props\]), and if $-\delta \not \in \Box$ then $P_\delta$ is an interior point. Hence, if $-\delta \in \Box$ and $-\delta' \not \in \Box$ for some $\delta' \in \mathbb{F}_q^{\times}$, then $L_\delta$ and $L_{\delta'}$ are not $K$-equivalent. We now use this observation to show that $L_1$ and $L_\gamma$ are not $K$-equivalent if $\gamma \not \in \Box$, by considering separately the cases where $-1 \in \Box$ and $-1 \not \in \Box$. If $-1 \in \Box$ then the point $P_1$ on $L_1$ is an exterior point, but $-\gamma \not \in \Box$ since $\gamma \not \in \Box$ and $-1 \in \Box$, so the point $P_\gamma$ on $L_\gamma$ is an interior point. Similarly, if $-1 \not \in \Box$ then $P_1$ is an interior point, but $-\gamma \in \Box$ since $\gamma \not \in \Box$ and $-1 \not \in \Box$, and hence $P_\gamma$ is an exterior point. Now suppose that $q$ is even. If $\alpha \neq 0$ then, because $\gamma \in \mathbb{F}_q^{\times}$ is a square, say $\gamma = \delta^2$, ${\mathrm{PG}}(BM_{13})$ is $K$-equivalent to ${\mathrm{PG}}(M)$ for $$M=\left[ \begin{matrix} \cdot & \alpha x & \cdot \\ \alpha x & \alpha x + y & \cdot \\ \cdot & \cdot & y \end{matrix} \right]_{x,y} = D \left[ \begin{matrix} \cdot & x & \cdot \\ x & \alpha x + y & \cdot \\ \cdot & \cdot & \gamma y \end{matrix} \right]_{x,y} D^\top, \quad \text{where} \quad D = \left[ \begin{matrix} \alpha & \cdot & \cdot \\ \cdot & 1 & \cdot \\ \cdot & \cdot & \delta^{-1} \end{matrix} \right].$$ Relabelling $\alpha x$ as $x$, it follows that ${\mathrm{PG}}(BM_{13})$ is $K$-equivalent to $$L = {\mathrm{PG}}\left( \left[ \begin{matrix} \cdot & x & \cdot \\ x & x + y & \cdot \\ \cdot & \cdot & y \end{matrix} \right]_{x,y} \right).$$ If, on the other hand, $\alpha=0$, then ${\mathrm{PG}}(BM_{13})$ is $K$-equivalent to $L_\gamma$ with $\gamma=1$. It remains to show that $L$ and $L_1$ are not $K$-equivalent. To see this, observe that $L$ does not intersect the nucleus plane, while $L_1$ intersects the nucleus plane in a point. Tensor orbit $o_{14}$ {#tensor-orbit-o_14 .unnumbered} --------------------- This tensor orbit has representative $$e_1 \otimes (e_1 \otimes e_1 + e_2 \otimes e_2) + e_2 \otimes (e_2 \otimes e_2 + e_3 \otimes e_3).$$ The first contraction space is $M_{14} = \langle e_1 \otimes e_1 + e_2 \otimes e_2 , e_2 \otimes e_2 + e_3 \otimes e_3 \rangle$, with rank distribution $[0,3,q-2]$. We see that ${\mathrm{PG}}(M_{14})$ is contained in $\langle \mathcal{V}_3(\mathbb{F}) \rangle$ (it is the representative in the second column of Table \[table:main\]), and that in order for ${\mathrm{PG}}(BM_{14})$ to be contained in $\langle \mathcal{V}_3(\mathbb{F}) \rangle$, we must have $B = \operatorname{diag}(\alpha,\beta,\gamma)$ for some $\alpha,\beta,\gamma \in \mathbb{F}_q^{\times}$. Hence, $$BM_{14} = \left[ \begin{matrix} \alpha x & \cdot & \cdot \\ \cdot & \beta x + \beta y & \cdot \\ \cdot & \cdot & \gamma y \end{matrix} \right]_{x,y}.$$ Relabelling, we see that ${\mathrm{PG}}(BM_{14})$ is equal to the line $$L_\gamma = {\mathrm{PG}}\left( \left[ \begin{matrix} x & \cdot & \cdot \\ \cdot & \gamma (x + y) & \cdot \\ \cdot & \cdot & y \end{matrix} \right]_{x,y} \right) \quad \text{for some} \quad \gamma \in \mathbb{F}_q^{\times}.$$ The rest of the argument is essentially the same as in the $o_{13}$ case. If $q$ is even then every line of the form $L_\gamma$ is $K$-equivalent to $L_1$ because every $\gamma \in \mathbb{F}_q^{\times}$ is a square; that is, $L_1 = D L_\gamma D^\top$ for $D=\text{diag}(1,\delta^{-1},1)$ with $\delta^2=\gamma$. If $q$ is odd then there are two $K$-orbits. Indeed, fixing some $\gamma \not \in \Box$, we find that ${\mathrm{PG}}(BM_{14})$ is $K$-equivalent to either $L_1$ or $L_\gamma$, and that $L_\gamma$ is $K$-equivalent to $L_{\gamma'}$ for every $\gamma' \not \in \Box$. Moreover, $L_1$ and $L_\gamma$ are not $K$-equivalent, because for an arbitrary $\delta \in \mathbb{F}_q^{\times}$, the rank-$2$ points on the line $L_\delta$ that correspond to $x=0$ and $y=0$, respectively, are exterior or interior points according to whether $-\delta \in \Box$ or not. Tensor orbit $o_{15}$ {#tensor-orbit-o_15 .unnumbered} --------------------- In this case the tensor orbit representative is $$e_1 \otimes (e_1 \otimes e_1 + e_2 \otimes e_2 + e_3 \otimes e_3 + ue_1 \otimes e_2) + e_2 \otimes (e_1 \otimes e_2 + ve_2 \otimes e_1),$$ where $v\lambda^2 + uv\lambda -1 \neq 0$ for all $\lambda \in \mathbb{F}_q$, namely condition ($$) in Tables \[pointTable\] and \[table:main\]. The first contraction space is $$M_{15} = \langle e_1 \otimes e_1 + e_2 \otimes e_2 + e_3 \otimes e_3 + ue_1 \otimes e_2 , e_1 \otimes e_2 + ve_2 \otimes e_1 \rangle,$$ with rank distribution $[0,1,q]$. Observe first that ${\mathrm{PG}}(B'M_{15})$ is contained in $\langle \mathcal{V}_3(\mathbb{F}) \rangle$ for $$B' = \left[ \begin{matrix} \cdot & 1 & \cdot \\ 1 & \cdot & \cdot \\ \cdot & \cdot & 1 \end{matrix} \right].$$ Let us therefore relabel $B'M_{15}$ as $M_{15}$, and also relabel $(u,v)$ as $(s,t)$, so that $$M_{15} = \left[ \begin{matrix} ty & x & \cdot \\ x & sx+y & \cdot \\ \cdot & \cdot & x \end{matrix} \right]_{x,y}.$$ Arguing as in previous cases, we find that in order for ${\mathrm{PG}}(BM_{15})$ to be contained in $\langle \mathcal{V}_3(\mathbb{F}) \rangle$, we must have $$B = \left[ \begin{matrix} \alpha & t\beta & \cdot \\ \beta & \alpha+st\beta & \cdot \\ \cdot & \cdot & \gamma \end{matrix} \right] \quad \text{for some } \alpha,\beta,\gamma \in \mathbb{F}_q.$$ In particular, $\gamma \neq 0$, and since the action is determined up to a non-zero scalar we may put $\gamma=1$. Hence, we may assume that $$B = \left[ \begin{matrix} \alpha & t\beta & \cdot \\ \beta & \alpha+st\beta & \cdot \\ \cdot & \cdot & 1 \end{matrix} \right],$$ which yields ${\mathrm{PG}}(BM_{15}) = {\mathrm{PG}}(M)$, where $$M = \left[ \begin{matrix} ty' & x' & \cdot \\ x' & sx'+y' & \cdot \\ \cdot & \cdot & x \end{matrix} \right]_{x,y} \quad \text{with} \quad x' = \beta ty + \alpha x + st \beta x \quad \text{and} \quad y' = \beta x + \alpha y.$$ For the sake of presentation, let us now formally state (and prove) the following claim. \[o15claim\] ${\mathrm{PG}}(BM_{15})$ is $K$-equivalent to the line $$L(u,v) = {\mathrm{PG}}\left( \left[ \begin{matrix} vy & x & \cdot \\ x & ux+y & \cdot \\ \cdot & \cdot & x \end{matrix} \right]_{x,y} \right)$$ for some $u,v \in \mathbb{F}_q$ satisfying condition $()$ in Tables \[pointTable\] and \[table:main\]. First suppose that $\alpha \neq 0$, and consider the matrix $$D_1 = \left[ \begin{matrix} 1 & \cdot & \cdot \\ -\beta\alpha^{-1} & 1 & \cdot \\ \cdot & \cdot & 1 \end{matrix} \right].$$ Then, with the vector subspace $M$ defined as above, we have $$D_1MD_1^\top = \left[ \begin{matrix} ty' & \delta' \alpha^{-1} x & \cdot \\ \delta' \alpha^{-1} x & \delta'\alpha^{-1}(\alpha^{-1}y'+(s-2\beta\alpha^{-1})x) & \cdot \\ \cdot & \cdot & x \end{matrix} \right]_{x,y}, \quad \text{where} \quad \delta' = \alpha^2 + \alpha\beta st - \beta^2t.$$ Since $t\lambda^2 + st\lambda - 1 \neq 0$ for all $\lambda \in \mathbb{F}_q$, we may take $\lambda=\beta\alpha^{-1}$ to verify that $\delta' \neq 0$, and hence we may write $\delta^{-1} = \delta'\alpha^{-1}$ for some $\delta \in \mathbb{F}_q^{\times}$. Setting $D_2 = \operatorname{diag}(1,\delta,1)$, we therefore have $$D_2(D_1MD_1^\top)D_2^\top = \left[ \begin{matrix} ty' & x & \cdot \\ x & \delta(\alpha^{-1}y'+(s-2\beta\alpha^{-1})x) & \cdot \\ \cdot & \cdot & x \end{matrix} \right]_{x,y}.$$ If we now write $u=\delta(s-2\beta\alpha^{-1})$ and $v=t\alpha\delta^{-1}$, and relabel $\delta \alpha^{-1} y'$ as $y$ (abusing notation in the sense that this is not the same $y$ as above), then we see that ${\mathrm{PG}}(BM_{15})={\mathrm{PG}}(M)$ is $K$-equivalent to $L(u,v)$. We also see that condition ($$) holds: each matrix with $x \neq 0$ must have rank $3$, so in particular if we set $x=1$ then the $(3,3)$ minor $vy^2+uvy-1$ must be non-zero for each $y \in \mathbb{F}_q$. Now suppose that $\alpha =0$ and $s \neq 0$. Then $$D_1MD_1^\top = \left[ \begin{matrix} s^2t\beta x & st\beta y & \cdot \\ st\beta y & \beta x - st\beta y & \cdot \\ \cdot & \cdot & x \end{matrix} \right]_{x,y}, \quad \text{where} \quad D_1 = \left[ \begin{matrix} s & \cdot & \cdot \\ -s & 1 & \cdot \\ \cdot & \cdot & 1 \end{matrix} \right].$$ Writing $\delta = s^2t$ and relabelling $st\beta y$ as $y$ then shows that ${\mathrm{PG}}(BM_{15})$ is $K$-equivalent to ${\mathrm{PG}}(M')$, where $$M' := \left[ \begin{matrix} \beta\delta x & y & \cdot \\ y & \beta x - y & \cdot \\ \cdot & \cdot & x \end{matrix} \right]_{x,y}.$$ Noting that $\beta \neq 0$ (because $B$ is non-singular) and taking $$D_2 = \left[ \begin{matrix} \cdot & 1 & \cdot \\ \beta^{-1} & \beta^{-1} & \cdot \\ \cdot & \cdot & 1 \end{matrix} \right],$$ we now find that $$D_2M'D_2^\top = \left[ \begin{matrix} -\beta^2 y' & x & \cdot \\ x & \beta^{-1}(\delta+2)x+y' & \cdot \\ \cdot & \cdot & x \end{matrix} \right]_{x,y}, \quad \text{where} \quad y':=\beta^2 y' + \beta x,$$ and relabelling $y'$ as $y$ shows that ${\mathrm{PG}}(BM_{15})$ is $K$-equivalent to $L(u,v)$ with $u=\beta^{-1}(\delta+2)$ and $v=-\beta^2$. Finally, suppose that $s=\alpha=0$. Then relabelling $\beta t y$ as $y$ gives $$M = \left[ \begin{matrix} t\beta x & y & \cdot \\ y & \beta x & \cdot \\ \cdot & \cdot & x \end{matrix} \right]_{x,y}.$$ Moreover, $q$ must be odd, because if $s=0$ then $t\lambda^2 \neq 1$ for all $\lambda \in \mathbb{F}_q$, and if $q$ is even then we can take $\lambda$ such that $\lambda^2=t^{-1}$ to yield a contradiction. Similarly, $t \neq 1$, because $t=1$ would imply that $\lambda^2 \neq 1$ for all $\lambda \in \mathbb{F}_q$. Now, we have $D_1MD_1^\top = M''$, where $$M'' = \left[ \begin{matrix} \beta(1+t) x - 2y & \beta(1-t)x & \cdot \\ \beta(1-t)x & \beta(1+t) x + 2y & \cdot \\ \cdot & \cdot & x \end{matrix} \right]_{x,y} \quad \text{and} \quad D_1 = \left[ \begin{matrix} -1 & 1 & \cdot \\ 1 & 1 & \cdot \\ \cdot & \cdot & 1 \end{matrix} \right].$$ Since $t \neq 1$ and $\beta \neq 0$, we may write $\delta^{-1} = \beta(1-t)$ for some $\delta \in \mathbb{F}_q$, so that $$D_2M''D_2^\top = \left[ \begin{matrix} \delta^2(\beta(1+t) x - 2y) & x & \cdot \\ x & \beta(1+t) x + 2y & \cdot \\ \cdot & \cdot & x \end{matrix} \right]_{x,y} \quad \text{for} \quad D_2 = \left[ \begin{matrix} \delta & \cdot & \cdot \\ \cdot & 1 & \cdot \\ \cdot & \cdot & 1 \end{matrix} \right].$$ Writing $y' = 2y-\beta(1+t)x$, we therefore see that ${\mathrm{PG}}(BM_{15})={\mathrm{PG}}(M)$ is $K$-equivalent to $${\mathrm{PG}}\left( \left[ \begin{matrix} -\delta^2y' & x & \cdot \\ x & 2\beta(1+t) x + y' & \cdot \\ \cdot & \cdot & x \end{matrix} \right]_{x,y} \right).$$ Relabelling $y'$ as $y$ shows that this is the line $L(u,v)$ with $v=-\delta^2$ and $u=2\beta(1+t)$. Let us now apply Claim \[o15claim\]. First suppose that $q$ is odd. Then there are at least two $K$-orbits of lines of the form $L(u,v)$, because the unique point of rank $2$ is either exterior or interior depending on whether $-v \in \Box$ or not. We claim that there are [exactly]{} two $K$-orbits, as per Table \[table:main\]. Consider two such lines $\langle P_2,P_3\rangle$ and $\langle P'_2,P'_3\rangle$, where $P_2$ and $P_2'$ are of rank $2$, $P_3$ and $P'_3$ are of rank $3$, and $P_2$ and $P'_2$ are either both exterior points or both interior points with respect to the unique conic plane $\pi$ in which they are contained (see facts (F1) and (F2) in Section \[V3props\]). We may assume that these two lines are $L(u,v)$ and $L(u',v')$, where either both $-v,-v' \in \Box$ or both $-v,-v' \not \in \Box$. One then sees that both of the points $P_3$ and $P'_3$ lie on a line through the point $P_1$ corresponding to $\langle e_3\otimes e_3\rangle$ and a point of rank $2$ in the conic plane $\pi$. In fact, the planes $\langle P_1,P'_2,P'_3\rangle$ and $\langle P_1,P_2,P_3\rangle$ are two planes on the point $P_1$ intersecting $\pi$ in an external line to the conic. Call these two external lines $L$ and $L'$. Since the stabiliser of $P_1$ and $\pi$ inside $K$ acts transitively on external lines to the conic in $\pi$ (compare with the case $o_{10}$), we may assume that $L=L'$. Since the stabiliser of an external line $M$ inside the group of a conic also acts transitively on both the set of interior points on $L$ and on the set of exterior points on $L$, we may also assume that $P_2=P_2'$, and therefore $v=v'$ (these points correspond to $x=0$). Restricting our coordinates to the conic plane $\pi$ corresponding to the top–left $2\times 2$ sub-matrix, and verifying that the point with coordinates $(0,1,u')$ lies on the line $L$ with equation $-X_0-uvX_1+vX_2=0$, we obtain $u=u'$. We conclude that $(u,v)=(u',v')$, proving the claim. If $q$ is even then there is only one $K$-orbit. The proof is essentially the same as in the $q$ odd case, except that now the stabiliser of an external line $L$ inside the group of a conic acts transitively on the points of $L$. Tensor orbit $o_{16}$ {#tensor-orbit-o_16 .unnumbered} --------------------- This tensor orbit has representative $$e_1 \otimes (e_1 \otimes e_1 + e_2 \otimes e_2 + e_3 \otimes e_3) + e_2 \otimes (e_1 \otimes e_2 + e_2 \otimes e_3).$$ The first contraction space is $M_{16} = \langle e_1 \otimes e_1 + e_2 \otimes e_2 + e_3 \otimes e_3, e_1 \otimes e_2 + e_2 \otimes e_3 \rangle$, with rank distribution $[0,1,q]$. The line ${\mathrm{PG}}(BM_{16})$ is contained in $\langle \mathcal{V}_3(\mathbb{F}) \rangle$ if and only if $$B = \left[ \begin{matrix} \cdot & \cdot & \alpha \\ \cdot & \alpha & \beta \\ \alpha & \beta & \gamma \end{matrix} \right]$$ for some $\alpha,\beta,\gamma \in {\mathbb{F}}_q$ with $\alpha \neq 0$, and this yields $$BM_{16} = \left[ \begin{matrix} \cdot & \cdot & \alpha x \\ \cdot & \alpha x & \beta x + \alpha y \\ \alpha x & \beta x + \alpha y & \gamma x + \beta y \end{matrix} \right]_{x,y}.$$ After a suitable relabelling, we may write $$BM_{16} = \left[ \begin{matrix} \cdot & \cdot & x \\ \cdot & x & y \\ x & y & \alpha x + \beta y \end{matrix} \right]_{x,y}.$$ If $q$ is odd then ${\mathrm{PG}}(BM_{16})$ is $K$-equivalent to the representative $$L_1 = {\mathrm{PG}}\left( \left[ \begin{matrix} \cdot & \cdot & x \\ \cdot & x & y \\ x & y & \cdot \end{matrix} \right]_{x,y} \right)$$ given in Table \[table:main\], because $$\left[ \begin{matrix} \cdot & \cdot & x \\ \cdot & x & y \\ x & y & \cdot \end{matrix} \right]_{x,y} = D \left[ \begin{matrix} \cdot & \cdot & x \\ \cdot & x & y \\ x & y & \alpha x + \beta y \end{matrix} \right]_{x,y} D^\top, \quad \text{where} \quad D = \left[ \begin{matrix} 1 & \cdot & \cdot \\ \tfrac{\beta}{2} & 1 & \cdot \\ -\frac{1}{2}\left(\alpha+\frac{\beta^2}{4}\right) & -\frac{\beta}{2} & 1 \end{matrix} \right].$$ Now suppose that $q$ is even. If $\beta=0$ then, because $\alpha \in \Box$, ${\mathrm{PG}}(BM_{16})$ is $K$-equivalent to $$L_1 = {\mathrm{PG}}\left( \left[ \begin{matrix} \cdot & \cdot & x \\ \cdot & x & y \\ x & y & \cdot \end{matrix} \right]_{x,y} \right).$$ We now claim that if $\beta \neq 0$, then ${\mathrm{PG}}(BM_{16})$ is $K$-equivalent to $$L_2 = {\mathrm{PG}}\left( \left[ \begin{matrix} \cdot & \cdot & x \\ \cdot & x & y \\ x & y & y \end{matrix} \right]_{x,y} \right).$$ To see this, first observe that $$D \left[ \begin{matrix} \cdot & \cdot & x \\ \cdot & x & y \\ x & y & \alpha x + \beta y \end{matrix} \right]_{x,y} D^\top = \left[ \begin{matrix} \cdot & \cdot & \beta^2 x \\ \cdot & \beta^2 x & \beta y \\ \beta^2 x & \beta y & \alpha x + \beta y \end{matrix} \right]_{x,y}, \quad \text{where} \quad D = \left[ \begin{matrix} \beta^2 & \cdot & \cdot \\ \cdot & \beta & \cdot \\ \cdot & \cdot & 1 \end{matrix} \right],$$ and then relabel to obtain $$BM_{16} = \left[ \begin{matrix} \cdot & \cdot & x \\ \cdot & x & y \\ x & y & \delta x + y \end{matrix} \right]_{x,y}, \quad \text{where} \quad \delta=\alpha\beta^{-2}.$$ If $\delta=0$ then the projective space obtained from the right-hand side above is equal to $L_2$, so assume now that $\delta \neq 0$. Since $BM_{16}$ is spanned by $$P = \left[ \begin{matrix} \cdot & \cdot & 1 \\ \cdot & 1 & \cdot \\ 1 & \cdot & \delta \end{matrix} \right] \quad \text{and} \quad Q = \left[ \begin{matrix} \cdot & \cdot & \cdot \\ \cdot & \cdot & 1 \\ \cdot & 1 & 1 \end{matrix} \right],$$ it is also spanned by $Q$ and $$P + \delta Q = \left[ \begin{matrix} \cdot & \cdot & 1 \\ \cdot & 1 & \delta \\ 1 & \delta & \cdot \end{matrix} \right].$$ Therefore, $$BM_{16} = \left[ \begin{matrix} \cdot & \cdot & x \\ \cdot & x & \delta x + y \\ x & \delta x + y & y \end{matrix} \right]_{x,y},$$ and so ${\mathrm{PG}}(BM_{16})$ is $K$-equivalent to $L_2$ because $$D \left[ \begin{matrix} \cdot & \cdot & x \\ \cdot & x & \delta x + y \\ x & \delta x + y & y \end{matrix} \right]_{x,y} D^\top = \left[ \begin{matrix} \cdot & \cdot & x \\ \cdot & x & y \\ x & y & y \end{matrix} \right]_{x,y}, \quad \text{where} \quad D = \left[ \begin{matrix} 1 & \cdot & \cdot \\ -\delta & 1 & \cdot \\ \cdot & \cdot & 1 \end{matrix} \right].$$ This completes the proof of the claim. It remains to show that $L_1$ and $L_2$ are not $K$-equivalent. To see this, observe that $L_1$ contains the rank-$2$ point corresponding to the matrix $$\left[ \begin{matrix} \cdot & \cdot & \cdot \\ \cdot & \cdot & 1 \\ \cdot & 1 & \cdot \end{matrix} \right],$$ and $L_2$ contains the rank-$2$ point corresponding to the matrix $$\left[ \begin{matrix} \cdot & \cdot & \cdot \\ \cdot & \cdot & 1 \\ \cdot & 1 & 1 \end{matrix} \right].$$ Since the first point lies in the nucleus plane of ${\mathcal{V}}({\mathbb{F}}_q)$ but the second does not, $L_1$ and $L_2$ represent distinct $K$-orbits (when $q$ is even). Tensor orbit $o_{17}$ {#tensor-orbit-o_17 .unnumbered} --------------------- The lines corresponding to the tensor orbit $o_{17}$ are constant rank-3 lines, that is, they have rank distribution $[0,0,q+1]$. We show that there is a unique $K$-orbit of such lines (see Proposition \[prop3.5\]). Throughout the proof, we refer to some results proved in Section \[sec:stabs\]; we remark that the arguments used to prove those results do not in turn depend on any of the arguments given here. In particular, we need the following lemma, which counts the number of symmetric representatives of the line orbits arising from $o_{13}$ and $o_{14}$. \[lem:nrs\] The total numbers of symmetric representatives of the line orbits corresponding to the tensor orbits $o_{13}$ and $o_{14}$ are, respectively, $\frac{\|K\|}{q-1}$ and $\frac{\|K\|}{6}$. See the arguments for $o_{13}$ and $o_{14}$ in Section \[sec:stabs\], where (in particular) the stabilisers inside $K$ of lines arising from these tensor orbits are determined. Now, the tensor orbit $o_{17}$ has representative $$e_1 \otimes (e_1 \otimes e_1 + e_2 \otimes e_2 + e_3 \otimes e_3) + e_2 \otimes (e_1 \otimes e_2 + e_2 \otimes e_3 + e_3 \otimes (\alpha e_1 + \beta e_2 + \gamma e_3)),$$ where $\lambda^3 + \gamma \lambda^2 - \beta \lambda + \alpha \neq 0$ for all $\lambda \in \mathbb{F}_q$, namely condition ($)$ in Tables \[pointTable\] and \[table:main\]. Note that $\alpha \neq 0$, for otherwise taking $\lambda=0$ would violate condition ($$). The first contraction space is $$M_{17} = \left[ \begin{matrix} x & y & \cdot \\ \cdot & x & y \\ \alpha y & \beta y & x + \gamma y \end{matrix} \right]_{x,y},$$ with rank distribution $[0,0,q+1]$ (as noted above). Setting $$B = \left[ \begin{matrix} \alpha & \cdot & \cdot \\ \cdot & -\gamma & 1 \\ \cdot & 1 & \cdot \end{matrix} \right]$$ gives $$BM_{17} = \left[ \begin{matrix} \alpha^{-1}x & y & \cdot \\ y & \beta y - \gamma x & x \\ \cdot & x & y \end{matrix} \right]_{x,y},$$ so that we can take ${\mathrm{PG}}(BM_{17})$ as a $K$-orbit representative of lines in $\langle \mathcal{V}_3(\mathbb{F}_q) \rangle$, as in Table \[table:main\]. We now show that there is only one orbit. \[lem:1\] The number of constant rank-3 lines in $\langle {\mathcal{V}}_3({\mathbb{F}}_q)\rangle$ is $\frac{\|K\|}{3}$. Assume that $q>2$, and recall that $\|K\|=\|{\mathrm{PGL}}(3,{\mathbb{F}}_q)\|=q^3(q^3-1)(q^2-1)$. Fix a point $P$ of rank 3. Through $P$ there are $q^2+q+1$ lines that contain exactly one point of ${\mathcal{V}}_3({\mathbb{F}}_q)$. According to Table \[pointTable\], the remaining lines through $P$ have rank distributions $[0,i,q+1-i]$ for $i\in \{0,1,2,3\}$. Let $n_0,\ldots,n_3$ denote the corresponding numbers of such lines through $P$. Then $$\label{eqn:1} \sum_{i=0}^3 n_i=\frac{q^5-1}{q-1}-(q^2+q+1),$$ and counting pairs $(R,L)$ where $R$ is a point of rank 2 and $L=\langle P,R\rangle$ is a line disjoint from ${\mathcal{V}}_3({\mathbb{F}}_q)$, we obtain $$\label{eqn:2} \sum_{i=1}^3 in_i=q^2(q^2+q),$$ since by Lemma \[lem:counting\] there are $q^2$ points of rank 2 on the lines through $P$ and a point of ${\mathcal{V}}_3({\mathbb{F}}_q)$. The lines with rank distribution $[0,2,q-1]$ are symmetric representatives of lines arising from the tensor orbit $o_{13}$. By Lemma \[lem:nrs\], there are $\frac{\|K\|}{q-1}$ such representatives. Let $\mathcal{P}_3$ denote the set of rank-3 points in $\langle {\mathcal{V}}_3({\mathbb{F}}_q) \rangle$. Then $\|\mathcal{P}_3\|=q^5-q^2$, as noted in Section \[V3props\]. Since $K$ acts transitively on $\mathcal{P}_3$, we have $$\frac{\|{\mathcal{P}}_3\| \cdot n_2}{q-1}=\frac{\|K\|}{q-1},$$ which implies that $n_2=q^3-q$. The lines with rank distribution $[0,3,q-2]$ are symmetric representatives of lines arising from the tensor orbit $o_{14}$. By Lemma \[lem:nrs\], there are $\frac{\|K\|}{6}$ such representatives. Since we are assuming that $q \neq 2$, we have $$\frac{\|{\mathcal{P}}_3\|\cdot n_3}{q-2}=\frac{\|K\|}{6},$$ which implies that $n_3=\tfrac{1}{6}(q^3-q)(q-2)$. Substituting the expressions for $n_2$ and $n_3$ into (\[eqn:1\]) and (\[eqn:2\]), we obtain $n_1 = \tfrac{1}{2}(q^4+q^2+2q)$ and hence $n_0 = \tfrac{1}{3}(q^4+q^3-q^2-q)$. The total number of constant rank-3 lines is therefore $$\frac{\|{\mathcal{P}}_3\|\cdot n_0}{q+1}=\frac{q^3(q^3-1)(q^2-1)}{3} = \frac{\|K\|}{3}.$$ If $q=2$ then the same proof works except that now there are no symmetric representatives of lines corresponding to the tensor orbit $o_{14}$ passing through $P$; that is, $n_3=0$ in this case. In Proposition \[lem:2\], we determine the stabiliser inside $K$ of a symmetric representative of a line orbit arising from the tensor orbit $o_{17}$. It turns out that this stabiliser has order $3$. Together with Lemma \[lem:1\], this implies the following result. \[prop3.5\] There is a unique $K$-orbit of constant rank-3 lines in $\langle {\mathcal{V}}_3({\mathbb{F}}_q)\rangle$. Immediate from Lemma \[lem:1\] and Proposition \[lem:2\]. This concludes the classification of the $K$-orbits of the symmetric representatives of lines arising from the tensor orbits $o_4, \ldots, o_{17}$. The results are summarised in the Table \[table:main\]. The number of $K$-line orbits is $15$ for all finite fields ${\mathbb{F}}_q$. Three tensor orbits, namely $o_4$, $o_7$ and $o_{11}$, do not yield a symmetric line orbit representative, and so these are omitted from the table. Four orbits split under the action of $K$ for $q$ even but not for $q$ odd, two orbits split for $q$ odd but not for $q$ even, and two orbits split for both $q$ even and odd. Line stabilisers for ${\mathbb{F}}$ a finite field {#sec:stabs} ================================================== In this section we compute the stabilisers of each of the $K$-orbits of lines in $\langle {\mathcal{V}}_3({\mathbb{F}}) \rangle$ determined in Section \[sec:orbits\]. As in that section, we assume here that ${\mathbb{F}}$ is a finite field ${\mathbb{F}}_q$. The line stabilisers are shown in Table \[table:stabs\]. The following (common) notation is used: $E_q$ is an elementary abelian group of order $q$, $C_k$ is a cyclic group of order $k$, and $\text{Sym}_n$ is the symmetric group on $n$ letters. Moreover, $A \times B$ denotes the direct product of groups $A$ and $B$, while $A:B$ denotes a split extension of $A$ by $B$, with normal subgroup $A$ and subgroup $B$. The corresponding total numbers of symmetric line-orbit representatives are readily deduced, and recorded in Table \[table:orbitsLengths\]. For the cases in which a line contains points of rank 2, we also determine for the $q$ odd case how many rank 2 points are exterior (or interior) points, and for the $q$ even case we determine how many rank 2 points lie in the nucleus plane. We remark that in some cases this information has already been obtained as part of the arguments in Section \[sec:orbits\]. ---------- ---------------------------------------------------------------------- ---------------------------------------------------------------------- ---------------------------------------------- Tensor orbit Common orbit (all $q$) $q$ odd $q$ even $o_5$ $E_q^2 : C_{q-1}^2 : C_2$ $o_6$ $E_q^{1+2}:C_{q-1}^2$ $o_8$ $C_{q-1} \times \text{O}^\pm(2,{\mathbb{F}}_q)$, $q \equiv \pm 1(4)$ $C_{q-1} \times \text{O}^\mp(2,{\mathbb{F}}_q)$, $q \equiv \pm 1(4)$ $E_q \times C_{q-1}$, $q$ even $C_{q-1} \times \text{SL}(2,{\mathbb{F}}_q)$ $o_9$ $E_q^2:C_{q-1}$ $o_{10}$ $E_q^2:\text{O}^-(2,{\mathbb{F}}_q)$ $o_{12}$ $\text{GL}(2,{\mathbb{F}}_q)$, $q$ odd $E_q^2:\text{GL}(2,{\mathbb{F}}_q)$, $q$ even $E_q^2:E_q:C_{q-1}$ $o_{13}$ $C_{q-1} \times C_2$, $q$ odd $C_{q-1} \times C_2$ $E_q:C_{q-1}$, $q$ even $E_q$ $o_{14}$ $C_2^2 : \text{Sym}_3$, $q\equiv 1 (4)$ $C_2^2 : C_2$, $q\equiv 1 (4)$ $C_2^2 : C_2$, $q\equiv 3 (4)$ $C_2^2 : \text{Sym}_3$, $q\equiv 3 (4)$ $\text{Sym}_3$, $q$ even $o_{15}$ $C_2^2$, $q$ odd $C_2^2$ $C_2$, $q$ even $o_{16}$ $E_q : C_{q-1}$, $q$ odd $E_q^2 : C_{q-1}$, $q$ even $E_q^2$ $o_{17}$ $C_3$ ---------- ---------------------------------------------------------------------- ---------------------------------------------------------------------- ---------------------------------------------- : The stabilisers of line orbits in $\langle \mathcal{V}_3(\mathbb{F}_q) \rangle$ under $K={\mathrm{PGL}}(3,{\mathbb{F}}_q)$, and the corresponding total numbers of symmetric representatives of lines arising from the associated tensor orbits. The layout of the table is consistent with that of Table \[table:main\]; that is, the groups in each column are the stabilisers of the orbit representatives shown in the corresponding column of Table \[table:main\]. For brevity, we write $q \equiv \pm 1(4)$ to mean $q \equiv \pm 1 \pmod 4$.[]{data-label="table:stabs"} ---------- -------------------------------- ------------------- ------------------------ -------------------------- Tensor $\sharp$ symmetric line-orbit $\sharp$ rank-$2$ $\sharp$ exterior $\sharp$ points in orbit representatives points points nucleus plane $o_5$ $\frac{1}{2}q(q+1)(q^2+q+1)$ $q-1$ $\tfrac{q-1}{2}$ $0$ $o_6$ $(q+1)(q^2+q+1)$ $q$ $q$ $q$ $o_8$ $q^4(q^2+q+1)$ $1$ $0$ or $1$ ($\dagger$) $0$ or $1$ ($\dagger$) $o_9$ $q(q^3-1)(q+1)$ $0$ – – $o_{10}$ $\frac{1}{2}q(q^3-1)$ $q+1$ $\tfrac{q+1}{2}$ $0$ $o_{12}$ $q^2(q^2+q+1)$ $q+1$ $q+1$ $1$ or $q+1$ ($\dagger$) $o_{13}$ $q^3(q^3-1)(q+1)$ $2$ $1$ or $2$ ($\dagger$) $0$ or $1$ ($\dagger$) $o_{14}$ $\frac{1}{6}q^3(q^3-1)(q^2-1)$ $3$ $1$ or $3$ ($\dagger$) $0$ $o_{15}$ $\frac{1}{2}q^3(q^3-1)(q^2-1)$ $1$ $0$ or $1$ ($\dagger$) $0$ $o_{16}$ $q^2(q^3-1)(q+1)$ $1$ $1$ $0$ or $1$ ($\dagger$) $o_{17}$ $\frac{1}{3}q^3(q^3-1)(q^2-1)$ $0$ – – ---------- -------------------------------- ------------------- ------------------------ -------------------------- : The total number of line orbit representatives in $\langle {\mathcal{V}}_3({\mathbb{F}}_q) \rangle$ corresponding to each tensor orbit in ${\mathbb{F}}_q^2 \otimes {\mathbb{F}}_q^3 \otimes {\mathbb{F}}_q^3$. Also shown is the total number of rank-$2$ points on each line (third column), and the total number of these that are exterior points (for $q$ odd, fourth column) or lie in the nucleus plane of ${\mathcal{V}}_3({\mathbb{F}}_q)$ (for $q$ even, fifth column). In some cases, indicated by ($\dagger$), the data in the fourth and/or fifth columns depends on the orbit and, for $q$ odd, (possibly) the parity of $q$ modulo $4$; in these cases, we refer the reader to the text for full details.[]{data-label="table:orbitsLengths"} Tensor orbit $o_5$ {#tensor-orbit-o_5-1 .unnumbered} ------------------ Here there is a single $K$-orbit, represented by the line $$L = {\mathrm{PG}}\left( \left[ \begin{matrix} x & \cdot & \cdot \\ \cdot & y & \cdot \\ \cdot & \cdot & \cdot \end{matrix} \right]_{x,y} \right),$$ which has rank distribution $[2,q-1,0]$. The two points of rank $1$ on $L$ determine a conic in ${\mathcal{V}}_3({\mathbb{F}}_3)$, and the stabiliser of $L$ must either fix or swap these points. In the preimage under the Veronese map $\nu_3$, this corresponds to the setwise stabiliser of two points, inside the stabiliser of a line of $\text{PG}(2,{\mathbb{F}}_q)$ in $\text{PGL}(3,{\mathbb{F}}_q)$. This group is isomorphic to $E_q^2:C_{q-1}^2:C_2$ (as shown in the second column of Table \[table:stabs\]), and so there are $$\frac{\|K\|}{2q^2(q-1)^2} = \frac{q^3(q^3-1)(q^2-1)}{2q^2(q-1)^2} = \frac{1}{2}q(q+1)(q^2+q+1)$$ lines in this $K$-line orbit (as shown in Table \[table:orbitsLengths\]). If $q$ is odd, then the rank-$2$ points on $L$ comprise $\tfrac{q-1}{2}$ exterior points and $\tfrac{q-1}{2}$ interior points; if $q$ is even, then all rank-$2$ points on $L$ lie outside the nucleus plane (see Table \[table:orbitsLengths\]). Tensor orbit $o_6$ {#tensor-orbit-o_6-1 .unnumbered} ------------------ Here there is a single $K$-orbit, represented by the line $$L = {\mathrm{PG}}\left( \left[ \begin{matrix} x & y & \cdot \\ y & \cdot & \cdot \\ \cdot & \cdot & \cdot \end{matrix} \right]_{x,y} \right),$$ which has rank distribution $[1,q,0]$. If we consider the unique point $P$ of rank $1$ on $L$, and any other point on $L$, then these two points determine a conic ${\mathcal{C}}$. The stabiliser of $L$ in $K$ is isomorphic to the stabiliser of the flag $(\nu_3^{-1}(P),\nu_3^{-1}({\mathcal{C}}))$ in $\text{PG}(2,{\mathbb{F}}_q)$. This group is isomorphic to $E_q^{1+2}:C_{q-1}^2$, where the group $E_q^{1+2}$ has centre $Z \cong E_q$ and $E_q^{1+2}/Z \cong E_q^2$. In particular, there are $(q+1)(q^2+q+1)$ lines in this orbit. Since $L$ is a tangent of ${\mathcal{C}}(P)$, every point on $L$ different from $P$ is an exterior point for $q$ odd. If $q$ is even, then every rank-$2$ point on $L$ lies in the nucleus plane. Tensor orbit $o_8$ {#tensor-orbit-o_8-1 .unnumbered} ------------------ Lines arising from the tensor orbit $o_{8}$ have rank distribution $[1,1,q-1]$. For $q$ odd, the unique point of rank $2$ can be an internal point or an external point to the unique conic that it determines. The stabiliser of an external point (respectively, internal point) inside the group of a non-degenerate conic has size $2(q-1)$ (respectively, $2(q+1)$). Let $L$ be a symmetric representative of the tensor orbit $o_8$, let $P_1$ be the point of rank $1$ on $L$, and let $P_2$ be the point of rank $2$. The stabiliser of $P_1$ inside the pointwise stabiliser of the conic plane $\pi=\langle {\mathcal{C}}(P_2)\rangle$ in $K$ has size $q-1$, as it corresponds to the group of homologies with common center and axis in ${\mathrm{PG}}(2,{\mathbb{F}}_q)$. The stabiliser of $L$ is therefore isomorphic to one of $C_{q-1} \times \text{O}^\pm(2,{\mathbb{F}}_q)$, and has size $2(q-1)(q\mp 1)$. Hence, the total number of lines in the two $K$-orbits arising from the tensor orbit $o_8$ is $$\frac{\|K\|}{2(q+1)(q-1)}+\frac{\|K\|}{2(q-1)^2}=q^4(q^2+q+1).$$ For $q$ even, the two orbits are characterised by whether or not the unique point of rank $2$ is the nucleus of the unique conic determined by it. The point of rank $2$ corresponding to $(x,y)=(0,1)$ in the orbit representative $L$ in the fourth column of Table \[table:main\] is the nucleus $N$ of the unique conic ${\mathcal{C}}$ determined by $N$. The stabiliser $K_L$ is therefore equal to the stabiliser of ${\mathcal{C}}$ and the unique point $R$ of rank $1$ on $L$. Note that $R$ is not on ${\mathcal{C}}$. In the preimage under the Veronese map, this corresponds to the stabiliser of an anti-flag. Hence, $K_L$ is isomorphic to $C_{q-1} \times \text{SL}(2,{\mathbb{F}}_q)$ and has size $$q(q^2-1)(q-1).$$ The other orbit is represented by the line $L_1$ in the second column of Table \[table:main\]. The unique point $P$ of rank $2$ on $L_1$ is not the nucleus of the conic ${\mathcal{C}}(P)$ that it determines, and therefore $K_{L_1}$ is equal to the stabiliser of ${\mathcal{C}}(P)$, the unique point of rank $1$ on $L$, and the point $Q$ on ${\mathcal{C}}(P)$ obtained by intersecting ${\mathcal{C}}(P)$ with the unique tangent to ${\mathcal{C}}(P)$ through $P$. If an element of $K_{L_1}$ fixes a point $Q'$ on ${\mathcal{C}}(P)\setminus\{Q\}$, then it fixes the intersection of the line through $Q'$ and $P$ with ${\mathcal{C}}(P)$, and therefore fixes ${\mathcal{C}}(P)$ pointwise. Since the pointwise stabiliser of ${\mathcal{C}}(P)$ inside $K_{L_1}$ corresponds to the group of perspectivities with centre not on the axis in the preimage under the Veronese map, it has size $q-1$. This implies that $K_{L_1}$ has size $q(q-1)$. Specifically, $K_{L_1} \cong E_q \times C_{q-1}$. The total number of lines in these two orbits is therefore $$\frac{\|K\|}{q(q-1)}+\frac{\|K\|}{q(q^2-1)(q-1)}=q^4(q^2+q+1),$$ as in the $q$ odd case. Tensor orbit $o_9$ {#tensor-orbit-o_9-1 .unnumbered} ------------------ Here there is a unique $K$-orbit, represented by the line $$L = {\mathrm{PG}}(\left( \left[ \begin{matrix} x & \cdot & y \\ \cdot & y & \cdot \\ y & \cdot & \cdot \end{matrix} \right]_{x,y} \right),$$ which has rank distribution $[1,0,q]$. Before we determine the stabiliser of $L$ in $K$, let us introduce some terminology and prove two lemmas. \[lem:counting\] A point of rank $3$ in $\langle {\mathcal{V}}_3({\mathbb{F}})\rangle$ lies on $q^2$ lines with rank distribution $[1,1,q-1]$ and on $q+1$ lines with rank distribution $[1,0,q]$. It follows from above treatment of the $o_8$ orbit that there are in total $q^4(q^2+q+1)$ lines with rank distribution $[1,1,q-1]$. Since there is just one orbit $\mathcal{P}_3$ of points of rank 3, each such point is on the same number, say $k$, of such lines. Counting pairs $(P,\ell)$ where $P$ is a point of rank $3$ and $\ell$ is a line with rank distribution $[1,1,q-1]$ containing $P$, we obtain $$\|{\mathcal{P}}_3\|\cdot k = q^4(q^2+q+1)(q-1),$$ which implies that $k=q+1$. The result follows because ${\mathcal{V}}_3({\mathbb{F}}_q)$ contains $q^2+q+1$ points and a line through a rank-3 point contains at most one point of ${\mathcal{V}}_3({\mathbb{F}}_q)$. Given a point $P$ of rank 3 in $\langle {\mathcal{V}}_3({\mathbb{F}})\rangle$, we denote by ${\mathcal{N}}(P)$ the set of points in ${\mathcal{V}}_3({\mathbb{F}})$ that together with $P$ span a line without rank-2 points. It follows from Lemma \[lem:counting\] that $\|{\mathcal{N}}(P)\|=q+1$. In the next lemma, we show that ${\mathcal{N}}(P)$ is a normal rational curve (NRC). The definition and properties of NRCs may be found in, for instance, [@Harris p. 10]. \[lem:nrc1\] If $P$ is a point of rank $3$ in $\langle {\mathcal{V}}_3({\mathbb{F}})\rangle$, then the set ${\mathcal{N}}(P)$ is a NRC of degree $4$ if $q$ is odd, and a conic if $q$ is even. Since $K$ acts transitively on rank-3 points, we may assume without loss of generality that $P$ corresponds to the matrix with ones on the anti-diagonal and zeroes everywhere else. Suppose first that $q$ is odd, and consider the set $S_P$ of images under the Veronese map of the points on the conic $X_0X_2+\frac{1}{2}X_1^2=0$ in ${\mathrm{PG}}(2,{\mathbb{F}}_q)$. Then $S_P$ is a NRC of degree 4, since it is the image of a non-degenerate conic (see fact (F4) of Section \[V3props\]). A straightforward calculation shows that each of the lines spanned by $P$ and a point of $S_P$ has rank distribution $[1,0,q]$; that is, ${\mathcal{N}}(P)=S_P$ is a NRC of degree 4. For $q$ even, consider the set $S_P$ of images under the Veronese map of the points on the line $X_1=0$. Then $S_P$ is a conic, and a straightforward calculation shows that each of the lines spanned by $P$ and a point of $S_P$ has rank distribution $[1,0,q]$; that is, ${\mathcal{N}}(P)=S_P$ is a conic. Now consider the line $L$ (as defined above), and let $R$ be the unique point of rank 1 on $L$. Suppose that $q$ is odd. The stabiliser of $R$ inside $K_L$ acts transitively on the points of rank 3 on $L$, since each point of rank 3 determines a NRC through $R$, each of these NRCs is the image of a non-degenerate conic (see fact (F4) of Section \[V3props\]) passing through $\nu_3^{-1}(R)$, and the stabiliser of $\nu_3^{-1}(R)$ inside ${\mathrm{PGL}}(3,{\mathbb{F}}_q)$ acts transitively on the non-degenerate conics containing $\nu_3^{-1}(R)$. Next, consider a point $P$ of rank 3 on $L$. Lemma \[lem:nrc1\] implies that the stabiliser of $P$ in $K$ equals the stabiliser of ${\mathcal{N}}(P)$ in $K$, which is isomorphic to ${\mathrm{PGL}}(2,{\mathbb{F}}_q)$, since ${\mathcal{N}}(P)$ is the image of a non-degenerate conic under the Veronese map. The stabiliser of $P$ inside $K_L$ fixes $P$ and $R=L\cap{\mathcal{N}}(P)$, and is therefore isomorphic to the stabiliser in ${\mathrm{PGL}}(2,{\mathbb{F}}_q)$ of a point in ${\mathrm{PG}}(1,{\mathbb{F}}_q)$, which is $E_q:C_{q-1}$. Explicitly, we have $DLD^\top=L$ if and only if $D \in \text{GL}(3,q)$ has the form $$D = \left[ \begin{matrix} d_{11} & d_{12} & d_{13} \\ \cdot & d_{22} & -d_{11}^{-1}d_{12}d_{22} \\ \cdot & \cdot & d_{11}^{-1}d_{22}^2 \end{matrix} \right].$$ The stabiliser of $L$ in $K$ therefore has order $$\frac{q^2(q-1)^2}{q-1} = q^2(q-1),$$ and is isomorphic to $E_q^2:C_{q-1}$. Indeed, the same is true for $q$ even, as can be seen from the explicit form of $D$ given above. In particular, the $K$-orbit of $L$ has size $q(q^3-1)(q+1)$ both for $q$ even and for $q$ odd. Tensor orbit $o_{10}$ {#tensor-orbit-o_10-1 .unnumbered} --------------------- A line $L$ in the orbit $o_{10}$ is a line in a conic plane $\pi$ disjoint from the conic ${\mathcal{C}}$ consisting of the rank-$1$ points in $\pi$. It follows that $\pi$ and ${\mathcal{C}}$ are fixed by $K_L$. The pointwise stabiliser inside $K$ of the plane $\pi$ corresponds to the pointwise stabiliser of a line in the projectivity group of ${\mathrm{PG}}(2,{\mathbb{F}}_q)$, and has size $q^2(q-1)$. The stabiliser in $K_\pi$ of an external line $L$ to a conic ${\mathcal{C}}$ has size $2(q+1)$. We therefore have $K_L \cong E_q^2:\text{O}^-(2,q)$. In particular, $\|K_L\|=2q^2(q-1)(q+1)$, and so the number of symmetric representatives of lines arising from the tensor orbit $o_{10}$ is $\tfrac{1}{2}q(q^3-1)$. Moreover, there are $q+1$ tangents to the conic ${\mathcal{C}}$, and every tangent meets the external line $L$. Since every point is on zero or two tangents, there are $\tfrac{q+1}{2}$ exterior points on $L$ if $q$ is odd (and the other $\tfrac{q+1}{2}$ points on $L$ are interior points). If $q$ is even, then all points on $L$ lie outside of the nucleus plane. Tensor orbit $o_{12}$ {#tensor-orbit-o_12-1 .unnumbered} --------------------- Here, for every $q$, there is a $K$-orbit represented by the line $$L = {\mathrm{PG}}(M) \quad \text{where} \quad M = \left[ \begin{matrix} \cdot & x & \cdot \\ x & \cdot & y \\ \cdot & y & \cdot \end{matrix} \right]_{x,y},$$ which has rank distribution $[0,q+1,0]$. If $q$ is odd then we have $DMD^\top=M$ for $D \in \text{GL}(3,{\mathbb{F}}_q)$ is and only if $D$ has the form $$D = \left[ \begin{matrix} d_{11} & \cdot & d_{13} \\ \cdot & d_{22} & \cdot \\ d_{31} & \cdot & d_{33} \end{matrix} \right].$$ Modulo scalars, these matrices comprise a subgroup of $K = \text{PGL}(3,{\mathbb{F}}_q)$ isomorphic to $\text{GL}(2,{\mathbb{F}}_q)$, of order $q(q^2-1)(q-1)$, and so there are $q^2(q^2+q+1)$ lines in this $K$-orbit when $q$ is odd. If $q$ is even then $DMD^\top=M$ if and only if $$D = \left[ \begin{matrix} d_{11} & \cdot & d_{13} \\ d_{21} & d_{22} & d_{23} \\ d_{31} & \cdot & d_{33} \end{matrix} \right].$$ Here there is no restriction on $d_{21}$ or $d_{23}$, so the stabiliser of $L$ in $K$ is isomorphic to $E_q^2:\text{GL}(2,{\mathbb{F}}_q)$ and has order $q^2\|\text{GL}(2,{\mathbb{F}}_q)\| = q^3(q^2-1)(q-1)$, so the orbit size is $q^2+q+1$. When $q$ is even, all points on $L$ lie in the nucleus plane of ${\mathcal{V}}_3({\mathbb{F}}_q)$. For $q$ odd, all of the points on $L$ are exterior points: If $q$ is odd and $\ell$ is a constant rank-$2$ line in $\langle {\mathcal{V}}({\mathbb{F}}_q)\rangle$ not contained in a conic plane of ${\mathcal{V}}({\mathbb{F}}_q)$, then every point on $\ell$ is an exterior point. Let $X_0,X_1,X_2$ (respectively $Y_0,\ldots,Y_5$) denote the homogeneous coordinates in ${\mathrm{PG}}(2,{\mathbb{F}}_q)$ (respectively ${\mathrm{PG}}(5,{\mathbb{F}}_q)$). There is a unique orbit of such lines, arising from the tensor orbit $o_{12}$. Each point $P_{x,y}$ on $\ell$ is in the unique conic plane $\langle \nu_3(\ell_{x,y})\rangle$, where $\ell_{x,y}$ is the line with equation $yX_0-xX_2=0$ in ${\mathrm{PG}}(2,q)$. The image $\nu_3(\ell_{x,y})$ is the conic with equation $Y_0Y_3-Y_1^2=0$ in the plane $\pi_{x,y}$ with equation $xY_2-yY_0=xY_4-yY_1=x^2Y_5-y^2Y_0=0$. The point $P_{x,y}$ is on the tangents $Y_0=0$ and $Y_3=0$ in $\pi_{x,y}$ to ${\mathcal{C}}(P_{x,y})$. If $q$ is even then there is a second $K$-orbit, represented by the line $$L_\text{e} = {\mathrm{PG}}(M_\text{e}) \quad \text{where} \quad M_\text{e} = \left[ \begin{matrix} \cdot & x & \cdot \\ x & x+y & y \\ \cdot & y & \cdot \end{matrix} \right]_{x,y}.$$ One may check that $DM_\text{e}D^\top=M_\text{e}$ if and only if $D$ has the form $$D = \left[ \begin{matrix} d_{11} & \cdot & d_{22}+d_{33} \\ d_{21} & d_{22} & d_{23} \\ d_{11}+d_{22} & \cdot & d_{33} \end{matrix} \right].$$ The stabiliser of $L_\text{e}$ in $K$ is therefore isomorphic to $E_q^2:E_q:C_{q-1}$, and has order $q^3(q-1)$, so $\|L_\text{e}^K\| = (q^3-1)(q+1)$. Hence, when $q$ is even the total number of lines arising from the tensor orbit $o_{12}$ is $$(q^2+q+1) + (q^3-1)(q+1) = q^2(q^2+q+1),$$ which is the same as in the $q$ odd case. A point on $L_\text{e}$ lies in the nucleus plane if and only if $x=y$, so $L_\text{e}$ intersects the nucleus plane in one point. Tensor orbit $o_{13}$ {#tensor-orbit-o_13-1 .unnumbered} --------------------- First consider the case where $q$ is odd. There are two orbits, each represented by $${\mathrm{PG}}\left( \left[ \begin{matrix} \cdot & x & \cdot \\ x & y & \cdot \\ \cdot & \cdot & \gamma y \end{matrix} \right]_{x,y} \right) \quad \text{for some} \quad \gamma \in {\mathbb{F}}_q^{\times}.$$ The rank-$2$ point corresponding to $y=0$ is always exterior. If $\gamma=1$, as in the second column of Table \[table:main\], then the rank-$2$ point corresponding to $x=0$ is exterior if $-1 \in \Box$ (that is, if $q \equiv 1 \pmod 4$) and interior otherwise (if $q \equiv 3 \pmod 4$). If $\gamma \not \in \Box$, as in the third column of Table \[table:main\], then the situation is reversed: the rank-$2$ point corresponding to $x=0$ is exterior if $q \equiv 3 \pmod 4$ and interior if $q \equiv 1 \pmod 4$. Now, let $L$ be the line spanned by the points $P_1$ and $P_2$ of rank 2 corresponding to $y=0$ and $x=0$, respectively, in the above matrix. The conics ${\mathcal{C}}(P_1)$ and ${\mathcal{C}}(P_2)$ (uniquely determined by $P_1$ and $P_2$) intersect in a point $Q$. The point $P_1$ is on the tangent line to ${\mathcal{C}}(P_1)$ through $Q$. The stabiliser of ${\mathcal{C}}(P_1)$ (as a subgroup of ${\mathrm{PGL}}(2,{\mathbb{F}}_q)$) fixing $Q$ and $P_1$ must also fix the other point of ${\mathcal{C}}(P_1)$ on a tangent to $P_1$, but acts transitively on the other points of ${\mathcal{C}}(P_1)$. It is isomorphic to $C_{q-1}$. On the other hand, the point $P_2$ is on a secant through $Q$. The subgroup of the stabiliser of ${\mathcal{C}}(P_2)$ fixing $Q$ and $P_2$ has order 2 (this is independent of the choice of $\gamma$). The stabiliser of $L$ is therefore isomorphic to $C_{q-1} \times C_2$, and so the total number of lines arising from the tensor orbit $o_{13}$ for $q$ odd is $$2 \cdot \frac{\|K\|}{2(q-1)}=q^3(q^3-1)(q+1).$$ For $q$ even, the first orbit is represented by the above line with $\gamma=1$. The rank-2 point corresponding to $y=0$ lies in the nucleus plane, and the other rank-2 point does not. The point $P_1$ is the nucleus of the conic ${\mathcal{C}}(P_1)$ and therefore the stabiliser of ${\mathcal{C}}(P_1)$, $Q$ and $P_1$ is isomorphic to $E_q:C_{q-1}$. The stabiliser of ${\mathcal{C}}(P_2)$, $Q$ and $P_2$ is trivial since $P_2$ is on a secant through $Q$. The other orbit is represented by $${\mathrm{PG}}\left( \left[ \begin{matrix} \cdot & x & \cdot \\ x & x+y & \cdot \\ \cdot & \cdot & y \end{matrix} \right]_{x,y} \right).$$ Neither rank-2 point lies in the nucleus plane. The point $P_3$ corresponding to $y=0$ is not the nucleus of ${\mathcal{C}}(P_3)$, but it is on the tangent through $Q$. The stabiliser of ${\mathcal{C}}(P_3)$, $Q$ and $P_3$ is therefore isomorphic to $E_q$. The stabiliser of ${\mathcal{C}}(P_2)$, $Q$ and $P_2$ is trivial since $P_2$ is on a secant through $Q$. We conclude that the total number of lines arising from the tensor orbit $o_{13}$ for $q$ even is $$\frac{\|K\|}{q(q-1)} + \frac{\|K\|}{q}=q^3(q^3-1)(q+1),$$ as in the $q$ odd case. Tensor orbit $o_{14}$ {#tensor-orbit-o_14-1 .unnumbered} --------------------- Consider the $K$-orbit represented by the line $$L_\gamma = {\mathrm{PG}}(\left( \left[ \begin{matrix} x & \cdot & \cdot \\ \cdot & \gamma( x+y) & \cdot \\ \cdot & \cdot & y \end{matrix} \right]_{x,y} \right) \quad \text{for some} \quad \gamma \in {\mathbb{F}}_q^{\times},$$ with rank distribution $[0,3,q-2]$. First suppose that $q$ is odd. The rank-2 point $P_\text{e}$ obtained for $(x,y)=(1,-1)$ is always an exterior point, while the other two rank-2 points, namely $P_1$ obtained for $(x,y)=(1,0)$ and $P_2$ obtained for $(x,y)=(0,1)$, are both exterior if $-\gamma \in \Box$ and both interior otherwise. In particular, if $\gamma=1$, as in the second column of Table \[table:main\], then there are three exterior points if $q \equiv 1 \pmod 4$, and one exterior point if $q \equiv 3 \pmod 4$. If $\gamma \not \in \Box$, as in the third column of Table \[table:main\], then the situation is reversed. Now, the conics ${\mathcal{C}}(P_\text{e})$ and ${\mathcal{C}}(P_i)$ meet in a point $Q_i$ ($i=1,2$), and the conics ${\mathcal{C}}(P_1)$ and ${\mathcal{C}}(P_2)$ meet in a point $Q_{12}$. For each $i\in\{1,2\}$, the point $P_i$ is on the secant through $Q_i$ and $Q_{12}$, and the point $P_\text{e}$ is on the secant through $Q_1$ and $Q_2$. The subgroup of the group of a conic stabilising two points on the conic and a third point on the secant through these two points has order $2$. By considering two of the three conic planes, this gives us a group of order $4$. By considering the conics in the preimage of the Veronese map, we obtain a triangle, from which one observes that the action on two of the sides determines the action on the third side. This implies that the action on two of the conic planes determines the action on the third conic plane. We conclude that the subgroup of $K_{L_\gamma}$ stabilising the points $P_\text{e}$, $P_1$ and $P_2$ has order $4$. Taking into account the permutations of the points $P_\text{e}$, $P_1$ and $P_2$ in the case where all three points are exterior, this amounts to a group of order $24$, isomorphic to $(C_2 \times C_2) : \text{Sym}_3$. In the other case, $K_{L_\gamma}$ is isomorphic to $(C_2 \times C_2) : C_2$ and has order $8$. Now consider the case where $q$ is even. Note that all points of rank 2 on $L_\gamma$ lie outside the nucleus plane. If $P_1$, $P_2$ and $P_3$ denote the points of rank 2 on $L_\gamma$, then each point $P_i$ is on the secant of the conic ${\mathcal{C}}(P_i)$ passing through the intersection points of ${\mathcal{C}}(P_i)$ with the other two conics ${\mathcal{C}}(P_j)$ and ${\mathcal{C}}(P_k)$, where $\{i,j,k\}=\{1,2,3\}$. The group fixing the conic ${\mathcal{C}}(P_i)$, two points on ${\mathcal{C}}(P_i)$ and a point $P_i$ on the secant passing though these two points is trivial, because for $q$ even this group also fixes the unique tangent through $P_i$. Taking into account the permutations of the points $P_1$, $P_2$ and $P_3$, we obtain $K_L\cong \text{Sym}_3$ and $\|K_L\|=6$. Since $\tfrac{1}{24}+\tfrac{1}{8}=\tfrac{1}{6}$, we conclude that the tensor orbit $o_{14}$ yields $$\frac{\|K\|}{6} = \frac{q^3(q^3-1)(q^2-1)}{6}$$ symmetric representatives of lines for every $q$. Tensor orbit $o_{15}$ {#tensor-orbit-o_15-1 .unnumbered} --------------------- Here every $K$-orbit is represented by a line $$L = {\mathrm{PG}}\left( \left[ \begin{matrix} vy & x & \cdot \\ x & ux+y & \cdot \\ \cdot & \cdot & x \end{matrix} \right]_{x,y} \right),$$ for some $v$. The rank distribution is $[0,1,q]$. For $q$ odd, the unique point of rank 2 is exterior when $-v \in \Box$ (as in the second column of Table \[table:main\]) and interior when $-v \not \in \Box$ (third column); for $q$ even, the unique point of rank 2 lies outside the nucleus plane. Let $\pi$ denote the plane containing the conic ${\mathcal{C}}$ uniquely determined by the point $R$ of rank 2 on $L$. The group $K_L$ fixing $L$ also fixes the point $P$ in ${\mathcal{V}}_3({\mathbb{F}}_q)$ corresponding to $e_3\otimes e_3$, and therefore also fixes the line $\ell$ obtained by projecting $L$ from $P$ on to $\pi$. This projection $\ell$ corresponds to the $2\times 2$ sub-matrix obtained by deleting the last row and the last column from the above matrix representation of $L$, and is therefore a line through the point $R$ that is external to the conic ${\mathcal{C}}$. The group $K_L$ is the stabiliser of $P$, ${\mathcal{C}}$, $\ell$ and $R$. The linewise stabiliser of $\ell$ in $K_L$ must fix the set $\{P,P^q\}$ of two conjugate points over the quadratic extension of ${\mathbb{F}}_q$. This implies that the stabiliser $K_L$ must fix $R$ and $\{P,P^q\}$, and must therefore have order twice the order of the pointwise stabiliser of $\ell$ in $K_L$, which has order $2$ for $q$ odd and is trivial for $q$ even. We conclude that $K_L \cong C_2^2$ if $q$ is odd, and $K_L \cong C_2$ if $q$ is even. Since the $G$-line orbit arising from the tensor orbit $o_{15}$ splits into two $K$-orbits for $q$ odd, there are in total $\frac{\|K\|}{2}$ symmetric representatives of lines for both $q$ even and $q$ odd. Tensor orbit $o_{16}$ {#tensor-orbit-o_16-1 .unnumbered} --------------------- Here for every $q$, there is a $K$-orbit represented by the line $$L = {\mathrm{PG}}(M) \quad \text{where} \quad M= \left[ \begin{matrix} \cdot & \cdot & x \\ \cdot & x & y \\ x & y & \cdot \end{matrix} \right]_{x,y}.$$ The rank distribution is $[0,1,q]$, so if $DMD^\top=M$ for $D = (d_{ij}) \in \text{GL}(3,{\mathbb{F}}_q)$ then the rank-$2$ point corresponding to $(x,y)=(0,1)$ must be fixed, so $d_{12}=d_{13}=0$ and $$\left[ \begin{matrix} d_{22} & d_{23} \\ d_{32} & d_{33} \end{matrix} \right] \left[ \begin{matrix} \cdot & 1 \\ 1 & \cdot \end{matrix} \right] \left[ \begin{matrix} d_{22} & d_{23} \\ d_{32} & d_{33} \end{matrix} \right]^\top = \left[ \begin{matrix} \cdot & \alpha \\ \alpha & \cdot \end{matrix} \right] \quad \text{for some} \quad \alpha \in {\mathbb{F}}_q^{\times}.$$ This makes the $(1,2)$ entry of $DMD^\top$ equal to $d_{11}d_{23}$, which forces $d_{23}=0$ because $D$ must be invertible. In particular, $D$ must be lower triangular. If $q$ is odd then by considering again the image of the unique rank-$2$ point on $L$, we deduce that $d_{32}=0$. By considering the image of an arbitrary point on $L$, we then see that $d_{31}=0$ and $d_{22}^2=d_{11}d_{33}$, so that $$D = \left[ \begin{matrix} d_{11} & \cdot & \cdot \\ d_{21} & d_{22} & \cdot \\ \cdot & \cdot & -d_{11}^{-1}d_{22}^2 \end{matrix} \right].$$ These matrices comprise a group of order $q(q-1)^2$, and upon quotienting out by the centre of $\text{GL}(3,{\mathbb{F}}_q)$ we see that the stabiliser of $L$ in $K$ has order $q(q-1)$, so that the orbit of $L$ has size $q^2(q^3-1)(q+1)$. Now suppose that $q$ is even, and consider the image of the point corresponding to $(x,y)=(1,0)$, under a lower-triangular matrix $D$. The $(3,3)$ entry is $2d_{31}d_{33}+d_{32}^2=d_{32}^2$, so we again deduce that $d_{32}=0$ (as in the $q$ odd case), but we do [not]{} need $d_{31}=0$. We also have $d_{22}^2=d_{11}d_{33}$ in the $q$ even case, so that $D$ has the same form as above, except with no restriction on $d_{31}$. The stabiliser of $L$ in $K$ therefore has order $q^2(q-1)$, and so the orbit has size $q(q^3-1)(q+1)$. If $q$ is even then we also have a second $K$-orbit, represented by the line $$L_\text{e} = {\mathrm{PG}}(M_\text{e}) \quad \text{where} \quad M_\text{e} = \left[ \begin{matrix} \cdot & \cdot & x \\ \cdot & x & y \\ x & y & y \end{matrix} \right]_{x,y}.$$ If $DM_\text{e}D^\top=M_\text{e}$ for $D = (d_{ij}) \in \text{GL}(3,{\mathbb{F}}_q)$, then again the rank-$2$ point corresponding to $(x,y)=(0,1)$ must be fixed. This forces $D$ to be lower triangular with $d_{33}=d_{22}$. By then considering an arbitrary point on $L_\text{e}$, we deduce that $D$ must have the form $$D = \left[ \begin{matrix} d_{11} & \cdot & \cdot \\ d_{11}^{-1}d_{32}^2-d_{32} & d_{11} & \cdot \\ d_{31} & d_{32} & d_{11} \end{matrix} \right].$$ These matrices comprise a subgroup of order $q^2(q-1)$ in $\text{GL}(3,{\mathbb{F}}_q)$, so the stabiliser of $L_\text{e}$ in $K$ has order $q^2$ and hence the orbit has size $q(q^3-1)(q^2-1)$. Therefore, in total there are $q^2(q^3-1)(q+1)$ lines in $\langle {\mathcal{V}}({\mathbb{F}}_q) \rangle$ arising from the tensor orbit $o_{16}$, whether $q$ is even or odd. When $q$ is odd, the unique rank-$2$ point is always exterior; when $q$ is even, the unique rank-2 point lies in the nucleus plane for the line $L$ but not for the line $L_\text{e}$. Tensor orbit $o_{17}$ {#tensor-orbit-o_17-1 .unnumbered} --------------------- For this final case, we show that the line stabiliser has order $3$. Recall that, by Lemma \[lem:nrc1\], each point $P$ of rank three defines a NRC ${\mathcal{N}}(P)$ contained in ${\mathcal{V}}_3({\mathbb{F}}_q)$. \[lem:polarity\] If $q$ is odd then the map $\rho~:~{\mathcal{P}}_3\rightarrow {\mathrm{PG}}(5,{\mathbb{F}}_q)$ given by $\rho(P) = \langle {\mathcal{N}}(P)\rangle$ defines the polarity $(Y_0,Y_1,\ldots,Y_5)\mapsto \frac{1}{2}Y_0+\frac{1}{2}Y_1+\frac{1}{2}Y_2+Y_3+Y_4+Y_5=0$ in ${\mathrm{PG}}(5,{\mathbb{F}}_q)$. Straightforward calculation. \[lem:one\_point\] If $P$ and $P'$ are two distinct points on a constant rank-3 line in $\langle {\mathcal{V}}_3({\mathbb{F}})\rangle$, then ${\mathcal{N}}(P)$ and ${\mathcal{N}}(P')$ intersect in at most one point. If $q$ is even then the statement follows immediately from the fact that each two conics on the quadric Veronesean intersect in a point. Now let $q$ be odd and suppose that $W={\mathcal{N}}(P)\cap {\mathcal{N}}(P')$ contains two distinct points $R$ and $Q$. Then $\langle {\mathcal{C}}(R,Q)\rangle$ intersects $\langle W\rangle$ in at least one line, and there exists a hyperplane through $W$ that contains two conics of the Veronesean. Since the map $\rho$ defined in Lemma \[lem:polarity\] is a polarity, this hyperplane is the image of a point $S$ on the line through $P$ and $P'$. However, since the hyperplane $S^\rho$ contains two conics, it is not a NRC, and therefore $S$ does not have rank 3, a contradiction. \[lem:2\] The linewise stabiliser in $K$ of a constant rank-$3$ line has order $3$. The linewise stabiliser $K_L$ inside $K$ of a constant rank-3 line $L$ in $\langle {\mathcal{V}}_3({\mathbb{F}}_q)\rangle$ must fix the set $\{P,P^q,P^{q^2}\}$ of three conjugate points of rank 2 on the line $\overline{L}$ defined over the cubic extension of ${\mathbb{F}}_q$. Also, no element of $K$ can fix one of these three points unless it acts as the identity on the line $\overline{L}$. For instance, if $g\in K_L$ fixes $P$, then $g$ must fix $P^q+P^{q^2}$ and $P+P^q+P^{q^2}$, which implies that $g$ fixes a frame of $\overline{L}$ and must therefore fix every point of $\overline{L}$. Next we prove that the pointwise stabiliser of $L$ is trivial. If $q$ is odd then, by Lemmas \[lem:nrc1\] and \[lem:one\_point\], any projectivity $\varphi$ fixing $L$ pointwise must fix $q+1$ NRCs pairwise intersecting in a point. If $q$ is even then the same lemmas imply that $\varphi$ must fix $q+1$ conics pairwise intersecting in a point. In both cases, the set of intersection points contains the image of a frame of ${\mathrm{PG}}(2,{\mathbb{F}}_q)$ under the Veronese map, and so $\varphi$ is the identity. It follows that $K_L$ has order 3. Algebraically closed fields and the real numbers {#sec:otherF} ================================================ In this section, we explain how the arguments from the case where ${\mathbb{F}}$ is a finite field can be modified to treat algebraically closed fields and the case ${\mathbb{F}}=\mathbb{R}$. When ${\mathbb{F}}$ is algebraically closed, the orbits $o_{10}$, $o_{15}$ and $o_{17}$ do not occur in the classification of tensors in ${\mathbb{F}}^2\otimes {\mathbb{F}}^3\otimes {\mathbb{F}}^3$ given in [@LaSh2015], and so in particular we do not obtain the corresponding $K$-line orbits in $\langle {\mathcal{V}}({\mathbb{F}}) \rangle$. On the other hand, unlike in that classification, in the study of the symmetric representation of the corresponding line orbits we need to distinguish between the cases $\operatorname{char}({\mathbb{F}})=2$ and $\operatorname{char}({\mathbb{F}})\neq 2$. Algebraically closed fields ${\mathbb{F}}$ with $\operatorname{char}({\mathbb{F}})\neq 2$ {#algebraically-closed-fields-mathbbf-with-operatornamecharmathbbfneq-2 .unnumbered} ----------------------------------------------------------------------------------------- The orbits listed in the third column of Table \[table:main\] (the ‘additional orbit, $q$ odd’ column) do not arise, because these depend on the existence of a non-square in ${\mathbb{F}}$. The orbits in the fourth column also do not arise, because their representatives are $K$-equivalent to the corresponding representatives in the second column for $\operatorname{char}({\mathbb{F}}) \neq 2$. Hence, the only tensor orbits that yield lines with symmetric representatives are $o_5$, $o_6$, $o_8$, $o_9$, $o_{12}$, $o_{13}$, $o_{14}$ and $o_{16}$, and none of these eight orbits splits under $K$. Algebraically closed fields ${\mathbb{F}}$ with $\operatorname{char}({\mathbb{F}})=2$ {#algebraically-closed-fields-mathbbf-with-operatornamecharmathbbf2 .unnumbered} ------------------------------------------------------------------------------------- In this case the representatives in the fourth column of Table \[table:main\] [do]{} occur, because they essentially depend on the existence of a nucleus of a non-degenerate conic in ${\mathrm{PG}}(2,{\mathbb{F}})$, a property that holds whenever $\operatorname{char}({\mathbb{F}})=2$. We therefore obtain the same eight $K$-orbits from the $\operatorname{char}({\mathbb{F}})\neq 2$ case, plus the four extra $K$-orbits corresponding to the representatives in the fourth column (for $o_8$, $o_{12}$, $o_{13}$ and $o_{16}$). The real numbers {#the-real-numbers .unnumbered} ---------------- Finally, consider the case ${\mathbb{F}}=\mathbb R$. Observe first that the orbit $o_{17}$ does not yield any lines with symmetric representatives, because every cubic polynomial with real coefficients has at least one real root, and so condition ($$) in Table \[table:main\] cannot hold. The line orbits corresponding to $o_{8}$, $o_{13}$ and $o_{14}$ split, with representatives as in the second and third columns of Table \[table:main\], as the existence of the representatives in the third column depends only on the existence of a non-square $\gamma\in {\mathbb{R}}$ (so one can take $\gamma<0$). However, the line orbit corresponding to $o_{15}$ does [not]{} split, because condition ($$) is equivalent to $u^2v^2+4v=v(u^2v+4)$ being negative, and this implies that $v$ is negative, so the case $-v \not \in \Box$ does not occur. In summary, we have a total of 13 $K$-line orbits: one arising from each of the tensor orbits $o_6$, $o_9$, $o_{10}$, $o_{12}$, $o_{15}$, and $o_{16}$, with representatives as in the second column of Table \[table:main\]; and two arising from each of $o_8$, $o_{13}$ and $o_{14}$, with representatives as in the second and third columns of Table \[table:main\]. The classification of pencils of conics in ${\mathrm{PG}}(2,{\mathbb{F}}_q)$ {#sec:pencils} ============================================================================ In this section we address the connection between our work and previous work of Jordan [@Jordan1906; @Jordan1907], Dickson [@Dickson1908] and Campbell [@Campbell1927]. The classification of lines in the ambient space of the quadric Veronesean under the group $K\cong {\mathrm{PGL}}_3({\mathbb{F}})$ is equivalent to the classification of pencils of conics in ${\mathrm{PG}}(2,{\mathbb{F}})$, that is, one-dimensional subspaces of ternary quadratic forms over ${\mathbb{F}}$. This classification was obtained by Jordan over the reals and over the complex numbers in [@Jordan1906; @Jordan1907]. There are 13 and 8 orbits, respectively, in accordance with our results in Section \[sec:otherF\]. In the case of a finite field of odd characteristic, the classification was obtained by Dickson [@Dickson1908], who determined all equivalence classes of pairs of ternary quadratic forms over finite fields of odd characteristic, resulting in an exhaustive list of 15 such pairs, consistent with our results. In the first paragraph of [@Dickson1908], Dickson anticipates that: [“The main difficulty lies in the case in which the family contains no binary forms, and that in which the binary forms are all irreducible. Neither of these cases occur when the field is ${\mathbb{C}}$ or ${\mathbb{R}}$, so that the problem is quite simple for these fields.”]{} Indeed, 15 of the 18 pages of [@Dickson1908] are dedicated to the classification of these two cases. The proof gives explicit coordinate transformations in order to reduce the families of ternary quadratic forms to canonical representatives of the associated equivalence classes, and can at times be quite tedious. The cases $q \equiv 0$, $1$ and $2 \pmod 3$ are treated separately in the proof of the case in which the family contains no binary forms. This case also relies on knowledge of the number of irreducible cubics of a given form and refers to Dickson’s treatise [Linear Groups]{} [@Dickson????]. Our results are in agreement with Dickson’s, although our proof is quite different. An advantage of our proof is that we do not need to treat the cases $q \equiv 0$, $1$ and $2 \pmod 3$ separately. In fact, our proof works for both even and odd characteristic. Despite the different approaches, in the case of lines without points of rank 3 (which corresponds to families without binary forms), both Dickson’s proof and our proof are based on counting arguments. This seems unavoidable. Interestingly enough, the proof in [@LaSh2015 Section 3.3] of the fact that there is a single orbit of constant rank-3 lines in ${\mathrm{PG}}({\mathbb{F}}_q^3\otimes {\mathbb{F}}_q^3)$ is [also]{}, seemingly unavoidably, based on a counting argument. All three arguments are, however, counting different objects. The even characteristic case was studied by Campbell [@Campbell1927], who provided a list of inequivalent classes of pencils of conics in ${\mathrm{PG}}(2,{\mathbb{F}}_q)$, $q$ even. However, unlike Dickson, Campbell did not obtain a full classification, and this fact seems to have been overlooked in the literature. For instance, in [@Hirschfeld1998] the classification is stated without proof as Theorem 7.31, with references to [@Campbell1927] and [@Dickson1908]. On the other hand, Campbell himself was aware of the incompleteness of his classification, stating on the first page of his paper [@Campbell1927] that: [“If there is an arbitrary coefficient in the typical pencil we say this pencil represents a set of classes, whenever different values of this coefficient may give nonequivalent pencils and so represent distinct classes.”]{} These “sets of classes” are listed on page 406 in Campbell’s paper as Set 10, Set 14, Set 15, Set 16 and Set 17. This also explains why Campbell’s paper is so short: the main difficulties in what would have been be a complete classification are not addressed. In particular, the pencils without binary forms (which correspond to our case $o_{17}$) are not classified. Our results complete the classification of pencils of conics in ${\mathrm{PG}}(2,{\mathbb{F}}_q)$. Representatives of each equivalence class of pencils (or one-dimensional subspaces of ternary quadratic forms) are given in Table \[pencilTable\]. There are 15 equivalence classes both for $q$ even and for $q$ odd. ---------- -------------------------------------------------------- ----------------------------------- -------------------- Tensor orbit Common class (all $q$) $q$ odd $q$ even $o_5$ $(X^2,Y^2)$ $o_6$ $(X^2,XY)$ $o_8$ $(X^2,Y^2+Z^2)$ $(X^2,Y^2+\gamma Z^2)$ $(X^2,YZ)$ $o_9$ $(X^2,YZ)$ $o_{10}$ $(vX^2+Y^2,XY+uY^2)$ $o_{12}$ $(XY,YZ)$ $(XY+Y^2,Y^2+YZ)$ $o_{13}$ $(XY,Y^2+Z^2)$ $(XY,Y^2+\gamma Z^2)$ $(XY+Y^2,Y^2+Z^2)$ $o_{14}$ $(X^2+Y^2,Y^2+Z^2)$ $(X^2+\gamma Y^2,\gamma Y^2+Z^2)$ $o_{15}$ $(XY+uY^2+Z^2,v_1X^2+Y^2)$ $(XY+uY^2+Z^2,v_2X^2+Y^2)$ $o_{16}$ $(XZ+Y^2,YZ)$ $(XZ+Y^2,YZ+Z^2)$ $o_{17}$ $(\frac{1}{\alpha}X^2-\gamma Y^2+YZ,XY+\beta Y^2+Z^2)$ ---------- -------------------------------------------------------- ----------------------------------- -------------------- : The equivalence classes of pencils of conics in ${\mathrm{PG}}(2,{\mathbb{F}}_q)$, where the parameters $\alpha, \beta, \gamma, u,v,v_1,v_2 \in {\mathbb{F}}_q$ correspond to those in Table \[table:main\]. []{data-label="pencilTable"} [1]{} A. D. Campbell, “Pencils of conics in the Galois fields of order $2^n$”, (1927) 401–406. L. E. Dickson, [Linear groups: with an exposition of the Galois field theory]{}, B. G. Teubner, Leipzig, 1901. L. E. Dickson, “On families of quadratic forms in a general field”, (1908) 316–333. J. Harris, [Algebraic geometry: a first course]{}, Springer-Verlag, New York, 1992. J. W. P. Hirschfeld, [Projective geometries over finite fields]{}, second edition, Oxford University Press, Oxford, 1998. J. W. P. Hirschfeld and J. A. Thas, [General Galois geometries]{}, Springer-Verlag, London, 2016. C. Jordan, “Réduction d’un réseau de formes quadratiques ou bilinéaires: première partie", (1906) 403–438. C. Jordan: “Réduction d’un réseau de formes quadratiques ou bilinéaires: deuxième partie", Gauthier-Villars, (1907) 5–51. M. Lavrauw and J. Sheekey, “Canonical forms of $2 \times 3 \times 3$ tensors over the real field, algebraically closed fields, and finite fields”, (2015) 133–147. M. Lavrauw and J. Sheekey, “Classification of subspaces in ${\mathbb{F}}^2\otimes {\mathbb{F}}^3$ and orbits in ${\mathbb{F}}^2\otimes {\mathbb{F}}^3\otimes {\mathbb{F}}^r$", (2017) 5–23. [^1]: [Acknowledgements.]{} This work forms part of the Australian Research Council Discovery Grant DP140100416, which funded the second author’s previous appointment at The University of Western Australia (UWA). The second author is also indebted to the Centre for the Mathematics of Symmetry and Computation at UWA for partially funding his visit to the University of Padua in June 2016, during which this work was initiated, and to the University of Padua for their hospitality.
--- abstract: 'We present results from high-resolution semi-global simulations of neutrino-driven convection in core-collapse supernovae. We employ an idealized setup with parametrized neutrino heating/cooling and nuclear dissociation at the shock front. We study the internal dynamics of neutrino-driven convection and its role in re-distributing energy and momentum through the gain region. We find that even if buoyant plumes are able to locally transfer heat up to the shock, convection is not able to create a net positive energy flux and overcome the downwards transport of energy from the accretion flow. Turbulent convection does, however, provide a significant effective pressure support to the accretion flow as it favors the accumulation of energy, mass and momentum in the gain region. We derive an approximate equation that is able to explain and predict the shock evolution in terms of integrals of quantities such as the turbulent pressure in the gain region or the effects of non-radial motion of the fluid. We use this relation as a way to quantify the role of turbulence in the dynamics of the accretion shock. Finally, we investigate the effects of grid resolution, which we change by a factor 20 between the lowest and highest resolution. Our results show that the shallow slopes of the turbulent kinetic energy spectra reported in previous studies are a numerical artefact. Kolmogorov scaling is progressively recovered as the resolution is increased.' author: - \| David Radice, Christian D. Ott, Ernazar Abdikamalov, Sean M. Couch,\ Roland Haas, and Erik Schnetter bibliography: - 'bibliography/bh\_formation\_references.bib' - \| % bibliography/gw\_references.bib - \| % bibliography/sn\_theory\_references.bib - \| % bibliography/grb\_references.bib - \| % bibliography/nu\_obs\_references.bib - \| % bibliography/methods\_references.bib - \| % bibliography/eos\_references.bib - \| % bibliography/NSNS\_NSBH\_references.bib - \| % bibliography/stellarevolution\_references.bib - \| % bibliography/nucleosynthesis\_references.bib - \| % bibliography/gr\_references.bib - \| % bibliography/nu\_interactions\_references.bib - \| % bibliography/sn\_observation\_references.bib - \| % bibliography/populations\_references.bib - \| % bibliography/pns\_cooling\_references.bib - \| % bibliography/spectral\_photometric\_modeling\_references.bib - \| % bibliography/cs\_hpc\_references.bib - \| % bibliography/numrel\_references.bib - \| % bibliography/gw\_data\_analysis\_references.bib - \| % bibliography/gw\_detector\_references.bib - \| % bibliography/radiation\_transport\_references.bib - \| % bibliography/fluid\_dynamics\_references.bib - \| % bibliography/cosmology\_references.bib - \| % bibliography/privatebibs/radice.bib title: 'Neutrino-Driven Convection in Core-Collapse Supernovae: High-Resolution Simulations' --- Introduction {#sec:introduction} ============ The gravitational collapse of the iron core to a marks the last stage of evolution of stars with zero-age main-sequence masses in excess of $\sim 8\ M_\odot$. A small ($\sim$ few %) fraction of the enormous amount of gravitational binding energy released in this process ($\sim \textrm{few } \times 10^{53}\, \mathrm{erg}$) is somehow deposited in the outer layers (with mass-coordinate $\gtrsim 1.5\, M_\odot$) of the star and powers some of the most energetic explosions in nature, core-collapse supernovae (CCSNe). However, the exact details of the mechanism responsible for re-processing the available energy, which is mostly released as neutrinos streaming out of the , are still uncertain [@janka:07; @janka:12b; @burrows:13a; @foglizzo:15]. In the standard scenario the gravitational collapse of the iron core is halted by the repulsive component of the nuclear force at densities of a few $\times 10^{14}\ \mathrm{g}\, \mathrm{cm}^{-3}$, the inner core bounces back and launches a strong shock wave in the supersonically infalling outer part of the iron core. However, this initial shock wave does not propagate all the way out of the core. Instead, it loses energy due to neutrinos and photo-dissociation of iron-group nuclei and succumbs to the ram pressure of the infalling outer core material within tens of milliseconds. It turns into a stalled accretion shock at a radius of $\sim 100-200\ \mathrm{km}$. To launch an explosion a mechanism must be operating that revives the stalled shock. The most commonly proposed mechanism to achieve shock revival is the delayed neutrino mechanism [@bethewilson:85]. In this mechanism, neutrinos are absorbed in the “gain” layer behind the shock. This is thought to provide the necessary energy to revive and accelerate the shock in a run-away process [@bethe:90; @burrows:93; @pejcha:12a]. Whether this mechanism is the one powering e is still uncertain. It is now well established that for most progenitors the mechanism does not work in spherical symmetry [@ramppjanka:00; @liebendoerfer:01b; @thompson:03; @liebendoerfer:05; @sumiyoshi:05]. However, successful explosions have been obtained in multiple dimensions thanks to the development of non-spherical fluid instabilities such as the [@blondin:03; @foglizzo:07] and neutrino-driven convection [@herant:95; @bhf:95; @janka:96; @foglizzo:06]. These instabilities reduce the critical neutrino luminosity needed for explosion in various ways (more on this below). Neutrino-driven convection, in particular, seems to be the instability most commonly found for exploding or close-to-exploding models in 3D [@dolence:13; @murphy:13; @ott:13a; @couch:13b; @couch:14a; @takiwaki:14a; @abdikamalov:15; @melson:15a; @lentz:15; @melson:15b], however 3D SASI-dominated explosions have also been reported at least in simulations employing simplified physics [@hanke:13; @fernandez:15a; @cardall:15]. In this context, turbulence generated by buoyancy, SASI [@blondin:07; @endeve:12] and/or perturbations in the accretion flow [@couch:13d; @mueller:15; @couch:15b], is expected to have an important role by providing additional effective pressure support behind the shock [@bhf:95; @murphy:13; @couch:15a; @radice:15a]. At the same time, a full understanding of neutrino-driven convection is still missing. Previous studies were limited either because they were in 2D, ([e.g., ]{}@murphy:11 [@fernandez:14]), or because they did not have a sufficient resolution to fully resolve the turbulent dynamics ([e.g., ]{}@hanke:12 [@takiwaki:12; @dolence:13; @couch:14a; @handy:14; @couch:15a; @abdikamalov:15; @melson:15a; @cardall:15]). The former are probably affected by artefacts related to the symmetry assumptions due to the unphysical inverse cascade in 2D turbulence. The latter might instead be affected by systematic errors that are difficult to quantify without a resolution study spanning a large range of resolutions. The studies of @abdikamalov:15 and @radice:15a suggest that convection in current simulations is under-resolved and dominated by the so-called bottleneck effect, a phenomenon that arises when numerical viscosity suppresses some of the non-linear interactions of the energy cascade and results in the accumulation of kinetic energy at large scale [@yakhot:93; @she:93; @falkovich:94; @verma:07; @frisch:08]. This could result in low-resolution simulations being artificially more prone to explosion, as also observed in previous studies [@hanke:12]. In this study, we aim at increasing the understanding of the role of turbulent neutrino-driven convection in e and at identifying the key effects responsible for the global dynamics of the accretion flow in a controlled environment and with well resolved simulations. We develop a neutrino driven convection model that is simple enough to allow us to perform 3D simulations at unprecedented resolution, while including all of the basic physics ingredients of a realistic model: an accretion shock, the converging radial geometry, gravity, neutrino cooling, and neutrino heating. The rest of the paper is organized as follows. First, in Section \[sec:methods\], we present the details of our neutrino-driven convection model and a description of the numerical methods we employ for our numerical investigation. The general evolution and features of our runs are discussed in Section \[sec:results.general\]. There, we focus on the dynamics of large scale quantities, such as the average shock radius and entropy profiles. In Section \[sec:results.convection\], we study the dynamics of convection. In particular, we focus on the role of convection in transporting energy and momentum through the gain region. Section \[sec:results.turbulence\] is dedicated to the turbulent energy cascade and to the role of turbulence in providing an effective additional pressure support in the postshock region. We discuss the turbulent cascade and the kinetic energy spectrum of neutrino-driven turbulent convection in Section \[sec:results.cascade\]. Finally, we summarize and conclude in Section \[sec:conclusions\]. The appendices contain additional technical details of our model. Appendix \[sec:nuclear.dissociation\] describes our treatment of nuclear dissociation at the shock and Appendix \[sec:standing.accretion.shock\] contains the details of the construction of our initial conditions. Throughout this paper we use a system of units such that $G = c = M_{\mathrm{PNS}} = 1$, $M_{\mathrm{PNS}}$ being the gravitational mass. Where CGS values are quoted, it is to be intended that they correspond to the fiducial case with $M_{\mathrm{PNS}} = 1.3\ M_\odot$. Methods {#sec:methods} ======= In the following, we present the details of our approach and of the employed numerical methods. We note that the aim of our work is not to develop a realistic explosion model. Rather we want to construct a controlled setup containing all of the most important ingredients present in nature and in state-of-the-art global simulations. Neutrino-Driven Convection Model -------------------------------- Our initial conditions describe a stalled shock in the core of a massive star at a given radius $r_s$. We study the accretion flow in a 3D spherical wedge domain with a $90^\circ$ opening angle. The is excised and replaced by an inner boundary condition at a fixed radius, $r_{\mathrm{PNS}}$. The accretion flow is described by the equations of general relativistic hydrodynamics, $$\begin{aligned} \label{eq:hydro} \nabla_\mu J^\mu = 0\,, && \nabla_\nu T^{\mu\nu} = L^\mu\,,\end{aligned}$$ where $$\begin{aligned} J^\mu = \rho u^\mu\,, && T^{\mu\nu} = [\rho (1 + \epsilon) + p] u^\mu u^\nu + p g^{\mu\nu}\,,\end{aligned}$$ and $L^\mu$ is a term that we include to model neutrino heating and cooling (see below). $\rho$, $u^\mu$, $p$, $\epsilon$ and $g^{\mu\nu}$ denote the fluid rest-mass density, four-velocity, pressure, specific internal energy and the spacetime metric. The that we employ is a modified gamma-law $$\label{eq:eos.base} p = (\gamma - 1) \rho \tilde{\epsilon}\,,$$ where $\gamma=4/3$ is appropriate for a radiation-pressure dominated gas and $\tilde{\epsilon}$ represents the amount of specific “thermal energy” available after nuclear binding energy has been removed from $\epsilon$ for dissociated nuclei. We account for nuclear dissociation energy in a parametrized way similar to @fernandez:09a [@fernandez:09b]. See Appendix \[sec:nuclear.dissociation\] for the details of our implementation. The specific entropy for our equation of state is defined up to a constant, so we exploit this to choose the zero of entropy following @foglizzo:06 $$\label{eq:entropy} s = \frac{1}{\gamma - 1}\log\bigg[\frac{p}{p_1} \Big(\frac{\rho_1}{\rho}\Big)^\gamma \bigg],$$ where $\rho_1$ and $p_1$ are, respectively, the initial postshock density and pressure (see Appendix \[sec:standing.accretion.shock\]). In this way, $s$ is exactly zero at the location of the shock in the initial data. The gravity of the is included, while self-gravity of the accretion flow is neglected, [i.e. ]{}we use the Cowling approximation, so that the spacetime metric is constant in time and given by $$\label{eq:metric} {\mathrm{d}}\, s^2 = - \alpha^2(r)\, {\mathrm{d}}t^2 + A^2(r)\, {\mathrm{d}}r^2 + r^2 {\mathrm{d}}\Omega^2,$$ where ${\mathrm{d}}\, s^2$, not to be confused with the entropy, denotes the spacetime line element, ${\mathrm{d}}\Omega^2$ is the line element of the two-sphere and $$\label{eq:lapse} \alpha^2 = A^{-2} = 1 - \frac{2M_{\mathrm{PNS}}}{r}.$$ Neutrino heating and cooling is modeled using the light-bulb scheme introduced by @houck:92 [@janka:01] and later used in many studies of e, the most recent being @cardall:15. The functional form of $L^\mu$ that we use is similar to that of @fernandez:09b, with the appropriate general-relativistic corrections: $$\label{eq:lightbulb} L^\mu = u^\mu \mathcal{L} = u^\mu C \rho \bigg[ f_{\mathrm{heat}} \big(K p_1\big)^{3/2} \Big(\frac{r_s}{r}\Big)^2 - p^{3/2} \bigg] e^{-\big([s + s_{\mathrm{ref}}]_-\big)^2}\,,$$ where $C$ is an overall normalization constant, $p_1$ is the post-shock pressure, $K$ measures the strength of the heating[^1], $r_s$ is the shock radius and we use the notation $$\label{eq:negative.part} [X]_- = \begin{cases} \|X\|, & \textrm{if } X < 0, \\ 0, & \textrm{otherwise.} \end{cases}$$ $f_{\mathrm{heat}}$ is set to one for most simulations and when computing the initial conditions. We run some additional models with $f_{\mathrm{heat}} = 0.9, 0.95, 1.05$, and $1.1$ (see Table \[tab:runs\] for more details). In Equation we use a Gaussian cutoff of the heating/cooling term to avoid catastrophic cooling on the surface of the and to suppress heating ahead of the shock. The reference entropy $s_{\mathrm{ref}}$ is chosen to ensure that heating is not switched off when the shock expands and $s$ becomes slightly negative. In our simulations we find (empirically) $$s_{\mathrm{ref}} = \frac{1}{\gamma - 1} \ln 2$$ to perform well and avoid any artificial suppression of the heating in the gain region. Note that our heating prescription neglects the non-linear feedback between accretion and neutrino luminosity. As such, our scheme might not be appropriate in regimes where the accretion rate at the base of the flow shows significant variations. However, it should be reasonably adequate for the study of nearly steady-state neutrino-driven convection we perform here. [lccc]{} Run & $f_{\mathrm{heat}}$ & $\Delta r\ [\mathrm{m}]$ & $\Delta \theta = \Delta \varphi\ [\mathrm{deg}]$\ Ref. & $1.0\phantom{0}$ & $3839$ & $1.8\phantom{0}$\ 2x & $1.0\phantom{0}$ & $1919$ & $0.9\phantom{0}$\ 4x & $1.0\phantom{0}$ & $\phantom{0}960$ & $0.45$\ 6x & $1.0\phantom{0}$ & $\phantom{0}640$ & $0.3\phantom{0}$\ 12x & $1.0\phantom{0}$ & $\phantom{0}320$ & $0.15$\ 20x[^2] & $1.0\phantom{0}$ & $\phantom{0}191$ & $0.09$\ F0.9-Ref. & $0.9\phantom{0}$ & $3839$ & $1.8\phantom{0}$\ F0.95-Ref. & $0.95$ & $3839$ & $1.8\phantom{0}$\ F1.05-Ref.[^3] & $1.05$ & $3839$ & $1.8\phantom{0}$\ F1.1-Ref. & $1.1\phantom{0}$ & $3839$ & $1.8\phantom{0}$\ F1.1-2x & $1.1\phantom{0}$ & $1919$ & $0.9\phantom{0}$\ F1.1-6x & $1.1\phantom{0}$ & $\phantom{0}640$ & $0.3\phantom{0}$\ F1.1-12x & $1.1\phantom{0}$ & $\phantom{0}320$ & $0.15$\ 1D & $1.0\phantom{0}$ & $\phantom{0}640$ & $-$\ F1.1-1D & $1.1\phantom{0}$ & $\phantom{0}640$ & $-$\ Neutrino heating and cooling is consistently included in the generation of the initial conditions with $f_{\mathrm{heat}}$ set to one. Our initial model is uniquely identified by the radius $r_{\mathrm{PNS}}$, the initial shock position $r_s$, the accretion rate $\dot{M}$ and the heating parameter $K$. $C$ is fixed by the condition $\upsilon^r(r_{\mathrm{PNS}}) = 0$, where $\upsilon^i$ is the fluid three-velocity. Note that, since our is scale free, the mass, $M_{\mathrm{PNS}}$, scales out of the problem and our results can be applied to any mass with the proper rescaling. The results that we quote are for the fiducial case $M_{\mathrm{PNS}} = 1.3\ M_\odot$. In particular, the parameters used in this work are $r_{\mathrm{PNS}} = 30$ ($\simeq 57\ {\mathrm{km}}$), $r_s = 100$ ($\simeq 191\ {\mathrm{km}}$) and $\dot{M} = 10^{-6}$ ($\simeq 0.2\ M_\odot\, \mathrm{s}^{-1}$). $K$ is set to $9$ and, correspondingly the equilibrium $C$ is found to be $C = 9 \times 10^9$, which, for our models, corresponds to a luminosity in both the electron or anti-electron neutrinos of[^4] $$L_\nu \simeq 1.22 \times 10^{52} \left( \frac{12\ \mathrm{MeV}}{T_\nu} \right)^2 \frac{\mathrm{erg}}{\mathrm{s}}\,,$$ where $T_\nu$ is the temperature at the neutrinosphere in $\mathrm{MeV}$. $T_\nu$ needs not to be specified by our heating/cooling prescription, because our heating prescription depends only on the total neutrino luminosity and not separately on the neutrino number fluxes and average energies as would have been the case for a real transport scheme. Finally, a small random perturbation with relative amplitude $10^{-6}$ is added to the density field to break the symmetry. The details of the construction of the initial conditions are given in Appendix \[sec:standing.accretion.shock\]. An important parameter for quantifying the convective (in)stability of the initial conditions is the Brunt-Väisäla frequency, $\Omega_{\mathrm{BV}}$, which we write in terms of the quantity [@foglizzo:06] $$\label{eq:bvfrequency2} C_{\mathrm{BV}} = \frac{\gamma-1}{\gamma} g \partial_r s\,,$$ where $g$ is the gravitational acceleration, which we approximate as $M_{\mathrm{PNS}} / r^2$. We define $$\label{eq:bvfrequency} \Omega_{\mathrm{BV}} = \sqrt{\|C_{\mathrm{BV}}\|} \mathrm{sign}(C_{\mathrm{BV}})\,.$$ With our convention, negative values of $\Omega_{\mathrm{BV}}$ correspond to unstable stratification and $\|\Omega_{\mathrm{BV}}\|$ gives the growth rate of radial perturbations. In the case of e, an additional condition for convective instability is that the growth rate of perturbations should be high enough so that they can reach non-linear amplitudes and become buoyant before being advected out of the gain region by the radial background flow [@foglizzo:06]. This can be quantified by measuring the ratio between the two timescales, $$\label{eq:foglizzo.chi} \chi = \int \frac{[\Omega_{\mathrm{BV}}]_-}{\|\upsilon^r\|}\ {\mathrm{d}}r\,,$$ where the integral is extended over the gain region and we have once again used the notation of Equation . [@foglizzo:06] showed that if $\chi \gtrsim 3$ perturbations have enough time to develop large-scale convection. Our simulations have an initial value of $\chi = 5.33$, so we expect them to develop large-scale convection. Simulation Setup ---------------- The Equations are solved on a uniform spherical grid in flux-conservative form [@banyuls:97], using the 5th order MP5 finite difference high-resolution shock-capturing [@suresh:97] scheme as implemented in the `WhiskyTHC` code [@radice:12; @radice:14b]. `WhiskyTHC` employs a linearized flux-split method with carbuncle and entropy fix that makes full use of the characteristic structure of the general relativistic hydrodynamics equations with very small numerical dissipation. Our computational domain covers the region $57\ \mathrm{km} \lesssim r \lesssim 442\ \mathrm{km}$ ($825\ \mathrm{km}$ for models with $f_{\mathrm{heat}} \geq 1.05$), $\pi/4 < \theta < 3 \pi/4$ and $-\pi/4 < \varphi < \pi/4$. We use reflecting boundary conditions at the inner boundary, inflow conditions at the outer boundary, and impose periodicity in the angular directions. To ensure a constant accretion rate through the shock, we add artificial dissipation, using a standard 2nd order prescription, close to the outer boundary, always outside of the shock front, to some of the low-resolution runs. Dissipation is found not to be necessary for the 4x, 12x, and 20x runs. In the other simulations, instead, we find dissipation to be necessary to prevent oscillations in the fluid quantities close to the outer boundary that, if not suppressed, can alter the accretion rate by a few percent. The grid spacing for the reference resolution is $\Delta r \simeq 3.8\ \mathrm{km}$, $\Delta \theta = \Delta \varphi = 1.8^\circ$. The reference resolution is similar to the one employed in recent radiation-hydrodynamics simulations [@melson:15a; @lentz:15]. For the other resolutions we refined the grid relative to the reference run by factors $2, 4, 6, 12, 20$, [i.e. ]{}up to $\Delta r \simeq 190\ \mathrm{m}$ and $\Delta \theta = \Delta \varphi = 0.09^\circ$ for the 20x run. All of the simulations are carried out until $\simeq 640\ \mathrm{ms}$, apart from the 20x and the F1.1 runs. We start the 20x at $\simeq 317\ \mathrm{ms}$ from a snapshot of the 12x run and follow it for only $\simeq 60\ \mathrm{ms}$ due to its high computational cost. We stop the F1.1-Ref., F1.1-6x, and F1.1-12x runs at times $t \simeq 384\ \mathrm{ms}$, $t \simeq 560\ \mathrm{ms}$ and $t \simeq 566\ \mathrm{ms}$, when the shock reaches the outer boundary of the computational domain. Finally, we perform two additional runs in spherical symmetry at the same (radial) resolution as the 6x resolution. The main characteristics of our runs are summarized in Table \[tab:runs\]. Overall dynamics {#sec:results.general} ================ Shock evolution {#sec:results.shock} --------------- The overall dynamics of our runs is best summarized by the average shock radius evolution, shown in Figures \[fig:shock.radius\] and \[fig:shock.radius.high.fheat\]. The dynamics consist in an initial transient lasting $\simeq 25\ \mathrm{ms}$ where the shock radius first expands and then recedes. This transient is triggered by waves reflecting on the surface of the , where the initial conditions are necessarily an approximation to the real steady state solution, which predicts infinite density and zero velocity at the surface of the (this is an artifact arising due to the assumption of stationarity, see Appendix \[sec:standing.accretion.shock\]). ![\[fig:shock.radius\] Average shock radius evolution for all runs with $f_{\mathrm{heat}}=1$. After an initial transient, the shock radius expands as convection develops. For the fiducial model, the growth slows down significantly and quasi-periodic oscillations appear when the convective plumes start to interact non-linearly with the shock front. The black dashed line shows the shock radius for a reference 1D run.](fig1.pdf){width="0.98\columnwidth"} ![\[fig:shock.radius.high.fheat\] Average shock radius evolution for runs with enhanced heating. After the initial transient, the shock radius immediately starts to expand. The expansion is not significantly accelerated when the shock reaches large radii (as it would be in a full-physics simulations) partly because of our very simplified treatment of nuclear dissociation, with a constant specific energy loss for each fluid element crossing the shock. In a more realistic simulation, the amount of energy loss drops with radius and this leads to an accelerated expansion. The deviations between the reference resolution and the high resolution simulations are much more pronounced than for the case without enhanced heating ([c.f. ]{}Figure \[fig:shock.radius\]). The black dashed line shows the shock radius for a reference 1D run.](fig2.pdf){width="\columnwidth"} For the fiducial case with $f_{\mathrm{heat}} = 1.0$, after the initial transient, turbulence starts to develop: the initial seed perturbations trigger the formation of small buoyant plumes at the base of the gain layer. These plumes grow as they find their way to the shock and convection gains strength. After $t \simeq 150\ \mathrm{ms}$ the entropy perturbations are strong enough to cause large deformations of the shock front. When the plumes start to interact strongly with the shock at $t \gtrsim 150\ \mathrm{ms}$, the dynamics becomes fully non-linear and characterized by a slow growth of the shock radius and quasi-periodic oscillations with period of the order of the advection timescale (see below). Until this point the different runs appear to be monotonically convergent, with high-resolution simulations having smaller average shock radii. However, as soon as the dynamics becomes fully non-linear their shock radius evolutions lose point-wise convergence, although the evolutionary tracks of all of the runs are broadly consistent with each other. By comparison, the evolution of our 1D run, also shown in Figure \[fig:shock.radius\], is rather uneventful. The 1D run shows the same initial transient as the 3D data, but afterwards it starts oscillating around its original position and shows only a modest secular growth, which is mainly driven by the accumulation of material in the gain region and continues for the whole duration of the simulation. This shows that the growth of the shock radius after $t \simeq 75\ \mathrm{ms}$ and up to $t \simeq 100\ \mathrm{ms}$ is due to the initial development of convection, which is well captured by our runs. The dynamics of the shock and its behavior with resolution change rather drastically for models with enhanced heating. This can be seen in Figure \[fig:shock.radius.high.fheat\], where we show the average shock radius for the simulations with $f_{\mathrm{heat}} = 1.1$. The reference resolution simulation starts to diverge from the 6x and 12x resolutions as soon as the initial transient is over. The 2x resolution seems to be closer to the higher resolution runs, which appear to be converged, but eventually also diverges away after $t \simeq 300\ \mathrm{ms}$. Finally, the 6x and 12x resolutions appear to be consistent with each other for the entire simulated time. ![image](fig3.pdf){width="\textwidth"} Going back to the fiducial case without enhanced heating, the qualitative differences between resolutions are particularly evident in the visualizations of the fluid entropy. This is shown in Figure \[fig:entropy.xz\], where we display the color coded entropy in the $xz-$plane at a representative time ($t \simeq 365\ \mathrm{ms}$). Compared with the other resolutions, the reference resolution shows larger plumes and higher entropies. At this resolution, the dynamics is characterized by the motion of few large structures, while, at higher resolutions, the dynamics appears to be characterized by smaller structures evolving on shorter timescales. Note that the appearance of large scale coherent plumes is typically observed at the onset of explosion [@dolence:13; @fernandez:14; @mueller:15; @lentz:15]. This suggests that, as has been also observed by [@hanke:12; @abdikamalov:15] and consistently with what we find for the simulations with enhanced heating, low resolution could artificially ease the explosion (see also @couch:13b). On the other hand, note that the 12x resolution, which is the highest for which we carry out a long term evolution, is also the one showing the highest average shock radius growth rate (Figure \[fig:shock.radius\]), suggesting that turbulence has a more complex role than simply destroying large-scale plumes. As the resolution increases, first, secondary instabilities in the flow drive down the size of the typical plumes and create more complex flow structures. Second, at the highest resolutions (12x and 20x), plumes start to lose their coherence due to the presence of small-scale turbulent mixing. Instead of being characterized by entropy “bubbles” with sharp entropy gradients, as in the reference resolution, the flow in high resolution simulations appears to be dominated by the appearance and disappearance of large hot “clouds”, [i.e. ]{}entropy structures with a complex topology. An animation of the entropy on the equatorial plane for the Ref., 2x, 4x, and 12x resolutions is included in the online supplemental materials. ![\[fig:timescales\] Top panel: ratio of the advection timescale (Equation \[eq:advection.timescale\]) to heating timescale (Equation \[eq:heating.timescale\]). Middle panel: heating timescale. Bottom panel: advection timescale. $f_{\mathrm{heat}}=1.0$ for all of the runs shown in this figure. As it is the case for the shock radius evolution, also in the heating efficiency our simulations appear to be closely convergent until $t \simeq 100\ \mathrm{ms}$. Afterwards convergence is not monotonic, but all the runs are still in good agreement with each other, especially for $\tau_{\mathrm{adv}}$. The heating timescale shows a somewhat larger spread.](fig4.pdf){width="\columnwidth"} Another commonly employed diagnostic in simulations is the ratio between the timescale for the advection of a fluid element through the gain layer and the time necessary for it to absorb enough energy from neutrinos to become unbound. Specifically, the advection timescale is typically defined as [@fernandez:12; @mueller:12a] $$\label{eq:advection.timescale} \tau_{\mathrm{adv}} = \frac{M_{\mathrm{gain}}}{\dot{M}}\,,$$ where $\dot{M}$ is the accretion rate, $M_{\mathrm{gain}}$ is the total mass in the gain region $$M_{\mathrm{gain}} = \int \rho W \sqrt{\gamma} {\mathrm{d}}V\,,$$ where $W$ is the Lorentz factor, $\sqrt{\gamma}$ is the spatial volume form and the integral is extended over the gain region. The heating timescale is defined as ([e.g., ]{}@fernandez:12) $$\label{eq:heating.timescale} \tau_{\mathrm{heat}} = \frac{\|E_{\mathrm{bind}}\|}{\dot{Q}_{\mathrm{net}}}\,,$$ where $E_{\mathrm{bind}}$ is the binding energy of the gain region, which we compute as in [@mueller:12a]: $$E_{\mathrm{bind}} = \int \big\{\alpha [\rho (1 + \tilde{\epsilon} + p/\rho) W^2 - p] - \rho W^2\big\} \sqrt{\gamma} {\mathrm{d}}V\,.$$ $\dot{Q}_{\mathrm{net}}$ is the net heating/cooling rate $$\dot{Q}_{\mathrm{net}} = \int W \mathcal{L} \sqrt{\gamma}\, {\mathrm{d}}V\,,$$ where $\sqrt{\gamma}$ is the determinant of the spatial metric and the integrals in the previous equations are extended over the gain region. The advection and heating timescales as well as their ratio for the fiducial model ($f_{\mathrm{heat}} = 1.0$) are shown in Figure \[fig:timescales\]. In a similar way to what we see for the average shock radius, we find that the different runs are monotonically convergent during the first $\sim 100\ \mathrm{ms}$. Afterwards, the various simulations have consistent trends, but there is no point-wise convergence. After the initial transient, starting from $t\simeq 50\ \mathrm{ms}$, the advection timescale grows by roughly a factor $2$ as convection develops. Then, starting from $t\simeq 100\ \mathrm{ms}$ the advection timescale shows a secular growth, due to the increase of the mass in the gain region[^5] and it reaches $\sim 80\ \mathrm{ms}$ toward the end of the simulations. At the same time, the heating timescale remains roughly constant, especially at high resolution, and the runs slowly approach the approximate condition for explosion, $\tau_{\mathrm{adv}} \lesssim \tau_{\mathrm{heat}}$ [@murphy:08; @marek:09; @fernandez:12; @mueller:12a]. If we consider the ratio $\tau_{\mathrm{adv}}/\tau_{\mathrm{heat}}$ as a way to measure the proximity of the simulations to explosion, we can see from Figure \[fig:timescales\] that, for the first $\sim 300\ \mathrm{ms}$, high resolution simulations are indeed further away from explosion than low resolution simulations as observed by @hanke:12 and @abdikamalov:15 (although this trend seems to be reversed at very high resolutions). Whether this results in explosions being triggered artificially or not at low resolution will likely depend on how close the models are to explosion. For instance, @abdikamalov:15 found finite-resolution effects to be small for non-exploding models and comparatively large for exploding models. Similarly, in our simplified setup we also find the evolution of models with enhanced heating to be more sensitive to resolution (compare Figures \[fig:shock.radius\] and \[fig:shock.radius.high.fheat\]). The recent results by @melson:15b suggest that full-physics CCSN simulations are close to the critical threshold for explosion. One might speculate that, near criticality, relatively small differences as those documented in Figure 4 could lead to dramatic consequences for some progenitors. Dynamics of Convection {#sec:results.convection} ====================== Convective Energy Transport {#sec:results.transport} --------------------------- One of the characteristics of convection is that it provides a way to transport energy. In the context of neutrino-driven convection in e it is interesting to consider the role of convection in transporting energy from the bottom of the gain layer, where neutrino deposition is the strongest, outwards, toward the shock and, if an explosion is ultimately launched, by means of the latter, toward the envelope of the star. ![image](fig5a.pdf){width="\columnwidth"} ![image](fig5b.pdf){width="\columnwidth"} To analyze the efficiency of neutrino-driven convection for energy transport, we consider the angular-integrated energy equation. Our analysis can be considered as the general-relativistic analog of that of [@meakin:07b; @murphy:11], with some minor differences. Our starting point is the angle-averaged energy equation on the Schwarzschild background metric, Equation $$\label{eq:energy.evolution} \partial_t \langle A\, r^2 E \rangle + \partial_r \langle r^2 [E + p] \upsilon^r \rangle = \mathcal{G}_E\,,$$ where $\mathcal{G}_E$ describes heating/cooling by neutrinos and gravity $$\mathcal{G}_E = r^2 \rho W \mathcal{L} - \big[r^2 (E + p) \upsilon^r \big] \partial_r \log \alpha\,.$$ The energy density is $$\label{eq:energy} E = \rho h W^2 - p\,.$$ The total energy density can be decomposed as $$E = \rho h W (W - 1) + (\rho \epsilon + p) W + \rho W - p\,,$$ where we can distinguish the relativistic kinetic energy density $$K = \rho h W (W - 1)\,,$$ the “Newtonian” enthalpy density $$H = (\rho \epsilon + p) W\,,$$ and the rest-mass energy density $$D = \rho W\,.$$ The associated radial fluxes are $F_K = r^2 K\,\upsilon^r$, $F_H = r^2 H\,\upsilon^r$ and $F_D=r^2 D\,\upsilon^r$. We can rewrite Equation as $$\label{eq:energy.evolution2} \partial_t \langle A r^2 E \rangle + \partial_r \langle F_K + F_H + F_D \rangle = \mathcal{G}_E\,.$$ Furthermore, we decompose the radial velocity into a mean part and a “turbulent” part as $$\upsilon^r = \langle \upsilon^r \rangle + \delta\upsilon^r\,.$$ More in general we define the turbulent velocity to be $$\label{eq:turb.velocity} \delta\upsilon^i = \upsilon^i - \langle \upsilon^r \rangle \delta_{r}^{\phantom{r}{i}}\,.$$ Note that our definition of turbulent velocity is not the standard definition in the turbulence literature, since $\langle \delta \upsilon^{\theta} \rangle$ and $\langle \delta \upsilon^\phi \rangle$ are not necessarily exactly zero. On the other hand, since we consider only non-rotating models, it is natural to consider any non-radial fluid motion to be related to turbulence. Moreover, since our background model is spherically symmetric, we expect the ensemble averages of $\delta \upsilon^\phi$ and $\delta \upsilon^\theta$ to vanish. In the same way, we can split the fluxes into a mean and turbulent component as $$F_u = \bar{F}_u + F_u'\,,$$ where $u = K, H$ or $D$, and $$\begin{aligned} \bar{F}_u = \langle r^2 u \rangle \langle \upsilon^r \rangle\,, && F_u' = \langle r^2 u \delta\upsilon^r \rangle\,.\end{aligned}$$ Finally, the equation for the energy transported by convection can be written as $$\label{eq:energy.transport} \partial_t \langle A r^2 E \rangle + \partial_r [\bar{F}_K + \bar{F}_H + \bar{F}_D] + \partial_r [F_K' + F_H' + F_D'] = \mathcal{G}_E\,.$$ Note that $F_D$ is also the flux of the angle-averaged continuity equation $$\label{eq:continuity} \partial_t \langle A r^2 D \rangle + \partial_r \langle F_D \rangle = 0\,,$$ so that $\bar{F}_D$ and $F_D'$ can also be interpreted as mean and turbulent contributions to the mass transport: $$\label{eq:mass.transport} \partial_t \langle A r^2 D \rangle + \partial_r \bar{F}_D + \partial_r F_D' = 0\,.$$ Combining Equation with Equation , we obtain an equation for the energy density minus the rest-mass energy density, $$\label{eq:energy.net} \partial_t \langle A r^2 [E - D] \rangle + \partial_r \langle F_K + F_H \rangle = \mathcal{G}_E\,.$$ The quantity $E - D$ can be considered as the generalization of the sum of the Newtonian internal and kinetic energy densities of the fluid. To identify the important terms in the energy equation, we study the radial profiles of the angle-averaged mean and turbulent fluxes of Equation . To this end, we re-map the fluxes to be a function of the normalized radius $$\label{eq:mapped.radius} r_\star = \frac{r - r_g}{r_s - r_g}\,,$$ where $r_g$ and $r_s$ are the average gain and shock radius respectively and we defined the average gain radius, $r_g$, to be the radius at which neutrino heating becomes larger than neutrino cooling in an angle-averaged sense. This way the extent of the gain region in terms of the re-scaled radius is $0 \leq r_\star \leq 1$ and we do not have to worry about secular changes in the shock radius when averaging in time. Next, we average the re-mapped fluxes using data starting at $t = 30\,000\ M_{\mathrm{PNS}} \simeq 192\ \mathrm{ms}$ to exclude the initial transient and the development phase of convection and include only the later quasi-steady phase. The result of this analysis is portrayed by Figure \[fig:energy.fluxes\], where we show the angle-averaged total (mean + turbulent) and turbulent kinetic and enthalpy fluxes. In each panel, the gray shaded area shows the standard deviation of the 12x resolution. The other runs show variations of very similar magnitude and we do not show their standard deviations to avoid overcrowding the figure. The large extent of the gray region in the plots is indicative of the fact that the angle-averaged fluxes, total and turbulent for both kinetic energy and enthalpy, show large variations in time. These are large-scale oscillations that correlate with the quasi-periodic oscillations we see in the shock radius (Figure \[fig:shock.radius\]). The reference resolution also shows large spatial oscillations in the cooling layer where density and pressure have a steep gradient which is not sufficiently resolved at low resolution. These oscillations are present also for the 2x resolution, but are confined to a much deeper layer close to the and disappear at higher resolution. The angle-averaged fluxes are shown in the left panels of Figure \[fig:energy.fluxes\]. Both the enthalpy (top) and the kinetic (bottom) energy fluxes are negative. This means that, despite the presence of convection, the energetics of the flow are dominated by advection and there is no net transfer of energy upwards from the gain region to the shock. Note that this might change if the flow transitions to an explosion, see, [e.g., ]{}[@abdikamalov:15]. In the stalled shock phase, however, turbulence can only act in such a way as to decrease (in absolute value) the mean fluxes and favor the accumulation of mass and energy in the gain region, which is a necessary condition for shock expansion [@janka:01]. Finally, from the amplitude of the fluxes upstream of the shock, we can also see that, as expected, kinetic energy is the main form with which energy is accreted through the shock, but most of it is converted into thermal energy by the shock. The turbulent fluxes are shown in the right panels of Figure \[fig:energy.fluxes\]. The angle-averaged turbulent enthalpy flux is positive at the base of the gain region, while the angle-averaged turbulent kinetic energy flux is negative everywhere. This is the result of buoyant plumes driving thermal energy upwards and displacing lower entropy gas which is pushed downwards in a process that converts thermal energy back into kinetic energy. An important point that we can deduce from Figure \[fig:energy.fluxes\] is that the total amount of energy transported by turbulence is not particularly large compared to that of the background flow. Even at the base of the gain region, where heating is stronger and the convective enthalpy fluxes are more intense, the turbulent angle-average enthalpy flux is at most a few $\times 10^{50}\ \mathrm{erg}\ \mathrm{s}^{-1}$, which is only of order $10\%$ of the total enthalpy flux. Turbulence does contribute significantly to the total kinetic energy flux, with the turbulent angle-averaged fluxes being $\sim 80\%$ of the total, but kinetic energy is dwarfed by the thermal energy in the energy budged downstream from the shock. Obviously, these values are specific to our accretion model and, for instance, they change by a few percent as we vary $f_{\mathrm{heat}}$ from $0.9$ to $1.1$. However, we do not expect qualitative differences to appear for other models during the stalled accretion shock phase. Despite the violence of convection and the fact that buoyant plumes impinge violently onto the shock, the total energy fluxes are still dominated by the radial advection flow. This shows that the larger shock radius in multi-dimensional simulations with respect to one-dimensional simulations is not mainly due to the direct transport of energy by convection, which has a measurable, but small overall impact. Turbulence, instead, acts in a more indirect way by slowing down the drain of energy from the region close to the shock. This effect is analogous to, but distinct from, another well known consequence of neutrino-driven convection: the enhancement of the absorption efficiency due to the increased dwelling time of fluid elements in the gain region ([e.g., ]{}@bhf:95 [@murphy:08; @fernandez:09b]). We remark that the large oscillations shown in Figure \[fig:energy.fluxes\] in the angle averaged turbulent fluxes at the location of the shock wave are an artefact of our decomposition arising from the fact that the angle averaged velocity picks up values both upstream and downstream of the shock, so that the turbulent velocity, computed according to Equation , is artificially large. Obviously, this is only a limitation of our analysis and nothing “special” happens at the location of the shock. This can be confirmed by looking at the total fluxes in the left panels of Figure \[fig:energy.fluxes\]. Finally, concerning the behavior with resolution, we see that there is no clear monotonic trend with resolution in the fluxes. The low resolution runs (Ref. and 2x) tend to show more vigorous convection (as measured from the magnitude of the turbulent fluxes) than high resolution runs (4x and 6x). However, at very high resolution (12x) convection becomes again as strong as for the low resolution simulations. This is consistent with the behavior of $\tau_{\mathrm{adv}} / \tau_{\mathrm{heat}}$ shown in Figure \[fig:timescales\]. Note, however, that, given the large time variations of the fluxes, these differences are not at a sufficient level to draw strong conclusions concerning the behavior with resolution. Also, as for the timescales, the differences in the energy fluxes with resolution might become more pronounced for models that are closer to the explosion threshold. ![\[fig:entropy\] Time- and angle-averaged entropy profiles for the $f_{\mathrm{heat}}=1.0$ simulations. $r_g$ and $r_s$ are the gain and shock radius respectively. The time average excludes the first $t \simeq 192\ \mathrm{ms}$ and it is carried out until the end of the simulation. The zero of the entropy is chosen according to Equation . Multi-dimensional convection tends to flatten the average entropy profile, with respect to the initial conditions ([c.f. ]{}Figure \[fig:init\_data\_entropy\_omega\]) or the 1D spherically symmetric simulation. However, convection is not efficient enough to completely cancel out the entropy gradient.](fig6.pdf){width="\columnwidth"} We note that @yamasaki:06 constructed an analytic model to study the effects of convection on the critical luminosity needed for explosion [@burrows:93] and found, instead, turbulent energy transport due to convection to have a very significant effect. The discrepancy between our results and the model of @yamasaki:06 is due to the fact that in their model @yamasaki:06 estimated the turbulent energy fluxes assuming convection to be efficient enough to cancel the unstable gradient in the angle-averaged radial entropy profile. This is, however, not what it is found in simulations. The time- and angle-averaged entropy profiles from our simulations are shown in Figure \[fig:entropy\]. As for the fluxes, we remap the data to be a function of $r_\star$ and then average in time. We find that multi-dimensional convection is able to stabilize (and actually over-stabilize) the average entropy gradient at the base of the gain region and flatten it over most of the gain layer, as compared to the initial data ([c.f. ]{}Figure \[fig:init\_data\_entropy\_omega\]), or to the 1D simulation. However, convection is not efficient enough to completely remove the unstable stratification and the average entropy profile still has a negative radial gradient over most of the gain region. Similar entropy profiles have also been reported in other simulations with varying degree of sophistication, ([e.g., ]{}@murphy:08 [@hanke:12; @dolence:13; @ott:13a]). This means that @yamasaki:06 overestimated the turbulent enthalpy fluxes induced by neutrino-driven convection, which, according to simulations are not as large as to cancel the entropy gradient. As a consequence, the model of @yamasaki:06 overestimates the effects of the turbulent energy transport on the critical luminosity. Momentum transport {#sec:results.turbpress} ------------------ In the light of our previous discussion, we can conclude that thermal energy transport by turbulence appears to be only a $\sim 10\%$ effect. On the other hand, the fact that turbulence dominates the kinetic energy balance suggests that turbulence, and, in particular, turbulent pressure, might have a more important role in the momentum equation. This is indeed what was already suggested in various other studies (@murphy:13 [@couch:15a; @radice:15a]). ![\[fig:turb.press\] Time- and angle-averaged ratio between the radial turbulent pressure and the thermal pressure for the $f_{\mathrm{heat}}=1.0$ runs. $r_g$ and $r_s$ are the gain and shock radius respectively. The time average excludes the first $t \simeq 192\ \mathrm{ms}$ and it is carried out until the end of the simulation. Turbulence provides a significant contribution (of the order of $\simeq 30\%$) to the total pressure over most of the gain region and up to the shock, although $R_r^{\protect\phantom{r}r} / \langle r^2 p\rangle$ shows large variations close to the shock where the standard deviation of the pressure becomes of the order of the pressure due to the pressure jump across the shock.](fig7.pdf){width="\columnwidth"} To analyze this effect in detail, similarly to what we have done for the energy Equation , we consider the angle averaged radial momentum equation, $$\label{eq:momentum} \partial_t \langle r^2 A S_r \rangle + \partial_r \langle r^2 S_r \upsilon^r \rangle = - \partial_r \langle r^2 p \rangle + \mathcal{G}_S\,,$$ where the radial momentum is $S_r = \rho\, h\, W^2\, \upsilon_r$ and $\mathcal{G}_S$ is the term containing geometric, gravitational, and neutrino source terms in the momentum equation $$\begin{split} \mathcal{G}_S = A^2 r^2 W^2 (\upsilon^r)^2 \mathcal{L} - \big[ E + p + S_r \upsilon^r \big] A^2 + \frac{R_{\theta}^{\phantom{\theta}\theta} + R_{\phi}^{\phantom{\phi}\phi}}{r} + 2 p r\,, \end{split}$$ where $R_i^{\phantom{i}j}$ is the Reynolds stress tensor, which we define to be $$\label{eq:reynolds.stress} R_i^{\phantom{i}j} = \langle r^2 \rho h W^2 \upsilon_i\, \delta\upsilon^j \rangle.$$ The reason for the $r^2$ factor in this definition is that this simplifies the notation when considering spherically averaged equations. The flux term $F_S = \langle r^2 S_r\, \upsilon^r \rangle$ in the LHS of can also be decomposed in a mean and a turbulent part $F_S = \bar{F}_S + F_S' = \langle r^2 S_r \rangle \langle \upsilon^r \rangle + \langle r^2 S_r\, \delta \upsilon^r \rangle$. So that the momentum equation can be rewritten as $$\label{eq:momentum.turb} \partial_t \langle r^2 A S_r \rangle + \partial_r \bar{F}_S + \partial_r F_S' = - \partial_r r^2 \langle p \rangle + \mathcal{G}_S.$$ We note that $F_S' = R_r^{\phantom{r}{r}}$ is the radial component of the Reynolds stress tensor. Beside gravity, the two most important components of equation are the turbulent pressure $R_r^{\phantom{r}{r}}$ and the thermal pressure $\partial_r \langle r^2 p \rangle$. We find the mean angle-averaged momentum flux $\bar{F}_S$ to be contributing only $\sim 10\%$ of the total momentum flux. The remainder is carried by turbulence $F_S' = R_r^{\phantom{r}{r}}$ over most of the gain region. In Figure \[fig:turb.press\] we show the time- and angle-averaged ratio of the turbulent pressure $R_r^{\phantom{r}{r}}$ and the thermal pressure $r^2 \langle p \rangle$ for our runs. As for the energy fluxes, we time-average the data starting at $t \simeq 192\ \mathrm{ms}$ rescaling them as a function of $r_\star$ (Equation \[eq:mapped.radius\]). The shaded area shows the standard deviation (in time) of the 12x simulation. We find the turbulent pressure to provide roughly $\sim 30\%$ of the total pressure support over most of the gain region and close to $\sim 20\%$ at the location of the shock in a time-average sense. As highlighted by the shaded region in Figure \[fig:turb.press\], the ratio of turbulent to thermal pressure does, however, show significant (tens of %) deviations in time. These variations are particularly large close to the shock, because there the pressure has variations of order 1 (given that the pre-shock pressure is negligible). Turbulent pressure support drops near the base of the gain region and in the cooling layer, where turbulence is suppressed by the strong stable stratification near the surface. ![\[fig:turb.press.fheat\] Time- and angle-averaged ratio between the radial turbulent pressure and the thermal pressure as a function of the heating factor $f_{\mathrm{heat}}$ for the reference resolution. $r_g$ and $r_s$ are the gain and shock radius respectively. The time average excludes the first $t \simeq 192\ \mathrm{ms}$ and it is carried out until the end of the simulation. As the simulations approach the threshold for explosion the turbulent contribution to the total pressure becomes increasingly important. ](fig8.pdf){width="\columnwidth"} We point out that the ratio of the effective turbulent pressure to the thermal pressure is very sensitive to $f_{\mathrm{heat}}$: as $f_{\mathrm{heat}}$ changes from $0.9$ to $1.1$ at the reference resolution, the maximum of the time-averaged ratio grows from $\sim 20\%$ and saturates at the $\sim 40\%$ level as shown in Figure \[fig:turb.press.fheat\]. There we show the time- and angle-averaged ratio of turbulent to thermal pressure for simulations with different heating factors $f_{\mathrm{heat}}$ (see Equation \[eq:lightbulb\]) at the reference resolution. The time-average window is the same as for Figure \[fig:turb.press\]. What we find is consistent with what was found by @couch:15a who also find the ratio between turbulent pressure and pressure to be significant. In their simulations, at the transition to explosion, the effective pressure support from turbulence exceeds $50\%$ of the thermal pressure. The behavior with resolution for the ratio between turbulent and thermal pressure is in line with what we find for the energy fluxes or the characteristic timescales. Turbulent support initially appears to decrease with resolution, but rises again at the highest resolution (12x). Obviously, the same caveats discussed in the case of the energy fluxes hold here. The time variations of the ratio between turbulent pressure and pressure, with typical amplitudes of the order of $30\%$, are such that we cannot draw strong conclusions concerning their behavior with resolution based only on the differences we observe. There is, however, a clear correlation between shock radii, enthalpy fluxes and radial Reynolds stresses: large shock radii are found in simulations with high turbulent enthalpy flux and turbulent pressure. ![\[fig:reynolds.stress\] Time- and angle-averaged radial profiles for radial and tangential Reynolds stresses (Equation \[eq:reynolds.stress\]) for the $f_{\mathrm{heat}} = 1.0$ runs. The Reynolds stresses show large time and spatial deviations from their average value. The gray shaded region shows the standard deviation of the 12x run. Turbulence is anisotropic with $R_r^{\protect\phantom{r}r} \simeq 2 R_{\theta}^{\protect\phantom{\theta}\theta}$ over most of the gain region.](fig9.pdf){width="\columnwidth"} The time- and angle-averaged profiles for the radial and angular components of the Reynolds stresses are shown in Figure \[fig:reynolds.stress\]. We show $R_r^{\phantom{r}r}$ and $R_\theta^{\phantom{\theta}\theta}$ as a function of the normalized radius $r_\star$ (note that since our background model is non-rotating, $R_\phi^{\phantom{\phi}\phi} \sim R_\theta^{\phantom{\theta}\theta}$). The turbulence is highly anisotropic with $R_r^{\phantom{r}r} \sim 2 R_\theta^{\phantom{\theta}\theta} \sim 2 R_\phi^{\phantom{\phi}\phi}$ over most of the gain region, with the important exception of the regions close to the shock where there is near equipartition between $R_r^{\phantom{r}r}$ and $R_\theta^{\phantom{\theta}\theta}$ as also observed in other simulations, see [e.g., ]{}[@murphy:13; @couch:15a]. The angular components of the Reynolds stress also become dominant in the cooling layer, where radial motions are strongly suppressed by the steep stratification. The Effects of Turbulence {#sec:results.turbulence} ========================= We have seen that the effective pressure from turbulence can contribute a significant fraction of the total pressure support in the gain region. However, it is not a-priori clear how to translate this into terms of the global evolution of a . For example, one may ask the important question of how much the effective pressure from turbulence contributes to the evolution of the shock radius. We address this question in the following by means of a model explaining the shock radius evolution in terms of measurable flow quantities in the gain region. Momentum Balance Equation {#sec:results.turbulence.mom} ------------------------- We are going to derive an equation explaining the influence of turbulence on the shock radius evolution, which will be an extension of the approach introduced by @murphy:13. They considered the Rankine-Hugoniot conditions for a standing accretion shock in a supernova core. In our notation $$\label{eq:rankine.hugoniot.classic} [F_S + r^2 p]_d = [F_S + r^2 p]_u\,,$$ where $u$ and $d$ denote upstream and downstream values, respectively. They showed how these equations could be modified to account for the turbulent pressure. They formally decomposed the momentum flux in a turbulent and average part in Equation to obtain, assuming a purely radial accretion flow upstream from the shock, $$\label{eq:rankine.hugoniot.turb} [\bar{F}_S + R_r^{\phantom{r}r} + r^2 p]_d = [\bar{F}_S + r^2 p]_u\,.$$ This equation is not entirely rigorous because it uses averaged quantities inside a non-averaged equation[^6], however it has been shown to be well reproduced by the numerical simulations of @murphy:13 and @couch:15a. In particular, @couch:15a found that the turbulent pressure expressed in this fashion can be up to $50\%$ of the thermal pressure, making a very significant contribution to the momentum balance in Equation . @radice:15a pointed out that the effective adiabatic index of turbulence, which is related to the efficiency with which turbulent energy density is converted into thermal support, is $\gamma_{\mathrm{turb}} \simeq 2$, which is much larger than the $\gamma = 4/3$ of a radiation pressure dominated gas. This makes turbulent energy more “valuable” than thermal energy in the sense that, per unit specific internal energy, turbulent energy contributes a greater effective pressure than thermal energy. More recently, @murphy:15 extended Equation using the integral form of the momentum and energy equations with the goal of developing a new explosion condition, but they did not include the effects of turbulence in their analysis. See also @gabay:15 for an alternative approach for the derivation of an explosion condition based on the use of a virial-like relation for the moment of inertia of the accretion layer around the . Here, we extend the approach of @murphy:13 in a way similar to [@murphy:15], but with the different goal of finding a way to quantify the effects of turbulence on the explosion and not of constructing an explosion diagnostic, which would be inappropriate given the limitations of our model. Similarly to @murphy:15, our starting point is Equation . Let us consider two spheres with radius $r_1$ and $r_0$, with $r_1 \geq r_0$. Then Equation can be integrated between $r_0$ and $r_1$ to yield $$\label{eq:momentum.integrated} \begin{split} \langle F_S(r_1) + r_1^2 p(r_1) \rangle = \langle & F_S(r_0) + r_0^2 p(r_0) \rangle + \\ &\int_{r_0}^{r_1} \langle \mathcal{G}_S \rangle {\mathrm{d}}r - \int_{r_0}^{r_1} \partial_t \langle r^2 A S_r \rangle {\mathrm{d}}r\,, \end{split}$$ where $\langle F_S \rangle = \bar{F}_S + R_r^{\phantom{r}r}$. Note that for stationary solutions, in spherical symmetry and in the limit $r_0 \to r_s^-$ and $r_1 \to r_s^+$, $r_s$ being the shock radius, Equation reduces to Equation . In the spherically symmetric, but unsteady case and in the same limit ($r_0 \to r_s^-$ and $r_1 \to r_s^+$), Equation yields the explosion condition derived by @murphy:15. This is so, because, in the unsteady case, one finds $$\label{eq:shock.velocity} \lim_{\epsilon \to 0^+} \int_{r_s-\epsilon}^{r_s+\epsilon} \partial_t r^2 A S_r = \upsilon_s r_s^2 A(r_s) \big[ S_r(r_s^-) - S_r(r_s^+) \big]\,,$$ where $\upsilon_s$ is the shock velocity. Since our goal is to derive an equation for the shock radius directly and not an explosion condition, we proceed differently from @murphy:15. Our starting point is the observation that in the case in which $r_1 > r_{s,\max}$, the LHS of Equation is well approximated by the ram pressure of a free falling gas, [i.e. ]{}$\langle F_S(r_1) + r_1^2 p(r_1) \rangle \simeq F_S(r_1) \propto r_1^{-1/2}$. This suggests that an equation for $\sim r_1^{1/2}$ can be formally derived by integrating Equation again with respect to $r_0$. Since we are interested in the dynamics of neutrino-driven turbulent convection, we extend the second integral over the whole gain region and up to $r_1$. This yields $$\label{eq:momentum.integrated.2} \begin{split} (r_1 - r_g)& \big\langle F_S(r_1) + r_1^2 p(r_1)\big\rangle = \\ & \int_{r_g}^{r_1} \langle F_S + r^2 p \rangle {\mathrm{d}}r + \int_{r_g}^{r_1} {\mathrm{d}}r \int_{r}^{r_1} \langle \mathcal{G}_S \rangle {\mathrm{d}}r' - \\ & \qquad\qquad \int_{r_g}^{r_1} {\mathrm{d}}r \int_{r}^{r_1} \partial_t \langle (r')^2 A S_r \rangle {\mathrm{d}}r'\,, \end{split}$$ where $r_g$ is the gain radius. The maximum shock radius is implicitly determined by this equation as being the smallest value of $r_1$ for which Equation holds when the expressions for the unperturbed pre-shock accretion shock momentum flux and pressure are used as the LHS. If $r_1$ is chosen to be the maximum shock radius $r_{s,\max}$, Equation can be used to measure the relative importance of the different terms of the momentum equation on the shock radius. In steady state, the RHS of Equation contains terms describing the influence of 1. the background momentum flow $$\label{eq:shock.evo.mom.laminar} \int_{r_g}^{r_{s,\max}} \bar{F}_S {\mathrm{d}}r\,,$$ 2. thermal pressure support $$\label{eq:shock.evo.press} \int_{r_g}^{r_{s,\max}} \langle r^2 p \rangle {\mathrm{d}}r + 2 \int_{r_g}^{r_{s,max}} {\mathrm{d}}r \int_{r}^{r_{s,\max}} \langle p r'\rangle {\mathrm{d}}r'\,,$$ 3. turbulent pressure $$\label{eq:shock.evo.turb} \int_{r_g}^{r_{s,\max}} R_r^{\phantom{r}r} {\mathrm{d}}r\,,$$ 4. momentum deposition by neutrinos (in the approximation of our simplified prescription) $$\label{eq:shock.evo.neu} \int_{r_g}^{r_{s,\max}} {\mathrm{d}}r \int_{r}^{r_{s,\max}} A^2 (r')^2 W^2 (\upsilon^r)^2 \mathcal{L} {\mathrm{d}}r'\,,$$ 5. gravity $$\label{eq:shock.evo.grav} - \int_{r_g}^{r_{s,\max}} {\mathrm{d}}r \int_{r}^{r_{s,\max}} [E + p + S_r \upsilon^r] A^2 {\mathrm{d}}r'\,,$$ and 6. centrifugal support $$\label{eq:shock.evo.rotation} \int_{r_g}^{r_{s,max}} {\mathrm{d}}r \int_{r}^{r_{s,\max}} \frac{R_{\theta}^{\phantom{\theta}\theta} + R_{\phi}^{\phantom{\phi}\phi}}{r'}{\mathrm{d}}r'\,.$$ We denote the sum of all of these terms as $$\label{eq:shock.evo.rhs} \mathcal{F}(t) = \int_{r_g}^{r_1} \langle F_S + r^2 p \rangle {\mathrm{d}}r + \int_{r_g}^{r_1} {\mathrm{d}}r \int_{r}^{r_1} \langle \mathcal{G}_S \rangle {\mathrm{d}}r'\,,$$ while the term containing the time derivative of the momentum, which we compute as the residual of the stationary equation, will be denoted as $$\label{eq:shock.evo.residual} \mathcal{R}(t) = - \int_{r_g}^{r_1} {\mathrm{d}}r \int_{r_0}^{r_1} \partial_t \langle (r')^2 A S_r \rangle {\mathrm{d}}r'\,.$$ Both $\mathcal{F}(t)$ and $\mathcal{R}(t)$ are only functions of time. $\mathcal{F}(t)$ encodes the flow of momentum across the gain region, while the residual $\mathcal{R}(t)$ encodes time variations of the flow in the gain region and the shock velocity through Equation . If we set $r_1 = r_{s,\max}$ and use the expressions for the unperturbed upstream accretion flow, $$g(r_{s,\max}) = (r_{s,\max} - r_g) \big[ F_S + r_{s,\max}^2 p\big]_{\mathrm{free fall}}\,,$$ Equation can be written as a formal equation for the shock radius $$\label{eq:momentum.integrated.3} r_{s,\max}^{1/2}(t) \sim g\big(r_{s,\max}(t)\big) = \mathcal{F}(t) + \mathcal{R}(t)\,.$$ ![image](fig10a.pdf){width="\columnwidth"} ![image](fig10b.pdf){width="\columnwidth"} We show all terms of the RHS of Equation for the 12x run in Figure \[fig:shock.radius.equation\]. All of the terms are normalized by the sum of all terms, $\mathcal{F}$. For the purpose of making the figure easier to read, we separate these terms into two groups: terms associated with the “background” flow, shown in the left panel of the figure, and terms associated with the turbulent motion of the fluid, shown in the right panel. Thermal pressure and gravity are the two largest terms in magnitude in Equation . The laminar part of the momentum flux (Equation \[eq:shock.evo.mom.laminar\]) gives only a minor contribution and momentum deposition by neutrinos (Equation \[eq:shock.evo.neu\]) is unsurprisingly negligible. Note that this does not mean that neutrino heating is negligible, but only that the direct deposition of momentum by neutrinos is small. The fact that thermal pressure term dominates in Figure \[fig:shock.radius.equation\] is not in contrast with what we find for the ratio of turbulent pressure to thermal pressure (Figure \[fig:turb.press\]). Namely that turbulent pressure contributes significantly to the pressure balance. The reason for this apparent discrepancy is that most of the pressure support to the flow in Equation comes from the bottom of the gain region, where turbulent pressure is only $\sim 5\%$ of the thermal pressure. One of the first things that one notes from Figure \[fig:shock.radius.equation\] is that pressure and gravity nearly cancel each other. This means that, as a very first approximation, the flow can be considered to be quasi-stationary. The cancellation between pressure and gravity is, however, far from being exact, as can be seen from the red graph in the right panel of Figure \[fig:shock.radius.equation\], which summarizes the net effect of all background flow terms. This implies that, although the background flow is quasi-stationary, it is not static, but undergoes a secular evolution (mainly driven by the accumulation of mass and energy in the gain region; Section \[sec:results.general\]). The presence of secular changes in the flow is confirmed by the fact that $\mathcal{R}(t)$, which measures the rate of change of the total momentum of the flow, always has a positive sign and oscillates around a constant value for most of the simulation. Turbulent convection can be seen as a perturbation on top of this slowly evolving background. The amplitude of the terms associated with turbulence in the right panel of Figure \[fig:shock.radius.equation\] is small if compared to that of those associated with gravity and thermal pressure. This means that the turbulent eddies are not strong enough to drastically alter the overall settling of the accretion flow. However, turbulent fluctuations are rather large on the scale associated with that of the secular changes of the accretion flow, as can be seen from the fact that turbulence terms in the right panel of Figure \[fig:shock.radius.equation\] are as large as the residual or the net background flow. This is not too surprising in the light of our discussion in Section \[sec:results.convection\] on the energy and momentum equations. There we showed that turbulence produces large scale changes in the energy and momentum fluxes in space and time, but that the advection flow still dominates the overall energetics of the flow. Figure \[fig:shock.radius.equation\] provides a more quantitative and well defined way to measure this contribution. Finally, it is worth pointing out that centrifugal support from non-radial motion produced by turbulence also provides a significant contribution to the dynamics of the flow. It provides of order $\sim 25\%$ of the turbulent pressure support. This is a factor that has been neglected in previous studies. A Model for the Shock Evolution ------------------------------- So far, we have constructed a formal equation for the shock radius Equation via momentum conservation. Then, we used this equation as a way to measure the relative importance of different terms in giving support to the shock. The following question arises naturally: is the equation we are using merely a trivial identity involving the shock radius? Or do the quantities $\mathcal{F}$, $\mathcal{R}$ and $r_s$ have deeper connections? Clearly, in the first case the analysis presented above would be of little value. The results of our simulations suggest that this is not the case and that $\mathcal{F}$ and $\mathcal{R}$ are relevant quantities determining the evolution of $r_s$. In particular, we find that, given $\mathcal{F}$, $\mathcal{R}$, and the shock radius at a given time, it is possible to predict $r_{s,\mathrm{avg}}$ over a fraction of the advection timescale $\simeq 0.3 \tau_{\mathrm{adv}}$ (Equation \[eq:advection.timescale\]) with reasonable accuracy using a simple linear model based on Equation $$\label{eq:shock.evolution} \left[\frac{r_{s,\mathrm{avg}}(t + 0.3 \tau_{\mathrm{adv}})}{1\ \mathrm{km}}\right]^{1/2} = A + B \mathcal{F}(t) + C \mathcal{R}(t)\,,$$ where $A$, $B$ and $C$ are coefficients that we fit using a least-squares procedure. For the 12x model, we find $A \simeq 4.84$, $B \simeq 8.54 \times 10^{-44}\, \mathrm{s}^2\ \mathrm{g}^{-1}$ and $C \simeq 1.60 \times 10^{-43}\, \mathrm{s}^2\ \mathrm{g}^{-1}$. We find similar values also for the other resolutions. However, we do not expect these values to be in any way universal. Actually, we find them to change by factors of order of a few when varying the heating factor $f_{\mathrm{heat}}$ in Equation . More on this below. Finally, we note that it is also possible to construct a similar model for $r_{s,\max}$. We focus on $r_{s,\mathrm{avg}}$ because its time evolution is smoother, while $r_{s,\max}$ necessarily “jumps” in steps that are multiple of the grid spacing, since our analysis is not able to identify the location of the shock to better than a single cell. ![\[fig.shock.radius.model\] Average shock radius evolution from the 12x run ($f_{\mathrm{heat}}=1.0$) at the retarded time $t + 0.3 \tau_{\mathrm{adv}}$ and its predicted value from the shock evolution Equation . For comparison we also show the average shock radius at the time when the prediction is made. The average shock radius can be accurately predicted over a fraction of the advection timescale (Equation \[eq:advection.timescale\]) from the knowledge of $\mathcal{F}$ (Equation \[eq:shock.evo.rhs\]) and $\mathcal{R}$ (Equation \[eq:shock.evo.residual\]).](fig11.pdf){width="\columnwidth"} The results obtained with this simple model for the shock radius are shown in Figure \[fig.shock.radius.model\]. There, we show the prediction for the average shock radius at time $t + 0.3 \tau_{\mathrm{adv}}$ computed using the data available at time $t$, $\mathrm{Model}(t)$, the actual value of the average shock radius at the retarded time, $r_{s,\mathrm{avg}}(t + 0.3 \tau_{\mathrm{adv}})$, and the average shock radius at the time when the prediction is made, $r_{s,\mathrm{avg}}(t)$. As can be seen from the figure, the model specified by Equation is able to predict both the shock radius oscillations and the secular trend of the shock radius with high accuracy. Note that we did not include any explicit term to model this trend in our fit: the entire shock evolution is contained in $\mathcal{F}$ and $\mathcal{R}$. It is important to stress the fact that $\mathcal{F}$ and $\mathcal{R}$ encode information concerning the current shock position, as well as the flow in the gain region at the time when they are computed. They cannot be computed without resorting to a fully non-linear simulation. In this sense, Equation is not predictive. The intriguing aspect of Equation is that it suggests that $\mathcal{F}$ and $\mathcal{R}$ also encode information concerning the future shock position in a form which is easily extracted. This provides a validation to our interpretation of the different components of $\mathcal{F}$ and of their role in shaping the shock evolution (Section \[sec:results.turbulence.mom\]). It is interesting to consider whether our simple model is able to predict the onset of a runaway explosion. This is difficult to fully assess with our simplified simulations, because we neglect important effects leading to the explosion, such as the sudden drop in the accretion rate following the accretion of the $\mathrm{Si/Si\, O}$ interface [@buras:06b; @mezzacappa:07; @ugliano:12] or asphericities in the accretion flow (@couch:13d [@mueller:15]). We also neglect the feedback of accretion on the neutrino luminosity and we do not include all of the necessary microphysics for a fully quantitative study. \[sec:shock.evo.and.fheat\] ![\[fig:shock.radius.model.fheat\] Shock radius at the retarded time $t + 0.3 \tau_{\mathrm{adv}}$ (dots) and its predicted value from the shock evolution (Equation \[eq:shock.evolution\]) (solid lines) for the reference resolution, but using different heating factors.](fig12.pdf){width="\columnwidth"} As a first step to study the reliability of our model for exploding simulations, we carry out a preliminary study were we obtain shock expansion / contraction by changing the value of $f_{\mathrm{heat}}$ in Equation and fit the resulting shock evolutions using Equation . We perform these simulations at the reference resolution. As anticipated above, the fitting coefficients vary across the different runs. Nevertheless, we find Equation to be well verified by all simulations. We show the results of this analysis in Figure \[fig:shock.radius.model.fheat\]. As can be seen from this figure, the agreement between the predicted shock radius evolution from Equation and the retarded average shock radius is reasonably good even as the heating factor is changed to the point that the model is starting to explode. The agreement is not as good as for the 12x model, possibly due to the higher numerical noise present in the reference resolution data in the cooling layer and in the first few grid points at the base of the gain region, where density and pressure increase steeply. This noise can contaminate $\mathcal{F}$ and $\mathcal{R}$ which are the two building blocks of Equation . Turbulent Cascade {#sec:results.cascade} ================= Following, [e.g., ]{}@hanke:12 [@couch:13b; @dolence:13; @couch:14a; @couch:15a] and @abdikamalov:15, we consider the power spectrum of the turbulent velocity $\delta \upsilon^i$ (Equation \[eq:turb.velocity\]). Differently from most previous studies, however, we do not consider the spherical harmonics decomposition of the turbulent velocity on a sphere, but study the actual 3D spectrum of the turbulence. During the evolution, at each time starting from $t \simeq 192\ \mathrm{ms}$ ($320\ \mathrm{ms}$ for the 20x resolution), we restrict our attention to the largest cubic region, $B$, entirely contained in the convectively unstable gain region and interpolate the turbulent velocity from the spherical grid used in our simulations to a uniform Cartesian grid defined on this region. Then, we compute the specific turbulent energy spectrum as the Fourier transform of the two-point correlation function $$\label{eq:turb.energy.spec} \tilde{E}(k) = \frac{1}{2} \int_{\mathbb{R}^3} \delta(\|\mathbf{k}\| - k) \widehat{\delta \upsilon_i^}(\mathbf{k}) \widehat{\delta \upsilon^i}(\mathbf{k}) {\mathrm{d}}\mathbf{k} \,,$$ where $\delta(\cdot)$ is the Dirac delta, $\cdot^$ denotes the complex conjugation and $\widehat{\delta \upsilon^i}$ is computed as $$\label{eq:fourier.transform} \widehat{\delta \upsilon^i}(\mathbf{k}) = \int_B W(\mathbf{x}) \delta \upsilon^i(\mathbf{x}) \exp\left(2 \pi {\mathrm{i}}\mathbf{k}\cdot\frac{\mathbf{x}}{L}\right) {\mathrm{d}}\mathbf{x}\,.$$ $W$ is a windowing function that smoothly goes to zero at the boundary of the box (more on this later), $L$ is the box size, and we neglected general-relativistic corrections in computing $\widehat{\delta \upsilon_i^}$. Finally, in order to account for secular oscillations in the total turbulent energy, we normalize the spectrum to have unit integral $$\label{eq:turb.energy.spec.norm} E(k) = \left[\int_0^\infty \tilde{E}(k) {\mathrm{d}}k\right]^{-1} \tilde{E}(k) \,,$$ and we average in time. Windowing in the definition of the Fourier transform is required because our data is not periodic. In particular, $W(\mathbf{x})$ is computed as $$W(\mathbf{x}) = W\left(\frac{x-x_0}{x_1-x_0}\right) W\left(\frac{y-y_0}{y_1-y_0}\right) W\left(\frac{z-z_0}{z_1-z_0}\right)\,,$$ where the $(x_0, y_0, z_0)/(x_1, y_1, z_1)$ are the minimum/maximum values of, respectively, $x,y$ and $z$ in the cube where we compute the spectra, $$W(x) = \begin{cases} \exp\left[\frac{-1}{1 - \left(\frac{x-\Delta}{\Delta}\right)^2}\right] & \textrm{if } x < \Delta\,, \\ 1 & \textrm{if } \Delta \leq x \leq 1 - \Delta\,, \\ \exp\left[\frac{-1}{1 - \left(\frac{x-(1-\Delta)}{\Delta}\right)^2}\right] & \textrm{if } x > 1 - \Delta\,, \\ 0 & \textrm{otherwise}\,, \end{cases}$$ and $\Delta$ is the grid spacing for the Cartesian grid, which we take to be equal to $\Delta r$. ![image](fig13a.pdf){width="0.98\columnwidth"} ![image](fig13b.pdf){width="\columnwidth"} The normalized specific turbulent energy spectrum is shown in the left panel of Figure \[fig:spectrum\] for various resolutions. The right panel shows the spectra compensated ([i.e.* ]{}multiplied) by $k^{5/3}$ to highlight regions with Kolmogorov scaling, which we could expect on the basis of previous high-resolution local simulations [@radice:15a]. The shaded regions around each spectrum denote the standard deviation of the spectrum during the time-average window. The slope of the spectra of low resolution simulations is rather shallow and consistent with a $k^{-1}$ scaling. The low resolution spectra are comparable to the ones reported at low-resolution by previous studies [@dolence:13; @couch:14a; @couch:15a; @abdikamalov:15]. As argued by, @abdikamalov:15 and @radice:15a, the $k^{-1}$ slope is due to the bottleneck effect artificially trapping turbulent energy at large scale. As the resolution increases, the spectra become progressively steeper, but even the 12x resolution still shows a shallow $-4/3$ slope, indicative of the fact that even at this resolution the turbulence cascade is probably dominated by the bottleneck effect. However, the 20x resolution, which has over $1100$ points covering the gain region in the radial direction ($\sim 15$ times more than previous high-resolution simulations; @abdikamalov:15), finally shows an extended region compatible with the $k^{-5/3}$ scaling of Kolmogorov’s theory. This is particularly evident in the right panel of Figure \[fig:spectrum\] where we show the compensated spectrum. This shows unambiguously that the shallow slopes reported in the literature are a finite-resolution effect, as previously argued by @abdikamalov:15 and @radice:15a. It also give credence to the idea that the turbulent cascade of kinetic energy in neutrino-driven convection is well described by Kolmogorov theory, despite the presence of non-classical effects such as anisotropy, the geometric convergence of the flow, and mild compressibility. Unfortunately, the computational costs of running at the 20x resolution are prohibitive even for our simplified setup and we could not run the 20x simulation for more than $\simeq 60\ \mathrm{ms}$, which is roughly equal to $\tau_{\mathrm{adv}}$. This is enough to study the energy spectrum at intermediate and small scales (including the inertial range), which we find to have reached a new equilibrium already $\sim 3\ \mathrm{ms}$ after the mapping from the 12x run. It is also sufficient to serve as benchmark data for future validation of turbulence models. Generating such benchmark data is one of the goals of the present work. It is not enough evolution time, however, to assess whether the structure of the gain layer and the general dynamics of the simulation model changes once the inertial range starts to be resolved. Conclusions {#sec:conclusions} =========== Multi-dimensional instabilities are expected to play a fundamental role in the mechanism powering most e ([e.g., ]{}@murphy:11 [@murphy:13; @hanke:13; @couch:13b; @takiwaki:14a; @couch:15a; @melson:15a; @lentz:15; @melson:15b]). Neutrino-driven convection, in particular, is most commonly associated with post-collapse evolutions having strong neutrino heating and, in general, conditions that are most favorable for explosion [@dolence:13; @murphy:13; @ott:13a; @couch:13b; @couch:14a; @takiwaki:14a; @abdikamalov:15], however it is not excluded that dominated e could also explode [@hanke:13; @fernandez:15a; @cardall:15]. Despite its central role in the context of the delayed neutrino mechanism, neutrino-driven convection has not been studied in a systematic way before. Previous studies were either performed in 2D, [e.g., ]{}[@murphy:11; @fernandez:14], or spanned a relatively small range in resolution [@hanke:12; @dolence:13; @couch:14a; @couch:15a]. Hence, it is difficult to assess to what level they are affected by finite-resolution effects. @abdikamalov:15 and @radice:15a showed that, at the resolutions typically used in 3D simulations the dynamics of the turbulent cascade of energy from large to small scale is severely affected by numerical viscosity. This artificially prevents kinetic energy from decaying to small scales and leads to an unphysical accumulation of energy at the largest scales, a phenomenon known as the bottleneck effect [@yakhot:93; @she:93; @falkovich:94; @verma:07; @frisch:08]. This large scale energy, in turn, results in an additional pressure support to the accreting flow [@radice:15a]. Considering the fact that turbulent pressure was found to be crucial in triggering explosions [@couch:15a], having artificially large turbulent pressures, might result in a qualitative change in the evolution of a simulation. For this reason, it is important to quantify finite-resolution effects in simulations. In the present study, we performed a series of semi-global neutrino-driven convection simulations with the goal of understanding the dynamics of neutrino-driven convection and the effects of finite resolution in simulations. Our simulations are rather unsophisticated when compared to state-of-the-art radiation-hydrodynamics simulations, [e.g., ]{}[@melson:15b; @lentz:15]. However, they include most of the basic physics ingredients relevant for neutrino-driven turbulent convection and they have the advantage of being completely under control. The converging geometry, the advection of gas through an accretion shock toward a central , gravity, photo-dissociation of heavy-nuclei at the shock and neutrino/heating cooling are all included in a completely controlled way. The main limitations of our model are that we neglect the non-linear feedback between accretion and neutrino luminosity, which we assume to be constant, and that we fix the amount of specific energy lost to nuclear dissociation. These approximations would be particularly limiting in the study of the transition to explosion of our models. Such a study would require us to follow the shock as it develops large radial displacements with respect to its initial conditions and correctly account for significant changes in the accretion rate. However, this is not the aim of this work. Our current approximations are expected to be adequate for the study of the nearly stationary neutrino-driven convection we report here. We considered a constant accretion rate and analytic, stationary initial conditions so as to be able to perform long term evolutions and collect well-resolved statistics of the turbulent flow. We employed high-order, low dissipation numerical methods, a grid adapted to the problem (a spherical wedge) and we varied the grid scaling across different simulations by a factor 20, achieving unprecedented resolutions for this kind of study, with a radial spacing of $191\ \mathrm{m}$ and an angular resolution of $0.09^\circ$ in the gain layer. We find that, as resolution increases, the qualitative dynamics of the flow changes drastically. At low resolution, the dynamics is characterized by the presence of large, slowly evolving, high-entropy plumes. At higher resolution, the dynamics is dominated by smaller structures evolving on a faster timescale. Given that the transition to explosion seems to be preceded by the formation of large, long-lived, high-entropy plumes [@fernandez:14; @mueller:15; @lentz:15], this is a first indication that low resolution might be artificially favouring explosion. At high resolution, turbulent mixing is very effective at smoothing out sharp interfaces between high and low entropy regions: high-entropy plumes lose their coherence due to small scale mixing and they resemble more “clouds” than “bubbles”. This means that the separation of the flow into very well defined high and low entropy regions seen in most simulations is also an artifact of low resolution. This is not too surprising: the physics of neutrino-driven convection is not that of a multiphase flow. This calls into question the usefulness of arguments describing neutrino-driven convection in terms of an ensemble of “bubbles” moving through the accretion flow. Despite these large qualitative changes with resolution, but as predicted by @radice:15a, we find large scale quantities to be consistent among the different resolutions for our fiducial model. In other words, we find global quantities, such as the shock radius, the typical timescales for advection and heating, to be consistent across all of the resolutions and to be even monotonically convergent in the first $100\ \mathrm{ms}$, a phase in which convection is developing, but the buoyant plumes have not yet managed to strongly interact with the shock. Note, however, that we also find that this picture changes drastically for models where we induce an expansion of the shock by artificially increasing the heating rate. For these models, we find low and high resolution simulation to be diverging after the first $50\ \mathrm{ms}$ and low resolution simulations showing earlier shock expansion. We also find, in agreement with @hanke:12 and @abdikamalov:15, that low resolution typically yields more favorable conditions for explosion, especially at early times. These differences are rather modest for our fiducial set of simulations, but are more pronounced for simulations that are closer to or at the transition to explosion. Given that some of the current full-physics models appear to be on the verge of explosion [@melson:15b], our results serve as an additional reminder that a resolution study is necessary to confirm any result. At the same time, we think, in the light of our findings, that some cautious optimism can be justified in the sense that many quantities of interest in e appear to be well converged at modest resolution, even though others, like the velocity spectra $E(k)$, are affected by serious artifacts until very high resolution is reached. Furthermore, it is interesting to note that our results suggest that the flow dynamics and the resulting CCSN evolution may change quantitatively and qualitatively at very high resolution when turbulence begins to be resolved. The simulation at twelve times our fiducial resolution exhibits a reversal of the just discussed trend with resolution: its shock radius evolution has the steepest slope. Explosion diagnostics such as the ratio of advection to heating timescales suggest that it is approaching explosion faster than less resolved simulations. We cannot fully understand this trend until it becomes possible to carry out even higher-resolution long-term simulations. However, we speculate that our finding may be a consequence of the increasing non-linear coupling of a greater range of scales and the development of strong intermittency as the inertial range of turbulence begins to be resolved. The effects of fully resolved turbulence (perhaps captured by a sub-grid model) may ultimately be beneficial for explosion. In order to better quantify the importance of turbulent convection for explosions, we studied the efficiency with which neutrino-driven convection transports energy and momentum across the gain region. We find the energy balance in the flow to be dominated by the thermal energy and the overall energetics to be driven by the background advective flow. Turbulence opposes the overall negative (down-flowing) radial enthalpy fluxes, but it is only able to contribute a small $\sim 10\%$ correction to the overall thermal energy flow. The kinetic energy evolution, on the other hand, is dominated by turbulence, which provides $\sim 80\%$ of the kinetic energy flux and $\sim 90\%$ of the advective part of the momentum flux (the part of the momentum flux not containing the pressure gradient). We also find the effective pressure support provided by turbulence to be significant and of the order of $\sim 30\%-40\%$ of the thermal pressure in our simulations. According to our results neutrino-driven turbulent convection plays a more important role in the evolution of the momentum than in the evolution of the energy. This suggest that the large differences in, [e.g., ]{}shock radii, between turbulent multi-dimensional and one-dimensional simulations can be mostly accounted for by the effects of turbulence in the momentum equation in agreement with [@couch:15a]. In this respect, we showed that it is possible to derive an equation that can explain and even predict the shock radius evolution over a fraction of the advection timescale starting from integrals of the terms appearing in the radial momentum equation. This new diagnostic generalizes and refines the approach by @murphy:13 in which the shock position was derived starting from an approximated angle-averaged shock jump condition. With our approach it is possible to quantify, in a rigorous way, the relative importance of different terms in providing pressure support to the shock. Our analysis suggests that turbulence plays an indirect role in the revival of the shock. Rather than directly transporting energy to the shock, turbulence acts as an effective barrier slowing down the drain of energy from the shock by the radial advection. We studied the turbulent energy cascade in the gain region by means of the 3D power-spectrum of the turbulent velocity $E(k)$. We find conclusive evidence that the shallow spectra reported by many investigations are the result of the numerical bottleneck effect, as previously suggested by @abdikamalov:15. In particular, we observe that as resolution increases, the spectra become progressively steeper. At the highest resolution, the spectrum has a slope compatible with the $k^{-5/3}$ slope predicted by the classical theory of Kolmogorov, [e.g., ]{}[@pope:00], and as suggested by local simulations [@radice:15a]. Unfortunately, resolving the inertial range of neutrino-driven convection requires resolutions that are not even affordable for a full simulation in our simplified setup and that could only be employed to simulate a relatively short time frame ($\simeq 60\ \mathrm{ms}$) starting from a lower resolution simulation. This was enough to be able to measure the spectrum of the turbulent kinetic energy, which we find to have already reached a new equilibrium after $\sim 3\ \mathrm{ms}$. However, these $60\ \mathrm{ms}$ of evolution are not enough to fully assess the ramifications of not resolving the inertial range in a global simulation. Our simulations appear to be already converged at large scale, but the difference between resolving and not resolving the inertial range could become more substantial for models close to the explosion threshold. At the moment, achieving a resolution sufficient to fully resolve the inertial range dynamics in global simulations seems to be impossible. At the same time, the numerical schemes currently adopted for simulations show rather poor performance for under-resolved turbulent flows [@radice:15a]. It is thus our opinion that performing qualitatively and quantitatively accurate simulations will require the use of some form of turbulent closure. In future work, we plan to use the simulation data presented here as a basis to guide the construction of numerical turbulent closures specialized for applications. Acknowledgements {#acknowledgements .unnumbered} ================ We acknowledge helpful discussions with W. D. Arnett, A. Burrows, C. Meakin, P. Mösta, J. Murphy, and L. Roberts. This research was partially supported by the National Science Foundation under award nos. AST-1212170 and PHY-1151197 and by the Sherman Fairchild Foundation. The simulations were performed on the Caltech compute cluster Zwicky (NSF MRI-R2 award no. PHY-0960291), on the NSF XSEDE network under allocation TG-PHY100033, and on NSF/NCSA BlueWaters under NSF PRAC award no. ACI-1440083. A. Parametrized Nuclear Dissociation Treatment {#sec:nuclear.dissociation} ============================================== Nuclear dissociation is included in a parametrized way using an approach similar to that of [@fernandez:09a; @fernandez:09b], but with some important differences discussed here. [@fernandez:09a] suggested to parametrize the amount of specific internal energy lost to nuclear dissociation, $\epsilon_{\mathrm{ND}}$, as a fraction, $\bar{\epsilon}$, of the free-fall kinetic energy at the initial location of the shock: $$\epsilon_{\mathrm{ND}} = \frac{1}{2} \bar{\epsilon}\, \upsilon_{\mathrm{FF}}^2\,,$$ where $\upsilon_{\mathrm{FF}}$ is the free-fall velocity at the initial location of the shock. In the relativistic case this translates to $$\epsilon_{\mathrm{ND}} = \bar{\epsilon} ( W_{\mathrm{FF}} - 1 )\,,$$ where $W_{\mathrm{FF}}$ is the free-fall Lorentz factor (see Appendix \[sec:standing.accretion.shock\]). A typical range of values for $\bar{\epsilon}$ is $0.2 - 0.4$ [@fernandez:09b]. [@fernandez:09a; @fernandez:09b] used the nuclear burning module of the FLASH code [@fryxell:00] to simulate nuclear dissociation with the inclusion of an energy sink term. This approach is perfectly viable in classical hydrodynamics, but not in the relativistic case, because, in relativistic hydrodynamics, the inertia (and momentum) of the fluid depends on the enthalpy and, for this reason, a sink term in the energy equation would result in an inconsistency with the shock jump conditions. In our implementation, instead, nuclear dissociation is included in the equation of state as follows. We model the effect of the thermal energy lost to nuclear dissociation with a modified gamma-law of the form $$\label{eq:eos} p = (\gamma - 1) \rho \big(\epsilon - \epsilon_{\mathrm{ND}}^\ast(\epsilon)\big)\,,$$ where $$\epsilon^\ast_{\mathrm{ND}}(\epsilon) = \begin{cases} 0\,, & \textrm{if } \epsilon \leq \epsilon_{\mathrm{ND}}\,, \\ \eta (\epsilon - \epsilon_{\mathrm{ND}})\,, & \textrm{if } \epsilon_{\mathrm{ND}} < \epsilon \leq \epsilon_{\mathrm{ND}} \big(\frac{\eta + 1}{\eta}\big)\,, \\ \epsilon_{\mathrm{ND}}\,, & \textrm{if } \epsilon > \epsilon_{\mathrm{ND}} \big(\frac{\eta + 1}{\eta}\big)\,, \end{cases}$$ and $\eta = 0.95$ is an efficiency parameter needed to ensure that $p(\rho,\cdot)$ is a one-to-one function (this is needed for the recovery of $\rho, \upsilon^i$ and $\epsilon$ from the evolved variables at the end of each iteration during the evolution). Another advantage of this approach, as compared to the one of @fernandez:09a [@fernandez:09b], is that it does not involve possibly stiff cooling terms that can give rise to numerical problems. The results presented in this paper are obtained with $\bar{\epsilon} = 0.3$, which corresponds to a value of $\epsilon_{\mathrm{ND}} = 0.003$ ($\simeq 2.7\times 10^{18}\ \mathrm{erg}\ \mathrm{g}^{-1}$) for a shock stalled at $100$ Schwarzschild radii of the ($\simeq 191\ {\mathrm{km}}$). We remark that our results are sensitive to the choice of $\bar{\epsilon}$, because the flow becomes stable against the development of convection when $\bar{\epsilon}$ is sufficiently small. The reason is that smaller $\bar{\epsilon}$ result in larger radial velocities immediately downstream from the shock, which, in turn, prevent buoyant instabilities from growing into fully-developed convection before being advected out of the gain region [@foglizzo:06]. For more details, we refer to the studies of @fernandez:09a [@fernandez:09b], and @cardall:15 that showed the impact of nuclear dissociation on the development of neutrino-driven convection and . B. Relativistic Standing Accretion Shock Solution {#sec:standing.accretion.shock} ================================================= The initial conditions of our simulations represent a stationary standing accretion shock. Our model is similar to the accretion models of [@chevalier:89; @houck:92] and the model of [@janka:01], which has been used in many studies of and convection, [e.g., ]{}@blondin:03 [@foglizzo:06; @cardall:15]. To construct the initial conditions, we solve the equations of relativistic hydrodynamics on top of a fixed gravitational background (Equation \[eq:metric\]) time-independently. Heating, cooling and nuclear dissociation are also taken into account in the same way as for the subsequent numerical evolution. The initial conditions are specified by choosing values for the radius, $r_{\mathrm{PNS}}$, the shock radius, $r_s$, the accretion rate $\dot{M}$, and the heating coefficient $K$. The heating/cooling normalization coefficient $C$ is then tuned so that the velocity vanishes at the radius. Pre-Shock Flow -------------- The pre-shock flow is assumed to be cold and free falling, so that the pre-shock Lorentz factor $W_0$ can be computed from the lapse function at the location of the shock: $W_0 = \frac{1}{\alpha_0}$. The pre-shock density is computed by fixing the accretion rate $\dot{M}$: $$\label{eq:preshock.rho} \rho_0 = \frac{\dot{M}}{4 \pi r_s^2 W_0 \| \upsilon_0^r \|}\,,$$ where $\upsilon_0^r$ is the radial component of the pre-shock velocity. In practice, for numerical reasons, to minimize disturbances in the upstream flow, our initial conditions have a small, but non-zero, pre-shock internal energy $\epsilon_0$, which we compute from the requirement that the Mach number of the upstream flow should be equal to $100$. Shock Jump Conditions --------------------- The post-shock density, pressure and velocity can be computed from the Rankine-Hugoniot conditions of a stationary shock \[eq:rankine.hugoniot\] $$\begin{aligned} \rho_1 W_1 \upsilon_1^r &= \rho_0 W_0 \upsilon_0^r\,, \\ \rho_1 h_1 W_1^2 (\upsilon_1^r)^2 + p_1 \alpha^2 &= \rho_0 h_0 W_0^2 (\upsilon_0^r)^2 + p_0 \alpha^2\,, \\ \rho_1 h_1 W_1^2 \upsilon_1^r &= \rho_0 h_0 W_0^2 \upsilon_0^r\,,\end{aligned}$$ where the indices $0$ and $1$ refer to pre- and post-shock quantities, respectively, and we made use of Equation . Note that the effect of dissociation is automatically included in these jump conditions, since it is accounted for in the equation of state. In the strong shock limit, $\epsilon_0 \ll 1$, and for small post-shock velocities, $W_1 \simeq 1$, they can be simplified as \[eq:ss.rankine.hugoniot\] $$\begin{aligned} \rho_1 \upsilon_1^r &= \rho_0 W_0 \upsilon_0^r\,, \\ \rho_1 h_1 (\upsilon_1^r)^2 + p_1 \alpha^2 &= \rho_0 W_0^2 (\upsilon_0^r)^2\,, \\ \rho_1 h_1 \upsilon_1^r &= \rho_0 W_0^2 \upsilon_0^r\,.\end{aligned}$$ These can easily be solved for the post-shock density, $$\rho_1 = \rho_0 W_0 \frac{\upsilon_0^r}{\upsilon_1^r}\,,$$ specific internal energy $$\label{eq:postshock.eps} \epsilon_1 = \frac{W_0 - 1 + (\gamma - 1)\epsilon_{\mathrm{ND}}}{\gamma}\,,$$ and velocity $$\upsilon_1^r = \frac{\upsilon_0^r + \sqrt{(\upsilon_0^r)^2 - 4 \psi}}{2}\,,$$ where $$\psi = \alpha^2 \frac{\gamma-1}{\gamma} \frac{W_0-1-\epsilon_{\mathrm{ND}}}{W_0}\,.$$ Note that Equation is valid as long as $\bar{\epsilon}$ is sufficiently small so that $$\bar{\epsilon}\bigg(\frac{\eta+1}{\eta} - \frac{\gamma-1}{\gamma}\bigg) \leq \frac{1}{\gamma}.$$ For $\gamma = 4/3$ and $\eta = 0.95$ this means $\bar{\epsilon} \lesssim 0.42$, which is satisfied since we take $\bar{\epsilon} = 0.3$. Post-Shock Flow --------------- The initial post-shock flow is computed by looking for a time-independent version of Equation , with boundary-conditions given by the post-shock density, velocity and specific internal energy. Note that, as we show later, the equations are singular at the point where $\upsilon^r=0$ and no boundary condition is required downstream of the shock. Instead, the normalization coefficient of the heating/cooling source, $S$, has to be adjusted so that $\upsilon^r$ vanishes at the surface of the . The first condition that we use is the continuity equation (first part of Equation \[eq:hydro\]), that, in the stationary, spherically symmetric case, is simply $$\label{eq:trivial.continuity} r^2 \rho W \upsilon^r = \frac{1}{4\pi}\dot{M}\,.$$ The energy equation is rewritten in non-conservation form, by projecting the second part of Equation along the velocity four-vector, to yield (see e.g., @gourgoulhon:06 for a detailed derivation) $$u^\mu \nabla_\mu ( \rho \epsilon ) = - (\rho \epsilon + p)\nabla_\mu u^\mu + \mathcal{L}\,.$$ In the stationary, spherically symmetric case, this becomes $$\label{eq:ode.energy} \frac{\dot{M}}{4\pi} \partial_r \epsilon = - 2 p r W \upsilon^r - p r^2 \partial_r (W \upsilon^r) + \mathcal{L}\,.$$ Similarly, the momentum equation is also rewritten in non-conservation form, by projecting the second part of Equation perpendicularly to $u^\mu$ to obtain the relativistic Euler equations, $$\rho h a_\mu = - \nabla_\mu p - u^\nu \nabla_\nu p u_\mu\,,$$ where $a_\mu$ is the relativistic 4-acceleration vector $$a_\mu = u^\nu \nabla_\nu u_\mu = u^\nu (\partial_\nu u_\mu - \Gamma^{\alpha}_{\phantom{\alpha}\nu\mu} u_\alpha)\,,$$ and $\Gamma^{\alpha}_{\phantom{\alpha}\nu\mu}$ are the Christoffel symbols of the Levi-Civita connection. In our case, the Euler equation reduces to $$\label{eq:ode.momentum} \rho h W \upsilon^r A^2 \partial_r (W \upsilon^r) = \frac{1}{A^4}\frac{2M}{r^2} \rho h W^2 (\upsilon^r)^2 - \big[1 + A^2 W^2 (\upsilon^r)^2\big] \partial_r p - \frac{A^2}{r^2} M W^2 \big[1 + A^2 (\upsilon^r)^2\big] \rho h\,.$$ Note that this equation has a manifest singularity for $\upsilon^r = 0$. In practice, $\upsilon^r$ is never exactly equal to zero, although tuning $C$ yields very small values of $\|\upsilon^r\|$ close to the surface of the , and we find that the ODE integrator of our choice, the implicit multistep “MSBDF” method implemented in the GNU Scientific Library [@galassi:09], is sufficiently robust to handle these equations. Finally we can substitute the derivative of the pressure in the right-hand-side of Equation using the to find $$\label{eq:ode.momentum2} \begin{split} \Big[\rho h W \upsilon^r A^2 - \frac{\gamma p}{W \upsilon} \big(1 + A^2 W^2 (\upsilon^r)^2\big) \Big]& \partial_r (W \upsilon^r) = \\ & \frac{1}{A^4}\frac{2M}{r^2} \rho h W^2 (\upsilon^r)^2 + \frac{1 + A^2 W^2 (\upsilon^r)^2}{r^2 W \upsilon^r} \big[ 2 \gamma r W \upsilon^r p - (\gamma - 1) \mathcal{L} \big] - \frac{A^2}{r^2} M W^2 \big[1 + A^2 (\upsilon^r)^2\big] \rho h\,. \end{split}$$ ![Initial data profiles. Left panel: density and velocity. Right panel: entropy and Brunt-Väisäla frequency $\Omega_{\mathrm{BV}}$ computed as in Equations and .[]{data-label="fig:init_data_entropy_omega"}](fig14a.pdf "fig:"){width="0.495\columnwidth"} ![Initial data profiles. Left panel: density and velocity. Right panel: entropy and Brunt-Väisäla frequency $\Omega_{\mathrm{BV}}$ computed as in Equations and .[]{data-label="fig:init_data_entropy_omega"}](fig14b.pdf "fig:"){width="0.495\columnwidth"} Equations , and are the ones that we solve numerically, together with the , to generate our initial conditions. Figure \[fig:init\_data\_entropy\_omega\], shows the profile of the entropy and the Brunt-Väisäla frequency for the initial conditions used throughout this study. [^1]: $K p_1$ is the equilibrium pressure at the location of the shock when neglecting advection, [i.e. ]{}in the limit of instantaneous heating and cooling. [^2]: Run for $\simeq 60\ \mathrm{ms}$ starting from 12x at $t \simeq 317\ \mathrm{ms}$. [^3]: Run with extended domain: $r_{\max} \simeq 825\, \mathrm{km}$.\[footnote:F105\] [^4]: The luminosity can be obtained by recasting the heating term in Equation into Eq. (28) of @janka:01. [^5]: Note that the accretion rate is constant, so that the advection timescale is proportional to the mass in the gain region. [^6]: Note that, for instance, the averaged equations do not formally have shocks in their solution, because the angle average smooths all of the transitions, unless the shock is perfectly spherical.
--- abstract: 'Computing partition function is the most important statistical inference task arising in applications of Graphical Models (GM). Since it is computationally intractable, approximate methods have been used to resolve the issue in practice, where mean-field (MF) and belief propagation (BP) are arguably the most popular and successful approaches of a variational type. In this paper, we propose two new variational schemes, coined Gauged-MF (G-MF) and Gauged-BP (G-BP), improving MF and BP, respectively. Both provide lower bounds for the partition function by utilizing the so-called gauge transformation which modifies factors of GM while keeping the partition function invariant. Moreover, we prove that both G-MF and G-BP are exact for GMs with a single loop of a special structure, even though the bare MF and BP perform badly in this case. Our extensive experiments, on complete GMs of relatively small size and on large GM (up-to 300 variables) confirm that the newly proposed algorithms outperform and generalize MF and BP.' author: - \| Sungsoo AhnMichael ChertkovJinwoo Shin\ School of Electrical Engineering,\ Korea Advanced Institute of Science and Technology, Daejeon, Korea\ $^{1}$ Theoretical Division, T-4 & Center for Nonlinear Studies,\ Los Alamos National Laboratory, Los Alamos, NM 87545, USA,\ $^{2}$Skolkovo Institute of Science and Technology, 143026 Moscow, Russia\ `{sungsoo.ahn, jinwoos}@kaist.ac.kr``chertkov@lanl.gov`\ title: Gauging Variational Inference --- Introduction ============ Preliminaries {#sec:pre} ============= Gauge optimization for approximating partition functions {#sec:alg} ======================================================== Experimental results ==================== Conclusion and future research ============================== We explore the freedom in gauge transformations of GM and develop novel variational inference methods which result in significant improvement of the partition function estimation. In this paper, we have focused solely on designing approaches which improve the bare/basic MF and BP via specially optimized gauge transformations. In terms of the path forward, it is of interest to extend this GT framework/approach to other variational methods, e.g., Kikuchi approximation [@kikuchi1951theory], structured/conditional MF [@saul1996exploiting; @carbonetto2007conditional]. Furthermore, G-BP and G-MF were resolved in our experiments via a generic optimization solver (IPOPT), which was sufficient for the illustrative tests conducted so far, however we expect that it might be possible to develop more efficient distributed solvers of the BP-type. Finally, we plan working on applications of the newly designed methods and algorithms to a variety of practical inference applications associated to GMs. [10]{} Robert Gallager. Low-density parity-check codes. , 8(1):21–28, 1962. Frank R. Kschischang and Brendan J. Frey. 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--- abstract: 'By applying the closed-time-path Green function formalism to the chiral dynamical model based on an effective Lagrangian of chiral quarks with the nonlinear-realized meson fields as bosonized auxiliary fields, we then arrive at a chiral thermodynamic model for the meson fields after integrating out the quark fields. Particular attention is paid to the spontaneous chiral symmetry breaking and restoration from the dynamically generated effective composite Higgs potential of meson fields at finite temperature. It is shown that the minimal condition of the effective composite Higgs potential of meson fields leads to the thermodynamic gap equation at finite temperature, which enables us to investigate the critical behavior of the effective chiral thermodynamical model and to explore the QCD phase transition. After fixing the free parameters in the effective chiral Lagrangian at low energies with zero temperature, we determine the critical temperature of the chiral symmetry restoration and present a consistent prediction for the thermodynamical behavior of several physically interesting quantities, which include the vacuum expectation value $v_o(T)$, quark condensate $<\bar{q}q>(T)$, pion decay constant $f_\pi(T)$ and pion meson mass $m_{\pi}(T)$. In particular, it is shown that the thermodynamic scaling behavior of these quantities becomes the same near the critical point of phase transition.' author: - Da Huang - 'Yue-Liang Wu' title: 'Chiral Thermodynamic Model of QCD and its Critical Behavior in the Closed-Time-Path Green Function Approach' --- Introduction ============ Thermodynamics of quantum chromodynamics (QCD) has been attracted a lot of attention during the last three decades. Many interesting physical phenomena are related to it, such as the equation of state of quark gluon plasma(QGP), chiral symmetry breaking and restoration, deconfinement phase transition, and so on. Thus, the study of the QCD thermodynamics becomes a basic problem in our understanding of the strong interaction. Particularly, the deconfined QGP is expected to be formed in ultrarelativistic heavy-ion collisions(HIC) [@Shuryak2004; @Gyulassy2005; @Shuryak2005; @Arsene2005; @Back2005; @Adams2005; @Adcox2005; @Blaizot2007] and many experiments like those at the Relativistic Heavy Ion Collision (RHIC) and at the Large Hadron Collider (LHC) have been built to explore the nature of the QGP and to search for the critical point of the phase transition, which enables many theoretical ideas testable and makes the research area more exciting. In this paper, we are going to study the thermodynamic properties of the effective chiral dynamical model(CDM)[@Dai:2003ip] of low energy QCD with spontaneous symmetry breaking via the dynamically generated effective composite Higgs potential of meson fields. The main assumption in such an effective CDM was based on an effective Lagrangian of chiral quarks with effective nonlinear-realized meson fields as the bosonized auxiliary fields, which may be resulted from the Nambu-Jona-Lasinio(NJL) four quark interaction[@Nambu:1961tp] due to the strong interactions of nonperturbative QCD at the low energy scale, then after considering the quantum loop contributions of quark fields by integrating over the quark fields with the loop regularization (LORE) method[@wu1; @Wu:2003dd], the resulting effective chiral Lagrangian for the meson fields in the CDM[@Dai:2003ip] has been turned out to provide a dynamically generated spontaneous symmetry breaking mechanism for the $SU(3)_L\times SU(3)_R$ chiral symmetry. The key point for deriving the dynamically generated spontaneous symmetry breaking mechanism is the use of the LORE method which keeps the physically meaningful finite quadratic term and meanwhile preserves the symmetries of original theory. More specifically, the advantage of the LORE method is the introduction of two intrinsic energy scales without spoiling the basic symmetries of original theory, such intrinsic energy scales play the role of the characteristic energy scale $M_c$ and the sliding energy scale $\mu_s$. Here $M_c$ is the characterizing energy scale of nonperturbative QCD below which the effective quantum field theory becomes meaningful to describe the low energy dynamics of QCD, and $\mu_s$ reflects the low energy scale of QCD on which the interesting physics processes are concerned. As a consequence, it was shown that the resulting effective CDM of low energy QCD can lead to the consistent predictions for the light quark masses, quark condensate, pseudoscalar meson masses and the lowest nonet scalar meson masses as well as their mixing at the leading order[@Dai:2003ip]. Based on the success of the effective CDM at zero temperature, we will show in this paper how the CDM can be extended to an effective chiral thermodynamic model (CTDM) at finite temperature by applying the closed-time-path Green function(CTPGF) approach[@Schwinger1961; @Keldysh1965; @Chou1985; @Zhou1980; @Rammer2007; @Calzetta2008], and how the CTDM enables us to describe the critical behavior of the low energy dynamics of QCD, and in particular to determine the critical temperature of the chiral symmetry breaking and restoration. This comes to our main motivation in the present paper. For our present purpose, we will investigate the CTDM with considering only two flavors and ignoring the possible instanton effect which is known to account for the the anomalous $U(1)_A$ symmetry breaking. Following the almost same procedure in deriving the CDM for the composite meson fields, we will arrive at an effective CTDM at finite temperature by adopting the CTPGF formalism. The key step in the derivation is the replacement of the zero-temperature propagator for the quark field with its finite temperature counterpart in the CTPGF formalism which has been shown to be more suitable for characterizing the nonequilibrium statistical processes[@Chou1985; @Zhou1980; @Rammer2007; @Calzetta2008]. We then obtain the dynamically generated effective composite Higgs potential at finite temperature, its minimal condition leads to the gap equation at finite temperature and enables us to explore the critical behavior and temperature for the chiral symmetry breaking and restoration. After fixing the free parameters in the effective Lagrangian, we present our numerical predictions for the temperature dependence of physically interesting quantities, such as the vacuum expectation value $v_o(T)$, the pion decay constant $f_\pi(T)$, and the masses of pseudoscalar and scalar mesons. The resulting critical temperature is found to be $T_c \simeq 200$ MeV, which is consistent with the NJL model prediction[@Klevansky1992; @Hatsuda1994; @Alkofer1996; @Buballa2005]. Outline on Chiral Dynamical Model of Low Energy QCD {#sec2} =================================================== Before exploring the thermodynamic behavior of the chiral dynamical model, it is useful to have a brief review on its derivation of the effective chiral Lagrangian at zero-temperature. For our present purpose with paying attention to the investigation of the chiral symmetry restoration at finite temperature, we shall consider the simple case with only two flavors of light quarks $u$ and $d$, and ignore the instanton effects. For simplicity, we also assume the exact isospin symmetry for two light quarks $q=(u,d)^T$ by taking the equal mass $m_u=m_d=m$. Our discussion here is mainly following the previous paper by Dai and Wu[@Dai:2003ip]. Let us begin with the QCD Langrangian for two light quarks $$\label{QCD} {\cal L}_{QCD}=\bar{q}\gamma^\mu(i\partial_\mu+g_s G^a_\mu T^a)q - \bar{q}M q-\frac{1}{2}tr G_{\mu\nu}G^{\mu\nu}$$ where $q=(u,d)^T$ denotes the SU(2) doublet of two light quarks and the summation over color degrees of freedom is understood. $G^a_\mu$ are the gluon fields with SU(3) gauge symmetry and $g_s$ is the running coupling constant. $M$ is the light quark mass matrix $M= diag(m_1,m_2)\equiv diag(m_u,m_d)$. In the limit $m_i\to0$ ($i=1,2$), the Lagrangian has the global $U(2)_L\times U(2)_R$ chiral symmetry. Due to the instanton effect, the chiral symmetry $U(1)_L\times U(1)_R$ is broken down to the diagonal $U(1)_V$ symmetry. This instanton effect can be expressed by the effective interaction[@'tHooft1; @'tHooft2] $$\label{instanton} {\cal L}^{inst}=\kappa_{inst}e^{i\theta_{inst}}\det(-\bar{q}_R q_L)+h.c.$$ where $\kappa_{inst}$ is the constant containing the factor $e^{-8\pi^2/g^2}$. Obviously, such an instanton term breaks the $U(1)_A$ chiral symmetry. As our present consideration is paid to the phenomenon of the $SU(2)_L\times SU(2)_R$ chiral symmetry breaking and its restoration at finite temperature, we will switch off the instanton effect by simply setting $\kappa_{inst}=0$ in the following discussions. The basic assumption of the CDM is that at the chiral symmetry breaking scale ($\sim 1 GeV$) the effective Lagrangian contains not only the quark fields but also the effective meson fields describing bound states of strong interactions of gluons and quarks. After integrating over the gluon fields at high energy scales, the effective Lagrangian at low energy scale is expected to have the following general form when keeping only the lowest order nontrivial terms: $$\begin{aligned} \label{Lag1} {\cal L}_{eff}(q,\bar{q},\Phi)&= &\bar{q}\gamma^\mu i\partial_\mu q+\bar{q}_L\gamma_\mu {\cal A}^\mu_L q_L+\bar{q}_R\gamma_\mu {\cal A}_R^\mu q_R -[\bar{q}_L(\Phi-M)q_R+h.c.]\nonumber\\ && +\mu_m^2 tr(\Phi M^\dagger+M\Phi^\dagger)-\mu^2_f tr \Phi\Phi^\dagger\end{aligned}$$ where $\Phi_{ij}$ are the effective meson fields which basically correspond to the composite operators $\bar{q}_{Rj}q_{Li}$. ${\cal A}_L$ and ${\cal A}_R$ are introduced as the external source fields. It is noticed in Eq.(\[Lag1\]) that the effective meson fields $\Phi_{ij}$ are the auxiliary fields in the sense that there is no kinetic term for them, which may explicitly be seen by integrating out $\Phi_{ij}$, we then obtain the following effective Lagrangian of quarks: $$\begin{aligned} {\cal L}^{NJL}_{eff}(q,{\bar q}) &=& \bar{q}\gamma^\mu i\partial_\mu q+\bar{q}_L\gamma_\mu {\cal A}^\mu_L q_L+\bar{q}_R\gamma_\mu {\cal A}_R^\mu q_R\nonumber\\ &&- (\frac{\mu^2_m}{\mu_f^2}-1)({\bar q}_L M q_R +\bar{q}_R M^\dagger q_L)+\frac{1}{\mu_f^2}{\bar q}_L q_R \bar{q}_R q_L\end{aligned}$$ which arrives exactly at the Nambu-Jona-Lasinio(NJL) model[@Nambu:1961tp] of effective four-quark interaction with the quark mass matrix $(\frac{\mu_m^2}{\mu_f^2}-1)M$. When the mass term is understood as the well-defined current quark mass term in the QCD Lagrangian Eq.(\[QCD\]), it is then clear that $\mu_m^2/\mu_f^2 = 2$. Note that the high order terms with the dimension above two for the effective meson fields are not included, which are assumed to be small and generated in loop diagrams, so only the lowest order nontrivial fermionic interaction terms are taken into account after integrating over the gluon field. For our present purpose, we only focus on the scalar and pseudoscalar mesons while the vector and axial vector sectors will not be considered here. As the pseudoscalar mesons are known to be the would-be Goldstone bosons, the effective chiral field theory is naturally to be realized as a nonlinear model. Thus we may express the effective meson fields $\Phi(x)$ into the following $2\times 2$ complex matrix form: $$\begin{aligned} \label{meson_def} && \Phi(x)\equiv \xi_L(x)\phi(x)\xi_R^\dagger(x),\qquad U(x)\equiv \xi_L(x)\xi_R^\dagger(x)=\xi_L^2(x)=e^{i\frac{2\Pi(x)}{f}}\nonumber\\ && \phi^\dagger(x)=\phi(x)=\sum^{3}_{a=0}\phi^a(x)T^{a},\qquad \Pi^\dagger(x)=\Pi(x)=\sum^{3}_{a=0}\Pi^a(x)T^a\end{aligned}$$ where $T^a$ ($a=0,1,2,3$) with $[T^a,T^b]=if^{abc}T^c$ and $2 tr T^a T^b=\delta_{ab}$ are the four generators of $U(2)$ group. The fields $\Pi^a(x)$ represent the pseudoscalar mesons and $\phi^a(x)$ the corresponding scalar chiral partners. Where $f$ is known as the decay constant with mass dimension. The Lagrangian Eq.(\[Lag1\]) with the definition of composite meson fields $\Phi_{ij}(x)$ Eq.(\[meson\_def\]) is our starting point for the derivation of the effective chiral Lagrangian for mesons. The Lagrangian can be obtained by integrating over the quark fields and the procedure for the derivation can be formally expressed in terms of the generating functionals via the following relations $$\begin{aligned} \label{formal} \frac{1}{Z}\int [d G_\mu] [d q] [\bar{q}] e^{i\int d^4x {\cal L}_{QCD}}& = & \frac{1}{\bar Z} \int [d\Phi] [d q] [d\bar{q}] e^{i\int d^4x {\cal L}_{eff}(q,{\bar q},\Phi)} \nonumber \\ & = & \frac{1}{Z_{eff}} \int[d\Phi] e^{i\int d^4x {\cal L}_{eff}(\Phi)}\end{aligned}$$ Let us first demonstrate the derivation of ${\cal L}_{eff}(\Phi)$. In order to obtain the effective chiral Lagrangian for the meson fields, we need to integrate over the quark fields (which is equivalent to calculate the Feynman diagrams of quark loops) from the following chiral Lagrangian $$\label{Lag2} {\cal L}^q_{eff}(q,{\bar q}) = \bar{q}\gamma^\mu i\partial_\mu q+\bar{q}_L\gamma_\mu {\cal A}^\mu_L q_L+\bar{q}_R\gamma_\mu {\cal A}_R^\mu q_R -[\bar{q}_L(\Phi-M)q_R+h.c.]+\chi\bar{q}q$$ where we have introduced the real source field $\chi(x)$ for the composite operator $\bar{q}q$ and the source term $\chi\bar{q}q$ which will be useful for the derivation of the chiral thermodynamic model, while eventually the source field is taken to be zero $\chi =0$. With the method of path integral, the effective Lagrangian of the meson fields is evaluated by integrating over the quark fields $$\int [d\Phi]exp\{i\int d^4x{\cal L}^M\}=Z_0^{-1}\int[d\Phi][dq][d\bar{q}]exp\{i\int d^4x {\cal L}^q_{eff}\}$$ The functional integral of the right hand side is known as the determination of the Dirac operator $$\int[dq][d{\bar q}]exp\{i\int d^4 x {\cal L}^q_{eff}\}=\det(i{\cal D}^\chi)$$ To obtain the effective action, it is useful to go to Euclidean space via the Wick rotation $$\gamma_0\to i\gamma_4,\quad G_0\to i G_4,\quad x_0\to -i x_4$$ and to define the Hermitian operator $$\begin{aligned} S^{M}_E &=& \int d^4x_E {\cal L}^M_E = \ln \det(i{\cal D}^\chi_E)\nonumber\\ &=& \frac{1}{2}[\ln\det(i{\cal D}^\chi_E)+\ln\det(i{\cal D}^\chi_E)^\dagger]+\frac{1}{2}[\ln\det(i{\cal D}^\chi_E)-\ln\det(i{\cal D}^\chi_E)^\dagger]\nonumber\\ &\equiv& S^{M}_{Re}+S^{M}_{Im}\end{aligned}$$ with $$\begin{aligned} S^{M}_{E Re} &=& \int d^4x_E {\cal L}^M_{Re} = \frac{1}{2}\ln\det(i{\cal D}^\chi_E(i{\cal D}^\chi_E)^\dagger)\equiv \frac{1}{2}\ln\det \Delta^\chi_E-\ln Z_0\label{ChEF}\\ S^{ M}_{E Im} &=& \int d^4 x_E {\cal L}^M_{Im} = \frac{1}{2} \ln\det(i{\cal D}^\chi_E/(i{\cal D}^\chi_E)^\dagger)\equiv \frac{1}{2}\ln\det\Theta_E\end{aligned}$$ where the imaginary part ${\cal L}^M_{Im}$ appears as a phase which is related to the anomalous terms and will not be discussed in the present paper. The operators in the Euclidean space are given by $$\begin{aligned} i{\cal D}^\chi_E &=& -i\gamma\cdot\partial-\gamma\cdot{\cal A}_L P_L -\gamma\cdot{\cal A}_R P_R+\hat{\Phi}P_R+\hat{\Phi}^\dagger P_L+\chi\nonumber\\ &=& i{\cal D}_E +\chi \nonumber\\ (i{\cal D}^\chi_E)^\dagger &=& i\gamma\cdot\partial+\gamma\cdot{\cal A}_R P_L +\gamma\cdot{\cal A}_L P_R+\hat{\Phi}^\dagger P_R+\hat{\Phi}P_L+\chi\nonumber\\ &=& (i{\cal D}_E)^\dagger+\chi\end{aligned}$$ with $\hat{\Phi}=\Phi-M$ and $P_{\pm}=(1\pm\gamma_5)/2$. $i{\cal D}^\chi_E$, $(i{\cal D}^\chi_E)^\dagger$ and $\Delta^\chi_E$ are regarded as matrices in coordinate space, internal symmetry space and spin space. Noticing the following identity $$\ln\det O= Tr\ln O$$ with $Tr$ being understood as the trace defined via $$Tr O = tr\int d^4 x <x\|O\|y>\|_{x=y}$$ Here $tr$ is the trace for the internal symmetry space and $<x\|O\|y>$ is the coordinate matrix element defined as $$\begin{aligned} <x\|O_{ij}\|y>=O^k_{ij}(x)\delta^4(x-y),\quad \delta^4(x-y) = \int^{\infty}_{-\infty}\frac{d^4k}{(2\pi)^4}e^{ik\cdot(x-y)}\end{aligned}$$ For the derivative operator, one has in the coordinate space $$<x\|\partial^\mu\|y>=\delta^4(x-y) (-ik^\mu+\partial^\mu_y)$$ With the above definitions, the operators $i{\cal D}^{k}_E$, $(i{\cal D}^{k}_E)^\dagger$ and $\Delta^{k}_E$ in the Euclidean space are given by $$\begin{aligned} i{\cal D}^{k}_E &=& -\gamma\cdot k -i\gamma\cdot\partial-\gamma\cdot{\cal A}_L P_L -\gamma\cdot{\cal A}_R P_R+\hat{\Phi}P_R+\hat{\Phi}^\dagger P_L+\chi\nonumber\\ &=& -\gamma\cdot k+i{\cal D}^\chi_E\\ (i{\cal D}_E^{ k})^\dagger &=& \gamma\cdot k+ i\gamma\cdot\partial+\gamma\cdot{\cal A}_R P_L +\gamma\cdot{\cal A}_L P_R+\hat{\Phi}^\dagger P_R+\hat{\Phi}P_L+\chi\nonumber\\ &=& \gamma\cdot k+(i{\cal D}^\chi_E)^\dagger\\ \Delta_E^{k} &=& k^2+\hat{\Phi}\hat{\Phi}^\dagger P_R +\hat{\Phi}^\dagger\hat{\Phi}P_L-i\gamma\cdot D_E\Phi P_L -i\gamma\cdot D_E \Phi^\dagger P_R\nonumber\\ && -\sigma_{\mu\nu}{\cal F}_{R\mu\nu}P_L-\sigma_{\mu\nu}{\cal F}_{L\mu\nu}P_R+(iD_{E\mu})(iD_{E\mu})+2k\cdot(iD_E)\nonumber\\ && -i\gamma\cdot\partial\chi + \chi [i{\cal D}_E + (i{\cal D}_E)^\dagger ] +\chi^2\nonumber\\ &=& k^2+\Delta_E^\chi = k^2+\Delta_E+ \Delta_{\chi}\end{aligned}$$ where $$\begin{aligned} \label{partial} iD_E\Phi &=& i\partial\Phi+{\cal A}_L\Phi-\Phi{\cal A}_R\\ iD_E &=& i\partial+{\cal A}_R P_L+{\cal A}_L P_R\end{aligned}$$ and $$\begin{aligned} \Delta_E &\equiv& \hat{\Phi}\hat{\Phi}^\dagger P_R +\hat{\Phi}^\dagger\hat{\Phi}P_L-i\gamma\cdot D_E\Phi P_L -i\gamma\cdot D_E \Phi^\dagger P_R\nonumber\\ && -\sigma_{\mu\nu}{\cal F}_{R\mu\nu}P_L-\sigma_{\mu\nu}{\cal F}_{L\mu\nu}P_R+(iD_{E\mu})(iD_{E\mu})+2k\cdot(iD_E)\\ \Delta_{\chi} &\equiv& -i\gamma\cdot\partial\chi + \chi [i{\cal D}_E + (i{\cal D}_E)^\dagger ] +\chi^2 \\ \Delta_E^{\chi}&\equiv& \Delta_E+ \Delta_{\chi}\end{aligned}$$ Thus the effective action is obtained as $$\label{ChEF2} S^{M}_{E Re}=\frac{N_c}{2}\int d^4 x_E\int\frac{d^4 k}{(2\pi)^4} e^{ik(x-y)} tr_{SF}\ln(k^2+\Delta^\chi_E)\|_{x = y}-\ln Z_0$$ where the subscripts $SF$ refer to the trace over the spin and flavor indices and $N_c$ is the color number. Before proceeding, we would like to mention some formulae which will be useful for the derivation of chiral thermodynamic model late on. By taking the functional derivative of Eq.(\[ChEF2\]) with respect to the source field $\chi$ at $\chi=0$, we have $$\begin{aligned} \label{ChEFk} \frac{\delta S^{M}_{E Re}}{\delta \chi(x)}\|_{\chi =0} & = & \frac{N_c}{2}\int d^4 x_E tr_{SF}\int\frac{d^4 k}{(2\pi)^4} e^{ik(x-y)} \left(-\gamma\cdot k + [i{\cal D}_E + (i{\cal D}_E)^\dagger ] \right) \frac{1}{k^2+\Delta_E}\|_{x = y} \\ & = & \frac{N_c}{2}\int d^4 x_E tr_{SF}\int\frac{d^4 k}{(2\pi)^4} \left(-\gamma\cdot k + \gamma\cdot( {\cal A}_R -{\cal A}_L)\gamma_5 + \hat{\Phi} + \hat{\Phi}^{\dagger} \right) \frac{1}{k^2+\Delta_E} \nonumber\end{aligned}$$ from the right hand side one may pick up the quark propagator as $\chi$ is the source field for the quark operator $\bar{q}(x)q(x)$. This can explicitly be shown by differentiating the effective action with respect to $\chi$, which gives the coinciding limit of the quark propagator $\lim_{x\to y}tr_{SF}{\langle}T[q(x) \bar{q}(y)]{\rangle}$. Alternatively, if taking the functional derivative of Eq.(\[ChEF2\]) with respect to the operator $\Delta_{\chi}$, we obtain another form of the formulae $$\label{ChEFd} \frac{\delta S^{ M}_{E Re}}{\delta (\Delta_{\chi})_{ij}} = \frac{N_c}{2}\int\frac{d^4k}{(2\pi)^4} \left(\frac{1} {k^2+\Delta_E+ \Delta_{\chi}}\right)_{ji}$$ In other word, when functionally integrating over $\Delta_{\chi}$ and summing over the flavor and spin degrees of freedom, we are led to the effective action Eq.(\[ChEF2\]). The physical action is yielded by taking the source field to be zero $\chi(x)=0$ $$\label{ChEF3} S^M_{E Re}=\frac{N_c}{2}\int d^4 x_E\int\frac{d^4 k}{(2\pi)^4} tr_{SF}\ln(k^2+\Delta_E)-\ln Z_0$$ To derive the effective chiral Lagrangian for the meson fields, we shall make the following redefinition for $\Delta_E^k\equiv k^2+\Delta_E$ $$\begin{aligned} \label{separation} \Delta_E^k&=&\Delta_0+\tilde{\Delta}_E \\ \Delta_0&\equiv&k^2+\bar{M}^2\\ \tilde{\Delta}_E&\equiv& (\hat{\Phi}\hat{\Phi}^\dagger-\bar{M}^2)P_R +(\hat{\Phi}^\dagger\hat{\Phi}-\bar{M}^2)P_L -i\gamma\cdot D_E\Phi P_L -i\gamma\cdot D_E\Phi^\dagger P_R\nonumber\\ &&-\sigma_{\mu\nu}{\cal F}_{R\mu\nu}P_L-\sigma_{\mu\nu}{\cal F}_{L\mu\nu}P_R +(iD_{E\mu})(iD_{E\mu})+2k\cdot(iD_E)\end{aligned}$$ where $\bar{M}$ is the supposed vacuum expectation values (VEVs) of $\hat{\Phi}$, i.e., $<\hat{\Phi}>=\bar{M}=diag.(\bar{m}_u,\bar{m}_d)$ with $\bar{m}_u=\bar{m}_d=\bar{m}$ under the exact isospin symmetry. Here $\bar{m}_i=v_i-m_i$ is regarded as the dynamical quark masses, and $v_i$ is supposed to be the VEVs of the scalar fields, i.e., $<\phi>=V=diag.(v_1,v_2)$ which will be determined from the minimal conditions of the effective potential in the effective chiral Lagrangian ${\cal L}_{eff}(\Phi)$. With this convention, it is seen that the minimal conditions of the effective potential are completely determined by the lowest order terms up to the dimension four $(\hat{\Phi}\hat{\Phi}^\dagger-\bar{M}^2)^2$ in the effective chiral field theory of mesons. By regarding $\tilde{\Delta}_E$ as the perturbative interaction term and taking $Z_0=(\det \Delta_0)^{\frac{1}{2}}$, the effective action in the Euclidean space can be written as $$\begin{aligned} S^M_{ERe} &=& \frac{N_c}{2}\int d^4 x_E \int\frac{d^4 k}{(2\pi)^4} tr_{SF} [\ln(\Delta_0+\tilde{\Delta}_E)-\ln\Delta_0]\nonumber\\ &=& \frac{N_c}{2}\int d^4 x_E \int\frac{d^4k}{(2\pi)^4}tr_{SF}\ln(1+\frac{1}{\Delta_0}\tilde{\Delta}_E)\nonumber\\ &=& \frac{N_c}{2}\int d^4 x_E \int\frac{d^4k}{(2\pi)^4}tr_{SF}\sum^\infty_{n=1}\frac{(-1)^{n+1}}{n}(\frac{1}{\Delta_0}\tilde{\Delta}_E)^n \nonumber\\ &\simeq& \frac{N_c}{2}\int d^4 x_E \int\frac{d^4k}{(2\pi)^4}tr_{SF} (\frac{1}{\Delta_0}\tilde{\Delta}_E-\frac{1}{2}\frac{1}{\Delta_0^2}\tilde{\Delta}_E^2)\end{aligned}$$ Note that we only keep the first two terms in the expansion over $\tilde{\Delta}_E$ since these are the only divergent terms in the integration over the internal momentum $k$. Particularly, the divergence degree of the first integral is quadratical while the second logarithmic. In order to maintain the gauge invariance and meanwhile keep the divergence behavior of the integral, we adopt the loop regularization (LORE) method proposed in [@wu1; @Wu:2003dd] for the momentum integral $$\begin{aligned} I_2 &=& \int\frac{d^4k}{(2\pi)^4}\frac{1}{k^2+\bar{M}^2}\to I^R_2=\frac{M_c^2}{16\pi^2} L_2(\frac{\mu^2}{M_c^2})\\ I_0 &=& \int\frac{d^4k}{(2\pi)^4}\frac{1}{(k^2+\bar{M}^2)^2}\to I^R_0=\frac{1}{16\pi^2}L_0(\frac{\mu^2}{M_c^2})\end{aligned}$$ with the consistent conditions for the tensor type divergent integrals $$I^R_{2\mu\nu}=\frac{1}{2}g_{\mu\nu}I^R_2,\quad I^R_{0\mu\nu}=\frac{1}{4}g_{\mu\nu}I^R_0$$ where $$\begin{aligned} I_{2\mu\nu} = \int\frac{d^4k}{(2\pi)^4}\frac{k_{\mu}k_{\nu}}{(k^2+\bar{M}^2)^2},\qquad I_{0\mu\nu} = \int\frac{d^4k}{(2\pi)^4}\frac{k_{\mu}k_{\nu}}{(k^2+\bar{M}^2)^3}\end{aligned}$$ The two diagonal matrices $L_0=diag.(L^{(1)}_0,L^{(2)}_0)$ and $L_2=diag.(L^{(1)}_2,L^{(2)}_2)$ are given by the following form: $$\begin{aligned} \label{intg1} L^{(i)}_0&=&\ln\frac{M_c^2}{\mu_i^2}-\gamma_\omega+y_0(\frac{\mu_i^2}{M_c^2})\\ L^{(i)}_2&=&1-\frac{\mu_i^2}{M_c^2}[\ln\frac{M_c^2}{\mu_i^2}-\gamma_\omega+1+y_2(\frac{\mu_i^2}{M_c^2})]\end{aligned}$$ with $$\begin{aligned} y_0(x) &=& \int^x_0 d\sigma \frac{1-e^{-\sigma}}{\sigma},\quad y_1(x) = \frac{1}{x} (e^{-x}-1+x),\quad y_2(x) = y_0(x)-y_1(x)\end{aligned}$$ Note that $M_c^2$ is the characteristic energy scale from which the nonperturbative QCD effects start to play an important role and the effective chiral field theory is considered to be valid below the scale $M_c$. We have also introduced the definitions $$\mu_i^2=\mu_s^2+\bar{m}_i^2, \quad \bar{m}_i=v_i-m_i$$ with $\mu_s^2$ the sliding energy scale. It is usually taken to be at the energy scale at which the physical processes take place, which is expected to be around the QCD scale $\Lambda_{QCD}$ for our present consideration. With these analysis, the effective chiral Lagrangian can be systematically obtained to be $$\begin{aligned} \label{Lag_meson} S^M_{ERe} &=& \frac{N_c}{16\pi^2}\int d^4 x_E tr_F \{M_c^2L_2[(\hat{\Phi}\hat{\Phi}^\dagger-\bar{M}^2)+(\hat{\Phi}^\dagger\hat{\Phi}-\bar{M}^2)]\nonumber\\ &&-\frac{1}{2} L_0 [D_E\hat{\Phi}\cdot D_E\hat{\Phi}^\dagger+ D_E\hat{\Phi}^\dagger\cdot D_E\hat{\Phi}+ (\hat{\Phi}\hat{\Phi}^\dagger-\bar{M}^2)^2+(\hat{\Phi}^\dagger\hat{\Phi}-\bar{M}^2)^2]\}\end{aligned}$$ where the trace over the spin indices give the factor $2$ since our quark fields defined in Eq.(\[Lag1\]) are Weyl fermion fields, each of which has 2 degrees of freedom. By transforming back to the Minkowski spacetime signature and adding the extra terms in Eq.(\[Lag1\]), we finally arrive at the following effective chiral Lagrangian at zero temperature $$\begin{aligned} {\cal L}_{eff}(\Phi) &=& \frac{1}{2}\frac{N_c}{16\pi^2}tr_F L_0[D_\mu\hat{\Phi}^\dagger D^\mu\hat{\Phi} +D_\mu\hat{\Phi}D^\mu\hat{\Phi}^\dagger -(\hat{\Phi}^\dagger\hat{\Phi}-\bar{M}^2)^2 -(\hat{\Phi}\hat{\Phi}^\dagger-\bar{M}^2)^2]\nonumber\\ &&+\frac{N_c}{16\pi^2}M_c^2 tr_F L_2[(\hat{\Phi}^\dagger\hat{\Phi}-\bar{M}^2) +(\hat{\Phi}\hat{\Phi}^\dagger-\bar{M}^2)]\nonumber\\ &&+\mu_m^2 tr_F(\Phi M^\dagger+M\Phi^\dagger)-\mu_f^2 tr\Phi\Phi^\dagger.\end{aligned}$$ The derivation of the above effective chiral Lagrangian in an equivalent rotated basis[@Espriu:1989ff] is given in Appendix \[appendix\], which may be more transparent for the spontaneous symmetry breaking with the composite Higgs-like scalar mesons. Derivation of Chiral Thermodynamic Model of QCD {#title} =============================================== After a brief outline for the derivation of the effective chiral Lagrangian of the CDM for mesons at zero temperature, it is straightforward to incorporate the finite temperature effects into the effective Lagrangian. The method for the derivation of finite temperature effective Lagrangian is similar by applying for the closed-time-path Green function(CTPGF) formalism. The CTPGF formalism, developed by Schwinger [@Schwinger1961] and Keldysh [@Keldysh1965], has been used to solve lots of interesting problems in statistical physics and condensed matter theory [@Chou1985]. It is generally believed that this technique is quite efficient in investigating the nonequilibrium and finite temperature dynamical systems as this formalism simultaneously incorporates both the statistical and dynamical properties[@Chou1985; @Zhou1980]. It has also been used to treat a system of self-interacting bosons described by $\lambda\phi^{4}$ scalar fields [@Calzetta1988]. A brief introduction to this formalism is given in Appendix A. Readers who are not familiar with the CTPGF formalism are refered to the excellent review articles [@Chou1985; @Das:2000ft] and monographs [@Calzetta2008; @Rammer2007]. As shown in the Appendix A, the main step for the derivation of the effective action with finite temperature in the CTPGF formalism is to replace the field propagator with its finite temperature counterpart[@Zhou1980; @Chou1985; @Quiros:1999jp; @Das:2000ft]. Applying the CTPGF formalism to the propagator of the quark fields in Eq.(\[ChEFk\]), we arrive at the following result $$\begin{aligned} & & \frac{\delta S^{M}}{\delta \chi(x)}\|_{\chi=0} = \frac{N_c}{2}\int\frac{d^4k}{(2\pi)^4} tr_{SF} \left[-\gamma\cdot k + \left(i{\cal D}_E + (i{\cal D}_E)^\dagger \right) \right] \nonumber \\ & & \left [\left( \begin{array}{cc} \frac{1}{k^2+\Delta_E} & 2\pi i\theta(-k_4)\delta(k^2+\Delta_E) \\ 2\pi i\theta(k_4)\delta(k^2+\Delta_E) & -\frac{1}{k^2+\Delta_E} \\ \end{array} \right) - 2\pi i n_F(\omega)\delta(k^2+\Delta_E) \left( \begin{array}{cc} 1 & 1\\ 1 & 1 \end{array}\right)\right]\end{aligned}$$ where $\omega$ is defined as the effective energy $\omega\equiv\sqrt{\vec{k}^2+\Delta_E}$ and $n_F(\omega)$ represents the Fermi-Dirac distribution function $$n_F(\omega)\equiv \frac{1}{e^{\beta\omega}+1}$$ For our present purpose, we only need to calculate the first component of the effective action since it is the only one which is related to the causal propagation[@Quiros:1999jp; @Das:2000ft]. For convenience, we may adopt the following alternative formula which is similar to Eq.(\[ChEFd\]) at zero temperature $$\begin{aligned} \frac{\delta S^{ M}_{E Re}}{\delta \Delta_{\chi}}&=&\frac{N_c}{2}\int\frac{d^4 k}{(2\pi)^4} [\frac{1}{k^2+\Delta_E+ \Delta_{\chi} }-2\pi i n_F(\omega)\delta(k^2+\Delta_E+ \Delta_{\chi} )]\nonumber\\ &=& \frac{N_c}{2} [\int\frac{d^4 k}{(2\pi)^4} \frac{1}{k^2+\Delta_E+ \Delta_{\chi} } -\int\frac{d^3 k}{(2\pi)^3}n_F(\omega)\frac{1}{\sqrt{\vec{k}^2+\Delta_E+ \Delta_{\chi} }} ]\end{aligned}$$ By functionally integrating over $\Delta_{\chi}$ and summing over the spin and flavor indices, we then obtain the effective action for the CTDM $$\begin{aligned} S^M_{E Re} &=& \frac{N_c}{2}\int d^4 x_E tr_{SF}[ \int\frac{d^4 k}{(2\pi)^4}\ln(k^2+\Delta_E)+\frac{1}{\beta}\int \frac{d^3 k}{(2\pi)^3}\ln(1+e^{-\beta\sqrt{\vec{k}+\Delta_E}})] \nonumber\\ &&-\ln Z_0\end{aligned}$$ where we have put the source field $\chi(x)=0$ in the end of the calculation. Separating $\Delta_E^k$ as in Eq.(\[separation\]) and identifying $\tilde{\Delta}_E$ as the perturbation part, we then make the expansion in terms of $\tilde{\Delta}_E$. Here we only keep the lowest order terms as they are the only ones relevant to our present discussion. Also, we take $$\ln Z_0= \frac{N_c}{2}\int d^4 x_E tr_{SF}[ \int\frac{d^4 k}{(2\pi)^4}\ln(k^2+\bar{M}^2)+\frac{1}{\beta}\int \frac{d^3 k}{(2\pi)^3}\ln(1+e^{-\beta\sqrt{\vec{k}+\bar{M}^2}})]$$ which provides the cancelation for the infinite zero-point energy. Thus, the effective action with leading terms in the Euclidean space can be written as $$\begin{aligned} \label{ChEF8} &S^M_{E Re}& \simeq \frac{N_c}{2}\int d^4 x_E tr_{SF}\{[\int\frac{d^4 k}{(2\pi)^4}\frac{1}{k^2+\bar{M}^2}-\int\frac{d^3 k}{(2\pi)^3}\frac{1}{\sqrt{\vec{k}^2+\bar{M}^2}(e^{\beta\sqrt{\vec{k}^2+\bar{M}^2}}+1)}] \tilde{\Delta}_E\nonumber\\ &&- \frac{1}{2}[\int\frac{d^4k}{(2\pi)^4}\frac{1}{(k^2+\bar{M}^2)^2} -\int\frac{d^3k}{(2\pi)^3}\frac{\beta\sqrt{\vec{k}^2+\bar{M}^2} e^{\beta\sqrt{\vec{k}^2+\bar{M}^2}}+e^{\beta\sqrt{\vec{k}^2+\bar{M}^2}}+1}{2(\vec{k}^2+\bar{M}^2)^{3/2} (e^{\beta\sqrt{\vec{k}^2+\bar{M}^2}}+1)^2}]\tilde{\Delta}^2_E\}\nonumber\\ &&= \frac{N_c}{16\pi^2} \int d^4 x_E tr_F\{M_c^2 L_2(T)[(\hat{\Phi}\hat{\Phi}^\dagger-\bar{M}^2)+(\hat{\Phi}^\dagger\hat{\Phi}-\bar{M}^2)]\nonumber\\ &&-\frac{1}{2} L_0(T) [D_E\hat{\Phi}\cdot D_E\hat{\Phi}^\dagger+ D_E\hat{\Phi}^\dagger\cdot D_E\hat{\Phi}+ (\hat{\Phi}\hat{\Phi}^\dagger-\bar{M}^2)^2+(\hat{\Phi}^\dagger\hat{\Phi}-\bar{M}^2)^2]\}\end{aligned}$$ where we have defined $$\begin{aligned} \label{tt} L_0(T) &\equiv& L_0(\frac{\mu^2}{M_c^2})-\frac{1}{\pi}\int d^3 k \frac{\beta\sqrt{\vec{k}^2+\bar{M}^2} e^{\beta\sqrt{\vec{k}^2+\bar{M}^2}}+e^{\beta\sqrt{\vec{k}^2+\bar{M}^2}}+1}{(\vec{k}^2+\bar{M}^2)^{3/2} (e^{\beta\sqrt{\vec{k}^2+\bar{M}^2}}+1)^2} \nonumber \\ L_2(T) &\equiv& L_2(\frac{\mu^2}{M_c^2})-\frac{4}{\pi M_c^2}\int d^3 k \frac{1}{\sqrt{\vec{k}^2+\bar{M}^2}(e^{\beta\sqrt{\vec{k}^2+\bar{M}^2}}+1)}\end{aligned}$$ and used the results $$\begin{aligned} tr_{S}\tilde{\Delta}_E &=& 2 [(\hat{\Phi}\hat{\Phi}^\dagger-\bar{M}^2)+(\hat{\Phi}^\dagger\hat{\Phi}-\bar{M}^2)]\\ tr_{S}\tilde{\Delta}_E^2 &=& 2[D_E\hat{\Phi}\cdot D_E\hat{\Phi}^\dagger+ D_E\hat{\Phi}^\dagger\cdot D_E\hat{\Phi}+ (\hat{\Phi}\hat{\Phi}^\dagger-\bar{M}^2)^2+(\hat{\Phi}^\dagger\hat{\Phi}-\bar{M}^2)^2]\end{aligned}$$ Finally, transforming the action to the Minkowski spacetime and adding the extra terms in Eq.(\[Lag1\]), we arrive at the following effective chiral Lagrangian at finite temperature for the composite meson fields $$\begin{aligned} {\cal L}_{eff}(\Phi) &=& \frac{1}{2}\frac{N_c}{16\pi^2}tr_F L_0(T) [D_\mu\hat{\Phi}^\dagger D^\mu\hat{\Phi} +D_\mu\hat{\Phi}D^\mu\hat{\Phi}^\dagger -(\hat{\Phi}^\dagger\hat{\Phi}-\bar{M}^2)^2 -(\hat{\Phi}\hat{\Phi}^\dagger-\bar{M}^2)^2]\nonumber\\ &&+\frac{N_c}{16\pi^2}M_c^2 tr_F L_2(T) [(\hat{\Phi}^\dagger\hat{\Phi}-\bar{M}^2) +(\hat{\Phi}\hat{\Phi}^\dagger-\bar{M}^2)]\nonumber\\ &&+\mu_m^2(T) tr_F(\Phi M^\dagger+M\Phi^\dagger)-\mu_f^2(T) tr\Phi\Phi^\dagger\end{aligned}$$ where $L_0(T) $ and $L_2(T) $ are given in Eq.(\[tt\]). Note that the initial mass scale $\mu_f$ ($\mu_m$) characterizes the nonperturbative gluon effect at zero temperature. At the finite temperature, it is expected that the mass scale $\mu_f$($\mu_m$) is in general temperature dependent, which will be seen more clear below. Dynamical Symmetry Breaking and Thermodynamic Properties of CTDM ================================================================ Let us now focus on the dynamically generated effective composite Higgs potential of meson fields, which can be reexpressed as the following general form $$\begin{aligned} V_{eff}(\Phi) &=& -tr_F \hat{\mu}_m^2(T)(\Phi M^\dagger+ M \Phi^\dagger)+\frac{1}{2}tr_F\hat{\mu}^2_f(T) (\Phi\Phi^\dagger+\Phi^\dagger \Phi)\nonumber\\ &&+\frac{1}{2}tr_F\lambda(T)[(\hat{\Phi}\hat{\Phi}^\dagger)^2+(\hat{\Phi}^\dagger\hat{\Phi})^2]\end{aligned}$$ with $\hat{\mu}^2_f(T)$, $\hat{\mu}_m^2(T)$ and $\lambda(T)$ the three diagonal matrices $$\begin{aligned} \hat{\mu}_f^2(T) &\equiv& \mu_f^2(T)-\frac{N_c}{8\pi^2}(M_c^2L_2(T)+\bar{M}^2L_0(T))\\ \hat{\mu}_m^2(T) &\equiv& \mu_m^2(T)-\frac{N_c}{8\pi^2}(M_c^2L_2(T)+\bar{M}^2L_0(T))\\ \lambda(T) &\equiv& \frac{N_c}{16\pi^2}L_0(T)\end{aligned}$$ Taking the nonlinear realization $\Phi(x)=\xi_L(x)\phi(x)\xi_R^\dagger(x)$ with supposing that the minimal of the above effective potential occurs at the point $<\phi>=V(T)=diag.(v_1(T),v_2(T))$, we can write the scalar fields as follows $$\phi(x)=V(T)+\varphi(x)$$ where the VEVs may be written in terms of the following general form: $$v_i(T)=v_o(T)+\beta_o m_i\quad\quad i=1,2 \quad or \quad i=u,d$$ For the equal mass $m_u=m_d=m$ considered in our present case, it leads to the general VEVs $v_1(T)=v_2(T)=v(T)$ and the single form $v(T)=v_o(T)+\beta_o m$. By differentiating the effective composite Higgs potential of the scalar meson field at the VEV $v(T)$, we then obtain the minimal conditions: $$\label{mini cond} -\hat{\mu}^2_f(T)_i v(T)_i+ \hat{\mu}^2_m(T)_i m_i-2\lambda(T)_i \bar{m}^3(T)_i=0$$ with equal mass of two flavor quarks, it reduces to one minimal condition. For convenience of discussions, it is helpful to decompose $\mu^2(T)$, $\hat{\mu}^2_f(T)$, $\hat{\mu}_m^2(T)$ and $\lambda(T)$ into two parts with one part independent of the current quark mass $m$. Practically, it can be done by making an expansion with respect to the current quark masses $$\begin{aligned} && \mu^2(T)=\mu_o^2(T)+2(\beta_o-1)v_o(T) \tilde{m},\quad \mu_o^2(T)=\mu_s^2+v_o^2(T),\nonumber\\ && \tilde{m}(T)=m[1+(\beta_o-1)m/(2v_o(T))]\\ && \hat{\mu}_f^2(T) = \bar{\mu}_f^2(T)+2\mu_{fo}(T)\tilde{m}(T)[1+\sum_{k=1}\alpha_k(T)(\frac{\tilde{m}(T)}{\mu_o(T)})^k(\beta_o-1)^k]\\ && \hat{\mu}_m^2(T) = \bar{\mu}_m^2(T)+2\mu_{fo}(T)\tilde{m}(T)[1+\sum_{k=1}\alpha_k(T)(\frac{\tilde{m}(T)}{\mu_o(T)})^k(\beta_o-1)^k]\\ && \lambda(T)=\bar{\lambda}(T)-\lambda_o \sum_{k=1}\beta_k(T)(\frac{\tilde{m}(T)}{\mu_o(T)})^k(\beta_o-1)^k, \quad \lambda_o=\frac{N_c}{16\pi^2}\end{aligned}$$ By keeping only the nonzero leading terms in the expansion of current quark masses, we then obtain the following constraints from the minimal condition Eq.(\[mini cond\]) $$\label{mini1} \bar{\mu}_f^2(T)+2\bar{\lambda}(T)v_o^2(T)=0$$ Here the temperature-dependent parameters $\bar{\mu}_m^2(T)$, $\bar{\mu}_f^2(T)$ and $\bar{\lambda}(T)$ are related to the initial parameters in the effective potential and the characteristic energy scale via the following relations $$\begin{aligned} \bar{\mu}_f^2(T) &=& \mu_f^2(T)-\frac{N_c}{8\pi^2}(M_c^2\bar{L}_2(T)+v_o^2(T)\bar{L}_0(T))\\ \bar{\mu}_m^2(T) &=& \mu_m^2(T)-\frac{N_c}{8\pi^2}(M_c^2\bar{L}_2(T)+v_o^2(T)\bar{L}_0(T))\\ \bar{\lambda}(T) &=& \frac{N_c}{16\pi^2}\bar{L}_0(T)\end{aligned}$$ where $\bar{L}_0(T)$ and $\bar{L}_2(T)$ represent the leading order expansion of $L_0(T)$ and $L_2(T)$ with respect to $m$. Explicitly, they are given by $$\begin{aligned} \bar{L}_0(T) &\equiv& L_0(\frac{\mu_o^2(T)}{M_c^2})-\frac{1}{\pi}\int d^3 k \frac{\beta\sqrt{\vec{k}^2+v_o^2(T)} e^{\beta\sqrt{\vec{k}^2+v_o^2(T)}}+e^{\beta\sqrt{\vec{k}^2+v_o^2(T)}}+1}{(\vec{k}^2+v_o^2(T))^{3/2} (e^{\beta\sqrt{\vec{k}^2+v_o^2(T)}}+1)^2}\\ \bar{L}_2(T)&\equiv& L_2(\frac{\mu_o^2(T)}{M_c^2})-\frac{4}{\pi M_c^2}\int d^3 k \frac{1}{\sqrt{\vec{k}^2+v_o^2(T)}(e^{\beta\sqrt{\vec{k}^2+v_o^2(T)}}+1)}\end{aligned}$$ With these definitions of parameters, the minimal condition Eq.(\[mini1\]) is transformed into the following form $$\begin{aligned} \label{gap eqn} \mu_f^2(T) &=& \frac{N_c}{8\pi^2}M_c^2 \bar{L}_2(T)\nonumber\\ &=& \frac{N_c}{8\pi^2}[M_c^2-\mu_o^2(T)(\ln\frac{M_c^2}{\mu_o^2(T)}-\gamma_\omega+1+y_2(\frac{\mu_o^2(T)}{M_c^2}))]\nonumber\\ && -\frac{2N_c}{\pi^2}\int^\infty_0 dk\frac{k^2}{\sqrt{k^2+v_o^2(T)}(e^{\beta\sqrt{k^2+v_o^2(T)}}+1)}\end{aligned}$$ which is the gap equation at finite temperature. In order to obtain the critical temperature, let us make a simple assumption for the temperature dependence of the mass scale $\mu_f^2(T)$ $$\label{mu_f} \mu_f^2(T) = \gamma_o v_o^2(T)$$ with $\gamma_o$ a temperature independent constant. The reason for this assumption will become manifest below from the thermodynamic property of the pion meson mass. In fact, recall that the appearance of $\mu_f^2$ in the effective Lagrangian Eq.(\[Lag1\]) can be traced back to the integration over the gluon fields. In principle, it can be calculated from QCD and given in terms of the QCD parameters $g_s(\mu)$ and $\Lambda_{QCD}$. So the thermodynamic property of $\mu_f^2$ is expected to be obtained from the detailed analysis of gluon dynamics at finite temperature. The above simple assumption means that the gluon thermodynamics has the same temperature dependence as the chiral thermodynamics of quark condensate. With such an assumption and based on the chiral thermodynamic gap equation, we are able to calculate the critical temperature for the chiral symmetry restoration. Suppose that at the critical temperature the VEV $v_o(T)$ approaches to vanish, so does the $\mu_f^2(T)$, then the gap equation becomes $$\begin{aligned} 0 &=& \frac{N_c}{8\pi^2}[M_c^2-\mu_s^2(\ln\frac{M_c^2}{\mu_s^2}-\gamma_\omega+1+y_2(\frac{\mu_s^2}{M_c^2})]-\frac{2N_c}{\pi^2}\int^\infty_0 d k \frac{k}{e^{\beta k}+1}\nonumber\\ &=& \frac{N_c}{8\pi^2}[M_c^2-\mu_s^2(\ln\frac{M_c^2}{\mu_s^2}-\gamma_\omega+1+y_2(\frac{\mu_s^2}{M_c^2})] -\frac{2N_c}{\pi^2}T^2\int^\infty_0 d k^\prime \frac{k^\prime}{e^{k^\prime}+1}\nonumber\\ &=&\frac{N_c}{8\pi^2}[M_c^2-\mu_s^2(\ln\frac{M_c^2}{\mu_s^2}-\gamma_\omega+1+y_2(\frac{\mu_s^2}{M_c^2})] -\frac{2N_c}{12}T_c^2\end{aligned}$$ where $k^\prime$ is defined as $k^\prime=\beta k$ and we have used the result $$\int^\infty_0 d k^\prime \frac{k^\prime}{e^{k^\prime}+1}=\frac{\pi^2}{12}$$ Thus, the critical temperature for the chiral symmetry restoration is given by $$\label{Tc} T_c = \sqrt{\frac{3}{4\pi^2}[M_c^2-\mu_s^2(\ln\frac{M_c^2}{\mu_s^2}-\gamma_\omega+1+y_2(\frac{\mu_s^2}{M_c^2}))]}$$ which shows that the critical temperature is characterized by the quadratic term evaluated in the LORE method, which differs from the dimensional regularization where the quadratic term is in general suppressed. So far, we have explicitly shown the mechanism of dynamical spontaneous chiral symmetry breaking and its restoration at finite temperature. We are now going to present the explicit expressions for the masses of the scalar mesons, pseudoscalar mesons and/or light quarks. To be manifest, let us first write down the scalar and pseudoscalar meson matrices $$\begin{aligned} \sqrt{2}\varphi= \left( \begin{array}{cc} \frac{a_0^0}{\sqrt{2}}+\frac{\sigma}{\sqrt{2}} & a_0^+ \\ a_0^- & -\frac{a_0^0}{\sqrt{2}}+\frac{\sigma}{\sqrt{2}} \\ \end{array} \right)\end{aligned}$$ and $$\begin{aligned} \sqrt{2}\Pi= \left( \begin{array}{cc} \frac{\pi^0}{\sqrt{2}}+\frac{\eta'}{\sqrt{2}} & \pi^+ \\ \pi^- & -\frac{\pi^0}{\sqrt{2}}+\frac{\eta'}{\sqrt{2}} \\ \end{array} \right)\end{aligned}$$ Keeping to the leading order of the current quark masses, we have $$\begin{aligned} \label{pscalar} m_{\pi^{0,\pm}}^2(T)=m_{\eta'}^2(T)\simeq \frac{2\mu_P^3(T)}{f_\pi^2(T)}(m_u+m_d)=\frac{4\mu_P^3(T)}{f_\pi^2(T)}m\end{aligned}$$ for the pseudoscalar mesons, and $$\label{scalar} m_{a_0^{0,\pm}}^2(T) = m_{\sigma}^2(T) \simeq 3(\bar{m}_u^2(T)+\bar{m}_d^2(T))=6\bar{m}^2(T)$$ for the scalar mesons. Where $\mu_P^3$ is given by $$\label{mu_P} \mu_P^3(T)=(\bar{\mu}^2_m(T)+2\bar{\lambda}(T)v_o^2(T))v_o(T)=\mu_f^2(T) v_o(T)=\gamma_o v_o^3(T)$$ where we have used the minimal condition Eq.(\[mini1\]) and the relation Eq.(\[mu\_f\]). Note that in obtaining the above results for the scalar and pseudoscalar meson masses, the SU(2) triplet and singlet mesons have the common masses: $m_{a_0}^2=m_{\sigma}^2$ and $m^2_{\pi}=m^2_{\eta'}$, which is the reflection of the present assumption of the exact $U(1)_{A}$ symmetry. However, as we discussed in Sec. 2, in the real world such a symmetry gets quantum anomalous and will be broken down by the instanton effects, which is ignored in our present consideration. Predictions with Input Parameters at Low Energy and Critical Temperature in CTDM ================================================================================ In order to make numerical predictions for the temperature dependence of the masses of the light scalar and pseudoscalar mesons, it needs to fix the values of input parameters in the effective chiral Lagrangian with finite temperature. There are in general five parameters: $\mu_f^2$ ($\mu_m^2$), $M_c^2$, $\mu_s^2$, $v_o$, and a universal current quark mass $m$. To fix the parameters, we shall use the constraints at low energy with zero temperature. In general, the minimal condition Eq.(\[mini cond\]) with different quark masses will lead to two constraints by expanding the equation with respect to the current quark mass up to the order of $m^2$. For the equal mass case, we get the following minimal condition $$\begin{aligned} && \bar{\mu}_f^2+2\bar{\lambda}v_o^2=0\label{mini cond1}\end{aligned}$$ with $$\begin{aligned} & & \frac{\lambda_o}{\bar{\lambda}}[(\frac{2v_o^2}{\mu_o^2}-1)(1-\frac{v_o^2}{3\mu_o^2}) -\frac{2v_o}{3\mu_o}\alpha_1(1-r)+r]=1\label{mini cond2} \\ && r\equiv\frac{\mu_s^2}{\mu_o^2}-\frac{\mu_o^2}{M_c^2}[1-\frac{\mu_s^2}{\mu_o^2}+O(\frac{\mu_o^2}{M_c^2})]\\ && \alpha_1(1-r) \equiv \frac{2v_o}{\mu_o}[\frac{\mu_s^2}{2\mu_o^2}+O(\frac{\mu_o^2}{M_c^2})]\end{aligned}$$ Note that in obtaining the above result one needs to keep to the order of $m^2$ in the current quark mass expansion. As we have shown in Sec.II that in order to have well-defined QCD current quark masses, it requires that $$\label{mm} (\frac{\mu_m^2}{\mu_f^2}-1)M=M,\quad i.e.\quad \mu_m^2=2\mu_f^2$$ which fixes the parameter $$\label{beta} \beta_o=\frac{\mu_m^2}{\mu_f^2}=2$$ Also, from the original Lagrangian of chiral dynamical model Eq.(\[Lag1\]), the auxiliary fields $\Phi_{ij}$ are found to be given by the quark fields as follows $$\Phi_{ij}=-\frac{1}{\mu^2_f}\bar{q}_{Rj}q_{Li}+\frac{\mu_m^2}{\mu^2_f}M_{ij}$$ By assuming that the quark condensation is almost flavor independent, i.e., $<\bar{u}u>\simeq<\bar{d}d>$, and combining the condition Eq.(\[mm\]), then the dynamical quark masses take the simple form $$\bar{m}=v-m=v_o+(\beta_o-1)m=v_o+m$$ which may be identified with the expected constituent quark masses after dynamically spontaneous symmetry breaking, and $v_o$ is caused by the quark condensation $$\label{cond} v_o=-\frac{1}{2\mu_f^2}{\langle}\bar{q}q{\rangle},\quad q=u,d$$ To determine the remaining parameters, we consider the following constraints. There are two constraints arising from the pseudoscalar sector. One is from the normalization of the kinetic terms. $$\label{pi decay} \bar{\lambda}v_o^2={f_\pi^2\over 4}$$ After some manipulation, the equation can be transformed into the following form $$\label{ps kine} L_0(\frac{\mu_o^2}{M_c^2})v_o^2=\frac{(4\pi f_\pi)^2}{4N_c}\equiv \bar{\Lambda}_f^2\simeq (340 MeV)^2$$ where we have used the pion decay constant $f_\pi\simeq 94$MeV. The other comes from the current quark mass and pion mass via the relation Eq.(\[pscalar\]). Taking the pion mass $m_{\pi^{0,\pm}}\simeq 139$ MeV and the VEV $v_o = 350^{+20}_{-20}$ MeV or alternatively the current quark mass $m=4.76_{+0.08}^{-0.04}$MeV as the inputs, we obtain the following relation $$\begin{aligned} \label{condensation} 2\mu_f^2 v_o =-{\langle}\bar{q}q{\rangle}=\frac{m_{\pi^{0,\pm}}^2 f_{\pi}^2}{2m}=(262^{+1}_{-2} MeV)^3\end{aligned}$$ With the above relations and constraints Eqs.(\[mini cond1\]), (\[mini cond2\]), (\[mm\]), (\[cond\]),(\[ps kine\]), (\[condensation\]), all the parameters can be completely determined $$\begin{aligned} \label{det} && v_o\simeq 350^{+20}_{-20}MeV\nonumber\\ && M_c\simeq 881_{+57}^{-32}MeV,\quad \mu_s\simeq 312_{+11}^{-3}MeV\nonumber\\ && \mu_m^2=2\mu_f^2=(226_{+7}^{-5}MeV)^2\nonumber\\ && \beta_o=2, \quad \gamma_o={\mu_f^2\over v_o^2}=0.209_{+0.041}^{-0.031}\nonumber\\ && {\langle}\bar{q}q{\rangle}= -(262^{+1}_{-2} MeV)^3\end{aligned}$$ With these parameters, we can immediately obtain the critical temperature for the chiral symmetry restoration $$\label{Tc} T_c = \sqrt{\frac{6}{8\pi^2}[M_c^2-\mu_s^2(\ln\frac{M_c^2}{\mu_s^2}-\gamma_\omega+1+y_2(\frac{\mu_s^2}{M_c^2}))]}\simeq 200_{+15}^{-9}MeV$$ which is consistent with NJL model prediction [@Hatsuda1994; @Klevansky1992; @Alkofer1996; @Buballa2005]. Chiral Symmetry Restoration and Critical Phase Transition of Low Energy QCD ============================================================================= In this section, we will present numerical predictions based on the CTDM. Especially we will show the thermodynamic behavior of the VEV $v_o(T)$, pion decay constant $f_\pi(T)$, the quark condensate ${\langle}\bar{q}q{\rangle}(T)$ and the masses of pseudoscalar mesons $m_{\pi^{0,\pm}}(T)$. Their temperature dependence and the properties of critical phase transition are plotted in all the diagrams with adopting the central values of the quantities listed in Eq.(\[det\]). From the gap equation Eq.(\[gap eqn\]), we can numerically solve the vacuum expectation value $v_o(T)$ at any finite temperature until the critical temperature where $v_o(T)$ approaches to vanish. The result is shown in Fig.(\[v01\]). ![Temperature dependence of VEV $v_o$[]{data-label="v01"}](v01.eps "fig:")\ By the normalization of kinetic terms of pseudoscalar sector Eq.(\[pi decay\]), we can obtain the expression determining the pion decay constant at finite temperature $$f_\pi(T) = \sqrt{4\bar{\lambda}(T)v_o^2(T)}=2v_o(T)\sqrt{\frac{N_c}{16\pi^2}\bar{L}_0(T)}$$ which is presented in Fig.(\[fpi1\]) ![Temperature dependence of the pion decay constant[]{data-label="fpi1"}](fpi1.eps "fig:")\ Furthermore, the quark condensate ${\langle}\bar{q}q{\rangle}(T)$ is given by: $${\langle}\bar{q}q{\rangle}(T) =-2\mu_f^2(T) v_o(T)=-2\gamma_o v_o^3(T)$$ its variation with respect to temperature is displayed in Fig.(\[cond2\]) ![Temperature dependence of the quark condensate[]{data-label="cond2"}](cond1.eps "fig:")\ The leading order approximation of the pseudoscalar meson mass $m_{\pi^{0,\pm}}$ with respect to current quark mass $m$ are expressed in Eq. (\[pscalar\]), which is shown in Fig.(\[mpi1\]). ![Temperature dependence of the pion mass[]{data-label="mpi1"}](mpi3.eps "fig:")\ Let us now turn to thermodynamic property of the pseudoscalar meson mass Eq.(\[pscalar\]): $$\begin{aligned} m_{\pi^{0,\pm}}^2(T) &\simeq& \frac{4\mu_P^3(T)}{f_\pi^2(T)}m = \frac{4 v_o(T) \mu_f^2(T) }{f_\pi^2(T)}m = \frac{\mu_f^2(T)}{\bar{\lambda}(T)v_o(T)}m\end{aligned}$$ which explicitly shows that when keeping the mass scale $\mu_f^2$ to be a temperature-independent constant, the thermodynamic mass of the pseudoscalar meson becomes divergent near the critical temperature $T_c$ as $v_o(T_c) =0$, which is obviously contrary to our intuition. This is a manifest reason why we should make an assumption for the temperature dependence of $\mu_f^2(T)$ given in Eq.(\[mu\_f\]), which can lead to the expected thermodynamic behavior for the pseudoscalar meson mass near the critical point $$\begin{aligned} m_{\pi^{0,\pm}}^2(T) = \frac{\gamma_o v_o(T)}{\bar{\lambda}(T)} m,\end{aligned}$$ which is shown in Fig.(\[mpi1\]). According to the above derivation, we see that the phase transition of chiral symmetry restoration is second order in our simple model. Thus, it is natural to determine the critical behavior of all the quantities discussed previously. Now we would like to find the the scaling behavior of the vacuum expectation value $v_o(T)$ near the critical point. By expanding our gap equation Eq.(\[gap eqn\]) around the critical temperature $T_c$ with respect to the small value of VEV $v_o(T)^2$ up to the order of $v_o(T)^2$, we can obtain: $$\begin{aligned} \frac{1}{6}N_c(T_c^2-T^2)-C v_o(T)^2=0,\end{aligned}$$ where $C = \frac{N_c}{8\pi^2}\gamma(0,\frac{\mu_s^2}{M_c^2})+\frac{1}{4}-\gamma_o$ and $\gamma(s,x)\equiv \int^x_0t^{s-1}e^{-t}dt$ is the lower incomplete gamma function. Given above equation, we can easily obtain: $$\begin{aligned} v_o = \sqrt{\frac{N_c}{6C}}(T_c^2-T^2)^{\frac{1}{2}}\propto (T_c-T)^{\frac{1}{2}}.\end{aligned}$$ Thus, the critical dimension is $\beta=0.5$. Other quantities such as $f_\pi(T)$, $m_{\pi^{0,\pm}}(T)$ and $(-{\langle}\bar{q}q{\rangle}(T))^{1/3}$ all have the same scaling behavior. Such a critical behavior is not accidental, which can actually be understood from the fact that near the critical temperature all these quantities are proportional to the VEV $v_o(T)$ with $\bar{\lambda}(T)$ keeping fixed to $\bar{\lambda}(T_c)\neq 0$. Conclusions and Remarks ======================= In this paper, we have extended the chiral dynamical model to the chiral thermodynamic model by adopting the CTPGF approach. The resulting effective chiral Lagrangian for the composite meson fields at finite temperature is similar to the one of the CDM, but all the couplings and mass scales become temperature dependent. We have discussed in detail the finite temperature behavior of CTDM. Much attention has been paid to the thermodynamic chiral symmetry breaking and its restoration at finite temperature. After fixing the free parameters in the effective chiral Lagrangian, we have determined the critical temperature for the chiral symmetry restoration, its value has been found to be around $T_c \simeq 200$ MeV which is consistent with other predictions based on the NJL model[@Hatsuda1994; @Klevansky1992; @Alkofer1996; @Buballa2005]. We have also explicitly presented the thermodynamic behavior of several interesting quantities which include the vacuum expectation value VEV $v_o(T)$, the pion decay constant $f_\pi(T)$, the quark condensate ${\langle}\bar{q}q{\rangle}(T)$ and the pseudoscalar meson mass $m_{\pi^{0,\pm}}(T)$, they all display the property of the chiral symmetry restoration at the critical temperature $T_c$. From the numerical calculations, we have shown that they all have the same scaling behavior near the critical point. It is interesting to note that the mass scale $\mu_f$ for the four quark interaction in the NJL model should be temperature dependent at finite temperature as expected from the gluon thermodynamics, its thermodynamic behavior near the critical point is required to be same as the one of the chiral symmetry breaking. As a consequence, we are led to the assumption that $\mu_f^2(T)=\gamma_o v_o^2(T)$ in order to yield the expected thermodynamic behavior of the pion meson mass and to avoid the divergent behavior near the critical point of phase transition. Finally, we would like to remark that as limited from our main purpose in the present paper we have only considered two flavor quarks and ignored the important instanton effects and U(1)$_A$ anomalous effect, which prevents us to discuss some other interesting properties, such as the large strange quark mass effects and the anomalous $U(1)_A$ symmetry restoration at finite temperature, we shall investigate those interesting effects elsewhere. [Acknowledgement]{} The authors would like to thank L.X. Cui and Y.B. Yang for useful discussions. This work was supported in part by the National Science Foundation of China (NSFC) under Grant \#No. 10821504, 10975170 and the Project of Knowledge Innovation Program (PKIP) of the Chinese Academy of Science. Brief Outline on Closed-Time-Path Green Function (CTPGF) Formalism ================================================================== The formalism used in zero-temperature quantum field theory is suitable to describe observables (e.g. cross-sections) measured in empty space-time, as particle interactions in an accelerator. However, at high temperature, the environment has a non-negligible density of matter which makes the assumption of zero-temperature field theories inapplicable. Namely, under those circumstances, the methods of zero-temperature field theories are not sufficient any more and should be replaced by others, which is closer to thermodynamics where the background state is a thermal bath. Therefore we shall develop quantum field theory with finite temperature which is extremely useful to study all phenomena due to the collective effects, such as: phase transitions, early universes, etc. There are several approaches for the finite temperature field theories, in this appendix we shall focus on the closed-time-path Green function (CTPGF) formalism which is simply applicable in our case. The CTPGF formalism, developed by Schwinger [@Schwinger1961] and Keldysh [@Keldysh1965], has been used to solve lots of interesting problems in statistical physics and condensed matter theory [@Chou1985]. It is generally believed that this technique is quite efficient in investigating the nonequilibrium and finite temperature dynamical systems, this is because such a formalism naturally incorporates both the statistical and dynamical information[@Chou1985; @Zhou1980]. Excellent review articles [@Chou1985; @Das:2000ft] and monographs [@Calzetta2008; @Rammer2007] have described different aspects of these issues. In this appendix, we will briefly outline the main method with the Schwinger-Keldysh propagators and the universal Feynman rules for the general theory. Let us begin by the general discussion of statistical physics. A dynamical system can be characterized by its Hamiltonian $H$ and a statistical ensemble of this system in equilibrium at a finite temperature $T=\frac{1}{\beta}$ (in units of Boltzmann constant) is described in terms of a partition function $$Z(\beta)={\ensuremath{\mathop{\mathrm{Tr}}}}\rho(\beta)={\ensuremath{\mathop{\mathrm{Tr}}}}e^{-\beta\mathcal {H}}$$ Here $\rho(\beta)$ is known as the density matrix operator and $\mathcal{H}$ can be thought of as the generalized Hamiltonian of the system. For example, for a canonical ensemble in which the system can only exchange energy with the heat bath, $\mathcal{H}$ is defined as: $$\mathcal{H}=H$$ while for a grand canonical ensemble in which the system can not only exchange energy with the heat bath but also exchange particles with the reservoir, ${\mathcal H}$ is taken as $$\mathcal{H}=H-\mu N$$ where $\mu$ is the chemical potential and $N$ represents the particle number operator. A observable in a statistical ensemble is the ensemble average for any operator $$\langle\mathcal{O} \rangle_\beta=\frac{1}{Z(\beta)}{\ensuremath{\mathop{\mathrm{Tr}}}}\rho(\beta)\mathcal{O}$$ Since the partition function and ensemble averages involve a trace operation, this feature leads to the famous KMS (Kubo-Martin-Schwinger) relation. $$\begin{aligned} \langle\mathcal{O}_1(t)\mathcal{O}_2(t^\prime)\rangle_\beta &=& \frac{1}{Z(\beta)}{\ensuremath{\mathop{\mathrm{Tr}}}}e^{-\beta\mathcal{H}}\mathcal{O}_1(t)\mathcal{O}_2(t^\prime)\nonumber\\ &=& \frac{1}{Z(\beta)}{\ensuremath{\mathop{\mathrm{Tr}}}}e^{-\beta\mathcal{H}}\mathcal{O}_2(t^\prime) e^{-\beta\mathcal{H}} \mathcal{O}_1(t) e^{\beta\mathcal{H}}\nonumber\\ &=& \frac{1}{Z(\beta)}{\ensuremath{\mathop{\mathrm{Tr}}}}e^{-\beta\mathcal{H}}\mathcal{O}_2(t^\prime) \mathcal{O}_1(t-i\beta)\nonumber\\ &=& \langle \mathcal{O}_2(t^\prime)\mathcal{O}_1(t-i\beta)\rangle_\beta\end{aligned}$$ Note that KMS relation only rely on the trace operation and does not depend on any periodicity property of operators along the temperature (imaginary time) interval. Now it is easily seen that the operator $e^{-\beta\mathcal{H}}$ in the definition of the partition function is very similar to the time evolution operator in the imaginary time axis $e^{-i(-i\beta)\mathcal{H}}$ [@Bloch1958]. We promote this similarity and analytically extend the time variable to the complex plane. So the operator $e^{-\beta\mathcal{H}}$ would live on the line interval which is parallel to the negative imaginary time-axis, with its length $\beta$. The analogy implies that we can define our theory on some certain contour on the complex t-plane. The contour should satisfy the following several requirement: (i) The two endpoints of the contour must be fixed in an interval in a line parallel to the imaginary axis with its length $\beta$. The stating point A (corresponding to time $t_i$) and the ending point B(corresponding to $t_f=t_i-i\beta$) are identified, and one requires that $\mathcal{O}\|_B=\mathcal{O}\|_A$ if $\mathcal{O}$ is bosonic and and $\mathcal{O}\|_B=-\mathcal{O}\|_A$ if $O$ is fermionic; (ii) For a system whose spectrum of the Hamiltonian is semi-positive (at least bounded below due to the stability of the system), the contour needs to have a monotonically decreasing or constant imaginary part for the reason of the convergence of the complete partition function.The same result can also be obtained by analyzing the convergence of the two-point Green function.[@Quiros:1999jp]; (iii) The contour needs to pass the whole real axis of t-plane on which the field operators $\mathcal{O}_i(t)$ are defined (Real-time Formalism. Otherwise, like the imaginary time formalism, the field operators need to be analytically extended to the imaginary axis first). The particular family of such real time contours is depicted in Fig. \[realcontour\] ![Contour used in the real time formalism[]{data-label="realcontour"}](fig05.eps "fig:")\ where the contour ${\cal C}$ is ${\displaystyle {\cal C}=C_1\bigcup C_2 \bigcup C_3 \bigcup C_4}$. The contour $C_1$ goes from the initial time $t_i$ to the final time $t_f$, $C_3$ from $t_f$ to $t_f-i\sigma$, with $0\le\sigma\le\beta$, $C_2$ from $t_f-i\sigma$ to $t_i-i\sigma$, and $C_4$ from $t_i-i\sigma$ to $t_i-i\beta$. Different choices of $\sigma$ lead to an equivalence class of quantum field theories at finite temperature. Our preferred choice is the Schwinger-Keldysh one with $\sigma=0$. With this specific contour, the action of a field configuration is the sum of contributions from the three parts, $$\label{action} S = \int\limits_{\cal C}\!dt\, L(t) = \int\limits_{{t_{\mathrm{i}}}}^{{t_{\mathrm{f}}}}\!dt\, L(t) -\int\limits_{{t_{\mathrm{i}}}}^{{t_{\mathrm{f}}}}\!dt\, L(t) -i\!\int\limits_0^\beta\!d\tau\, L({{t_{\mathrm{i}}}}- i\tau)\,$$ where $$L(t) = \int\!d\vec{x}\, \mathcal{L}[\phi(t,\vec{x})]\,,$$ and $\mathcal{L}$ is the Lagrangian density. In the following we will take the theory of a scalar field $\phi(x)$ as an example. However, in the limit $t_i\to -\infty$ and $t_f\to\infty$, it can be shown that the third branch gets decoupled from the other two (the factors in the propagators connecting such branches are asymptotically damped). Consequently, in this limit, we are effectively dealing with two branches leading to the name “closed time path formalism”. In this contour, then, the time integration has to be thought of as $$\label{contourC} \int_{\cal C} dt = \int^\infty_{-\infty} d t_+ -\int^{\infty}_{-\infty} dt_-$$ where the relative negative sign arises because time is decreasing in the second branch of the time contour. The advantage of introducing the contour ${\cal C}$ is that one can introduce the sources coupled to the field $\phi$ which is not vanishing on the two Minkowski parts of the contour. This procedure would give us the generating functional $$\label{gen-func} Z[J_1,J_2] = \int\!{\cal D}\phi\,\exp\left(iS +i\!\int\limits_{-\infty}^\infty\!dt_+\!\int\!d\vec{x}\,J_1(x)\phi_1(x) -i\!\int\limits_{-\infty}^\infty\!dt_-\!\int\!d\vec{x}\,J_2(x)\phi_2(x) \right) \ .$$ Here $J_{1,2}$ and $\phi_{1,2}$ are the sources and fields on the two Minkowski parts of the contour, i.e., $$\begin{aligned} J_1(t,\vec{x}) = J(t_+,\vec{x})\,, & \qquad & \phi_1(t,\vec{x}) = \phi(t_+,\vec{x})\,,\\ J_2(t,\vec{x}) = J(t_-,\vec{x})\,, & \qquad & \phi_2(t,\vec{x}) = \phi(t_-,\vec{x})\,.\end{aligned}$$ By taking second variations of $Z$ with respect to the source $\phi$ one finds the Schwinger-Keldysh propagator $$iG_{ab}(x-y) = \frac 1{i^2}\,\frac{\delta^2\ln Z[J_1,J_2]} {\delta J_a(x)\,\delta J_b(y)} = i \left(\begin{array}{cc} G_{11} & -G_{12}\\-G_{21} & G_{22} \end{array}\right) \ .$$ In the operator formalism, the Schwinger-Keldysh propagator corresponds to the contour-ordered correlation function. In the single time representation[@Chou1985], this means: $$\label{Gab-oper} \begin{split} iG_{11}(t,{{\vec x}}) = {\langle}T \phi_1(t,{{\vec x}})\phi_1(0){\rangle}_{\beta}\,,\qquad & iG_{12}(t,{{\vec x}}) = {\langle}\phi_2(0)\phi_1(t,{{\vec x}}){\rangle}_{\beta}\,,\\ iG_{21}(t,{{\vec x}}) = {\langle}\phi_2(t,{{\vec x}})\phi_1(0){\rangle}_{\beta}\,,\qquad & iG_{22}(t,{{\vec x}}) = {\langle}\bar T \phi_2(t,{{\vec x}})\phi_2(0){\rangle}_{\beta}\,. \end{split}$$ where $T$ ($\bar{T}$) denotes normal (reversed) time ordering, and $$\begin{aligned} \phi_1(t,{{\vec x}}) &=& e^{iHt-i{{\vec P}}\cdot{{\vec x}}} \phi(0) e^{-iHt+i{{\vec P}}\cdot{{\vec x}}}\,,\\ \phi_2(t,{{\vec x}}) &=& e^{iH(t-i\sigma)-i{{\vec P}}\cdot{{\vec x}}} \phi(0) e^{-iH(t-i\sigma)+i{{\vec P}}\cdot{{\vec x}}}\,.\end{aligned}$$ Let us now consider the free real scalar theory. If one goes to the momentum space and by inserting the complete set of states into the definitions (\[Gab-oper\]), one finds the explicit form of the previously defined Schwinger-Keldysh propagator. $$\begin{aligned} \label{SKprop} iG_{11}(k) &=& \frac{i}{k^2-m^2+i\epsilon}+2\pi n_B(\omega)\delta(k^2-m^2)\,,\qquad \omega\equiv\|k_0\|\\ iG_{12}(k) &=& 2\pi [n_B(\omega)+\theta(-k_0)]\delta(k^2-m^2)\,,\\ iG_{21}(k) &=& 2\pi [n_B(\omega)+\theta(k_0)]\delta(k^2-m^2)\,,\\ iG_{22}(k) &=& -\frac{i}{k^2-m^2+i\epsilon}+2\pi n_B(\omega)\delta(k^2-m^2)\,.\end{aligned}$$ Or in the matrix form: $$\begin{aligned} iG_\beta(k) &=& \left(\begin{array}{cc}\frac{i}{k^2-m^2+i\epsilon} & {2\pi\theta(-k_0)\delta(k^2-m^2)} \\ {2\pi\theta(k_0)\delta(k^2-m^2)} & -\frac{i}{k^2-m^2-i\epsilon}\end{array}\right)\nonumber\\ &+& 2\pi n_B(\omega)\delta(k^2-m^2)\left(\begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array}\right)\end{aligned}$$ where $n_B(\omega)\equiv\frac{1}{e^{\beta\omega}-1}$ stands for the Bose-Einstein distribution function. Note that the propagator is a $2\times2$ matrix, a consequence of the doubling of the degrees of freedom. However, the propagators (12), (21) and (22) are unphysical since at least one of their time arguments is on the negative branch. They are required for the consistency of the theory. The only physical propagator is the (11) component shown in Eq.(\[SKprop\]). For perturbative calculations we need to know the complete Feynman rules besides of the propagators. From the generating functional Eq.(\[gen-func\]) and the action Eq.(\[action\]) defined on contour ${\cal C}$ Eq.(\[contourC\]), we see that the complete theory contains two types of vertices- type-1 for the original fields $\phi_1(x)$ while type-2 for the doubled fields $\phi_2(x)$. The vertices for the partner fields will have a relative negative sign corresponding to the original vertices, because time is decreasing in the negative branch. The four possible propagators, (11), (12), (21) and (22) defined above connect them. All of them have to be considered for the consistency of the theory. The golden rule is that: physical legs must always be attached to type 1 vertices[@Quiros:1999jp]. For other Feynman rules, including the integration measure, the symmetry factors involved in Feynman diagrams, the topology of the Feynman diagrams, etc. are all the same as the zero-temperature field theory. For the application to the present paper, we also need to know the Schwinger-Keldysh propagator for fermions as the quark fields here are represented as the chiral fermion fields $$\begin{aligned} iS_\beta(k) &=&(k\sla+m)[ \left(\begin{array}{cc}\frac{i}{k^2-m^2+i\epsilon} & {2\pi\theta(-k_0)\delta(k^2-m^2)} \\ {2\pi\theta(k_0)\delta(k^2-m^2)} & -\frac{i}{k^2-m^2-i\epsilon}\end{array}\right)\nonumber\\ && - 2\pi n_F(\omega)\delta(k^2-m^2)\left(\begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array}\right)]\end{aligned}$$ where $n_F(\omega)\equiv\frac{1}{e^{\beta\omega}+1}$ stands for the Fermi-Dirac distribution function. When transforming into the Euclidean spacetime the Schwinger-Keldysh propagator defined above becomes $$\begin{aligned} iS_{E\beta}(k) &=&(-i)(k\sla_E+m)[ \left(\begin{array}{cc}\frac{1}{k_E^2+m^2} & {2\pi i\theta(-k_{E4})\delta(k_E^2+m^2)} \\ {2\pi i\theta(k_{E4})\delta(k_E^2+m^2)} & -\frac{1}{k_E^2+m^2}\end{array}\right)\nonumber\\ && - 2\pi i n_F(\omega)\delta(k_E^2+m^2)\left(\begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array}\right)]\end{aligned}$$ In Sec. 3 the factor $-i$ is canceled by the factor $i$ in the integration measure transformation $d^4k\to i d^4 k_E$. Derivation of Chiral Dynamical Model in the Chiral Rotated Basis {#appendix} ================================================================ In this appendix, we will derive the effective chiral Lagrangian for mesons in the so-called chiral Rotated Basis“[@Espriu:1989ff]. Although the obtained effective chiral Lagrangian will not change, it is more transparent to see the chiral symmetries and their spontaneous breaking in this derivation. Let us begin with the effective Lagrangian $$\label{Lag3} {\cal L}^q_{eff}(q,{\bar q}) = \bar{q}\gamma^\mu i\partial_\mu q+\bar{q}_L\gamma_\mu {\cal A}^\mu_L q_L+\bar{q}_R\gamma_\mu {\cal A}_R^\mu q_R -[\bar{q}_L(\Phi-M)q_R+h.c.]$$ where the auxiliary meson fields $\Phi(x)$ is defined as in Eq.(\[meson\_def\]) $$\begin{aligned} && \Phi(x)\equiv \xi_L(x)\phi(x)\xi_R^\dagger(x),\qquad U(x)\equiv \xi_L(x)\xi_R^\dagger(x)=\xi_L^2(x)=e^{i\frac{2\Pi(x)}{f}}\nonumber\\ && \phi^\dagger(x)=\phi(x)=\sum^{3}_{a=0}\phi^a(x)T^{a},\qquad \Pi^\dagger(x)=\Pi(x)=\sum^{3}_{a=0}\Pi^a(x)T^a,\end{aligned}$$ where $\Pi(x)$ and $\phi(x)$ represent the pseudoscalar and scalar mesons respectively. Note that except for the mass term or the source term, the Lagrangian is invariant under the transformation of the local $U(2)_L\times U(2)_R$ chiral symmetry: $$\begin{aligned} &&q_L(x)\to g_L(x)q_L(x),\quad q_R(x)\to g_R(x)q_R(x);\quad \Phi(x)\to g_L(x)\Phi(x)g_R^\dagger(x),\nonumber\\ && {\cal A}_{L\mu}\to g^\dagger_L {\cal A}_{L\mu}g_L(x) -i g^\dagger_L\partial_\mu g_L(x), \quad {\cal A}_{R\mu} \to g^\dagger_R {\cal A}_{R\mu}g_R(x) -i g^\dagger_R\partial_\mu g_R(x),\end{aligned}$$ The transformation for $\Phi(x)$ can also be written in terms of the fields $\phi(x)$ and $\xi_L(x)$ as: $$\begin{aligned} \phi(x) \to h(x)\phi(x)h^\dagger(x),\quad \xi_L(x)\to g_L(x)\xi_L(x)h^\dagger(x)=h(x)\xi_L(x)g^\dagger_R(x).\end{aligned}$$ Let us now introduce new quark fields, which is referred to the chiral Rotated Basis” in[@Espriu:1989ff]. $$\begin{aligned} q_L = \xi_L Q_L,&\quad& \bar{q}_L=\bar{Q}_L \xi_L^\dagger,\nonumber\\ q_R = \xi_L^\dagger Q_R,&\quad& \bar{q}_R=\bar{Q}_R\xi_L.\end{aligned}$$ With this new quark basis, we can rewrite the Lagrangian Eq.(\[Lag3\]) in the following form $$\begin{aligned} \label{Lag4} {\cal L}^Q_{eff}(Q,{\bar Q}) = \bar{Q}\gamma^\mu i\partial_\mu Q+\bar{Q}_L\gamma^\mu L_\mu Q_L+\bar{Q}_R\gamma^\mu R_\mu Q_R -[\bar{Q}_L(\phi-{\cal M})Q_R+h.c.],\end{aligned}$$ where the fields $L_\mu, ~R_\mu$ and ${\cal M}$ are defined as $$\begin{aligned} \label{transform} L_\mu &\equiv& \xi_L^\dagger{\cal A}_{L\mu}\xi_L+i\xi_L^\dagger\partial_\mu\xi_L,\quad R_\mu \equiv \xi_L{\cal A}_{R\mu}\xi^\dagger_L+i\xi_L\partial_\mu\xi^\dagger_L, \nonumber\\ {\cal M} &\equiv& \xi_L^\dagger M \xi_L^\dagger, \quad {\cal M}^\dagger \equiv \xi_L M^\dagger \xi_L.\end{aligned}$$ In the above rotated basis“, the quark fields $Q_{L(R)}(x)$ transform only under the diagonal $U_V(2)$ symmetry: $$\begin{aligned} Q_L(x) \to h(x) Q_L(x), \quad Q_R(x) \to h(x) Q_R(x).\end{aligned}$$ Thus, the quark fields $Q_{L(R)}(x)$ are much like the constituent quark” defined in the nonrelativistic quark model [@Manohar:1983md]. When the chiral symmetry is spontaneous breaking and the meson field $\phi(x)$ acquires the vacuum expectation value (VEV) $<\phi(x)>=V$, $Q_{L(R)}$ will obtain a mass term $\bar{Q}_L (V-{\cal M}) Q_R + h.c.$. In the case where each quark flavor possesses the universal current mass $m$, the VEV matrix is diagonal $V=v\cdot I$, here $I$ is the identity matrix in the flavor space. The mass term is $(v-m)\bar{Q}_L Q_R+h.c.$, namely the mass of quarks $Q(x)_{L(R)}$ is the dynamical quark mass $\bar{m}= v-m$ defined in Sec.\[sec2\]. Note that the Lagrangian Eq.(\[Lag4\]) has the same structure as the original one Eq.(\[Lag3\]) except for the definition of the mass and gauge fields. Thus, we may expect that the effective chiral Lagrangian for the meson fields has the same structure as Eq.(\[Lag\_meson\]). Indeed, by integrating over the quark fields following the procedure from Eq.(\[Lag2\]) to Eq.(\[ChEF3\]), we obtain $$\begin{aligned} \label{ChEF4} S^M_{E Re} = \frac{N_c}{2}\int d^4 x_E \int \frac{d^4k}{(2\pi)^4} tr_{SF} \ln(k^2+ \Delta^\prime_E)-\ln Z_0,\end{aligned}$$ where ${\Delta}^\prime_E$ above is defined as $$\begin{aligned} \Delta^\prime_E &=& \hat{\phi}\hat{\phi}^\dagger P_R + \hat{\phi}^\dagger \hat{\phi} P_L - i\gamma\cdot D^\prime_E \phi P_L - i\gamma\cdot D^\prime_E\phi^\dagger P_R\nonumber\\ && -\sigma_{\mu\nu}{\cal F}^\prime_{R\mu\nu} P_L - \sigma_{\mu\nu} {\cal F}^\prime_{L\mu\nu} P_R +(i D^\prime_{E\mu})(iD^\prime_{E\mu})+2 k\cdot(i D^\prime_E),\end{aligned}$$ and $$\begin{aligned} i D^\prime_{E\mu} \phi &=& i\partial_\mu \phi + L_\mu \phi - \phi R_\mu,\\ i D^\prime_{E\mu} &=& i\partial_\mu + R_\mu P_L + L_\mu P_R,\\ \hat{\phi} &\equiv& \phi-{\cal M}.\end{aligned}$$ In order to derive the effective action for meson field, we redefine $\Delta^{k\prime}_E\equiv k^2+ \Delta^\prime_E$ to the following two terms: $$\begin{aligned} \Delta^{k\prime}_E \equiv k^2+ \Delta^\prime_E = \Delta_0 +\tilde{\Delta}^\prime_E,\end{aligned}$$ with $$\begin{aligned} \Delta_0 &=& k^2+\bar{M}^2,\nonumber\\ \tilde{\Delta}^\prime_E &=& [(\hat{\phi}\hat{\phi}^\dagger-\bar{\cal M}\bar{\cal M}^\dagger) P_R +(\hat{\phi}^\dagger\hat{\phi}-\bar{\cal M}^\dagger\bar{\cal M})P_L-i\gamma\cdot D_E^\prime \phi P_L -i\gamma\cdot D^\prime_E\phi^\dagger P_R \nonumber\\ && -\sigma_{\mu\nu} {\cal F}^\prime_{R\mu\nu}P_L-\sigma_{\mu\nu}{\cal F}^\prime_{L\mu\nu} P_R + (i D^\prime_{E\mu})(i D^\prime_{E\mu}) + 2k\cdot(iD_E^\prime)] \nonumber\\ && +(\bar{\cal M}\bar{\cal M}^\dagger - \bar{M}^2)P_R + (\bar{\cal M}^\dagger\bar{\cal M}-\bar{M}^2)P_L,\end{aligned}$$ where $\bar{M}$ is supposed vacuum expectation values (VEVs) of $\hat{\Phi}$, i.e., $<\hat{\Phi}> = \bar{M}$ which is real and $\bar{\cal M} \equiv \xi^\dagger_L\bar{M}\xi^\dagger_L$. As we will see that the terms in the third line of the definition of $\tilde{\Delta}^\prime_E$ is crucial to prove the equivalence of the obtained effective action for mesons in this rotated basis to the one given in Eq.(\[Lag\_meson\]). Now if we regard $\tilde{\Delta}_E^\prime$ as the perturbation and take $Z_0=(\det \Delta_0)^{\frac{1}{2}}$ as before, we can expand the effective action Eq.(\[ChEF4\]) according to $\tilde{\Delta}_E^\prime$: $$\begin{aligned} \label{Lag5} S^M_{E Re} &=& \frac{N_c}{2} \int d^4 x_E \int\frac{d^4k}{(2\pi)^4} tr_{SF}[\ln(\Delta_0+\tilde{\Delta}^\prime_E)-\ln\Delta_0]\nonumber\\ &=& \frac{N_c}{2} \int d^4 x_E \int\frac{d^4k}{(2\pi)^4} tr_{SF}\ln(1+\frac{1}{\Delta_0}\tilde{\Delta}^\prime_E)\nonumber\\ &=& \frac{N_c}{2} \int d^4 x_E \int\frac{d^4k}{(2\pi)^4} tr_{SF} \sum^\infty_{n=1}\frac{(-1)^{n+1}}{n}(\frac{1}{\Delta_0}\tilde{\Delta}^\prime_E)^n.\end{aligned}$$ If we only keep the leading two terms in the expansion as in the previous sections, we can obtain: $$\begin{aligned} \label{ChEF5} S^M_{E Re} &\approx& \frac{N_c}{2}\int d^4 x_E \int \frac{d^4 k}{(2\pi)^4} tr_{SF}[\frac{1}{\Delta_0}\tilde{\Delta}^\prime_E-\frac{1}{2}\frac{1}{\Delta_0^2}(\tilde{\Delta}^\prime_E)^2]\nonumber\\ &=& \frac{N_c}{16\pi^2}\int d^4 x_E tr_F \{M_c^2L_2[(\hat{\phi}\hat{\phi}^\dagger-\bar{\cal M}\bar{\cal M}^\dagger)+ (\hat{\phi}^\dagger\hat{\phi}-\bar{\cal M}^\dagger\bar{\cal M})]\nonumber\\ &&-\frac{1}{2} L_0 [D^\prime_E\hat{\phi}\cdot D^\prime_E\hat{\phi}^\dagger+ D^\prime_E\hat{\phi}^\dagger\cdot D^\prime_E\hat{\phi}+ (\hat{\phi}\hat{\phi}^\dagger-\bar{\cal M}\bar{\cal M}^\dagger)^2+(\hat{\phi}^\dagger\hat{\phi}-\bar{\cal M}^\dagger\bar{\cal M})^2]\nonumber\\ && +M_c^2L_2[(\bar{\cal M}\bar{\cal M}^\dagger-\bar{M}^2)+(\bar{\cal M}^\dagger\bar{\cal M}-\bar{M}^2)]\nonumber\\ && -\frac{1}{2}L_0[(\hat{\phi}\hat{\phi}^\dagger-\bar{\cal M}\bar{\cal M}^\dagger)(\bar{\cal M}\bar{\cal M}^\dagger-\bar{M}^2)+(\hat{\phi}^\dagger\hat{\phi}-\bar{\cal M}^\dagger\bar{\cal M})(\bar{\cal M}^\dagger\bar{\cal M}-\bar{M}^2)\nonumber\\ &&+(\bar{\cal M}\bar{\cal M}^\dagger-\bar{M}^2)(\hat{\phi}\hat{\phi}^\dagger-\bar{\cal M}\bar{\cal M}^\dagger)+(\bar{\cal M}^\dagger\bar{\cal M}-\bar{M}^2)(\hat{\phi}^\dagger\hat{\phi}-\bar{\cal M}^\dagger\bar{\cal M})\nonumber\\ && (\bar{\cal M}\bar{\cal M}^\dagger-\bar{M}^2)^2+(\bar{\cal M}^\dagger\bar{\cal M}-\bar{M}^2)^2]\},\end{aligned}$$ where the matrix $L_0$ and $L_2$ are defined in Eq.(\[intg1\]). It is easy to see that the last four lines of terms vanish if the different flavors of quarks have the same current mass which leads to the same vacuum expectation value for each flavor: $$\begin{aligned} \bar{\cal M}\bar{\cal M}^\dagger-\bar{M}^2 &=& (\xi_L^\dagger \bar{M} \xi_L^\dagger)(\xi_L\bar{M}\xi_L)-\bar{M}^2 \nonumber\\ &=& \xi_L^\dagger \bar{M}^2\xi_L-\bar{M}^2=0.\end{aligned}$$ The last line is valid since the mass matrix $\bar{M}$ is diagonal with the same eigenvalues and $\bar{M}$ can commute with the SU(2) matrix $\xi_L$. Note also that in the case of 2 flavors with the universal current quark mass, after the chiral symmetry $SU_L(2)\times SU_R(2)$ is broken to $SU_V(2)$ and the field $\phi(x)$ acquires vacuum expectation value (VEV) $V=v\cdot I$, we have: $$\begin{aligned} iD_{E\mu}^\prime \hat{\phi} &\approx& (v-m)[L_\mu-R_\mu]\nonumber\\ &=& (v-m)i[\xi_L^\dagger(\partial_\mu-i{\cal A}_{L\mu})\xi_L-\xi_L(\partial_\mu-i{\cal A}_{R\mu})\xi^\dagger_L]\nonumber\\ &=& (v-m)i\xi_L^\dagger(D_{E\mu} U)\xi_L^\dagger = -(v-m)i\xi_L(D_{E\mu} U^\dagger)\xi_L.\end{aligned}$$ In order to prove the last two equalities, we have to use the definition of $U\equiv \xi_L^2$ and the identity $\partial_\mu \xi_L =-\xi_L(\partial_\mu\xi_L^\dagger)\xi_L$. Therefore, from the kinetic terms for $\phi(x)$ in Eq.(\[ChEF5\]), we can obtain the kinetic term for the pseudoscalar meson field $U\equiv e^{i2\Pi(x)/f_\pi}$: $$\begin{aligned} -\frac{N_c(v-m)}{16\pi^2}\int d^4 x_E tr_E [L_0(D_{E\mu}U)(D_{E\mu }U^\dagger)].\end{aligned}$$ However, when the chiral symmetry is restored, as discussed in the context of CDTM, this term will disappear as the VEV of $\phi(x)$ vanishes. Thus, the discussion in this rotated basis" gives a more transparent picture of Goldstone boson character of the pseudoscalar mesons. Next we would like to prove the equivalence between the effective chiral Lagrangians Eq.(\[Lag\_meson\]) and Eq.(\[ChEF5\]). According to the definition of the transformations Eq.(\[transform\]), we can easily obtain the following relations: $$\begin{aligned} D^\prime_{E\mu} \hat{\phi} = \xi^\dagger_L(D_{E\mu} \hat{\Phi})\xi^\dagger_L \quad D^\prime_{E\mu} \hat{\phi}^\dagger = \xi_L(D_{E\mu} \hat{\Phi}^\dagger)\xi_L,\end{aligned}$$ where $D_{E\mu}\hat{\Phi}$ and $D_{E\mu}\hat{\Phi}^\dagger$ are defined as in Eq.(\[partial\]). Thus, the first two lines of terms in Eq.(\[ChEF5\]) can be written in a form with respect to $\Phi$: $$\begin{aligned} \label{ChEF6} S^M_{E Re} &=& \frac{N_c}{2}\int d^4 x_E \text{tr}_F \{M_c^2 L_2[\xi^\dagger_L(\hat{\Phi}\hat{\Phi}^\dagger-\bar{M}^2)\xi_L +\xi_L(\hat{\Phi}^\dagger\hat{\Phi}-\bar{M}^2)\xi_L^\dagger]\nonumber\\ && -\frac{1}{2}L_0[\xi_L^\dagger(D_E\hat{\Phi}\cdot D_E\hat{\Phi}^\dagger)\xi_L + \xi_L(D_E\hat{\Phi}^\dagger\cdot D_E\hat{\Phi})\xi_L^\dagger \nonumber\\ && + \xi_L^\dagger(\hat{\Phi} \hat{\Phi}^\dagger-\bar{M}^2)^2\xi_L + \xi_L(\hat{\Phi}^\dagger \hat{\Phi}-\bar{M}^2)^2\xi_L^\dagger]\}\nonumber\\ &=& \frac{N_c}{2}\int d^4 x_E \text{tr}_F \{M_c^2 [(\xi_L L_2\xi^\dagger_L)(\hat{\Phi}\hat{\Phi}^\dagger-\bar{M}^2) +(\xi_L^\dagger L_2\xi_L)(\hat{\Phi}^\dagger\hat{\Phi}-\bar{M}^2)]\nonumber\\ && -\frac{1}{2}\{(\xi_LL_0\xi_L^\dagger)[D_E\hat{\Phi}\cdot D_E\hat{\Phi}^\dagger+(\hat{\Phi} \hat{\Phi}^\dagger-\bar{M}^2)^2] \nonumber\\ &&+ (\xi_L^\dagger L_0\xi_L)[D_E\hat{\Phi}^\dagger\cdot D_E\hat{\Phi}+(\hat{\Phi}^\dagger \hat{\Phi}-\bar{M}^2)^2]\}\}.\end{aligned}$$ In general the matrices $\xi_L$ and $L_0$ ($L_2$) do not commute with each other when different flavors do not have the same current masses. So the part of effective chiral Lagrangian shown in Eq.(\[ChEF6\]) is in general not equivalent to Eq.(\[Lag\_meson\]) by just comparing to the same truncated terms. In fact, by taking into account the higher-order terms in Eq.(\[Lag5\]) and the last four lines of terms in Eq.(\[ChEF5\]), it is expected that the extra terms would cancel the unwanted terms in Eq.(\[ChEF6\]) due to non-commutativity of $\xi_L$ and $L_0$ ($L_2$). In our present case, as we only consider two flavors with the same current quark mass, $\xi_L$ can commute with $L_0$ and $L_2$, there is no such extra terms. Furthermore, as mentioned before, the last four lines of terms in Eq.(\[ChEF5\]) will vanish due to commutativity of $\xi_L$ and $\bar{M}$. Thus, we arrive at the following effective chiral Lagrangian: $$\begin{aligned} S^M_{E Re} &=& \frac{N_c}{16\pi^2}\int d^4 x_E tr_F \{M_c^2L_2[(\hat{\Phi}\hat{\Phi}^\dagger-\bar{M}^2)+(\hat{\Phi}^\dagger\hat{\Phi}-\bar{M}^2)]\nonumber\\ &&-\frac{1}{2} L_0 [D_E\hat{\Phi}\cdot D_E\hat{\Phi}^\dagger+ D_E\hat{\Phi}^\dagger\cdot D_E\hat{\Phi}+ (\hat{\Phi}\hat{\Phi}^\dagger-\bar{M}^2)^2+(\hat{\Phi}^\dagger\hat{\Phi}-\bar{M}^2)^2]\},\end{aligned}$$ which is exactly agree with Eq.(\[Lag\_meson\]) obtained in the original unrotated basis. Following the procedure in Sec.\[title\], we can derive the effective chiral Lagrangian at finite temperature in the chiral thermodynamic model (CTDM), which is: $$\begin{aligned} \label{ChEF7} S^M_{E Re} &\approx& \frac{N_c}{16\pi^2}\int d^4 x_E tr_F \{M_c^2L_2(T)[(\hat{\phi}\hat{\phi}^\dagger-\bar{\cal M}\bar{\cal M}^\dagger)+ (\hat{\phi}^\dagger\hat{\phi}-\bar{\cal M}^\dagger\bar{\cal M})]\nonumber\\ &&-\frac{1}{2} L_0(T) [D^\prime_E\hat{\phi}\cdot D^\prime_E\hat{\phi}^\dagger+ D^\prime_E\hat{\phi}^\dagger\cdot D^\prime_E\hat{\phi}+ (\hat{\phi}\hat{\phi}^\dagger-\bar{\cal M}\bar{\cal M}^\dagger)^2+(\hat{\phi}^\dagger\hat{\phi}-\bar{\cal M}^\dagger\bar{\cal M})^2]\nonumber\\ && +M_c^2L_2(T)[(\bar{\cal M}\bar{\cal M}^\dagger-\bar{M}^2)+(\bar{\cal M}^\dagger\bar{\cal M}-\bar{M}^2)]\nonumber\\ && -\frac{1}{2}L_0(T)[(\hat{\phi}\hat{\phi}^\dagger-\bar{\cal M}\bar{\cal M}^\dagger)(\bar{\cal M}\bar{\cal M}^\dagger-\bar{M}^2)+(\hat{\phi}^\dagger\hat{\phi}-\bar{\cal M}^\dagger\bar{\cal M})(\bar{\cal M}^\dagger\bar{\cal M}-\bar{M}^2)\nonumber\\ &&+(\bar{\cal M}\bar{\cal M}^\dagger-\bar{M}^2)(\hat{\phi}\hat{\phi}^\dagger-\bar{\cal M}\bar{\cal M}^\dagger)+(\bar{\cal M}^\dagger\bar{\cal M}-\bar{M}^2)(\hat{\phi}^\dagger\hat{\phi}-\bar{\cal M}^\dagger\bar{\cal M})\nonumber\\ && (\bar{\cal M}\bar{\cal M}^\dagger-\bar{M}^2)^2+(\bar{\cal M}^\dagger\bar{\cal M}-\bar{M}^2)^2]\}.\end{aligned}$$ By using the same argument as the one for the zero-temperature chiral dynamical model (CDM) analyzed in the text, we can prove the equivalence between Eq.(\[ChEF7\]) and Eq.(\[ChEF8\]). Thus, based on the equivalent Lagrangians, the resulting physics, especially the spectrum of mesons, will not be changed. [99]{} E. V. Shuryak, Prog. Part. Nucl. Phys. 53 (2004) 273. M. Gyulassy, L. McLerran, Nucl. Phys. A 750 (2005) 30. E. V. Shuryak, Nucl. Phys. A 750 (2005) 64. I. Arsene et al, Nucl. Phys. A 757 (2005) 1. B. B. Back et al, Nucl. Phys. A 757 (2005) 28. J. Adams et al, Nucl. Phys. A 757 (2005) 102. K. Adcox et al, Nucl. Phys. A 757 (2005) 184. J.-P. Blaizot, J. Phys. G 34 (2007) S243. Y. -B. Dai, Y. -L. Wu, Eur. Phys. J. [C39]{}, S1 (2004). \[hep-ph/0304075\]. Y. Nambu, G. Jona-Lasinio, Phys. Rev. [122]{}, 345-358 (1961). Y. L. Wu, Int. J. Mod. Phys. A [18]{}, 5363 (2003) \[arXiv:hep-th/0209021\]. Y. L. Wu, Mod. Phys. Lett. A [19]{}, 2191 (2004) \[arXiv:hep-th/0311082\]. D. Espriu, E. de Rafael and J. Taron, Nucl. Phys. B [345]{}, 22 (1990) \[Erratum-ibid. B [355]{}, 278 (1991)\]. J. Schwinger, J. Math. Phys. 2 (1961) 407. L. V. Keldysh, Sov. Phys. JETP 20 (1965) 1018. G. Z. Zhou, Z. B. Su, B. L. Hao, L. Yu, Phys. Rev. B 22 (1980) 3385. K. C. Chou, Z. B. Su, B. L. Hao, L. Yu, Phys. Rep. 118 (1985) 1. E. Calzetta and B. L. Hu, NonEquilibrium Quantum Field Theory (Cambridge University Press, 2008) J. Rammer, Quantum Field Theory of Non-equilibrium States (Cambridge University Press, 2007) S. P. Klevansky, Rev. Mod. Phys. 64 (1992) 649. T. Hatsuda, T. Kunihiro, Phys. Lett. B 145 (1984) 7; Phys. Rep. 247 (1994) 221. R. Alkofer, H. Reinhardt, H. Weigel, Phys. Rep. 265 (1996) 139. M. Buballa, Phys. Rep. 407 (2005) 205. G. ’t Hooft, Phys. Rev. Lett. [37]{} 8(1976), Phys. Rev. [ D14]{} 3432(1976); Err. Phys. Rev. [D18]{} (1978) 2199. G. ’t Hooft, \[hep-ph/9903189\], 1999 E. Calzetta, B. L. Hu, Phys. Rev. D 37 (1988) 2878. A. Das, In \Mitra, A.N. (ed.): Quantum field theory\ 383-411. \[hep-ph/0004125\]. M. Quiros, talk given at ICTP Summer School In High-Energy Physics And Cosmology, 29 Jun - 17 Jul 1998, Miramare, Trieste, Italy \[hep-ph/9901312\]. C. Bloch, Nuc. Phys. 7 (1958) 451. A. Manohar and H. Georgi, Nucl. Phys. B [234]{}, 189 (1984).
--- abstract: \| Intrusion detection systems (IDSs) generate valuable knowledge about network security, but an abundance of false alarms and a lack of methods to capture the interdependence among alerts hampers their utility for network defense. Here, we explore a graph-based approach for fusing alerts generated by multiple IDSs (e.g., Snort, OSSEC, and Bro). Our approach generates a weighted graph of alert fields (not network topology) that makes explicit the connections between multiple alerts, IDS systems, and other cyber artifacts. We use this multi-modal graph to identify anomalous changes in the alert patterns of a network. To detect the anomalies, we apply the role-dynamics approach, which has successfully identified anomalies in social media, email, and IP communication graphs. In the cyber domain, each node (alert field) in the fused IDS alert graph is assigned a probability distribution across a small set of roles based on that node’s features. A cyber attack should trigger IDS alerts and cause changes in the node features, but rather than track every feature for every alert-field node individually, roles provide a succinct, integrated summary of those feature changes. We measure changes in each node’s probabilistic role assignment over time, and identify anomalies as deviations from expected roles. We test our approach using IDS alerts generated from a network of 24 virtual machines (workstations, data and print servers, DHCP and DNS servers), virtual switches, and a virtual server that approximates connections to the internet. The simulation includes three weeks of normal background traffic, as well as cyber attacks that occur near the end of the simulations. The network includes installations of Snort and OSSEC, which generated alerts throughout the experiment. A NetFlow sensor also captured the network traffic during the simulation. This paper presents a novel approach to multi-modal data fusion and a novel application of role dynamics within the cyber-security domain. Our results show a drastic decrease in the false-positive rate when considering our anomaly indicator instead of the IDS alerts themselves, thereby reducing alarm fatigue and providing a promising avenue for threat intelligence in network defense. author: - Anthony Palladino - 'Christopher J. Thissen' bibliography: - 'bibliography.bib' title: 'Cyber Anomaly Detection Using Graph-node Role-dynamics' --- <ccs2012> <concept> <concept\_id>10002978.10002997</concept\_id> <concept\_desc>Security and privacy Intrusion/anomaly detection and malware mitigation</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010257.10010321</concept\_id> <concept\_desc>Computing methodologies Machine learning algorithms</concept\_desc> <concept\_significance>300</concept\_significance> </concept> </ccs2012>
--- abstract: 'We study how the presence of a background magnetic field, of intensity compatible with current observation constraints, affects the linear evolution of cosmological density perturbations at scales below the Hubble radius. The magnetic field provides an additional pressure that can prevent the growth of a given perturbation; however, the magnetic pressure is confined only to the plane orthogonal the field. As a result, the “Jeans length” of the system not only depends on the wavelength of the fluctuation but also on its direction, and the perturbative evolution is anisotropic. We derive this result analytically and back it up with direct numerical integration of the relevant ideal magnetohydrodynamics equations during the matter-dominated era. Before recombination, the kinetic pressure dominates and the perturbations evolve in the standard way, whereas after that time magnetic pressure dominates and we observe the anisotropic evolution. We quantify this effect by estimating the eccentricity $\epsilon$ of a Gaussian perturbation in the coordinate space that was spherically symmetric at recombination. For a perturbations at the sub-galactic scale, we find that $\epsilon = 0.7$ at $z=10$ taking the background magnetic field of order $10^{-9}$ gauss.' address: - 'Dipartimento di Fisica, “Sapienza” Università di Roma, P.le A. Moro 5 (00185), Roma, Italy.' - 'Physics Department, University of Oxford, OX1 3RH Oxford, United Kingdom.' - 'Dipartimento di Fisica G. Occhialini, Università Milano-Bicocca and INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy.' - 'Dipartimento di Fisica, Università di Ferrara and INFN, sezione di Ferrara, Polo Scientifico e Tecnologico - Edificio C Via Saragat, 1, I-44122 Ferrara Italy' - 'ENEA, C.R. Frascati (Rome), UTFUS-MAG, Italy.' - 'INFN, Sezione Roma1, Italy' author: - Massimiliano Lattanzi - Nakia Carlevaro - Giovanni Montani title: 'Gravitational instability of the primordial plasma: anisotropic evolution of structure seeds' --- Cosmological perturbations ,Cosmological magnetic fields ,Structure formation ,Magnetohydrodynamics. General Remarks =============== Our theoretical knowledge of the Universe is based on the Standard Cosmological Model, that provides a convenient framework to satisfactorily explain the majority of cosmological observations, like the anisotropy pattern of the cosmic microwave background (CMB) [@Komatsu:2010fb; @Larson:2010gs; @Dunkley:2010ge], the large-scale structure of the Universe [@Tegmark:2004; @Cole:2005sx; @Tegmark:2006az], the Hubble diagram of distant type Ia supernovae [@Riess:1998cb; @Perlmutter:1998np; @Frieman:2008sn] and the abundances of light elements [@Iocco:2008va]. The Standard Cosmological Model relies on the assumption that the Universe, at least at large scales, is highly homogeneous and isotropic, and its geometry is thus described by the Robertson-Walker metric. In fact, the distribution of luminous red galaxies shows that the present Universe is homogeneous on scales greater than $\sim 100$ Mpc [@Hogg:2004vw], while the isotropy of the CMB itself (which has a black-body distribution at $T=2.73\,$K with temperature fluctuations of order $10^{-5}$ or less) is an indication of the isotropy of the Universe as a whole and a strong evidence for homogeneity at the time of hydrogen recombination (nearly 400.000 years after the Big Bang, corresponding to a cosmological redshift $z_\mathrm{rec}=1100$). On the other hand, below the “homogeneity scale” of 100 Mpc, the distribution of matter is definitely inhomogeneous. Such a dichotomy between the smoothness in the matter-energy distribution at $z={z_{rec}}$ and the clumpiness of the recent Universe (for $z\lesssim1$) below a certain scale is explained by the mechanism of gravitational instability: the structures we observe today have been formed through the growth of tiny density perturbation seeds that, accordingly to the currently accepted model, were created in the early Universe during a phase of inflationary expansion. The presence of a large scale ([i.e., ]{}coherent over a Hubble length), strong magnetic field is forbidden by the observed isotropy, as it would naturally single out a preferred spatial direction. However, a background and uniform magnetic field could be present at cosmological scales provided that its intensity is small enough. In particular, upper limits on the present field intensity of order $\sim 10^{-9}$ G have been derived from observations of the CMB temperature anisotropies [@BFS97; @Paoletti:2010rx] and of its temperature-polarization correlation [@Scannapieco:1997mt; @Komatsu:2010fb][^1]. Smaller scale fields, on the other hand, could be as strong as $10^{-6}$ G. The effects of large-scale magnetic fields on the evolution of cosmological structures have been studied extensively in the literature (for a complete review, see Ref. [@BMT07] and references therein), where both Newtonian and general relativistic treatments, the latter often using covariant and gauge-invariant techniques, are present. It is known that, among others, magnetic fields slow down (and possibly prevent) the growth of perturbations and can produce vorticities and shape distortions in the density field [@VTP05; @Tsagas:1999ft]. In this Letter, our goal is to revisit the issue of the existence of a “magnetic Jeans length” and of its dependence on the direction along which the perturbation propagates, other than to give a realistic estimate of its value in a matter-dominated Universe. The presence of a magnetic Jeans length has been discussed in several papers [@BMT07; @VTP05; @Tsagas:1999ft; @RR71; @Kim:1994zh] both in a Newtonian and in general relativistic framework (for an analysis of the standard Jeans mechanism in the presence of dissipative effect see also [@CQG; @IJMPD; @MPLA]), but some of the analyses failed to recognize its angular dependence. Here, we present a neat derivation of the relevant instability scales bases on the equations of magnetohydrodynamics (MHD) on an expanding Universe. We discuss a simple generalization of the magnetic Jeans length suited for two-fluid systems, and clarify some misunderstandings that are present in the literature. We also show the results of the numerical integration of the coupled MHD and Poisson equations. Finally, we numerically study the distortion introduced in the density field by the anisotropy in the critical length. The Letter is organized as follows. In Sec. 2, we characterize the Universe as a plasma. In Sec. 3, we introduce the basic equations, and in Sec. 4 we carry out the linearization procedure. We derive the existence of the magnetic Jeans length from analytical considerations in Sec. 5, while in Sec. 6 we show some numerical results. Finally, we draw our conclusions in Sec. 7. The plasma features of the pre- and post-recombination Universe =============================================================== In this Section, we aim at characterizing the plasma features of the cosmological fluid. Between the time of $e^+ e^-$ annihilation ([i.e., ]{}for a temperature[^2] $T\simeq m_e$, corresponding to a redshift $z\sim 10^9$) and the present ($z=0$), the matter-energy content of the Universe is provided by electrons, protons and neutrons (the three species being collectively referred to as baryons in the cosmological jargon), photons, neutrinos, and two elusive components dubbed dark matter and dark energy, that presently account for more than 99% of the total energy budget of the Universe. However, dark energy has been subdominant for most of the past history of the Universe and can be safely neglected at redshifts $z>1$. Dark matter and neutrinos interact only gravitationally with the other components and can be neglected as long as the plasma properties of the fluid are concerned. The cosmological baryon-to-photon ratio is extremely small and equal to $n_b/n_\gamma\simeq6.1\times 10^{-10}$, where $n_\gamma$ and $n_b$ are the photon and baryon number densities, respectively. Both $n_b$ and $n_\gamma$ scale with redshift as $(1+z)^3$; their present values are $n_{b}(z=0) \simeq2.5\times 10^{-7} {\,\mathrm{cm}}^{-3}$ and $n_\gamma(z=0)\simeq 410 {\,\mathrm{cm}}^{-3}$. Most of the baryons in the Universe are in the form of $^1\mathrm H$ nuclei (i.e., isolated protons) so that, for simplicity, in the following we assume $n_b=n_p$ ($n_p$ being the proton number density). Furthermore, $n_p$ is also equal to the electron number density $n_e=n_p$, because of the charge neutrality of the Universe. We study separately the properties of the cosmological fluid before and after the time of hydrogen recombination occurring at $z={z_{rec}}=1100$ ($T\simeq 0.25{\,\mathrm{eV}}$). For $z>{z_{rec}}$, the protons and electrons are free and thus one deals with a fully ionized plasma. In this regime, photons and baryons are tightly coupled due to Thomson scattering, and share a common temperature $T(z)=T_\gamma(z)=T_\gamma^0 (1+z)$, where the present photon temperature $T_\gamma^0=2.73\,{\,\mathrm{K}}$. After recombination ($z<{z_{rec}}$), most of the electrons and protons exist in the form of neutral hydrogen atoms, and only a small residual ionized fraction $x_e=2.5\times 10^{-4}$ survives, making the fluid a weakly ionized plasma. In this regime, the photon temperature still scales as $(1+z)$, while that of baryons evolves in the same way only until $z=100$, due to residual scatterings that keep them in thermal equilibrium with photons; after that time, their temperature decreases faster, as $(1+z)^{2}$. The baryonic fluid remains neutral until the time of reionization, when the UV radiation produced by the first stars ionizes again the hydrogen present in the cosmological medium. This is likely to have happened around $z\simeq 10$, however the precise details of the reionization history are still largely unknown, and for this reason we limit our analysis to redshifts $z>10$. In the following, we will assume the presence of a background homogeneous magnetic field ${\boldsymbol{B}}(z)$, whose contribution to the total energy density of the Universe can be considered negligible. We recall that the field intensity $B(z)$ scales as $(1+z)^2$ and, unless otherwise stated, we take its value at the present time to be $B(z=0)=10^{-9}$ G. The pre-recombination Universe ------------------------------ A fundamental quantity characterizing a plasma is the Debye length $\lambda_D$, namely the length over which electrons screen out electric fields in a plasma. It defines the length scale over which a system can consistently considered to be a plasma. The Debye length of a hydrogen plasma at temperature $T$ is $$\lambda_D=\sqrt{\frac{T}{4\pi n_e e^2}}\simeq(6.9{\,\mathrm{cm}})\sqrt{\frac{T/\mathrm{K}}{n_e/\mathrm{cm}^{-3}}}\;,$$ where $e$ is the proton charge. Using $T=T_\gamma$ and the values given above, one gets $$\lambda_D(z)=\frac{2.3\times 10^4 {\,\mathrm{cm}}}{(1+z)}\;. \label{eq:ldz}$$ The redshift dependence of $\lambda_D$ implies that the comoving Debye length $\bar\lambda_D\equiv\lambda_D (1+z)$ is constant during the cosmological evolution and equal to $\bar\lambda_D\simeq 2 \times 10^4{\,\mathrm{cm}}$. Plasma effects can be important in a system when its physical dimension $L$ is much larger than the Debye length. For the Universe, the relevant length is the Hubble radius $L=l_H\equiv H^{-1}$, where $H$ is the Hubble parameter. This length represents the maximum scale at which microphysical processes can operate in order to establish the thermodynamical equilibrium. Today, $l_H\simeq 10^{28}{\,\mathrm{cm}}$; during the matter-dominated era $l_H\propto (1+z)^{-3/2}$, while in the radiation-dominated era $l_H\propto(1+z)^{-2}$. From the analysis of both these scales, it is evident that $l_H\gg\lambda_D$ turns out in the period considered here. Moreover, under the same hypotheses, the baryonic matter $M_D$ within a Debye sphere is also constant and given by $$M_D = \tfrac{4}{3}\,\pi m_p n_b\lambda_D^3\simeq 10^{-50} M_{\odot}\;,$$ where $m_p$ is the proton mass. This “Debye mass” clearly results to be much smaller than any other of cosmological interest and we can conclude that the cosmological fluid can be considered as neutral at all relevant scales. Another meaningful index is the so-called plasma parameter $N_D$, [i.e., ]{}the number of particles within a Debye sphere: $$\label{eq:nd} N_D= \tfrac{4}{3}\,\pi n_b\lambda_D^3 \;.$$ The dependence of $\lambda_D$ and $n_p$ on the redshift $z$ implies that also $N_D$ is a constant. In particular, since $N_D\simeq 10^7\gg1$, the cosmological fluid results to be a weakly coupled plasma. In order to provide a complete characterization of the cosmological plasma, we now turn our attention to the plasma dissipative properties, starting from the plasma resistivity $\eta$. For an electron-proton plasma, this is given by $\eta=m_e \nu_{ei}\;/\;n_e e^2$, where $m_e$ is the electron mass and $\nu_{ei}$ is the electron-ion collision frequency. For the case under consideration, $\nu_{ei}$ is well approximated by the electron-electron collision frequency $\nu_{ee}$ [@plasmaf], [i.e., ]{}$$\nu_{ei}\simeq \nu_{ee} \simeq(2.91\times 10^{-6} \,\mathrm{s}^{-1}) \left(\frac{n_e}{{\,\mathrm{cm}}^{-3}}\right)\left(\frac{T}{{\,\mathrm{eV}}}\right)^{-3/2}\ln \Lambda_C\;, \label{eq:nu_ei}$$ where $\ln\Lambda_C$ is the Coulomb logarithm, introduced to quantify the effects that small-angle-diffusion collisions have in the Coulomb scattering. A simply estimate of $\Lambda_C$ in a plasma is given by $\Lambda_C\simeq12\pi N_D$, so that for the cosmological fluid, the Coulomb logarithm is $\simeq 20$. Substituting [Eq.(\[eq:nu\_ei\])]{} into the expression for the resistivity given above, we get $$\label{etaz} \eta(z)\simeq 1.6\times \left(\frac{1+z}{1+{z_{rec}}}\right)^{-3/2} \Omega \,{\,\mathrm{cm}}\;.$$ Close to recombination, the cosmological plasma has an electric resistivity equal to $\eta({z_{rec}})\simeq1.6 \,\Omega {\,\mathrm{cm}}$, [i.e., ]{}a conductivity $\simeq 0.6$ siemens ${\,\mathrm{cm}}^{-1}$, a value typical of a semiconductor \[in Gaussian units, $\eta({z_{rec}}) = 1.8\times 10^{-12} {\,\mathrm{s}}$\]. Let us now turn our attention to the viscous properties of the plasma. The shear viscosity coefficient of matter strongly coupled with radiation can be expressed as $$\eta_v=\frac{4}{15}\;a_{SB} T^4 \tau\;,$$ where $a_{SB}\simeq 5.7\times10^{-8}\,\mathrm{W}\,\mathrm{K}^{-4}\mathrm{m}^{-2}$ is the Stefan-Boltzmann constant, while $\tau$ denotes the mean collision time between particles and can be estimated as $\tau\simeq(n_\gamma\sigma_T v)^{-1}$ (here, $v\simeq c$ and we have introduced $\sigma_T\simeq6.6\times10^{-29}\mathrm{m}^2$ as the cross section for Thomson scattering). For a photon gas at equilibrium at temperature $T$, $n_\gamma\simeq2.0\times10^7(T/\mathrm{K})^3\mathrm{m}^{-3}$ and then $$\eta_v(T) \simeq 1.2 \times 10^{-4}\left(\frac{T}{\mathrm{K}} \right) \frac{\mathrm{kg}}{\mathrm{m\,s}}\;.$$ The resistivity and viscosity coefficients enter the MHD equations through the following diffusion coefficients $\bar\eta\equiv\eta/4\pi$ and $\bar\eta_v\equiv\eta_v/\rho$, where $\rho$ is the density of the fluid. Taking $\rho=\rho_b=m_pn_b \simeq 4.2\times 10^{-28}(1+z)^3$ kg m$^{-3}$ and using $T=T_\gamma^0(1+z)$, we get $$\begin{aligned} &\bar\eta_v \simeq (6.4 \times 10^{17}\,\mathrm{m^2\,s^{-1}}) \left(\frac{1+z}{1+z_{rec}}\right)^{-2}\;, \\ &\bar\eta \simeq (1.3 \times 10^4\,\mathrm{m^2\,s^{-1}})\; \left(\frac{1+z}{1+z_{rec}}\right)^{-3/2}\;.\end{aligned}$$ The relative magnitude of the viscous and magnetic diffusion rates can be parameterized through the magnetic Prandtl number $Pr_m \equiv \bar\eta_v/\bar \eta$. Using the expression above for $\bar \eta_v$ and $\bar\eta$, we obtain $$Pr_m \simeq 5.0\times 10^{13} \left(\frac{1+z}{1+z_{rec}}\right)^{-1/2}\;,$$ so that $Pr_m \gg 1$, [i.e., ]{}viscous diffusion is more important than resistive diffusion, at recombination and indeed always at the redshifts under consideration. Having discussed the relative importance of viscosity- and resistivity-driven dissipative effects, we now analyze at which scales these effects are relevant. In a magnetized plasma, a useful parameter is the Lundquist number $S \equiv L v_A/\eta$, where $L$ is a typical length scale and $v_A=(B/4\pi\rho)^{1/2}$ is the Alfvén velocity. The Lundquist number is basically the ratio between the resistive diffusion timescale $\tau_r= L^2/\bar\eta$ and the Alfvén crossing timescale $\tau_A = L/v_A$. Assuming $B(z=0)=10^{-9}\,\mathrm{G}$ and $\rho=\rho_b$, we obtain $v_A\simeq1.2\times10^{5}\,\mathrm{m/s}\,[(1+z)/(1+z_{rec})]^{1/2}$ and thus $$S = \frac{\tau_r}{\tau_A}=2.8\times 10^{23} \left(\frac{L}{{\,\mathrm{Mpc}}}\right)\left(\frac{1+z}{1+z_{rec}}\right)^2\,.$$ We also compare $\tau_A$ with the viscous diffusion timescale defined as $\tau_v = L^2/\bar\eta_v$. The ratio $S_v\equiv\tau_v/\tau_A$ can be thought as a viscous analogous to the Lundquist number: $$S_v = \frac{\tau_v}{\tau_A}=5.7\times 10^{9} \left(\frac{L}{{\,\mathrm{Mpc}}}\right)\left(\frac{1+z}{1+z_{rec}}\right)^2\;.$$ The baryonic mass (at the average background density) contained in a sphere of radius equal to the length where $S\sim1$ can be calculated to be $\sim 10^{-51} M_{\odot} [(1+z)/(1+z_{rec})]^{-3}$, while the analogous quantity for $S_v$ is $\sim 10^{-10} M_\odot [(1+z)/(1+z_{rec})]^{-9/2}$. Both mass scales are well below the values of cosmological relevance at the redshifts of interest. The discussion above shows how the following hierarchy among the relevant time scales holds for all mass range and redshifts of interest: $$\tau_A \ll \tau_v \ll \tau_r \;, \label{eq:hierdiss}$$ meaning that viscosity always dominates over resistivity, and that both dissipative effects can indeed be neglected when studying the propagation of Alfvén waves. The post-recombination Universe \[ssec:post\_uni\] -------------------------------------------------- After the time of recombination $z_{\textrm{rec}}=1100$, the cosmological plasma exists in a weakly ionized state, the neutral and ionized components having densities $\rho_n\simeq \rho_b$ and $\rho_i=x_e\rho_b\ll \rho_n$, respectively. We take the residual ionization fraction $x_e$ constant and equal to $2.5\times 10^{-4}$ in the range ${z_{rec}}>z>10$. The results of the previous Subsection can be generalized to show that the hierarchy [(\[eq:hierdiss\])]{} holds also in this regime. In spite of the small value of the ionization fraction, the magnetic field could still affect the dynamics of the whole system in view of the interactions between neutral and charged particles. In particular, the magnetic forces acting on the charged particles can be communicated to the neutrals through collisions. However, if the coupling is not tight enough, the neutrals feel the magnetic field but drift with respect to the ions in a process termed ambipolar diffusion. Its relevance at a given length scale $L$ is quantified by the ambipolar Reynolds number ${R_{\mathrm{amb}}}$ [@MS56; @Sh83; @BJ04; @LM06] $${R_{\mathrm{amb}}}(L) \equiv \frac{v \,\gamma_\mathrm{in}\,x_e\,\rho_n}{v_A^2} L=\frac{L}{{L_{\mathrm{amb}}}}\; , \label{eq:Ramb}$$ where $\gamma_\mathrm{in}=1.9\times 10^{-9} \mathrm{cm}^3\,\mathrm{s}^{-1}$ [@DR83] is the ion-neutral drag coefficient due to collisions between the two species, $v_A^2=B^2/4\pi \rho_n$ is the Alfvén velocity in the tightly coupled limit, $v$ is the characteristic velocity of the fluid, and ${L_{\mathrm{amb}}}$ is the ambipolar length, i.e., the scale where ${R_{\mathrm{amb}}}=1$. The ambipolar Reynolds number is just the ratio between the ambipolar diffusion timescale ${\tau_{\mathrm{amb}}}=L^2/(\tau_\mathrm{ni} v_A^2)=\tau_A^2/\tau_\mathrm{ni}$ (where $\tau_\mathrm{ni}=(\gamma \rho_i)^{-1}$ is the neutral collision timescale) and some characteristic timescale $\tau=L/v$. If ${R_{\mathrm{amb}}}\ll 1$, the neutrals are uncoupled from the plasma. On the contrary, when ${R_{\mathrm{amb}}}\gtrsim 1$ the dynamics of the two components can be described through ordinary single-fluid MHD with an additional dissipative term [@BJ04]. This term becomes progressively less important as ${R_{\mathrm{amb}}}$ grows and can be neglected in the limit ${R_{\mathrm{amb}}}\gg 1$, or $L\gg {L_{\mathrm{amb}}}$. Assuming that the evolution of the fluid is driven by Alfvénic phenomena, i.e., $v \sim v_A$, the ambipolar length is given by $${L_{\mathrm{amb}}}= (1.2\,\mathrm{Mpc})\,(1+z)^{-5/2}\;, \label{eq:lamb}$$ and ${R_{\mathrm{amb}}}={\tau_{\mathrm{amb}}}/\tau_A$. Thus, ${R_{\mathrm{amb}}}\gg 1$ if and only if the tight-coupling condition $\tau_\mathrm{ni}\ll\tau=\tau_A$ is satisfied. ![Mass contained (at the background baryon density) within the scale ${L_{\mathrm{amb}}}$ defined in [Eq.(\[eq:lamb\])]{} as a function of redshift $z$. Above line, the condition ${\tau_{\mathrm{amb}}}>\tau_A$ holds.\[fig:lamb\]](mamb){width="0.9\hsize"} It is straightforward to check that for $1100>z>10$, it is always ${R_{\mathrm{amb}}}>1$ at scales larger than a few tens of comoving kiloparsecs, meaning that the following hierarchy holds: $$\tau_\mathrm{ni} \ll \tau_A \ll {\tau_{\mathrm{amb}}}\,,$$ so that the ions and neutrals are tightly coupled and ambipolar diffusion can be safely neglected. In order to better illustrate this point, in Figure \[fig:lamb\] we plot the mass contained in a sphere of radius ${L_{\mathrm{amb}}}$ at the background baryon density as a function of redshift. It is evident that, in the redshift range considered, ${R_{\mathrm{amb}}}\gg 1$ for all scales $M\gg 10^6 M_\odot$. In this regime, we can therefore neglect the dissipative term mentioned above and use single-fluid ideal MHD. We conclude that ambipolar diffusion does not affect the dynamics of the cosmological plasma after recombination. Basic equations =============== In this Section, we derive the basic equations describing the linear evolution of instabilities in the cosmological fluid, modeled as a magnetized plasma. In this respect, we underline that the full investigation of the perturbative dynamics of the Universe would require a general-relativistic treatment, in order to correlate the matter and geometrical fluctuations [^3]. However, as long as one is interested in scales much smaller than the Hubble radius, [i.e., ]{}$L\ll H^{-1}$, a Newtonian treatment provides a consistent description of the dynamics. Nonetheless, in this scenario the expansion of the Universe can be accounted as the bulk background motion of the fluid . The starting point of our treatment is the Eulerian set of equations governing the fluid motion, on which one can develop a perturbative theory by adding small fluctuations to the unperturbed cosmological background solution. The zeroth-order dynamics is derived by considering a flat homogeneous and isotropic Universe whose energy density is dominated by non-relativistic matter, and correctly describes the expansion of the Universe. We assume that a background magnetic field is present, whose contribution to the total energy density of the Universe can be considered negligible. Let us now start by briefly recalling the basic equations of non-relativistic, ideal and single fluid MHD, which govern the plasma motion. The mass conservation and the Newtonian gravitational field are described by the continuity and Poisson equations, the single-fluid dynamics is described by the Euler equation in presence of a magnetic field ${\boldsymbol{B}}$ and, finally, the electromagnetic interaction can be summarized by the frozen-in and the Gauss laws. Such equations read \[initial-system\] $$\begin{aligned} {{\partial}_t}\rho+{{\boldsymbol{\nabla}}}\cdot\rho{\boldsymbol{v}}=0&\;,\label{continuity}\\ {\nabla^2}\Phi-4\pi G\rho=0&\;,\label{poisson}\\ \rho{{\partial}_t}{\boldsymbol{v}}+\rho({\boldsymbol{v}}\cdot{{\boldsymbol{\nabla}}}){\boldsymbol{v}}+{{\boldsymbol{\nabla}}}P+{\qquad}{\qquad}{\qquad}{\qquad}&\nonumber\\ +\rho{{\boldsymbol{\nabla}}}\Phi-({{\boldsymbol{\nabla}}}\times{\boldsymbol{B}})\times{\boldsymbol{B}}/4\pi=0&\;,\label{eq:Euler}\\ {{\partial}_t}{\boldsymbol{B}}-{{\boldsymbol{\nabla}}}\times({\boldsymbol{v}}\times{\boldsymbol{B}})=0&\;,\label{eq:base2}\\ {{\boldsymbol{\nabla}}}\cdot{\boldsymbol{B}}=0&\;,\label{eq:Gauss} \end{aligned}$$ respectively, where $\rho$ is the mass density, ${\boldsymbol{v}}$ is the velocity field, $\Phi$ is the gravitational potential and $G$ is Newton constant. This system constitutes the base of our perturbative approach. To derive the zeroth-order dynamics, we assume the usual Robertson-Walker metric, [i.e., ]{}$ds^2=dt^2 - a^2(t)\,d\ell^2$, where $a=a(t)$ represents the cosmological scale factor, and a perfect fluid energy-momentum tensor as the matter source of the gravitational field, [i.e., ]{}${T_{\mu}}^{\nu}=\mbox{diag}\,[\,{\rho_0},\,-{P_0},\,-{P_0},\,-{P_0}\,]$, with ${\rho_0}={\rho_0}(t)$. In this scheme, the behavior of the mass density with time is obtained from the energy-momentum conservation law $T_{0;\,\nu}^{\,\nu}=0$ and from the Friedmann equation, [i.e., ]{}$$\begin{aligned} &\dot{\rho}_0+3H({\rho_0}+{P_0})=0\;,\quad\quad \label{EMT}\\ &\dot{a}^2+\mathcal{K}-\tfrac{8}{3}\pi G{\rho_0}a^2=0\;, \label{Friedmann}\end{aligned}$$ respectively (the dot ($\dot{\;\;}$) denotes the total derivative with respect to synchronous time). Here $H=\dot{a}/a$ is the Hubble parameter and $\mathcal{K}=const.$ is the curvature factor. Setting the matter-dominated Universe equation of state (EoS) ${P_0}\sim0$ (${P_0}\ll{\rho_0}$) in [Eq.(\[EMT\])]{}, the zeroth-order solution of the system [(\[initial-system\])]{} turns out to be $$\begin{aligned} \label{zeroth-order-sol} {\rho_0}=\frac{{\bar{\rho}}}{a^3}\;,\;\;\; {{\boldsymbol{v}}_0}=H{{\boldsymbol{r}}}\;,\;\; {{\boldsymbol{B}}_0}=\frac{{\bar{{\boldsymbol{B}}}_0}}{a^2}\;,\;\;\; {{\boldsymbol{\nabla}}}{\Phi_0}=\tfrac{4}{3}\pi G {\rho_0}{{\boldsymbol{r}}}\;,\end{aligned}$$ where ${\bar{\rho}}$ and ${\bar{{\boldsymbol{B}}}_0}$ are dimensional constants, ${{\boldsymbol{r}}}$ ($r=\mid\!\!{{\boldsymbol{r}}}\!\mid$) denotes the radial coordinate vector and, of course, $a(t)$ satisfies [Eq.(\[Friedmann\])]{}. We observe how this non-stationary solution characterizing the background dynamics is not affected by the so-called “Jeans swindle” proper of the static solution . To obtain now the explicit time dependence of the unperturbed quantities involved in the model, we restrict the analysis to the flat case, [i.e., ]{}$\mathcal{K}=0$. From the Friedmann equation [(\[Friedmann\])]{} and using the solution for ${\rho_0}$, one readily obtains \[time-parameters\] $$\begin{aligned} a&=\Big(6\pi G{\bar{\rho}}\Big)^{1/3} t^{2/3}\;, \\ {\rho_0}&=\frac{1}{6\pi G t^{2}}\; .\label{rho_t}\end{aligned}$$ Finally, we recall that the adiabatic sound speed is defined by $v_s=\sqrt{\partial P/\partial \rho}$. For a general specific heat ratio $\gamma$, we assume that the pressure varies as $P=K \rho^{\gamma}$, so that the speed of sound is given by $$\label{time-vs} v_s^{2}=\gamma\,K \,{\rho_0}^{\gamma-1}=\frac{\gamma\,K}{(6\pi\, G)^{\gamma-1}}\,t^{-2\gamma+2}\;.$$ Perturbation scheme =================== In order to analyze the implications that the physics of an ideal magnetized plasma can have on the structure formation, we will follow the standard perturbation approach. In this respect, we consider small perturbations around the zeroth-order cosmological solution derived above, [i.e., ]{}we write $\rho= {\rho_0}+{\rho_1}$ (with ${\rho_1}\ll{\rho_0}$) and similarly for the other quantities $P$, ${\boldsymbol{v}}$, $\Phi$ and ${\boldsymbol{B}}$. Substituting the perturbed quantities in [Eqs.(\[initial-system\])]{} and keeping only terms up to first order, one gets \[perturbation-system\] $$\begin{aligned} {{\partial}_t}{\rho_1}+3H{\rho_1}+H({\boldsymbol{r}}\cdot{{\boldsymbol{\nabla}}}){\rho_1}+{\rho_0}{{\boldsymbol{\nabla}}}\cdot{{\boldsymbol{v}}_1}&=0\\ {\nabla^2}{\Phi_1}-4\pi G{\rho_1}&=0\\ {{\partial}_t}{{\boldsymbol{v}}_1}+ H{{\boldsymbol{v}}_1}+ H({\boldsymbol{r}}\cdot{{\boldsymbol{\nabla}}}){{\boldsymbol{v}}_1}+v_s^2{{\boldsymbol{\nabla}}}{\rho_1}/{\rho_0}+\;\;\nonumber\\ +{{\boldsymbol{\nabla}}}{\Phi_1}-({{\boldsymbol{\nabla}}}\times{{\boldsymbol{B}}_1})\times{{\boldsymbol{B}}_0}/(4\pi{\rho_0})&=0\\ {{\partial}_t}{{\boldsymbol{B}}_1}+2H{{\boldsymbol{B}}_1}+H({\boldsymbol{r}}\cdot{{\boldsymbol{\nabla}}}){{\boldsymbol{B}}_1}+\quad{\qquad}{\qquad}\quad\nonumber\\ +{{\boldsymbol{B}}_0}({{\boldsymbol{\nabla}}}\cdot{{\boldsymbol{v}}_1})-({{\boldsymbol{B}}_0}\cdot{{\boldsymbol{\nabla}}}){{\boldsymbol{v}}_1}&=0\\ {{\boldsymbol{\nabla}}}\cdot{{\boldsymbol{B}}_1}&=0\end{aligned}$$ where, as already discussed, the pressure and density perturbations have been related through the adiabatic sound speed, [i.e., ]{}${P_1}=v^2_s{\rho_1}$. We are assuming that $B_0^2/4\pi \rho_0=v_A^{2}\ll 1$, where $B_0=\|{{\boldsymbol{B}}_0}\|$, in order to preserve the isotropy of the background flow. In the following, we replace ${{\boldsymbol{B}}_1}$ with the dimensionless magnetic fluctuation ${{\boldsymbol{b}}_1}\equiv{{\boldsymbol{B}}_1}/B_0$. Moreover, the analysis of the system above can be simplified by Fourier-transforming the spatial dependence of the involved quantities, [i.e., ]{}using perturbations in the form of plane waves, taking $$\begin{aligned} \phi_1({{\boldsymbol{r}}},t)=\tilde{\phi}_1(t)e^{i {{\boldsymbol{k}}}\cdot{{\boldsymbol{r}}}}\;,\end{aligned}$$ with $\phi_1 = \left\{{\rho_1},\,{{\boldsymbol{v}}_1},\,{\Phi_1},\,{{\boldsymbol{B}}_1}\right\}$ and ${{\boldsymbol{k}}}$ is the physical wavenumber scaling as $1/a(t)$. It is convenient to consider also the comoving wavenumber ${\boldsymbol{q}} = a{{\boldsymbol{k}}}$, that stays constant during the expansion. The evolution for a given harmonic can be obtained by the equations in real space with the substitutions $\phi_1\to\tilde\phi_1$, ${{\boldsymbol{\nabla}}}\to i{{\boldsymbol{k}}}$ and ${{\partial}_t}\to {{\partial}_t}- i H ({{\boldsymbol{k}}}\cdot{{\boldsymbol{r}}})$. In the following, for the sake of simplicity, we will drop the tilde over the Fourier transformed variables. Then, the system [(\[perturbation-system\])]{} reduces to (hats denote unit vectors): $$\begin{aligned} \dot{\rho}_1+3H{\rho_1}+i{\rho_0}({{\boldsymbol{k}}}\cdot{{\boldsymbol{v}}_1})&=0\;,\\[0.2cm] \dot{{\boldsymbol{v}}}_1+H{{\boldsymbol{v}}_1}+i\left[\frac{v_s^2}{{\rho_0}}-\frac{4\pi G }{k^2}\right]{\rho_1}{{\boldsymbol{k}}}+iv_A^2\,{\hat{{\boldsymbol{B}}}_0}\times({{\boldsymbol{k}}}\times{{\boldsymbol{b}}_1})&=0\;,\\[0.2cm] \dot{{\boldsymbol{b}}}_1+ i{\hat{{\boldsymbol{B}}}_0}({{\boldsymbol{k}}}\cdot{{\boldsymbol{v}}_1})-i({\hat{{\boldsymbol{B}}}_0}\cdot{{\boldsymbol{k}}}){{\boldsymbol{v}}_1}&=0\;,\end{aligned}$$ where we have already eliminated ${\Phi_1}$ by means of the Poisson equation in $k-$space, [i.e., ]{}$k^2{\Phi_1}=-4\pi G{\rho_1}$. It is understood that the constraint ${{\boldsymbol{k}}}\cdot{{\boldsymbol{b}}_1}=0$ always hold. Decomposing now ${{\boldsymbol{v}}_1}$ in its components ${v_{1}^{{\scriptscriptstyle{\parallel}}}}$ and ${{\boldsymbol{v}}_{1}^{{\scriptscriptstyle{\perp}}}}$ parallel and orthogonal to the direction of ${\boldsymbol{q}}$ respectively, [i.e., ]{}${{\boldsymbol{v}}_1}={v_{1}^{{\scriptscriptstyle{\parallel}}}}\,{\hat{{\boldsymbol{q}}}}+{{\boldsymbol{v}}_{1}^{{\scriptscriptstyle{\perp}}}}$ (where ${{\boldsymbol{v}}_{1}^{{\scriptscriptstyle{\perp}}}}\cdot{\hat{{\boldsymbol{q}}}}=0$), and introducing the following scalar variables: $$\begin{aligned} \delta&\equiv{\rho_1}/{\rho_0}\;,\phantom{({{\boldsymbol{b}}_1}\cdot{\hat{{\boldsymbol{B}}}_0})} \theta\equiv i ({{\boldsymbol{k}}}\cdot{{\boldsymbol{v}}_1}) = i k {v_{1}^{{\scriptscriptstyle{\parallel}}}}\;,\\ {\bar{b}}&\equiv\,({{\boldsymbol{b}}_1}\cdot{\hat{{\boldsymbol{B}}}_0})\;,\phantom{{\rho_1}/{\rho_0}}{\bar{v}}\equiv i\,k\, ({{\boldsymbol{v}}_{1}^{{\scriptscriptstyle{\perp}}}}\cdot{\hat{{\boldsymbol{B}}}_0})\;,\end{aligned}$$ we finally get a further simplified system: \[master-system\] $$\begin{aligned} \dot{\delta}+\theta=0\;,\\ \dot{\theta}+2H\theta-\omega_0^2\delta-\omega_A^2{\bar{b}}=0\;,\\ \dot{{\bar{b}}}+ (1-\mu^2)\theta - \mu {\bar{v}}=0\;,\\ \dot{{\bar{v}}}+2H{\bar{v}}+\mu\omega_A^2{\bar{b}}=0\;,\end{aligned}$$ where we have defined $$\begin{aligned} \mu\equiv{\hat{{\boldsymbol{B}}}_0}\cdot{\hat{{\boldsymbol{q}}}}\;,\qquad \omega_A^{2}\equiv v_A^2 k^2\;,\qquad \omega_0^{2}\equiv v_s^{2}k^2-4\pi G{\rho_0}\;,\end{aligned}$$ and, of course, $0\leqslant\mu\leqslant1$. We stress that $\omega_0^2$ is not positive definite. Evolution of the density contrast and conditions for collapse ============================================================= The form [(\[master-system\])]{} of the evolution equations has the advantage that it clearly expresses the relationship between the physical quantities involved, other that being very well suited for numerical integration. Some further analytical insight can however be gained by reducing it to an unique higher-order equation for the variable $\delta(t)$. Considering the case of a matter-dominated Universe, and using the explicit time dependence of the quantities involved in the model in that case, [i.e., ]{}[Eqs.(\[time-parameters\])]{}, with some algebra one can derive the following fourth-order differential equation for $\delta(t)$: $$\begin{aligned} \label{master-delta} &9 t^4 \delta^{(4)}+ 60 t^3 \delta^{(3)}+ \left[9{\Lambda_2}+76+9{\Lambda_1}t^{-2\nu}\right]\,t^2\,\delta^{(2)}+\nonumber\\ &\quad+\left[(12{\Lambda_2}+8)+12{\Lambda_1}(1-3\nu)\,t^{-2\nu}\right]\,t\,\delta^{(1)}+\\ &\qquad+\left[-6{\Lambda_2}\mu^2+3{\Lambda_1}(3{\Lambda_2}\mu^2+12\nu^2-2\nu)\,t^{-2\nu}\right]\,\delta=0\;,\nonumber\end{aligned}$$ where $\delta^{(\ell)}$ denotes the $\ell^{th}$ derivative of $\delta$ with respect to time, and we have defined the following constants: $$\begin{aligned} \label{Lambdas} \nu\equiv \gamma-4/3\;,\quad {\Lambda_1}=v_s^{2}k^{2}t^{2\gamma-2/3}\;,\quad {\Lambda_2}=\omega_A^{2}t^{2}\;.\end{aligned}$$ We recall that $\gamma$ is the specific-heat ratio ($P\sim\rho^{\gamma}$) and that $\gamma\geqslant4/3$, [i.e., ]{}$\nu\geqslant0$. The most general solution of [Eq.(\[master-delta\])]{} for $\delta$ is found to be the superposition of four independent solutions $\delta_i$ $(i=1,\dots,4)$, given by: $$\begin{aligned} \label{delta_sol} \delta_i= A_i\;t^{x_i}\;\; _{2}\mathcal{F}_{3}\Big[(a_{1i},\,a_{2i});(\,b_{1i},\,b_{2i},\,b_{3i}); \;-\frac{\;{\Lambda_1}t^{-2\nu}\,}{4\nu^{2}}\;\Big]\;,\end{aligned}$$ where $_{p}\mathcal{F}_{q}[(a_{1},...,a_{p});(\,b_{1},...,b_{q});\,z]$ denotes the generalized hypergeometric function of argument $z$, the $A_i$’s are arbitrary integration constants and $$\begin{aligned} &x_1=(-1+\sqrt{\Delta_-})/6\;,\qquad x_2=(-1-\sqrt{\Delta_-})/6\;,\\ &x_3=(-1+\sqrt{\Delta_+})/6\;,\qquad x_4=(-1-\sqrt{\Delta_+})/6\;,\\ &\phantom{\int_0^1}\!\!\!\!\!\Delta_\pm=13-18{\Lambda_2}\pm6\sqrt{(3{\Lambda_2}-2)^{2}+24\mu^{2}{\Lambda_2}\;}\;.\end{aligned}$$ The constant coefficients $a$ and $b$ depend, in general, on $\nu$, ${\Lambda_2}$, $\mu$ and we report their complete expressions in \[Hyp\_ab\]. We are now interested in discussing the asymptotic behavior of the hypergeometric functions in the limit of very small or very large argument, [i.e., ]{}${\Lambda_1}/4\nu^{2}t^{2\nu}$$\gg$$1$ or $\ll$$1$. As in the non-magnetic case discussed in Ref. , we restrict the analysis to the range $0\leqslant\nu\leqslant1/3$, [i.e., ]{}treating the standard regime $4/3\leqslant\gamma\leqslant5/3$. From the asymptotic expansion of the $\mathcal{F}$ functions in the case of large argument, [i.e., ]{}${\Lambda_1}/4\nu^{2}t^{2\nu}\gg1$, the density contrast always shows a damped oscillating behavior with time. In fact, in this regime it always exists at least one asymptotic solution proportional to positive power of the argument ${\Lambda_1}/4\nu^{2}t^{2\nu}$, which results to be the leading term of the solution superposition. In this case, $\delta$ decreases with time. On the other hand, in the limit ${\Lambda_1}/4\nu^{2}t^{2\nu}\to0$, the asymptotic expansion of the solutions [(\[delta\_sol\])]{}, can be written as $$\begin{aligned} \delta_i \sim t^{x_i}+\mathcal{O}({\Lambda_1}/4\nu^{2}t^{2\nu})\;.\end{aligned}$$ In order for the gravitational collapse to occur, at least one of the modes has to be growing, [i.e., ]{}$x_i>0$. It is fairly easy to show that $x_1$, $x_2$ and $x_4$ are always negative, whereas the sign of $x_3$ depends on $\mu$ and ${\Lambda_2}$. In particular, when $\mu\neq 0$, we obtain $x_3>0$ irregardless of the value of ${\Lambda_2}$, while when $\mu=0$, $x_3$ is positive if ${\Lambda_2}< 2/3$. This means that, on the plane orthogonal to the magnetic field ($\mu=0$), a new stability condition arise if the magnetic field is strong enough. The threshold value related to $\Lambda_1$ that should discriminates the two regimes of growing and decreasing density contrast can be set as $\Lambda_1=1$ . Remembering that $\rho_0 = 1/6\pi G t^2$, such condition rewrites in terms of the wave number as $$k \gtrless k_J \equiv \sqrt{\frac{24\pi G\, \nu^{2}\, {\rho_0}}{v_s^{2}}}\;, \label{eq:Jeans}$$ which is substantially the same as the usual Jeans condition for gravitational instability. In fact, in the non-magnetic case, (to which our analysis reduces for $\omega_A=0$) this is the only criterion that separates the growing and the decaying modes . In a similar way, the new threshold ${\Lambda_2}=2/3$ yields to the following condition $$k \gtrless k_A \equiv \sqrt{\frac{4\pi G\,{\rho_0}}{v_A^{2}}}= \sqrt{\frac{16\pi^2 G \,{\rho_0}^2}{B_0^{2}}}\;. \label{eq:JeansM}$$ Summarizing, we find that the presence of a background magnetic field introduces an anisotropy in the stability criterion. While outside the plane orthogonal to ${{\boldsymbol{B}}_0}$, the stability of the perturbations is dictated only by the standard Jeans condition $k\gtrless k_J$, on that plane the unstable modes are those for which the conditions $k < k_J$ and $k<k_A$ both hold[^4]. In other words, if $k_A<k_J$ (basically equivalent to $v_A > v_s$), there are Jeans-unstable modes (those in the window $k_A < k <k_J$) that, in the orthogonal plane, are stabilized by the magnetic pressure. The window of stable modes gets wider for larger values of the ambient magnetic field, as expected. We underline that these results are qualitatively the same as those obtained for a static and uniform background [@PCLMB11]. A similar analysis was carried on by the authors of Ref. [@VTP05] obtaining a similar results. However, their derivation contained a mistake when separating the real and imaginary components of the evolution equations . For this reason they find a second-order differential equation instead than the fourth-order one discussed here. Numerical Analysis ================== In the previous Section, we have gained an important insight on the effect of a background magnetic field on the evolution of density perturbations. We now show some results obtained through the direct numerical integration of the differential system [(\[master-system\])]{}. ![Top panel: Alfvén (red solid line) and sound (blue dashed line) speed as functions of redshift, for $B_0(z=0)=10^{-9}$ G. The discontinuity at $z=1100$ corresponds to the recombination on neutral hydrogen. Bottom panel: magnetic (red solid line) and standard (blue dashed line) Jeans mass as a function of redshift. The mass contained inside the Hubble radius (black dotted line) is also shown for comparison. \[fig:soundspeed\]](soundspeed "fig:"){width="0.75\hsize"} ![Top panel: Alfvén (red solid line) and sound (blue dashed line) speed as functions of redshift, for $B_0(z=0)=10^{-9}$ G. The discontinuity at $z=1100$ corresponds to the recombination on neutral hydrogen. Bottom panel: magnetic (red solid line) and standard (blue dashed line) Jeans mass as a function of redshift. The mass contained inside the Hubble radius (black dotted line) is also shown for comparison. \[fig:soundspeed\]](jeansmass "fig:"){width="0.75\hsize"} Preliminaries ------------- We will focus on the period of the cosmological evolution that goes from the onset of matter domination ($z\simeq 3000$) to the time of reionization ($z\simeq 10$). We start from matter domination because, before that time, the growth of density perturbation was slowed down and practically frozen by the rapid expansion of the Universe. In the matter-dominated Universe, $a\propto t^{2/3}$ and $H=2/3t$. We can ignore the presence of a dark energy component since this is sub-dominant until very recent times. The time period that we consider can be divided into two distinct phases, [i.e., ]{}before and after the recombination of hydrogen occurring at ${z_{rec}}=1100$. Before recombination, the baryons are completely ionized and they are tightly coupled to photons, at least at scales larger than the comoving photon mean free path $\lambda_\gamma\simeq 1.8 [(1+z)/(1+{z_{rec}})]^{-2}$. At these scales, the total pressure of the fluid is given by radiation pressure. After recombination, most of the protons and electrons are in the form of neutral hydrogen atoms, leaving a small ionized fraction $x_e\simeq 2.5\times10^{-4}$. At the scales of interest, the neutral and ionized components are tightly coupled by collisions (see Sec. \[ssec:post\_uni\]) and can be treated as a single fluid. However, photons are now free streaming so that the baryon pressure is given just by kinetic pressure, dropping down by several orders of magnitude with respect to its pre-recombination value. ![image](delta_dwarf){width="0.33\hsize"} ![image](delta_gal){width="0.33\hsize"} ![image](delta_clus){width="0.33\hsize"} In view of this, we take the speed of sound of the cosmological fluid before and after recombination to be $$\label{vsvs} v_s^2\|_{z>{z_{rec}}}=\frac{1}{3}\;\frac{k_B T_b \sigma}{m_p + k_B T_b \sigma}\;,\qquad v_s^2\|_{z<{z_{rec}}}=\frac{5}{3}\;\frac{k_B T_b}{m_p}\;,$$ respectively, where $\sigma = 4 a_{SB} T^3/3n_b k_B\simeq 1.5\times 10^9$ is the specific entropy. We recall that $T_b=T_\gamma=T_\gamma\|_{z=0}(1+z)$ for $z>100$, while afterwards $T_b\propto (1+z)^2$. The expression above for the sound speed is rigorously valid only for scales corresponding to baryonic masses $\gtrsim 10^{11} M_\odot$, that stay above the photon mean free path until the time of recombination. At smaller scales, baryons lose the radiation support before recombination, roughly when the given scale goes below the photon mean free path, and it is at this time that the switch between the two expressions in [Eq.(\[vsvs\])]{} should take place. In the following, we shall consider masses nearly as small as $10^8\,M_\odot$, for which the photon decoupling effectively takes place at $z\simeq 3000$ (very close to the time of matter-radiation equality). However, we choose to switch between the two expressions for the sound speed at $z={z_{rec}}$ irregardless of scale and shall comment later on how we expect this choice to affect our results. Following the discussion in Sec. \[ssec:post\_uni\], the Alfvén velocity is taken to be $$v_A=\sqrt{\frac{B_0^2}{4\pi\rho_b}}\;,$$ where $\rho_b$ should always be intended as the total baryon density, both before and after recombination. A plot of the Alfvén and sound speeds as functions of redshift is shown in the upper panel of Figure \[fig:soundspeed\]. The sudden drop in the sound speed at recombination is due to the sharp decrease of baryon pressure after photon decoupling. In a detailed model, the presence of different uncoupled components making up the matter content of the Universe should be taken into account. In fact, most of the matter ($\sim 80\%$) is in the form of cold dark matter (CDM), interacting with the baryon-photon fluid only through the gravitational force. Thus, a proper treatment should rely on a two-fluid description. In the following, we shall ignore perturbations in the CDM component, however we argue that we can still draw meaningful conclusions about the perturbations in the baryonic component. In fact, in the pre-recombination era, the large radiation pressure prevents baryons to fall into the potential wells created by CDM; this is known to be true in the non-magnetic case but we expect it to hold also in the case under consideration since, as seen in the last Section, the magnetic field only acts to increase stability. Thus in this regime the CDM and baryon perturbations are effectively decoupled. After recombination, the baryon density perturbations will take some time to catch up with those in the CDM component and we expect our treatment will rigorously remain valid for some time. Before showing the results of the numerical integration, we illustrate in the lower panel Figure \[fig:soundspeed\] the evolution of the standard Jeans wavenumber [(\[eq:Jeans\])]{} and of its “magnetic” counterpart [(\[eq:JeansM\])]{}. In order to take out the change in $k_J$ and $k_A$ due to the expansion, we follow the convention to express the results in terms of the mass contained inside the corresponding length scales $1/k_{J,A}$. In particular, we consider the total baryonic mass (irrespective of the ionization state), contained inside a sphere of radius $2\pi /k_{J,A}$. It can be seen that the window of modes that are made stable by the magnetic field, [i.e., ]{}those between the red dashed and the black solid line, spans, right after recombination, five orders of magnitude in mass. As noted in the introduction, the existence of a magnetic Jeans length has been studied previously and all expressions for the critical wavenumber agree, apart from numerical factors, with expression [(\[eq:JeansM\])]{}. However, the numerical estimates of this and associated quantities that are found in the literature sometimes differ from our results. The reason seems to be that often the density $\rho_0$ that appears in [Eq.(\[eq:JeansM\])]{} is taken to be the present critical density $\rho_{c}\simeq 9\times 10^{-27}$ kg m$^{-3}$. This yields at the present time a magnetic Jeans length $\lambda_A \sim 1/k_A \sim 10$ kpc for $B_0(z=0)=10^{-9}$G and $\lambda_A\sim 1$ Mpc for $10^{-7}$G [^5]. Using instead the baryon density for $\rho_0$ will yield values of $\lambda_A$ a factor $\rho_c/\rho_b=\Omega_b^{-1}\simeq 20$ larger, i.e., $\lambda_A\sim 0.2\,(20)$ Mpc for $B_0(z=0)=10^{-9}\,(10^{-7})$ G. This amounts to a factor $20^3\simeq 10^4$ difference in the corresponding mass scale[^6]. Results ------- ![image](delta_real_dwarf){width="\hsize"} We now discuss the results of the direct numerical integration of [Eqs.(\[master-system\])]{}. The initial conditions for the integration have been chosen using the fact that power-law solutions for $\delta$ can be found in the limit $t\to 0$. There are four distinct solutions of this kind, but only one corresponds to a growing mode. We have matched the initial conditions to the asymptotic growing solution at the initial time of integration. The latter has been chosen so that all the modes of interest were outside the horizon at that time. Even if the initial time falls in what would be the radiation-dominated era, nevertheless we always consider a matter-dominated Universe. All results have been normalized to the initial value of the density contrast. ![image](eccentricity){width="0.8\hsize"} In Figure \[fig:deltak\], we show the evolution of the density contrast for three different wavenumbers $k=(17,\,1.7,\,0.36)\,\mathrm{Mpc}^{-1}$ (normalized at the present time), [i.e., ]{}for the following baryonic masses $M=(1.7\times 10^{8},\,1.7\times 10^{11},\,1.7\times 10^{13})\, M_\odot$. These masses roughly correspond to the scale of a dwarf galaxy, of a galaxy and of a galaxy cluster respectively. For each mode, we show the evolution in both the direction parallel to the background magnetic field ($\mu = 1$) and orthogonally to that direction ($\mu=0$). In all cases, the perturbation is initially growing but then starts to oscillate once the Jeans mass (that is growing) becomes larger than the mass of the perturbation. This happens earlier for smaller scales. In this phase the magnetic pressure does not play any role, as the much larger radiation pressure is actually providing the force that prevents the collapse. In fact, there is no difference in the evolution parallel and orthogonal to the field, as the radiation pressure is isotropic. The situation changes dramatically after recombination, when the baryon pressure drops and only the magnetic pressure can possibly oppose the growth, at least in the plane orthogonal to the field. Thus, perturbations in the direction of the field can grow unhindered, while the perturbations that are orthogonal can be stabilized. As it can be seen from Figure \[fig:deltak\], this is what happens for perturbations at the dwarf galaxy scale: at $z=10$, the relative growth of parallel perturbations with respect to orthogonal ones is of order 100. For perturbations at the galactic scale and larger, instead, the evolution is basically the same in all directions. This can be understood by looking at the lower panel of Figure [(\[fig:soundspeed\])]{}, from which it is clear that the pressure induced by a magnetic field of $10^{-9}$ G can only stabilize perturbations with mass $\lesssim 10^{10}\, M_\odot$. We recall that we have neglected the fact that, at the dwarf galaxy scale, the support of radiation pressure is not lost at recombination but some time before (see previous Subsection). From the discussion above, it is clear that, had we taken into account this fact, the evolution of orthogonal and parallel perturbations would have begun to differentiate earlier. This goes in the direction of enhancing the anisotropic growth of perturbation and the “squeezing” effect studied in the following. The fact that, after recombination, the evolution of the density contrast in the presence of a magnetic field changes for different directions leads to the reasonable expectation that some degree of anisotropy will be generated even in initially symmetric structures. In order to show this, we consider a Gaussian density fluctuation with standard deviation $\sigma$ in coordinate space at recombination: $$\delta({\boldsymbol{x}},\,t_{rec})= \delta({\boldsymbol{x}}=0,\,t_{rec}) e^{-\frac{\|{\boldsymbol{x}}\|^2}{2\sigma^2}}\;,$$ where the ${\boldsymbol{x}}$ are comoving coordinates centered at the maximum of the perturbation. After Fourier-transforming, we separately evolve the different harmonics in momentum space using [Eqs.(\[master-system\])]{} and we finally transform back to obtain the perturbation in coordinate space at a later time. In Figure \[fig:deltax\] we show, for a background magnetic field directed along the $y$ axis and with a present intensity of 1 nG, the evolution of a perturbation with $\sigma = 0.05$ kpc at recombination (so that the $3\sigma$ region encloses a mass $M\simeq1.5\times 10^{8} M_\odot$ at the mean baryonic density, [i.e., ]{}roughly the mass of a dwarf galaxy). In particular, we show equal density contours at $z=1000,\,100$ and $10$. It is evident from the figure how the perturbation becomes progressively squeezed along the direction orthogonal to the magnetic field. In order to quantify the anisotropy in the perturbation, we consider the isodensity contour corresponding to half the value at the peak and calculate its eccentricity $ \epsilon = \sqrt{1-b^2/a^2}$, where $a$ and $b$ are the lengths of the semi-major and semi-minor axis of the contour, respectively. In Figure \[fig:ecc\], we show how the eccentricity changes with redshift; for the parameters used above, we get $\epsilon \simeq 0.7$ at $z=10$. Conclusions =========== We have studied the effect of a background magnetic field on the linear evolution of cosmological density perturbations at scales well below the Hubble length, where a Newtonian treatment can be used, focusing on the matter-dominated era. The conditions that allow for the growth of small density perturbations have been clearly stated. In particular, we have found that in the plane orthogonal to the ambient magnetic field, a new critical length appears, related to the presence of the magnetic pressure, while everywhere else outside that plane the stability is dictated by the standard Jeans criterion. This is also confirmed through a direct numerical integration of the relevant MHD equations during the matter-dominated era, and this effect is shown to be possibly important after recombination, when the magnetic pressure of baryons is much larger than their the kinetic pressure. Finally, it has been shown how the dependence of the critical scale on the angle between the perturbation wavevector and the magnetic field could lead to a sizable anisotropy in the perturbations at sub-galactic scales at the onset of non-linearity. Our analysis has relied on some approximations: in particular, we have ignored the gravitational effects of dark matter perturbations. We have argued that this approximation limit the validity of our treatment to some time after recombination. We defer a more detailed and fully general relativistic analysis, also taking into account the different fluid components, to a future work. [Acknowledgment: N.C. gratefully acknowledges the CPT - Université de la Mediterranée Aix-Marseille 2 and the financial support from “Sapienza” University of Roma. M.L. acknowledges support from a joint Accademia dei Lincei / Royal Society fellowship for Astronomy. The work of M.L. has been supported by MIUR - Ministero dell’Istruzione, dell’Università e della Ricerca through the PRIN grants “Matter-antimatter asymmetry, Dark Matter and Dark Energy in the LHC era” (contract number PRIN 2008NR3EBK-005) and “Galactic and extragalactic polarized microwave emission” (contract number PRIN 2009XZ54H2-002).]{} Hypergeometric Coefficients {#Hyp_ab} =========================== In following, we write the complete form of the coefficients of the hypergeometric function of [Eq.(\[delta\_sol\])]{}. They read: $$\begin{aligned} a_{1(^{1}\;_{2})}&=1\mp\sqrt{\Delta_{-}\;}/12\nu-\sqrt{1-36\mu^{2}{\Lambda_2}\;}/12\nu\;,\\ a_{1(^{3}\;_{4})}&=1\mp\sqrt{\Delta_{+}\;}/12\nu-\sqrt{1-36\mu^{2}{\Lambda_2}\;}/12\nu\;,\\ a_{2(^{1}\;_{2})}&=1\mp\sqrt{\Delta_{-}\;}/12\nu+\sqrt{1-36\mu^{2}{\Lambda_2}\;}/12\nu\;,\\ a_{2(^{3}\;_{4})}&=1\mp\sqrt{\Delta_{+}\;}/12\nu+\sqrt{1-36\mu^{2}{\Lambda_2}\;}/12\nu\;,\end{aligned}$$ and $$\begin{aligned} b_{1(^{1}\;_{2})}&=1\mp\sqrt{\Delta_{-}\;}/6\nu\;,\\ b_{2(^{1}\;_{2})}&=1\mp\sqrt{\Delta_{-}\;}/12\nu-\sqrt{\Delta_{+}\;}/12\nu\;,\\ b_{3(^{1}\;_{2})}&=1\mp\sqrt{\Delta_{-}\;}/12\nu+\sqrt{\Delta_{+}\;}/12\nu\;,\\ b_{1(\;_{3})}&=1-\sqrt{\Delta_{+}\;}/6\nu\;,\\ b_{2(\;_{3})}&=1-\sqrt{\Delta_{-}\;}/12\nu-\sqrt{\Delta_{+}\;}/12\nu\;,\\ b_{3(\;_{3})}&=1+\sqrt{\Delta_{-}\;}/12\nu-\sqrt{\Delta_{+}\;}/12\nu\;,\\ b_{1(\;_{4})}&=1-\sqrt{\Delta_{-}\;}/12\nu+\sqrt{\Delta_{+}\;}/12\nu\;,\\ b_{2(\;_{4})}&=1+\sqrt{\Delta_{-}\;}/12\nu+\sqrt{\Delta_{+}\;}/12\nu\;,\\ b_{3(\;_{4})}&=1+\sqrt{\Delta_{+}\;}/6\nu\;.\end{aligned}$$ [0]{} E. 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Roberge, A. Dalgarno, Astrophys. J. 264, 485 (1983). I.Y. Dodin, N.J. Fisch, Phys. Plasmas 17, 112118 (2010). E.W. Kolb, M.S. Turner, [The Early Universe]{} (Addison-Wesley, 1990). G. Montani, M.V. Battisti, R. Benini, G. Imponente, Primordial Cosmology (World Scientific, 2011). D. Pugliese, N. Carlevaro, M. Lattanzi, G. Montani, R. Benini, Physica D 241, 721 (2012). [^1]: It has been argued (see, e.g., [@Adamek:2011pr]) that the limits obtained from the CMB temperature data can be significantly relaxed in the presence of free-streaming neutrinos. However this does not affect the limits from the polarization. [^2]: All throughout the paper, we use natural units with $c=\hbar=k_B=1$. [^3]: For a complete derivation of the Vlasov theory on curved spacetime, see Ref. [@DF10]. [^4]: It is easy to verify how the physical meaning of the condition is that the timescale for gravitational collapse $\tau_c \sim L/v_c \sim \sqrt{L^3/GM}\sim\sqrt{1/G{\rho_0}}$ is much shorter than both the acoustic and Alfvén timescales $\tau_s \sim L/v_s \sim 1/k v_s$ and $\tau_A \sim L/v_A \sim 1/k v_A$. [^5]: The latter value has also been said to be of the order of the scale of a galaxy cluster, while in effect it is closer to the scale of a galaxy, as it can be seen by the fact that the mass enclosed inside a sphere of 1 Mpc radius at the critical density is $\sim 10^{11} M_\odot$. The reason why a galaxy is much smaller than 1 Mpc is that it has detached from the Hubble flow and undergone non-linear evolution, so that its density is much larger than the cosmological average . [^6]: The discussion so far has ignored the gravitational action of dark matter; this can be roughly taken into account by using the total matter density $\rho_m$ at the numerator of [Eq.(\[eq:JeansM\])]{}, while keeping the same expression for $v_A$. This makes the value of $\lambda_A$ roughly twice smaller than in the case in which only baryons are considered, i.e., $\lambda_A\sim 0.1\,(10)$ Mpc for $B_0(z=0)=10^{-9}\,(10^{-7})$ G.
--- author: - 'A. Matter[^1][^2]' - 'L. Labadie' - 'J. C. Augereau' - 'J. Kluska' - 'A. Crida' - 'A. Carmona' - '[J.F. Gonzalez]{}' - 'W. F. Thi' - 'J.-B. Le Bouquin' - 'J. Olofsson' - 'B. Lopez' bibliography: - 'biblioHD139614.bib' title: 'Inner disk clearing around the Herbig Ae star HD139614: Evidence for a planet-induced gap ?[^3]' --- Introduction ============ Viscous accretion, photoevaporation, and dynamical clearing are processes through which disks are thought to shape and dissipate most of their mass, which lies in the gas content . Dust also plays an important role since it dominates the disk opacity and provides the raw material for building the rocky planets and the giant-planet cores. Dust is affected by processes stemming from its coupling to the gas such as dust trapping, radial migration, and grain growth , or from the stellar irradiation . Such evolution processes can produce specific signatures such as gaps, inner holes, and asymmetries . Identifying them in the inner disk regions ($\sim 0.1-10$ au) is essential since these regions are the expected cradle of telluric planets [@Righter27062011] and the location of the photoevaporation onset [@2014prpl.conf..475A]. The emission deficit in the infrared (IR) SED of pre-transitional and transitional disks [e.g., @1989AJ.....97.1451S; @2010ApJ...712..925C; @2010ApJ...718.1200M] has been commonly interpreted as a clearing of their inner regions, which make these objects relevant laboratories for observing the signatures of disk evolution and dissipation processes. Infrared interferometry can specifically probe these inner disk regions and detect dust radial evolution [@2004Natur.432..479V], brigthness asymmetries [e.g., @2013ApJ...768...80K], and dust clearing .\ The Herbig star HD139614 (see Table \[tab:star\]) is of particular interest here. It is associated with the Upper Centaurus Lupus (UCL) region of the Sco OB2 association [located at $140\pm27$ pc [@2008hsf2.book..235P and references therein]]{}. Its group-I SED presents pretransitional features (near-infrared (NIR) excess with mid-infrared (MIR) emission deficit at 6–7 $\mu$m) and weak MIR amorphous silicate features [@2010ApJ...721..431J], which suggests significant dust evolution in the inner regions. High-resolution imaging with T-RECS/Gemini only resolved the MIR continuum emission at 18 $\mu$m [17$\pm$4 au, @2011ApJ...737...57M], while MIR interferometry probed the inner N-band emitting region and revealed a narrow gap-like structure (located from $\sim$ 2.5 to 6 au) depleted in warm $\mu$m-sized dust . Recent works showed or found hints of the presence of gaps in Group I disks , and HD139614 is a rare, if not unique, case of object for which a small au-sized gap ($\sim 3$ au) has been spatially resolved. Previous studies, mainly based on sub-mm interferometry [e.g., @2011ApJ...732...42A; @2014ApJ...783L..13P] and NIR imaging [@2013ApJ...766L...2Q; @2014ApJ...790...56A], have focused on objects with large cavities ($\sim 10$–100 au). Their origin is ambiguous and possibly combines, for example, photoevaporation [@2013MNRAS.430.1392R], magnetorotational instability [@2007NatPh...3..604C], and/or multiple unseen planets [@2011ApJ...729...47Z]. Single Jovian planets are expected to open au-sized gaps while inducing a gas and dust surface density decrease in the inner disk regions [e.g., @2007MNRAS.377.1324C]. Knowing that spectroscopic observations have proven the presence of gas and gas tracers like PAHs [@2010ApJ...718..558A] in the HD 139614 disk, this object constitutes a unique opportunity to characterize the early stages of inner disk dispersal and [potentially]{} witness planet-disk interaction. Spatially resolved IR observations are thus required.\ We report the first multiwavelength analysis of HD139614, combining MIR VLTI/MIDI data with new NIR interferometric data obtained with VLTI/PIONIER and VLTI/AMBER , and far-IR photometry with Herschel[^4]/PACS . We aim to obtain new and robust constraints on the innermost $\sim 0.1$–1 au dust structure. We also aim to use radiative transfer to refine the constraints of the previous analytical modeling of the MIDI data (e.g., gap characteristics, outer disk properties). This is essential to determining the degree of differentiation of the inner and outer disk regions, on either side of the gap, and identify viable mechanisms for the inner disk clearing around HD 139614.\ Section 2 summarizes the new observations that complement the previous VLTI/MIDI and optical/IR SED data used in . Section 3 presents the analysis and geometrical modeling of the PIONIER and AMBER data. Section 4 describes the radiative transfer modeling of the broadband SED and the complete set of interferometric data. Section 5 discusses the modeling results against the mechanisms possibly responsible for the gap structure. This includes a comparative study with hydrodynamical simulations of gap opening by a planet. Finally, Sect. 6 summarizes our work and outlines the perspectives. [$d^{(1,2)}$ \[pc\]]{} [$A_{\rm V}^{(3)}$ \[mag\]]{} [SpTyp$^{(4)}$]{} [$M_^{(4)}$ \[M$_{\sun}$\]]{} [$R_^{(4)}$ \[R$_{\sun}$\]]{} [$\log T_^{(4)}$ \[K\]]{} [$\log g^{(5)}$]{} [Age$^{(6)}$ \[Myr\]]{} ------------------------ ------------------------------- ------------------- -------------------------------- -------------------------------- ---------------------------- -------------------- -------------------------- [140$\pm 27$]{} [0.09 ]{} [A7V]{} [1.7$\pm 0.3$]{} [1.6]{} [3.895]{} [4.0]{} [$8.8_{-1.9}^{+4.5}$ ]{} \[tab:star\] Observations and data processing ================================ [ccccccc]{} &[UT]{}&[Baseline]{}&[Calibrator]{}&[Seeing ()]{}&[Airmass]{}&[Label]{}\ \ & [08:55:41]{}&[A1-G1-K0-I1]{}&[HD145191]{} & [1.3]{}&[1.0]{}& [N/A]{}\ [06/06/2013]{}& [01:27:00]{}&[A1-G1-J3-K0]{} &[HD137598]{} & [1.1]{}&[1.1]{}& [N/A]{}\ &&&[HD141702]{} &&&\ [16/06/2013]{}& [23:31:00]{}&[D0-G1-H0-I1]{} &[HD137598]{} & [1.0]{}&[1.3]{}&[N/A]{}\ [03/07/2013]{}& [23:58:00]{}&[A1-B2-C1-D0]{} &[HD137598]{} & [1.1]{}&[1.1]{}&[N/A]{}\ &&&[HD141702]{} &&&\ \ & [05:27:00]{}&[UT1-UT2-UT4]{} &[HD141702]{}&[1.0]{}&[1.0]{}&[1,2,3]{}\ [09/05/2012]{}& [06:27:00]{}&[UT1-UT2-UT4]{} &[HD140785]{}&[0.6]{}&[1.1]{}&[4,5,6]{}\ \[tab:log\] VLTI/PIONIER observations ------------------------- We observed HD139614 with the 1.8 m Auxiliary Telescopes (AT) in the frame of a large program on Herbig stars (ID: 190.C-0963) conducted with VLTI/PIONIER (see Table \[tab:log\]). It combines the light from four telescopes in [H]{} band, and provides six squared visibilities, noted as $V^2$, and four closure phases per ATs configuration. Our observations were obtained at low spectral resolution ($R\sim 40$) providing three spectral channels centered at 1.58 $\mu$m, 1.67 $\mu$m, and 1.76 $\mu$m. [The calibration stars were selected with the SearchCal tool from the Jean-Marie Mariotti Center (JMMC). Each HD 139614 observation was bracketed by two calibrator observations, and the estimated transfer function was interpolated at the time of observation.]{} The projected baseline lengths range from 10.3 m to 139.8 m ($\lambda/2B \simeq 17$ mas to 1 mas). The PIONIER field of view (FOV) is $\sim 200$ mas ($\sim 25$ au at 140 pc). We performed a standard data reduction using the pipeline “pndrs”, described in . Figure \[fig:uv\] shows the UV coverage and calibrated data. [The final uncertainties include the statistical errors and the uncertainty on the transfer function.]{} VLTI/AMBER observations ----------------------- HD139614 was observed with AMBER (program 0.89.C-0456(A)) using the 8-m telescope triplet UT1-UT2-UT4. AMBER can combine three beams and provides spectrally dispersed $V^2$ and closure phases. Our observations were obtained in low spectral resolution ($R\sim 35$) in the [H]{} and [K]{} bands and included two interferometric calibrators (see Table \[tab:log\]) chosen using the SearchCal tool. The data consisted of two sets of eight and five exposures of 1000 frames. The data were reduced with the JMMC “amdlib” package (release 3.0.5). For each exposure, a frame selection was made to minimize the impact of the instrumental jitter and the non-optimal light injection into the optical fibers. Twenty percent of the frames with the highest fringe S/N provided the smallest errors on the resulting $V^2$. Despite this, the [H]{}-band data showed significant variability and low S/N ($\sim 1.5$) probably because of the [H]{}$=7.3$ magnitude of HD139614 that was close to the AMBER limiting magnitude ([H]{}$_{\rm corr}=7.5$). Moreover, the low-resolution closure phases are affected by a strong dependency on the piston, which is not reduced by stacking frames (see AMBER manual, ESO doc. [VLT-MAN-ESO-15830-3522]{}). Therefore, from the the AMBER observations, we only kept the [K]{}-band $V^2$ measurements. The instrumental transfer function was calibrated using the closest calibrator in time. The calibrated $V^2$ errors include the statistical error obtained when averaging the individual frames and the standard deviation of the transfer function over the calibrator observations. Our final dataset consists of six dispersed $V^2$ in the \[2.0–2.5\] $\mu$m range. The projected baseline lengths range from 52.0 m to 127.7 m (5 mas to 1.8 mas at $\lambda=2.2$ $\mu$m). The AMBER FOV is $\sim 60$ mas ($\sim 9$ au at 140 pc). Figure \[fig:uv\] shows the UV coverage and calibrated data. Photometric data ---------------- We complemented the SED used in with [Herschel]{}/PACS observations acquired on 7 March 2011 in the frame of the Key Program GASPS [@2013PASP..125..477D]. Data reduction, flux extraction, and error estimation were performed as in , and lead to $(9.66\pm0.96)\times10^{-13}$ W.m$^{-2}$ at 70 $\mu$m, $(4.93\pm0.01)\times10^{-13}$ W.m$^{-2}$ at 100 $\mu$m, and $(2.43\pm0.01)\times10^{-13}$ W.m$^{-2}$ at 160 $\mu$m. We also included measurements at 800 $\mu$m and 1.1 mm taken from @1996MNRAS.279..915S. Except for the recent Herschel data, we assumed a 10% relative uncertainty on the SED points, which is conservative, especially for the 2MASS data with formal uncertainties $\sim$ 5% [@2006AJ....131.1163S]. The broadband SED is shown in Fig.\[fig:radmcsed\]. Stellar parameters and spectrum ------------------------------- Table \[tab:star\] shows the stellar parameters used for HD 139614. Following @2008hsf2.book..235P, we adopt a distance of $140\pm 27$ pc hereafter. For the stellar flux, we use the same Kurucz spectrum ($T_{\rm eff}=7750$ K, $\log g=4.0$, Fe/H=-0.5) as in . Given the temperature quoted in Table 1 ($T_{\rm eff}\sim7850$ K), this may be slightly too cold but remains consistent with other temperature estimates [e.g., @2013MNRAS.429.1001A]. Observational analysis and geometrical modeling =============================================== Observational results --------------------- As shown in Fig. \[fig:uv\], the PIONIER (top) and AMBER (bottom) $V^2$ shows circumstellar emission that is well resolved at a level of a few mas. The PIONIER $V^2$ data present an exponential-like profile with a steep decrease at low frequencies ($\leq 10$ m/$\mu$m) that may suggest a fully resolved emission. The PIONIER FOV ($\sim 25$ au) partly encompasses the outer disk, which starts at $\sim 6$ au . NIR scattering by sub- and micron-sized grains at the outer disk’s surface may thus contribute to this steep decrease. At high frequencies ($\geq 40$ m/$\mu$m), the $V^2$ reach an asymptotic level between 0.5 at 1.58 $\mu$m and 0.35 at 1.76 $\mu$m, which translates to a visibility level between 0.7 and 0.6, respectively. This is close to the stellar-to-total flux ratio (STFR) evaluated to 0.7 at 1.65 $\mu$m using the stellar Kurucz spectrum of Sect. 2.5 and the NIR SED (2MASS measurements). This indicates that the circumstellar emission is fully resolved by PIONIER at high spatial frequencies. In the [K]{} band, the AMBER $V^2$ measurements range from 0.1 at 2 $\mu$m to 0.3 at 2.5 $\mu$m for the longest baselines (UT2-UT4 and UT1-UT4) and from 0.15 to 0.35 for the shortest one (UT1-UT2). The corresponding average $V$ level is 0.4 (UT2-UT4 and UT1-UT4) and 0.5 (UT1-UT2) across the [K]{} band, which is close to the STFR evaluated to 0.4 at 2.2 $\mu$m. The inner disk is thus fully resolved in the [K]{} band, at least for the longest baselines. This suggests a NIR-emitting region that probably spreads over at least one au ($\sim 7$ mas at 140 pc), as already suggested by .\ Our observations also show that $V^2$ depends on wavelength across both spectral bands. Such a chromatic effect has been studied, for instance, in the frame of IR image reconstruction . At high spatial frequencies, where the disk is fully resolved, the chromatic variation in the asymptotic level of visibility is $\sim 0.1$ (expressed in $V$) across the [H]{} band and $\sim 0.2$ across the [K]{} band. This reveals the decrease in the unresolved stellar contribution combined with a positive chromatic slope due to the emission from the hottest dust grains. Assuming a blackbody law for this hottest component, we could reproduce the NIR SED (from $\lambda=1.25$ $\mu$m to $\lambda=3.4$ $\mu$m) and the STFR chromaticity across the [H]{} band ($\sim 0.1$) and the [K]{} band ($\sim 0.2$), with dust grain temperatures from 1300 to 1500 K, which is close to the sublimation temperature for silicates. We note that the hottest emission at 1500 K induces a slightly lower STFR ($\sim 0.65$) at 1.65 $\mu$m than the ratio induced by the blackbody emission at 1300 K ($\sim 0.7$). The latter is closer to the STFR derived from the 2MASS measurement at 1.65 $\mu$m, namely $0.7$.\ The PIONIER closure phases do not show a clear departure from zero, hence noticeable signatures of brightness asymmetries. We do note a dispersion of $\lesssim 2^{\circ}$ for the yellow and red measurements and $\lesssim 5^{\circ}$ for the noisier blue and green ones (see Fig.\[fig:uv\]). Since the closure phase produced by a binary system is, at the first order, proportional to the flux ratio between the components [@2006MNRAS.367..825V], a closure phase dispersion of $\pm 2-5^{\circ}$ would translate to an upper limit of $\sim 3-8\times10^{-2}$ on the flux ratio. A flux ratio of $\sim 10^{-2}$ would imply an upper mass limit for the hypothetical companion of about 0.11 M$_{\odot}$, using the recent BT-SETTL atmospheric models for very low-mass stars, brown dwarfs, and exoplanets of @2012RSPTA.370.2765A. Based on the available PIONIER data, it is thus unlikely that HD139614 is actually a tight binary system hosting a companion that is more massive than 0.11 M$_{\odot}$ in its nearby environment ($\sim$ au). Geometrical modeling -------------------- We used a geometrical approach to derive the basic characteristics of the NIR-emitting region. Given the exponential-like profile of PIONIER $V^2$ measurements, we considered a Lorentzian-like brightness distribution to represent the NIR emission, independently in [H]{} and [K]{} bands. This centrally peaked brightness profile has broader tails than the usual Gaussian profile. It can be used to estimate a characteristic size for a spatially resolved emitting region that presents a gradually decreasing brightness profile with smooth outer limits. This is relevant for HD 139614, given its expected spatially extended NIR-emitting region. The squared visibility of a Lorentzian-like circumstellar emission can be written as $$V^2_{\rm circ}(u,v)=\left\|\exp\left(-2\pi \frac{r_{\scriptscriptstyle 50\%}}{\sqrt{3}}\frac{B_{\rm eff}(i,PA)}{\lambda}\right)\right\|^2,$$ where $r_{\scriptscriptstyle 50\%}$ is defined as the angular radius of half-integrated flux (containing 50% of the total flux), and $B_{\rm eff}(i,PA)$ is the effective baseline with $i$ and $PA$ the inclination and position angle of the circumstellar component. Then, the total visibility $V^2_{\rm tot}$ is [$V^2_{\rm tot}(u,v)=\|f_(\lambda)V_(u,v)+(1-f_(\lambda))V_{\rm circ}(u,v)\|^2$]{}, with $V_=1$ the visibility of the unresolved star. Here, $f_(\lambda)$ is the star-to-total flux ratio that is estimated at each wavelength using the Kurucz spectrum of Sect. 2.5 for the star and a single-temperature blackbody for the circumstellar contribution. To limit the number of free parameters, we only consider two STFRs $f_{,1300}$ and $f_{,1500}$, calculated with a 1300 K and a 1500 K blackbody emission (see Sect. 3.1). The free parameters of the model are $r_{\scriptscriptstyle 50\%}$ (in mas), $i$, and $PA$. ### PIONIER Considering the complete set of $V^2$ data, we computed a grid of models by scanning the parameters space in 70 steps and, from the $\chi^2$ calculated from the measured and modeled $V^2$, derived the Bayesian probability ($\exp \left[-\chi^2/2 \right]$) for each of the parameters. These marginal probability distributions are the projection of $\exp \left[-\chi^2/2 \right]$ along the three dimensions of the parameters space. The range of explored values is \[1–20\] mas (0.1–2.8 au at 140 pc) for $r_{\scriptscriptstyle 50\%}$, \[0$^{\circ}$–70$^{\circ}$\] for $i$, and \[0$^{\circ}$–175$^{\circ}$\] for $PA$. It appears that low $i$ values ($\lesssim 35^{\circ}$) are favored, while no clear constraint is obtained on $PA$. Indeed, with $i \lesssim 35^{\circ}$, all $PA$ values between $0^{\circ}$ and $150^{\circ}$ fit the data almost equally well ($\chi^2_{\rm red} \simeq 2.3-2.6$). Since our available PIONIER dataset does not seem to highlight a significant difference in position angle and inclination relative to the outer disk, we adopted the values derived by for the outer disk: $112\pm9^{\circ}$ and $20\pm2^{\circ}$ (with $1-\sigma$ uncertainties), respectively. We again explored the same range of values for $r_{\scriptscriptstyle 50\%}$ for the two STFRs. The best-fit solution is represented by $r_{\scriptscriptstyle 50\%}=3.9\pm 0.1$ mas ($0.55\pm0.01$ au at 140 pc) for $f_{,1300}$, and by $r_{\scriptscriptstyle 50\%}=2.7\pm 0.2$ mas ($0.4\pm0.02$ au at 140 pc) for $f_{,1500}$. The 1-$\sigma$ uncertainties correspond to the 68% confidence interval derived directly from the marginal probability distribution. Using the $f_{,1500}$ ratio ($f_{,1500} < f_{,1300}$ in [H]{} band) leads to a better fit ($\chi^2_{\rm red}=2.6$) than the $f_{,1300}$ ratio ($\chi^2_{\rm red}=6.6$). As shown in Fig.\[fig:lorentzmodels\], the $V^2$ decrease at low spatial frequencies is reproduced in both cases, while the modeled $V^2$ for $f_{,1300}$ significantly overestimates the $V^2$ measurements at the highest spatial frequencies. This suggests a discrepancy between the STFR predicted by the asymptotic $V^2$ level and the one estimated from the 2MASS measurements and the Kurucz spectrum. Possible causes are 1) the uncertainties on the 2MASS measurements [$\sim 5\%$ @2006AJ....131.1163S] and on the stellar parameters of the Kurucz model; 2) a change in the STFR between the time of the 2MASS and PIONIER observations, which however, appears unlikely given the absence of significant visible or MIR variability and the face-on orientation of the disk ; and 3) the degradation of $V^2$ measurements at long baselines, where the coherent flux is lower. HD 139614 has a $H$ mag=7.3 that is close to the PIONIER sensitivity limit . ![[Best-fit Lorentzian model (solid line) for PIONIER and AMBER overplotted on their measured $V^2$, as a function of the effective baseline (see Eq.1). For clarity, we only plot the AMBER $V^2$ at $\lambda=2.0$ $\mu$m, 2.2 $\mu$m, and 2.5 $\mu$m.]{}[]{data-label="fig:lorentzmodels"}](vis_lorentzian_bestfit.eps) Finally, no fully resolved emission was needed to reproduce the steep $V^2$ decrease at low spatial frequencies. ### AMBER Considering the complete set of AMBER dispersed $V^2$ data, we followed the same procedure as for PIONIER (same parameter space and same derivation of the formal errors). It quickly appeared that our AMBER interferometric dataset is too sparse in UV coverage (one baseline triplet covering only projected baselines lengths longer than 50 m) to constrain the inclination and the position angle of the [K]{}-band emitting region. Therefore, assuming again that the inner component is coplanar with the outer disk, we explored a broad range of $r_{\scriptscriptstyle 50\%}$ values (1–20 mas) and found a best-fit solution represented by $r_{\scriptscriptstyle 50\%}=4.3\pm 0.1$ mas ($0.60 \pm 0.02$ au at 140 pc) for $f_{,1300}$, and by $r_{\scriptscriptstyle 50\%}=4.2\pm 0.1$ mas ($0.58 \pm 0.02$ au at 140 pc) for $f_{,1500}$. As shown in Fig.\[fig:lorentzmodels\], both solutions agree well with the measured $V^2$ ($\chi^2_{\rm red}\sim1.6$), since $f_{,1300}(\lambda) \simeq f_{,1500}(\lambda)$ across the [K]{}-band.\ Our new NIR data spatially resolved the innermost region of HD 139614 and suggest that hot dust material is located around the expected dust sublimation radius of micron-sized silicate dust ($\simeq 0.2$ au). We do not rule out the presence of refractory dust material within this radius. Moreover, the inner dust component probably extends out to 1 au or 2 au. Indeed, our derived $r_{\scriptscriptstyle 50\%}$ values imply that, for instance, 80% of the flux in such Lorentzian profiles would then be contained within $\sim 1.1$ au in the [H]{} band, and $\sim 1.7$ au in the [K]{} band. Considering the uncertainty on the stellar distance ($\pm 27$ pc), the $r_{\scriptscriptstyle 50\%}$ values would vary from 0.35 au to 0.45 au in the [H]{} band and from 0.5 au to 0.7 au in the [K]{} band. These values are still consistent with our conclusions. The slight difference in size estimate between the [H]{} and [K]{} bands suggests a disk chromaticity that is probably related to a temperature gradient. Finally, the data do not allow us to constrain a difference in inclination and position angle between the inner and outer components and are consistent with a coplanarity. Radiative transfer modeling =========================== Disk model ---------- Based on , we consider a two-component model composed of an inner and an outer dust component spatially separated by a dust-depleted region. Although we first assume an empty gap when performing the model’s grid computation and $\chi^2$ minimization, an upper limit on the dust mass in the gap will then be estimated. We mention that the results of , which showed an incompatibility of the IR SED and the mid-IR visibilities with a continuous disk structure, confirmed previous results based on SED modeling. Notably, the possibility of a partially self-shadowed continuous disk was explored by . Using a passive disk model with a puffed-up inner rim, they could not reproduce both the NIR excess and the rising MIR spectrum of HD139614 (see their Fig.4). The artificial increase in the inner rim scale height, which is required to match the NIR excess, implied a strong shadowing of the outer disk. This led to decreasing and too low MIR emission. We performed radiative transfer modeling tests with a continuous disk, including a puffed-up and optically thick inner rim to induce self-shadowing over the first few au. However, this model systematically produced too much flux in the 5–8 $\mu$m region, and decreasing and too weak emission at $\lambda > 8$ $\mu$m. This implied MIR visibilities without sine-like modulation, in contrast with the MIDI data. Moreover, the puffed-up inner rim systematically induced a too spatially confined NIR-emitting region that is incompatible with the NIR interferometric data. ### RADMC3D We used the radiative transfer code RADMC3D to produce disk images and SED . Its robustness and accuracy were validated through benchmark studies . This code can compute the dust temperature distribution using the Monte-Carlo method of @2001ApJ...554..615B with improvements like the continuous absorption method of . We considered an axially symmetric two-dimensional disk in a polar coordinate system ($r$, $\theta$) with a logarithmic grid spacing in $r$ and $\theta$. An additional grid refinement in $r$ was applied to the inner edge of both components to ensure that the first grid cell is optically thin. The radiation field and temperature structure computed by the Monte Carlo runs were used to produce SEDs and images by integrating the radiative transfer equation along rays (ray-tracing method). Isotropic scattering was included in the modeling. While the thermal source function is known from a first Monte Carlo run, the scattering source function is computed at each wavelength through an additional Monte Carlo run prior the ray-tracing. ### Disk structure Each dust component is represented by a parameterized model of passive disk with a mass $M_{\rm dust}$ and inner and outer radii, $r_{\rm in}$ and $r_{\rm out}$. Assuming it is similar to the gas density distribution in hydrostatic equilibrium, the dust density distribution is given by $$\rho(r,z)=\frac{\Sigma(r)}{H(r)\sqrt{2\pi}}exp\left[-\frac{1}{2}\left(\frac{z}{H(r)}\right)^2\right], \label{eq:dustdensity}$$ where $z \simeq (\pi/2-\theta)r$ is the vertical distance from the midplane, in the case of a geometrically thin disk ($z >> r$). The dust surface density $\Sigma(r)$ and scale height $H(r)$ are parameterized as $\Sigma(r) = \Sigma_{\rm out}\left(r/r_{\rm out}\right)^p$ and $H(r) = H_{\rm out}\left(r/r_{\rm out}\right)^{1+\beta}$, where $\beta$ is the flaring exponent, $\Sigma_{\rm out}$ and $H_{\rm out}$ are the dust surface density and scale height at $r_{\rm out}$. To enable a smooth decrease in density after $r_{\rm out}$, we apply another $p$ exponent to the surface density profile. We also include the possibility of rounding off or “puffing up” the inner rim of each component. For that, we artificially reduce or increase the dust scale height $H(r)$ to reach a chosen value $\hat{H}_{ \rm in}$ at $r_{\rm in}$. The width $\epsilon r_{\rm in}$ over which this reduction or increase is done – from no change at $r_1 = (1+\epsilon)r_{\rm in}$ to the dust scale height $\hat{H}_{ \rm in}$ – sets the “sharpness" of the rim. Following , the modified dust scale height $\hat{H}(r)$, between $r_{\rm in}$ and $r_1$, writes as $$\hat{H}(r)=\left(1-\frac{r-r_{\rm in}}{r_1-r_{\rm in}}\right)\hat{H}_{\rm in}+\left(\frac{r-r_{\rm in}}{r_1-r_{\rm in}}\right)H(r).$$ ### Dust properties The dust grain properties are described by their optical constants taken from the Jena database[^5]. The mass absorption and scattering coefficients (in cm$^{2}$.g$^{-1}$), $\kappa_{\rm abs}(\lambda),$ and $\kappa_{\rm sca}(\lambda)$ are then computed using the Mie theory. The spherical shape approximation is usually safe for amorphous or featureless dust species for which the extinction properties are much less sensitive to grain shape effects than to crystalline material . Assuming a grain size distribution $n(a)\propto a^{-3.5}$ [@1977ApJ...217..425M] with minimum $a_{\rm min}$ and maximum $a_{\rm max}$ grain sizes, the size-averaged mass absorption/scattering coefficient is obtained by adding the mass absorption/scattering coefficients of each grain size times their mass fraction. Then, the global $\kappa_{\rm abs}(\lambda)$ and $\kappa_{\rm sca}(\lambda)$ are derived by adding the size-averaged mass absorption/scattering coefficients of each dust species times their abundance to form a single “composite dust grain” that is the mix of the constituents. This averaged approach has already been justified in other disk studies and implicitly assumes a thermal coupling between the grains. This is expected since dust grains in disks are likely to be in the form of mixed aggregates in thermal contact. Each disk component has a homogeneous composition in the radial and vertical directions. No dust settling or radial segregation is considered here. Modeling approach ----------------- We split our modeling approach into several steps, considering the mutual radiative influence between the different disk components and the associated dataset. For the star emission, we use the synthetic spectrum detailed in Sect. 2.5 and Table \[tab:star\]. ### Inner disk We address first the inner disk to reproduce the NIR SED and the PIONIER and AMBER $V^2$. Following Sect. 3.2, we set the inner disk’s inclination and position angle to $i_{\rm disk}=20^{\circ}$ and $PA_{\rm disk}=112^{\circ}$. The inner rim is modeled as a vertical wall.\ For the radial structure, $r_{\rm in}$ is first set to 0.2 au . We vary the surface density profile exponent $p$ to explore different dust radial distributions. We also prevent $\Sigma(r)$ from decreasing too abruptly after $r_{\rm out}$ by setting $p=-10$. Greater $p$ values (e.g., $\sim -5$) would induce too smooth a decrease, keeping $r_{\rm out}$ from representing the inner disk’s size.\ For the vertical structure, the inner disk scale height at $r_{\rm out}$ is assigned a broad range of values , namely $H_{\rm out}/r_{\rm out}=[0.03,0.05,0.1.0.2,0.3]$. This encompasses the dust scale height values that are typically inferred or considered (0.1–0.15) for disks .\ ---------------- ------------------ ---------------------------- ---------------------------- --------------- [Dust setup]{} [Dust species]{} [$a_{\rm min}$ $(\mu$m)]{} [$a_{\rm max}$ $(\mu$m)]{} [Abundance]{} [Olivine]{} [0.1]{} [20]{} [80%]{} [Graphite]{} [0.05]{} [0.2]{} [20%]{} [Olivine]{} [5]{} [20]{} [80%]{} [Graphite]{} [0.05]{} [0.2]{} [20%]{} [3]{} [Olivine]{} [0.1]{} [20]{} [100%]{} [4]{} [Olivine]{} [5]{} [20]{} [100%]{} [5]{} [Graphite]{} [0.05]{} [0.2]{} [100%]{} [6]{} [Olivine]{} [0.1]{} [3000]{} [100%]{} ---------------- ------------------ ---------------------------- ---------------------------- --------------- : Description of the dust setups \[tab:dustsetup\] We consider a dust mix of silicates and graphite. Including graphite is justified by its being a major constituent of the interstellar grains and by recent disks modeling that shows the importance of featureless and refractory species like graphite to explain the observations . We considered three graphite mass fractions (see Table \[tab:dustsetup\]) to evaluate its impact on the modeling. We assumed the same graphite composition as in and considered grains with $a_{\rm min}=0.05$ $\mu$m and $a_{\rm max}=0.2$ $\mu$m to maximize the IR opacity. Since graphite is a featureless species, this choice is not critical. For the silicate content, we assumed a pure iron-free olivine composition [see @2010ApJ...721..431J] with $a_{\rm min}=0.1$ $\mu$m and 5 $\mu$m to keep or suppress the silicate feature. We fixed $a_{\rm max}=20$ $\mu$m since the optical and NIR opacity contribution from mm grains is negligible.\ The inner disk free parameters are $p$, $r_{\rm out}$, and $M_{\rm dust}$. The last is a lower limit since we only include $\mu$m-sized and sub-$\mu$m-sized grains. Then, the modeling steps are - computing a grid of $10\times10\times10$ models on $M_{\rm dust}$, $p$, $r_{\rm out}$ for each model setup (combination of a dust setup and a $H_{\rm out}/r_{\rm out}$ value; see Table \[tab:freeparams\]). We then calculate a reduced $\chi^2_{\rm r}$ separately for each dataset: the NIR SED (5 points from 1.25 $\mu$m to 4.5 $\mu$m), the dispersed PIONIER $V^2$, and dispersed AMBER $V^2$. The synthetic $V^2$ are computed from the RADMC3D images multiplied by a 2D Gaussian with a FWHM equal to the instrument FOV. The best-fit model of each model setup is found by minimizing the sum of the reduced $\chi^2_r$ values. - for each model setup, computing a grid of $8\times8\times8$ models around the global minimum $\chi^2_r$ and calculating the marginal probability distribution ($\exp[-\chi^2/2]$) to determine the best-fit value and 1-$\sigma$ uncertainty (68% confidence interval) on $M_{\rm dust}$, $p$, and $r_{\rm out}$. The best-fit model with the lowest reduced $\chi^2_r$ value is chosen as the global best-fit solution. - Once the global best-fit solution is found, slightly varying $r_{\rm in}$ around its initial value ($0.2$ au) to further improve the fit to the NIR visibilities at high spatial frequencies. ### Outer disk We then address the outer disk to find a solution that is consistent with the Herschel/PACS data and the sub-mm SED. Based on the model of , we set $r_{\rm in}=5.6$ au and $r_{\rm out}=150$ au and assume a pure-olivine composition for the silicate content, with a grain size distribution from 0.1 $\mu$m to 3 mm to account for the weak silicate emission feature in the [N]{} band and the low sub-mm spectral index [@1996MNRAS.279..915S]. We set the surface density profile exponent to $p=-1$, as is typically observed in the outer regions of disks [e.g., @2007ApJ...659.705A; @2010ApJ...723.1241A]. Then, two typical flaring profiles for irradiated disks are explored, i.e., $\beta=1/7$ and $\beta=2/7$ [e.g., @1997ApJ...490..368C], and the dust scale height $H_{\rm out}/r_{\rm out}$ is varied between 0.1 and 0.15, which is typical of disks . We also vary the graphite abundance (from 0 to 20%) and the dust mass between 10$^{-6}$ M$_{\odot}$ and 10$^{-4}$ M$_{\odot}$. ![[: observed SED (black diamonds) and SED of the best-fit RADMC3D model (red line). The [SPITZER]{}/IRS spectrum is indicated with a black line, and the stellar contribution as a dotted line.]{}[]{data-label="fig:radmcsed"}](SED_results_bestfit.eps) ### Outer disk inner edge We then focus on the inner edge to reproduce the MIDI data and improve the fit to the MIR SED. Both trace the geometry and flux fraction intercepted by the outer disk’s inner edge, hence its scale height and radial position. Since the SED measurements at 4.8 $\mu$m, 12 $\mu$m, and 25 $\mu$m are consistent with the [Spitzer]{} spectrum, modeling and fitting them will be sufficient and lead to a decent fit of the [Spitzer]{} spectrum. Reproducing the full [Spitzer]{} spectrum in detail is beyond the scope of this paper. Using the global best-fit solution of the inner disk, we compute a grid of $8\times8\times8$ models on the outer disk inner radius $r_{\rm in}$, the modified scale height $\hat{H}_{\rm in}$ at $r_{\rm in}$, and the rounding-off parameter $\epsilon$ (see Table \[tab:freeparams\]). We then calculate a reduced $\chi^2_{\rm r}$ separately for each dataset, namely the dispersed MIDI visibilities (47 data points between 8 $\mu$m and 13 $\mu$m) and the MIR SED (at 4.8 $\mu$m, 12 $\mu$m, and 25 $\mu$m). The best-fit model is found by minimizing the sum of the reduced $\chi^2_{\rm r}$ values. The parameters’ best-fit values and 1-$\sigma$ uncertainties are computed in the same way as for the inner disk. [ccccc]{}\ &\ &[$H_{\rm out}/r_{\rm out}$]{}&[$r_{\rm out}$ (au)]{}&[$M_{\rm dust}$ (M$_{\odot}$)]{}&[$p$]{}\ [1]{}&&&&\ [2]{}&&&&\ [3]{}&&&&\ [4]{}&&&&\ [5]{}&&&&\ \ &\ &[$\hat{H}_{\rm in}$]{}&$\epsilon$&[$r_{\rm in}$ (au)]{}\ &&&\ \[tab:freeparams\] Results ------- \ ### A tenuous and extended inner disk ---------------- ------------------------------- ---------------------------- --------------------------------------- ------------------------- ---------------------- [Dust setup]{} [$H_{\rm out}/r_{\rm out}$]{} [$r_{\rm out}$ (au)]{} [$M_{\rm dust}$ (M$_{\odot}$)]{} [$p$]{} [$\chi^2_{\rm r}$]{} \[1mm\] [0.05]{} [$1.89^{+0.14}_{0.13}$]{} [$2.8^{+0.8}_{-0.8}\times10^{-10}$]{} [$1.6^{+0.3}_{-0.3}$]{} 3.9 \[1mm\] [0.1]{} [$2.55^{+0.13}_{-0.13}$]{} [$8.8^{+4.2}_{-3.7}\times10^{-11}$]{} [$0.6^{+0.3}_{-0.3}$]{} 4.0 \[1mm\] [0.03]{} [$1.10^{+0.11}_{0.11}$]{} [$9.5^{+3.1}_{-3.1}\times10^{-11}$]{} [$3.4^{+0.3}_{-0.3}$]{} 4.4 \[1mm\] [0.05]{} [$2.20^{+0.10}_{-0.10}$]{} [$4.2^{+1.2}_{-1.2}\times10^{-11}$]{} [$1.1^{+0.3}_{-0.3}$]{} 4.0 \[1mm\] ---------------- ------------------------------- ---------------------------- --------------------------------------- ------------------------- ---------------------- : Best-fit model setups for the inner disk. \[tab:inner\] Table \[tab:inner\] shows the results of the inner disk modeling. Several models with different $r_{\rm out}$ values for the inner disk fit the NIR data equally well. Since our MIDI data partly resolve the inner disk , we can use them to break out the degeneracy in $r_{\rm out}$ and identify the global best-fit solution. For each inner disk solution, we computed a grid of $8\times8\times8$ models on the outer disk’s inner edge parameters, and kept the model giving the lowest reduced $\chi^2_{\rm r}$ on the MIR SED and visibilities. #### Size and location Only the inner disk models with $r_{\rm out} > 2$ au are compatible with the MIR visibilities, especially from 8 to 9.5 $\mu$m. We constrained the inner disk’s outer radius to $r_{\rm out}=2.55^{+0.13}_{-0.13}$ au. With an uncertainty of ($\pm 27$ pc) on the stellar distance, $r_{\rm out}$ varies from 2.0 to 2.9 au, which is still consistent with our conclusions. This small a variation in $r_{\rm out}$ would not significantly modify the conditions of irradiation of the inner disk and thus its temperature and brightness profile. From the best-fit solution, we varied $r_{\rm in}$ by steps of 0.05 au around $r_{\rm in}=0.2$ au to optimize the fit to the PIONIER and AMBER $V^2$. The best agreement is nevertheless found for $r_{\rm in}=0.2$ au. The $r_{\rm in}<0.2$ au values slightly increased the amount of unresolved inner disk emission and induced too high a [K]{}-band $V^2$ at low spatial frequencies. The $r_{\rm in}>0.2$ au values induced too prominent a lobe in the [H]{}-band $V^2$ profile at high spatial frequencies (see Appendix). As shown in Figs.\[fig:radmcsed\] and \[fig:radmcmodel\], our model agrees well with the NIR SED and the PIONIER and AMBER $V^2$. Nevertheless, the best-fit [H]{} band $V^2$ are still slightly too high at high spatial frequencies, which probably suggests that the inner rim has a smoother shape than the assumed vertical wall. The modeled closure phases in the [H]{} band appears consistent with the measured ones. Figure \[fig:imagemodel\] shows the best-fit model images at $\lambda=1.6$ $\mu$m and $\lambda=2$ $\mu$m. #### Dust composition Our modeling favored the dust setup “2” (80% olivine + 20% graphite, see Table \[tab:dustsetup\]). This implies a temperature for the hottest composite grains of $T\simeq1660$ K at $r_{\rm in}=0.2$ au, which is on the warm side of sublimation temperatures for $\mu$m-sized silicates [@1994ApJ...421..615P]. We also tested the effect of computing the temperature distribution separately for the two dust species. This led to olivine grains that were too cold ($\sim 1150$ K) to be responsible for the NIR excess and to very hot graphite grains ($\sim 1900$ K). The latter would induce too much flux at $\lambda < 2$ $\mu$m and would disagree with our conclusions in Sect. 3.1. Therefore small graphite grains (or any refractory, featureless and efficient absorber/emitter) in thermal contact with the silicate grains seems required to induce enough heating and emission in the inner disk’s outer parts ($> 1$ au) and to produce a spatially extended NIR-emitting region. The solution with a 100% graphite composition and $r_{\rm out}=2.20^{+0.10}_{-0.10}$ au is also compatible with the MIR visibilities. The dust compositions including smaller (sub-$\mu$m-sized) olivine grains (dust setups “1” and “3”) induce high inner rim temperatures ($\sim 1550$ K), for which these small grains may sublimate, and overpredict the strength of the 10 $\mu$m feature in the [Spitzer]{} spectrum. This suggests that the inner disk does not contain a detectable amount of sub-$\mu$m-sized silicate grains and that the weak silicate feature originates in the outer disk. #### Surface density profile We constrained the surface density profile to $p=0.6^{+0.3}_{-0.3}$ (see Fig.\[fig:surfacedensity\]). This positive profile suggests a radially increasing distribution of dust grains, and was required to create the extended NIR-emitting region predicted by the interferometric data. Such a radial distribution induces less shadowing from the inner rim and sufficient heating and re-emission from the outer parts of the inner disk. The radially integrated midplane optical depth (at $\lambda=0.55$ $\mu$m) reaches $\tau \simeq 1$ only at $r \simeq 0.6$ au, as shown in Fig.\[fig:taudisk\]. All the models with $p<0$ induced too strong a flux contribution from the inner disk rim in the SED at $\lambda \lesssim 2$ $\mu$m and therefore a NIR-emitting region that is too spatially confined. This implied [H]{}-band and [K]{}-band $V^2$ that are too high at low spatial frequencies, and too low at high spatial frequencies since the STFR is underestimated (see Appendix). #### Dust mass and surface density level We estimated a mass of $M_{\rm dust}=8.8^{+4.2}_{-3.7}\times10^{-11}$ M$_{\odot}$ in small dust grains, which dominate the optical/IR extinction efficiency, and set the NIR emission level. The NIR absorption efficiency is dominated by the small graphite grains with a mass absorption coefficient 100 times larger than for the $\mu$m-sized olivine grains. Therefore the total dust mass estimate largely depends on the graphite fraction, and a significant amount of silicate grains could be hidden in the inner disk. To estimate an upper limit on the dust mass, we extended the olivine grain size distribution up to 3000 $\mu$m (as in the outer disk) and kept the same absolute mass in graphite grains of our best-fit model. We then increased the mass in olivine grains (by steps of $10^{-10}$ M$_{\odot}$) until the modeled NIR emission (at 2.2 $\mu$m, 3.4 $\mu$m, and 4.8 $\mu$m) and/or NIR visibilities deviated by more than 3-$\sigma$ from the observed ones. As a result, we found an upper limit of about $6\times10^{-10}$ M$_{\odot}$. Higher mass values implied a deviation of more than 3-$\sigma$ with respect to several AMBER $V^2$ measurements (all across the [K]{} band) and to the observed SED at 4.8 $\mu$m. We show in Fig.\[fig:surfacedensity\] the best-fit dust surface density and overplot its upper limit (dotted line). Relative to the outer disk, the inner disk seems strongly depleted by at least $\sim 10^3$. However, this dust depletion may be biased by a difference in dust composition between the inner disk that contains graphite grains and the outer disk that only contains olivine grains (see Table \[tab:dustsetup\]). However, the NIR mass absorption coefficient of the inner disk’s dust mixture is larger, by a factor 5, than that of the outer disk. In the MIR, the two coefficients are equal. The difference in dust composition thus cannot account for the dust surface density ratio of $\sim 10^3$ between the two components. #### Dust scale height A dust scale height of $H_{\rm out}\simeq 0.25$ au ($H_{\rm out}/r_{\rm out}=0.1$) at $r_{\rm out}=2.55$ au is favored. This translates to $H_{\rm in}\simeq 0.01$ au ($H_{\rm in}/r_{\rm in}=0.044$) at the inner rim. We recall that $H(r)$ is the height from the midplane at which the dust density has decreased by a factor ${\rm e}^{-0.5}$ (see Eq.\[eq:dustdensity\]). Assuming a perfect dust-gas coupling, this would equal the pressure scale height of the gas in hydrostatic equilibrium. With a midplane dust temperature of 1660 K at $r_{\rm in}$, the gas pressure scale height would be smaller than the dust scale height ($H_{\rm in,gas}=0.006$ au, i.e. $H_{\rm in,gas}/r_{\rm in} \simeq 0.028$). If dust grains are coupled to the gas, this suggests that 1) gas is actually hotter (with a required temperature of about 5000 K), as expected in the inner disk regions from thermo-chemical modeling , and/or that 2) gas is vertically supported by additional sources, such as magnetic forces arising from the inner disk accretion driven by Magneto-rotational turbulence [@2014ApJ...780...42T]. With this dust scale height, the inner disk does not cast a significant shadow on the inner rim of the outer disk. Indeed, while the total integrated optical depth (at $\lambda=0.55$ $\mu$m) in the inner disk midplane is $\tau \simeq 4$, it decreases quickly to $\tau < 1$, above the midplane, so that the outer disk’s inner rim can intercept a large fraction of the stellar irradiation (see Fig.\[fig:taudisk\]). The impact can be clearly seen in the emission increase at $\lambda \leq 8$ $\mu$m in the best-fit model SED (see Fig.\[fig:radmcsed\]) and in the ring-like morphology of the outer disk’s rim in the synthetic image at 10 $\mu$m (see Fig.\[fig:imagemodel\]). ------------------------ ---------------- ------------------------------- ------------------------- ---------------------------- --------------------------------------- ------------------------- ------------- ----------------------------------- ---------------------------- ----------------------------------- ------------------------------------ [Dust setup]{} [$H_{\rm out}/r_{\rm out}$]{} [$r_{\rm in}$ \[au\]]{} [$r_{\rm out}$ \[au\]]{} [$M_{\rm dust}$ \[M$_{\odot}$\]]{} [$p$]{} [$\beta$]{} [$\hat{H}_{\rm in}/r_{\rm in}$]{} [$\epsilon$]{} [$i_{\rm disk}$ \[$^{\circ}$\]]{} [$PA_{\rm disk}$ \[$^{\circ}$\]]{} \[1mm\] [Inner disk]{} [2]{} [0.1]{} [0.20]{} [$2.55^{+0.13}_{-0.13}$]{} [$8.9^{+3.4}_{-2.5}\times10^{-11}$]{} [$0.6^{+0.3}_{-0.3}$]{} [2/7]{} [N/A]{} [0]{} [20]{} 112 \[1mm\] [Outer disk]{} [6]{} [0.135]{} [$5.7^{+0.3}_{-0.2}$]{} [150]{} [$1.3\times10^{-4}$]{} [$-1$]{} [1/7]{} [$0.04^{+0.02}_{-0.02}$]{} [$0.30^{+0.15}_{-0.05}$]{} [20]{} [112]{} ------------------------ ---------------- ------------------------------- ------------------------- ---------------------------- --------------------------------------- ------------------------- ------------- ----------------------------------- ---------------------------- ----------------------------------- ------------------------------------ \[tab:global\] ![image](images_RT_global_wgap_narrow_annotated.eps){width="130mm" height="45mm"} ### An au-sized gap As shown in Table \[tab:global\], the outer disk starts at 5.7 au, and the gap, depleted in warm small grains, is au-sized ($\sim$ 3.2 au), which confirms the results of . Considering the stellar distance uncertainty ($\pm 27$ pc), the outer disk’s inner radius varies between 4.7 au and 6.7 au. The corresponding temperature variation is less than 5 K in the disk midplane and $\sim 10$ K in the higher disk surface. The gap width would range from 2.7 au to 3.8 au, which is still in the au-sized range. Therefore, we do not expect significant changes in the emission profile of the outer disk’s inner edge, hence no impact on the modeled MIR SED and visibilities. The best-fit [N]{}-band visibilities are shown in Fig.\[fig:radmcmodel\]. Both the level and the shape (sine-like modulation) in the measured visibilities are reproduced by our model. Figure \[fig:imagemodel\] shows the best-fit model image at $\lambda=10$ $\mu$m (right panel), which shows the extended inner disk and the ring-like morphology of the outer disk’s inner rim. Then, we estimate an upper limit on the dust mass that could lie within the gap. Gaps opened by substellar companions can in principle filter dust grains partially decoupled from the gas [e.g., @2006MNRAS.373.1619R]. Therefore, the dust in the gap and the inner disk likely differs from the outer disk. For the gap, we thus assumed the same dust composition (size distribution from 5 $\mu$m to 3000 $\mu$m for olivine grains and graphite fraction of 3%) and scale height profile as for the inner disk and considered the simplest case of a constant dust surface density profile ($p=0$). We then varied the amount of dust in the gap until the modeled MIR SED (at 4.8 $\mu$m, 12 $\mu$m and 25 $\mu$m) and/or the MIR visibilities start deviating by more than 3-$\sigma$ relative to the observations. As a result, we estimated an upper limit of $\sim 2.5\times10^{-10}$ M$_\odot$. Above this value, the modeled SED overestimated the observed SED at 4.8 $\mu$m by more than 3-$\sigma$, while two of the modeled MIR visibilities became lower, by more than 1-$\sigma$, than the MIDI visibilities between 11.5 and 13 $\mu$m. As the gap is being filled in, the amplitude of the sine-like modulation also decreases in the modeled MIR visibilities. Interestingly, the mass estimate inside the gap is comparable to the upper limit found for the inner disk. We recall that other mass reservoirs could lie within the gap, such as cold mm-sized grains, pebbles, or minor bodies, which do not significantly contribute to the disk IR emission. ### An outer disk with a “smooth” inner edge A $1.3\times10^{-4}$ M$_{\odot}$ outer dust disk with a flaring index $\beta=1/7$ and a scale height $H_{\rm out}/r_{\rm out}=0.135$ at $r_{\rm out}$ consistently reproduces the HERSCHEL/PACS and sub-mm photometry measurements. A 100% olivine dust composition with 0.1 $\mu{\rm m}<a<3$ mm was sufficient to account for the [SPITZER]{}/IRS measurements and its weak 10 $\mu$m silicate feature, along with the sub-mm spectral slope. Adding graphite in our model was not required. The best-fit broadband SED is presented in Fig.\[fig:radmcsed\].\ Our MIDI data favored a slightly rounded shape for the outer disk’s inner rim over a purely vertical wall. A reduction of the scale height of the dusty disk from $1.2 \times r_{\rm in}=7.4$ au to $r_{\rm in}=5.7$ au ($\epsilon=0.3$), leading to $H_{\rm in}/r_{\rm in}=0.04$ (instead of $H_{\rm in}/r_{\rm in}=0.085$), was needed to reproduce the modulation in the measured MIDI visibilities. A purely vertical wall induced too strong a modulation with the apparition of a pronounced lobe for the third visibility measurement. The rim shape could be explored further in a future study using hydrodynamical simulations to infer the mass of an hypothetical substellar companion, as previously done for HD100546 . Caveats and limitations ----------------------- First, we only explored a limited set of dust compositions. Considering crystalline silicates [@2010ApJ...721..431J] or other featureless species (e.g., iron) may influence the dust mass and change the disk temperature and brightness profile. However, the available data do not allow us to disentangle more complex dust compositions.\ Our best-fit solution assumed a radially and vertically homogeneous composition. Considering dust radial segregation or settling will affect the disk opacity profile and may lead to different scale heights and surface density profiles that are compatible with the data. However, the available dataset on HD 139614 currently lacks resolved mm observations that probe larger grains and cannot allow us to investigate dust segregation.\ With a disk orientation close to face-on, the assumption of isotropic scattering may have induced too much NIR flux being scattered in our line of sight and biased our estimation of the inner disk dust mass. However, this should be limited since the NIR scattering opacity is strongly dominated by the sub-$\mu$m-sized grains of our disk model. These grains are still in the Rayleigh regime, in the IR, and scatter almost isotropically.\ The 2-D axisymmetric geometry we assumed is relevant for identifying the main disk structural aspects but not possible asymmetries. However, this is not problematic here since the available closure phases do not suggest [strong]{} brightness asymmetries. We also considered a parameterized dust density that follows the prescription of a gaseous disk in hydrostatic equilibrium. A self-consistent computation of the disk vertical structure, as in , may have modified our derived dust scale height. However, this approach still relies on a thermal coupling between dust and gas and does not necessarly allow exploring vertical structures departing from the “hydrostatic” prescription. Finally, the power-law surface density we used is an approximation that allows trends in the radial dust distribution to be identified. It remains very simplified compared to the dust density profiles derived from hydrodynamical simulations coupling gas and dust even though the latter can approach power-law profiles in some cases . Discussion ========== A group I Herbig star with a pre-transitional disk -------------------------------------------------- Our new results on the gapped disk of HD139614 support the idea that most of the group-I Herbig objects are in the disk-clearing stage, as suggested by and confirmed recently by from a global analysis of all the MIDI data obtained on Herbig stars. Interestingly, find that Group I objects known to harbor very large gaps of tens up to a hundred au, such as HD 142527 [e.g., @2006ApJ...636L.153F], would also have a small gap in the inner region ($\leq 10~$au). Moreover, several supposedly gapless and settled Group II disks show hints of a very narrow gap in their inner ($\sim $ au) dust distribution. [HD139614 stands out here]{} since it is a rare, if not unique, case of Herbig star’s disk for which an au-sized gap ($\sim 3$ au) has been clearly spatially resolved in the dust distribution [see @2014prpl.conf..497E for a review on gap sizes]. MIR gaps correspond to local depletion in small sub-$\mu$m to $\mu$m-sized grains, which are good tracers of the gas . As a result, HD 139614 is a relevant laboratory for investigating gap formation and disk dispersal in the inner regions, which are only reachable by IR interferometry.\ ![[Radially integrated optical depth profile, at $\lambda=0.55$ $\mu$m (stellar emission peak), of the best-fit model. For clarity, we articially cut the integrated optical depth profile of the inner disk at $r_{\rm out}=2.55$ au.]{}[]{data-label="fig:taudisk"}](tau_disk.eps){width="70mm" height="44mm"} As shown in Sect. 4, self-shadowing by the inner disk does not convincingly reproduce the pretransitional features of the HD 139614’s SED and the interferometric data. Among the main disk clearing mechanisms, two could explain a narrow gapped structure: photoevaporation and planet-disk interaction [@2014prpl.conf..497E]. The photoevaporation scenario may be relevant for HD 139614 given its accretion rate , which is in the range of predicted photoevaporative mass-loss rates $ 10^{-10}$–$10^{-8}$ M$_{\odot}$/yr [@2014prpl.conf..475A]. Moreover, the photoevaporative wind is expected to first open a gap at the critical radius where the mass-loss rate is at its maximum [@2014prpl.conf..475A]. In the extreme-UV regime, this critical radius was estimated to be $R_{c,EUV} \simeq 1.8M_/M_{\odot}$ au. For HD 139614 ($M_=1.7$ M$_{\sun}$), $R_{c,EUV} \simeq 3.1$ au, which is consistent with the gap location. However, several aspects of the HD 139614 disk hardly appears consistent with this scenario: - the detection of warm molecular gas in the 0.3–15 au region from observations of the rovibrational emission of $^{12}$CO and its isotopologue lines with CRIRES (Carmona et al., in prep.). Preliminary results suggest at least 0.4 M$_{\oplus}$ and up to 2 M$_{\rm Jup}$ of gas within 6.5 au. Moreover, the CO lines present a clear double-peaked profile that is indicative of a gaseous disk in Keplerian rotation, which is not consistent with the presence of a photoevaporative molecular disk wind; - the non-detection of the \[OI\] line at 6300 $\AA$, which suggests the absence of a disk wind ; - the short timescale ($< 10^{5}$ yrs) expected for the inner disk dissipation (in gas and dust) after the photoevaporative wind has opened a gap [@2014prpl.conf..475A]. For these reasons, we then focus on the planet-induced gap scenario. Single massive planets are expected to carve a few au wide gap in the gas; the precise width and depth depending on the planet mass and disk viscosity [e.g., @2014prpl.conf..667B]. A similar clearing is expected for the gas-coupled small dust grains. Moreover, the tenuous HD 139614 inner disk suggests a drastic evolution and differentiation of the inner regions, which is possibly reminiscent of dust filtration by a planet on the gap’s outer edge [e.g., @2006MNRAS.373.1619R]. Interestingly, @Carmona2014 find similar differences for the HD135344B transition disk. To determine whether our observational constraints on the dust can sustain the planet-induced gap scenario, we performed a comparative study with hydrodynamical simulations of gap opening by a giant planet, adapted to the case of HD 139614. Investigation of disk-planet interaction ---------------------------------------- ### Hydrodynamical setup We use the code FARGO-2D1D[^6], which is designed to study the global evolution of a gaseous disk perturbed by a planet . Our planet is on a circular orbit of radius $a_{\rm pl}$ and does not migrate. We consider planet masses of 1, 2, and $4\times 10^{-3}\,M_$, with $M_$ the mass of the star. For HD 139614, this corresponds to 1.7, 3.4, and 6.8 M$_{\rm Jup}$. The 2D grid extends from $r=0.35\,a_{\rm pl}$ to $2.5\,a_{\rm pl}$ with $N_r=215$ rings of $N_s=628$ cells arithmetically spaced so that the resolution is ${\rm d}\phi={\rm d}r/r=0.01$ at the planet location, where $\phi$ is the azimuth. The 1D grid extends from $0.04\,a_{\rm pl}$ to $20\,a_{\rm pl}$ and has open boundary conditions to allow for disk spreading. The simulation uses a locally isothermal equation of state: $P={c_S}^2\Sigma$ with $P$ the pressure, $\Sigma$ the gas surface density, and $c_s=H\Omega$ the sound speed. Here, $c_s$ is fixed such that $H/r$ is constant, with $H$ the gas pressure scale height and $\Omega$ the Keplerian angular velocity. We set $H/r=0.04$ to be consistent with the $H/r$ value inferred for the dust at the outer disk’s inner edge. The disk scale height influences the gap’s depth but is not a critical parameter here, given our limited constraint on the dust gap depth. The initial density profile is $\Sigma=\Sigma_0(r/a_{\rm pl})^{-1}$. The gas viscosity is $\nu=10^{-5}\sqrt{GM_a_{\rm pl}}$, with $G$ the gravitational constant; at the planet location, the Reynolds number is $R=r^2\Omega/\nu=10^5$, which gives $\alpha=6.25\times 10^{-3}$ in the prescription, but here the viscosity is assumed to be independent of time and space. With this equation of state and the planet on a fixed orbit, the units are arbitrary: Regardless of the physical value of $a_{\rm pl}$ and $\Sigma_0$, the results scale accordingly. For easier comparison with the case of HD139614, we have set $a_{\rm pl}=4.5$ au, and $\Sigma=15$ g cm$^{-2}$ at 10 au as extrapolated from the dust surface density at 10 au ($\Sigma_{\rm dust}=0.15$ g cm$^{-2}$), assuming a gas-to-dust ratio of 100. Figure \[fig:surfacedensity\] shows the gas density profiles. ### Simulated gas density profile With the considered disk viscosity and scale height, a $\sim 3$ M$_{\rm Jup}$ planet at 4.5 au produces a gas gap between $\sim$ 3 and 6 au. This is consistent with the dust gap width, the inner disk’s outer radius ($r_{\rm out}=2.55\pm0.13$ au), and the outer disk’s inner radius ($r_{\rm in}=5.7$ au) of our best-fit dust model. The gas gap appears shallower than in the dust, which is expected since gas pressure minima are strongly depleted in dust [e.g., @2006MNRAS.373.1619R]. The 1.7 and 6.8 M$_{\rm Jup}$ planets produced a gap that was either too narrow ($\lesssim 2$ au) or too wide ($\geq 4$ au).\ Another noticeable aspect is the surface density profile. From the radiative transfer, we highlighted a radially increasing dust surface density profile in the inner disk, while the usual profile in $r^{-1}$ was kept for the outer disk. The gas surface density profile appears very consistent with this. In the inner disk, the gas surface density increases from $0$ at the inner edge until it reconnects to the original profile in $r^{-1}$ at $r>5$ au. Moreover, if fitted by a power law, the gas profile’s exponent in the inner disk ($\sim 0.6$) is similar to the dust profile exponent ($0.64$). This radially increasing gas surface density, which we highlighted for the small dust grains, too, is intrinsic to the disk viscous evolution. As shown in @1974MNRAS.168..603L and @2007MNRAS.377.1324C, such a profile naturally appears in the inner region of a viscously spreading gaseous disk, which has a finite inner edge that is not too close to the star (possibly $4\%$ of the planet’s orbital radius for HD 139614). Interestingly, a radially increasing gas density profile was required to describe the CO emission within the cavity of the HD135344B disk [@Carmona2014].\ After a gap has been opened, the gas surface density ratio between the inner and outer components is mainly ruled by the amount of gas that can cross the gap. Moreover, if the gap is opened close to the disk’s innermost edge, where the density profile is radially increasing, the inner disk’s surface density will be lower than in the outer disk [see Fig.6 of @2007MNRAS.377.1324C]. Figure \[fig:surfacedensity\] shows an inner disk surface density deficit of a factor 10 in gas, relative to the outer disk. This does not vary much as the whole disk viscously spreads and the gas density decreases as a whole. Even after 1 Myr, the inner disk is much less depleted in gas ($\sim 10$) than in dust ($\sim 10^3$). This suggests that a planet opening a gap consistent with our observations cannot affect the accretion flow enough to the inner regions. With the 6.8 M$_{\rm Jup}$ planet or more viscous disks, we could not deplete the inner disk any more. Having the planet on a fixed orbit actually helps the inner disk’s depletion since a planet migrating in type-II migration follows the disk viscous spreading without hampering it. Also, when the disk is less massive than the planet, no migration occurs, and the situation is similar to our simulation [see Eq.11 of @2007MNRAS.377.1324C]. However, during the disk evolution, dust grains grow and decouple from the gas, or fragment and stay coupled. The inner disk depletion may thus differ for the small dust grains and the gas, as shown hereafter. ### Grain growth and fragmentation in the inner disk The outer edge of a gap opened by a planet is a pressure maximum and can trap the partially gas-decoupled dust grains. In the inner disk, these grains drift inward and are eventually lost to the star. The radial drift velocity is highest when the Stokes number $\mathrm{St}=\Omega\rho_\mathrm{d}a/\rho_\mathrm{g}c_\mathrm{s} \sim 1$, where $\Omega$ is the Keplerian angular velocity, $\rho_\mathrm{d}$ the intrinsic dust density, $a$ the grain size, $\rho_\mathrm{g}$ the gas density, and $c_\mathrm{s}$ its sound speed. The optimal size for radial migration is $a_\mathrm{opt}=\Sigma_\mathrm{g}/\sqrt{2\pi}\rho_\mathrm{d}$ in the midplane . Since the gap acts as a dust filter, the dust-to-gas ratio in the outer disk is likely to be close to the interstellar value of 0.01, leading to $\Sigma_{\rm g}\sim10$ gcm$^{-3}$ at the gap’s outer edge (see Fig. \[fig:surfacedensity\]). Our simulations indicate a gas surface density that is ten times lower in the inner disk, which implies $a_\mathrm{opt} \simeq 1$ mm for typical values of $\rho_{\rm d}$ (1–3 gcm$^{-3}$). Grain growth is expected to be efficient in the inner disk regions and to bring small grains to mm or cm sizes. Once grains have grown up to sizes $\sim a_\mathrm{opt}$ in the inner disk, they migrate inwards quickly and sublimate when approaching the star. This could occur on a timescale of $10^2$ yr for cm-sized bodies at 1 au in a typical accreting disk . Only the small grain ($\mu$m-sized or smaller) reservoir, partly replenished by fragmentation, would stay coupled to the gas and remain in the inner disk longer. This depletion of the small grains reservoir will lead to a tenuous inner disk with reduced optical depth. This is supported by hydrodynamical simulations of a disk of gas and dust containing a planet, where dust grain growth, fragmentation, and dynamics are treated self-consistently [@Gonzalez2015]. Although the disk and planet parameters are different from ours, they show an inner disk that depletes more efficiently in dust than in gas. For most of the considered fragmentation velocities ($<20$ ms$^{-1}$), growth is fast for small grains but more difficult as they keep growing. Fragmentation either prevents grains from growing above $a_\mathrm{opt}$, promoting their fast migration, or breaks them down again to smaller sizes, thereby preventing their fast migration and partly replenishing the small grain reservoir. The inner dust disk thus drains out on timescales shorter than the viscous disk evolution and only keeps the remaining small grains and the larger bodies that could have overcome the radial-drift and fragmentation barriers. After $10^5$ yr, the inner-to-outer disk dust density ratio has dropped to $\sim10^{-3}$ and leads to a tenuous inner disk with a reduced optical depth, which agrees with our observational results. Quantitatively reproducing our observed dust disk with similar simulations is beyond the scope of the paper. Nevertheless, the qualitative agreement supports the scenario of a planet-induced gap in the HD139614 dust disk. Conclusion ========== This first multiwavelength modeling of the dust disk around HD 139614 provided the following results: - We confirmed a gap structure, between 2.5 and 5.7 au, depleted [(but not necessarily empty)]{} in warm sub-$\mu$m- and $\mu$m-sized grains. HD 139614 appears to be a rare case of a Herbig star with a narrow au-sized gap in its dust disk. - The NIR-emitting region was found to be spatially extended from 0.2 to 2.5 au, with a radially increasing surface density profile ($p>0$), a dust scale height of $H\sim0.01$ au at $r=0.2$ au, and a strong dust depletion of at least $\sim 10^3$ (in surface density) relative to the outer disk. This suggests a drastic evolution of the inner regions. - With a dispersion of $\lesssim 2-5^{\circ}$ around zero, the PIONIER closure phases does not suggest significant brigthness asymmetries. Notably, HD139614 is unlikely to be a tight binary system with a companion more massive than about 0.11 M$_{\odot}$. Currently available data and modeling do not support self-shadowing from the inner disk or photoevaporation as the origin of the dust gap in the HD 139614 disk. We thus tested whether the mechanism of disk/planet interaction could be viable against our constraints obtained on the dust from the radiative transfer. Assuming that small dust grains, probed by IR interferometry, are coupled to the gas, we performed hydrodynamical simulations of gap formation by a planet in a gaseous disk using the code FARGO-2D1D. It appears that: - For typical values of disk viscosity and scale height, a 3 $M_{\rm jup}$ planet, [fixed]{} at $\sim$4.5 au, could produce a gap in the gas consistent with the dust gap, although a bit shallower. - The radially increasing dust surface density in the regions interior to the gap is reproduced in the gas. The original gas density profile ($\propto r^{-1}$) is maintained in the outer disk for a long time. The radially increasing profile is predicted analytically and represents the accretion profile of the gas (and coupled dust grains) close to the inner disk rim. To our knowledge, it is the first time that this behavior is observationally identified in the innermost dust distribution of a disk. - The inner disk dust depletion ($\sim 10^{-3}$) is not reproduced in gas ($\sim 0.1$). However, this can be explained by the radial drift and growth and fragmentation processes affecting dust and is predicted by hydro simulations coupling gas and dust. All of this supports the hypothesis of a planet-induced gap in the HD139614’s dust disk and reinforces the idea that disks around Group-I Herbig stars are already in the disk-clearing transient stage. Confirming the planet-induced gap scenario will require further observations such as upcoming MIR imaging of the gap with the second-generation VLTI instrument MATISSE [@2014Msngr.157....5L]. Moreover, ALMA will be able to probe the mm grains and obtain a robust estimate of their mass at $r < 10$ au, while mapping the gas density distribution and dynamics inside and outside the gap. Although the HD139614 gap seems out of reach for VLT/SPHERE, direct imaging of the outer regions could be attempted to detect other signatures of disk-planet interaction. HD139614 will also constitute an exciting target for the future E-ELT/MICADO, PCS, and METIS instruments. Acknowledgments {#acknowledgments .unnumbered} =============== We thank the anonymous referee for the comments that helped to improve this manuscript significantly. A. Matter acknowledges financial support from the Centre National d’[É]{}tudes Spatiales [(CNES)]{}. J.-F. Gonzalez is grateful to the LABEX Lyon Institute of Origins [(ANR-10-LABX-0066)]{} of the Université de Lyon for its financial support within the program “Investissements d’Avenir” [(ANR-11-IDEX-0007)]{} of the French government operated by the ANR. This work is supported by the French ANR POLCA project [(Processing of pOLychromatic interferometriC data for Astrophysics, ANR-10-BLAN-0511)]{}. This work was supported by the Momentum grant of the MTA CSFK Lendület Disk Research Group. Parameter effects in radiative transfer modeling ================================================ In this section, we illustrate the individual effect of the main disk parameters on our radiative transfer modeling. From our best-fit model shown in Table \[tab:global\] and Figs. \[fig:radmcsed\] and \[fig:radmcmodel\], we vary each parameter separately around its best-fit value, and show the impact on the modeled observables (SED, NIR $V^2$, and MIR visibilities). In this way, we illustrate the effects of the main disk parameters and how much they can be constrained by our available dataset. Inner disk’s inner radius ------------------------- We show here the effect of varying the inner radius of the inner disk on the SED and the [H]{}-band and [K]{}-band $V^2$. As mentioned in Section 4.3.1, $r_{\rm in} < 0.2$ au values slightly increase the amount of unresolved inner disk emission. As shown in Fig. \[fig:rin\], this induces higher modeled [H]{}-band $V^2$ combined with a more gradual $V^2$ decrease as a function of spatial frequency. However, the modeled [K]{}-band $V^2$ become overestimated at the lowest spatial frequencies (smallest baselines). The $r_{\rm in} > 0.2$ au values induce too prominent a lobe in the PIONIER $V^2$ profile at high spatial frequencies. The modulation amplitude thus increases too much as we increase $r_{\rm in}$ above 0.2 au. Surface density $p$ exponent ---------------------------- We examine here the effect of varying the power-law exponent $p$ of the inner disk’s dust surface density on the NIR SED and $V^2$. As mentioned in Section 4.3.1 and shown in Fig. \[fig:p\], the models with $p \leq 0.3$ start inducing too strong a flux contribution from the inner disk rim and an overestimated NIR emission at $\lambda \lesssim 2$ $\mu$ in the modeled SED. This too spatially confined NIR-emitting region implies [H]{}-band and [K]{}-band $V^2$ that is too high at low spatial frequencies and $V^2$ that is too low at high spatial frequencies since the stellar-to-total flux ratio is lower than in the case of our best-fit model, especially at $\lambda \lesssim 2$ $\mu$. Inner disk’s outer radius ------------------------- \ We examine here the effect of varying the outer radius $r_{\rm out}$ of the inner disk. In addition to the NIR SED and $V^2$, we also consider that the [N]{}-band visibilities illustrate the constraint provided by the MIDI data on the extension of the inner disk. Figure \[fig:rout\] shows that decreasing or increasing the size of the inner disk by at least 0.5 au, relative to our best-fit model, has a noticeable effect on all the observables. A smaller inner disk will induce too high [N]{}-band visibilities especially between 8 and 9.5 $\mu$m, while larger and over-resolved inner disks ($r_{\rm out} \geq 3$ au) will produce [N]{}-band visibilities that are too low and flatter. The effect on the [H]{}-band and [K]{}-band $V^2$ is directly correlated to the change in the NIR emission level in the modeled SED. Indeed, a smaller inner disk ($r_{\rm out} \lesssim 2.0$ au) is more optically thick (for the same dust mass) and will produce more NIR emission. This implies an overestimation of the disk contribution and consequently a lower stellar-to-total flux ratio, which will then induce lower NIR $V^2$ (see Fig. \[fig:rout\]), especially at long baselines where the inner disk is fully resolved. For a larger inner disk ($r_{\rm out} \geq 3$ au), the effect on the observables is the opposite. Inner disk scale height ----------------------- We show in Fig. \[fig:h\] the effect of varying the dust scale height $H_{\rm out}$ of the inner disk at its outer radius $r_{\rm out}$. A slight departure from the best-fit value $H_{\rm out}/r_{\rm out}=0.1$ has a noticeable impact on all the NIR observables. Indeed, increasing the scale height increases the amount of stellar flux captured and reprocessed by the inner disk and thus the NIR emission level. This implies a decrease in the stellar-to-total flux ratio and therefore lower NIR $V^2$, especially at high spatial frequencies. We thus quickly underestimate the $V^2$ level in [H]{} and [K]{} bands. On the other hand, slightly reducing the dust scale height to $H_{\rm out}/r_{\rm out} \simeq 0.07$ produces modeled SED and NIR $V^2$ that are still consistent with the observations. Outer disk’s inner radius ------------------------- Finally, we show in Fig. \[fig:h\] the effect of varying the inner radius of the outer disk $r_{\rm in}$ on the MIR visibilities, which are directly probing this region of the HD 139614 disk. Indeed, it appears that a deviation by more than $\sim 0.3$ au, which is the 1-$\sigma$ uncertainty on $r_{\rm in}$ (see Table \[tab:global\]), induces a noticeable shift in the sine-like modulation in the modeled MIR visibilities. The latter are no longer consistent with the measured MIDI visibilities, although the visibility level remains similar. No significant effect can be seen in the MIR SED. [^1]: Corresponding author : Alexis.Matter@oca.eu [^2]: Present address: Observatoire de la Côte d’Azur, boulevard de l’Observatoire, CS 34229, 06304 Nice, France. [^3]: Based on observations collected at the European Southern Observatory, Chile (ESO IDs : 385.C-0886, 087.C-0811, 089.C-0456, and 190.C-0963). [^4]: Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA. [^5]: Available at http://www.astro.uni-jena.de/Laboratory/OCDB/ [^6]: Available at `http://fargo.in2p3.fr/spip.php?rubrique16`
Introduction ============ Modulational instability of high-frequency nonlinear waves is a common process in a variety of circumstances involving wave propagation in continuous systems. Modulational processes can be seen to occur in a wide range of physical situations, from nonlinear waves in plasmas [@dter78] to nonlinear electromagnetic waves propagating in optical fibers [@mal97]. What usually happens in all those cases is that due to generic nonlinear interactions, the amplitude of a high-frequency carrier develops slow modulations in space and time. If the modulations are indeed much slower than the high-frequencies involved, one can obtain simplified equations describing the dynamics of the slowly varying amplitudes solely, the amplitude equations [@lili91; @novo99; @olig96; @riz98; @erich98]. In the present analysis we consider systems that become integrable in this modulational limit, a feature often displayed. Should this be indeed the case, no spatiotemporal chaos would be observed there. The basic interest then would be to see what happens when the approximations leading to modulational approximations cease to be satisfied. The paper is organized as follows: in §2 we introduce our model equation and discuss how and when it can be approximated by appropriate amplitude equations; in §3 we investigate the modulational process from the point of view of nonlinear dynamics; in §4 we perform full spatiotemporal simulations and compare the results with those obtained in §3; and in §5 we conclude the work. Model equation, modulational approximations, and amplitude equations ==================================================================== In the present paper we focus attention on a nonlinear variant of the Klein-Gordon equation (NKGE) to investigate the breakdown of modulational approximations in the context of nonlinear wave fields. The NKGE used here reads $$\partial_t^2 A(x,t) - \partial_x^2 A(x,t) + {\partial \Phi \over \partial A}=0, \label{eqzero}$$ ($\partial_t \equiv \partial / \partial t, \> \partial_x \equiv \partial / \partial x$) where we write the generalized nonlinear potential as $$\Phi(A) = \omega^2 \> {A(x,t)^2 \over 2} - {A(x,t)^4 \over 4} + {A(x,t)^6 \over 6}, \label{phi}$$ $\omega$ playing the role of a linear frequency which will set the fast time scale. The remaining coefficients on the right-hand-side were chosen to allow for modulational instability and saturation; we shall see that while the negative sign of the second fulfills the condition for modulational instability, the positive sign of the third provides saturation. The choice of their numerical values is arbitrary, but our results are nevertheless generic. The NKGE is known to describe wave propagation in nonlinear media and the idea here is to see how the dynamics changes as a function of the parameters of the theory: wave amplitude, and time and length scales. Let us first derive the conditions for slow modulations. We start by supposing that the field $A(x,t)$ be expressed in the form $$A(x,t) = \tilde A (x,t) e^{i \omega t} + {\rm complex\>\>conjugate}. \label{fatora}$$ Then, if one assumes slow modulations and discards terms like $\partial_t^2 \tilde A$ and the highest-order power of the potential $\Phi$, one obtains $$2 i \omega \partial_t \tilde A(x,t) - \partial_x^2 \tilde A(x,t) - 3 \|\tilde A(x,t)\|^2 \tilde A(x,t) = 0, \label{schroclassica}$$ which is, apart from some rescalings, the Nonlinear Schrödinger Equation - we shall refer to it as NLSE here - an integrable equation. This is no novelty; it is known that the modulational approximation is obtainable when there is a great disparity between the time scales of the high-frequency $\omega$ and the modulational frequency, we call it $\Omega$, such that terms of order $\Omega^2 \tilde A$, when compared to $\omega^2 \tilde A$, can be dropped from the governing equation. The magnitude of the modulational frequency can be estimated as follows. Consider Eq. (\[schroclassica\]) and suppose a democratic balance among the magnitude of its various terms; $\omega \partial_t \tilde A \sim \partial_x^2 \tilde A \sim {\tilde A}^3$. Then one obtains for $\partial_t \rightarrow \Omega$, $${\Omega \over \omega} \sim ({\tilde A \over \omega})^2. \label{condition}$$ It is thus clear that the modulational approximation is valid only when $\tilde A \ll \omega$, since this condition slows down the modulational process causing $\Omega \ll \omega$. The next question would be on what is to be expected when the modulational approach ceases to be valid. Before proceeding along this line, let us mention that a stability analysis can be performed on Eq. (\[schroclassica\]). One perturbs an homogeneous self-sustained state with small fluctuations of a given wavevector $k>0$ (we choose $k>0$ here, but the theory is invariant when $k \rightarrow - k$) and after some algebra one concludes that: (i) the field in the homogeneous state, let us call this field $A_h$, is given by $$A_h = a_o e^{- i {3 a_o^2 \over 2 \omega} t} \label{comp1}$$ where $a_o$ is an arbitrary amplitude parameter and where the exponential term should be seen as providing a small nonlinear correction to the linear frequency $\omega$; and (ii) the perturbation is unstable when $$k < k_{tr} \equiv \sqrt{6} a_o. \label{modinst}$$ with maximum growth rate at $$k_{max} \equiv {k_{tr} \over \sqrt{2}}. \label{maxgrowth}$$ When unstable, the homogeneous state typically evolves towards a state populated by regular structures which can be formed precisely because the underlying governing NLSE, Eq. (\[schroclassica\]), is of the integrable type as mentioned earlier. Beyond the modulational approximation ===================================== To advance the analysis beyond the modulational regimes we start from the basic equation, Eq. (\[eqzero\]), but do not use approximation $\partial_t^2 \tilde A \ll \omega^2 \tilde A$ leading to Eq. (\[schroclassica\]) and eventually to condition $\tilde A \ll \omega$. The idea is precisely to examine what happens as the ratio $\tilde A / \omega$ of Eq. (\[condition\]) grows from values much smaller, up to values comparable to the unit. Our first task is to examine how the regular and well known modulational instability analyzed in the previous section comes directly from Eq. (\[eqzero\]). To do that, let us write a truncated solution as the sum of an homogeneous term plus fluctuations with wavevector $k$, $A = A_h(t) + A_1(t) (e^{i k x} + e^{-i k x})$, $A_h$ and $A_1$ real. The truncation, that discards higher harmonics, is legitimate within linear regimes, but fortunately we shall see that it is not as restrictive as it might appear even in nonlinear regimes. The general idea favouring truncation here is that in the reasonable situation where modes with the fastest growth rates are more strongly excited, relation (\[maxgrowth\]) indicates that second harmonics are already outside the instability band since $2 k_{max} = \sqrt{2} k_{tr} > k_{tr}$. Under these circumstance one would be led to think that the most important modes would be the homogeneous and those at the fundamental spatial harmonic. We shall actually see that the truncation provides a nice and representative approach to the case of regular regimes. Note that we are already considering the amplitudes of the exponential functions as equal. This results from a simplified symmetrical choice of initial conditions and is totally consistent with the real character of Eq. (\[eqzero\]). After some lengthy algebra, one finds out that the coupled nonlinear dynamics of the fields $A_h$ and $A_1$ is governed by the Hamiltonian $$H={p_h^2 \over 2} + {\omega^2 q_h^2 \over 2} - {q_h^4 \over 2} + {2 q_h^6 \over 3} + {p_1^2 \over 2} + {\chi^2 q_1^2 \over 2} - {3 q_1^4 \over 4} + {5 q_1^6 \over 3} -$$ $$3 q_h^2 q_1^2 + 10 q_h^4 q_1^2 + 15 q_h^2 q_1^4, \label{hamiltonian}$$ where $\chi^2 \equiv \omega^2+k^2$, $q_h = A_h/\sqrt{2}$, $q_1 = A_1$, and where the $p$’s denote the two momenta conjugate to the respective $q$-coordinates. The Hamiltonian (\[hamiltonian\]) can be informative. As a first instance it can be used to determine the stability properties of the homogeneous pump, as mentioned before. To see this, assume that in average $q_h \gg q_1$ and solve the dynamics perturbatively. In zeroth order, one would have the following Hamiltonian governing the $(p_h,q_h)$ dynamics: $$h_o = {p_h^2 \over 2} + {\omega^2 q_h^2 \over 2} - {q_h^4 \over 2} + {2 q_h^6 \over 3}. \label{ho}$$ With help of action-angle variables ($\rho,\Theta$) for the zeroth-order part and conventional perturbative techniques the solution reads $$q_h = \sqrt{2 \rho \over \omega} \cos(\omega t - {3 \over 2} {\rho \over \omega^2} t), \label{ordemzero}$$ if $\rho$, the amplitude parameter, is not too large. Note that the oscillatory frequency undergoes a small nonlinear correction which is determined by the quartic term in $q_h$ of the Hamiltonian $h_o$. Next we consider the driven Hamiltonian controlling the dynamics of the canonical pair $(p_1,q_1)$ $$h_1 = {p_1^2 \over 2} + {\chi^2 q_1^2 \over 2} + 3\>q_h^2 q_1^2, \label{h1}$$ where we recall that the pair $(p_1,q_1)$ describes the inhomogeneity of the system. One again introduces action-angle variables $(I,\theta)$ to rewrite the linear Hamiltonian (\[h1\]) in the form $$h_1 = \chi I + 12 {\rho \over \omega} {I \over \chi} \cos^2 \theta \cos^2 (\omega t - {3 \over 2} {\rho \over \omega^2} t), \label{h11}$$ from which we obtain the resonant form $$h_{1,r} = {1 \over 2 \omega^2}(k^2 - 3\>\rho) I + {3\>\rho I \over 2 \omega^2} \cos (2 \phi), \label{hres}$$ with $\phi = \theta -(\omega - 3\rho / 2\omega^2)t$, and where use is made of the approximation $\chi \approx \omega + k^2/(2 \omega)$ valid when $k \ll \omega$. Now consider the stability of a small perturbation $I \sim 0$. For this initial condition $h_{1,r} \rightarrow 0$. Since $h_{1,r}$ is a constant of motion, solutions for arbitrarily large values of $I$, what would indicate instability, are possible only when $\|\cos(2 \phi)\| \le 1$. From the resonant Hamiltonian (\[hres\]), this demands $$k < \sqrt{6 \rho \over \omega}, \label{cond2}$$ and also indicates that maximum growth rate for $I$ occurs at $k_{max}=\sqrt{3 \rho / \omega}$. Comparisons of temporal dependence of Eqs. (\[fatora\]), (\[comp1\]) and (\[ordemzero\]) shows that $\rho / \omega = a_o^2$, and that conditions (\[modinst\]) and (\[cond2\]) are therefore one and the same as they should be. In other words, starting from the full nonlinear wave equation, we recover the typical results naturally yielded by the NLSE. But the Hamiltonian (\[hamiltonian\]) gives further information because apart the truncation, it is not an adiabatic approximation like Eq. (\[schroclassica\]); it can therefore tell us whether the reduced dynamics, if unstable, is of the regular or chaotic type. The interest on this issue comes from the fact that the reduced dynamics usually helps to determine the spatiotemporal patterns of the full system: while regular reduced dynamics is associated with regular structures - frequently a collection of solitons or soliton-like structures, chaotic reduced dynamics is associated with spatiotemporal chaos. The correlation has its roots on the so called stochastic pump model [@lili91]. The model states that intense chaos in a low-dimensional subsystem of a multidimensional environment can make the subsystem act like a thermal source, irreversibly delivering energy into others degrees-of-freedom. In the limit of a predominantly regular dynamics undergoing in the reduced system, irreversibility is greatly reduced and energy tends to remain confined within the subsystem. On the other hand, in the limit of deeply chaotic dynamics with no periodicity at all, energy flow out of the subsystem is fast and there is not even much sense in defining the subsystem as an approximately isolated entity. The intermediary cases are those more amenable to a description in terms of the stochastic pump [@lili91; @novo99]. We point out, however, that in all cases, even in the chaotic one, analysis of the reduced subsystem serves as an orientation on what to be expected in the full spatiotemporal dynamics. In our case, the appropriate subsystem is precisely the one we have been using. This is so because it is the smallest subsystem comprising the most important ingredients of the dynamics: the homogeneous and the only linearly unstable modes. We now examine these points in some detail. Let us consider $\omega = 0.1$. When $A_{h,o} \ll \omega$, $A_{h,o} \equiv A_h(t=0)$, the modulational approximation can be used to determine the instability range, $k < k_{tr}$. A surface of section plot based on the two-degrees-of-freedom Hamiltonian (\[hamiltonian\]) produces Fig. (\[fig1.ps\]), where we record the values of $(q_1,p_1)$ whenever $p_h = 0$ with $dp_h/dt > 0$, and where we take $q_1 = 0.01 q_h \ll q_h$ and $p_h = p_1 = 0$ to determine the total unique energy of the various initial conditions we launch in the simulations. Note that the particular “seed” initial condition introduced above is included in the ensemble of initial conditions launched and represents small perturbations to an homogeneous background. In Fig. (\[fig1.ps\]a) we take $A_{h,o}/\omega = 0.5$ and $k = 2 k_{max} > k_{tr}$. One lies outside the instability range and the figure reveals that the origin, which represents the purely homogeneous state $q_1=A_1=0$, is indeed stable. The remaining panels are all made for $k=k_{max}$ and increasing values of the ratio $A_{h,o}/\omega$. One sees that for such values of $k$ and $A_{h,o}$ not only the origin is rendered unstable, as it also becomes progressively surrounded by chaotic activity. The inner chaotic trajectory issuing from the origin - i.e., the trajectory representing the modulational instability - is encircled by invariant curves, but the last panel already shows that some external chaos, here represented by scattered points around the invariant curves, is also present. At some amplitude $A_{h,o} = A_{critical}$ slightly larger than the one used in panel (d), invariant curves are completely destroyed. Above the critical field, inner orbits are no longer restricted to move within confined regions of phase-space - these orbits are in fact engulfed by the external chaos seen in panel (d). External chaos here is a result of the hyperbolic point positioned at the local maximum of the generalized potential $\Phi(A)$, at $A_1 \sim 0.1$. One gross determination of the critical field based on simple numerical observation yields $A_{critical}/\omega \sim 1/1.62$, although the corresponding transition from order to disorder in the full simulations may not be so sharply defined. Yet, one may expect that in both low-dimensional and full simulations, the transition should occur at comparable amplitudes. Full spatiotemporal simulations =============================== To make the appropriate comparisons, we now look into the full simulation of Eq. (\[eqzero\]). Full simulations are made through a discretization of the spatial domain via a finite difference method. The dynamics is resolved temporally by means of a sympletic integrator, as was the purely temporal Hamiltonian. The results are quite robust and energy is conserved up to $10^{-6}$ parts in one. In Fig. (\[fig2.ps\]) we display the spacetime history of the quantity $Q = \sqrt{\omega^2 A(x,t)^2 + \dot A(x,t)^2}$ - we plot this quantity because it becomes a constant of motion in the limit where we discard nonlinearities and inhomogeneities. Initial conditions are the same as the seed initial condition used in Fig. (\[fig1.ps\]), but the control parameters differ. In the case of Fig. (\[fig2.ps\]a) where we choose $A_{h,o}/\omega = 0.1$ so as to safely satisfy $A_{h,o} \ll A_{critical}$ and $A_{h,o} \ll \omega$, it is seen that the spatiotemporal dynamics is very regular. The homogeneous state is unstable, but only periodic spatiotemporal spikes can be devised. This is the regular spatiotemporal dynamics so typical of the integrable NLSE and this kind of dynamics agrees very well with the low-dimensional predictions of the reduced Hamiltonian (\[hamiltonian\]). The fact that only one single structure can be found along the spatial axis at any given time, indicates that the dynamics is singly periodic (period-1) along this axis and thus basically understood in terms of the reduced number of active modes (homogeneous plus fundamental harmonics) of Hamiltonian (\[hamiltonian\]). We now move into the vicinity of the critical amplitude $A_{critical}$ discussed earlier. Under such conditions, one may expect to see the effects of spatiotemporal chaos. The value $A_{h,o}/\omega = 1/1.625$ is chosen in Fig. (\[fig2.ps\]b), where full simulations indeed display a highly disorganized state after a short regular transient. Regularly interspersed spikes can no longer be seen and spatial and temporal periodicities are lost, which characterizes spatiotemporal chaos. In this regime many modes become active (we shall return to this point later) and the reduced Hamiltonian (\[hamiltonian\]) fails to provide an accurate description of the full dynamics. However it still provides a good estimate on the point of transition. A more throughful examination of the transition in the full simulations suggests that the critical field there is a bit smaller - a value close to $\omega / 1.95$; for smaller values we have not observed noticeable signals of spatiotemporal chaos even for much longer runs than those presented here. The much larger oscillations executed by $Q$ in chaotic cases (see the legend of Fig. (\[fig2.ps\])) is a direct result of the destruction of the invariant curves seen in Fig. (\[fig1.ps\]). In the absence of invariant curves, initial conditions are no longer restricted to move near the origin. It is thus seen that there are limits to an integrable modulational description of the dynamics of a wave field. The limits are essentially set by the parameter $A_{h,o} / \omega$. If it is much smaller than the unit, the modulational description is valid and one can expect to see a collection of spatiotemporal periodic structures being formed as asymptotic states of the dynamics. On the other hand, as the parameter approaches the unit, nonintegrable features are likely to be seen. In particular, regularity does not survive for very long, and fluctuations with various length scales appear in the system. This is a regime of spatiotemporal chaos which fundamentally involves the presence of nonlinear resonances between the frequency of the carrier, $\omega$, and the intrinsic nonlinear modulational frequency, $\Omega$. Due to the presence of chaos, one is suggested that the transition involves an irreversible energy flow out of the reduced subsystem. If one computes the average number of active modes $$<N^2> \equiv {\sum_n n^2 \|A_n\|^2 \over \sum_n \|A_n\|^2}, \label{soma}$$ where the amplitudes are defined in the form $$A_n = \sum_j A(x_j,t) e^{i n k x_j}, \label{aene}$$ one obtains the plots shown in Fig. (\[fig3.ps\]). In Eq. (\[aene\]), $j$” is the discretization index. In Fig. (\[fig3.ps\]a) we use the same conditions as in Fig. (\[fig2.ps\]a). This panel shows that in rough terms, energy keeps periodically migrating between the homogeneous mode (when $\sqrt{<N^2>} \sim 0$) and the fundamental harmonic (when $\sqrt{<N^2>} \sim 1$). Conditions of Fig. (\[fig3.ps\]b) are the same as those of Fig. (\[fig2.ps\]b); one sees that as the ratio $A_{h,o}/\omega$ grows, periodicity is lost, and that energy flow out of the initial subsystem into other modes becomes clearly irreversible. In Fig. (\[fig3.ps\]c) we use the same previous conditions with exception of $A_{h,o}$ which we take $A_{h,o} = \omega / 1.95$. This slightly smaller, but not too small, value of the initial amplitude allows to observe the slow diffusive transit of energy during the initial stages of the corresponding simulations. During this stage one can actually look at the subsystem as an energy source adiabatically delivering energy into other modes - the concept of the stochastic pump applies more appropriately in those situations. Final conclusions ================= To summarize, in this paper we have studied the breakdown of modulational approximations in nonlinear wave interactions. We have analyzed a nonlinear Klein-Gordon equation to draw the following conclusions. Adiabatic or modulational approximations are accurate while the high-frequency of the carrier wave keeps much larger than the modulational frequency. Under these circumstances the full spatiotemporal patterns are regular as is the dynamics in the reduced subsystem where the system energy is initially injected. There is no net flow of energy out of the reduced subsystem into the remaining modes. On the other hand, when both frequencies become of the same order of magnitude, the reduced subsystem undergoes a transition to chaos. Correspondingly, the spatiotemporal patterns of the full system become highly disordered and energy spreads out over many modes. The correlation between the low-dimensional and high-dimensional spatiotemporal chaos has its roots on the stochastic pump model [@lili91]. According to the model, a low-dimensional chaotic subsystem may act like a thermal source, delivering energy in a irreversible fashion to other degrees-of-freedom of the entire system. Spectral simulations performed here indicates that this seems to be the case with the present setting. The transition to chaos involves a noticeable increase in terms of wave amplitude. This takes place when the invariant curves of Fig. (\[fig1.ps\]) are destroyed, allowing for the merge of the external and internal chaotic bands into one extended chaotic sea. This merging of chaotic bands is actually a result of reconnections involving the manifolds of the unstable fixed point at the origin, and other hyperbolic points associated with the curvature of the generalized potential $\Phi$ [@corso98]. When both manifolds reconnect, inner trajectories issuing from the origin start to execute the large and irregular oscillations seen in Fig. (\[fig2.ps\]). More detailed studied of the process is under current analysis. This work was partially supported by Financiadora de Estudos e Projetos (FINEP), Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), and Fundação da Universidade Federal do Paraná - FUNPAR, Brazil. S.R. Lopes wishes to express his thanks for the hospitality at the Plasma Research Laboratory, University of Maryland. Part of the numerical work was performed on the CRAY Y-MP2E at the Supercomputing Center of the Universidade Federal do Rio Grande do Sul. S.G. Thornhill and D. ter Haar, Phys. Reports [43]{}, 43 (1978). B. Malomed, D. Anderson, M. Lisak, and M.L. Quiroga-Teixeiro, Phys. Rev. E [55]{} 962 (1997). A.J. Lichtenberg and M.A. Lieberman, [Regular and Chaotic Motion]{}, Springer (1991). S.R. Lopes and F.B. Rizzato, Phys. Rev. E [Nonintegrable Dynamics of the Triplet-Triplet Spatio-Temporal Interaction]{} submitted (1999). G.I. de Oliveira, L.P.L. de Oliveira, and F.B. Rizzato, Phys. Rev. E [54]{}, 3239 (1996). F.B. Rizzato, G.I. de Oliveira, and R. Erichsen, Phys. Rev. E [57]{}, 2776 (1998). R. Erichsen, G.I. de Oliveira, and F.B. Rizzato, Phys. Rev. E [58]{}, 7812 (1998). S.R. Lopes and F.B. Rizzato, Physica D [117]{}, 13 (1998). G. Corso and F.B. Rizzato, Phys. Rev. E [58]{}, 8013 (1998).
--- author: - 'T. Ryabchikova' - 'M. Sachkov' - 'O. Kochukhov' - 'D. Lyashko' date: 'Received / Accepted ' subtitle: Resolving the third dimension in peculiar pulsating stellar atmospheres title: 'Pulsation tomography of rapidly oscillating Ap stars[^1]' --- Introduction {#intro} ============ About 10% to 20% of upper main sequence stars are characterised by remarkably rich line spectra, often containing numerous unidentified features. Compared to the solar case, overabundances of up to a few dex are often inferred for some iron-peak and rare-earth elements, whereas some other chemical elements are found to be underabundant (Ryabchikova et al. [@RNW04]). Some of these chemically peculiar (Ap) stars also exhibit organised magnetic fields with a typical strength of a few kG. Chemical peculiarities are believed to result from the influence of the magnetic field on the diffusing ions, possibly in combination with the influence of a weak, magnetically-directed wind (e.g., Babel [@Babel92]). More than 30 cool Ap stars exhibit high-overtone, low-degree, non-radial $p$-mode pulsations with periods in the range of 6–21 minutes (Kurtz & Martinez [@KM00]), with their observed pulsation amplitudes modulated according to the visible magnetic field structure. These rapidly oscillating Ap (roAp) stars are key objects for asteroseismology, which presently is the most powerful tool for testing theories of stellar structure and evolution. Recent progress in observational studies of roAp stars has been achieved by considering high time-resolution spectroscopy in addition to the classical high-speed photometric measurements. High-quality time-resolved measurements of magnetic pulsators have revealed a surprising diversity in the pulsational behaviour of different lines in the roAp spectra (e.g., Kanaan & Hatzes [@KH98]; Kurtz et al. [@KEM06] and references therein). Detailed analyses of the bright roAp star $\gamma$ Equ (Savanov et al. [@SMR99]; Kochukhov & Ryabchikova [@KR01a]) demonstrate that spectroscopic pulsational variability is dominated by the lines of rare-earth ions, especially those of Pr and Nd, which are strong and numerous in the roAp spectra. On the other hand, light and iron-peak elements do not pulsate with amplitudes above 50–100 [ms$^{-1}$]{}, which is at least an order of magnitude lower in comparison with the 1–5 [kms$^{-1}$]{} variability observed in the lines of rare-earth elements (REE). Many other roAp stars were proven to show very similar overall pulsational behaviour (e.g., Kochukhov & Ryabchikova [@KR01b]; Balona [@Balona02]; Mkrtichian et al. [@MHK03]; Kurtz et al. [@KEM05a]), with an exceptional case in 33 Lib, which shows small-amplitude variations in Fe lines, and possibly two other stars, HD 12932 and HD 19918, where RV variations in the Fei 5434.52 Å line have been marginally detected (Kurtz et al. [@KEM05b]). The early spectroscopic studies speculated that the unusual diversity of the pulsation signatures in roAp spectra can be attributed to an interplay between the $p$-mode pulsation geometry and inhomogeneous horizontal or vertical element distributions (see discussions in Kanaan & Hatzes [@KH98]; Savanov et al. [@SMR99]; Baldry & Bedding [@BB00]; Mkrtichian et al. [@MHP00]). However, none of these studies present models capable of explaining pulsations in real stars. Peculiar characteristics of the $p$-mode pulsations in a roAp star were finally clarified by Ryabchikova et al. ([@RPK02]), who were the first to empirically determine vertical stratification of chemical elements and relate chemical profiles to pulsational variability. In their study of the atmospheric properties of $\gamma$ Equ, Ryabchikova et al. show that the light and iron-peak elements are enhanced in the lower atmospheric layers, whereas REE ions are concentrated in a cloud with a lower boundary at $\log\tau_{5000}\la-4$ (Mashonkina et al. [@MRR05]). Thus, high-amplitude pulsations observed in REE lines occur in the upper atmosphere, while lines of elements showing no significant variability form in the lower atmosphere. This leads to the following general picture of roAp pulsations: we observe a signature of a magnetoacoustic wave, propagating outwards through the chemically stratified atmosphere with increasing amplitude. In addition to remarkable pulsational behaviour, the REE lines formed in the upper atmospheric layers of roAp stars exhibit an extra broadening, corresponding to a macroturbulent velocity =10 [kms$^{-1}$]{}, which cannot be attributed to the chemical stratification or magnetic field effects (Kochukhov & Ryabchikova [@KR01a]). In the recent detailed line-profile variability study, Kochukhov et al. ([@KR07]) have presented evidence for the existence of peculiar asymmetric oscillation patterns in the broad REE lines of several roAp stars. It was demonstrated that the inferred pulsation signatures can be reproduced with the spectrum synthesis calculations, which contain an extra pulsational line width variability in addition to the usual velocity perturbations. These results suggest that a turbulence zone modulated by pulsations probably exists in the upper atmospheres of roAp stars. The presence of significant phase shifts between pulsation radial velocity (RV) curves of different REEs (Kochukhov & Ryabchikova [@KR01a]) or even lines of the same element (Mkrtichian et al. [@MHK03]) can be attributed to the chemical stratification effects and, possibly, to a short vertical length of running magnetoacoustic wave. However, it is not immediately clear whether all spectroscopic observations of roAp stars can be fitted into this simple picture and to what extent magnetoacoustic pulsation theories can explain these observations. The wide diversity of pulsation signatures (in particular, phases and bisector variability) of the REE lines probing similar atmospheric heights and, especially, the presence of pulsation node in the atmosphere of 33 Lib (Mkrtichian et al. [@MHK03]) are inexplicable in the framework of the non-adiabatic pulsation models developed by Saio & Gautschy ([@SG04]) and Saio ([@S05]). These calculations correctly predict an increase in pulsation amplitude with height, but show neither nodes nor rapid phase changes in the REE line-forming region. To resolve this discrepancy it is therefore imperative to analyse in detail pulsational variations of many different ions in the spectra of representative sample of roAp stars. Only in this way can we derive meaningful observational constraints for pulsation theories and search for regular patterns in the pulsation characteristics of different roAp stars. First general results for roAp stars were presented by Kurtz at al. ([@KEM05a]). They considered bisector behaviour of the H$\alpha$ core and the Nd6145 Å line in 10 roAp stars. Based on the H$\alpha$ core measurements, the authors point out that the increasing amplitude with height in the atmosphere is a common characteristic of all stars in their sample. Kurtz et al. ([@KEM05a]) also note that in some stars, e.g. HD 12932, pulsations have a standing-wave behaviour, while more complex pulsations are observed in some other stars. These results were obtained from the analysis of bisector variability of one particular line, Nd6145 Å. This feature is not optimal for the bisector analysis because of the different blending effects both in the line core and the line wings that depend on the effective temperature and/or chemical anomalies. In this study we have embarked on the task of obtaining a detailed vertical cross-section of the roAp pulsation modes. We used the unique properties of roAp stars, in particular chemical stratification, to resolve the vertical structure of $p$-modes and to study propagation of pulsation waves at a level of detail, that was previously only possible for the Sun. Our pulsation tomography approach consists of characterising the pulsational behaviour of a carefully chosen, but extensive, sample of spectral lines including weak ones and subsequently interpretating the observations in terms of pulsation wave propagation. Therefore, our sample was limited to slowly rotating roAp stars. The aim of this analysis is to derive observationally a complete picture of how the amplitude and phase of magnetoacoustic waves depend on depth and to correlate resulting vertical mode cross-sections with other pulsational characteristics and with the fundamental stellar parameters. Furthermore, we envisage that unique 3-D maps of roAp atmospheres and non-radial pulsations can be derived by combining pulsation tomography results presented here with the horizontal pulsation maps obtained with Doppler imaging (Kochukhov [@K04]) or moment analysis (Kochukhov [@K05]). Our paper is organised as follows. Section \[observ\] describes acquisition and reduction of roAp time-series spectra. The choice of targets and physical properties of the stellar sample are discussed in Sect. \[parameters\]. Radial velocity measurements and period analysis are presented in Sect. \[RV\]. Pulsation tomography results are summarised in Sect. \[ph-a\]. Bisector variability is analysed in Sect. \[bis\]. Results of our study are summarised and discussed in Sect. \[disc\]. Observations and data reduction {#observ} =============================== ---------- ------------ ------------ ----------- ---------- ---------- ------- -------------------------- Star Start HJD End HJD Number of Exposure Overhead Peak Telescope/Instrument (2450000+) (2450000+) exposures time (s) time (s) $S/N$ (observing mode) HD9289 2920.54506 2920.62881 111 40 25 90 VLT/UVES (600 nm) HD12932 2921.62234 2921.70532 69 80 25 90 VLT/UVES (600 nm) HD19918 2921.52607 2921.60905 69 80 25 100 VLT/UVES (600 nm) HD24712 3321.65732 3321.74421 92 50 22 300 VLT/UVES (390+580 nm) HD101065 3071.67758 3071.76032 111 40 25 180 VLT/UVES (600 nm) HD122970 3069.70977 3069.79359 111 40 25 160 VLT/UVES (600 nm) HD128898 3073.80059 3073.88262 265 1.5 25 250 VLT/UVES (600 nm) HD134214 3070.77571 3070.85848 111 40 25 260 VLT/UVES (600 nm) HD137949 3071.76312 3071.84598 111 40 25 350 VLT/UVES (600 nm) HD201601 2871.46470 2871.56295 70 80 42 80 SAO 6-m/NES (425–600 nm) HD201601 2186.70618 2186.80296 64 90 43 250 Gecko/CFHT(654–6660 nm) HD201601 2186.82456 2186.92308 65 90 43 230 Gecko/CFHT(662–673 nm) ---------- ------------ ------------ ----------- ---------- ---------- ------- -------------------------- The main observational dataset analysed in our study consists of 958 observations of 8 roAp stars, obtained with the UVES instrument at the ESO VLT between October 8, 2003 and March 12, 2004 in the context of the observing programme 072.D-0138 (Kurtz et al. [@KEM06]). The ESO Archive facility was used to search and retrieve science exposures and the respective calibration frames. Observations of each target covered 2 hours and consisted of an uninterrupted high-resolution spectroscopic time-series with a total number of exposures ranging from 69 to 265. The length of individual exposures was 40$^{\rm s}$ or 80$^{\rm s}$, except for the brightest roAp star HD128898 ($\alpha$ Cir), for which a 1.5$^{\rm s}$ exposure time was used. The ultra-fast (625kHz/4pt) readout mode of the UVES CCDs allowed us to limit overhead to $\approx$$20^{\rm s}$, thus giving a duty cycle of 70–80% for the majority of the targets. The signal-to-noise ratio (S/N) of individual spectra is between 90 and 350, as estimated from the dispersion of the stellar fluxes in the line-free regions. Detailed description of observations for each target is presented in Table \[Table\_UVES\_Log\]. Columns of the table give the stellar name, heliocentric Julian dates for the beginning and the end of spectroscopic monitoring, number of observations, exposure and overhead times, peak signal-to-noise ratio of individual spectra, and information about the telescope and instrument where the data were obtained. The red arm of the UVES spectrometer was configured to observe the spectral region 4960–6990 Å (central wavelength 6000 Å). The wavelength coverage is complete, except for a 100 Å gap centred at 6000 Å. Observations were obtained with the high-resolution UVES image slicer (slicer No. 3), providing an improved radial velocity stability and giving maximum resolving power for $\lambda/\Delta\lambda\approx115\,000$. All spectra were reduced and normalised to the continuum level with a routine especially developed by one of us (DL) for a fast reduction of spectroscopic time-series observations. A special modification of the Vienna automatic pipeline for échelle spectra processing (Tsymbal et al. [@tsymbal]) was developed. All bias and flat field images were median-averaged before calibration. The scattered light was subtracted by using a 2-D background approximation. For cleaning cosmic ray hits, we used an algorithm that compares the direct and reversed spectral profiles. To determine the boundaries of échelle orders, the code used a special template for each order position in each row across dispersion axes. The shift of the row spectra relative to template was derived by a cross-correlation technique. Wavelength calibration was based on a single ThAr exposure, recorded immediately after each stellar time series. Calibration was done by the usual 2-D approximation of the dispersion surface. An internal accuracy of 30–40 [ms$^{-1}$]{} was achieved by using several hundred ThAr lines in all échelle orders. The final step of continuum normalisation and merging of the échelle order was carried out by transformation of the flat field blaze function to the response function in each order. The global continuum normalisation was improved by iteratively fitting a smoothing spline function to the high points in the average spectrum of each roAp star. With this procedure we corrected an underestimate of the continuum level, unavoidable in analysis of small spectral regions of the crowded spectra of cool Ap stars. Correct determination of the absolute continuum level is important for retrieving unbiased amplitudes of radial velocity variability when the centre-of-gravity method is used. In addition to the global continuum correction, spectroscopic time series were post-processed to ensure homogeneity in the continuum normalization of individual spectra. Extracted spectra were divided by the mean, the resulting ratio was heavily smoothed, and then it was used to correct continua in individual spectra. Without this correction a spurious amplitude modulation of pulsation in variable spectral lines may arise due to an inconsistent continuum normalisation. The red 600 nm UVES dataset was complemented by the observations of HD24712 obtained on November 11, 2004 in the DDT program 274.D-5011. Ninety-two time-resolved spectra were acquired with the UVES spectrometer, configured to use the 390+580 nm dichroic mode (wavelength coverage 3300–4420 and 4790–6750 Å). A detailed description of the acquisition and reduction of these data is given by Ryabchikova et al. ([@RSW06]). For the roAp star HD201601 ($\gamma$ Equ), we analysed 70 spectra obtained on August 19, 2003 with the NES spectrograph attached to the 6-m telescope of the Russian Special Astrophysical Observatory. These échelle spectra cover the region 4250–6000 Å and have typical $S/N$ of $\approx$$80$. The data were recorded by Kochukhov et al. ([@KRP04]), who searched for rapid magnetic field variability in $\gamma$ Equ. We refer readers to that paper for the details on the acquisition and reduction of the time-series observations at the SAO 6-m telescope. Post-processing of the échelle spectra of HD24712 and HD201601 was done consistently with the procedure adopted for the main dataset. In addition, the time-resolved observations of HD201601 obtained in 2001 using the single-order $f/4$ Gecko coudé spectrograph with the EEV1 CCD at the 3.6-m Canada-France-Hawaii telescope were used. Observations have a resolving power of about 115000, determined from the widths of a number of ThAr comparison lines. The reduction is described in Kochukhov et al. ([@KR07]). Fundamental parameters of programme stars {#parameters} ========================================= -------- ------------------- ------ ------ ------------------ -------------------------------- --------------------------- ------------------------------- HD Other [$\langle B_{\rm s}\rangle$]{} P Reference number name (K) ([kms$^{-1}$]{}) (kG) (min) 9289 BW Cet 7840 4.15 10.5 2.0 [10.522]{} this paper 12932 BN Cet 7620 4.15 3.5 1.7 [11.633]{} this paper 19918 BT Hyi 8110 4.34 3.0 1.6 [11.052]{} this paper 24712 DO Eri, HR 1217 7250 4.30 5.6 3.1 [6.125]{}, 6.282 Ryabchikova et al. ([@RLG97]) 101065 Przybylski’s star 6600 4.20 2.0 2.3 [12.171]{} Cowley et al. ([@CRK00]) 122970 PP Vir 6930 4.10 4.5 2.3 [11.187]{} Ryabchikova et al. ([@RSH00]) 128898 $\alpha$ Cir 7900 4.20 12.5 1.5 [6.802]{}, 7.34 Kupka et al. ([@KRW96]) 134214 HI Lib 7315 4.45 2.0 3.1 [5.690]{} this paper 137949 33 Lib 7550 4.30 $\le$2.0 5.0 [8.271]{},4.136, 9.422 Ryabchikova et al. ([@RNW04]) 201601 $\gamma$ Equ 7700 4.20 $\le$1.0 4.1 [12.20 ]{} Ryabchikova et al. ([@RPK02]) -------- ------------------- ------ ------ ------------------ -------------------------------- --------------------------- ------------------------------- Fundamental parameters of the programme stars are given in Table \[tbl2\]. For six stars, effective temperatures , surface gravity , and mean surface magnetic fields [$\langle B_{\rm s}\rangle$]{} were taken from the literature. For 4 remaining stars, HD9289, HD12932, HD19918, and HD134214, atmospheric parameters were derived using Strömgren photometric indices (Hauck & Mermilliod [@HM98]) with the calibrations by Moon & Dworetsky ([@MD85]) and by Napiwotzki et al. ([@N93]) as implemented in the [TEMPLOGG]{} code (Rogers [@R95]). In addition Geneva photometric indices (Burki et al. [@B05])[^2] with the calibration of Künzli et al. ([@KNK97]) was used for effective temperature determination. The colour excesses were estimated from the reddening maps by Lucke ([@L78]). In Table \[tbl2\] we present average values of the effective temperatures derived with three different calibrations. A typical dispersion is $\pm$150 K. In all stars but HD101065, rotational velocities were estimated by fitting line profiles of the magnetically insensitive Fei 5434.5, 5576.1 Å lines. Magnetic spectral synthesis code [SYNTHMAG]{} (Piskunov [@P99]; Kochukhov [@synthmag06]) was used in our calculations. Atomic parameters of spectral lines were extracted from the [VALD]{} (Kupka et al. [@vald299]) and [DREAM]{} (Biémont et al. [@dream99]) databases, supplemented with the new oscillator strengths for La (Lawler et al. [@La2]), Nd (Den Hartog et al. [@Nd2-03]), Nd (Ryabchikova et al. [@RRKB06]), Sm (Lawler et al. [@Sm2]), and Gd (Den Hartog et al. [@Gd2]). We confirmed rotational velocities derived previously for HD24712, HD122970, HD128898, HD137949, and HD201601. High spectral resolution of the present data allows us to improve the value of the projected rotational velocity for HD101065 (Przybylski’s star) using partially resolved Zeeman patterns in numerous lines of the rare-earth elements. The value of the magnetic field modulus, 2.3 kG (Cowley et al. [@CRK00]), was confirmed. We note that the derived values of in very sharp-lined roAp stars depend strongly on the Fe stratification, which is known for some of them (e.g. $\gamma$ Equ, see Ryabchikova et al. [@RPK02]) and is typically characterised by the Fe overabundance below $\log\tau_{5000}=-1$ and Fe depletion in the outer layers. Rotational velocities in Table \[tbl2\] are derived for a homogeneous Fe distribution. However, in all programme stars we see observational evidence of Fe stratification, such as anomalous strength of the high-excitation lines compared to the low-excitation lines (see Ryabchikova et al. [@RWL03]). We tested one of the most peculiar stars, 33 Lib, for possible influence of stratification on the derived and found that including stratification may decrease the inferred rotational velocity to 1.5 [kms$^{-1}$]{} in comparison to 2.5–3.0 [kms$^{-1}$]{} obtained with a homogeneous Fe abundance distribution. The lower value of seems to fit the observed resolved and partially resolved Zeeman components better. In three programme stars, HD9289, HD12932, and HD19918, mean magnetic modulus was estimated for the first time from differential magnetic broadening/intensification. Two spectral regions with a pair of lines having different magnetic sensitivity were synthesised for a set of magnetic field strengths. The first region contain well-known Fe 6147.7 Å (=0.83) and 6149.3 Å(=1.35) lines, while Fei 6335.3 Å (=1.16) and 6336.8 Å (=2.00) lines were analysed in the second spectral region. The derived values of the magnetic fields were confirmed by fitting other magnetically sensitive lines, for example, Fe 6432.7 Å (=1.82) and Eu 6437.6 Å (=1.76). Radial velocity measurements {#RV} ============================ To perform a meaningful study of the pulsational amplitudes in spectral lines of different chemical elements/ions, one has to be very careful in the choice of lines for pulsation measurements. For this purpose we have synthesised the observed spectral region for each star with the model atmosphere parameters and magnetic field values from Table \[tbl2\]. Abundances for HD 24712, HD 101065, HD 122970, HD 128898, HD 137949, and HD 201601 were taken from the papers cited in the last column of Table \[tbl2\]. For the remaining four stars, a preliminary abundance estimate was obtained in this paper. The radial velocities were measured with a centre-of-gravity technique. We used only unblended or minimally blended lines. In some cases where the line of interest was partially overlapping with the nearby lines, only the unblended central part of the line was considered; therefore, some lines were not measured between the continuum points. This usually leads to lower pulsation amplitudes if we have strong variations in the pulsation signal across a spectral line (see $\gamma$ Equ – Sachkov et al. [@sach04], and HD 99563 – Elkin et al. [@EKM05]). Bisector radial velocity measurements were performed for H$\alpha$ core and for a subset of Y , Eu, Nd, Nd, Pr, Pr, Tb, and Thspectral lines. The pulsational RV variability of the two strongest Th lines at $\lambda\lambda$ 5376.13 and 6599.48 Å was investigated for several stars in our sample for the first time. A detailed frequency analysis of the RV data for 8 stars from our sample was carried out by Kurtz et al. ([@KEM06]). These authors used the same observations as we do and, despite the relatively short time span (2 hours) of the spectroscopic time series, they claimed to resolve several frequencies for each star and to find amplitude modulation that was not observed in photometry. Although a detailed frequency analysis is not the primary goal of our paper, we did repeat time-series analysis for all lines. First we applied the standard discrete Fourier transformation (DFT) to the RV data. The period corresponding to the highest pulsation amplitude value was then improved by the sine-wave least-square fitting of the RV data with pulsation period, amplitude, and phase treated as free parameters. This fit was removed from the data and then Fourier analysis was applied to the residuals. This procedure was repeated for all frequencies with the S/N above 5. To verify our analysis against the results of Kurtz et al. ([@KEM06]), we applied it to the set of Pr lines (to each line separately and to the average Pr RV) and compared in detail the resulting solution for HD 134214. Our frequency solution agrees perfectly with Kurtz et al. ([@KEM06]), although our RV amplitudes are systematically lower. We believe that the primary reason for this discrepancy, which is present for other stars as well, is our systematically higher continuum placement, leading to lower pulsation amplitude when the centre-of-gravity method is used. As explained in Sect. \[observ\], we have rectified observations with a spline-fit over wide wavelength regions, whereas Kurtz et al. ([@KEM06]) have probably assigned continuum points to the high spectrum points in the immediate vicinity of the line considered. For each spectral line, we estimated the probability that the detected periodicity is not due to noise (Horne & Baliunas [@HB86]). Then, for the lines with probabilities higher than 0.999, we calculated a weighted average value of the pulsation period. All amplitudes and phases were then recalculated keeping the period fixed. This information is summarised in Table \[tbl3\] (available online only) for all lines studied in each star. Table \[tbl3\] contains amplitudes (first line) and phases (second line), together with the corresponding errors for the dominant pulsation period. When no pulsation signal was detected, we give the formal amplitude solution for the fixed period without phase information. If a star had one dominant period, the RV analysis was done with this period. In the cases of more than one dominant period, a simultaneous fit was performed with up to three periods. RV variation was approximated with the expression $$\langle V \rangle = V_0 (t-t_0) + \sum_{i=1}^3 V_{i} \cos{\{2 \pi [(t-t_0)/P_{i} - \varphi_{i}]\}}. \label{cosfit}$$ Here the first term takes possible drift of the spectrograph’s zero point into account. For all stars but HD 24712, HJD of the first exposure of the star at a given night was chosen as a reference time $t_0$. For HD 24712, HJD=2453320.0 was used as a reference time. Both $V_{i}$ and $\varphi_{i}$ are, respectively, amplitude and phase of the RV variability with the $i$th period ($i_{\rm max}=3$). With the minus sign in front of $\varphi_{i}$, a larger phase corresponds to a later time of RV maximum. This phase agreement is natural when discussing effects of the outward propagation of pulsation waves in the atmospheres of roAp stars. The periods employed for determining of the RV amplitudes and phases are given in the seventh column of Table \[tbl2\]. When more than one period was inferred in the fitting procedure, the main period for which pulsation amplitudes and phases are given in Table \[tbl3\] (online material) is highlighted with italics. Bisector measurements {#bis} ===================== Radial velocity analysis was complemented by studying bisector variability. Figure \[ha\] shows bisector amplitudes (left panel) and phases (right panel) across the core of [H$\alpha$]{} line calculated with the pulsation periods from Table \[tbl2\]. Where two periods or the main period and its first harmonic are resolved, simultaneous fit with two frequencies was done. In all programme stars, an increase in bisector amplitude by two or more times from the transition region to the deepest part of the core is observed. Thus we supported the conclusion made by Kurtz et al. ([@KEM05a]). This change is gradual in all but two stars: HD 9289 and HD 19918, which have the lowest S/N. It is, therefore, unclear if small jumps of RV amplitude across the core are real or caused by the low S/N of the spectroscopic data. RV changes are accompanied by tiny phase changes. Only in four stars do phase changes exceed the error bars. These are the stars with the shortest pulsation periods close to the acoustic cut-off frequencies: HD 24712, HD 128898, HD 132214, and HD 137949 (33 Lib). The NLTE calculations show that, in the atmosphere of the star with between 7000 and 8000 K, the [H$\alpha$]{}core is formed at $-5\la\log\tau_{5000}\la-2$ (Mashonkina, private communication). It is applied to a normal atmosphere; however, a core-wing anomaly is present in all programme stars (Cowley et al. [@CWA]), which was attributed to a peculiar atmospheric stratification containing a region of increased temperature below $\log\tau_{5000}=-4$ (Kochukhov et al. [@cwaT]). This change in the atmospheric structure may lead to an upward shift in formation depth of the base of the H$\alpha$ core (Mashonkina, private communication). More care should be taken in the choice of metal spectral lines for the bisector measurements than for the centre-of-gravity RV analysis. Blends with non-pulsating lines change the run of bisector pulsation amplitude across the line profiles, while blends with pulsating lines (weak REE lines in far wings, for example) may change both amplitudes and phases. These artifacts often lead to the wrong conclusion about roAp pulsational behaviour, especially when these conclusions are based on analysis of only one line. For instance, the well-known Nd 6145.068 Å line, analysed in many pulsation studies, is blended with the Sii 6145.016 Åline and with the Ce 6144.853 Å. The latter feature makes a non-negligible contribution to the spectrum of Przybylski’s star. The strongest Nd 5294.11 Å line is blended with the Fei 5293.96 Å line in the blue wing and with the high-excitation Mn 5294.32 Å line in the red wing. Both lines are normally weak, but they are strengthened in the stratified atmospheres of roAp stars with the effective temperatures higher than 7000–7200 K. This blending results in a drop of the bisector velocity amplitudes starting from some intensity points in the line profile. Zeeman splitting may also change velocity amplitude distribution across the line profile. Figure \[HD134214\_Nd\] illustrates the influence of blends and Zeeman effect on the bisector amplitudes and phases of Nd- and Pr lines in HD 134214. Pairs of Nd lines with similar intensities and Zeeman structure have identical bisector phases independent of any blending. Thus, phases may be considered as a more reliable indicator of the pulsational characteristics. One may see that, up to the residual intensity 0.85, the Nd5294 Å line shows the same bisector velocity amplitudes as another strong Nd line at 5204 Å. However, after this intensity point, the two amplitude curves start to deviate: in the Nd 5294 Å line, the amplitude drops towards the line wings, probably due to blends. Also blends decrease bisector amplitudes in the Nd6145 Åcompared to the Nd6327 Å line, which has the same intensity and Zeeman structure. Three out of the four Pr lines shown in Fig. \[HD134214\_Nd\] are practically free of blends in most of the programme stars, and additionally, these lines have identical Zeeman splitting, which results in similar bisector pulsational behaviour. As shown in Section\[ph-a\], Tb lines exhibit a particularly interesting pulsation behaviour, so their blending should be investigated in detail. The Tb lines are not as strong and numerous as Pr and Nd. From 10 lines only the strongest one, Tb 5505 and three others, 5847, 6323, and 6687 Å, may be used for bisector measurements. Most often we measured the Tb5505.408 Å line. This line is blended by the Ce 5505.204 Åin the blue wing and by Ce 5505.588 Å in the red wing. The Tb $\lambda$ 6323.619 Å line is free of blends but may be partially blended with the weak atmospheric O$_2$ line at $\lambda$ 6323.75 Å. Figure \[Tb3\] shows a spectral region around the Tb 5505.408 Å line in HD 101065, where the blending problem is the most severe. For comparison we show the region with the Tb 6323.619 Å line; therefore, wavelength scale is given in [kms$^{-1}$]{} relative to the centre of each line. We also show synthetic spectrum calculated with the highest possible Ce abundance and magnetic field strength appropriate for HD101065 (see Table \[tbl2\]). Although the full intensity of the calculted feature is equal to the intensity of the observed one, for demonstration of the blending effects we did not convolve synthetic spectrum with the instrumental and rotational profiles. Note that the red wing of the Tb 6323.619 Å line is blended with the atmospheric O$_2$ line 6323.75, therefore no bisector measurements were carried out above intensity level 0.9. The blue wing asymmetry in both Tb lines is rather an effect of the hyperfine splitting, than the real influence of blends, because the same asymmetry is observed in other stars where the Ce contribution is negligible. Taking into account the similarity of the RV amplitudes and, in particular, phases in the bisector measurements of both lines we conclude that even in HD 101065 the blending does not affect seriously the measured pulsation characteristics. Phase-amplitude diagrams {#ph-a} ======================== Empirical understanding and detailed theoretical interpretation of the pulsation phenomenon in roAp stars requires construction of the vertical cross-section of pulsation modes. Chemical stratification enables this difficult task by separating formation layers of the spectral lines of different chemical elements far apart, thus allowing us to resolve the vertical dimension of pulsating Ap-star atmospheres, as can be done for no other type of pulsating stars. However, the key information needed for such vertical pulsation tomography – formation depth of REE spectral lines – is difficult to obtain. Early studies (e.g., Kanaan & Hatzes [@KH98]) have suggested that the line intensity may be used as a proxy of the relative formation heights. However, it is now understood that the chemical stratification effects are dominant in the atmospheres of cool Ap stars and, therefore, formation region of weak lines of one element is not necessarily located deeper than the layer from where the strong lines of another element originate. This is why a comparison of the intensities of different pulsating lines is only meaningful for the absorption features of the same element/ion. The study by Ryabchikova et al. ([@RPK02]) presented the first comprehensive analysis of the vertical stratification of light, iron-peak, and rare-earth elements in a roAp star, demonstrating that only by taking height-dependent abundance profiles into account can one calculate correct formation depths of pulsating spectral lines and meaningfully interpret the results of time-resolved spectroscopy. In a later paper, Mkrtichian et al. ([@MHK03]) study the vertical profiles for $p$-mode oscillations in 33 Lib (HD137949) using pulsation centre-of-gravity measurements for individual lines. To establish the vertical atmospheric coordinate, the authors used the $W_{\lambda} - \log \tau$ transformation scale in 33 Lib’s atmosphere by assuming homogeneous elemental distribution. This simplified approach was sufficient for detecting the node in the Nd, Nd line-forming region – a discovery later confirmed by Kurtz et al. ([@KEM05b]) and by the present paper. However, as Mkrtichian et al. ([@MHK03]) themselves acknowledge, their method is too simplified because it ignores the vertical and horizontal inhomogeneous distribution of chemical abundances. Indeed, their technique cannot be used for simultaneous interpretation of the pulsational variability of different elements since Fe, as well as Ca, Cr and all REE elements, are strongly stratified in the atmosphere of 33 Lib and other roAp stars (see Ryabchikova et al. [@RNW04]), and stratification is not the same for different elements. Mkrtichian et al. ([@MHK03]) suggest that in future studies some of these difficulties can be alleviated by the “multi-frequency tomographic approach” that will use the phase and amplitude profiles of different oscillations modes to establish a link to the atmospheric geometric depth scale. In general, the detailed pulsation tomography analysis of roAp stars should be based on sophisticated atmospheric modelling including chemical stratification, NLTE, magnetic field, and eventually, pulsation effects on the shape and intensity of spectral lines with substantial RV variability. Such modelling is very demanding in terms of the quality of observations, required input data, and computer resources. This is why only two roAp stars, $\gamma$ Equ and HD24712, have been studied with this method up to now (Mashonkina et al. [@MRR05]; Ryabchikova et al. [@RMR06]). Here we suggest a different approach to the pulsation tomography problem. In the framework of the outward propagating magnetoacoustic wave, one expects a continuous amplitude versus phase relation for pulsation modes. Lines showing later RV maximum should originate higher in the atmosphere. Thus, new insight into the roAp pulsation modes structure can be obtained by inferring and interpreting the trend in pulsation velocity amplitude as a function of pulsation phase. Such phase-amplitude diagrams offer a possibility of tracing the vertical variation in the mode structure without tedious assignment of the physical depth to each pulsation measurement. We note that this paper is not the first one to use the phase-amplitude diagrams for roAp stars. Baldry et al. ([@BB98]) and Baldry & Bedding ([@BB00]) produced similar diagrams in the pulsational studies of $\alpha$ Cir and HR 3831. However, their diagrams described pulsational properties not for individual lines with a proper identification but for small spectral regions often containing several spectral lines, sometimes without correct identification. As a result, the most important information on wave propagation was lost (see Sect. \[alcir\]). For each programme star we have produced phase-amplitude diagrams using a set of lines of representative chemical species, including Y, Eu, the H$\alpha$ core, La, Dy, Dy, Nd, Nd, Pr, Pr, Tb, and Th. We considered velocity amplitudes and phases derived with both the centre-of-gravity RV measurements and the bisector analysis. The line blending varies significantly from star to star due to different effective temperatures and chemical anomalies, therefore we cannot use identical set of spectral lines for all stars. For each star we tried to employ unblended or minimally blended lines, in particular, for the bisector measurements. In addition, the line broadening expressed in the terms of macroturbulent velocity was estimated for a few representative lines of each chemical species. This information is useful for assessing the isotropic velocity component at the formation heights of REE lines. Below we present the results for individual stars. HD 101065 (Przybylski’s star) ----------------------------- The atmosphere of HD 101065 is known to be very rich in REE and underabundant in most other elements, including the Fe-peak species. Due to the low effective temperature, slow rotation, and abundance anomalies, most lines in the spectrum of HD 101065 are strong and sharp. As a result, one can achieve impressive accuracy of pulsation measurements, typically $\sim$5–8 [ms$^{-1}$]{}, for moderately strong lines. Although it is not easy to find unblended Fe lines in the forest of the REE lines, we managed to measure a few of them and to detect no pulsational variability above 30 [ms$^{-1}$]{}. Pulsation amplitudes at the level of 6 to 17 [ms$^{-1}$]{} were detected in the cores of very strong Ba $\lambda\lambda$ 6142, 6496 Å lines, which gives us an idea about pulsation amplitudes close to the photospheric layers. The centre-of-gravity and bisector phase-amplitude diagrams for HD101065 are shown in Fig. \[HD101065\]. The centre-of-gravity measurements give us a general idea of the pulsation wave propagation in roAp atmosphere. Pulsations are characterised by the amplitude $\sim$40 [ms$^{-1}$]{} at the levels of the Y, La , and strong Eu line formation. Then they pass, with the gradually increasing amplitude, the layers where the Dy, Dy, Nd, Pr, Nd, the H$\alpha$ core and Pr lines are formed. The phase does not change by more than 0.15 of the pulsation period (less than 1 radian) from Eu to Pr lines. After that, a rapid change in pulsation phase occurs in Tb and, next, in Th lines. We measured two unblended Thlines at $\lambda$ 5376.13 and 6599.48 Å. HD 101065 is the only star in our sample that shows similar pulsation signatures (bisector amplitude versus pulsation phase) for the [H$\alpha$]{} core, Nd, Nd, Pr and weak Pr lines. For all these features we find an increase in both pulsation amplitude and phase from the line wings to the line core. Kurtz et al. ([@KEM05a]) have obtained similar amplitudes and phases for the H$\alpha$ core and, based on the bisector measurements of the single Nd 6145 Å line, they argue that the formation layers of Nd lines start above the layers of formation of the deepest part of the H$\alpha$ core. Our Fig. \[HD101065\] provides strong evidence that most of REE lines, including Pr and weaker Pr lines, are formed in the same layers as the H$\alpha$ core. In addition we analysed the broadening of pulsating lines. All low-amplitude Ce lines do not exhibit any extra broadening above the adopted value of the projected rotation velocity. All Eu lines and Dy and Nd lines in both ionisation states show an extra broadening corresponding to between 2.5 and 4 [kms$^{-1}$]{}. Macroturbulent velocity grows from 5 [kms$^{-1}$]{} in weak Pr 5765 Å line to 7–8 [kms$^{-1}$]{} in the strong Pr lines $\lambda\lambda$ 5300, 6707 Å. We need =10 [kms$^{-1}$]{} to fit the observed profiles of the unblended Tb 6832 Å and Th5376 Å lines. It is worth noting that this rapid increase in the line broadening accompanies the change in the line profile variability pattern from the normal symmetric shape to the blue-to-red running waves, which become noticeable for the strongest Pr lines and, in particular, for Tb lines (Kochukhov et al. [@KR07]). An extra broadening of the Pr lines cannot be attributed to hyperfine structure. Considering the amplitude-phase diagrams, we see a change in the pulsation behaviour for Tb lines and, even stronger, for the Th 5376.13 Å line. In contrast to the variability of the H$\alpha$ core and Nd lines, we observe a rapid increase in the amplitude and in the phase from the line core to line wings in the lines of Tb and Th (see Fig. \[HD101065\_prof\] - Online only). This phenomenon cannot be explained by the pulsation wave propagating outwards. In general, over the large fraction of the Przybylski’s star atmosphere, up to the layers of the Nd and weak Pr line formation, pulsations are represented well by non-radial non-adiabatic [p]{}-modes (Saio [@S05]) with an amplitude increasing outward. Above these layers, pulsation characteristics change in a way that is not forseen by the theoretical models. HD 122970 --------- Pulsational observations of this star were obtained at the rotational phase 0.75, between the maximum and minimum of the magnetic field (see Fig. 3 in Ryabchikova et al. [@RWA05]). The centre-of-gravity and bisector phase-amplitude diagrams for HD122970 are presented in Fig. \[HD122970\] (Online only). For a few chosen lines we also show the variability of the pulsational characteristics along the line profiles in Fig. \[HD122970\_int\]. The overall pulsational behaviour in the atmosphere of this star is similar to HD 101065: low-amplitude pulsations are seen in Y and Eu lines, the amplitude increases slightly in the layers where the H$\alpha$ core forms, and then it increases further in the Dy and Nd line formation layers. A rapid growth of the amplitude occurs in the region of the Pr line formation. Up to these layers, pulsations have a standing-wave character, with almost constant phases for all lines of Y, Eu, La, Dy, Nd, and Pr. The phase shifts become noticeable for Tb and Th lines, as in HD 101065, and, again, in these lines pulsation amplitude and phase increase from the line core to line wing, contrary to predicted effect of the outward-running wave. While in HD 101065 the regions of the H$\alpha$ core, Nd, and Pr line formation are the same or, at least, partially overlap, in HD122970 the phase-amplitude diagrams show a significantly more stratified atmospheric structure. Ryabchikova et al. ([@RSH00]) noted that the projected rotational velocities derived from the REE lines are lower than those obtained from Fe-peak lines. More precise present observations confirm this result. We determined $\approx$3.5 [kms$^{-1}$]{}from the REE line fitting, whereas the Fe-peak elements show broadening corresponding to =4.5 [kms$^{-1}$]{}. A 3.877 d rotation period obtained by Ryabchikova et al. ([@RWA05]) for HD122970, together with the low , suggests a nearly pole-on geometry. Therefore, the difference in the apparent rotational velocities of the Fe-peak and REE lines may indicate a concentration of the REEs near the visible stellar rotation pole. No extra broadening is required to fit the Nd, Nd, Pr, and weak Pr lines, while =2–3 [kms$^{-1}$]{} and 6 [kms$^{-1}$]{} are needed to fit the strong Pr lines and the Th 5376 Å line, respectively. The Y lines, which show the smallest detectable RV amplitudes, have the same pulsation phases and extra broadening as strong Pr lines. The lines of the elements with Z$\le$38 do not show any pulsation amplitude above 7–20 [ms$^{-1}$]{}. Two measured Zr lines have amplitudes of $\approx$200 [ms$^{-1}$]{}. HD 24712 -------- According to the ephemeris given by Ryabchikova et al. ([@RWA05]), our pulsational observations were obtained at the rotational phase 0.944, i.e. close to the magnetic maximum. Detailed observational analysis of the spectroscopic pulsation signatures in HD 24712 was recently presented by Ryabchikova et al. ([@RSW06]). HD 24712 is the only roAp star for which NLTE line formation calculations of [H$\alpha$]{}, Pr, and Nd were carried out (Mashonkina et al. [@MRR05]; Ryabchikova et al. [@RMR06]). The observed intensities of the Nd and Pr lines in the first and the second ionisation states are explained by the stratified atmosphere with a step-like enhancement of these elements just above the NLTE formation depth of the [H$\alpha$]{} core ($\log\tau_{5000}$=$-4$). Up to this atmospheric level, the observed distribution of pulsation phases agrees rather well with the predictions of the non-adiabatic pulsation models (see Fig. 1 in Kochukhov [@K06] and Fig. 3 in Sachkov et al. [@SR06]), which is supported by the amplitude-phase diagrams for HD 24712 shown in Fig. \[HD24712\] (Online only). In contrast to the standing-wave dominated behaviour of HD 101065 and HD 122970, we see a running wave in HD 24712 from the low atmosphere, where the first pulsation signal is detected. Due to the availability of the simultaneous spectroscopic and photometric monitoring, the sequence of the pulsation phase changes is established with high accuracy in this star (Ryabchikova et al. [@RSW06]). Pulsation follows the same order as in HD 101065 and in HD 122970: Eu, La, [H$\alpha$]{} core, Nd, Pr, Tb lines. The phase distribution with the optical depth, as well as our amplitude-phase diagrams, shows the phase shifts between [H$\alpha$]{} and Nd lines and between Pr and Nd lines. The NLTE calculations do not predict a significant difference in the location of the Pr and Nd line formation regions, therefore the observed phase shift is probably caused not by the vertical distribution of chemical elements, but by the horizontal abundance inhomogeneities. According to the pulsation phases, Nd lines have to be formed at the same layers as the deeper parts of the [H$\alpha$]{} core. At the same time, we caution that the existing NLTE calculations of [H$\alpha$]{}, Pr, and Nd lines are very preliminary, and they do not include potentially important effects of stratified abundance distribution on the atmospheric structure and neglect an influence of the magnetic desaturation on the line formation. The bisector velocity amplitudes of HD24712 are either constant along the line profile, or they slightly increase from the line wings to the line core, except for the Tb and, possibly, Ylines, in which bisector amplitude increases from the line core to the line wings (Fig. \[HD24712\_int\] - Online only). Pulsation phases are nearly constant along the profiles of most lines, but show a tendency to grow from the line centre to the line wings for the strongest Pr lines and for Tb lines. Both Th lines show pulsational amplitudes close to the detection limit. A very interesting result is obtained for Y lines. Yttrium is the only element with Z$\le$40 whose lines have a measurable pulsation amplitude in HD24712 (see Ryabchikova et al. [@RSW06]). These lines are strong enough to provide precise centre-of-gravity and bisector measurements, therefore pulsation phases can be determined with high accuracy. While RV amplitudes do not exceed 50–100 [ms$^{-1}$]{}for the strongest Y 5087 Å line, the bisector phase distribution across the line profile coincides with the phase distribution across the profile of the strongest Pr 5300 Åline, which has an RV amplitude above 300 [ms$^{-1}$]{}. The estimated depth of formation of Y lines in chemically homogeneous atmosphere lies around $\log\tau_{5000}=-1.5$ to $-2.5$, where the core of [H$\alpha$]{}line starts forming. Note that Doppler imaging of HD 24712 shows a similarity between yttrium and REE surface distributions (Lüftinger et al. [@MDI06]). The observed phase difference between Y and the [H$\alpha$]{} and the coincidence of the phases in Y and strong Pr lines, on the one hand, and the large difference in amplitudes, on the other, is difficult to explain. We will see that the same pulsational behaviour of Y lines is observed for some other roAp stars. Although the projected rotational velocity of HD 24712 is rather high, spectral synthesis clearly shows that we need an extra broadening, equivalent to =6 [kms$^{-1}$]{}, to fit strong Prlines, as well as Tb and Th lines, whereas no extra broadening is required for Nd lines. Interestingly, one needs =4 [kms$^{-1}$]{} to reproduce the profile of the Y 5087 Åline. HD 134214 --------- This star has one of the shortest pulsation periods, 5.7 min, which is close to the acoustic cut-off frequency calculated for HD 134214 by Audard et al. ([@AKM98]). The centre-of-gravity and bisector amplitude-phase diagrams are shown for HD134214 in Fig. \[HD134214\] (Online only). The amplitude-phase diagrams for this star are similar to those for HD 24712. In both stars we observe the running wave from the regions of Eu line formation to Tb, Th line-forming regions. In HD 134214, all lines pulsating with high amplitude show an increase both in amplitude and in phase from the line core to the line wings (see Fig. \[HD134214\_Nd\]). The low rotational velocity allows us to partially resolve components of the Zeeman split lines, which makes the line profile analysis easier and, in particular, allows us to estimate an extra broadening accurately. An extra broadening of $\sim$1 [kms$^{-1}$]{} is needed to fit the low-amplitude pulsating lines of Eu. Then the inferred increases gradually: 4 [kms$^{-1}$]{} (Nd, Dy), 5–6 [kms$^{-1}$]{} (Nd), 8–10 [kms$^{-1}$]{}(Pr), 10 [kms$^{-1}$]{} (Tb), and more than 10 [kms$^{-1}$]{} (Th). Just as in HD 24712, Y lines have small detectable RV amplitudes, but the same bisector phases as strong Pr lines do. We measured $\sim$6 [kms$^{-1}$]{} for the Y 5087 Å line. In both stars the bisector amplitude-phase diagrams for Nd and weaker Nd lines are overlapping, which supports the model of the Nd stratification in roAp atmospheres proposed by Mashonkina et al. ([@MRR05]). According to this model the strongest lines of Nd are formed at the same layers as weak Nd lines, therefore their pulsational characteristics should be similar. No definite pulsation signatures are detected in the lines with Z$\le$38 and also in Zr and Ba lines. The upper RV amplitude limit is 10 [ms$^{-1}$]{}. HD 128898 ($\alpha$ Cir) {#alcir} ------------------------ The main photometric period of $\alpha$ Cir, one of the short-period roAp stars, is 6.83 min, which is close to the acoustic cut-off frequency. The amplitude-phase diagram for HD128898 (Fig. \[alcir\_diagr\]) appears to be similar to those constructed for HD 24712 and HD 134214 and indicates a running wave. Based on an amplitude-phase diagram made for short spectral regions containing groups of unresolved lines, Baldry et al. ([@BB98]) suggested a high-overtone standing wave with a velocity node in the atmosphere of $\alpha$ Cir. This contradicts our results obtained from analysis of pulsations in isolated lines. Again, similar to HD 24712 and HD 134214, Y lines have the smallest RV amplitudes, but the same phases as the strongest Nd and Pr lines. The bisector analysis shows that bisector amplitude rapidly decreases from the line centre to the line wings with small changes in phase. This behaviour may be explained by a combination of non-radial pulsations and surface chemical inhomogeneity, which clearly manifests itself in the rotational modulation of the line profiles in $\alpha$ Cir (Kochukhov & Ryabchikova [@KR01b]). The average spectrum used in the present study corresponds to the rotation phase 0.44 (minimum of the photometric pulsational amplitude) if we apply the ephemeris from Kurtz et al. ([@acir]). All REE lines in our $\alpha$ Cir spectrum exhibit two-component profiles, which represents another proof of chemical spots. HD 12932 {#HD12932} -------- Kurtz et al. ([@KEM06]) suggest that HD 12932 is an example of the star with a standing wave behaviour in the layers of the [H$\alpha$]{}core formation based on the constancy of the bisector pulsation phases. Our measurements are presented in Fig. \[hd12932\] (Online only). While for Nd and the [H$\alpha$]{} core the bisector phases are almost constant for a given line, the phases differ from line to line. Phase shifts are small, but reliable, taking the high accuracy of the phase determination into account (Fig. \[HD12932\_int\] - Online only). The bisector phase measurements show that, if we attribute a given phase to the particular atmospheric height, many Nd lines should originate in extremely thin separate layers, which is impossible to explain. Due to the slow rotation and a rather weak magnetic field, HD 12932 is useful for demonstrating the effect of differential extra broadening of the pulsating lines. Figure \[hd12932\_prof\] presents a comparison between the line profiles of different elements/ions, scaled to the same central line depth. The wavelength scale is given in [kms$^{-1}$]{}, with line centre at zero velocity. The two iron lines with zero and large Zeeman broadening are shown to assess the expected magnetic field broadening effect. These lines, Fei 5434 Å and Fe 6432 Å, do not require any additional broadening other than Zeeman, instrumental, and rotational ones to fit the observed line profiles. The Eu line broadening is explained by the combined Zeeman and hyperfine splitting. Our spectral synthesis shows a growth in the from Nd lines (3–5 [kms$^{-1}$]{}), to Nd lines (6–9 [kms$^{-1}$]{}), Pr lines (7–10 [kms$^{-1}$]{}), and finally, to Tb lines (10–11 [kms$^{-1}$]{}). The stronger line of the same ion, the larger is required to fit the line profile. The Th lines in HD12932 are too weak for reliable measurements. The Y 5087 Å line exhibits an extra broadening similar to Nd lines, although no definite pulsation signal was detected in all measured Y lines, as well as in the lines of Na, Mg, Si, Ca, Sc, Cr, Fe, Co, Ni, and Sr. No variability is detected in the Fei 5434 Å line (RV amplitude 7$\pm$15 [ms$^{-1}$]{}). Thus, we do not confirm results by Kurtz et al. ([@KEM05a]), who claim a 3$\sigma$ detection of pulsations in this line. We looked at 47 other lines of Fei and Fe, and we detected a pulsation signal exceeding the 3$\sigma$ significance level only in seven features. For most of these lines weak variability is explained by blending with weak REE lines. Three out of six measured Ti lines show a pulsation signal with the RV amplitudes 80–100 [ms$^{-1}$]{}, while two Ti lines did not reveal any variability. All three Balines show RV amplitudes of $\approx$200 [ms$^{-1}$]{}. HD 201601 ($\gamma$ Equ) {#gequ} ------------------------ The amplitude-phase diagrams for the centre-of-gravity measurements in $\gamma$ Equ are shown in Fig. \[gequ-rv\] (Online only; left panel). Rather low spectral resolution of the $\gamma$ Equ observations used for this type of pulsation measurements does not allow us to make a precise bisector study. For the bisector analysis we used the time-series observations of $\gamma$ Equ carried out with the Gecko spectrograph at CFHT (see Kochukhov et al. [@KR07]). The bisector amplitude-phase diagrams are displayed in Fig. \[gequ-rv\] (right panel). If in the previous star, HD 12932, the RV amplitude remained nearly constant above a certain atmospheric height, in $\gamma$ Equ we observed a continuous decrease of RV amplitude outward. The bisector amplitude-phase diagrams are fairly similar in both stars, showing the same constancy in the phase measured across Nd spectral lines and a phase shift between lines (Fig. \[HD201601\_int\] - Online only). This phase shift is larger in $\gamma$ Equ. Again, Y lines have very small RV amplitudes, but their pulsation phases are the same as in the strongest Nd and Pr lines. Extra broadening of the REE lines in $\gamma$ Equ was discussed by Kochukhov & Ryabchikova ([@KR01a]), who showed that =10 [kms$^{-1}$]{} is needed for fitting strong Nd and Pr lines. HD 19918 {#HD19918} -------- Pulsational characteristics of HD 19918 (Fig. \[hd19918-rv\]) are similar to both HD 12932 and, in particular, to $\gamma$ Equ. An even more rapid decrease in the RV amplitude is observed in the atmosphere above a certain layer. The three stars show the largest RV amplitudes and the largest variation in the bisector velocity amplitudes from the line centre to the line wings (Fig. \[HD19918\_int\] - Online only). In all these stars we find first an increase in the RV amplitudes with approximately constant phases (standing wave), and after that the pulsation wave transforms into a running one. As in HD 12932, differential extra broadening is required to fit line profiles: =0 [kms$^{-1}$]{} (Fe), =2–3 [kms$^{-1}$]{}(Y), =4 [kms$^{-1}$]{} (Eu), =7–8 [kms$^{-1}$]{}(Nd), =8–10 [kms$^{-1}$]{} (Nd and Pr), and =11 [kms$^{-1}$]{} (Tb). Note that a smaller broadening is necessary to fit the line core than the line wings. Most line profiles show a triangular shape. Kurtz et al. ([@KEM05a]) report a detection of weak RV oscillations in the Fei5434 Å line. Our measurements confirm this result and also provide the following pulsation RV amplitudes for light and iron-peak elements: 30–50 [ms$^{-1}$]{} for Na, Ca, Cr, Fe lines; $\approx$100 [ms$^{-1}$]{} for Ba lines; 150–200 [ms$^{-1}$]{} for Tilines. HD 9289 {#HD9289} ------- The amplitude-phase diagrams for the centre-of-gravity measurements in the spectrum of HD 9289 are shown in Fig. \[hd9289-rv\]. Because of the low S/N and significant rotation compared to other stars, we did not perform bisector measurements. The amplitude-phase diagrams for this star are similar to those for HD 12932. HD 137949 (33 Lib) {#33Lib} ------------------ 33 Lib shows the most complex pulsational behaviour among all roAp stars. Mkrtichian et al. ([@MHK03]) find nearly anti-phase pulsations of Nd and Nd lines, which they attribute to the presence of pulsation node high in the atmosphere of 33 Lib. This was confirmed by Kurtz at al. ([@KEM05b]), who also find that in some REE lines the main frequency, corresponding to 8.27 min, and its harmonic have almost equal RV amplitudes. The high accuracy of the present observations allows us to study pulsational characteristics of the 33 Lib atmosphere in more detail. Figure \[33Lib-int\] shows RV amplitudes (top) and phases (bottom) measured for the main period as a function of the central intensity of spectral lines. The NLTE Nd stratification analysis by Mashonkina et al. ([@MRR05]) for HD 24712 shows that central intensities of both Nd and Nd lines are roughly proportional to there centre-of-gravity depth formation. Taking into account that the REE anomaly on which Nd stratification was based is the common feature for all roAp stars (Ryabchikova et al. [@RSMK01], [@RNW04]), we can expect similar REE gradients in the atmosphere of 33 Lib, hence, the similar dependence of the central intensities on the optical depth. The phase jump is defined very well by Nd lines and is also indicated by the measurements of Pr, Tb, Dy lines. Our results show that the phase jump (radial node) does not separate the formation regions of the lines in consecutive ionisation states as is claimed by Kurtz at al. ([@KEM05b]). Nd, Pr, and Th lines are observed from both sides of the jump. Moreover, a position of the phase jump relative to line depth provides some evidence that abundance distributions for at least Pr, Nd, Tb, Dy are similar, hence one may expect abundance jumps at approximately the same optical depths in 33 Lib atmosphere. Bisector measurements support the results obtained for the main frequency from centre-of-gravity measurements and allow us to carry out a more detailed study of the pulsation properties in 33 Lib. Figure \[33Lib-harm\] shows bisector amplitudes (top) and phases (bottom) measured in a few representative spectral lines for the main period 8.27 min (left panels) and its harmonic (right panels). We have to emphasise here a major difficulty in the bisector measurements for the lines splitted in strong magnetic field. Figure \[33Lib-prof\] shows intrinsic theoretical profiles of the lines from Fig. \[33Lib-int\], calculated with the abundances that reproduce the observed line intensity. Only the Nd 5320 Åline does not exhibit resolved Zeeman components in the intrinsic profile, therefore this line may provide the ‘purest’ pulsation distribution across the line profile. Unfortunately, no strong Nd or Pr line has negligible Zeeman splitting. The best and strongest Pr 5300 Å and Nd 5294 Å lines both have significant splitting and rather different Zeeman patterns. Moreover, the Nd 5294 Å line is blended in both wings, and these blends may distort the observed pulsation effects. In particular, it concerns the RV amplitudes and, to a lesser extent, pulsation phases. Therefore we chose another Nd 6327 Å line as a representative of Nd strong lines. Note that spectral lines with similar Zeeman patterns and similar intensities (Nd5677, 5802, 5851 Å) have the same bisector velocity amplitude and phase distributions. Pulsation phases are still similar for the lines with similar Zeeman patterns but with different intensities (Nd 6327 Å and Pr 5300 Å). Therefore, here we discuss mainly pulsation phase distribution in the atmosphere of 33 Lib. Across the profiles of all strong Pr and Nd lines and the Nd 5320 Å line, one phase jump is observed for the main period and two phase jumps are detected for the first harmonic. Kurtz et al. ([@KEM05b]) find only one phase jump for the first harmonic using bisector measurements of the Nd6145 Å line. The best pulsation distribution is defined by the bisector measurements in the Pr5300 Å line, where it is clearly seen that the position of the phase jumps corresponds to nearly zero RV amplitudes, indicating two radial nodes. Due to the diversity of the pulsational characteristics, it is difficult to compare pulsation phases for the lines of different species. However, a phase distribution for the main period (Fig. \[33Lib-int\], bottom left panel) suggests the existence of the $\pi$-radian jump between Nd and Nd lines, which was found earlier by Mkrtichian et al. ([@MHK03]), and another $\pi$-radian jump between the Eu 6645 Å and Dy 5730 Ålines. Comprehensive analysis of the variability of many REE lines in 33 Lib allowed us to obtain a refined picture of the pulsation node in high atmospheric layers. In particular, we show (see Fig. \[33Lib-int\], upper panel) that Nd and Nd lines do not simply show a $\pi$-radian phase difference, but a continuous phase trend exists in the RV curves of Nd lines, with some of the doubly ionised Nd lines (e.g., 5286 and 6690 Å) pulsating with phases typical of Nd. This appears to be the first demonstration of the existence of a pulsation node within the formation region of the lines belonging to the same REE ion. Another remarkable observation that we made is that harmonic RV variations are the strongest in the lines that form close to the position of the node. Thus, the mechanism exciting the first harmonic is probably directly related to the presence of a radial node in the upper atmosphere of 33 Lib. The phases of the [H$\alpha$]{} core lie between the phases of the Euline and the wing of the Dy 5730 Å line. Interestingly, the very core of [H$\alpha$]{} has the highest RV amplitude, 500 [ms$^{-1}$]{}, and the last measured point in the Dy line wing has very high RV amplitude, too, although this measurement is not accurate enough. Nevertheless, this gives us a hint of the line formation depths in the atmosphere of 33 Lib. Without detailed analysis of the chemical structure of the atmosphere, it is difficult to suggest a pulsational scenario for the star, but it is evident that we observe several waves over the whole atmosphere. The phase distribution across the Pr and Nd spectral lines allows us to estimate the pulsation wave speed. Assuming that we have one full harmonic wave over the Nd and Pr lines and considering the geometric size of the line-formation region in the stratified case (Mashonkina et al. [@MRR05]; Ryabchikova et al. [@RMR06]), we estimated the pulsation wave speed as less than 6 [kms$^{-1}$]{} for the harmonic period 4.136 min. It is lower than the sound speed in the 33 Lib atmosphere in adiabatic approximation and is similar to the pulsation wave speed in another roAp star, HD 24712 (Ryabchikova et al. [@RSW06]). The magnetic field, which is stronger in 33 Lib than in any other programme star, may be a reason for the difference in the observed distribution of the pulsational characteristics over the atmospheres of 33 Lib and HD 24712. What conclusions can be made from these results? Calculations of linear and non-linear radial pulsations (for example, Fadeev & Fokin [@FF85]) in adiabatic approximation show that the wave speed may differ from the sound speed because of the reflection conditions and of the finite amplitude. Therefore, our results cannot give any proof of the non-adiabaticity of the pulsations in roAp stars. However, that the pulsation wave speed is close to the sound speed strongly supports an acoustic nature for the pulsations and an absence of the shock waves. While no extra broadening is required to fit the Fe-peak lines, 3–4 [kms$^{-1}$]{} is necessary for Y, Eu, Gd, and Er lines. For Nd and weak Nd and for Pr lines, one needs 6–8 [kms$^{-1}$]{}, and, finally, for the strong Nd and Pr lines we have to introduce =10–12 [kms$^{-1}$]{} to fit the observed profiles. Rather high amplitude pulsations were detected in both Th lines with the phases corresponding to those for Tb lines. The RV amplitudes of the main frequency derived for the lines of Na, Mg, Si, Ca, Sc Cr, Ti, Fe, Ni lie in the 30–50 [ms$^{-1}$]{} range, while the amplitude of the harmonic is below the detection limit of $\sim$10 [ms$^{-1}$]{}. The Ba lines also reveal pulsations at the main frequency, but with higher RV amplitudes (70–100 [ms$^{-1}$]{}). The Ylines have similar amplitudes, which seem to grow with the line intensity. The Y lines also show detectable harmonic amplitudes. All these lines show pulsation phases in the range of pulsation phases for the singly ionised REEs. Summary and discussion {#disc} ====================== We have analysed spectroscopic pulsational variability of ten roAp stars in detail. For each object, several hundred spectral lines were measured and the resulting time series of radial velocity and bisector variation of selected lines were interpreted with the help of the least-square fitting technique. We confirm results of the previous studies, which suggest that non-radial oscillations are primarily detectable in the lines of heavy elements, especially in the doubly ionised REE lines. This phenomenon is attributed to the extreme chemical stratification present in the atmospheres of cool Ap stars. Due to the segregation of elements under the influence of radiative diffusion and due to lack of mixing, a thin layer enriched in heavy elements is created in the upper atmosphere, where rather weak roAp non-radial oscillations attain significant amplitudes. Stellar atmosphere modelling that includes stratification, NLTE, and magnetic field effects is an extremely complex problem. Up to now only the first preliminary models of different element distributions over stellar atmosphere based on the spectroscopic observations were calculated for a few cool Ap stars. Self-consistent diffusion calculations (LeBlanc & Monin [@LM04]) provide the next step in our understanding of the Ap star phenomenon. However, with the diffusion calculations for 39 elements from He to La, LeBlanc & Monin’s models do not include REE elements, which show the most outstanding stratification signatures in roAp atmospheres. The predicted stratification of La is far from the observed one because a small number of the ionisation states (two) is taken into account and NLTE effects are not considered. Empirically derived Nd abundance distribution shows directly that NLTE effects play a crucial role in the REE stratification studies (Mashonkina et al. [@MRR05]). Based on the empirical element distribution in the atmospheres of two roAp stars, HD 24712 and $\gamma$ Equ, we conclude that pulsation amplitudes and, in particular, phases derived from the lines of different elements correlate with the optical depth. Therefore, in the absence of the proper atmospheric models, the amplitude-phase diagrams proposed in the present paper are a powerful tool in the study of vertical structure of $p$-modes in the atmospheres of roAp stars. Our pulsation analysis of the radial velocity variations demonstrates similarity in the atmospheric pulsation characteristics. With the help of amplitude-phase diagram analysis we find that pulsation waves exhibit either a constant phase and amplitude changing with height or depth-dependence of both parameters. We interpret the former as a signature of standing pulsation wave and the latter as evidence of a travelling (running) wave in stellar atmosphere. Thus, in general, atmospheric pulsational fluctuations in roAp stars can be represented by a superposition of standing and running waves. According to our results, pulsation waves in three roAp stars, HD 24712, HD 134214, and $\alpha$ Cir, with the pulsation frequency close to or below the acoustic cut-off limit, have the running-wave character from the low atmospheric heights. In the longer period stars, standing waves are observed up to some of the atmospheric layers, defined by the formation depth of the lines of specific elements, while running waves dominate higher in the atmosphere. We also find that the change from the standing to running character of pulsation waves depends on the effective temperature of roAp star. Hotter stars seem to develop the running wave deeper in the atmosphere than the cooler stars with the same pulsation periods. In all stars but 33 Lib, independent of the atmospheric and pulsation parameters, pulsation measurements reveal waves travelling through the layers defined by the same sequence of chemical species. The lowest amplitudes are observed for Eu lines, then pulsations propagate in the layers where H$\alpha$ core, Nd, and Pr lines originate. Pulsation amplitude reaches maximum around these atmospheric heights and then decreases outwards in most stars. The RV extremum of the second REE ions is always observed later in time relative to the variation of singly ionised REEs. The largest phase shifts and amplitudes are often detected in Tb and Thlines. Pulsational variability in the latter lines is detected here for the first time. Similarity of the pulsation wave propagation signatures in the studied roAp stars suggests that layers enriched in different REE species are arranged in approximately the same vertical order in all stars. For two stars, 33 Lib and HD19918, we find weak but definite RV variability in the lines of iron-peak elements, confirming previous results by Mkrtichian et al. ([@MHK03]) and Kurtz et al. ([@KEM05b]), respectively. In all stars, spectral lines with the highest RV amplitudes have an additional broadening varying from 4 to 10–12 [kms$^{-1}$]{} in terms of the macroturbulent velocity required to reproduce the observed line profile shapes. The running wave characteristics usually appear in the lines with $\ge$10 [kms$^{-1}$]{}. According to Kochukhov et al. ([@KR07]), these are the lines where the usual symmetric pulsational line profile variability is transformed into an asymmetric blue-to-red moving pattern. These authors propose pulsational variability of the line widths, arising from the periodic expansion and compression of the turbulent layers in the upper atmospheres of roAp stars as an explanation for the asymmetric line profile variability pattern. The present study shows that the position of the turbulent layer in the roAp-star atmospheres is defined by the formation depth of Pr lines in a cooler part of the programme stars and by the formation depth of Nd lines in hotter stars. This turbulent layer, which is probably related to the REE abundance gradients in the upper atmosphere (Mashonkina et al. [@MRR05]), seems to be the key element in the depth-dependence of the spectroscopic pulsation characteristics in roAp stars. Abundance gradients may cause a non-standard temperature structure in Ap-star atmospheres. This is supported by an atmospheric anomaly that required to explain the peculiar core-wing transition of the Balmer lines (Kochukhov et al. [@cwaT]). Therefore, the comprehensive investigation of the roAp-star atmosphere, including chemical diffusion and magnetohydrodynamic modelling of the interaction of convection, pulsation and magnetic field, is necessary to fully understand the full variety of the phenomena associated with a turbulent REE-rich cloud in cool Ap stars. One of the sample stars, 33 Lib, differs from all other roAp stars in that, at a given atmospheric heights, it shows a comparable RV amplitude of the main frequency and its first harmonic. Comprehensive analysis of the REE lines of several elements in both ionisations stages shows that harmonic oscillation emerges close to the position of the pulsation node located within the REE-rich high atmospheric layer. This follows from the observation that the lines showing large double-wave variation are all located close to the minimum amplitude and to a $\pi$-radian phase jump of the main frequency. Thus, the physical mechanism giving rise to harmonic spectroscopic variability in 33 Lib must be closely related to the existence of the radial node. Based on the observation of the two radial nodes for the harmonic variation across the profiles of a few strongest Nd and Pr lines, we can estimate a radial wavelength and pulsation wave speed. The latter is below the adiabatic sound speed in the atmosphere of 33 Lib. The same result was obtained earlier for another roAp star, HD 24712, using a different approach (Ryabchikova et al. [@RSW06]). These results support an acoustic nature for the pulsations and reject the idea of the shock wave proposed by Shibahashi et al. ([@SKK04]) for interpreting of the blue-to-red pulsation pattern, because this model requires supersonic pulsation motions. 33 Lib is the only star showing definite and direct evidence of radial nodes in the atmosphere. But the absence of a measurable pulsation signal in the lines of the elements with Z$\le$38 in most of our programme stars poses a question about the existence of another node in the lower photospheric layers. Abundance stratification analyses of cool Ap stars (Ryabchikova et al. [@RPK02], [@RLK05]; Kochukhov et al. [@KTR06]), as well as the model atmosphere with self-consistent diffusion calculations (LeBlanc & Monin [@LM04]), show that light and iron-peak elements have a tendency to concentrate in the deeper atmosphere, below $\log\tau_{5000}=-1$, and to be strongly depleted in the outer atmospheric layers. Therefore, practically all Fe-peak lines observed in the optical spectral region are formed in the layers $-1.5<\log\tau_{5000}<0.0$. The non-adiabatic theoretical pulsation model for HD 24712 supports the existence of the nodal region near the photosphere ($\log\tau_{5000}\sim0$) (see Fig. 3 in Sachkov et al. [@SR06]). The position of the radial node depends on the frequency (or the radial order) of the mode. As the frequency increases, the node shifts outwards (Saio, private communication). According to this model, in the roAp stars HD 24712, HD 134214, and $\alpha$ Cir, the node is located close to, or slightly above, the continuum-forming layers. On the other hand, in the longer-period stars the node should be located below the photosphere (see Figs. 3, 4 in Saio & Gautschy [@SG04]; Fig. 8 in Saio [@S05]). For some intermediate pulsation frequencies, we should detect pulsation signal in the region of the iron-peak line formation, but this is not observed. Perhaps, the pulsation amplitude in the lower atmosphere is diminished to such an extent that pulsations are undetectable even in anti-node regions. A discovery of the low-amplitude pulsations in Y lines in phase with the highest-amplitude Pr lines delineates another problem. In the three stars with the shortest pulsation periods, HD 24712, HD 134214, and $\alpha$ Cir, the Y lines produce a secondary minimum in the amplitude-phase diagrams. These lines have the lowest detected RV amplitudes and are out of phase by $\pi$-radian with some other low-amplitude lines, for instance, Eu. Does this indicate a radial node near the Eu line formation heights or do we see a double-wave structure in the amplitude-phase diagram depending on the chemical structure of stellar atmospheres? The amplitude-phase diagrams derived for as many elements and spectral lines as possible are proven to be an extremely powerful tool for investigating the pulsation properties of roAp atmospheres. Detailed inferences about the vertical mode structure obtained in our study call for in-depth theoretical investigation of the propagation of pulsation waves in magnetically dominated and chemically stratified atmospheres. The most important and difficult theoretical challenge is to recognise and model the physical processes that are responsible for remarkable observation of pulsation wave transformation in the REE-rich layer. Why does the dominant character of pulsational perturbation changes from standing to running wave? What causes pulsation amplitude to diminish about certain atmospheric height? Why does isotropic turbulence increase dramatically in this layer? And finally, what is the origin of the complex bisector behaviour observed in several roAp stars? Substantial theoretical developments are needed to resolve these issues. We are thankful to A. Fokin and H. Saio for very useful discussions on modelling pulsations in stellar atmospheres. Resources provided by the electronic databases (VALD, SIMBAD, NASA’s ADS) are acknowledged. This work was supported by the research grants from RFBI (04-02-16788a, 06-02-16110a), from the RAS Presidium (Program “Origin and Evolution of Stars and Galaxies”), from the Swedish Kungliga Fysiografiska Sällskapet and Royal Academy of Sciences (grant No. 11630102), and from Austrian Science Fund (FWF-P17580). Audard, N., Kupka, F., Morel, P., Provost, J., & Weiss, W.W. 1998, , 335, 954 Babel, J. 1992, , 258, 449 Baldry, I. K., Bedding, T.R., Viskum, M., Kjeldsen, H., & Frandsen, S. 1998, , 295, 33 Baldry, I. K., & Bedding, T. R. 2000, , 318, 341 Balona, L. A. 2002, , 337, 1059 Biémont, E., Palmeri, P., & Quinet, P. 1999, , 635, 2691. Burki, G. et al. 2005, GENEVA photometric database, Geneva Observatory Cowley, C. R., Hubrig, S., Ryabchikova, T., et al. 2001, , 367, 939 Cowley, C. R., Ryabchikova, T., Kupka, F., et al. 2000, , 317, 299 Den Hartog, E. A., Lawler, J. E., Sneden, C., & Cowan, J. J. 2003, , 148, 543 Den Hartog, E. A., Lawler, J. E., Sneden, C., & Cowan, J. 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Ser., 310, 208 Sachkov, M., Ryabchikova, T., Bagnulo., et al. 2006, MemSAI, 77, 397 Saio, H. 2005, , 360, 1022 Saio, H., & Gautschy, A. 2004, , 350, 485 Savanov, I. S., Malanushenko, V. P., & Ryabchikova, T. R. 1999, Astron. Lett., 25, 802 Shibahashi H., Kurtz D. W., Kambe E., Gough D. O. 2004, in [IAU Symposium No. 224, The A-star Puzzle]{}, eds. J.Zverko, J.Žižňovský, & S.J.Adelman, W.W.Weiss, Cambridge University Press, IAUS 224, 829 Tsymbal, V., Lyashko, D., & Weiss, W. W. 2003, in [Modelling of Stellar Atmiiospheres]{}, IAU Symp. No. 210, eds. N. Piskunov, W.W.Weiss, D.F. Gray, ASP, E49 [^1]: Based on observations made with the SAO 6-m telescope, with the Canada-France-Hawaii Telescope, and the ESO VLT (DDT programme 274.D-5011 and programme 072.D-0138, retrieved through the ESO archive). [^2]: [http://obswww.unige.ch/gcpd/ph13.html]{}
--- abstract: 'With the advent of the next generation wide-field cameras it became possible to survey in an unbiased mode galaxies spanning a variety of local densities, from the core of rich clusters, to compact and loose groups, down to filaments and voids. The sensitivity reached by these instruments allowed to extend the observation to dwarf galaxies, the most “fragile” objects in the universe. At the same time models and simulations have been tailored to quantify the different effects of the environment on the evolution of galaxies. Simulations, models, and observations consistently indicate that star-forming dwarf galaxies entering high-density environments for the first time can be rapidly stripped from their interstellar medium. The lack of gas quenches the activity of star formation, producing on timescales of ${\sim}$1 Gyr quiescent galaxies with spectro-photometric, chemical, structural, and kinematical properties similar to those observed in dwarf early-type galaxies inhabiting rich clusters and loose groups. Simulations and observations consistently identify ram pressure stripping as the major effect responsible for the quenching of the star-formation activity in rich clusters. Gravitational interactions (galaxy harassment) can also be important in groups or in clusters whenever galaxies have been members since early epochs. The observation of clusters at different redshifts combined with the present high infalling rate of galaxies onto clusters indicate that the quenching of the star-formation activity in dwarf systems and the formation of the faint end of the red sequence is a very recent phenomenon.' author: - Alessandro Boselli - Giuseppe Gavazzi date: 'Received: 10/08/2014 / Accepted: 27/08/2014' title: 'On the origin of the faint-end of the red sequence in high density environments. ' --- [example.eps]{} gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore Introduction {#s1} ============ In 2006, the present authors reviewed the “Environmental effects on late-type galaxies (LTGs) in nearby clusters” (Boselli & Gavazzi 2006). Is there a compelling urgency for a new review on such a short time lag? The answer is yes, because the DR7 release of the Sloan Digital Sky Survey (SDSS, Abazajian et al. 2009) which disclosed the complete northern sky to photometric and spectroscopic observations to limits as faint as 17.7 ($r$) mag, was yet to come in 2006. Beside the SDSS, several panoramic multifrequency surveys of large stretches of the local Universe became available after 2006 (we will review them in Sect. \[surveys\]). The crucial novelty of these surveys is that they allowed for the first time to sample clusters of galaxies [at large]{}, embedded in surrounding regions of relatively low galactic density, making it possible to contrast directly, not only by statistical means, the galaxy properties in ambients of significantly different density: cluster cores, cluster outskirts, loose groups, filaments, and voids. Indeed, many other significant progresses in this field were achieved after 2006, concerning both the observations and the simulations which brought a deeper conviction that the environment—[nurture]{}—plays a relevant role on the evolution of galaxies, especially at the low-mass end. It is on the new evidences that the present review is focused. As in Boselli & Gavazzi (2006), Virgo, Coma, and A1367 and their surrounding superclusters, being among the best studied clusters, will dominate the discussion, and will be treated as representative of the local Universe. The key motivation for environmental issues on galaxy evolution is that, no matter if galaxies spend most of their life in relatively low-density regions of space (the filaments of the cosmic web), a large fraction of them is sooner or later processed through denser (groups) and denser (clusters) environments. This picture clearly emerges from numerical simulations in the $\Lambda {CDM}$ cosmology (e.g. the Millennium simulation, Springel et al. 2005), where satellite galaxies continuously feed regions of higher density (central galaxies, groups, clusters) at the intersection of multiple filaments. Even at $z=0$ there is substantial evidence for high-velocity infall of “healthy” (star forming) galaxies into rich clusters, which also arises from the anisotropy of the velocity distribution of LTGs compared to early-type galaxies (ETGs) (see Fig. 14 of Boselli & Gavazzi 2006). Whereas ETGs obey to a Gaussian distribution, LTGs tend to populate the wings of the distribution at high and low velocity. Considering the time scales for the various transformations, Boselli et al. (2008a) and Gavazzi et al. (2013a,b) were able to estimate that infall on the Virgo (Coma) cluster occurred at a rate of approximately 300–400 (100) galaxies with mass $M_{\rm star} \gtrsim 10^9\;\hbox{M}_{\odot}$ per Gyr in the last 2 (7.5) Gyrs. ![([Top]{}) The SDSS distribution of galaxies in the central region ($40\times 25 $ arcmin) of A1367. [Boxes]{} highlight galaxies showing evidence of high-velocity first pass through the cluster IGM. The [color inset]{} contains a 1.4 GHz VLA continuum map showing extended trailing emission behind three galaxies CGCG 97073, 97079 and 97087 (adapted from Gavazzi et al. 1995). The [rectangle labeled]{} BIG lies close to the X-ray cluster center. It contains a compact group of galaxies infalling onto the cluster (Cortese et al. 2006a, b). ([Bottom]{}) The head-tail galaxy FGC1287 (VLA-B) and CGCG 97026 (VLA-A) are found just outside one virial radius of A1367 (adapted from Scott et al. 2012). X-ray emission (ROSAT) from the hot ICM is indicated with cyan contours. Reproduced with permission of Oxford University Press[]{data-label="f1"}](fig1.pdf){width="100.00000%"} ![Contours of the smoothed density distribution of SDSS galaxies in a $9\times 3.5^\circ$ region around A1367. The large-scale distribution of galaxies in the Coma supercluster form a connected structure elongated in the E–W direction. Galaxies 97073 and 97079 (with radio and $\hbox{H}\alpha$ tails) and the head–tail radio galaxy 97127 near the center of the main cluster are highlighted. The location of the newly discovered HI head–tail galaxies FGC 1287 and CGCG 97026 at the W periphery of the cluster are also marked, as well as the wide-angle tail CGCG 98040 (in the NGC 4065 group at the E of the cluster) []{data-label="f3"}](fig2.pdf){width="100.00000%"} We wish to begin and to close this review with Abell 1367, the prototypical laboratory for the study of galaxies under the influence of hydrodynamical processes. Figure \[f1\] shows the distribution of its central galaxies from the SDSS. Also marked are two faint $(M_{\rm star} \sim 10^{9.5}\;\hbox{M}_{\odot}$) CGCG galaxies 97073, 97079 next to 97087 that have projected distances less than one Mpc from the cluster X-ray center (well inside the virial radius of A1367 that was estimated 2.1 Mpc by Boselli & Gavazzi 2006). Gavazzi & Jaffe 1985 discovered their “head–tail” radio continuum emission (see color inset from Gavazzi et al. 1995) and Gavazzi et al. (2001) detected $\hbox{H}\alpha$ emission trailing behind them for ${\sim}$75 Mpc, an evident sign of high-velocity interaction with the ICM. The inset labeled “BIG” contains a compact group of blue galaxies infalling at $2{,}500\,\hbox{km}\,\hbox{s}^{-1}$ into the main cluster (Sakai et al. 2002; Gavazzi et al. 2003b; Cortese et al. 2006a,b). Next to BIG, the elliptical galaxy NGC3862 (CGCG 97127), the brightest cluster member, harbors the head–tail radio galaxy 3C 264 (not shown in Fig.\[f1\]). The striking new feature of A1367 is the discovery by Scott et al. (2012) of two “HI head-tail” galaxies FGC1287 and CGCG 97026 at the cluster periphery (just outside the virial radius) reproduced in Fig.\[f1\]. This is perhaps not too surprising, as recent hydrodynamical simulations (Tonnesen & Bryan 2009; Bahe et al. 2013; Cen et al. 2014) claim that ram-pressure is effective out to 2–3 times the virial radius of a cluster. Moreover Book & Benson (2010), confirmed by observational studies of satellite galaxy SFR versus cluster-centric radius, suggest that quenching of the SFR relative to the field takes place at similarly large clustercentric projected distances (Balogh et al. 2000; Verdugo et al. 2008; Braglia et al. 2009). Moreover, there is evidence that ram pressure becomes effective at lower density than previously assumed (Bekki 2009). See, for example, Freeland et al. (2010) who studied the effects of the IGM in the poor group NGC4065 belonging to the Coma supercluster, visible just $4^\circ$ to the E of A1367 in Fig.\[f3\]. FGC1287 ($M_{\rm star} \sim 10^{9.9}\;\hbox{M}_{\odot}$) lies a little over one virial radius away from the cluster center, i.e. approximately one Mpc away from the hot gas in A1367, as mapped by [ROSAT]{}. We remind, however, that the map reproduced in Fig. \[f1\] underestimates the real extent of the X-ray emission from A1367, as a deeper [XMM]{}-Newton observation reveals (Finoguenov private communication). Environmental effects (gas deficiency and star-formation quenching) in the Virgo cluster are detected more than one virial radius away from M87 (Gavazzi et al. 2012). It is difficult to disentangle whether the HI-tail of FGC1287 is triggered by the IGM associated with A1367 or with its hosting group. Looking at the smoothed distribution of SDSS galaxies shown in Fig.\[f3\] it appears that galaxies around A1367 form a continuum structure with contiguous groups, elongated in the E–W direction generally traced by the Coma Supercluster as a whole. One group to the W contains 97026, the newly discovered HI tail, and another group to the E (NGC4065) contains a well-known wide-angle radio galaxy associated with 98040 (Jaffe & Gavazzi 1986). What is perhaps more surprising is that even evolved clusters like Coma, as soon as their galaxies are observed with sufficiently long exposures, reveal the presence of trailing emission (star-forming trails or ionized gas) (Yagi et al. 2010; Yoshida et al. 2012; Fossati et al. 2012), witnessing profound ongoing environmental transformations affecting many dwarf and some massive LTGs. Summarizing, adding the aforementioned cases to other well-known examples in the Virgo cluster (e.g. Vollmer et al. 2001) there is multiple evidence that today’s clusters harbor many actively star-forming galaxies in their first-time high-velocity pass through the dense and hot IGM. Our aim is to show that they are being quickly transformed into passive systems under the influence of the dynamical pressure. We notice that most of them have stellar masses $M_{\rm star} \lesssim 10^{10}\;\hbox{M}_{\odot}$, which is tentatively assumed hereafter as an empirical separation between dwarf and giant galaxies. Dwarf galaxies, the most common objects in the universe, have a very important role for understanding the processes that gave birth to the local evolved stellar systems. Models of galaxy evolution consistently indicate that they are the building blocks of massive objects, formed by subsequent merging events. Only recently these objects became accessible to systematic observations outside the Local Group. Given their shallow potential wells, these “fragile” stellar systems provide us with a sensitive probe of their environment. Up to approximately 10 years ago, dwarf galaxies were supposed to belong to two main sequences: Magellanic Irregular (Im) and blue-compact-dwarf (BCD), actively star forming, dominating the field on one hand, and quiescent dwarf elliptical (dE) and spheroidal (dS0), abundant in clusters on the other. The advent of large panchromatic and kinematic surveys allowed us to realize that some Im found in clusters are in fact completely quiescent. An example is VCC1217 (IC1318) (see Fig.\[f1217\]), located approximately $1.5^\circ$ south of M87 in the Virgo cluster. In this dwarf ($M_{\rm star} \sim 10^{8.9}\;\hbox{M}_{\odot}$) irregular LSB galaxy the star formation is absent from the disk, being probably truncated recently, as testified by its PSB-like spectrum. This object received recent attention due to the presence of a long tail of star-forming blobs first revealed by GALEX, and trailing behind it, a clear signature of an ongoing ram-pressure stripping event (Hester et al. 2010; Fumagalli et al. 2011; Kenney et al. 2014). ![[Left panel]{} RGB image of VCC1217 in the Virgo cluster obtained with $u,g,i$ NGVS images with superposed contours obtained combining two long GALEX NUV exposures of 16,000 and 4,500 s, respectively. The integrated spectrum of the galaxy ([right panel]{}) is typical of a post-starburst[]{data-label="f1217"}](fig3a.pdf "fig:"){width="60.00000%"} ![[Left panel]{} RGB image of VCC1217 in the Virgo cluster obtained with $u,g,i$ NGVS images with superposed contours obtained combining two long GALEX NUV exposures of 16,000 and 4,500 s, respectively. The integrated spectrum of the galaxy ([right panel]{}) is typical of a post-starburst[]{data-label="f1217"}](fig3b.pdf "fig:"){width="40.00000%"} On the other hand, only a minority of dEs can be recognized as the low-mass counterparts of giant elliptical galaxies. Some have a complex morphology (e.g. pseudobulges, inner disks, spiral arms), some rotate (Toloba et al 2009; 2011; 2012), some contain dust, gas, and star formation in their center. Recent observations (Boselli et al. 2008a,b; Gavazzi et al. 2010) suggest that dEs, and ultimately the faint-end of the red sequence, can result from the recent migration of faint star-forming galaxies through the “green-valley”. This is sketched for the Virgo cluster on the color–stellar mass relation shown in Fig.\[modellisam\] using the most recent spectrophotometric models for the evolution of galaxies in rich environments. Two such cases are offered by VCC1491 and VCC1499 reproduced in Fig.\[e1499\]. They are located within $1^\circ$ projected distance from M87 in the Virgo cluster. Seen on the exquisite B-band photographic plate taken by Binggeli et al. (1985) for the construction of the Virgo Cluster Catalog (VCC) they look morphologically identical and they were both classified as dEs. In spite of their similar stellar mass \[$M_{\rm star}(1491) = 10^{8.62}\; \hbox{M}_{\odot}, M_{\rm star}(1499) = 10^{8.23}\;\hbox{M}_{\odot}$\], when seen on multiband CCD images they appear dramatically different in color: $(g-i)_{1491}=1.06; (g-i)_{1499}=0.59$, and spectroscopically (see Fig.\[e1499\], right panel). While VCC1491 has a spectrum typical of a passive galaxy, VCC1499 has a post-star-burst (PSB) spectrum (both spectra are integrated over the whole galaxy). Unfortunately for neither galaxies kinematical measurements are available, but we would not be surprised if VCC1499 was a fast rotator (Cappellari et al. 2011b), i.e. a LTG recently converted into a ETG. ![[Left panel]{} RGB image of VCC1491 ([right-red]{}) and VCC1499 ([left-blue]{}) in the Virgo cluster obtained combining UV and optical NGVS images. Their integrated spectra ([right panel]{}) are characteristic of a red, passive galaxy (VCC1491, [top]{}) and of a PSB (VCC1499, [bottom]{})[]{data-label="e1499"}](fig4a.pdf "fig:"){width="60.00000%"} ![[Left panel]{} RGB image of VCC1491 ([right-red]{}) and VCC1499 ([left-blue]{}) in the Virgo cluster obtained combining UV and optical NGVS images. Their integrated spectra ([right panel]{}) are characteristic of a red, passive galaxy (VCC1491, [top]{}) and of a PSB (VCC1499, [bottom]{})[]{data-label="e1499"}](fig4b.pdf "fig:"){width="40.00000%"} Boselli et al. (2008a) and Gavazzi et al. (2010) argued that PSB (or $k+a$) galaxies (Poggianti et al. 2004) showing the characteristic blue continuum and strong Balmer lines in absorption might consist of galaxies undergoing the fast transition across the “green valley” due to an abrupt truncation of the SFR by ram-pressure. As remarked by these authors they come exclusively under the form of dwarfs (i.e. with stellar masses $M_{\rm star} \lesssim 10^{10}\;\hbox{M}_{\odot}$) in the outskirts of local rich clusters of galaxies. The present paper is conceived for reviewing the works done in the past decade on galaxy evolution in relation to the environment (we apologize for the missing references). We hope this primarily observational review will contribute at convincing the reader that environmental transformations should be taken into higher consideration as drivers of galaxy evolution. The advent of large-scale cosmological simulations including “gastrophysics” will allow to constrain the environmental effects in detail from a theoretical point of view as well, complementing the more acknowledged stellar and AGN feedback processes. Here we will skip a detailed discussion on the physical processes, as they were extensively treated in Boselli & Gavazzi (2006). We will focus instead mainly on observations (Sects. \[s2\], \[s3\]) leaving some room for comparison with models (Sects. \[s4\], \[s5\]) and concluding with evidences of evolution as a function of lookback time and density. Recent blind and pointed surveys of nearby clusters {#surveys} =================================================== \[s2\] Large-scale surveys ------------------- The study of environmental effects on the evolution of galaxies took advantage from several recent multifrequency surveys covering large portions of the sky.[^1] Among these the one that had certainly the major impact is the SDSS (York et al. 2000). Thanks to its photometric and spectroscopic mode in the optical domain, the SDSS allowed the observation of millions of galaxies in a vast luminosity interval, located in regions spanning a wide range of environments, from local voids to the core of the richest clusters. Early SDSS releases have been used to study the dependence of the structural, spectrophotometric, and star-formation properties of galaxies as a function of galaxy density, resulting, however, in controversial results on the role of the environment on galaxy evolution. On the one hand Kauffmann et al. (2004) found that “For galaxies in the range $10^{10} - 3\times 10^{10} M_\odot$ the median specific star-formation rate decreases by more than factor of 10 as the population shifts from predominantly star-forming at low density to predominantly inactive at high densities”. Conversely Hogg et al. (2004), beside confirming the morphology-density effect, do not find any further environmental difference once ETGs are separated from LTGs on the basis of their Sersic index.[^2] Similarly, Balogh et al. (2004) do not see a progressive reddening of galaxies as a function of local density. In other words, they do not find evidence for objects crossing the green valley in their way from the blue to the red sequence. The DR7 of the SDSS was released in 2009 (Abazajian et al. 2009) and included the complete photometry in the northern galactic cap; the DR9 came out in 2012 (Ahn et al. 2012) based on a new photometric pipeline; the DR10 in 2013 (Ahn et al. 2014), based on a renewed spectral pipeline. We remind that the SDSS spectral database is complete to $r=17.77$ mag (Strauss et al. 2002), except for “shredding” and fiber conflict effects (Blanton et al. 2005a,b). To avoid large galaxies whose photometry is uncertain because of these problems, most statistical studies that came out from the SDSS are limited to $z>0.05$. Other authors made a different use of the SDSS data: by directly analyzing the images and extracting magnitudes by hand, independently of the SDSS pipeline. With this approach Gavazzi et al. (2010, 2012, 2013a) took advantage of the SDSS superior material to directly compare the properties of galaxies in the centers of Coma, A1367, and Virgo with their isolated counterparts taken in the outskirts of these clusters. The GALEX mission (Martin et al. 2005) covered the entire sky in the far (FUV, $\lambda_{\rm eff}$ 1,539 Å) and near ultraviolet (NUV, $\lambda_{\rm eff}$ 2,316 Å) down to a limit of $\simeq$21 mag with an angular resolution of $\sim$5 arcsec. Sensitive to the emission of the youngest stars, GALEX provided the census of the star-formation activity in galaxies in the local universe. The sensitivity of the instrument in targeted observations combined with its large field of view (${\sim} 1\,\hbox{deg}^2$) allowed the detection of low surface brightness systems such as dwarf galaxies and tidal streams in several nearby clusters, making it an ideal instrument for the study of the effects of the environment in the local universe (Gil de Paz et al. 2007; Boselli et al. 2014a,b). The gaseous component, primary feeder of the star-formation process, was observed thanks to the Arecibo Legacy Fast Arecibo L-band Feed Array (ALFALFA),[^3] first announced by Giovanelli et al. (2005) and completed in 2012. This survey has mapped ${\sim}7{,}000\,{\rm deg}^2$ of the high galactic latitude sky visible from Arecibo (i.e in the declination strip 0$^\circ$–30$^\circ$), providing a HI line spectral database covering the redshift range between $-1{,}600\,\hbox{km}\,\hbox{s}^{-1}$ and 18,000 km $\hbox{s}^{-1}$ with 5 km $\hbox{s}^{-1}$ resolution at a sensitivity of 2.3 mJy. Exploiting Arecibo’s large collecting area and relatively small beam size ($3.5'$), ALFALFA was specifically designed to probe the faint end of the HI mass function in the local universe down to $M({\rm HI}) \simeq 10^7\;\hbox{M}_{\odot}$ (Martin et al. 2010). Haynes et al. (2011) presented the current catalog of 21 cm H I line sources extracted from ALFALFA over $\sim$2,800 deg$^2$ of sky: the $\alpha.40$ catalog. Covering 40% of the final survey area, the $\alpha.40$ catalog contains 15,855 sources in the spring sky, in two declination strips: 4$^\circ$–16$^\circ$ and 24$^\circ$–28$^\circ$. These include large stretches of the Local and of the Coma superclusters. ![Coverage of different surveys of the Virgo cluster region in between $12~{\rm h} < {\rm R.A.}({\rm J}2000) < 13~{\rm h}$ and $0^\circ < \hbox{dec} < 20^\circ$. The footprint of the Virgo Cluster Catalogue (VCC, Binggeli et al. 1985) is indicated by the [black dashed line]{}, the Next Generation Virgo Cluster Survey (NGVS, Ferrarese et al. 2012) by the [yellow solid line]{}, and the Herschel Virgo Cluster Survey (HeViCS, Davies et al. 2010) by the [cyan solid line]{}. The [light and dark gray areas]{} indicate the regions observed during the GALEX Ultraviolet Virgo Cluster Survey (GUViCS, Boselli et al. 2011) in the NUV (2,316 Å) band at two different depths (adapted from Boselli et al. 2014a). [Red, green, and blue empty circles]{} indicate Virgo cluster galaxies belonging to the red sequence, green valley, and blue cloud. The [size of the symbols]{} is proportional to the galaxy stellar mass. The [black contours]{} indicate the X-ray diffuse emission of the cluster, from Böhringer et al. (1994). Courtesy of ESO[]{data-label="GUVICS"}](fig5.pdf){width="80.00000%"} In the infrared domain, the all sky surveys undertaken with the AKARI (Murakami et al. 2007) and WISE (Wright et al. 2010) space missions provided photometric data for hundreds of thousands of galaxies in the spectral range 2–180 $\mu\hbox{m}$. With a significantly better sensitivity and angular resolution than IRAS, AKARI and WISE are crucial for quantifying the contribution of the old stellar populations dominating the emission of galaxies for $\lambda \lesssim 5~\mu\hbox{m}$ and that of the warm and cold dust components at longer wavelengths. Dust emission is critical for quantifying the attenuation of the stellar light in the UV and optical domain and is thus crucial for a correct determination of the present and past star-formation activity of galaxies. Blind panoramic surveys of clusters at $z=0$ and their host superclusters ------------------------------------------------------------------------- Because of the limited sensitivity of the instruments used in all sky surveys, the study of the dwarf galaxy population was mainly restricted to the very nearby universe. Different representative regions have been the target of dedicated studies. Among these, the most studied are certainly the Virgo and the Coma clusters. The Virgo cluster is the largest concentration of galaxies within 35 Mpc. Virgo is one of the closest rich clusters, whose distance (of only 16.5 Mpc, Gavazzi et al. 1999; Mei et al. 2007) allows us to study galaxies spanning a wide range in morphology and luminosity, from giant spirals and ellipticals down to dwarf irregulars, BCDs, dEs, and dS0. Furthermore, Virgo is still in the process of being assembled so that a wide range of processes (ram-pressure stripping, tidal interactions, harassment, and pre-processing) are still taking place. The Virgo cluster has been the target of several blind multifrequency surveys covering the whole range of the electromagnetic spectrum. The GALEX UV Virgo Cluster Survey (GUViCS; Boselli et al. 2011)[^4]. covered ${\simeq}120\,\hbox{deg}^2$ centered on the cluster in the FUV and NUV bands providing UV data for more than 1,200,000 sources, out of which $\sim$850 identified as cluster members (Voyer et al. 2014; see Fig. \[GUVICS\]). The sensitivity of the survey, reached typically with $\sim$1,500 s (one orbit) exposures, allowed the detection of low surface brightness features of 27.5–28 mag $\hbox{arcsec}^{{-}2}$ such as dwarf galaxies, including quiescent systems (Boselli et al. 2005a) and tidal features produced during the mutual interaction of galaxies (Boselli et al. 2005b; Arrigoni Battaia et al. 2012). The UV observations are of paramount importance for a large number of studies. In star-forming galaxies, the present day star-formation activity can be measured from the UV flux emitted by the youngest stellar population (Kennicutt 1998; Boselli et al. 2001, 2009), provided that dust extinction can be accurately determined (e.g. using the far-IR to UV flux ratio, Cortese et al. 2006a, 2008a; Hao et al. 2011). In quiescent galaxies, the UV emission can help to date the last generation of stars (on a few 100 Myr timescale) or is associated to very old populations (UV upturn; O’Connell 1999; Boselli et al. 2005a). The Virgo cluster has been mapped in four optical bands ($u^, g', i', z'$) by the Next Generation Virgo cluster Survey (NGVS, Ferrarese et al. 2012).[^5]. Carried out with the 1 deg$^2$ MegaCam instrument on the Canadian French Hawaii Telescope, the survey covered 104 deg$^2$ of the Virgo cluster, from the dense core out to the virial radius (see Fig. \[GUVICS\]). Designed to study at the same time point-like and extended, low surface brightness sources, it has a sensitivity of 25.9 $g$ magnitudes for point-sources and a surface brightness limit of $\mu_{g}$ 29 mag $\hbox{arcsec}^{-2}$. The survey has detected $\sim 3 \times 10^7$ sources, including hundreds of low surface brightness Virgo cluster members down to absolute magnitudes of $M_{\rm g} \sim {-}6$, and thousands of globular clusters associated with the massive galaxies. Sensitive to the stellar emission, the survey has been designed to study the luminosity and mass function of cluster galaxies, as well as the color–magnitude and the most important structural and photometric scaling relations down to this absolute magnitude limit. The Virgo cluster has been observed in the far infrared by Spitzer* and Herschel. The VIRGOFIR program (Fadda et al., in prep.) mapped $30\,\hbox{deg}^2$ of the Virgo cluster at 24 and $70\,\mu\hbox{m}$ with Spitzer, while the Herschel Virgo Cluster Survey (HeViCS; Davies et al. 2010, 2012)[^6]. $64\,{\rm deg}^2$ with PACS (100, 160 $\mu\hbox{m}$) and SPIRE (250, 350, 500 $\mu\hbox{m}$) on board of Herschel (see Fig. \[GUVICS\]). The HeViCS survey is confusion limited at $250\;\mu\hbox{m}$ (${\sim}1\,\hbox{MJy\,sr}^{-1}$) and has an angular resolution spanning from $6''$ at $100\,\mu\hbox{m}$ up to $36''$ at $500\,\mu\hbox{m}$ (diffraction limited). It is thus perfectly suited for studying the cold dust properties, one of the most important phases of the ISM in galaxies. Far infrared data are fundamental for accurately correcting for dust attenuation the UV and optical emission of galaxies and are thus crucial for quantifying and studying the present day star-formation activity, and the past star-formation history of cluster galaxies. At the distance of the Virgo cluster the ALFALFA survey provided HI masses for galaxies down to $M({\rm HI}) \simeq 10^{7.5}\;\hbox{M}_{\odot}$ (Giovanelli et al. 2007; Kent et al. 2008; Haynes et al. 2008). Slightly deeper HI data ($M({\rm HI}) \simeq 10^{7}\;\hbox{M}_{\odot}$) have been obtained by the Arecibo Galaxy Environment Survey (AGES; Taylor et al. 2012, 2013) on the cluster along two radial strips covering M49 $(10 \times 2\,\hbox{deg}^2$) and east of M87 $(5\,\hbox{deg}^2)$. Finally, X-ray data relative to the emission of the hot ICM are available thanks to ROSAT (Böhringer et al. 1994) and ASCA (Shibata et al. 2001), including spectroscopy from XMM-Newton (Urban et al. 2011). ![Coverage of available surveys in the Coma cluster region. The deep [GALEX]{} fields centered on the core of the cluster (Smith et al. 2010) and on NGC 4839 (Hammer et al. 2010b) are indicated with large (1 deg diameter) [solid circles]{}, the shallower one obtained by Cortese et al. (2008b) with [dotted circles]{}. The [blue rectangle]{} indicates the area covered with [Herschel]{} by Hickinbottom et al. (2014). The [red polynomial]{} region centered on the core of the cluster shows the area covered by deep $\hbox{H}_{\alpha}$ observations of Yagi et al. (2010). The footprints of the ACS Coma survey (Carter et al. 2008a) are indicated by [small squares]{}. [Red]{}, [green]{}, and [blue dots]{} indicate SDSS galaxies belonging to the red sequence, green valley, and blue cloud. The lowest contour of the X-ray emission from XMM is given. The patchy black contur in the center of the image shows the X-ray emission of the hot IGM (Briel et al. 2001)[]{data-label="coma"}](fig6.pdf){width="100.00000%"} The Coma cluster has also been the target of several multifrequency blind surveys ([see Fig.]{} \[coma\]). Located at a distance of $\sim$96 Mpc, Coma is a relaxed, spiral poor cluster characterized by a strong X-ray emission (Sarazin 1986; Briel et al. 2001). Nine square degree of the cluster, corresponding to ${\sim} 25\,\hbox{Mpc}^2$, have been observed in the UV with GALEX by Cortese et al. (2008b), and one extra field centered on the infalling region 1.6 Mpc south-west from the cluster core with a deep observation by Hammer et al. (2010a), and another one by Smith et al. (2010) centered on the cluster core. The Coma cluster has been also the target of a dedicated Hubble/ACS blind survey designed to cover $740\,\hbox{arcmin}^2$ but unfortunately not completed because of the failure of the ACS camera (Carter et al. 2008a; Hammer et al. 2010b).[^7]. The central 4 deg$^2$ of the Coma cluster have been observed in the infrared domain by Spitzer at 24 and 70 $\mu$m (Bai et al. 2006). The Coma cluster is also included in the H-ATLAS survey (Eales et al. 2010)[^8]. and has thus been observed by Herschel at low sensitivity ($5\sigma$ sensitivity for point-like sources $\simeq$50 to 100 mJy) in the PACS and SPIRE bands at 100, 160, 250, 350, and $500\,\mu\hbox{m}$. The cluster has been also the target of a deeper, dedicated observation with Herschel/PACS at 70, 100, and 160 $\mu$m on an area of $1.75\times1.0^\circ$ encompassing the core and the southwest infalling region (Hickinbottom et al. 2014). Spectroscopic observations of dwarf galaxies in the Coma cluster have been also obtained by Smith et al. (2008, 2009). Worth mentioning are also the impressive narrow band $\hbox{H}\alpha$ imaging observations of large portions of the Coma cluster done with the Suprime-Cam on the Subaru telescope by Yagi et al. (2007, 2010) and Yoshida et al. (2008) and the following spectroscopic observations of Yoshida et al. (2012). We remind that X-ray data of the cluster are available thanks to ROSAT (Briel et al. 1992), XMM (Briel et al. 2001; Arnaud et al. 2001; Neumann et al. 2001; Finoguenov et al. 2004a), Chandra (Vikhlinin et al. 2001; Churazov et al. 2012; Andrade-Santos et al. 2013), and INTEGRAL (Renaud et al. 2006; Eckert et al. 2007). Other recent blind surveys of very nearby clusters worth mentioning are those of A1367 in the UV and HI bands (Cortese et al. 2005, 2008a,b), of the Shapley supercluster in the optical (Mercurio et al. 2006), near-infrared (Merluzzi et al. 2010), far infrared and UV bands (Haines et al. 2011), including optical spectroscopy (Smith et al. 2007), and X-ray (Bonamente et al. 2001; Akimoto et al. 2003; Finoguenov et al. 2004b), and of the Fornax cluster in the PACS and SPIRE Herschel bands (Davies et al. 2013), and in X-ray by XMM (Murakami et al. 2011), while a ultra deep optical survey (FOCUS) is under way. Pointed observations -------------------- In the recent years a growing effort has been also devoted in targeted multifrequency observations of nearby cluster and field galaxies, including dwarfs, with the specific purpose of understanding the physical processes at the origin of the red sequence in high-density regions. A particular attention has been paid in gathering high-resolution spectroscopy data necessary to constrain the kinematical properties of the observed galaxies. The SMAKCED (Stellar content, Mass and Kinematics of Cluster Early-type Dwarfs) project (Janz et al. 2012)[^9]. has been designed to obtain medium resolution ($R=3{,}800$), long slit spectroscopy and deep near infrared photometry of $\sim$100 dwarf elliptical galaxies in the Virgo cluster. This project is a continuation of the study of the kinematic and spectrophotometric properties of dE started a few years before within the MAGPOP collaboration by Toloba and collaborators (Toloba et al. 2009, 2011, 2012). The $\hbox{ATLAS}^{\rm 3D}$ survey (Cappellari et al. 2011a),[^10]. although limited to relatively massive objects ($M_{\rm star} \gtrsim 6 \times 10^9\;\hbox{M}_{\odot}$), was designed to observe with SAURON on the William Herschel Telescope a volume limited sample ($D < 42$ Mpc) of 260 ETGs in the local universe using kinematic and spectrophotometric data at different frequencies. The $\hbox{ATLAS}^{\rm 3D}$ survey, which was originally defined to study 2D intermediate resolution spectroscopy data, extended in data quality and in statistics the SAURON project (Bacon et al. 2001). Another recent survey of nearby galaxies based on 2D-integral field spectroscopy at intermediate/low resolution ($R$ 850 and 1,650 in the spectral range 3,700–7,000 Å) data is CALIFA (Calar Alto Legacy Integral Field Area survey; Sanchez et al. 2012),[^11]. a project designed to observe with PPAK at the 3.5-m telescope of Calar Alto and study $\sim$600 galaxies in the local universe $(0.005< z < 0.03)$. Although mainly limited to massive galaxies, the sample includes galaxies spanning a wide range in morphological type and environment, including Coma and A1367, and is thus perfectly suited to study any possible physical process able to transform star forming into quiescent systems in high-density regions. High-quality optical images of dwarf elliptical galaxies in the Virgo cluster have been obtained with the ACS camera on the HST during the ACS Virgo cluster survey (ACSVCS; Cote et al. 2004).[^12]. The survey covered 100 ETGs, out of which 35 dE, dEN or dS0, in the F475W and F850LP bandpasses, which roughly correspond to the SDSS $g$ and $z$ bands. The same team has also undertaken the ACS Fornax cluster survey (ACSFCS; Jordan et al. 2007a,b), a similar survey of 43 ETGs in the Fornax cluster. In the optical domain it is worth mentioning $\hbox{H}\alpha 3$, a narrow-band $\hbox{H}\alpha$ imaging survey of dwarf, HI-detected star forming galaxies located in the surrounding regions of the Virgo and Coma/A1367 clusters (Gavazzi et al. 2012, 2013a,b). This survey extended previous observations of the clusters (Gavazzi et al. 1998, 2002, 2006a; Koopmann et al. 2001; Boselli et al. 2002; Boselli & Gavazzi 2002) to objects in low density regions at similar distances. Concerning the gaseous component we should mention the VLA survey of Virgo galaxies (VIVA) of Chung et al. (2009) that, although targeting bright spirals, allowed the detection of extraplanar HI gas stripped during the interaction with the hot ICM (Chung et al. 2007) where local episodes of star formation can give birth to new dwarf systems (see Sect. \[INDIVIDUAL\]). We must also mention the Herschel Reference Survey (HRS; Boselli et al. 2010),[^13] a project aimed at studying among other scientific topics, the effects of the cluster environment on the properties of the ISM in galaxies. Multifrequency data covering the whole electromagnetic spectrum, including new Herschel data, have been collected in the literature or thanks to dedicated observations. The sample is ideally designed for this purpose since it is a complete volume-limited $(15 < {\it Dist} < 25 \hbox{ Mpc})$, K-band-selected sample of galaxies spanning a wide range in morphological type, including at the same time field objects and Virgo cluster galaxies selected using consistent criteria. As selected, the sample includes spiral galaxies down to $M_{\rm star} \simeq 3 \times 10^8\;\hbox{M}_{\odot}$ and thus covers the relatively bright dwarf systems. Main observational results {#s3} ========================== Luminosity functions -------------------- The availability of large-scale multifrequency surveys such as SDSS and GALEX allowed the determination of the luminosity function in different photometric bands and the study of the variation of their characteristic parameters as a function of galaxy density (e.g. Blanton & Moustakas 2009). With respect to previous works for the major part based on the observations of the central regions of a few well-known clusters, these surveys have the advantage of extending the study to the periphery of the cluster, not always mapped by pointed observations. These regions are of great importance since they define the prevailing conditions first encountered by galaxies infalling in high-density regions (e.g. Boselli & Gavazzi 2006). Owing to the unprecedented statistical significance of datasets extracted from the SDSS, the role of the environments on the evolution of galaxies belonging to regions of different density, from the field to loose and compact groups up to massive clusters, has been significantly clarified. By combining SDSS data of 130 X-ray selected clusters at $z \simeq 0.15$ (Popesso et al. 2005) determined and studied the properties of the composite luminosity functions of nearby clusters. This work has shown that the luminosity function has a bimodal behavior, with an upturn and an evident steepening in the faint magnitude range in any SDSS band. Both the bright and the faint end can be fitted with a Schechter function, the former with a slope $\alpha \simeq {-}1.25$, the latter with ${-}2.1 \leq \alpha \leq {-}1.6$ (see Fig. \[LFottica\]). The same authors were also able to separate the contribution of early- and late-type galaxies using color indices (Popesso et al. 2006). They have shown that while the luminosity function of late-type systems can be fitted with a single Schechter function of slope $\alpha = {-}2.0$ in the $r$-band, the early-types require two Schechter functions, the faint one being responsible for the faint-end upturn observed in the global composite luminosity function of the cluster. The shape of the bright-end tail of the early-type luminosity function does not depend on the local galaxy density, while the faint-end shows a significant and continuous variation with the environment, with a clear flattening near the core of the cluster. A steepening of the faint-end of the luminosity function at the periphery of nearby clusters and a flattening in the core has been also observed by Barkhouse et al. (2007, 2009) using data on 57 low-redshift Abell clusters observed with the KPNO 0.9 m telescope. All these teams interpreted these results as an evidence of a combined effect of galaxy transformation from star forming to quiescent systems through harassment in the periphery and dwarf tidal disruption in the core (Popesso et al. 2006; Barkhouse et al. 2007; de Filippis et al. 2011). The presence of an upturn at faint magnitudes in the luminosity function, however, was questioned by de Filippis et al. (2011), while a significantly flatter faint end slope ($\alpha \sim {-}1.1$) has been found in the Hydra I and the Centaurus clusters down to $M_V \sim {-}10$ by Misgeld et al. (2008, 2009). Hansen et al. (2009) have shown that the shape of the luminosity function of satellites does not change with clustercentric distance. They have also shown that the luminosity function of both red and blue satellites is only weakly dependent on richness. Their ratio, however, dramatically changes. The average color of satellites is redder near the cluster centers. The properties of the SDSS luminosity function of cluster galaxies was extended to groups by Zandivarez et al. (2006), Zandivarez & Martinez (2011) and Robotham et al. (2010). Zandivarez et al. (2006) and Zandivarez & Martinez (2011) have shown a steepening of the faint end slope and a brightening of the characteristic magnitude as the mass of the system increases, while Robotham et al. (2010) concluded that the steepening at the faint end is mainly due to an increase of the quiescent galaxy population. Studies of very nearby clusters such as Virgo have the advantage of both detecting the dwarf galaxy population down to much fainter limits and determine cluster memberships according to different and independent criteria such as colors, morphology, surface brightness, unpractical at larger distances. Recently, Lieder et al. (2012) determined the $V$ and $I$ luminosity function of the central $4\,{\rm deg}^2$ of the Virgo cluster, deriving a faint end slope $\alpha = {-}1.50$. Using new spectra of galaxies in the direction of the Virgo cluster, Rines & Geller (2008) have shown that the faint-end of the $r$-band luminosity function has a slope consistent with that of the field $(\alpha={-}1.28)$ down to $M_r \simeq {-}13.5 \simeq M^* + 8$, thus significantly flatter than that generally determined for clusters using SDSS data (e.g. Popesso et al. 2005, 2006; see Fig. \[LFottica\]). The analysis of Rines & Geller (2008) has indicated that the difference is primarily due to the use of a statistical subtraction for the correction of the background contamination. They have shown that a simple separation in apparent magnitude versus surface brightness, originally proposed by Sandage and collaborators (Sandage et al. 1985; Binggeli et al. 1985) and later adopted by Boselli et al. (2011) in the UV bands, provides a powerful membership classification. They thus opened new ways for identifying cluster members, allowing the determination of the optical luminosity function in five photometric bands down to $M_g \simeq {-}6 (M^* + 15)$ on a region centered on the Virgo cluster as large as $104\,{\rm deg}^2$ using the NGVS data (Ferrarese et al. 2012). The preliminary results obtained using the NGVS data of the central $4\,\hbox{deg}^2$ of Virgo seem to indicate that the optical luminosity function of Virgo has a slope of $\alpha \simeq {-}1.4$ down to this absolute magnitude limit (Ferrarese, private communication). We recall that this value, which perfectly matches that obtained in the eighties by Sandage and collaborators (1985) using photographic plate material, is very similar to the one obtained for the field using the SDSS once low surface brightness objects as those detected in Virgo are considered (Blanton et al. 2005b). ![The $r$ band luminosity function of the Virgo cluster within 1 Mpc from M87, from Rines & Geller (2008) ([filled squares]{}) ([upper]{} ([lower]{}) scale $r$ absolute (apparent) magnitude). The fitted Schechter function ([black solid line]{}) is compared to the field luminosity function from the SDSS of Blanton et al. (2005b) ([dotted–dashed line]{}) and to the composite luminosity function of clusters of Popesso et al. (2006) ([dotted line]{}). The optical luminosity function of Virgo has a slope comparable to that of the field once low surface brightness objects are considered. © AAS. Reproduced with permission[]{data-label="LFottica"}](fig7.pdf){width="70.00000%"} ![The FUV ([upper panel]{}) and NUV ([lower panel]{}) luminosity functions of the Virgo cluster ([black symbols and solid line]{}) compared to that of the Coma cluster ([red]{}), A1367 ([green]{}), and the field ([blue]{}), from Boselli et al. (2011). The UV luminosity function of the Virgo cluster is similar to that of the field. Courtesy of ESO[]{data-label="LFUV"}](fig8.pdf){width="100.00000%"} Pointed observations with the GALEX satellite allowed the determination of the UV luminosity function of well-known nearby clusters such as Coma (Cortese et al. 2008b; Hammer et al. 2012), A1367 (Cortese et al. 2005), the Shapley supercluster (Haines et al. 2011), and the Virgo cluster (Boselli et al. 2011) in two photometric bands (FUV, $\lambda_{\rm eff} = 1{,}539$ Å; NUV, $\lambda_{\rm eff} = 2{,}316$ Å) (see Fig. \[LFUV\]). Steep slopes of the fitted Schechter functions ($\alpha \simeq {-}1.5/{-}1.6$) both in the NUV and FUV bands have been observed in Coma (Cortese et al. 2008b), A1367 (Cortese et al. 2005), and in the Shapley supercluster (Heines et al. 2011) down to UV absolute magnitudes of ${\simeq}{-}14$. These values of $\alpha$ are significantly steeper than those observed in the field by (Wyder et al. 2005; $\alpha \simeq {-}1.2$). By dividing galaxies according to their morphological type, these authors have also shown that the steepening of the observed UV luminosity function is due to the contribution of ETGs, becoming important at faint luminosities. We recall that in these quiescent objects the UV emission is not only related to star-forming events as in LTGs (e.g. Boselli et al. 2009), but rather to evolved stellar populations (UV upturn; e.g. O’Connell 1999; Boselli et al. 2005a). Slightly flatter slopes $(\alpha = {-}1.39)$ have been observed in the infalling region centered on NGC 4839 in Coma by Hammer et al. (2012) using a deep GALEX field including galaxies down to UV magnitudes of $-$10.5. The GUViCS survey allowed the determination of the UV luminosity function down to UV magnitudes of $-$10 ($M_{UV}^+ 7$) in the central $12\,{\rm deg}^2$ of the Virgo cluster (Boselli et al. 2011, Fig. \[LFUV\]). The faint-end slope of the fitted Schechter function determined in this work ($\alpha \simeq {-}1.1/{-}1.2$) is significantly flatter than the one observed in other clusters and very close to the one determined for the field. Boselli et al. (2011) interpreted the difference between the faint-end slope of the luminosity function of Virgo and other nearby clusters as due to two possible effects. The first is related to the small volume sampled by their study and thus to the small number of luminous objects used to constrain the shape of their luminosity function. They have also shown, however, that because of the higher distance of the other clusters, the determination of their luminosity functions suffers from significantly narrower dynamic range in luminosity (down to $-$14 vs. $-$10 in Virgo) and by the quite uncertain statistical corrections for the background contamination. This conclusion is consistent with the results of Rines & Geller (2008) obtained in the optical bands. The mild steepening of the faint-end of the luminosity function is primarily due to the population of evolved galaxies (E-S0-dE; Boselli et al. 2011). Given the similarity of the field and cluster luminosity functions, Boselli et al. (2011) concluded that their results are consistent with a transformation of star-forming dwarf galaxies in quiescent systems due to a ram pressure stripping event removing their gas content once galaxies fall into the cluster. Band Reference $\alpha$ ------ ------------------ ----------------------------- --------------- OPT 130 clusters Popesso et al. (2005) $-$2.0 OPT Hydra, Centaurus Misgeld et al. (2008, 2009) $-$1.1 OPT Virgo Lieder et al. (2012) $-$1.5 OPT Virgo Rines & Geller (2008) $-$1.28 OPT Virgo Ferrarese et al. (2012) $-$1.4 UV Coma/A1367 Cortese et al. (2005) $-$1.5/$-$1.6 UV Shapley Haines et al. (2011) $-$1.5 UV field Wyder et al. (2005) $-$1.2 UV Coma (N4839) Hammer et al. (2012) $-$1.39 UV Virgo Boselli et al. (2011) $-$1.1/$-$1.2 FIR Coma Bai et al. (2006, 2009) $-$1.4/$-$1.5 FIR Shapley Haines et al. (2011) $-$1.4/$-$1.5 \[tab1\] The study of the luminosity function of cluster galaxies has been extended to the far-infrared domain thanks to Spitzer* and Herschel. Bai et al. (2006, 2009), using 24, 70, and 160 $\mu{\rm m}$ MIPS data determined the luminosity function of the two clusters Coma and A3266 $(z=0.06)$ down to $L_{\rm IR} \sim 10^{42}\,\hbox{erg\,s}^{-1}$ and $L_{\rm IR} \sim 10^{43}\,\hbox{erg\,s}^{-1}$, respectively. These works have shown that the far-infrared luminosity function of these cluster galaxies $(\alpha={-}1.4/{-}1.5$ and $L^* \sim\,10.5 \hbox{L}_{\odot}$) is comparable to that observed in the field. They have also shown that both $L^$ and $\alpha$ fade close to the cluster core. Similar results have been obtained in the 24 and $70\,\mu\hbox{m}$ bands by Haines et al. (2011) in the Shapley supercluster. Haines et al. (2011) interpreted these results as an evidence that the LTG population, responsible for the far-infrared emission, has been only recently accreted in clusters. More recently Davies et al. (2012) determined the first far-infrared luminosity distribution in the Herschel* PACS (100–160$\mu\hbox{m}$) and SPIRE (250–350–500$\mu\hbox{m}$) bands for the central $64\,{\rm deg}^2$ of the Virgo cluster. They have shown that optically selected galaxies have a far-infrared luminosity distribution peaked at intermediate luminosities, showing a lack of both bright and low-luminosity systems. The faint-end slope of the various luminosity functions is summarized in Table \[tab1\]. Gas content ----------- The availability of HI blind surveys on large regions of the sky such as ALFALFA (Giovanelli et al. 2005) and AGES (Taylor et al. 2012, 2013) allowed the acquisition of homogeneous sets of data down to a well-defined sensitivity limit for extragalactic sources belonging to a large variety of environments. Indeed ALFALFA and AGES included nearby clusters such as Virgo (Giovanelli et al. 2007; Kent et al. 2008; Haynes et al. 2011), Coma and A1367 (Cortese et al. 2008c). These surveys are generally less deep than previous pointed observations of selected objects but cover simultaneously and without any a priori selection all kind of extragalactic sources. They mainly confirmed that the atomic gas content of galaxies decreases in high-density regions (e.g. Cayatte et al. 1990; Solanes et al. 2001; Gavazzi et al. 2005, 2006b). They also allowed the detection of a minority of ETGs, including dwarf systems, inside the Virgo cluster with gas contents of the order of a few $10^7\,\hbox{M}_{\odot}$ (di Serego Alighieri et al. 2007). These surveys led to the first robust determination of the mean structural and spectrophotometrical properties of HI-selected galaxies in different environments and to compare them with those of optically selected samples (Gavazzi et al. 2008; Cortese et al. 2008c). The main contribution of these surveys to the study of the role of the environment on galaxy evolution, however, comes from the combination of HI data with optical and UV data from SDSS and GALEX. Using a sample of $\sim$10,000 objects with multifrequency data, Huang et al. (2012) have shown that, for stellar masses below $M_{\rm star} \lesssim 10^{9.5}\,\hbox{M}_{\odot}$, galaxies follow a sequence along the color–magnitude relation (CMR) or the [SSFR]{} (star formation per unit stellar mass or specific star formation) vs. $M_{star}$ relation regulated by the available quantity of HI gas, which becomes the dominant barionic component in low mass systems. By studying this effect in different environments, from the field to the core of nearby rich clusters and including groups, Gavazzi et al. (2013a,b) confirmed that the distribution of galaxies along the CMR below $M_{\rm star} = 10^{9.5}\,\hbox{M}_{\odot}$ is regulated by their atomic gas content. Late-type galaxies located on the blue sequence have larger gas fractions than spirals in the green valley, or post starburst galaxies characterized by red colors (Boselli et al. 2014a,b). The galaxies mostly devoid of gas are the red early-type systems forming the red sequence. Gavazzi et al. (2003a, 2013b) have also shown that this sequence in colors or gas fraction is also related to the mean density of the environment where galaxies reside, with redder colors and gas-poor systems dominating high-density regions (see Fig. \[HaHI\]). These results, that extend previous analysis mainly based on massive galaxies (Hughes & Cortese 2009; Cortese & Hughes 2009), have been interpreted as a clear evidence that gas removal due to ram pressure stripping events quenches the activity of star formation, transforming gas-rich, star-forming systems into quiescent objects, as first proposed by Boselli et al. (2008a,b). The work of Gavazzi et al. (2013b), combined with that of Fabello et al. (2012) based on stacking analysis of HI data of a large sample of nearby low-mass galaxies and that of Catinella et al. (2013) for massive systems, however, have shown that gas removal and the relative quenching of the star-formation activity is already present in intermediate-density regions such as groups with halo masses $M > 10^{13}\,\hbox{M}_{\odot}$. ![[Upper panel]{}: the sky distribution of late-type galaxies within seven annuli of increasing radius from M87. Galaxies in each ring are given with a [different color]{}. Variation of the specific star formation rate [SSFR]{} ([middle upper]{}), of the HI mass per unit star formation rate [SFR]{} per unit HI mass ([middle lower]{}), and of the HI-deficiency parameter $\hbox{Def}_{{\rm HI}}$ ([lower]{}) as function of the projected angular separation from M87 (from Gavazzi et al. 2013a). Courtesy of ESO[]{data-label="HaHI"}](fig9.pdf){width="100.00000%"} Interestingly, all these evidences questioned the general assumption that in massive galaxies the quenching of the activity of star formation is mainly due to AGN feedback (e.g. Martin et al. 2007; Schawinski et al. 2009, see, however, Schawinski et al. 2014), proposing environmental effects as an alternative process for the origin of the green valley (Hughes & Cortese 2009; Cortese & Hughes 2009). The most recent CO surveys of cluster and field objects support this scenario. The analysis of a large K-band-selected, volume-limited sample of nearby field and cluster galaxies, the Herschel Reference Survey (Boselli et al. 2010), as well as the detailed study of a small sample of CO mapped galaxies, have both shown that also the molecular gas phase is stripped by the interaction of galaxies with the hot intergalactic medium permeating rich clusters such as Virgo (Fumagalli et al. 2009; Boselli et al. 2014b). The stripping process, however, is less efficient than that on the atomic phase because the molecular hydrogen is mainly located in the central regions of galaxies, where the gravitational restoring force is at its maximum. This evidence supports the idea that the responsible process for gas removal is ram pressure stripping (Fumagalli et al. 2009; Scott et al. 2013; Boselli et al. 2014b). All these works have been mostly limited to massive galaxies since the CO observation of metal-poor, low-luminosity objects is still challenging. We can expect, however, that given the shallower potential well of dwarf galaxies with respect to massive systems, the stripping of the gaseous component in all its phases is even more efficient in low-mass objects. ![The NGVS optical color ([top]{}) and the WISE 12 $\mu$m ([bottom]{}) images of the dwarf elliptical VCC 781 in the Virgo cluster. The dust lane present in the optical image is detected in the core of the galaxy in emission in the mid-infrared by WISE[]{data-label="VCC781"}](fig10a.pdf "fig:"){width="50.00000%"} ![The NGVS optical color ([top]{}) and the WISE 12 $\mu$m ([bottom]{}) images of the dwarf elliptical VCC 781 in the Virgo cluster. The dust lane present in the optical image is detected in the core of the galaxy in emission in the mid-infrared by WISE[]{data-label="VCC781"}](fig10b.pdf "fig:"){width="50.00000%"} The HI blind surveys of the nearby universe have also revealed the presence in clusters of HI clouds not associated with any stellar component (Kent et al. 2007, 2009; Kent 2010; Haynes et al. 2007 in ALFALFA; Taylor et al. 2012, 2013 in AGES). If some of the HI clouds are probably tidal debris harassed from massive Virgo cluster objects, such as the case of VIRGOHI21 and NGC 4254 (Haynes et al. 2007), others might have been formed from the stripped gas of spirals entering the cluster for the first time (Keent et al. 2009). Interferometric observations have revealed the presence of spectacular tails of HI gas in the Virgo cluster (Oosterloo & van Gorkom 2005; Chung et al. 2007). The typical cometary shape of these features strongly suggests that ram pressure stripping is the responsible process. The main results of these observations are that ram pressure stripping is acting well outside the virial radius of clusters, making this process the most efficient gas stripping process also in regions of relatively low density of the intergalactic medium. The lack of evident associated regions of star formation also indicate that, contrary to what predicted by models (Kepferer et al. 2009), the stripped gas can be hardly transformed into new stars and, eventually, give birth to dwarf galaxies (Boissier et al. 2012; see, however Yagi et al. 2013). Similar long tails of hot gas have been observed in X-rays with Chandra in two spiral galaxies in the cluster A3627 (Sun et al. 2007, 2010). These observations indicate that also the hot halo gas of galaxies can be stripped by ram pressure in high-density environments. Dust content ------------ The Spitzer and Herschel missions allowed a detailed analysis of the mid- and far-infrared properties of local galaxies in high-density environments. The spectral coverage (5–500 $\mu\hbox{m}$), combined with the high angular resolution (from a few to 36 arcsec at $500\,\mu\hbox{m}$) and the sensitivity of the different instruments were crucial for resolving the different dust components (PAHs, hot and cold dust) in galaxies. The analysis of the brightest Virgo cluster spirals has clearly shown that the cold dust component distributed over the disk is removed with the gaseous component during the interaction with the hot intracluster medium. Indeed, HI-deficient spiral galaxies have truncated dust disks compared to unperturbed field objects (Cortese et al. 2010). Their total dust content is also reduced with respect to normal, field galaxies (Cortese et al. 2012a). Concerning the dwarf galaxy population, de Looze et al. (2010, 2013) have shown the existence of a significant number of dE with presence of dust in their inner regions (Fig. \[VCC781\]). 36 % of the dwarf galaxies belonging to the green valley, identified by Boselli et al. (2008a) as galaxies migrating from the blue cloud to the red sequence (transition type galaxies), have been detected by Herschel. There is also evidence of the presence of a residual dust content in several dwarf ETGs located outside the diffuse X-ray emitting gas permeating the cluster (Boselli et al. 2014a). These observations are consistent with the idea that the dust associated with the gaseous phase is removed outside-in during the interaction with the intergalactic medium of galaxies infalling for the first time into the cluster. Their interstellar medium can be retained only in the inner regions, where the gravitational potential well of the galaxy is at its maximum. Star formation -------------- The statistical studies of the star-formation properties of cluster galaxies in the past years have been mainly focused on the bright galaxy population. Using a sample of 79 nearby clusters with available multifrequency data, Popesso et al. (2007) have shown that the cluster integrated star-forming properties do not change as a function of the cluster properties. They see, however, that the fraction of blue galaxies depends on the total X-ray luminosity of the clusters, suggesting that environmental processes linked to the presence of the hot X-ray emitting intracluster gas might affect the star-formation history of cluster galaxies. Using a sample of nine nearby clusters, including Coma and A1367, with new spectroscopic data, Rines et al. (2005) have shown that the fraction of star-forming bright galaxies, as determined from the presence of the Balmer $\hbox{H}\alpha$ emission line, increases with clustercentric distance and reaches that of the field at $\sim$2 to 3 virial radii. This work mainly confirms the first results obtained by Gomez et al. (2003) and Lewis et al. (2002) using the SDSS and 2dF surveys. The combination of $\hbox{H}\alpha$ and HI data has been crucial for understanding the very nature of the underlying physical process responsible for the quenching of the star-formation activity of cluster galaxies. This has been possible in the Virgo cluster and in the Coma/A1367 supercluster (Gavazzi et al. 2013a,b). These works have shown that the quenching of the star-formation activity follows the stripping of the atomic gas as indicated by the tight relation between the [SSFR]{} and the HI-deficiency parameter.[^14] Indeed, the specific star-formation rate decreases with the clustercentric distance exactly as the HI-deficiency, making LTGs redder (Gavazzi et al. 2013a,b).[^15] ![The relation between the ratio of the $\hbox{H}\alpha$ (star forming disk) to $r$-band (old stellar population) effective radii and the HI-deficiency parameter. [Big red dots]{} are average values along the $y$-axis in different bins of HI deficiency, from Fossati et al. (2013). The star-forming disk is truncated once the atomic gas is removed in cluster galaxies. Courtesy of ESO[]{data-label="Fossati"}](fig11.pdf){width="70.00000%"} An accurate study of the star-formation properties of low-luminosity cluster galaxies has been made possible owing to the narrow-band $\hbox{H}\alpha$ imaging data that our team has obtained in these past years. By comparing the $\hbox{H}\alpha$ morphology of galaxies in Virgo, Coma and A1367 to those of field galaxies selected according to similar criteria, Fossati et al. (2013) have shown that the star-forming disk of LTGs, including dwarf systems, is truncated once the galaxies are devoid of their gas (Fig. \[Fossati\]). The star-formation process is thus quenched outside-in, confirming previous results obtained for the bright galaxy population (Koopmann et al. 2006; Boselli & Gavazzi 2006; Boselli et al. 2006; Cortese et al. 2012b). The $\hbox{H}\alpha$ imaging of a few dE galaxies have also revealed the presence of an emitting nucleus, suggesting that, after a stripping process, there is still some retention of gas where the gravitational potential well of the galaxy is at its maximum.[^16] This gas is able to feed a nuclear star-formation activity (Boselli et al. 2008a). More in general, multizone chemo-spectrophotometric models of galaxy evolution especially tailored to reproduce the effects due to ram pressure stripping in cluster environments have shown that the observed outside-in truncation of the star formation in gas stripped galaxies can be responsible for the inversion of the color gradient observed in massive galaxies (Boselli et al. 2006) or in dwarf ellipticals (Boselli et al. 2008a). There is also evidence of some galaxies with a disturbed $\hbox{H}\alpha$ or UV morphology, witnessing and undergoing interaction with the hostile cluster environment (Smith et al. 2010). The $\hbox{H}\alpha$ and UV data have been also crucial for studying extraplanar star-formation events in galaxies with clear signs of an undergoing perturbation (see Sects. \[INDIVIDUAL\] and \[EXTRAPLANAR\]). ![The dependence of the quenching age, defined as the redshift for which the look-back time is equal to the age of dwarf galaxies determined using the Balmer lines, on the projected distance from the core of the Coma cluster, from Smith et al. (2008). Dwarf elliptical galaxies in the periphery of the Coma cluster stopped their activity of star formation only at recent epochs. Reproduced with permission of Oxford University Press[]{data-label="Smithspsage"}](fig12.pdf){width="100.00000%"} A systematic study of the star-formation properties of galaxies in 23 nearby ($z \sim 0.06$) groups based on GALEX data has been presented in Rasmussen et al. (2012a). This work has shown that, as in clusters, the fraction of star forming galaxies within groups is suppressed with respect to the field in the inner ${\sim} 2$ virial radii (${\sim}1.5$ Mpc). The same work has shown a suppression of the specific star-formation rate by a factor of ${\sim} 40$ % with respect to the field, quantifying the impact of the group environment on quenching the activity of star formation in infalling galaxies. At fixed galaxy density and stellar mass, this suppression is stronger in more massive groups. Rasmussen et al. (2012a) concluded that the average time scale for quenching the star-formation activity is $\gtrsim$2 Gyr and identified a combination of tidal encounters and starvation as the responsible process. Stellar populations ------------------- The study of the nature of the various stellar populations of dwarf galaxies in different environments is one of the topics that has improved the most in the past years owing to the advent of large photometric and spectroscopic surveys of nearby clusters. The most recent works have been mainly focused either on the optical (Haines et al. 2006b; Lisker et al. 2006a, 2008; Misgeld et al. 2009; Janz & Lisker 2009; Gavazzi et al. 2010; Scott et al. 2013; den Brok et al. 2011; McDonald et al. 2011) or on the optical-UV CMR (Boselli et al. 2005a,b, 2008a, 2014a; Lisker & Han 2008; Haines et al. 2008; Kim et al. 2010; Smith et al. 2012b), or based on the analysis of absorption line indices obtained from deep spectroscopic observations (Smith et al. 2006, 2008, 2009, 2012a; Haines et al. 2006, 2007; Michielsen et al. 2008; Toloba et al. 2009; Paudel et al. 2010a,b, 2011; Boselli et al. 2014a). The first works of Haines et al. (2006a,b) on the cluster A2199 and on the Shapley supercluster, along with that of Smith et al. (2006) on 94 clusters with spectroscopic data from the SDSS, have consistently shown a different evolution of massive and dwarf galaxies in high-density regions. While the fraction of massive ($M_{\rm r} < {-}20$) red galaxies dominated by an old population gradually decreases from $\sim$80% in the core to $\sim$40% in the periphery (3–4 virial radii) of the cluster, the radial variation for dwarfs (${-}19<M_{\rm r}<17.8$ mag) is much more pronounced, from $\sim$90% in the core to $\sim$20% at one virial radius (Haines et al. 2006a,b; Boselli et al. 2014a). Consistently, by studying the dispersion of the fundamental plane relation of early-type systems, Smith et al. (2006) have shown that, for a given velocity dispersion—thus total mass—galaxies in the periphery of clusters have stronger Balmer absorption lines, indicative of younger ages than those located in the cluster core (see Fig. \[Smithspsage\]). Consistent results have been obtained using new spectroscopic observations in the Coma cluster by Smith et al. (2008, 2009, 2012a). All these studies have indicated strong Balmer absorption lines and enhanced $\alpha/{\rm Fe}$ ratios in the red dwarf galaxy population in the outskirts of Coma compared to the core, where only the oldest galaxies reside, consistent with a relatively recent formation of the red sequence ($0.4 <z< 0.7$). The SW substructure of Coma is also composed of younger dwarf quiescent galaxies, probably formed between $0.1< z < 0.2$ (Smith et al. 2009). The mean age of the underlying stellar population is mainly related to the total mass of galaxies in massive objects (${M}_{\rm star} > 10^{10}\;\hbox{M}_{\odot}$) and only marginally on the environment, while the reverse holds for dwarf systems. Indeed, the mean age of the stellar population of dE galaxies is about a factor of two younger at one virial radius than in the core of the cluster (Smith et al. 2012a). The earlier work of Poggianti et al. (2004) has shown the presence of low luminosity post-starburst galaxies in the Coma cluster, consistently with the picture where dwarf galaxies abruptly truncated their star-formation activity and became red objects. These post-starburst galaxies are mainly low mass systems situated around massive clusters, as indicated by the recent spectroscopic survey of 48 nearby clusters ($0.04<z<0.07$) of Fritz et al. (2014). Similar trends between the mean age of the stellar populations and the clustercentric distance have been also observed in the Virgo cluster by Michielsen et al. (2008), Toloba et al. (2009, 2014b), and Paudel et al. (2010a, 2011). In particular, Toloba et al. (2009) have shown that dE galaxies characterized by the youngest stellar population not only are located in the periphery of the cluster, but also are generally rotationally supported systems. Thanks to the proximity of the Virgo cluster, spectroscopic studies have been used to investigate stellar population gradients within the disk of dwarf elliptical galaxies. The available works have consistently shown that, on average, the mean age of the stellar populations increases from the center to the outskirts of galaxies (Chilingarian et al. 2009; Koleva et al. 2009, 2011; Paudel et al. 2011) contrary to what generally happens in massive systems (Fig. \[Gavazzispsdensity\]). All these evidences are consistent with a scenario where low-luminosity star-forming galaxies recently entered the cluster environment, losing their gas content and quenching their star-formation activity, thus becoming dwarf ellipticals (Boselli et al. 2008a; Toloba et al. 2009; Gavazzi et al. 2010; Koleva et al. 2013). ![The $g-i$ vs. $M_i$ CMR for galaxies in the Coma/A1367 supercluster from very high density regions in the core of Coma and A1367 to very low density regions in the voids surrounding the clusters, clockwise from upper left. Red symbols indicate early-type galaxies in the red sequence, blue symbols late-type systems in the blue cloud, from Gavazzi et al. (2010). Star forming systems are virtually lacking in the densest regions in the core of the clusters, while the faint end of the red sequence is not present only in the lowest density regions dominated by star forming systems. Courtesy of ESO[]{data-label="Gavazzispsdensity"}](fig13a.pdf "fig:"){width="50.00000%"} ![The $g-i$ vs. $M_i$ CMR for galaxies in the Coma/A1367 supercluster from very high density regions in the core of Coma and A1367 to very low density regions in the voids surrounding the clusters, clockwise from upper left. Red symbols indicate early-type galaxies in the red sequence, blue symbols late-type systems in the blue cloud, from Gavazzi et al. (2010). Star forming systems are virtually lacking in the densest regions in the core of the clusters, while the faint end of the red sequence is not present only in the lowest density regions dominated by star forming systems. Courtesy of ESO[]{data-label="Gavazzispsdensity"}](fig13b.pdf "fig:"){width="50.00000%"}\ ![The $g-i$ vs. $M_i$ CMR for galaxies in the Coma/A1367 supercluster from very high density regions in the core of Coma and A1367 to very low density regions in the voids surrounding the clusters, clockwise from upper left. Red symbols indicate early-type galaxies in the red sequence, blue symbols late-type systems in the blue cloud, from Gavazzi et al. (2010). Star forming systems are virtually lacking in the densest regions in the core of the clusters, while the faint end of the red sequence is not present only in the lowest density regions dominated by star forming systems. Courtesy of ESO[]{data-label="Gavazzispsdensity"}](fig13c.pdf "fig:"){width="50.00000%"} ![The $g-i$ vs. $M_i$ CMR for galaxies in the Coma/A1367 supercluster from very high density regions in the core of Coma and A1367 to very low density regions in the voids surrounding the clusters, clockwise from upper left. Red symbols indicate early-type galaxies in the red sequence, blue symbols late-type systems in the blue cloud, from Gavazzi et al. (2010). Star forming systems are virtually lacking in the densest regions in the core of the clusters, while the faint end of the red sequence is not present only in the lowest density regions dominated by star forming systems. Courtesy of ESO[]{data-label="Gavazzispsdensity"}](fig13d.pdf "fig:"){width="50.00000%"}\ The study of the photometric properties of cluster galaxies has been primarily aimed at understanding whether genuine giant ellipticals and dwarf systems follow the same CMR, an indication that would suggest a common origin for the two families of objects. The works done so far do not bring fully consistent results. A unique CMR has been invoked by Misgeld et al. (2009) in the Centaurus cluster, by Hammer et al. (2010b) in Coma and more recently by Smith Castelli et al. (2013) by combining SDSS data of Virgo galaxies with those obtained by the ACS Virgo Cluster Survey. Conversely, variations of the CMR have been found using SDSS data of more than 400 Virgo objects by Janz & Lisker( 2009). These authors, however, concluded that this observational result does not rule out a common origin for the two populations. Lisker et al. (2008) looked for any systematic difference in the optical CMR of the different subclasses of dwarf elliptical galaxies. Their analysis has shown that, at relatively high luminosities, nucleated dwarf ellipticals have redder colors than normal dE, while the variations with local galaxy density, if present, are minor. The works of Haines et al. (2008) and of Gavazzi et al. (2010) have been fundamental to extend these interesting results to other environments such as the periphery or rich clusters or intermediate mass groups. Using the $NUV-i$ color index, much more sensitive to age variations than the optical colors, Haines et al. (2008) have shown that the red sequence is not formed in the field at low luminosities. The faint end of the red sequence, however, is already formed within the different substructures of the Virgo cluster characterized by a wide range in galaxy density (Boselli et al. 2014a). Consistently, Gavazzi et al. (2010) have shown that both the shape of the luminosity function determined for galaxies selected according to their color, and the $g-i$ CMR change as a function of galaxy density, indicating that the processes that gave birth to the faint end of the red sequence have been efficient also in relatively low-density environments. These results are fully consistent with those determined by analyzing statistically significant samples extracted from the SDSS indicating that red dwarf galaxies are extremely rare in the field (Geha et al. 2012). The work of Gavazzi et al. (2010) have also revealed the presence of low-luminosity galaxies with clear signs of a recent activity of star formation (post-starburst) in Coma and A1367. These galaxies are analogous to those found in the Virgo cluster by Boselli et al. (2008a) and originally defined as transition type galaxies as their spectrophotometric properties indicate a recent abrupt truncation of their star-formation activity (Koleva et al. 2013). The works of van den Bosch et al. (2008), Wilman et al. (2010), and Cibinel et al. (2013) studied the relationship between galaxy color and density in nearby groups using SDSS photometric data. These works have shown that the perturbation induced by the environment affects more the color than the morphology of galaxies (van den Bosch et al. 2008). They have also shown that galaxies become red only once they have been accreted on to halos of a certain mass (Wilman et al. 2010). The quenching of the star-formation activity is primarily in the outer disk of galaxies entering groups (Cibinel et al. 2013). ![Histogram of the number of dwarf elliptical galaxies in the Virgo cluster in bins of absolute B band magnitude (from the VCC; right y-axis). The light gray dashed line shows the distribution of all dEs, the dark gray area that of dEs with blue nuclei, the medium grey that of dEs with an underlying disky structure, the light gray that of all dEs galaxies after excluding those with disky structures and blue nuclei. The black solid line shows the fraction of the dEs with blue nuclei (left y-axis), from Lisker et al. (2006b). © AAS. Reproduced with permission[]{data-label="Lisker"}](fig14.pdf){width="70.00000%"} Structural parameters {#STRUCTURAL} --------------------- The study of the structural properties of dwarf galaxies in nearby clusters has enormously benefited from the SDSS. The first systematic study of the structural properties of dwarf elliptical and spheroidal galaxies in the Virgo cluster has been done by Lisker and collaborators (Lisker et al. 2006a,b, 2007, 2009; Janz & Lisker 2008; Lisker 2009). These works basically extended previous analyses done on the photographic plates taken at the Du Pont telescope at Las Campanas in the eighties by Binggeli and collaborators (Binggeli et al. 1985; Binggeli & Cameron 1991). Lisker et al. (2006a) have looked for unseen disky features such as spiral arms, bars, and edge-on disks in dwarf ellipticals by applying unsharp masks or subtracting the axisymmetric light distribution of each galaxy from the image in a complete sample of Virgo galaxies extracted from the SDSS. They have shown that dE galaxies with typical spiral features, as the one first discovered by Jerjen et al. (2000), are rather common objects. Among the brightest dwarf ellipticals, $\sim$50% of the objects possess these features, while their fraction decreases with decreasing luminosity (Fig. \[Lisker\]). Such galaxies have been later discovered in other clusters like Coma using the exquisite quality images obtained with the HST (Graham et al. 2003; den Brok et al. 2011; Marinova et al. 2012). Their spiral structure is generally grand design and not flocculent, and they have a flat shape suggesting that they are genuine disk galaxies (Lisker et al. 2006a; Lisker & Fuchs 2009). The best quality imaging material gathered thanks to the ACSVCS survey of Virgo has shown that these kind of features are very frequent in quiescent systems of intermediate luminosity (Ferrarese et al. 2006). The same survey has also indicated that dusty features observed in absorption are also quite frequent but mainly in the most massive objects (42%). There is also a population of dwarf ellipticals with blue nuclei probably due to a recent episode of star formation. The work of Lisker et al. (2007) has shown that galaxies with disky structures, including those with blue nuclei, are not distributed near the cluster core as are the bright ellipticals and lenticulars, or the other ordinary spheroidal dwarf ellipticals, but are rather distributed uniformly all over the cluster. Thanks to HST images it has also been shown that dwarf ETGs in the periphery of the Peresus cluster have more disturbed morphologies than those located in the core of the cluster (Penny et al. 2011). Lisker et al. (2009) have studied the properties of the nucleated dwarf elliptical galaxies without any sign of recent star formation (blue nuclei) or disky structure (spiral arms, bars, disks) located close to the bright central elliptical galaxies. By dividing the sample into fast- and slow-moving objects according to their velocity with respect to the mean recessional velocity of the cluster, they have shown that fast-moving objects have a projected axial ratio consistent with a flatter shape, while the slow-moving are roundish objects. Deep near-infrared images of dwarf elliptical galaxies have also revealed different structures in their 2D-stellar distribution (Janz et al. 2012, 2014). All these results have been interpreted as an evidence that dwarf elliptical galaxies have different origins. Those presenting disky structures, or having a flat shape (thus having a high-velocity with respect to the cluster) have been recently accreted as star-forming systems and have been transformed during their interaction with the cluster environment, while the roundish objects with a low-velocity dispersion have been in the cluster since a very early epoch, or might have even been formed within the cluster. In the recent years a huge effort has also been made at understanding whether dwarf elliptical galaxies are just the low luminosity extension of bright ellipticals or rather are a totally independent category of objects. Systematic differences in the two galaxy populations, and possible similarities with the properties observed in spiral galaxies, could be interpreted as a clear indication of a different formation scenario (Kormendy et al. 2009; Kormendy & Bender 2012). The debate is principally motivated by the possible existence of a few dwarf elliptical galaxies similar to M32 that, contrary to the general dE galaxy population, have a very compact structure characterized by a very high surface brightness. These objects might not follow the standard scaling relations depicted by the other quiescent systems. The seminal work of Graham & Guzman (2003) has clearly shown that the observed change in slope in the surface brightness vs. absolute magnitude relation observed between massive and dwarf ellipticals is naturally due to the shape of their Sersic light profile, with an index $n$ increasing with the luminosity. Continuity between E and dE in different scaling relations has been also reported by Gavazzi et al. (2005), Ferrarese et al. (2006), Misgeld et al. (2008, 2009), Misgeld & Hilker (2011), and Smith Castelli et al. (2013). On the contrary, there is evidence that the size–luminosity relation depicted by bright ellipticals is not followed by dwarf systems, that rather have an almost constant extension ($R_{\rm eff} \simeq 1$ kpc) regardless of their luminosity (Janz & Lisker 2009; Smith Castelli et al. 2008; Misgeld & Hilker 2011). This dispute will certainly come to an end once the NGVS survey (Ferrarese et al. 2012), that has covered homogeneously the whole Virgo cluster, thus including several hundreds of early-type systems in four ($u^,g',i',z'$) photometric bands, will be fully exploited. Thanks to its exquisite angular resolution, the ACSVCS (Cote et al. 2004) allowed the detailed analysis of the structural properties of the nuclei of ETG. The HST data have shown that ground based optical surveys generally underestimate the presence of nuclei in ETGs probably because of their limited angular resolution (Cote et al. 2006). The nuclei are generally resolved even in dwarf systems, ruling out any possible low-level AGN nature (Cote et al. 2006). The same data have also shown the lack of any strong evidence that nucleated galaxies are more centrally clustered than other non-nucleated objects. By fitting their radial light distribution, Cote et al. (2007) have shown that a Sersic profile generally overestimates the nuclear emission in bright and massive galaxies while it underestimates it in low-luminosity objects, with a difference that steadily varies with galaxy luminosity. They discussed the observed nuclear properties of ETGs in the context of gas infall in various formation scenarii. The comparison of the structural properties of red sequence and blue cloud galaxies in nearby groups has been carried out by van den Bosch et al. (2008) using a large sample of galaxy groups extracted from the SDSS. Studying their color and light concentration, their analysis has shown that the transformation mechanism operating on satellites affects more colors than morphology. They have also shown that the observed differences between satellite and central galaxies do not depend on the halo mass, suggesting that the process at the origin of the observed perturbation is the same in different environments. They thus concluded that the most probable process is starvation since they considered ram pressure stripping and galaxy harassment efficient only in high-mass systems such as clusters. Kinematics ---------- The kinematical properties of galaxies and their relations with the environment have been the subject of various works done by the SAURON (Bacon et al. 2001), $\hbox{ATLAS}^{\rm 3D}$ (Cappellari et al. 2011a), CALIFA (Sanchez et al. 2012) and SMAKCED (Toloba et al. 2014a) teams. While the first three projects were mainly focused on bright galaxies (see, however, Rys et al. 2013), with SAURON and $\hbox{ATLAS}^{\rm 3D}$ primarily on early-type systems, SMAKCED was devoted to the study of dwarf elliptical galaxies in the Virgo cluster. The final purpose of this project was that of understanding whether dE are rotationally supported systems, as first claimed by Pedraz et al. (2002), Geha et al. (2002, 2003) and van Zee et al. (2004). Based on very small samples, these pioneering works were mainly limited by the spectral resolution of the adopted instruments, barely sufficient to observe velocity dispersions of the order of 20–$30\,\hbox{km\,s}^{-1}$ typical of dwarf systems. Using a sample of 21 dwarf elliptical galaxies mainly located in the Virgo cluster, Toloba et al. (2009, 2011) have shown the existence of a large fraction of rotationally supported dE. Their analysis indicated that pressure-supported systems, generally characterized by old stellar populations, are located preferentially in the inner regions of the cluster, while rotationally supported objects, composed of younger stellar populations, are mainly located at the periphery of the cluster (Toloba et al. 2009). Furthermore, these works have shown that rotationally supported dE have rotation curves similar to those of LTGs of similar luminosity and follow the same Tully–Fisher relation as star forming systems (Toloba et al. 2011; Fig. \[Toloba\]). The analysis of the main scaling relations, such as the Faber–Jackson and the Fundamental Plane relations, consistently revealed that dE are different from massive ellipticals since have structural and kinematical properties closer to those of star-forming systems rather than to those of E and S0 (Toloba et al. 2012). Although the kinematical properties of dE might change from object to object (Rys et al. 2013), all these observational evidences are consistent with the picture where gas-rich star-forming systems entering the hostile cluster environment lose their gaseous component on relatively short time scales, stopping their activity of star formation and becoming quiescent systems. The angular momentum of the galaxy is conserved on relatively long time scales if the galaxy interacts with the hot intergalactic medium (ram pressure stripping), while the system is heated if the dominant process is gravitational (galaxy harassment). The observations indicate that the dynamical interactions with the IGM are dominant at the present epoch, producing the rotationally supported systems at the periphery of the cluster, while the gravitational one, dominant in the past, produced the hot systems mainly located in the virialized inner region of the cluster (Toloba et al. 2014b; Boselli et al. 2014a). Independent evidence of gravitational interactions comes also from the presence of dE with counter rotating cores observed by Toloba et al. (2014c) close to the Virgo cluster core. This evolutionary picture is consistent with that proposed for more massive galaxies by the $\hbox{ATLAS}^{\rm 3D}$ survey, where the most massive slow rotators are formed by major dry merging events, while fast rotators by the quenching of the star formation of spiral galaxies (Cappellari et al. 2011b; Cappellari 2013). The only difference with the fading model proposed for low luminosity systems is that, associated with the quenching of the star-formation activity of spiral disks in cluster, there is also a growth of the bulge (Cappellari 2013). Individual galaxies {#INDIVIDUAL} ------------------- As already introduced in Sect. \[s1\], a typical example of galaxies undergoing environmental transformations is the galaxy FGC 1287 at the periphery of the cluster Abell 1367, showing a cometary HI tail of 250 kpc projected length. Although the origin of this feature is still uncertain (Scott et al. 2012), its typical shape suggests that it might have been created by the interaction of its ISM with the hot and dense ICM of the cluster during a ram pressure stripping event. We recall that this result agrees with the presence of massive ram pressure stripped galaxies up to more than 1 Mpc away from the cluster core observed in several nearby clusters as summarized in Boselli & Gavazzi (2006), including a recent discovery in the Shapley supercluster (Merluzzi et al. 2013). It also agrees with the results of the most recent hydrodynamical simulations of gas stripping in cluster galaxies (see Sect. \[HYDRODYNAMICAL\]). Consistent with this scenario is also the discovery of cluster galaxies with long tails of ionized gas, first observed in A1367 by Gavazzi et al. (2001) and later in Virgo (Yoshida et al. 2004; Kenney et al. 2008), in Coma (Yagi et al. 2007, 2010; Yoshida et al. 2008, 2012; Fossati et al. 2012), and in A3627 (Zhang et al. 2013). If the $\hbox{H}\alpha$ filamentary structure associated with NGC 4388 in Virgo is located close to the cluster core (Yoshida et al. 2004; kenney et al. 2008), those relative to the galaxies in A1367 and Coma are much more in the periphery of the clusters, again indicating that ram pressure stripping is efficient not only in the cluster core. The observation of these peculiar objects gives also other important information on the physical process acting on cluster galaxies. They first indicate that the stripping process is also able to remove the ionized phase of the gas (Gavazzi et al. 2001). There are, however, other indications that part of the gas can be ionized in situ by shocks (Yoshida et al. 2012). They indicate that star formation in the stripped gas, if present, is generally very modest and located in small ($\sim$200 to 300 pc) defined knots of stellar mass $10^6$–$10^7\,\hbox{M}_{\odot}$ (Yoshida et al. 2008). The few star-forming blobs associated with NGC 4388 or with the interacting systems VCC 1249–M49 in Virgo are even smaller, with stellar masses ${\lesssim}10^{4.5}\,\hbox{M}_{\odot}$ (Arrigoni Battaia et al. 2012; Yagi et al. 2013). This result is consistent with the evidence that the efficiency of star formation in the stripped material is lower than the standard efficiency observed over spiral disks generally described by the Schmidt law (Boissier et al. 2012). This last result is of particular importance in constraining models of star formation from the stripped gas (see Sect. \[EXTRAPLANAR\]). Indeed, it indicates that the formation of dwarf galaxies in the stripped material, if possible, is not frequent and cannot produce any significant steepening of the faint end of the luminosity function. There are, however, a few cases where the star-formation process in the stripped gas is important. First discovered by Cortese et al. (2007) in two clusters at $z \sim 0.2$, there exist a few objects with long tails of star-forming regions prominent in UV images generally named “fireballs” or “jellyfish” (Smith et al. 2010). They are more frequent in massive clusters such as Coma than in small systems as Virgo. An exception to this rule is the spectacular IC 3418 (VCC 1217) close to the core of the Virgo cluster (Hester et al. 2010; Fumagalli et al. 2011; Jachym et al. 2013; Kenney et al. 2014). The galaxy, totally stripped of its gas, quenched its activity of star formation $\sim$300 to 400 Myr ago probably after a starburst activity. In the tail, the observed $\hbox{H}\alpha$ peaks are displaced from the UV emitting knots, suggesting that the gas clumps are continuously accelerated by ram pressure, leaving behind new stars decoupled from the gas (Kenney et al. 2014). The typical mass of these fireballs is of ${\sim} 10^5\,\hbox{M}_{\odot}$ (Fumagalli et al. 2011), thus still too small to make this process relevant for the formation of dwarf galaxies or for modifying the shape of the luminosity function. Another spectacular example with similar characteristics is the galaxy ESO 137-001 in the Norma cluster (A3627, Sun et al. 2007; Jachym et al. 2014). Multifrequency observations have been critical to demonstrate that also the hot X-ray emitting gas in the halo of galaxies can be stripped by ram pressure in cluster (Sun et al. 2007, 2010; Ehlert et al. 2013; Zhang et al. 2013) and group galaxies (Rasmussen et al. 2012b). The most evident case is the galaxy ESO 137-001 in A3627 (Sun et al. 2007; Fumagalli et al. 2014). While in clusters the hot and cold gas removal is primarily due to ram pressure stripping, in groups both ram pressure and tidal interactions contribute, as clearly indicated by the recent Chandra* and VLA observations of the nearby group of NGC 2563 (Rasmussen et al. 2012b). The importance of these results resides in the fact that they are the first observational justification that the gaseous halo of galaxies is removed in high-density environments, as generally assumed in cosmological and semi-analytic models of galaxy evolution. This assumption in models and simulations is crucial since it makes the feedback process of supernovae extremely efficient in expelling the disk gas, thus quenching the activity of star formation and transforming on very short time scales gas-rich systems into quiescent, red galaxies (see Sect. \[MODELLIspsCOSMOLOGICI\]). Although not frequent because of the high-velocity dispersion within clusters, tidal interactions can also be related to the formation and evolution of dwarf systems in high-density regions. There exists, indeed, a few representative and interesting cases in nearby clusters. One example is NGC 4254 in Virgo. Entering the cluster at high velocity for the first time, the galaxy was harassed of a fraction of its gas that is now forming a cloud (VIRGOHI21; Davies et al. 2004) apparently not associated with any other optical counterpart (Haynes et al. 2007; Duc & Burnaud 2008; Wezgowiec et al. 2012). The wide field and the high sensitivity to low surface brightness features of the NGVS survey allowed the identification of three dwarfs galaxies satellite of the bright NGC 4216 in Virgo undergoing a tidal disruption process (Paudel et al. 2013). The data obtained by NGVS and GUViCS have shown the presence of small star-forming complexes produced during the interaction of the dwarf, gas-rich Im galaxy VCC 1249 (UGC 7636) with the bright elliptical M49 (Arrigoni Battaia et al. 2012). Although the stellar mass of these objects is still very small $(10^4-10^5\,\hbox{M}_{\odot}$), it has been suggested that a similar process might have been at the origin of Ultra Compact Dwarf (UCD) galaxies, a population recently discovered in high-density regions (Drinkwater et al. 2004). We can also mention the discovery of a blue infalling group in A1367, the prototypal example of pre-processing in the nearby universe (Sakai et al. 2002; Gavazzi et al. 2003b; Cortese et al. 2006b). The low-velocity dispersion within a group infalling into the main cluster makes gravitational interactions very efficient in perturbing galaxies before they become real members of A1367. These perturbations produce small condensations of matter that might later evolve as independent entities and thus be progenitors of dwarf cluster galaxies. Modeling {#s4} ======== Cosmological simulations {#MODELLIspsCOSMOLOGICI} ------------------------ Cosmological models of galaxy evolution indicate that galaxies are formed from the condensation of gas within dark matter halos. In a hierarchical formation scenario, small structures are formed first and later merged to give birth to massive objects. By cooling, the gas conserves its angular momentum leading to the formation of rotating systems. The violent interactions associated with merging events heat the systems, forming bulges and ellipticals. In this bottom-up formation scenario, galaxies now belonging to rich clusters might have suffered the effects of different environments during their life (pre-processing; Dressler 2004). They might have been members of small groups where gravitational interactions with other members were important before entering the evolved cluster (Boselli & Gavazzi 2006). As described in De Lucia (2011), in cosmological simulations the physical evolution of galaxies is reproduced using various techniques. Hydrodynamical simulations mimic the evolution of the gaseous component modeling different physical processes such as gas cooling, star formation, and feedback. Because of this complex approach, these simulations are limited by spatial and mass resolution (e.g. Berlind et al. 2005). In semi-analytic models of galaxy formation (SAM), the link between the evolution of dark matter halos traced by high-resolution N-body simulations and the barionic matter is accomplished using simple physical prescriptions, without following explicitly the coupled evolution between structure formation and gas physics through the numerical integration of hydro/gravity equations. Nevertheless this (computationally cheap) approach has the advantage of covering a significantly larger range in stellar mass and spatial resolution and thus is well suited for the study of the evolution of dwarf systems in different environments as those analyzed in this work. Beside “harassment” (Moore et al. 1996), other investigations of the effects of the cluster environment in cosmological hydrodynamical simulations have been undertaken by McCarthy et al. (2008). In this work the authors simulated the stripping[^17] of the hot halo of satellite galaxies entering massive halos typical of groups and clusters. These simulations indicated that an important fraction of the hot gas of satellite galaxies ($\sim$30%) is still at place 10 Gyr after the beginning of the interaction. More recently, Bahe et al. (2012) and Wetzel et al. (2013) concluded that the confinement pressure exerted by the intracluster medium is not sufficient to significantly decrease the impact of the ram pressure exerted on the hot gas halo of satellite galaxies. Using the same cosmological simulations, the same team has also shown that ram pressure stripping exerted by the extended gas halo surrounding groups and clusters is sufficiently strong to strip the hot gas atmosphere of infalling galaxies up to $\sim$5 times the virial radius (Bahe et al. 2013). These results are consistent with the simulations of Cen et al. (2014). The most recent semi-analytic models of galaxy evolution arrive to reproduce massive and dwarf galaxies down to stellar masses of $\sim$10$^{7.5}$ (Guo et al. 2011, 2013) and sample a large range in environments, from the field to massive clusters analog to Coma in the local universe (some $10^{15}\,\hbox{M}_{\odot}$). Only some of the most recent models, however, have implemented tuned recipes such as those proposed by McCarthy et al. (2008) to accurately reproduce the effects induced by the group and cluster environment on galaxy evolution (Font et al. 2008; Kang & van den Bosch 2008; Kimm et al. 2009; Taranu et al. 2014; Weinmann et al. 2011; Lisker et al. 2013, the last two especially tailored to mimic the dwarf galaxy populations in local clusters such as Virgo and Coma). Indeed, in previous studies the effects of the environments were simulated just by instantaneously removing the whole hot-gas halo of galaxies once they became satellites. This assumption makes supernova feedback sufficient to totally sweep the cold gas disk that, in the lack of a surrounding halo, is permanently ejected in the intracluster medium. The lack of gas quenches the activity of star formation on very short time scales producing a large fraction of red galaxies (Kang & van den Bosch 2008; Font et al. 2008; Kimm et al. 2009; De Lucia 2011). For this reason Okamoto & Nagashima (2003) and Lanzoni et al. (2005) identified starvation as the main physical process responsible for the quenching of the star-formation activity of cluster galaxies. Despite the implementation of tuned recipes for mimicking the stripping of the hot gas, the results of the most recent semi-analytic models still overpredict the number and the colors of red objects (Fontanot et al. 2009; Wang et al. 2012), suggesting that the effects induced by the environments are overestimated (Weinmann et al. 2011; Fig. \[Weinmann\]). ![Fraction of red galaxies as a function of the projected distance from the cluster center (left panels), line-of-sight velocity with respect to the cluster center (central panels), and absolute magnitude (right panels), from Weinmann et al. (2011). Observations of the Virgo cluster (first row), of the Coma cluster (central row), and of the Perseus cluster (lower row) are shown with black lines and relative errorbars, while the results of the semi-analytic models are the colored lines. The semi-analytic models of galaxy evolution overpredict the number of red, quiescent galaxies especially at low stellar masses or in clusters still under formation (Virgo). This suggests that the gas stripping process and the subsequent quenching of the star formation activity as predicted by models, which happens mainly in groups during the pre-processing of galaxies, is too rapid (Weinmann et al. 2011). Reproduced with permission of Oxford University Press[]{data-label="Weinmann"}](fig16.pdf){width="100.00000%"} The same models, based on the MS-II simulations of Guo et al. (2011) including a physical prescription for the stripping of the hot gas and supernova feedback, reproduce well the velocity dispersion and the luminosity functions of nearby clusters, but overpredict the dwarf to giant fraction probably because of an incorrect prescription for tidal disruption (see however Henriques et al. 2013). Several works have also indicated that the evolution of cluster galaxies might have been affected during their previous membership to groups (pre-processing) (Book & Benson 2010; De Lucia et al. 2012; Bahe et al. 2013; Lisker et al. 2013; Taranu et al. 2014). For this reason, gas stripping and quenching of the star formation might have happened at earlier epochs and outside the virial radius of the evolved cluster (Bahe et al. 2013). The models also indicate that pre-processing might have been more important in low-mass systems than in massive galaxies (De Lucia et al. 2012). Models thus suggest that dwarf elliptical galaxies in local clusters might have been processed early and continuously in groups and cluster halos instead of being late-type objects recently transformed in quiescent systems (Lisker et al. 2013; Taranu et al. 2014). Hydrodynamical simulations of gas stripping {#HYDRODYNAMICAL} ------------------------------------------- Different teams developed their own hydrodynamical simulations to reproduce the effects induced by the hostile cluster or group environment on galaxy evolution. Tonnesen et al. (2007) tried to identify, using cosmological simulations, which, among the different processes acting on galaxies in rich cluster, is the dominant one. Their work has indicated that the interaction with the hot intergalactic medium (ram pressure stripping) is the most important at the present epoch. Models and simulations indeed show that, although dominant in the core of the cluster where the velocity of the galaxy and the density of the ICM are maximal, ram pressure stripping is an ongoing process eroding the gaseous component all over the orbit of the galaxy within the cluster (Roediger & Hensler 2005; Roediger & Bruggen 2006, 2007; Bruggen & De Lucia 2008; Kapferer et al. 2008). The simulations of Tonnesen et al. (2007) indeed indicate that ram pressure is an important process out to the virial radius, able to remove all the gas on time scales of the order of $\geq$1 Gyr. The limit on the cluster region where ram pressure stripping is active and efficient has been later extended to $\sim$3 virial radii by Cen et al. (2014), consistent with the most recent observations of head-tail galaxies in the periphery of nearby clusters (see Sect. \[INDIVIDUAL\]). Timescales of the order of 1.5 Gyr are also obtained using 3D hydrodynamical simulations by Roediger & Bruggen (2007) and Cen et al. (2014) for a substantial reduction of the total gas content of the perturbed galaxy via ram pressure. As mentioned before, however, these hydrodynamical simulations based on a single and homogeneous gas phase for the ISM generally underestimate the efficiency of ram pressure stripping (Tonnesen & Bryan 2009). Multiphase hydrodynamical simulations indicate that gas ablation in all its phases (from diffuse atomic to dense molecular gas) can take place at all galactic radii if ram pressure stripping is sufficiently strong, as it is the case in rich clusters of galaxies such as Virgo (Tonnesen & Bryan 2009) and as seen in the few extreme cases of ram-pressure known to date. Modeling the effects of ram pressure on a multiphase gas disk is still challenging since it requires to take into account several physical effects as to resolve hydrodynamic and thermal instabilities across a large range of physical scales, such as heating and cooling of the ISM, self-gravity, star formation, stellar feedback, and magnetic field. Given the fractal distribution of the gaseous component within the ISM, models must also predict simultaneously the gas distribution on different scales, from giant molecular clouds to the tails whose size can exceed the size of galaxies (Roediger 2009). Tonnesen & Bryan (2009) performed high-resolution (40 pc) 3D hydrodynamical simulations of galaxies undergoing a ram pressure stripping event. They have included in their code radiative cooling on a multiphase medium. This naturally produces a clumpy ISM with densities spanning six orders of magnitude, thus quite representative of the physical conditions encountered in normal, LTGs. Their simulations show that under these conditions the gas is stripped more efficiently up to the inner regions with respect to an homogeneous gas. They also show that all the low-density, diffuse gas is quickly stripped at all radii. When the ram pressure stripping is strong there is also less gas at high densities. The deficiency in high-density regions results from the lack of the diffuse component feeding giant molecular clouds (Tonnesen & Bryan 2009). The recent work of Ruszkowski et al. (2014) indicates that an accurate description of magnetic fields in models does not significantly change the efficiency of gas stripping, but only explains the formation of the filamentary structures observed in the stripped material. Smith et al. (2012c) studied the effects induced by ram pressure stripping on the stellar disk and dark matter halo of cluster galaxies. Their simulations indicate that, although the ISM–IGM interaction acts only on the gaseous component, thanks to their mutual gravitational interaction, the gas displacement perturbs the potential well of the galaxy, dragging the stellar disk and the cusp of the dark matter halo off center. The perturbation can also mildly deform and heat the stellar disk. The same team has also simulated the effects of ram pressure stripping on newly formed tidal dwarf galaxies (Smith et al. 2013). Because of their low dark matter content, tidal dwarfs are very fragile systems that can be easily perturbed and even totally destroyed through gas and stellar loss. Consistently with these works, Kronberger et al. (2008a) have shown that a ram pressure stripping event can affect the 2D velocity field of galaxies determined from emission lines. The perturbation is symmetric in a face-on interaction, while can displace the dynamical center of the galaxy in edge-on interactions. The interaction can also increase the activity of star formation in the inner regions where the compressed gas is located (Kronberger et al. 2008b). Bekki (2009) simulated the effects of halo gas stripping in galaxies of different mass belonging to different environments. This work has shown that halo gas stripping on Milky Way type galaxies is very efficient not only in massive clusters but also in small and compact groups. The removal of gas happens outside-in, producing truncated star forming radial profiles. The stripping process is more rapid in dense environments than in groups, and in low-mass systems with respect to massive galaxies. The same team has also shown that repeated slow encounters within groups are able to transform star forming systems into S0 (Bekki & Couch 2011). These gravitational interactions can trigger the formation of new stars through repetitive starbursts, and at the same time consume the gas reservoir producing gas-poor objects. The resulting systems have lower velocity rotations and higher velocity dispersions than their progenitors. The ram pressure stripping event can either enhance or reduce the activity of star formation depending on the mass of the galaxy, on the inclination of its disk with respect to the orbit and the environment in which the galaxy resides (Bekki 2014). Kawata & Mulchaey (2008) have simulated the effects of gas stripping in groups of virial mass $8\times10^{12}\,\hbox{M}_{\odot}$ and total X-ray luminosity $L_{\rm X} \simeq 10^{41}\,\hbox{erg s}^{-1}$ on a galaxy of $M_{\rm star} \simeq 3.4\times 10^{10}\,\hbox{ M}_{\odot}$. Their N-body/smoothed particle hydrodynamic simulations show that ram pressure stripping is not able to remove the cold gas over the disk of the galaxy, but is sufficient to remove the hot gas located in the halo on time scales of $\sim$1 Gyr. Because of the lack of new gas feeding star formation, the galaxy quenches its activity in $\sim$4 Gyr. These conclusions, however, are not confirmed by the simulations of Hester (2006) who identify ram pressure as the most efficient process in stripping the cold gas component even inside groups of galaxies. Stripping of the hot gaseous halos in clusters and groups galaxies have also been modeled using 3D-hydrodynamical simulations by McCarthy et al. (2008). These simulations indicate that a significant fraction of the hot gas, $\simeq$30%, can remain in place after a 10 Gyr interaction, thus in contradiction with other works. McCarthy et al. (2008) developed simple analytic models ideally constructed to be included in semi-analytic models of galaxy evolution to reproduce the gas stripping process in high-density environments. Formation of dwarf elliptical galaxies via galaxy harassment ------------------------------------------------------------ Early N-body simulations specially designed to follow the hierarchical growth of clusters and galaxy harassment and to study the possible formation of dwarf elliptical galaxies through the transformation of star forming disks were carried out by Mastropietro et al. (2005). These simulations are designed for a $\Lambda\hbox{CDM}$ cluster of $10^7$ particles with a total mass similar to the Virgo cluster. The simulations indicate that most of the galaxies undergo major structural modifications even at the outskirts of the cluster, with a large fraction of them transforming from late-type rotating systems into dwarf spheroidal hot systems on time scales of a few Gyrs. The effects are, however, most important in the inner 100 kpc of the cluster. The harassed galaxies are more compact and have comparable or higher surface brightness than unperturbed objects, probably because of the formation of bars or grand design spiral arms. The mass loss is important and induces the formation of round galaxies. Harassment also heats the systems, decreasing the rotational velocity-to-velocity dispersion $v/\sigma$ ratio. Rotation is totally lost only in the most perturbed objects. More recently, Aguerri & Gonzalez-Garcia (2009) have developed high-resolution N-body simulations to test the tidal stripping scenario for the formation of dE. These simulations studied the perturbation induced to disk galaxies with different bulge-to-disk ratios. They show that, while the bulge is only marginally affected, the disk and the dark matter halo are efficiently perturbed in the outer parts. The scale length of the stellar disk, for instance, can be reduced by 40–50%. After several fast encounters galaxies can lose up to 50–80% of their mass and 30–60% of their luminous matter. Prograde interactions produce stable bars, while retrograde encounters do not. The formation of bars is more important in the absence of bulges. The interaction heats the system decreasing its $v/\sigma$ ratio. In a recent study, Benson et al. (2014) simulated the effects of different environmental processes (starvation, ram pressure stripping, tidal stirring) on the evolution of dwarf galaxies in Virgo-like clusters. These simulations were principally focused in reproducing the galaxy kinematical properties. Benson et al. (2014) identified tidal stirring induced by the cluster halo on disk galaxies as the most probable process able to reproduce the observed gradient of the angular momentum of dEs with the clustercentric distance to M87 at the center of the Virgo cluster (see Fig. \[benson\]). ![The distribution of the angular momentum of dEs as a function of the projected distance from M87 (from Benson et al. 2014). © AAS. Reproduced with permission[]{data-label="benson"}](fig17.pdf){width="70.00000%"} Spectro-photometric models of galaxy transformation {#MODELspsSPECTRO} --------------------------------------------------- Boselli et al. (2006, 2008a,b) have developed 2D chemo-spectrophotometric models of galaxy evolution especially designed to reproduce the perturbations induced by the hostile environment on star forming disks infalling for the first time in rich clusters. These models reproduce two different kinds of perturbations: ram pressure stripping and starvation. The models are based on the chemo-spectrophotometric models of galaxy evolution of Boissier & Prantzos (2000) where disk galaxies are formed in a dark matter halo and form stars following a Schmidt law modulated by the rotation of the galaxy (Boissier et al. 2003). Calibrated on the Milky Way, the models have two free parameters, the spin parameter $\lambda$, which is a dimensionless measurement of the specific angular momentum, and the rotational velocity $V_{\rm C}$. Starvation is simulated simply stopping the infall of fresh gas, that the model requires for unperturbed systems to reproduce the observed color and metallicity gradients of nearby galaxies.[^18] A stripping event is modeled considering that ram pressure has an intensity varying along the orbit of the galaxy within the cluster, as predicted by the scenario of Vollmer et al. (2001) explicitly designed for Virgo. ![The extinction corrected $NUV-i$ (AB system) vs. $M_{star}$ relation for all galaxies of the sample, from Boselli et al. (2014a). Filled symbols are for slow rotators, crosses for fast rotators. Symbols are color coded for dividing objects in the blue cloud form the green valley and the red sequence. The large orange filled squares indicate the models of unperturbed galaxies of spin parameter $\lambda=0.05$ and rotational velocity 40, 55, 70, 100, 130, 170, and $220\;\hbox{km s}^{-1}$. The magenta lines indicate the starvation models. The black lines show the ram pressure stripping models. Different symbols along the models indicate the position of the model galaxies at a given look-back time from the beginning of the interaction. Courtesy of ESO[]{data-label="modellisam"}](fig18.pdf){width="70.00000%"} These models have been first tuned on a well-known anemic Virgo cluster galaxy undergoing a ram pressure stripping event, NGC 4569 (M90, Boselli et al. 2006), and then extended to dwarf galaxies in the Virgo cluster (Boselli et al. 2008a,b). They show that only ram pressure stripping can reproduce the truncated radial profiles observed in the gaseous component and in the young stellar populations of NGC 4569, ruling out the starvation scenario. They also show that ram pressure stripping can efficiently remove, on very short time scales ($\simeq$150 Myr) all the gas in low-mass star forming systems. The lack of gas induces a quenching of the star-formation activity, transforming gas-rich star-forming systems into gas-poor quiescent objects (Fig. \[modellisam\]). This transformation happens on time scales of the order of 0.8–1.7 Gyr. The chemical, structural, spectrophotometric properties of the transformed galaxies are very similar to those of dwarf elliptical galaxies (Boselli et al. 2008a,b). This transformation process would imply that the overall cluster and field luminosity functions are similar, with a mutual inversion of the faint end slope of blue, star forming systems, frequent in low-density regions, and red, quiescent objects, abundant in the core of the cluster, as indeed observed in Virgo and Coma. The models predictions are also consistent with the observed properties of the globular clusters of dwarf ellipticals and their expected progenitors (Boselli et al. 2008a; see however Sanchez-Janssen & Aguerri 2012). These results consistently indicate that ram pressure stripping is able to explain the formation of the faint end of the red sequence characterizing rich clusters of galaxies. ![Projections of the HI column density (left) and of the $\hbox{H}\alpha$ surface density (right) in the ram pressure stripped gas for a model galaxy with star formation and feedback, adapted from Tonnesen & Bryan (2012). A small amount of star formation can take place in the highest column density regions of the HI gas stripped during the interaction out to 80 kpc from the disk of the galaxy. Reproduced with permission of Oxford University Press[]{data-label="SFstrippedgas"}](fig19a.pdf "fig:"){width="30.00000%"} ![Projections of the HI column density (left) and of the $\hbox{H}\alpha$ surface density (right) in the ram pressure stripped gas for a model galaxy with star formation and feedback, adapted from Tonnesen & Bryan (2012). A small amount of star formation can take place in the highest column density regions of the HI gas stripped during the interaction out to 80 kpc from the disk of the galaxy. Reproduced with permission of Oxford University Press[]{data-label="SFstrippedgas"}](fig19b.pdf "fig:"){width="17.00000%"} ![Projections of the HI column density (left) and of the $\hbox{H}\alpha$ surface density (right) in the ram pressure stripped gas for a model galaxy with star formation and feedback, adapted from Tonnesen & Bryan (2012). A small amount of star formation can take place in the highest column density regions of the HI gas stripped during the interaction out to 80 kpc from the disk of the galaxy. Reproduced with permission of Oxford University Press[]{data-label="SFstrippedgas"}](fig19c.pdf "fig:"){width="30.00000%"} ![Projections of the HI column density (left) and of the $\hbox{H}\alpha$ surface density (right) in the ram pressure stripped gas for a model galaxy with star formation and feedback, adapted from Tonnesen & Bryan (2012). A small amount of star formation can take place in the highest column density regions of the HI gas stripped during the interaction out to 80 kpc from the disk of the galaxy. Reproduced with permission of Oxford University Press[]{data-label="SFstrippedgas"}](fig19d.pdf "fig:"){width="17.00000%"}\ Formation of dwarf galaxies in stripped material {#EXTRAPLANAR} ------------------------------------------------- N-body/hydrodynamic simulations of ram pressure stripping events on star forming galaxies in rich clusters show the formation of extended gas tails (350 kpc) in the direction opposite to the motion of the galaxy within the intergalactic medium (Kronberger et al. 2008b; Kapferer et al. 2009). These gas tails are similar to those observed in the periphery of A1367 and Coma by Gavazzi et al. (2001), Yoshida et al. (2008); Yagi et al. (2010), Fossati et al. (2012) in $\hbox{H}\alpha$, by Scott et al. (2012) in HI and in X-rays in A3627 (Sun et al. 2010). The same simulations also predict an increase of the total star-formation activity of the galaxy by an order of magnitude, 95% of which produced in the tail of diffuse gas (Kapferer et al. 2009). These results were later questioned by Tonnesen & Bryan (2012) who showed, using high-resolution adaptive mesh simulations, that gas stripping produces a truncation of the star forming disk on time scales of a few hundreds million years. They show that there is a moderate increase of the star formation in the bulge but without a starburst phase. They also show that the star formation in the tail is low, and any contribution to the intercluster light, if present, is likely to be very small (see Fig. \[SFstrippedgas\]). The difference in the results between Tonnesen & Bryan (2012) and Kapferer et al. (2009) comes mainly from the nature of the two different codes and only marginally on the adopted conditions (Tonnesen & Bryan 2012). Consistently with Tonnesen & Bryan (2012), Yamagami & Fujita (2011) predict that in the absence of magnetic field, Kelvin–Helmholtz instabilities destroy molecular clouds, preventing the formation of new stars in the tails of stripped gas. Observations seem more consistent with the prediction of Tonnesen & Bryan (2012) and Yamagami & Fujita (2011). Indeed, there are only a few examples of cluster galaxies with major ongoing star-formation events in the stripped material far from the galaxy disk. Typical examples are IC 3418 in Virgo (Hester et al. 2010; Fumagalli et al. 2011; Kenney et al. 2014), ESO137-001 in A3627 (Sun et al. 2007; Jachym et al. 2014; Fumagalli et al. 2014), and two massive objects in two clusters at $z \simeq 0.2$ (Cortese et al. 2007). The extended tails of ionized gas observed in the $\hbox{H}\alpha$ images of galaxies in nearby clusters (Gavazzi et al. 2001; Kenney et al. 2008; Yoshida et al. 2008; Yagi et al. 2010; Fossati et al. 2012; Jachym et al. 2014) are only marginally associated with major star-formation events in the tails. In the Virgo cluster there are also evident cases of galaxies with extended HI tails but without any associated star-formation event (Boissier et al. 2012), or objects where the outplanar star-formation process is happening only very close to the galaxy disk (Abramson et al. 2011). Observations vs. models: a new evolutionary picture {#s5} =================================================== The Virgo cluster ----------------- The comparison of the observational results obtained so far with model predictions can be used to reconstruct the evolutionary picture that gave birth to the dwarf quiescent galaxies inhabiting nearby clusters that form the faint end of the red sequence. Multifrequency data are critical for this purpose since they provide us with a complete and coherent view of the undergoing process and of its effect on galaxy evolution (e.g. Boselli 2011). We can first apply the exercise to the Virgo cluster. Here, indeed, multifrequency data spanning the whole range of frequencies are available for galaxies down to stellar masses of ${\sim} 10^7\,\hbox{M}_{\odot}$. At the same time, most of the available models and simulations have been carried out with the aim of reproducing the physical conditions undergone by galaxies located in clusters with characteristics similar to those of Virgo. Observations and models consistently indicate that dwarf elliptical galaxies might have been formed by the transformation of low-luminosity late-type systems recently entered the Virgo cluster once they have lost their gaseous content during the interaction with the hostile cluster environment, as first proposed by Boselli et al. (2008a,b). The most plausible physical process responsible for this transformation is the ram pressure $\rho_{\rm ICM} V^2$ exerted by the hot and dense intracluster medium (where $\rho_{\rm ICM}$ is its density) on galaxies moving at high velocity ($V$) within the cluster. Both models and observations indicate that this ram pressure is able to overcome the gravitational forces keeping the gas anchored to the potential well of the galaxy, $G \Sigma_{\rm gas} \Sigma_{\rm star}$, where $G$ is the gravitational constant and $\Sigma_{\rm gas}$ and $\Sigma_{\rm star}$ are the gaseous and stellar surface density, respectively (Gunn & Gott 1972). This is particularly true in dwarf systems, where the gravitational potential well is shallower than in massive galaxies. Ram pressure stripping easily removes both the atomic (e.g. Cayatte et al. 1990; Solanes et al. 2001; Gavazzi et al. 2005, 2013a) and in the strongest cases the molecular gas phases (Fumagalli et al. 2009; Boselli et al. 2014b), as well as the associated dust (Cortese et al. 2010, 2012a,b). Models and observations consistently indicate that ram pressure stripping is efficient even outside the virial radius of the cluster (Roediger & Hensler 2005; Roediger & Bruggen 2006, 2007; Boselli & Gavazzi 2006; Tonnesen & Bryan 2009; Cen et al. 2014). The higher dispersion in the velocity distribution of star-forming systems with respect to that of massive ellipticals definitely shows the presence of gas rich systems infalling in the Virgo cluster (Boselli et al. 2008a, 2014a). The stripping process is quite rapid since it is able to remove most of the gas content on time scales of $\sim$1.5 Gyr (Roediger & Bruggen 2007; Tonnesen & Bryan 2009). This time scale even reduces to 100–200 Myr in dwarf systems because of their weak restoring forces (Boselli et al. 2008a). Some gas retention, with the associated dust (de Looze et al. 2010, 2013), might be present in the core of the stripped galaxies where the gravitational potential well is at its maximum. This gas might feed star formation up to more recent epochs, and thus be at the origin of the blue centers observed in the most massive dwarf ellipticals (${\sim} 10^{9.5 - 10} M_\odot$) (Lisker et al. 2006b; Boselli et al. 2008a; Michielsen et al. 2008; Paudel et al. 2011). Indeed, gas removal is an outside-in process able to perturb only the gaseous component not affecting the stellar component, if not indirectly via the star-formation process (Boselli et al. 2006). The stripped galaxies often show in their morphology several remnants of their past spiral origin, such as grand design spiral arms, disks, and bars (Jerjen et al. 2000; Geha et al. 2003; Lisker et al 2006a,b; Lisker & Fuchs 2009). They also conserve their angular momentum, showing rotation curves similar to those observed in spiral galaxies of similar luminosity (Toloba et al. 2009, 2011, 2014b). Because of the lack of gas, the star formation stops, making galaxies redder and redder (Boselli et al. 2008a, 2014a; Cortese & Hughes 2009; Hughes & Cortese 2009; Gavazzi et al. 2013a). When plotted in a color magnitude diagram, they leave the blue sequence, cross the green valley and become red, quiescent systems. The time scale for this transformation is only slightly longer than the time scale for the gas stripping (Boselli et al. 2008a, 2014a). A rapid quenching of the star-formation activity is also indicated by the presence of several dwarf ellipticals such as VCC 1499 (see Sect. \[s1\]) in a post-starburst phase.[^19] The structural properties of the newly formed dwarf elliptical galaxies, as well as those of their relative globular cluster content, are fully consistent with those of their star forming progeny (Boselli et al. 2008a,b; see, however, Sanchez-Janssen & Aguerri 2012). There is strong and consistent evidence that the kinematical, structural, and spectrophotometric properties of dwarf ellipticals within Virgo tightly depend on their position within the cluster. Indeed, the most roundish (Lisker et al. 2009), pressure-supported systems (Toloba et al. 2009, 2011) dominated by old stellar populations in their center (Michielsen et al. 2008; Paudel et al. 2010a, 2011) are located close to the cluster core, while the disky, rotationally supported systems, dominated by younger stellar populations in their center, are predominantly situated in the outskirts of the Virgo cluster. The statistical properties of the Virgo cluster galaxies are also consistent with this picture. The shape of the optical and UV luminosity functions, and in particular their faint end slope, are very similar to those observed in the field once low surface brightness galaxies are considered (Rines & Geller 2008; Boselli et al. 2011; Ferrarese et al., in prep.). There is just an inversion of the relative contribution of star forming and quiescent galaxies, the former dominating in low-density regions, the latter typical of rich environments, as indeed expected in such a scenario. We recall, however, that both the GUViCS UV (Boselli et al. 2011) and the NGVS optical (Ferrarese et al., in prep.) luminosity functions have been determined for the central regions of the cluster and thus do not sample any possible radial variation (steepening of the faint end in the cluster periphery) already observed in other nearby clusters (Popesso et al. 2006; Barkhouse et al. 2007, 2009). The comparison of models with observations strongly favors a soft interaction of galaxies with the hot intergalactic medium (ram pressure) rather than more violent phenomena such as tidal interactions or galaxy harassment for several reasons. The most important one is that gravitational interactions remove a significant fraction of the stellar component, producing lower luminosity objects with systematically truncated stellar disks (Mastropietro et al. 2005; Aguerri & Gonzalez-Garcia 2009). The interaction would also significantly increase stochastic motions and reduce ordered motions, thus finally decreasing $v/\sigma$ (Mastropietro et al. 2005; Aguerri & Gonzalez-Garcia 2009). The observed structural properties of dwarf galaxies are more consistent with those of star-forming systems stripped of their gas by ram pressure stripping rather than those of harassed galaxies (Boselli et al. 2008b). Furthermore, efficient gravitational perturbations are quite rare given the high-velocity dispersion of the cluster. We thus expect that harassment requires long time scales for galaxy transformation, longer than the observed evolution with redshift of the faint end of the CMR (see Sect. \[EVOLUZIONEz\]). We can also expect that strong gravitational interactions, through tidal disruption and tidal galaxy formation, should significantly modify the shape of the luminosity function (Popesso et al. 2006; Barkhouse et al. 2007; de Filippis et al. 2011). As previously mentioned, the observations do not show any systematic difference in the luminosity function of Virgo and the field. At the same time, the detailed observations of the few cluster galaxies with clear signs of an undergoing perturbation do not show the formation of intermediate mass objects, but rather the formation in the stripped material of very small, compact blobs of stellar mass ${\sim} 10^3{-}10^6\,\hbox{M}_{\odot}$ (Yoshida et al. 2008; Fumagalli et al. 2011; Arrigoni Battaia et al. 2012; Yagi et al. 2013). This phenomenon of galaxy disruption/formation does not seem to be very frequent and is thus quite unlikely that it is able to modify the shape of the luminosity function in the observed stellar mass range. Furthermore, the output of this process are not low surface brightness, extended systems as those dominating the faint end of the luminosity function and of the CMR, but rather compact sources much more similar to UCD galaxies (yet of lower mass; Arrigoni Battaia et al. 2012). All the evidence described above, however, does not necessarily rule out the hypothesis that gravitational perturbations might play an important role on those galaxies that entered the cluster long time ago or have been even formed within it. On long time scales, galaxy harassment can heat the perturbed systems, producing roundish, pressure-supported objects such as those observed in the core of the cluster by Lisker et al. (2007, 2009). These galaxies, that have on average a velocity distribution similar to the Gaussian distribution drawn by the massive virialized ellipticals, are indeed characterized by older stellar populations (Michielsen et al. 2008; Paudel et al. 2010a, 2011) and lower $v/\sigma$ (Toloba et al. 2009, 2011; Boselli et al. 2014a) than their disky dominated counterparts at the periphery of the cluster. Furthermore, the beautiful systematic work of Lisker, Janz and collaborators, have undoubtedly shown that dwarf elliptical galaxies are not an homogeneous class of objects, but rather present different structures remnants of different possible origins (Lisker 2009; Janz et al. 2012, 2014). We can also add that chemo-spectrophotometric models of galaxy evolution expressly conceived to take into account the perturbations induced by the cluster environment ruled out the hypothesis that starvation, or strangulation, is at the origin of the observed properties of dwarf (and giant) gas-poor galaxies in clusters (Boselli et al. 2006, 2008a). We recall, however, that these models simulate starvation by stopping the infall of pristine gas, as already noted in Sect. \[MODELspsSPECTRO\]. In the original definition of Larson et al. (1980) starvation is a more aggressive process, where the interaction of galaxies with the hot and dense intracluster medium removes the hot gaseous halos of galaxies, a process that is much more similar to the ram pressure stripping considered here. A still open question is the range of stellar mass where this transformation process is efficient. Given the tight relation between stellar mass and restoring forces on galaxy disks, we expect the gas stripping process to be less efficient in massive objects (Boselli et al. 2008a). Here an important fraction of the atomic and molecular gas can still be retained on the inner disk, feeding star formation. Total gas stripping might occur only after several crossings of the cluster (Boselli et al. 2014b). It is thus possible in this scenario that the most massive ETGs inhabiting the cluster have been formed through major merging events, as indeed suggested by kinematical arguments (Cappellari et al. 2011a,b; Boselli et al. 2014a). It would be interesting to extend the studies of the main scaling relations and see whether the possible presence of strong discontinuities, as claimed by Kormendy et al. (2009) and Kormendy & Bender (2012), or continuity (Graham & Guzman 2003; Gavazzi et al. 2005; Ferrarese et al. 2006; Misgeld et al. 2008, 2009; Misgeld & Hilker 20011; Smith Castelli et al. 2013) can shade light on this specific point. Within this evolutionary picture we see only one apparent inconsistency. The cosmological simulations by Weinmann et al. (2011) and Lisker et al. (2013) especially tuned to reproduce the properties of the Virgo cluster seem to indicate that dwarf elliptical galaxies in local clusters such as Virgo might have been processed early and continuously in groups and in the cluster halo and are thus not LTGs recently transformed in quiescent systems. There are, however, a few indications that these semi-analytic models fails to reproduce several statistical properties of local clusters. Among these, the most evident is that they overestimate the color and the number of red objects, suggesting that in semi-analytic models the effects of the environment are still not optimized (Weinmann et al. 2011). Other clusters -------------- Are these results consistent with what observed in other nearby clusters? The multifrequency observations of Coma, A1367, and the Shapley supercluster confirm this scenario. All observations indicate that the fraction of gas-rich star-forming galaxies continuously increases with clustercentric distance up to a few virial radii (Rines et al. 2005; Gavazzi et al. 2013b), and that, on average, gas-poor late-type systems typical of high-density environments populate the green valley in between the blue cloud and the red sequence. The observations also indicate that dwarf elliptical galaxies in the infalling regions at the periphery of these rich clusters have, on average, younger stellar populations than those located close to the cluster core (Smith et al. 2006, 2008, 2009, 2012a), often with characteristic spectra indicating a recent abrupt truncation of their star-formation activity (Poggianti et al. 2004; Gavazzi et al. 2010). There is also evidence that some of these low-mass quiescent systems have spiral arms (Graham et al. 2003) and residual star formation in their center (Haines et al. 2008). The physical properties of star-forming and quiescent dwarf galaxies in other nearby clusters are thus similar to those observed in Virgo. As for Virgo, we expect that ram pressure stripping is the most probable process responsible for the observed trends. In rich clusters such as Coma the density of the intracluster medium is, on average, a factor of ten higher than in less relaxed clusters such as Virgo. The velocity dispersion of galaxies also increases with the mass of the cluster; it is thus natural that the efficiency of ram pressure stripping, which varies as $\rho_{\rm ICM} V^2$, is higher in these environments than in Virgo. On the contrary, the increase of the velocity dispersion of the cluster makes gravitational perturbations less efficient just because the time during which two galaxies can interact becomes shorter (Boselli & Gavazzi 2006). There are also a few spectacular observations that confirm this result. Besides the aforementioned FGC1287 in the outskirts of A1367 (Scott et al. 2012), the detection of several gas-rich galaxies at $\sim$ 1 virial radius with extended H$\alpha$ tails in Coma and A1367 (Gavazzi et al. 2001; Yagi et al. 2007, 2010; Yoshida et al. 2008, 2012; Fossati et al. 2012) has indeed shown that gas stripping via ram pressure can also remove the ionized gas component. At the same time, the X-ray and CO observations of ESO 137-001 in A3627 have indicated that the hot and the molecular gas components can also be removed (Sun et al. 2007; Jachym et al. 2014). Ram pressure stripping seems thus more important than previously thought. This statement is consistent with the most recent hydrodynamical simulations indicating that the ram pressure stripping process is more efficient whenever a multiphase ISM medium (atomic plus molecular gas) is considered (Tonnesen & Bryan 2009). Other simulations also indicate that ram pressure stripping is efficient to remove gas in galaxies up to $\sim$3 virial radii. Altogether these results have proven that ram pressure stripping is still active well outside the virial radius, making it the most plausible process responsible for the radial variation of the gas stripping and following quenching of the star-formation activity observed in nearby clusters. Not all pieces of evidence, however, rule out gravitational interactions as a possible process able to modify the evolution of cluster galaxies. In a recent work, for instance, Poggianti et al. (2013) discovered a population of compact objects analogue to those found at high redshift, three times more frequent in clusters than in the field, accounting for $\sim$12% of the massive ($M_{\rm star} > 10^{10}\,\hbox{M}_{\odot}$) galaxy population in nearby clusters. These objects, mainly lenticulars or ellipticals, have been probably shaped by gravitational interactions able to truncate stellar disks. There are also some inconsistencies with the observation of the optical luminosity function of nearby clusters. If the results of Popesso et al. (2005) and Cortese et al. (2008b), indicating that the cluster luminosity function is significantly steeper than in the field, (with a faint end slope in the fitted Schechter function of ${-}2.1 \leq \alpha \leq {-}1.6$) are confirmed, other gravitational processes such as harassment and tidal disruption should be invoked. We recall, however, that these results should still be confirmed observationally, in particular because they might suffer from the quite uncertain background subtraction technique (Rines & Geller 2008). In any case, it is also conceivable that, given the very different nature of Virgo and Coma or other massive clusters, the former being spiral rich and still under formation, the latter quite relaxed and spiral poor, the relative weight of the different physical process acting on galaxies might significantly change. If observations and simulations mainly suggest that the dynamical interactions between galaxies and the hot intracluster medium are probably modulating the evolution of galaxies in the present epoch, this might not have been the case in the past, when pre-processing was probably more important (e.g. Dressler 2004; Boselli & Gavazzi 2006). Gravitational perturbations in infalling groups are rare in the present epoch: the blue infalling group in A1367 is indeed the only known case in the local universe (Sakai et al. 2002; Gavazzi et al. 2003b; Cortese et al. 2006b). Furthermore, the luminosity function, in particular in the optical bands, gives a view of the cumulative evolution of galaxies and thus might be not representative of a recent evolution. Lower density environments -------------------------- It is now interesting to determine the range of galaxy density within which this evolutionary process holds. The works of Haines et al. (2008), Gavazzi et al. (2010, 2013b), and Rasmussen et al. (2012a) are crucial for this purpose. In particular, Gavazzi et al. (2010) have shown (see Fig. \[Gavazzispsdensity\]) that while the bright end of the red sequence is already fully defined in all kinds of environments, the faint end is present only for densities $\delta_{1,1{,}000} >4,$ where $\delta_{1,1{,}000}$ is the 3D density contrast defined as $${\delta_{1,1{,}000} = \frac{\rho -\langle \rho\rangle }{{\langle \rho\rangle }}}$$ $\rho$ is the local number density and $\langle \rho\rangle = 0.05$ gal ${\rm h}^{-1}\,\hbox{Mpc}^{-3}$ is the mean number density measured in the Coma/A1367 supercluster region.[^20] This threshold in local number density of $\delta_{1,1{,}000} > 4$ roughly corresponds to the density observed in groups with more than 20 objects and velocity dispersion of ${\sim }200\,\hbox{km s}^{-1}$. The lack of red sequence faint quiescent galaxies in the field has been later confirmed on strong statistical basis by Wilman et al. (2010) and Geha et al. 2012. Kilborn et al. (2009), Fabello et al. (2012), Gavazzi et al. (2013b), and Catinella et al. (2013) have shown that LTGs in medium-density environments such as groups also suffer from gas deficiency. The lack of gas quenches the activity of star formation, making, on average, redder galaxies (Gavazzi et al. 2013b). Wilman et al. (2010), using a large sample of SDSS galaxies in different density environments, have shown that galaxies become red only once they have been accreted into halos of a certain mass. Rasmussen et al. (2012a), on the other hand, have clearly shown that the quenching of the star-formation activity is stronger in galaxies with stellar mass $M_{\rm star} < 10^{9.2}\,\hbox{M}_{\odot}$. ROSAT and Chandra observations have shown that about half of the optically selected nearby groups are characterized by a diffuse X-ray emission, similar to clusters of galaxies (Mulchaey & Zabuldoff 1988; Mulchaey 2000; Mulchaey et al. 2003; Osmond & Ponman 2004; Rasmussen et al. 2008; Sun et al. 2009). The X-ray emission, which is typical of those groups dominated by an ETG, generally extends out to less than half of the virial radius. They are characterized by gas densities of $n_0 \sim 10^{-2}{-}10^{-3}\,\hbox{cm}^{-3}$ and velocity dispersions of $\sigma \sim 150{-}400\,\hbox{km\,s}^{-1}$ (Rasmussen et al. 2008). It is thus conceivable that ram pressure stripping is also acting on galaxies in these medium density systems (Kantharia et al. 2005; Verdes-Montenegro et al. 2007; Sengupta et al. 2007; McConnachie et al. 2007; Jeltema et al. 2008). Its efficiency, however, significantly drops since both the mean density of the intergalactic medium and the velocity dispersion are significantly smaller than in cluster galaxies (Kawata & Mulchaey 2008). In dwarf systems, however, the ram pressure stripping process can still be active given their shallow gravitational potential well, as suggested by the results of Rasmussen et al. (2012a). Indeed, through a statistical study of the spectroscopic properties of a large sample of nearby, ETGs, Thomas et al. (2010) have shown that the impact of the environment on the star-formation history of galaxies increases with decreasing galaxy mass. At the same time, in groups the role of gravitational perturbations can be more important than in rich systems just because the duration of the interactions is longer (Kilborn et al. 2009). The hydrodynamical simulations of Bekki & Couch (2011), indeed, have shown that repeated slow encounters within groups are able to transform star forming systems into lenticular galaxies. It is thus possible that the same process modulates the evolution of lower mass systems. On the other hand, the analysis of a large sample of galaxies in groups using SDSS data done by van den van den Bosch et al. (2008) seems to indicate starvation or strangulation as the most probable process quenching the activity of star formation of late-type systems and thus forming the red sequence. This last conclusion, however, is under debate since it is based on the assumption that harassment and ram pressure stripping are efficient only in massive systems, an assumption not fully supported by the results presented in this review. We recall, however, that the simulations of Kawata & Mulchaey (2008) indicate that the total hot halo gas of galaxies of intermediate mass can be removed on time scales of $\sim$1 Gyr. The lack of new gas feeding the star-formation process quenches the activity of these objects. We also remind that other processes have been proposed in the literature to remove the gas and quench the activity of star formation of dwarf systems in small groups of galaxies similar to the Local group. Among these, we can mention tidal stirring (Mayer et al. 2001a,b), or the combination of tidal interactions and ram pressure stripping exerted on dwarf satellites during their crossing of the halo of massive galaxies (Mayer et al. 2006). Regardless of the very nature of the gas stripping process, we expect that the lack of gas feeding star formation quenches the activity and makes galaxies in groups redder, thus populating the faint end of the red sequence. Dependence on redshift {#EVOLUZIONEz} ---------------------- The identification of the process at the origin of the faint end of the red sequence can be further constrained by comparing the time scale for quenching the activity of star formation, transforming blue galaxies into red systems, and the infall rate in nearby clusters, with the evolution of the red sequence as a function of redshift. This exercise has been done for the first time by Boselli et al. (2008a) and later revisited by Gavazzi et al. (2013a,b). Boselli et al. (2008a) used their own chemo-spectrophotometric models of galaxy evolution to calibrate different indicators necessary to quantify the lookback time since the beginning of the interaction. In this way they identified four different indices: the H$\alpha$ emission line equivalent width and the HI-deficiency parameter are sensitive to short time scales ($\sim$200 Myr), while the equivalent width of the H$\beta$ absorption line and the FUV-H colour index are sensitive to significantly longer lookback times ($\sim$1 Gyr). By determining the fraction of dwarf elliptical galaxies still undergoing the transformation process as indicated by these four indices in the whole dE Virgo population, Boselli et al. (2008a) deduced that the infall rate of dwarf galaxies in Virgo is of the order of 300 objects $\hbox{Gyr}^{{-}1}$. Following similar arguments, Gavazzi et al. (2013a) determined infall rates of $\sim$300 to 400 $\hbox{Gyr}^{-1}$ galaxies in Virgo and of 100 galaxies $\hbox{Gyr}^{-1}$ of mass $M_{\rm star} > 10^9\,\hbox{M}_{\odot}$ in Coma. These are important rates considering that the total number of dwarf Virgo members in the same luminosity range is $\sim$650 and implies that the faint end of the red sequence has been formed in the last $\sim$2 Gyr (corresponding to $z = 0.16$ in a $H_0 = 70$kms$^{-1}\,\hbox{Mpc}^{-1},\Omega_m = 0.3$, and $\Omega_{\lambda} = 0.7$ cosmology) if we assume that this rate did not change significantly with time (Boselli et al. 2008a). ![The variation in the red sequence of the dwarf to giant number ratio as a function of the redshift, from Stott et al. (2007). Dwarf elliptical galaxies have been formed mainly at recent epochs. © AAS. Reproduced with permission[]{data-label="StottspsRDTG"}](fig20.pdf){width="80.00000%"} The most recent studies of the CMR and of the luminosity function in clusters at different redshifts can be used for a direct comparison with this result. Since the work of Kodama et al. (2004) and De Lucia et al. (2004) there is a growing evidence that the fraction of luminous-to-faint galaxies on the red sequence significantly decreased since $z = 0.8$ (Fig. \[StottspsRDTG\]; De Lucia et al. 2007, 2009; Stott et al. 2007, 2009; Gilbank & Balogh 2008; see however Andreon 2008). Stott et al. (2007) determined that the number of dwarf galaxies on the red sequence increased by a factor of 2.2 since $z = 0.5$ (4 Gyr), a number roughly consistent with that determined from the infall rate of galaxies in Virgo (Boselli et al. 2008a; Gavazzi et al. 2013a). Jaffe et al. (2011), by analyzing the shape of the CMR traced by galaxies selected according to morphological criteria, have shown that the CMR was already formed in clusters at redshift $0.4 < z < 0.8$. They noticed, however, the presence of several low luminosity objects with bluer colors than those of the mean CMR, indicating that these galaxies have reached the red sequence later in time than more massive galaxies. By comparing the fraction of red dwarf galaxies with clear signs of a post-starburst activity (‘$k+a$’) with the infalling rate, De Lucia et al. (2009) have shown that not all dwarf galaxies have moved from the blue cloud to the red sequence after having passed a post-starburst phase. They have thus concluded that the transformation process that gave birth to the faint end of the red sequence is not always rapid, but can mildly and continuously change the galaxies properties with time. Using a different set of data of high redshift galaxies, Bolzanella et al. (2010) suggested that the environmental mechanisms of galaxy transformation start to be effective only below $z = 1$. Furthermore, they indicated that the migration from the blue cloud to the red sequence occurs on a shorter timescale than the transformation from disk-like morphologies to ellipticals, consistently with that observed in the local universe (presence of spiral arms in Virgo and Coma dE, see Sect. \[STRUCTURAL\]) (Giodini et al. 2012; Mok et al. 2013; Vulcani et al. 2013). It has been shown that the fraction of galaxies in dense environments at $z= 0.7$ located in between the red and the blue sequence of the color–magnitude relation, thus of objects probably undergoing the transformation process, increases with decreasing luminosity (Cassata et al. 2007), as expected in the proposed scenario. There are also several indications that the faint end of the luminosity function of red cluster galaxies has been formed only after $z = 0.6$ (Harsono & de Propris 2007; Gilbank et al. 2008; Rudnick et al. 2009, 2012; Lemaux et al. 2012; see, however Crawford et al. 2009; de Propris et al. 2013). More locally, the work of Hansen et al. (2009) based on the photometric properties of galaxies in group and clusters at redshift $0.1 < z < 0.3$ has shown that, while the luminosity function of blue and red satellites is only weakly dependent on the cluster richness for masses above $3 \times 10^{13}\,\hbox{ M}_{\odot}$, the mix of faint red and blue galaxies changes dramatically. Overall, although controversial results have been reported, most of the observational evidence suggests that the faint end of the red sequence has been formed only at relatively recent epochs. It might be more challenging to see whether the physical process at the origin of this transformation in the past was the same as the one identified in the local universe (ram pressure stripping). Simulations indicate that the present day rich clusters have been formed by the aggregation of smaller structures and that $\sim$40% of the galaxies in local clusters have been accreted through groups (Gnedin 2003; McGee et al. 2009; De Lucia et al. 2012). It is thus conceivable that galaxies have been processed during their membership to these systems before entering in the cluster (pre-processing, Dressler 2004; Vijayaraghavan & Ricker 2013; Dressler et al. 2013). There is also some evidence that clusters were denser at high $z$ than in the local universe (Poggianti et al. 2010). It is thus possible that gravitational interactions played a more important role in shaping dwarf galaxy evolution in the past than in the local universe. The short time scale for the assembly of the faint end of the red sequence ($\sim$4 Gyr), however, favors rapid processes such as ram pressure stripping rather than harassment, whose time scale is longer given that multiple encounters are required. The interesting works of Poggianti and collaborators also favor this scenario. By studying the spectral properties of cluster galaxies at redshift $0.4< z < 0.8$, Poggianti et al. (2009) have shown that the fraction of post-starburst galaxies increases with the velocity dispersion of the cluster, suggesting that the process at the origin of their transformation is related to the intracluster medium. Consistently with this scenario, Dressler and collaborators concluded that massive ellipticals have been formed by early major merging events within the groups accreting the cluster, while lenticulars more recently by less violent processes. This view is also consistent with the most recent multifrequency observations of the Virgo cluster (Boselli et al. 2014a). ![High contrast representation of the $\hbox{H}\alpha$ NET frame obtained with the Suprime-Cam at the Subaru telescope (4 hours exposure) zoomed on a 10x10 $\rm arcmin^2$ region containing galaxies 97073 and 97079 (compare with Figure \[f1\]), showing tails of ionized gas exceeding 100 kpc in length[]{data-label="subaru"}](fig21.pdf){width="100.00000%"} Concluding remark {#s6} ================= In spring 2014 we took a $1\,{\rm deg}^2$ deep (4 h exposure) $\hbox{H}\alpha$ image of A1367 using Suprime-Cam at the Subaru telescope, in collaboration with Michintoshi Yosida and Masafumi Yagi. Moreover, we obtained one field at the NW periphery of the Coma cluster which, added to the two fields previously obtained by Yagi et al. (2010) with the same instrument, brings to 1.5 sq. deg. the area surveyed in this cluster at similar depth. Figure \[subaru\] speaks for itself. It contains a detail of the A1367 field, zoomed on two of the galaxies highlighted in Fig. \[f1\]. The gray scale represents the intensity of the $\hbox{H}\alpha+\hbox{N[II]}$ lines after a preliminary subtraction of the stellar continuum. The figure shows ionized gas (no stars) trailing behind the two low-mass galaxies (${\textit{M}_{\rm star}} \ 10^{9.3, 9.5} \ \hbox{M}_{\odot}$) with approximately 75 (97073) and 100 kpc (97079) projected length. It cannot be excluded that the two galaxies suffered from a close encounter at the location where the two tails seem to cross each other. Nevertheless they dramatically witness gas loss due to ram-pressure. In total we covered 36 LTGs in A1367 and 28 in the Coma cluster. The preliminary result of the survey is that at least 24 (perhaps 27) of them, i.e. 40%, present a cometary $\hbox{H}\alpha$ tail. In other words, roughly one out of two of the star forming galaxies in the two clusters shows signs of an ongoing ram pressure interaction. This confirms that ram pressure is a quick phenomenon, that replenishment of fresh gas-rich galaxies is currently taking place in the clusters belonging to the Great Wall and that environmental transformations are indeed ubiquitous. If this rate is representative of other clusters, we predict that future $\hbox{H}\alpha$ surveys of similar depths (few $10^{-18} \,{\rm erg\,cm}^{-2}\,\hbox{s}^{-1}\,\hbox{arcsec}^{-2}$) of the Virgo cluster would lead to the detection of several hundred LTGs showing cometary structures, especially in dwarf systems. The recent availability of $\hbox{H}\alpha$ filters matching the field of view of MegaCam at the CFHT will make such survey feasible in the near future. Another major breakthrough for unveiling at the same time the kinematics and the chemistry of these systems will certainly be provided by the wide-field integral-field units that are becoming available at 10 m class telescopes (e.g. MUSE at VLT: Fumagalli et al. 2014 and KCWI at Keck) and by ALMA for disclosing the study of the gas in the molecular phase. We wish to thank Massimo Dotti, Matteo Fossati, Michele Fumagalli, and Elisa Toloba for their comments on the manuscript and Yannick Roehlly for his help in the preparation of the illustrations. G.G wishes to thank Michitoshi Yoshida and Masafumi Yagi for their permission to use the Halpha map in Fig.\[subaru\] prior to publication. The authors would like to thank L. Cortese, K. Rines, R. Smith, T. Lisker, E. Toloba, S. Tonnesen, and J. Stott for allowing reproducing their published figures. During the writing of this review we made extensive use of the GOLDMine database (Gavazzi et al. 2003a) and of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, and the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, The University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington. 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al (2000) AJ 120:1579 Yoshida M, Ohyama Y, Iye M et al (2004) AJ 127:90 Yoshida M, Yagi M, Komiyama Y et al (2008) ApJ 688:918 Yoshida M, Yagi M, Komiyama Y et al (2012) ApJ 749:43 Zandivarez A, Mart[í]{}nez HJ (2011) MNRAS 415:2553 Zandivarez A, Mart[í]{}nez HJ, Merch[á]{}n ME (2006) ApJ 650:137 Zhang B, Sun M, Ji L et al (2013) ApJ 777:122 [^1]: Most large-scale surveys are currently available for the northern hemisphere. However, several large survey of the southern sky are under way, e.g. the ESO/VST and the DES (Dark Energy Survey) at NOAO. [^2]: The absence of a significant environmental effect in Hogg et al. (2004) is due to a sensitivity bias: their analysis includes galaxies brighter than $M_i={-}20$, while significant environmental issues affect galaxies fainter than $M_i={-}19$. [^3]: http://egg.astro.cornell.edu/index.php/. [^4]: http://galex.lam.fr/guvics/ [^5]: https://www.astrosci.ca/NGVS/The\_Next\_Generation\_Virgo\_Cluster\_Survey/Home.html [^6]: http://wiki.arcetri.astro.it/bin/view/HeViCS/WebHome [^7]: http://astronomy.swin.edu.au/coma/index-mm.htm [^8]: http://www.h-atlas.org/ [^9]: http://smakced.net/ [^10]: http://www-astro.physics.ox.ac.uk/atlas3d/ [^11]: http://califa.caha.es/ [^12]: https://www.astrosci.ca/users/VCSFCS/Home.html [^13]: http://hedam.lam.fr/HRS/ [^14]: The HI-deficiency parameter is defined as the difference, on logarithmic scale, between the expected and the observed HI gas mass of each single galaxy. The expected atomic gas mass is the mean HI mass of a galaxy of a given optical size and morphological type determined in a complete sample of isolated galaxies taken as reference (Haynes & Giovanelli 1984). [^15]: Early claims of enhanced radio-continuum emission from LTGs in the A1367 and Coma clusters (Gavazzi & Jaffe 1985, 1986) are insufficient to infer a significant star-formation enhancement in clusters. It would not be surprising, however, if on a short time scale this would take place from the shocked material during the early phases of a ram pressure event. [^16]: Examples of ETGs with nuclear or circumnuclear $\hbox{H}\alpha$ emission in the Virgo cluster are given in Boselli et al. (2008a): VCC 450, 597, 710, 1175, 1617, 1855, or in Toloba et al. (2014b): VCC 170, 781, 1304, 1684. [^17]: There is a clear inconsistency in the definition of ram pressure and starvation in cosmological simulations and semi-analytic models with respect to the observation and the simulations of single galaxies in local clusters. Cosmologists indistinctly define ram pressure, starvation or strangulation the removal of the hot gaseous halo of satellite galaxies entering the extended gas halo surrounding groups and clusters. In the study of nearby objects, ram pressure stripping is generally referred to the stripping of the cold gas component exerted by the hot cluster intergalactic medium on galaxies moving at high velocity. In the same studies, starvation or strangulation refers to the gas consumption via star formation of galaxies once the infall of pristine cold gas is stopped. [^18]: We stress that this definition of starvation differs from the one originally proposed by Larson et al. (1980), where the galaxy quenches its activity of star formation once the gas of its halo, generally feeding the disk in unperturbed systems, is removed during the interaction with the hostile environment. In the Boselli et al. models, starvation is a passive process where star formation decreases after gas consumption because of the lack of the infall of pristine gas from the surrounding environment. [^19]: The definition of post-starburst does not necessarily imply there was a particularly acute starburst phase, but that a normal star-formation phase was abruptly interrupted. [^20]: The local number density $\rho$ around each galaxy is measured within a cylinder of radius 1 $h^{-1}$ Mpc and half-length $1{,}000\,\hbox{km\,s}^{-1}$.
--- abstract: \| Background: The semi-magic Sn ($Z = 50$) isotopes have been subject to many nuclear-structure studies. Signatures of shape coexistence have been observed and attributed to two-proton-two-hole (2p-2h) excitations across the $Z = 50$ shell closure. In addition, many low-lying nuclear-structure features have been observed which have effectively constrained theoretical models in the past. One example are so-called quadrupole-octupole coupled states (QOC) caused by the coupling of the collective quadrupole and octupole phonons. Purpose: Proton-scattering experiments followed by the coincident spectroscopy of $\gamma$ rays have been performed at the Institute for Nuclear Physics of the University of Cologne to excite low-spin states in $^{112}$Sn and $^{114}$Sn, to determine their lifetimes and extract reduced transitions strengths $B(\Pi L)$. Methods: The combined spectroscopy setup SONIC@HORUS has been used to detect the scattered protons and the emitted $\gamma$ rays of excited states in coincidence. The novel $(p,p'\gamma)$ DSA coincidence technique was employed to measure sub-ps nuclear level lifetimes. Results: 74 level lifetimes $\tau$ of states with $J = 0 - 6$ were determined. In addition, branching ratios were deduced which allowed the investigation of the intruder configuration in both nuclei. Here, $sd$ IBM-2 mixing calculations were added which support the coexistence of the two configurations. Furthermore, members of the expected QOC quintuplet are proposed in $^{114}$Sn for the first time. The $1^-$ candidate in $^{114}$Sn fits perfectly into the systematics observed for the other stable Sn isotopes. Conclusions: The $E2$ transition strengths observed for the low-spin members of the so-called intruder band support the existence of shape coexistence in $^{112,114}$Sn. The collectivity in this configuration is comparable to the one observed in the Pd nuclei, i.e. the 0p-4h nuclei. Strong mixing between the $0^+$ states of the normal and intruder configuration might be observed in $^{114}$Sn. The general existence of QOC states in $^{112,114}$Sn is supported by the observation of QOC candidates with $J \neq 1$. author: - 'M. Spieker' - 'P. Petkov' - 'E. Litvinova' - 'C. M[ü]{}ller-Gatermann' - 'S.G. Pickstone' - 'S. Prill' - 'P. Scholz' - 'A. Zilges' bibliography: - 'Sn\_prc.bib' title: 'Shape coexisistence and collective low-spin states in $^{112,114}$Sn studied with the $(p,p''\gamma)$ DSA coincidence technique' --- Introduction ============ The low-energy and low-spin level scheme of the semi-magic stable Sn isotopes has been considered as a “textbook” example of the seniority scheme, see, [e.g.]{}, Ref.[@Tal93a]. However, shell-model calculcations with a finite-range force pointed out that at least configurations with two broken pairs, i.e. seniority $\nu \leq 4$ are needed to fully account for the low-energy levels[@Bon85a]. To describe the excitation energy of the $3^-_1$ state, one particle - one hole neutron configurations had to be included which originated from excitations across the $^{100}$Sn inert core. In addition to these structures, low-energy two proton - two hole intruder states are observed in the Sn isotopes, see, [e.g.]{}, the review articles[@Wood92a; @Heyd11a] and references therein. These positive-parity states will likely mix with states of the “normal” configuration. It is, thus, not trivial whether rather pure collective quadrupole states of two- and three-phonon nature will be observed in the Sn isotopes. Experimental studies in $^{124}$Sn have identified candidates for members of the three-phonon multiplet[@Band05a]. In fact, identifying such structures in the semi-magic Sn nuclei has been named as an important step to answer the question as to whether pure vibrational modes can be observed in the $Z = 50$ region[@Garr10a].\ For years the Cd isotopes had been considered as prime examples exhibiting the vibrational character put forward by Bohr and Mottelson[@Bohr75a]. But this simple picture has been questioned[@Garr10a] also due to the existence of shape coexistence at low energies in the Cd isotopes, see, [e.g.]{}, the recent review article[@Garr16a] and references therein. In addition, a quintuplet of negative-parity states is expected due to the coupling of the collective quadrupole and octupole phonon in vibrational nuclei. Its study might further help to understand the concept of vibrational excitations in nuclei since Pauli blocking is expected to be smaller and since these states will not mix with positive-parity intruder states. The $1^-$ member of this multiplet has been studied systematically in $^{112,116-124}$Sn[@Bryss99a; @Pys06a].\ Many lifetimes in $^{112}$Sn are known from an $(n,n'\gamma)$ experiment performed at the University of Kentucky (USA)[@Kum05a; @Orce07a]. These inelastic neutron-scattering (INS) experiments use the $^{3}$H$(p,n){}^{3}$He reaction to generate neutrons of different energies by tuning the incident proton energy. To minimize feeding effects, several neutron energies are usually used to extract lifetimes with the INS Doppler-shift attenuation (DSA) technique. The scattered neutrons are, however, not detected in coincidence with the $\gamma$ rays. For more details on this method, see, [e.g.]{}, Refs.[@Pet13a; @Kum05a] and references therein. In contrast, the new $(p,p'\gamma)$ DSA coincidence technique with the SONIC@HORUS setup[@Pick17a] at the University of Cologne (Germany) detects the scattered protons in coincidence with the $\gamma$-rays emitted from the excited state to determine nuclear level lifetimes without feeding contributions[@Henn15a]. A further advantage of this method is that much less target material is needed compared to the $(n,n'\gamma)$ experiments. Thus, also less abundant isotopes such as $^{114}$Sn can be studied with this method.\ It is the purpose of this work to report on the two $(p,p'\gamma)$ experiments we performed to study excited states in $^{112,114}$Sn up to an excitation energy of 4MeV and determine the level lifetimes. experimental details ==================== The $^{112,114}$Sn$(p,p'\gamma)$ experiments were performed at the 10MV FN-Tandem facility of the University of Cologne where the protons were accelerated to an energy of $E_p = 8$MeV. The combined spectroscopy setup SONIC@HORUS used for the $p \gamma$-coincidence experiments consists of passivated implanted planar silicon (PIPS) detectors and up to 14 high-purity germanium (HPGe) detectors. Six of these can be equipped with BGO shields for active Compton suppression[@Pick17a]. The current on target accounted to about 5nA with a master-trigger rate of up to 25kHz, which corresponded to operating the silicon and germanium detectors at maximum count rates of about 11kHz and 15kHz, respectively. $p \gamma$-coincidence data for excited states up to about 4MeV were acquired by using XIA’s DGF-4C Rev.F modules[@Hubb99a; @Skul00a; @Warb06a]. A level-2 global first level trigger (GFLT) was set externally to record twofold and multifold coincidences as listmode data[@Elv11a; @Henn14a]. The SONIC chamber, i.e. SilicON Identification Chamber, which was used in these experiments was the second SONIC version housing seven silicon detectors in total[@Pick17a]. Four of these detectors are placed in tubes at angles of $\theta = \SI{122}{\degree},~ \phi = \SI{45}{\degree}, \SI{135}{\degree}, \SI{225}{\degree}, \SI{315}{\degree}$, while another three silicon detectors can be fixed to the chamber by using magnets at $\theta = \SI{114}{\degree},~ \phi = \SI{0}{\degree}, \SI{90}{\degree}$ and $\SI{180}{\degree}$, respectively.\ A precise energy calibration of the HPGe detectors is crucial for any DSA lifetime measurement. For the $(p,p'\gamma)$ DSA coincidence technique a $^{56}$Co calibration source is mounted in SONIC throughout the experiment to guarantee for this precise calibration. The energy calibration of the HPGe detectors is precise to a level of at least 0.2keV and, thus, $\gamma$-energy centroid shifts which are well below 1keV can be recognized. The energy calibration of the silicon detectors is performed by identifying specific excited states of the target nucleus in the proton spectra. The assignments in the proton spectra are cross checked by setting a gate onto $\gamma$ transitions in the HPGe detectors. The absolute photopeak efficiency of the setup was determined using a $^{226}$Ra source of known activity and a $^{56}$Co source whose activity was used as a scaling factor to obtain agreement with the $^{226}$Ra data, see Fig.\[fig:efficiency\]. It is, thus, known up to 3.5MeV and no significant extrapolation was needed to study excited states in $^{112,114}$Sn up to an energy of about 4MeV. ![\[fig:efficiency\](color online) Absolute photopeak efficiencies for the $^{112,114}$Sn$(p,p'\gamma)$ experiments. The empirical Wiedenh[ö]{}ver function (red dashed line) has been used to fit the experimentally measured efficiencies (black circles)[@Wieden94a].](efficiency.eps){width="0.75\linewidth"} Lifetime determination ====================== $\gamma$-energy centroid shifts and the Doppler-shift attenuation factor $F \left( \tau \right)$ ------------------------------------------------------------------------------------------------ Correlated $p \gamma$-coincidence matrices can be generated from the $p \gamma$-coincidence listmode data. For the case of the DSAM technique, these coincidence matrices can be determined unambiguously since the kinematics of the $(p,p'\gamma)$ allow clear correlations, see, [e.g.]{}, Refs.[@Henn14a; @Henn14b; @Henn15a; @Henn15b]. In total, there are three of such kinematically correlated DSA groups for excited states up to 4MeV in $^{112,114}$Sn when using the second SONIC version. Each group typically contains eleven subgroups characterized by their common Doppler angle $\Theta$, respectively. If statistics are not sufficient less subgroups can be considered, i.e. more Doppler angles $\Theta$ can be grouped into one subgroup resulting in larger overall $\cos(\Theta)$ uncertainties.\ Excitation-energy gates select specific excitation regions and exclude feeding contributions. Figs.\[fig:Sn\_shifts\][(b)]{}-[(e)]{} present the observed energy-centroid shifts of the $1^-$ state at $E_x = 3433$keV in $^{112}$Sn and of the $3^-_1$ states in $^{112,114}$Sn, respectively. From the slope of the linear fit, the Doppler-shift attenuation factor $F(\tau)$ can be determined: $$\begin{aligned} E_{\gamma}(\Theta,t) = E_{\gamma}^0 \left( 1+ F(\tau) \frac{v_0}{c} \cos \Theta \right) \nonumber \label{eq:DSAM_03}\end{aligned}$$ If $\gamma$-decay branching is observed, the Doppler-shift attenuation factor can be determined from $\gamma$-decays to different final states. Figs.\[fig:Sn\_shifts\][(f)]{} and [(g)]{} present the case of a $2^+$ state at 3185keV in $^{114}$Sn. As can been seen from the figure, the two $\gamma$-decay branches yield consistent Doppler-shift attenuation factors $F(\tau)$ within their statistical uncertainties. ![\[fig:Sn\_shifts\](color online) Method to extract $\gamma$-energy centroid shifts. [(a)]{} Excitation-energy spectrum, which corresponds roughly to the energy loss of the incident protons. Several excited states in $^{112}$Sn are marked. Gates can be applied to select the excitation of specific states. [(b)]{} The $\gamma$-energy centroid shifts are observed in the excitation-energy gated $\gamma$ spectra, [e.g.]{}, the DSAM subgroups at $\Theta = \SI{35}{\degree}, \SI{90}{\degree},$ and $\SI{145}{\degree}$, [(c)]{} from which a linear dependence on $\cos(\Theta)$ is extracted. The example of the $J^{\pi} = 1^-$ state at $E_x = 3433.4(2)$keV in $^{112}$Sn is presented. [(d)]{}, [(e)]{} observed energy-centroid shifts of the $3^-_1$ states in $^{112,114}$Sn which are below 1keV. [(f)]{}, [(g)]{} $\gamma$-energy centroid shifts for $\gamma$-decays to different final states of the $2^+$ at 3185.5(2)keV in $^{114}$Sn. The Doppler-shift attenuation factors $F(\tau)$ and the lifetimes $\tau$ determined from a comparison to a Monte-Carlo simulation are also shown in panels [(c)]{} to [(g)]{}.](method_DSAM.eps){width="1\linewidth"} Simulation of the stopping process ---------------------------------- ![\[fig:ftau\]Determination of the lifetime $\tau_{3^-_1}$ from the $F(\tau)$ curve in $^{112}$Sn. The grey band corresponds to the statistical uncertainties of the experimentally determined $F_{\mathrm{exp}}(\tau)$ value. The final lifetime $\tau$ of 280(20) fs is obtained by comparison to the Monte-Carlo simulation. The solid black line corresponds to the calculation using the stopping-power parameters of Table\[tab:targets\], whereas the dashed lines correspond to the variation of these parameters to estimate the systematic uncertainties.](ftau_3-_112Sn.eps){width="1\linewidth"} Nuclear level lifetimes $\tau$ were determined from a comparison of the experimentally extracted $F(\tau)$ values with the predictions of a Monte-Carlo simulation of the stopping process, see Fig.\[fig:ftau\] and Ref.[@Henn15a]. The Monte-Carlo simulation considers electronic and nuclear stopping according to the formalism of Lindhard, Scharff, and Schiøtt (LSS)[@Lind63a; @Lind68a]. The computer code[@Pet15a], which has been used in this work, is a modified version of the DESASTOP program by Winter[@Wint83a; @Wint83b] where the universal scattering function described in Ref.[@Curr69a] has been implemented. It has been further modified to be able to deal with multi-layered target compositions, i.e. especially alloy layers of different composition and also allows the implementation of details of the experimental setup to obtain a more constrained simulation. Besides the stopping powers in the target and stopper material, the areal densities of the materials have to be known as precisely as possible. Otherwise, severe errors in the lifetime calculation might be introduced as shown in, [e.g.]{}, Refs.[@Pet13a].\ Unfortunately, Rutherford Backscattering Spectrometry (RBS) experiments with 2MeV $^{4}$He$^+$ ions, performed at the RUBION facility of the Ruhr-Universit[ä]{}t Bochum in Germany, revealed that a Sn-Au alloy had formed, i.e. the target and stopper material were not completely separated. However, by using the RBS simulation software SIMNRA[@SIMNRA1; @SIMNRA2] and by introducing a thickness distribution in the different alloy layers, a reasonable description of the experimental RBS spectra could be obtained, see Fig.\[fig:Sn\_rbs\]. Thus, the areal densities and the relative contributions of Sn and Au to the different layers could be extracted. [ccccccc]{} Layer & $a_{\mathrm{Sn}}$ & $b_{\mathrm{Au}}$ & density alloy & areal density & $f_e$ & $p$\ & & & \[g/cm$^3$\] & \[mg/cm$^2$\] & &\ \ 1 & 0.50 & 0.50 & 13.1 & 0.87 & 0.61 & 0.61\ 2 & 0.64 & 0.36 & 11.4 & 0.14 & 0.64 & 0.60\ \ \ 1 & 0.55 & 0.45 & 12.7 & 0.70 & 0.63 & 0.61\ 2 & 0.37 & 0.63 & 14.9 & 0.13 & 0.59 & 0.61\ 3 & 0.42 & 0.58 & 14.3 & 0.14 & 0.60 & 0.61\ \ The alloy introduces some additional considerations which have to be made. First of all, the stopping powers for tin recoils in the alloy results as the sum of the relative contributions to the respective layer $i$, i.e. $$\begin{aligned} \left(\frac{dE}{dx}\right)_{\mathrm{alloy},i} = \left[ a \cdot \left(\frac{dE}{dx}\right)_{\mathrm{Sn}} + b \cdot \left(\frac{dE}{dx}\right)_{\mathrm{Au}} \right]_{i}, \nonumber\end{aligned}$$ which are determined from the stopping-power tables of Northcliffe and Schilling[@North70a] and corrections to these due to electronic structures of the respective material[@Zieg74a]. Here, $a$ and $b$ are known from the RBS data simulation. Second, effective charges $Z$ and effective masses $A$ are introduced to transform $E$ and $x$ to the dimensionless variables of the LSS theory[@Curr69a] by also using $a$ and $b$. And third, one introduces compound densities, which are not necessarily homogeneous and are not compatible with a simple averaging. The stopping power results including the RBS analysis are shown in Table\[tab:targets\]. Furthermore, the $^{112}$Sn and $^{114}$Sn targets were only enriched to 85.5$\%$ and 66.5$\%$, respectively. This had to be taken into account in the Monte-Carlo simulation since the density profile is affected. Due to these complications, a thorough check of the input parameters for the Monte-Carlo simulations by means of known lifetimes was necessary. Results will be presented in Sec.\[sec:tau\_comp\]. ![\[fig:Sn\_rbs\](color online) RBS spectra for the $^{112,114}$Sn+Au foils (black line) and the simulation (red dashed line) using the SIMNRA software[@SIMNRA1; @SIMNRA2]. See the text for more details.](rbs_112Sn_114Sn.eps){width="0.75\linewidth"} Since the lifetime measurement relies on the correct determination of the stopping powers, systematic uncertainties should be estimated as well. To do so, the $f_e$ parameter of the electronic stopping power was varied by 10$\%$ since the fit to the tabulated electronic stopping powers of Northcliffe and Schilling[@North70a] is characterized by a 10$\%$ uncertainty. Furthermore, calculations were performed with different screening factors $f_n$ for the nuclear potential of $f_n = 0.7, 0.85$, and $1.0$, i.e. approximately a 18$\%$ variation. The $f_n = 1.0$ value is the standard value of the LSS theory[@Curr69a], 0.85 has been used in this work, and $f_n = 0.7$ is commonly used. Within this parameter range, the combinations of $f_e$ and $f_n$ variation in the different layers leading to the largest variation of the lifetime $\Delta \tau_{sys.} = \|\tau_{\mathrm{sys.,}\pm} - \tau\|/\tau$ were calculated, where “+” indicates a longer lifetime and “-” a shorter lifetime. The results are shown in Fig.\[fig:sysun\] for different lifetimes in $^{112}$Sn determined in this work. The systematic uncertainties are conservatively estimated with $\Delta \tau_{sys.,-} \leq 19\,\%$.\ The lifetimes extracted in this work might appear slightly longer compared to previous results, even though there are several exceptions, see Table\[tab:112Sn\_tau\]. Due to the low recoil energies in the $(p,p'\gamma)$ reaction, i.e. $E_{\mathrm{rec.}} < 200$keV, in contrast to heavy-ion induced reactions, where the line-shape analysis is used, the proton-induced reaction is much more sensitive to the nuclear stopping power, i.e. the screening factor $f_n$[@Curr69a]. Despite the good agreement, it cannot be excluded that the different alloy layers are assumed too dense in the RBS simulation. This would also result in slightly longer lifetime values and explain the larger $f_n$ value, i.e. the larger nuclear-stopping contribution needed in our analysis to obtain agreement with known lifetimes. At present, no decision in favor of any of these two scenarios can be made due to missing sensitivity to the very details of the target+stopper alloy composition. ![\[fig:sysun\](color online) Systematic uncertainties of the lifetime measurements. The lifetime $\tau$ has been calculated with the stopping power parameters of Table\[tab:targets\]. The systematic uncertainties correspond to the variation of the lifetime $\tau$ in percent when these parameters are changed as described in the text. A mean uncertainty of about 19$\%$ is calculated.](sysuncer_112Sn.eps){width="0.9\linewidth"} experimental results ==================== In total 74 lifetimes and lifetime limits have been determined. Out of these, 39 lifetimes have been determined for the first time; 13 lifetimes in $^{112}$Sn and 26 lifetimes in $^{114}$Sn, respectively. The lifetimes are given in Tables \[tab:112Sn\_tau\] and \[tab:114Sn\_tau\]. Note that only statistical uncertainties are stated here. As has been mentioned in the previous section, systematic uncertainties should be considered at the 19$\%$ level or smaller. Besides the determination of nuclear level lifetimes, several new and also weak $\gamma$ decays of excited states were observed with SONIC@HORUS which are marked with $$. Some of these have relative $\gamma$-decay intensities $I_{\gamma}$ of smaller than $1\,\%$. For most cases, the lifetimes $\tau$ and $\gamma$-decay intensities $I_{\gamma}$ are in very good agreement with previously known and adopted values[@ENSDF]. However, discrepancies are observed. We have observed twelve new excited states and 116 new $\gamma$ transitions in $^{112}$Sn. In $^{114}$Sn, six new levels and 33 new $\gamma$ transitions were found. [p[0.10]{}p[0.09]{}p[0.09]{}p[0.09]{}p[0.08]{}p[0.09]{}p[.10]{}p[0.09]{}p[0.09]{}p[0.10]{}]{} \ $E_x$\[keV\]& $J^{\pi}_i$ & $J^{\pi}_f$ & $E_{\gamma}$\[keV\] & $I_{\gamma}$\[$\%$\] & $\delta$ & $\Pi L$ & $F(\tau)$ & $\tau$\[fs\] & $\tau_{\mathrm{lit.}}$\[fs\]\ \ $E_x$\[keV\]& $J^{\pi}_i$ & $J^{\pi}_f$ & $E_{\gamma}$\[keV\] & $I_{\gamma}$\[$\%$\] & $\delta$ & $\Pi L$ & $F(\tau)$ & $\tau$\[fs\] & $\tau_{\mathrm{lit.}}$\[fs\]\ \ \ \ 1256.5(2)& $2^+_1$ & $0^+_1$ & 1256.5(2)& 100 & -& $E2$ & 0.110(6) & 800(110) & 542(7)$^{\mathrm{a,b}}$\ 2150.5(3)& $2^+_2$ & $0^+_1$ & 2150.5(2)& 20(3) & -& $E2$ & & &\ & & & & & -0.28(6)& & & &\ & & & & & $7^{+3}_{-2}$& & & &\ 2190.5(2)& $0^+_2$ & $2^+_1$ & 934.0(2) & 100 & -& $E2$ & & & $\geq 3900$\ 2247.0(3) & $4^+_1$ & $2^+_1$ & 990.47(10) & 100 & & $E2$ & $< 0.03$ & $> 4400$& 4800(720)$^{\mathrm{a}}$\ 2353.7(2)& $3^-_1$ & $2^+_1$ & 1097.2(2)& 100 & 0.02(2) & $\centering E1$ & 0.217(10) & 280(20) & 310(20)$^{\mathrm{a}}$\ 2475.5(2)& $2^+_3$ & $0^+_1$ & 2475.5(2)& 100 & - & $E2$ & 0.049(6) & &\ & $2^+_3$ & $2^+_1$ & 1218.9(2)& 36(5) & -0.54(7) & $M1+E2$ & 0.039(12) & &\ & $2^+_3$ & $0^+_2$ & 284.9(2)& 0.70(10)& - & $E2$& & &\ 2520.5(2)& $4^+_2$ & $2^+_1$ & 1264.0(2)& 100 & -0.04(4) & $E2$ & 0.067(9) & 1600(300) & $>$1100\ 2548.6(2)& $6^+_1$ & $4^+_1$ & 301.6(2)& 100 & & $(E2)$ & & &19.81(12)ns$^{\mathrm{a}}$\ 2617.4(3)& $0^+_3$ & $2^+_1$ & 1360.9(3)& 100 & - & $E2$ & $< 0.03$& $> 4200$ & $> 580$\ & $0^+_3$ & $2^+_2$ & 466.8(4)$^$& 1.2(3) & - & $E2$ & & &\ 2720.6(2) & $2^+_4$ & $0^+_1$ & 2720.6(2)& 10.0(14) & - & $E2$& & &\ & $2^+_4$ & $2^+_1$ & 1464.1(2)& 100 & 0.17(10) & $M1+E2$ & 0.066(9) & 1500(300) & 1100$^{+\,1500}_{-\,450}$\ & $2^+_4$ & $2^+_2$ & 570.0(2)$^$& 1.4(2) & & & &\ & $2^+_4$ & $0^+_2$ & 529.7(3)$^$& 1.0(2) & - & $E2$ & & &\ & $2^+_4$ & $3^-_1$ & 366.6(3)$^$& 13(2)& & $(E1)$ & &\ 2755.2(3) & $3^+_1$ & $2^+_1$ & 1499.0(3) & 25(4) & 0.03(5) & $M1 + E2$ & & & $> 1150$\ & $3^+_1$ & $2^+_2$ & 604.8(2) & 4.4(7) & & & & &\ & $3^+_1$ & $4^+_1$ & 508.3(2) & 100 & 0.2(1) & & & &\ & $3^+_1$ & $3^-_1$ & 401.5(3) & 0.8(2) & & & & &\ & $3^+_1$ & $4^+_2$ & 234.7(6) & 1.4(3) & & & & &\ 2764.9(2)& $< 5$ & $2^+_1$ & 1508.5(2)& 100 & & & 0.055(8) & 2400(700) & $>$1500\ & $< 5$ & $2^+_2$ & 614.3(3)$^$& 1.9(5) & & & & &\ 2783.5(2)& $4^+$ & $2^+_1$ & 1527.0(2)& 100 & -0.06(4) & $E2$ & 0.127(10) & 580(70) & 450$^{+\,150}_{-\,90}$\ & $4^+$ & $4^+_1$ & 536.3(3)$^$& 2.7(5) & & & & &\ 2913.0(5)& $4^+$ & $2^+_1$ & 1656.2(2)& 91(13) & -0.11(11) & $E2$ & & &\ & $4^+$ & $2^+_2$ & 762.5(2)$^$ & 7(2) & & $(E2)$ & & &\ & $4^+$ & $4^+_1$ & 665.4(2) & 100 & & & 0.11(4) & &\ & $4^+$ & $3^-_1$ & 559.1(2)$^$ & 14(2) & & $(E1)$ & 0.18(5) & &\ 2917.0(2)& $(2^+,3,4^+)$ & $2^+_1$ & 1660.5(3)$^$& 2.1(4) & & & & & $> 1600$\ & $(2^+,3,4^+)$ & $2^+_2$ & 766.4(2) & 8.5(12) & & & & &\ & $(2^+,3,4^+)$ & $4^+_1$ & 669.8(2) & 100 & & & & &\ & $(2^+,3,4^+)$ & $4^+_2$ & 396.8(3)$^$ & 0.8(2) & & & & &\ 2926.6(4)& $6^+_2$ & $6^+_1$ & 377.4(2) & 100 & & & 0.50(12) & 80(40) & $> 300$\ 2945.0(7)& $4^+$ & $2^+_1$ & 1688.5(2)& 100 & & $(E2)$ & 0.036(10) & 3100(1000) & $> 1600$\ & $4^+$ & $2^+_2$ & 794.2(2) & 5.4(10) & & $(E2)$ & & &\ & $4^+$ & $4^+_1$ & 697.9(2)$^$ & $< 1.5$ & & & & &\ & $4^+$ & $2^+_3$ & 469.5(2) & 18(3) & & $(E2)$ & & &\ & $4^+$ & $4^+_2$ & 424.6(3)$^$ & 4.9(9) & & & & &\ & $4^+$ & $6^+_1$ & 396.4(4)$^$ & 2.3(5) & & $(E2)$ & & &\ & $4^+$ & $4^+$ & 161.4(2)$^$ & 9(2) & & & & &\ 2966.4(3)& $2^+$ & $0^+_1$ & 2966.4(4)& 61(9) & - & $E2$ & 0.061(11) & &\ & $2^+$& $2^+_1$ & 1709.7(3)& 100 & 0.3(4) & $M1 + E2$ & 0.069(13) & &\ & $2^+$& $4^+_1$ & 718.0(3)$^$& 3.8(6)& & $(E2)$ & & &\ & $2^+$& $3^-_1$ & 612.3(2)& 29(4)& & $(E1)$ & & &\ 2969.0(2)& $(1,3)$ & $2^+_1$ & 1712.5(2)& 100 & & & 0.131(12) & 610(70) & 430$^{+\,300}_{-\,130}$\ & $(1,3)$& $2^+_2$ & 818.3(2) & 7.5(11)& & & & &\ 2985.7(2) & $0^+$ & $2^+_1$ & 1729.2(2) & 100 & - & $E2$& 0.086(11) & 1100(190) & $>$2400\ & $0^+$& $2^+_2$ & 835.3(2)$^$ & 1.0(3)& - & $E2$ & & &\ 3077.8(3) & $3^+$ & $2^+_1$ & 1821.6(3) & 83(12) & -1.3$^{+0.3}_{-0.5}$ & $M1+E2$ & & & $> 1800$\ & & & & & 3.0(10) & & & &\ & & & & & 0.60$^{+0.10}_{-0.20}$ & & & &\ & $3^+$ & $4^+_1$ & 831.1(4) & 9(2) & & & & &\ & $3^+$ & $2^+_3$ & 601.8(5)$^$ & 9(2) & & & & &\ & $3^+$ & $4^+_2$ & 557.3(3) & 3.8(10) & & & & &\ & $3^+$ & $2^+$ & 357.2(2)$^$ & 10(2) & & & & &\ & $3^+$ & $3^+_1$ & 322.5(2)$^$ & 10(2) & & & & &\ 3092.4(2) & $2^+$ & $0^+_1$ & 3092.4(2) & 29(4) & - & $E2$ & 0.12(2) & &\ & $2^+$ & $2^+_1$ & 1835.8(2) & 100 & -1.5(10) & $M1+E2$ & 0.138(11) & &\ & $2^+$ & $0^+_2$ & 901.9(3)$^$ & 1.6(3) & - & $E2$ & & &\ 3113.2(2) & $(2^+,3,4^+)$ & $2^+_1$ & 1856.8(2)$^$ & 4.0(7) & & & & &\ & $(2^+,3,4^+)$ & $2^+_2$ & 962.7(2) & 100 & & & 0.06(3) & 1600(1000)& -\ & $(2^+,3,4^+)$ & $4^+_1$ & 866.0(2)$^$ & 17(2) & & & & &\ & $(2^+,3,4^+)$ & $3^-_1$ & 759.2(2)$^$ & 3.5(6) & & & & &\ & $(2^+,3,4^+)$ & $2^+_3$ & 637.7(2)$^$ & 2.1(5) & & & & &\ & $(2^+,3,4^+)$ & $2^+_4$ & 392.6(2)$^$ & 16(2) & & & & &\ & $(2^+,3,4^+)$ & $3^+_1$ & 357.2(3)$^$ & 4.7(8) & & & & &\ & $(2^+,3,4^+)$ & $4^+_3$ & 329.6(2)$^$ & 15(2) & & & & &\ & $(2^+,3,4^+)$ & $4^+_4$ & 200.5(4)$^$ & 1.2(3) & & & & &\ 3132.5(2) & $5^-_1$ & $4^+_1$ & 885.6(2) & 100 & -0.02$^{+0.01}_{-0.04}$& $E1$ & & & $> 1450$\ & $5^-_1$ & $3^-_1$ & 778.7(4) & 18(7) & & $(E2)$ & & &\ 3148.3(2) & $4^+$& $2^+_1$ & 1891.8(2) & 100 & 0.05(10) & $E2$ & & & 800$^{+\,1400}_{-\,300}$\ 3248.2(2) & $2^+$ & $0^+_1$ & 3248.2(2) & 100 & - & $E2$ & 0.072(10) & 1400(300) & $>$1600\ & $2^+$ & $2^+_1$ & 1991.7(2) & 21(3)& & & & &\ & $2^+$ & $3^-_1$ & 894.0(2) & $< 82$ & & $(E1)$ & & &\ & $2^+$ & $2^+_3$ & 772.7(2) & 15(2)& & & & &\ 3272.5(2) & $4^+$ & $2^+_1$ & 2016.0(2) & 100 & -0.0(1)& $E2$ & 0.11(2) & 730(200) & 430$^{+\,300}_{-\,130}$\ & $4^+$ & $2^+_2$ & 1122.1(2) & 2.5(8) & & $E2$ & & &\ & $4^+$ & $4^+_1$ & 1025.6(2)$^$ & 13(2) & & & & &\ & $4^+$ & $6^+_1$ & 723.7(3)$^$ & 1.9(7) & & $(E2)$ & & &\ & $4^+$ & $4^+$ & 488.9(4)$^$ & 2.8(11) & & & & &\ 3285.7(2) & $2^+$ & $0^+_1$ & 3285.7(2) & 100 & - & $E2$ & 0.19(2) & &\ & $2^+$ & $2^+_1$ & 2029.2(2) & 82(14) & & & 0.22(2) & &\ & $2^+$ & $2^+_2$ & 1135.5(3)$^$ & 4.3(8) & & & & &\ & $2^+$ & $3^-_1$ & 931.9(3)$^$ & 6(2) & & $(E1)$ & & &\ 3337.8(2) & $2^+$ & $2^+_1$ & 2081.3(2) & 100 & & & 0.15(2) & 470(90) & $>$480\ & $2^+$ & $2^+_2$ & 1187.3(2)$^$ & 20(3) & & & &\ 3352.8(2) & $2^+$ & $0^+_1$ & 3352.8(2) & 100 & - & $E2$ & 0.055(12) & 2600(700) & $>$2000\ & $2^+$ & $2^+_1$ & 2096.3(2) & 30(4) & & & &\ & $2^+$ & $2^+_2$ & 1202.5(2)$^$ & 23(3) & & & &\ & $2^+$ & $4^+_1$ & 1105.7(3)$^$ & 4.7(8) & & $(E2)$ & &\ & $2^+$ & $0^+_3$ & 735.4(5)$^$ & 1.7(6) & - & $E2$ & &\ & $2^+$ & $2^+$ & 631.7(3)$^$ & 6.0(11) & & & &\ 3378.3(2) & (1,$2^+$) & $2^+_1$ & 1227.8(2) & 100 & & & & &\ & (1,$2^+$) & $2^+_3$ & 903.0(3)$^$ & 6(2) & & & & &\ 3383.3(2) & $3^-$ & $2^+_1$ & 2126.8(2) & 100 & 0.1(5) & $E1$ & 0.197(14) & 310(20) & 260$^{+\,120}_{-\,70}$\ & $3^-$ & $2^+_2$ & 1232.9(2)$^$ & 4.8(11) & & & &\ & $3^-$ & $(2^+,3,4^+)$ & 466.5(2) & 13(2) & & & &\ 3396.6(2) & $2^{(-)}$ & $2^+_1$ & 2139.9(2)$^$ & 9(2) & & & & &\ & $2^{(-)}$ & $2^+_2$ & 1246.1(2) & 100 & & & 0.15(3) & 460(130) & 330$^{+\,140}_{-\,80}$\ & $2^{(-)}$ & $3^-_1$ & 1042.4(2) & 42(8) & 1.8(12) & & & &\ & $2^{(-)}$ & $2^+$ & 675.8(2)$^$ & 6.1(13) & & & & &\ 3412.7(2) & $6^+$ & $4^+_1$ & 1165.7(2) & 100 & & $(E2)$ & & &\ 3417.1(2) & $4^+$ & $2^+_1$ & 2160.6(2) & 100 & & $(E2)$ & $< 0.05$ & $> 2216$ & $> 500$\ & $4^+$ & $4^+_1$ & 1170.1(2)$^$ & 15(3) & & & & &\ 3433.4(2) & $(1^-)$ & $0^+_1$ & 3433.4(2) & 100 & - & $E1$ & 0.925(14) & 7.9(9) & 4.3(5)$^{\mathrm{c}}$\ 3455.7(2) & $2^+,3^+$ & $2^+_1$ & 2199.1(2) & 100 & 2.8(10) & $M1+E2$ & & & $> 940$\ & $2^+,3^+$ & $2^+_2$ & 1305.3(2)$^$ & 27(4) & & & & &\ & $2^+,3^+$ & $3^+_1$ & 700.2(2) & 33(5) & & & & &\ & $2^+,3^+$ & $2^+$ & 489.5(2)$^$ & 21(3) & & & & &\ 3497.9(2) & $5^-$ & $3^-_1$ & 1144.2(2) & 100 & & & 0.17(4) & 410(140) & 64$^{+\,64}_{-\,30}$\ & $5^-$ & $4^+$ & 977.1(2) & 38(8) & & & & &\ & $5^-$ & $4^+$ & 714.7(3)$^$ & $< 5$ & & & & &\ 3518.6(4) & $2^+$ & $2^+_2$ & 1368.4(2) & 100 & & & 0.20(4) & 310(90) & -\ & $2^+$ & $0^+_2$ & 1328.2(3)$^$ & 19(3) & - & $E2$ & & &\ & $2^+$ & $4^+_1$ & 1271.1(8)$^$ & 7.6(11) & & $(E2)$& & &\ & $2^+$ & $3^-_1$ & 1165.3(2) & 8.3(13) & & $(E1)$ & & &\ & $2^+$ & $2^+_3$ & 1042.8(8)$^$ & 17(2) & & & & &\ & $2^+$ & $2^+$ & 797.6(8)$^$ & 8.6(13) & & & & &\ & $2^+$ & $4^+_3$ & 735.1(5)$^$ & 1.5(5) & & $(E2)$ & & &\ 3524.0(4) & $2^+$ & $2^+_1$ & 2267.7(2) & 100 & -0.07(40) & $M1+E2$ & 0.08(2) & 1100(400) & -\ & $2^+$ & $4^+_1$ & 1276.8(4) & 37(6) & & $(E2)$ & & &\ & $2^+$ & $3^-_1$ & 1169.5(9)$^$ & 11(2) & & $(E1)$ & & &\ & $2^+$ & $3^+_1$ & 768.3(2)$^$ & 29(4) & & & & &\ & $2^+$ & $< 5$ & 759.4(4)$^$ & 29(7) & & & & &\ & $2^+$ & $0^+$ & 538.7(3)$^$ & 5(2) & - & $E2$ & & &\ & $2^+$ & $2^+$ & 431.5(4) & 5(2) & & & & &\ 3526.5(2)$^{\dagger}$ & $(1,2^+)$ & $0^+$ & 3526.5(2)$^$ & 100 & & & 0.25(2) & 230(30) & $>$180\ & $(1,2^+)$ & $2^+_2$ & 1375.6(2)$^$ & 40(6) & & & & &\ & $(1,2^+)$ & $2^+$ & 805.9(4)$^$ & 10(2) & & & & &\ 3529.2(3) & $(4^+)$ & $2^+_2$ & 1378.7(2) & 100 & & & & &\ & $(4^+)$ & $4^+_1$ & 1282.5(3) & 87$^{+13}_{-15}$ & & & & &\ & $(4^+)$ & $4^+_2$ & 1008.8(2) & 43(9) & & & & &\ & $(4^+)$ & $6^+_1$ & 980.1(3)$^$ & 41(9) & & & & &\ & $(4^+)$ & $4^+$ & 380.5(4) & 24(5) & & & & &\ 3553.2(2) & $(3)^-$ & $2^+_1$ & 2296.8(2) & 100 & & & 0.187(14) & 460(130) & 240$^{+\,160}_{-\,80}$\ & $(3)^-$ & $3^+_1$ & 797.7(3)$^$ & 20(4) & & & & &\ 3557.8(2) & $(4^+)$ & $3^-_1$ & 1204.1(2) & 100 & & & & &\ 3583.2(4) & $(2^+,4^+)$ & $2^+_1$ & 2326.9(2)$^$ & 23(6) & & & 0.11(5) & 770(540) & -\ & $(2^+,4^+)$ & $2^+_2$ & 1433.2(5)$^$ & 13(7) & & & & &\ & $(2^+,4^+)$ & $3^-_1$ & 1229.0(5)$^$ & 19(11) & & & & &\ & $(2^+,4^+)$ & $2^+_3$ & 1107.8(2)$^$ & 100 & & & & &\ & $(2^+,4^+)$ & $4^+$ & 669.8(5)$^$ & 72(13) & & & & &\ & $(2^+,4^+)$ & $2^+$ & 617.1(4)$^$ & 21(5) & & & & &\ 3601.6(2)$^{\dagger}$ & $2^+$ & $0^+_1$ & 3601.6(2)$^$ & 100 & - & $E2$ & 0.11(2) & 730(200) & -\ & $2^+$ & $2^+_1$ & 2345.5(3)$^$ & 63(10) & & & & &\ & $2^+$ & $0^+_2$ & 1411.4(2)$^$ & 29(8) & - & $E2$ & & &\ & $2^+$ & $4^+_2$ & 1081.8(2)$^$ & 29(5) & & $(E2)$ & & &\ & $2^+$ & $2^+$ & 881.1(3)$^$ & 15(3) & & & & &\ & $2^+$ & $4^+$ & 452.9(4)$^$ & 10(2) & & $(E2)$ & & &\ 3603.1(2) & $\leq 6$ & $4^+_1$ & 1356.1(3) & 100 & & & & &\ 3610.8(4) & $(2^+,3,4^+)$ & $2^+_1$ & 2354.2(2) & 100 & & & 0.44(7) & 90(30) & 111$^{+60}_{-34}$\ & $(2^+,3,4^+)$ & $2^+_2$ & 1459.9(5) & 73(12) & & & & &\ & $(2^+,3,4^+)$ & $4^+_1$ & 1364.2(6)$^$ & 15(6) & & & & &\ 3643.4(3)$^{\dagger}$ & $(2^+,3,4^+)$ & $4^+_1$ & 1396.4(2)$^$ & 36(7) & & & & &\ & $(2^+,3,4^+)$ & $4^+_2$ & 1122.9(2)$^$ & 73(12) & & & & &\ & $(2^+,3,4^+)$ & $2^+$ & 922.8(3)$^$ & 65(12) & & & & &\ & $(2^+,3,4^+)$ & $4^+$ & 726.4(2)$^$ & 100 & & & & &\ 3654.1(2) & $2^+$ & $0^+_1$ & 3654.1(2) & 100 & - & $E2$ & 0.36(3) & &\ & $2^+$ & $2^+_1$ & 2397.8(2) & 79(12) & & & 0.31(4) & &\ 3688.0(6)$^{\dagger}$ & $(1,2^+)$ & $0^+_1$ & 3688.3(3)$^$ & 60(11) & - & & & &\ & $(1,2^+)$ & $2^+_1$ & 2431.5(4)$^$ & 65(12) & & & & &\ & $(1,2^+)$ & $2^+_2$ & 1537.5(3)$^$ & 40(8) & & & & &\ & $(1,2^+)$ & $2^+$ & 721.8(2)$^$ & 100 & & & & &\ 3705.6(5)$^{\dagger}$ & $(2^+,3,4^+)$ & $2^+_2$ & 1554.8(3)$^$ & 52(8) & & & & &\ & $(2^+,3,4^+)$ & $4^+_1$ & 1459.0(3)$^$ & 100 & & & & &\ 3719.6(2)$^{\dagger}$ & $(2^+,3,4^+)$ & $2^+_1$ & 2462.9(2)$^$ & 100 & & & & &\ & $(2^+,3,4^+)$ & $4^+_1$ & 1472.7(3)$^$ & 24(6) & & & & &\ 3725.3(2) & $(1,2^+)$ & $2^+_1$ & 2468.8(2) & 100 & - & & & &\ & $(1,2^+)$ & $2^+_2$ & 1574.7(3)$^$ & 30(6) & & & & &\ & $(1,2^+)$ & $0^+_2$ & 1534.8(3)$^$ & 61(15) & - & & & &\ 3774.5(4)$^{\dagger}$ & $2^+$ & $0^+_1$ & 3774.3(5)$^$ & 39(8) & - & $E2$ & & &\ & $2^+$ & $2^+_1$ & 2518.0(2)$^$ & 100 & & & & &\ & $2^+$ & $4^+_1$ & 1527.3(2)$^$ & 56(10) & & $(E2)$ & & &\ & $2^+$ & $3^-_1$ & 1420.6(3)$^$ & 68(12) & & $(E1)$ & & &\ & $2^+$ & $2^+$ & 808.8(8)$^$ & 29(7) & & & & &\ 3781.0(5) & $(2^+,3,4^+)$ & $2^+_2$ & 1630.1(2) & 100 & & & & &\ & $(2^+,3,4^+)$ & $4^+$ & 997.8(7)$^$ & 54(11) & & & & &\ 3827.1(3) & $(1^-,2^+)$ & $0^+_1$ & 3827.1(2)$^$ & 100 & - & & 0.41(2) & &\ & $(1^-,2^+)$ & $2^+_1$ & 2570.8(2)$^$ & 50(8) & & & 0.26(4) & &\ & $(1^-,2^+)$ & $3^-_1$ & 1473.0(7)$^$ & 23(5) & & & & &\ 3873.4(3) & $(1,2^+)$ & $0^+_1$ & 3873.4(3)$^$ & 100 & - & & 0.95(4) & 5(3) & -\ 3913.5(2) & $ 2^+ $ & $0^+_1$ & 3913.5(2)$^$ & 100 & - & $E2$ & 0.38(3) & 120(20) & -\ & $ 2^+ $ & $4^+_2$ & 1392.4(3)$^$ & 23(4) & & $(E2)$ & & &\ 3925.5(8)$^{\dagger}$ & $(1,2^+)$ & $0^+_1$ & 3926.0(6)$^$ & 38(8) & - & & & &\ & $(1,2^+)$ & $2^+_1$ & 2668.4(10)$^$ & 100 & & & 0.16(2) & 410(70)& -\ 3984.7(3) & $(1^-,2^+)$ & $0^+_1$ & 3984.7(3)$^$ & 100 & - & & 0.53(5) & 64(12) & -\ & $(1^-,2^+)$ & $3^-_1$ & 1630.0(3)$^$ & 17(3) & & & & &\ & $(1^-,2^+)$ & $2^+_3$ & 1507.8(4)$^$ & 9(2) & & & & &\ 4019.1(9)$^{\dagger}$ & $(1,2^+)$ & $0^+_1$ & 4018.4(6)$^$ & 100 & - & & 0.54(6)& 60(20) & -\ & $(1,2^+)$ & $0^+_2$ & 1828.9(6)$^$ & 73(15) & - & & & &\ 4044.0(2)$^{\dagger}$ & $(1,2^+)$ & $0^+_1$ & 4044.2(8)$^$ & 21(5) & - & & & &\ & $(1,2^+)$ & $2^+_1$ & 2787.5(5)$^$ & 100 & & & & &\ & $(1,2^+)$ & $2^+_2$ & 1893.3(4)$^$ & 78(13) & & & 0.18(4) & 350(110) & -\ 4077.2(10)$^{\dagger}$ & $(1,2^+)$ & $0^+_1$ & 4076.6(5)$^$ & 87$^{+13}_{-17}$ & - & & & &\ & $(1,2^+)$ & $2^+_1$ & 2819.6(10)$^$ & 100 & & & & &\ & $(1,2^+)$ & $2^+_2$ & 1927.0(3)$^$ & 63(14) & & & & &\ & $(1,2^+)$ & $2^+_3$ & 1602.9(12)$^$ & 40(10) & & & & &\ 4086.5(2)$^{\dagger}$ & $(1,2^+)$ & $0^+_1$ & 4086.3(4)$^$ & 100 & - & & & &\ & $(1,2^+)$ & $2^+_1$ & 2829.9(5)$^$ & 78(19) & & & & &\ & $(1,2^+)$ & $2^+_2$ & 1936.1(3)$^$ & 48(14) & & & & &\ 4096.7(2) & $< 5$ & $2^+_1$ & 2840.2(2)$^$ & 100 & & & & &\ 4141.2(3) & $1^-$ & $0^+_1$ & 4141.2(3) & 100 & - & $E1$ & 0.55(2) & 59(5) & 39(9)$^{\mathrm{d}}$\ 4160.5(3) & $1^-$ & $0^+_1$ & 4160.5(3) & 100 & - &$E1$ & 0.77(5) & 23(6) & 14.9(14)$^{\mathrm{d}}$\ [p[0.10]{}p[0.09]{}p[0.07]{}p[0.10]{}p[0.07]{}p[0.12]{}p[.10]{}p[0.09]{}p[0.09]{}p[0.08]{}]{} \ $E_x$\[keV\]& $J^{\pi}_i$ & $J^{\pi}_f$ & $E_{\gamma}$\[keV\] & $I_{\gamma}$\[$\%$\] & $\delta$ & $\Pi L$ & $F(\tau)$ & $\tau$\[fs\] & $\tau_{\mathrm{lit.}}$\[fs\]\ \ $E_x$\[keV\]& $J^{\pi}_i$ & $J^{\pi}_f$ & $E_{\gamma}$\[keV\] & $I_{\gamma}$\[$\%$\] & $\delta$ & $\Pi L$ & $F(\tau)$ & $\tau$\[fs\] & $\tau_{\mathrm{lit.}}$\[fs\]\ \ \ 1299.7(2)& $2^+_1$ & $0^+_1$ & 1299.7(2)& 100 & -& $E2$ & 0.145(13) & 590(70) & 610(40)$^{\mathrm{a}}$\ 1952.9(2)& $0^+_2$ & $2^+_1$ & 653.2(2)& 100 & - & $E2$ & & & 9(3)ps$^{\mathrm{a}}$\ 2155.9(2)& $0^+_3$ & $2^+_1$ & 856.2(2)& 100 & -& $E2$ & & & $> 11$ps$^{\mathrm{a}}$\ 2187.3(3)& $4^+_1$ & $2^+_1$ & 887.6(2)& 100 & & $(E2)$ & & & 7.6(6)ps$^{\mathrm{a}}$\ 2238.6(2)& $2^+_2$ & $0^+_1$ & 2238.5(2) & 100 & - & $E2$ & $< 0.04$ & $> 2100$ & -\ & $2^+_2$ & $2^+_1$ & 938.9(2) & 81(12) & -7.1$^{+1.2}_{-1.9}$ & $M1+E2$ & & &\ & $2^+_2$ & $0^+_2$ & 286.5(10) & 0.9(3) & - & $E2$ & & &\ 2274.5(2)& $3^-_1$ & $2^+_1$ & 974.8(2) & 100 & & $(E1)$ & 0.123(11) & 700(80) & 520(30)$^{\mathrm{a}}$\ 2420.5(2)& $0^+_4$ & $2^+_1$ & 1120.8(2)& 100 & - & $E2$ & & &\ 2453.8(2)& $2^+_3$ & $0^+_1$ & 2453.7(2) & 28(4) & - & $E2$ & 0.04(2) & &\ & $2^+_3$ & $2^+_1$ & 1154.0(2) & 100 & -2.8$^{+1.8}_{-9.5}$ & $M1+E2$ & 0.05(2) & &\ & $2^+_3$ & $2^+_2$ & 215.4(4) & 1.3(3) & & & & &\ 2514.4(2)& $3^+_1$ & $4^+_1$ & 327.1(2) & 100 & 0.02$^{+0.02}_{-0.01}$ & $M1+E2$ & & &\ 2613.7(4)& $4^+_2$ & $2^+_1$ & 1314.5(2) & 100 & & $(E2)$ & 0.100(9) & 920(130) & 793(144)$^{\mathrm{b}}$\ & $4^+_2$ & $4^+_1$ & 426.0(4) & 1.6(6) & -0.24$^{+0.06}_{-0.05}$ & $M1+E2$ & & &\ & $4^+_2$ & $2^+_2$ & 375.2(3) & 1.8(6) & & $(E2)$ & & &\ 2764.9(5)& $4^+$ & $2^+_1$ & 1465.3(2) & 100 & & $(E2)$ & 0.042(19) & 2900(1600) & 808(433)$^{\mathrm{b}}$\ & $4^+$ & $4^+_1$ & 577.3(3)$^$ & 2.3(9) & & & & &\ & $4^+$ & $2^+_2$ & 525.7(2) & 1.1(6) & & $(E2)$ & & &\ & $4^+$ & $3^-_1$ & 490.3(3) & 1.4(7) & & $(E1)$ & & &\ & $4^+$ & $3^+_1$ & 251.1(3) & 4.3(10) & $-0.1^{+0.1}_{-4.2}$& $M1+E2$ & & &\ 2814.6(2)& $5^-_1$ & $4^+_1$ & 627.4(2) & 100 & & $(E1)$ & & & $> 2020$\ & $5^-_1$ & $3^-_1$ & 539.9(2) & 13(3) & & $(E2)$ & & &\ 2859.2(5)& $4^+$ & $2^+_1$ & 1559.7(2) & 100 & & $(E2)$ & 0.104(10) & 900(130) & -\ & $4^+$ & $4^+_1$ & 672.1(4)$^$ & 4.2(12) & & & & &\ & $4^+$ & $2^+_2$ & 619.8(3)$^$ & $< 1.5$ & & $(E2)$ & & &\ & $4^+$ & $2^+_3$ & 405.5(3)$^$ & $< 1.7$ & & $(E2)$ & & &\ 2904.9(3)& $3^-$ & $2^+_1$ & 1605.1(4) & 3.4(7) & & $(E1)$ & & &\ & $3^-$ & $4^+_1$ & 717.3(2) & 100 & $-0.7^{+0.2}_{-0.4}$ & $(E1)$ & 0.098(25) & 880(360) & -\ & $3^-$ & $3^+_1$ & 390.2(2) & 26(4) & & & & &\ & $3^-$ & $4^+_2$ & 290.3(4) & 1.4(5) & & $(E1)$ & & &\ 2915.6(2)& $2^+$ & $0^+_1$ & 2915.5(2) & 100 & - & $E2$ & 0.067(6) & &\ & $2^+$ & $2^+_1$ & 1615.8(2) & 29(4) & $0.08 < \delta < 1.7$ & $M1+E2$ & 0.06(2) & &\ 2943.4(2)& $2^+$ & $0^+_1$ & 2943.4(6) & 2.5(7) & - & $E2$ & & &\ & & & & & -0.61(15) & $M1+E2$ & & &\ & & & & & -7$^{+10}_{-3}$ & $M1+E2$ & & &\ & $2^+$ & $0^+_2$ & 990.3(3) & 7.7(13) & - & $E2$ & & &\ & $2^+$ & $2^+_2$ & 704.3(3) & 3.2(7) & & & & &\ & $2^+$ & $3^-_1$ & 668.3(2) & 87(12) & & $(E1)$ & & &\ & $2^+$ & $0^+_4$ & 522.4(5) & 1.0(5) & - & $E2$ & & &\ & $2^+$ & $2^+_3$ & 489.6(2) & 2.9(6) & & & & &\ 3024.9(2)& $2^+$ & $2^+_1$ & 1725.4(2) & 100 & & & & &\ & $2^+$ & $0^+_2$ & 1071.7(4)$^$ & 9(2) & - & $E2$ & & &\ & $2^+$ & $2^+_2$ & 786.4(2)$^$ & 13(2) & & & & &\ & $2^+$ & $2^+_3$ & 571.1(2)$^$ & 12(2) & & & & &\ 3028.1(2)& $0^+$ & $2^+_1$ & 1728.4(2) & 100 & - & $E2$ & 0.10(3) & 900(400) & -\ & $0^+$ & $2^+_2$ & 789.4(5)$^$ & 1.6(10) & - & $E2$ & & &\ & $0^+$ & $2^+_3$ & 574.1(3)$^$ & 3.8(9) & - & $E2$ & & &\ 3185.5(2)& $2^+$ & $0^+_1$ & 3185.5(2) & 58(8) & - & $E2$ & 0.25(2) & &\ & & & & & -0.27(7) & $M1+E2$ & & &\ & & & & & 7$^{+5}_{-2}$ & $M1+E2$ & & &\ 3206.6(6)& $4^+$ & $2^+_1$ & 1907.2(3) & 100 & & $(E2)$& & &\ & $4^+$ & $4^+_1$ & 1019.6(4) & 29(7) & & & & &\ & $4^+$ & $3^-_1$ & 932.3(2) & 8(3) & & $(E1)$& & &\ & $4^+$ & $4^+_2$ & 592.0(9) & 7(3) & & & & &\ 3211.3(2)& $(1,2^+)$ & $0^+_1$ & 3211.2(2) & 100 & & & 0.16(3) & &\ & $(1,2^+)$ & $2^+_1$ & 1911.8(2) & 38(6) & & & 0.14(4) & &\ & $(1,2^+)$ & $0^+_2$ & 1258.0(7)$^$ & 24(4) & & & & &\ & $(1,2^+)$ & $0^+_3$ & 1054.6(2)$^$ & 42(6) & & & 0.21(6) & &\ & $(1,2^+)$ & $2^+_3$ & 757.5(2)$^$ & $< 6$ & & & & &\ 3225.1(2)& $3^-$ & $2^+_1$ & 1925.4(2) & 100 & & $(E1)$ & 0.15(2) & 500(95) & -\ & $3^-$ & $2^+_3$ & 771.4(4) & 2.1(8) & & $(E1)$ & & &\ & $3^-$ & $3^-$ & 319.9(4) & 6.6(14) & & & & &\ 3308.5(2)& $0^+$ & $2^+_1$ & 2008.8(2) & 100 & - & $E2$ & & &\ 3326.4(2)& $2^+$ & $0^+_1$ & 3326.2(2)$^$ & 59(9) & - & $E2$ & 0.11(4) & 750(380)& -\ & $2^+$ & $2^+_1$ & 2026.4(2)$^$ & 100 & & & & &\ & $2^+$ & $0^+_2$ & 1373.2(3)$^$ & 24(4) & - & $E2$ & & &\ & $2^+$ & $4^+_1$ & 1139.0(3)$^$ & 13(2) & & $(E2)$ & & &\ 3356.3(7)& $4^+$ & $2^+_1$ & 2057.1(2) & 100 & & $(E2)$ & & &\ & $4^+$ & $4^+_1$ & 1168.6(6)$^$ & 6(3) & & & & &\ 3397.3(2)& $3^-$ & $2^+_1$ & 2097.6(2)$^$ & 72(11) & & & 0.24(6) & 250(90) & -\ & $3^-$ & $2^+_2$ & 1158.3(2)$^$ & 30(5) & & & & &\ & $3^-$ & $3^-_1$ & 1122.0(4) & 100 & -0.4$^{+0.2}_{-0.7}$ & $M1+E2$ & & &\ & $3^-$ & $2^+_3$ & 943.2(2)$^$ & 27(4) & & & & &\ 3422.0(3)& $0^+$ & $2^+_1$ & 2121.9(4) & 100 & - & $E2$ & & &\ & $0^+$ & $2^+_2$ & 1182.9(5)$^$ & 41(10) & - & $E2$ & & &\ & $0^+$ & $2^+_3$ & 968.2(4)$^$ & 21(7) & - & $E2$ & & &\ 3452.1(2)& $(1^-)$ & $0^+_1$ & 3452.1(2) & 100 & & $(E1)$ & 0.93(3) & 6(3) & -\ 3478.1(4)& $2^+$ & $0^+_1$ & 3478.1(4) & 10(2) & - & $E2$ & & &\ & $2^+$ & $2^+_1$ & 2178.5(2) & 100 & & & & &\ & $2^+$ & $0^+_2$ & 1524.4(3) & 13(3) & - & $E2$ & & &\ & $2^+$ & $2^+_2$ & 1240.0(2) & 83(13) & & & & &\ & $2^+$ & $3^-_1$ & 1203.3(2) & 67(10) & & $(E1)$ & & &\ & $2^+$ & $2^+_3$ & 1025.1(2)$^$ & 21(4) & & & & &\ & $2^+$ & $3^+_1$ & 962.9(3) & 52(8) & & & & &\ & $2^+$ & $4^+$ & 619.7(4) & 11(3) & & $(E2)$ & & &\ & $2^+$ & $3^-$ & 572.4(4) & 14(3) & & & & &\ 3483.9(4)$^{\dagger}$& $1^-,2^+$ & $2^+_1$ & 2184.1(2) & 100 & & & 0.16(3) & &\ & $1^-,2^+$ & $0^+_3$ & 1327.7(3)$^{\#}$ & 14(2) & & & & &\ & $1^-,2^+$ & $3^-_1$ & 1209.0(2) & 35(2) & & & 0.10(4) & &\ 3487.5(4)& $5^-$ & $3^-_1$ & 1213.3(4)$^$ & 40(13) & & $(E2)$ & & &\ & $5^-$ & $3^-$ & 582.3(2)$^$ & 100 & & $(E2)$ & & &\ 3494.3(3)$^{\dagger}$& $1,2^+$ & $0^+_1$ & 3494.2(3) & 81(13) & & & 0.18(5) & &\ & $1,2^+$ & $2^+_1$ & 2194.4(2) & 100 & & & 0.12(3) & &\ & $1,2^+$ & $0^+_2$ & 1540.5(3) & 26(5) & & & & &\ 3514.1(3)& $3^-$ & $2^+_1$ & 2214.4(2) & 100& & $(E1)$ & 0.27(8) & 206(93) & -\ & $3^-$ & $4^+_1$ & 1327.0(4)$^{\#}$ & 9(3) & & $(E1)$ & & &\ & $3^-$ & $2^+_2$ & 1275.0(2)$^$ & 23(4) & & $(E1)$ & & &\ 3524.4(2)& $3^-$ & $2^+_1$ & 2224.5(3)$^$ & 100 & & $(E1)$ & $< 0.09$ & &\ & $3^-$ & $4^+_1$ & 1337.0(2) & 27(5) & & $(E1)$ & & &\ & $3^-$ & $3^-_1$ & 1249.6(3) & 24(4) & & $(E2)$ & 0.10(5) & 900(690) & -\ & $3^-$ & $3^+_1$ & 1010.1(3) & 31(6) & & & & &\ 3547.6(2)& $0^+$ & $2^+_2$ & 1308.9(2)$^$ & 92$^{+8}_{-21}$ & - & $E2$ & & &\ & $0^+$ & $2^+_3$ & 1093.8(3)$^$ & 100 & - & $E2$ & & &\ 3560.8(2)& $2^+$ & $0^+_1$ & 3560.8(2) & 100 & - & $E2$ & 0.13(3) & 590(190)& -\ & $2^+$ & $2^+_1$ & 2261.1(3)$^$ & 19(4) & & & & &\ & $2^+$ & $0^+_3$ & 1404.9(4) & 14(3) & - & $E2$ & & &\ 3610.2(4)& $5^{(-)}$ & $4^+_1$ & 1422.9(3) & 100 & & $(E1)$ & 0.33(6) & 150(40) & -\ 3650.3(3)$^{\dagger}$& $1^-,2^+$ & $0^+_1$ & 3650.1(3) & 100 & & & 0.20(5) & 320(120)& -\ & $1^-,2^+$ & $2^+_1$ & 2350.3(3) & 26(5) & & & & &\ & $1^-,2^+$ & $0^+_3$ & 1493.7(3) & 21(4) & & & & &\ & $1^-,2^+$ & $3^-_1$ & 1374.6(2) & 82(13) & & & & &\ 3679.5(4)$^{\dagger}$& $1,2^+$ & $0^+_1$ & 3679.4(2) & 100 & & & 0.30(4) & &\ & $1,2^+$ & $2^+_1$ & 2379.5(3) & 81(12) & & & 0.37(5) & &\ 3692.5(3)& $2^+$ & $0^+_1$ & 3692.8(3)$^$ & 100 & - & $E2$ & 0.13(4) & 580(250) &\ & $2^+$ & $2^+_1$ & 2392.3(3)$^$ & 47(8) & & & & &\ 3722.5(3)& $(2^+)$ & $2^+_1$ & 2422.5(2)$^$ & 73(12) & & & 0.30(7) & 180(60) &\ & $(2^+)$ & $3^-_1$ & 1446.6(2) & 100 & & & & &\ 3792.2(3)& $1,2^+$ & $0^+_1$ & 3792.2(3)$^$ & 100 & & & & &\ & $1,2^+$ & $2^+_1$ & 2492.7(5)$^$ & 47(13) & & & & &\ 3869.4(5)$^{\dagger}$& $2^+$ & $0^+_1$ & 3869.2(5) & 100 & - & $E2$ & 0.33(8) & &\ & $2^+$ & $2^+_1$ & 2569.1(11) & 19(9) & & & & &\ & $2^+$ & $4^+_1$ & 1682.7(4) & 55(12) & & $E2$ & 0.40(6) & &\ & $2^+$ & $3^-_1$ & 1595.2(2) & 18(6) & & $(E1)$ & & &\ 3933.0(4)& $1,2^+$ & $0^+_1$ & 3933.0(4)$^$ & 100 & & & 0.79(8) & 19(9) & -\ & $1,2^+$ & $2^+_1$ & 2633.6(3)$^$ & 24(5) & & & & &\ 4022.4(3)$^{\dagger}$& $1,2^+$ & $0^+_1$ & 4022.4(3) & 100 & & & 0.81(4) & 18(5) & -\ Lifetimes of the $2^+_1$ and $3^-_1$ states {#sec:tau_comp} ------------------------------------------- In general, the lifetimes determined in this work are in excellent agreement with lifetimes reported in Refs.[@Kum05a; @Orce07a; @Jungcl11a; @Gabl01a] and also the lower limits found are in good agreement with previously known lifetimes, see, [e.g.]{}, $\tau \left( 2^+_2 \right)$ and $\tau \left( 4^+_1 \right)$ in Table \[tab:112Sn\_tau\].\ \ [$J^{\pi} = 2^+_1$:]{} A rather obvious inconsistency between lifetime measurements employing Doppler-shift methods[@Jungcl11a] and from Coulex experiments[@Allm15a; @Kum17a] has been observed for $\tau \left( 2^+_1 \right)$ in the stable even-even Sn isotopes and is apparently also seen for the unstable tin isotopes, see the recent work on $^{110}$Sn[@Kumb16a]. Our present $(p,p'\gamma)$ experiments might support the lifetimes, which have been reported in Ref.[@Jungcl11a] by A.Jungclaus [et al.]{}, see Tables\[tab:112Sn\_tau\] and \[tab:114Sn\_tau\]. It should be noted that due to the kinematics of the $(p,p'\gamma)$ reaction, our systematic uncertainties are dominated by the variation of the nuclear stopping power in contrast to Ref.[@Jungcl11a] where the electronic stopping dominates the systematic uncertainties. J.N. Orce [et al.]{} first reported a $\tau \left( 2^+_1 \right)$ of $750^{+125}_{-90}$fs in $^{112}$Sn using INS-DSAM and later revised their measured lifetime to $530^{+100}_{-80}$fs[@Orce07a], i.e. closer to the presently adopted value. The authors argued that one should introduce a correction to the recoil-velocity distribution when using neutrons with an energy “well above” the excitation threshold. In the light of the new data, this discussion might not be necessary since the inital recoil velocity can be determined precisely from the $p\gamma$ coincidence data and since feeding can be excluded due to the excitation-energy gate. We want to mention that $\tau \left( 2^+_1 \right)$ in $^{114}$Sn was determined using the $^{112}$Sn data set. Here, the $^{114}$Sn admixture to the target accounted to roughly 13$\%$. Due to a large $^{116}$Sn admixture ($\sim 10\,\%$) in the $^{114}$Sn target, it was not possible to unambiguously determine the energy-centroid shifts of the two close-lying $2^+$ states of $^{114}$Sn and $^{116}$Sn.\ \ Since some Coulex experiments and especially those with radioactive ion beams, see, [e.g.]{}, Refs.[@Jungcl11a; @Doorn08a; @Bad13a], rely on the normalization to “well-known” $B(E2)$ values in stable nuclei, it is important to resolve these discrepancies. The lifetime of the $2^+_1$ in $^{116}$Sn should be certainly remeasured as well.\ [$J^{\pi} = 3^-_1$:]{} The lifetime $\tau \left( 3^-_1 \right) = 280(20)$fs in $^{112}$Sn agrees nicely with the previously reported value of A.Jungclaus [et al.]{}[@Jungcl11a], while it is in conflict with the one reported by A.Kumar [et al.]{} of $510^{+200}_{-120}$fs[@Kum05a]. Possibly, the latter discrepancy might be attributed to feeding missed in the $(n,n'\gamma)$ experiment. The lifetime $\tau \left( 3^-_1 \right) = 700(80)$fs in $^{114}$Sn does, however, not confirm the value of 520(30)fs measured by A.Jungclaus [et al.]{}[@Jungcl11a]. It should be mentioned that also $\tau \left( 3^-_1 \right)$ could be estimated from the $^{112}$Sn data set for $^{114}$Sn. Both values measured in the $(p,p'\gamma)$ experiments are consistent. $\gamma$-decay intensities, newly observed and non-observed $\gamma$ decays --------------------------------------------------------------------------- The total photopeak efficiency of the combined SONIC@HORUS setup was already shown in Fig.\[fig:efficiency\]. An uncertainty of less than 10$\%$ due to the geometry of the $^{56}$Co source is included in the uncertainties given for the $\gamma$-decay intensities $I_{\gamma}$ in Tables \[tab:112Sn\_tau\] and \[tab:114Sn\_tau\]. If $\gamma$-decay branching of an excited state was observed, the respective $\gamma$-decay branching ratio could be calculated as follows: $$I_{\gamma} = \frac{A_i \varepsilon(E_{\gamma,j})}{A_j \varepsilon(E_{\gamma,i})}$$ where $A_i$ is the peak intensity of a $\gamma$ decay with decay energy $E_{\gamma,i}$ corrected by the corresponding detection efficiency $\varepsilon(E_{\gamma,i})$. Usually, one should also correct for the $p\gamma$-angular correlation, detector deadtimes, and the system deadtime. This correction was previously estimated to be less than 20$\%$[@Henn14a].\ A comparison of the $\gamma$-decay branching ratios determined in this work with adopted ratios[@ENSDF] showed that this statement is in general correct. Within the statistical uncertainties very good agreement was obtained. For instance, for the $2^+_2$ state of $^{114}$Sn at 2238.6(2)keV the following $\gamma$-decay intensities are adopted[@ENSDF]: 100$\%$ ($0^+_1$), 82(2)$\%$ ($2^+_1$) and 0.8(3)$\%$ ($0^+_2$) which are in perfect agreement with our results, see Table \[tab:114Sn\_tau\]. The same holds for the $2^+_3$ state of $^{114}$Sn and the $2^+_2$ state of $^{112}$Sn, compare Table \[tab:112Sn\_tau\], as well as for states with $J^{\pi} \neq 2^+$, [e.g.]{}, the $4^+$ state of $^{114}$Sn at 2764.9(5)keV or the $3^-$ state at 2904.9(3)keV. For these states the very weak $\gamma$-decays with intensities of about 1$\%$ were also observed in our experiment. However, discrepancies are already observed for the $2^+_3$ state of $^{112}$Sn where a $\gamma$-decay intensity of 20(2)$\%$ to the $2^+_1$ state was previously reported[@ENSDF; @Wig76a; @Kum05a]. In our experiment an intensity of 36(5)$\%$ was observed. We cannot comment on the efficiency calibration of Ref.[@Wig76a], however, Ref.[@Kum05a] used a $^{226}$Ra source for the efficiency calibration which only provides a reliable efficiency calibration up to 2.45MeV. Apparently, the efficiency has been underestimated since discrepancies for the $\gamma$-decay intensities are observed for decays with $E_{\gamma} > 2.45$MeV. For instance, Ref.[@Wig76a] reported an $I_{\gamma}$ of 15.9(13)$\%$ for the decay of the $2^+$ state of $^{112}$Sn at 2720.6(2)keV to the ground state while Ref.[@Kum05a] gave a value of 33(6)$\%$. Our analysis provides a value of 10.0(14)$\%$, see Table \[tab:112Sn\_tau\]. The efficiency-calibration problem of Ref.[@Kum05a] for $\gamma$-decays with $E_{\gamma} > 2.45$MeV might become even more obvious for the $2^+$ states at 2966.4(3)keV and 3092.4(2)keV. Our data support the adopted values, while Ref.[@Kum05a] provided completely opposite results, i.e. a stronger $\gamma$-decay intensity to the ground state. For $\gamma$-decays with $E_{\gamma} < 2.45$MeV, as already stated, our results do in general support the previous findings of Ref.[@Kum05a], [e.g.]{}, for the $3^+_2$ state at 3077.8(3)keV and the $5^-_1$ state at 3132.5(2)keV of $^{112}$Sn.\ ### $^{112}$Sn Many new levels and $\gamma$ transitions in $^{112}$Sn were reported in Ref.[@Kum05a]. We will shortly comment on those levels where conflicting results were observed. However, we also want to stress explicitly that the majority of new levels observed in Ref.[@Kum05a] is supported by our data, compare Table \[tab:112Sn\_tau\].\ [3141keV:]{} A new level was proposed in Ref.[@Kum05a] based on a $\gamma$ decay with $E_{\gamma} = 990.2(4)$keV. This $\gamma$-decay energy does in fact coincide with the one of the $\gamma$-decay of the $4^+_1$ to the $2^+_1$. Based on our data and a careful analysis of different excitation-energy gates we propose to reject this level assignment and claim that this $\gamma$-ray has been solely observed due to feeding in Ref.[@Kum05a]. Two excited states in the relevant energy range, i.e. at 3113.2(2)keV and 3132.5(2)keV decay to the $4^+_1$ state.\ [3288keV:]{} Also this level was proposed in Ref.[@Kum05a]. The assignment was based on the observation of a $\gamma$ transition with an energy of 1097.2(3)keV which coincides with the $\gamma$-decay energy of the $3^-_1$ state to the $2^+_1$, compare Table \[tab:112Sn\_tau\]. We propose to reject this assignment due to feeding. Two excited states at 3248.2(2)keV and 3285.7(2)keV decay to the $3^-_1$ state in the relevant energy interval. Both states were observed in the $(n,n'\gamma)$ experiment of Ref.[@Kum05a] as well.\ [3524keV:]{} While the $\gamma$ transition with an energy of 3524.2(10)keV was proposed to belong to a $J^{\pi} = 2^+$ state at 3524.3(3)keV in Ref.[@Kum05a], we propose that two states exist at 3524.0(4)keV and 3526.5(2)keV, respectively. This observation is supported by both the different level lifetimes observed and the $\gamma$ transitions depopulating the respective levels, see Table \[tab:112Sn\_tau\]. The lifetime limit of $\tau > 180$fs given in Ref.[@Kum05a] has, thus, to be attributed to the excited state at 3526.5(2)keV. For this state a lifetime of 229(27)fs has been determined from our data.\ ### $^{114}$Sn Both tin isotopes were studied before by means of the $(p,t)$ reaction[@Guaz04a; @Guaz12a]. Many excited states which were first observed in the $(p,t)$ experiments of Ref.[@Guaz04a] are now supported by the observation of $\gamma$ decays from these levels, see Table \[tab:114Sn\_tau\]. Here, we will only comment on the contradicting spin-parity assignments which were made in Ref.[@Guaz04a] and are partly adopted[@ENSDF].\ \ [2514keV:]{} Two levels have been adopted at an energy of 2514keV with $J^{\pi} = 3^-$ and $J^{\pi} = 3^+$, respectively. The $3^+$ assignment is based on the measurement of two multipole-mixing ratios $\delta$[@ENSDF]. Both $\gamma$-decays from and to the $3^+$ state were also observed in our experiment. We, thus, conclude that this state has been excited in our experiment. No signs of the $3^-$ state reported at 2510keV[@Guaz04a] were seen.\ [2576keV:]{} Based on the observation of a clear $L = 2$ transfer, a $J^{\pi} = 2^+$ state was proposed at an energy of 2576keV[@Guaz04a]. The level was not observed in our $p\gamma$ coincidence data. However, there is a weak $\gamma$ transition with $E_{\gamma} \sim 390$keV in coincidence with the $\gamma$-decay of the $4^+_1$ level seen in our $\gamma\gamma$-coincidence data. Consequently, we cannot exclude the possibility that this state might exist in $^{114}$Sn. However, since the $(p,n)$ channel is open ($Q(p,n) = -6.8$MeV), this state is rather populated in the $\beta^+$-decay of $^{114}$Sb to $^{114}$Sn than directly by the $(p,p')$ reaction.\ [3025keV:]{} Two excited states are adopted at an energy of 3025keV with a spin-parity assignment of $J^{\pi} = 2,3^+$ and $J^{\pi} = 0^+$, respectively. In addition, another close-lying state at 3028keV with $J^{\pi} = 2,3^+$ has been reported[@ENSDF]. In fact, Ref.[@Guaz04a] reported the $J^{\pi} = 0^+$ assignment for $E_x = 3028(3)$keV. Especially, the new $\gamma$ transition to the $0^+_2$ state for the state at 3024.9(2)keV excludes a $J^{\pi} = 0^+$ assignment.\ [3397keV:]{} Ref.[@Guaz04a] assigned $J^{\pi} = 6^+$ to the excited state at 3397(3)keV. In addition, a possible $4^-$ state is adopted at an energy of 3396.9(5)keV which was actually reported to have a $(3,4)^-$ assignment[@Schim94a; @Wir95a]. For the latter a $\gamma$-decay with $E_{\gamma} = 1122.3$keV to the $3^-_1$ state was observed to be of mixed $M1+E2$ character. New $\gamma$-decays of this level to the $2^+_1$, $2^+_2$ and $2^+_3$ state have been observed, which favor a $J^{\pi} = 3^-$ spin-parity assignment, see Table \[tab:114Sn\_tau\]. A comparison of the theoretical $L = 3$ and the experimentally measured angular distribution shown in Ref.[@Guaz04a] might also support a $J^{\pi} = 3^-$ assignment which is tentatively given in Table \[tab:114Sn\_tau\].\ [3452keV:]{} Two spin-parity assignments were previously reported for a possible state at $E_x \approx 3452$keV, either $6^+$[@ENSDF] or $0^+$[@Guaz04a]. The latter is based on a rather clear $L = 0$ transfer seen in the $(p,t)$ reaction. However, it is the $L = 0$ transfer with the smallest cross section observed in Ref.[@Guaz04a]. The decay of this state to the ground state of $^{114}$Sn has also been observed in an old $(n,n'\gamma)$ experiment[@ENSDF]. The $6^+$ assignment is, thus, odd. A $\gamma$ decay with an energy of 3452.1(2)keV leading to the ground state was also observed in our experiment. It is not possible to originate from a $J^{\pi} = 0^+$ state. Based on our data we propose a $J^{\pi} = 1^-$ spin-parity assignment. As will be discussed in Sec.\[sec:qoc\], this state is a suitable candidate for the quadrupole-octupole coupled (QOC) $1^-$ state. We cannot exclude the existence of a doublet at this energy and, thus, the existence of an additional $0^+$ state.\ [3781keV:]{} This $2^+$ state is strongly populated in the $\beta^+$-decay of $^{114}$Sb to $^{114}$Sn[@Wig76a], i.e. in the $(p,n)$ reaction and is, thus, also clearly seen in our $\gamma\gamma$-coincidence data. However, it is not observed at all in our $p\gamma$-coincidence data, i.e. it is not strongly excited in the $(p,p')$ reaction. discussion ========== Shape coexistence and multiphonon quadrupole states in $^{112,114}$Sn --------------------------------------------------------------------- ![\[fig:sc\_112Sn\]Normal and 2p-2h intruder states in $^{112}$Sn. The $B(E2)\downarrow$ values are given in W.u.. The data for the intruder $6^+$ state has been taken from Ref.[@Gangu01a]. The 0p-4h ($^{108}$Pd) and 4p-0h ($^{116}$Xe) Yrast sequences are also shown and have been shifted to the energy of the $0^+_2$ state of $^{112}$Sn. The $B(E2;2^+_1 \rightarrow 0^+_1)$ value is taken from Ref.[@Jungcl11a]. The data for $^{108}$Pd and $^{116}$Xe was taken from Ref.[@ENSDF].](level_coex_112Sn.eps){width="1\linewidth"} Shape coexistence has been discussed in the Sn isotopes for decades, see the review articles[@Wood92a; @Heyd11a], and its existence has been mainly attributed to proton 2p-2h excitations across the $Z = 50$ shell closure. Many experiments have been performed to study the positive-parity and negative-parity intruder bands in $^{112,114}$Sn, see, [e.g.]{}, Refs.[@Hara88a; @Schim92a; @Gabl01a; @Gangu01a]. We were now able to establish the $0^+$, $2^+$ and $4^+$ members of the positive-parity intruder configuration in $^{112,114}$Sn, see Figs.\[fig:sc\_112Sn\] and \[fig:sc\_114Sn\]. These are clearly identified in terms of their interband transitions which are by far the most collective $E2$ transitions. For instance, the $B(E2;2^+_3 \rightarrow 0^+_1)$ and $B(E2;2^+_3 \rightarrow 2^+_1)$ values amount only to 0.082(13)W.u. and 0.23(6)W.u. in $^{112}$Sn, respectively. The transition to the $2^+_1$ as well as the $B(E2)$ value to the $2^+_2$ are also weak for the intruder $4^+_{\mathrm{intr.}}$ state. However, strong $E2$ transitions are observed to both the $6^+_1$ and $4^+_2$ state indicating that these states might be structurally related, see Fig.\[fig:sc\_112Sn\]. It must be mentioned that for the latter transition a multipole-mixing ratio needs to be determined to make a final statement. Interestingly, a very collective $E2$ strength of $B(E2;6^+_{\mathrm{intr.}} \rightarrow 4^+_{2784\,\mathrm{keV}}) = 68(22)$W.u. is calculated based on the data of Refs.[@Gangu01a; @ENSDF]. The $B(E2)\downarrow$ to the $4^+_2$ is 9(4)W.u. and, thus, comparable to the $B(E2; 4^+_{\mathrm{intr.}} \rightarrow 6^+_1)$ value, see Fig.\[fig:sc\_112Sn\].\ ![\[fig:sc\_114Sn\]Same as Fig.\[fig:sc\_112Sn\] but for $^{114}$Sn. The $B(E2;2^+_1 \rightarrow 0^+_1)$ value is taken from Ref.[@Jungcl11a] and the data for the intruder $6^+$ state was taken from Ref.[@Gabl01a]. The $^{110}$Pd and $^{118}$Xe was taken from Ref.[@ENSDF].](level_coex_114Sn.eps){width="1\linewidth"} In $^{114}$Sn, the $4^+_{\mathrm{intr.}}$ state corresponds to the $4^+_2$ state at 2613.7(4)keV, see Fig.\[fig:sc\_114Sn\]. Therefore, the decays seen and discussed in $^{112}$Sn are not observed. However, in contrast to $^{112}$Sn, the $B(E2;4^+_{\mathrm{intr.}} \rightarrow 2^+_1) = 6.6(10)$W.u. is comparably large. Interestingly, this is also true for the $B(E2;2^+_{\mathrm{intr.}} \rightarrow 2^+_1)$ value which is $\leq 8$W.u.. The $6^+_{\mathrm{intr.}}$ state has been identified at an energy of 3188keV[@Gabl01a], see also Fig.\[fig:sc\_114Sn\]. The $B(E2; 6^+_{\mathrm{intr.}} \rightarrow 4^+_3) = 18.9(12)$W.u. is also comparably large. Still, it is at least a factor of 2 smaller than the value calculated for the corresponding transition to the $4^+$ state at 2783.5(2)keV in $^{112}$Sn.\ [cccccc]{} $J^{\pi}_i$ & $E_x$ & $E_{x,\mathrm{IBM}}$ & $J^{\pi}_f$ & $B(E2)_{\mathrm{exp.}}\downarrow$ & $B(E2)_{\mathrm{IBM}}\downarrow$\ & \[MeV\] & \[MeV\] & & \[W.u.\] & \[W.u.\]\ \ $2^+_1$ & 1.30 & 1.30 & $0^+_1$ & 11.1(7) & 11\ $4^+_1$ & 2.19 & 2.28 & $2^+_1$ & 5.9(5) & 19\ $0^+_2$ & 1.95 & 1.99 & $2^+_1$ & 23.2(8) & 21\ $2^+_3$ & 2.45 & 2.54 & $0^+_1$ & 0.023(9) & 0.004\ & & & $2^+_1$ & 3(2) & 17\ & & & $2^+_2$ & - & 8\ \ $0^+_3$ & 2.16 & 2.15 & $2^+_1$ & $\leq 5$ & 2\ $2^+_2$ & 2.24 & 2.46 & $0^+_1$ & $\leq 0.12$ & 0.04\ & & & $2^+_1$ & $\leq 8$ & 2\ & & & $0^+_2$ & $\leq 44$ & 31\ & & & $0^+_3$ & - & 27\ $4^+_2$ & 2.61 & 3.00 & $2^+_1$ & 6.6(10) & 0.2\ & & & $4^+_1$ & 1.6(10) & 0.06\ & & & $2^+_2$ & 62(25) & 85\ $6^+$ & 3.19 & 3.63 & $4^+_1$ & 1.68(9) & 1.5\ & & & $4^+_2$ & 97(5) & 93\ & & & $4^+_3$ & 18.9(12)& 0.7\ To test the mixing hypothesis between the normal and intruder configuration, we performed $sd$ IBM-2 calculations using the computer code NPBOS[@Otsu85a] for $^{114}$Sn, see Table\[tab:ibm\_114sn\]. As can be seen in Figs.\[fig:sc\_112Sn\] and \[fig:sc\_114Sn\], the observed reduced $E2$ transition strengths are closer to the corresponding quantities observed in the Pd isotopes, i.e. the 0p-4h nucleus. Similar observations were made for the $B(E2;2^+_{\mathrm{intr.}} \rightarrow 0^+_{\mathrm{intr.}})$ values in the Cd isotopes with $N = 62 - 68$ which are comparable to the $B(E2;2^+_1 \rightarrow 0^+_1)$ values observed in the corresponding Ru isotopes[@Hey92a]. We, therefore, adopted the IBM-2 parameters of Ref.[@Kim96a] determined for $^{110}$Pd to describe the intruder configuration. For the normal configuration we slightly adjusted the parameters which were reported in Ref.[@Singh97a], see the caption of Table \[tab:ibm\_114sn\]. As can be seen the $B(E2)$ strengths in the intruder band are nicely described by the model. Also the “interband” transitions are fairly well described. Of course, deviations are observed, see, [e.g.]{}, the $B(E2;4^+_1 \rightarrow 2^+_1)$ and $B(E2;2^+_3 \rightarrow 2^+_1)$ values. It is tempting to speculate that these deviations arise from mixing effects which are not covered by the simplified IBM approach or if the mixing parameter $\beta$ is kept at 0. Clearly, certain configurations will be outside of the $sd$ IBM-2 model space. In fact, indications of mixing effects between the two $4^+$ states at 2613.7(4)keV and 2764.9(5)keV were already proposed based on their excitation energies, see the review article[@Wood92a]. These are now further strengthened by reduced $E2$ transition strengths. Still, the decay rates of the $0^+_2$ and $0^+_3$ as well as the decay rate to the $0^+_2$ are perfectly described using the IBM approach. These two $0^+$ states are almost perfectly mixed. The $0^+_3$ has a slightly larger admixture of the intruder configuration. It is, thus, not surprising that the two $B(E2; 2^+_{\mathrm{intr.}} \rightarrow 0^+_{i})$ values ($i =2,3$) add up to the $B(E2;2^+_1 \rightarrow 0^+_1)$ value observed in $^{110}$Pd, compare Fig.\[fig:sc\_114Sn\]. Unfortunately, the decay $2^+_2 \rightarrow 0^+_3$ has not been observed in $^{114}$Sn so far. If the scenario drawn is true a $\gamma$-decay intensity $I_{\gamma}$ of about 0.002$\%$ would be expected. Indeed, the $2^+_2 \rightarrow 0^+_3$ was recently observed for the case of $^{116}$Sn[@Pore17a], which was populated through the $\beta^-$-decay of $^{116m1}$In. Here, an $I_{\gamma}$ of 0.0091(6)$\%$ was determined and a very collective $B(E2)\downarrow$ of 100(8)W.u. was calculated. Given the adopted value is correct, the $B(E2;2^+_1 \rightarrow 0^+_1)$ is 41(6)W.u. in $^{112}$Pd. A summed $B(E2; 2^+_{\mathrm{intr.}} \rightarrow 0^+_i)$ strength of 144(8)W.u. would consequently not be expected if the intruder configuration would solely result from $^{112}$Pd in $^{116}$Sn. Unfortunately, lifetimes of the $4^+_{\mathrm{intr.}}$ and $6^+_{\mathrm{intr.}}$ are not known. A stringent comparison is presently not possible. It is still interesting to note that in contrast to $^{112}$Sn the $B(E2;0^+_2 \rightarrow 2^+_1) = 18(2)$W.u. is similar to the one observed in $^{114}$Sn. For a deeper understanding of mixing between possible two-phonon quadrupole states and intruder states further investigations are clearly necessary.\ [cccc]{} $J^{\pi}_i$ & $E_x$ & $J_f^{\pi}$ & $B(E2)_{\mathrm{exp.}}$\ & \[keV\] & & \[W.u.\]\ \ $0^+$ & 2617.4(3) & $2^+_1$ & $\leq 2$\ & & $2^+_2$ & $\leq 7$\ $2^+$ & 2720.6(2) & $0^+_1$ & $\leq 0.02$\ & & $2^+_1$ & $0.06^{+0.08}_{-0.01}$\ & & $2^+_2$ & $\leq 4.3$\ & & $0^+_2$ & 3.3(12)\ $3^+$ & 2755.2(3) & $2^+_1$ & $\leq 0.004$\ & & $2^+_2$ & $\leq 12$\ & & $4^+_1$ & $\leq 45$\ & & $4^+_2$ & $\leq 0.2$\ $4^+$ & 2783.5(2) & $2^+_1$ & 5.1(6)\ & & $4^+_1$ & $\leq 35$\ \ $4^+$ & 2859.2(5) & $2^+_1$ & 2.8(4)\ & & $4^+_1$ & $\leq 10$\ & & $2^+_2$ & $< 5$\ & & $2^+_3$ & $< 46$\ $2^+$ & 2943.4(2) & $0^+_1$ & $< 0.001$\ & & $2^+_1$ & $\leq 0.3$\ & & $0^+_2$ & $\leq 0.4$\ & & $2^+_2$ & $\leq 0.9$\ & & $0^+_4$ & $\leq 1.6$\ & & $2^+_3$ & $\leq 5.2$\ $0^+$ & 3028.0(2) & $2^+_1$ & 1.7(7)\ & & $2^+_2$ & 1.4(10)\ & & $2^+_3$ & 16(8)\ As stressed in the introduction, candidates for three-phonon quadrupole states were identified in $^{124}$Sn[@Band05a]. Possible candidates in $^{112,114}$Sn are given in Table \[tab:multph\_sn\]. The experimentally calculated $B(E2)$ values are indeed similar to the values which were observed in $^{124}$Sn, i.e. the forbidden transitions are approximately weaker by one order of magnitude compared to the transitions leading to the two-phonon states. Note, that for most states only upper limits could be determined. We also have to keep in mind that the two-phonon states are not pure, compare Table \[tab:ibm\_114sn\]. In addition, at least one other configuration is present in $^{114}$Sn, i.e. $(3s_{1/2})^{-1}(1g_{7/2})^{1}$ leading to $J^{\pi} = 3^+$ and $4^+$. The $B(M1;4^+ \rightarrow 3^+_1) \approx 0.1$$\mu_N^2$ between the states at 2764.9(5)keV and 2514.4(2)keV is the largest value observed and might be caused by the corresponding spin-flip transition. Whether the proposed $3^+$ and $4^+$ three-phonon quadrupole members have the same admixture is not clear. Clearly, the structure of the ground state, i.e. the underlying single-particle structure changes from $^{112}$Sn to $^{114}$Sn. The main fragments of the single-particle levels in the odd-A Sn isotopes can be seen in Fig.\[fig:odd\_Sn\][(a)]{}. It will be discussed in connection with the negative-parity states. For unambiguous assigments, multipole-mixing ratios $\delta$ need to be determined in the future. For now, we can only conclude that the states discussed do not decay as expected for pure three-phonon states and that the situation in $^{114}$Sn seems to be even more complex. The latter might be attributed to the small $N = 64$ subshell gap which is also seen in Fig.\[fig:odd\_Sn\][(a)]{} and even more pronounced mixing effects. ![\[fig:odd\_Sn\](color online) [(a)]{} systematics of the low-lying single-particle states in the odd Sn isotopes. The data have been compiled from Ref.[@ENSDF]. The experimentally observed states might in good approximation reflect the corresponding single-particle levels, i.e. $1g_{7/2}$ (blue), $2d_{5/2}$ (red), $3s_{1/2}$ (black), $2d_{3/2}$ (purple), and $1h_{11/2}$ (green) . [(b)]{} energy difference between the $1h_{11/2}$ level and the $2d_{5/2}$ as well as the $2g_{7/2}$. [(c)]{} energy evolution of the $3^-_1$ (red diamonds) and $5^-_1$ state (green squares) in the stable even-even Sn isotopes[@ENSDF]. For the discussion, see text.](Sn_oddshells.eps){width="0.98\linewidth"} Quadrupole-octupole coupled states {#sec:qoc} ---------------------------------- ### The $J^{\pi} = 1^-$ candidate ![\[fig:e1\_Sn\](color online) Systematics for the two-phonon $1^-$ state in the stable Sn isotopes. [(a)]{} $B(E1)$$\uparrow$ values determined in Refs.[@Bryss99a; @Pys04a; @Pys06a; @Kum05a] and this work. The grey band corresponds to the $B(E2; 0^+_1 \rightarrow 2^+_1)$ values of Ref.[@Jungcl11a] scaled to the $E1$ strength of $^{116}$Sn, i.e. assuming $B(E1) \sim \beta_2$, while the light-blue band corresponds to the $B(E3; 0^+_1 \rightarrow 3^-_1)$ values of Ref.[@Jon81a] scaled in the same way, i.e. $B(E1) \sim \beta_3$. The RQTBA calculations are shown as well (red dashed line)[@Lit10a; @Lit13b]. Note that these results were scaled with a factor of 0.5, see text. [(b)]{} R(E1)$_{1^-/3^-}$=$B(E1;1^- \rightarrow 0^+_1)$/$B(E1;3^-_1 \rightarrow 2^+_1)$ and a band of expected values for a two-phonon structure in grey, see Ref.[@Piet99a]. [(c)]{} Experimentally determined excitation energies of the two-phonon $1^-$ candidates and the excitation energies of the constituent-phonon states, respectively.](E1_data.eps){width="0.75\linewidth"} The quadrupole-octupole coupled $1^-$ state has been systematically studied in $^{116-124}$Sn using the NRF technique[@Bryss99a]. Later on, $^{112}$Sn was added to the systematics[@Pys04a; @Pys06a] including the aforementioned $(n,n'\gamma)$ experiment[@Kum05a]. The last missing stable Sn isotope, i.e. $^{114}$Sn, was added in this work. Figs.\[fig:e1\_Sn\][(a)]{}-[(c)]{} present the existing and new data. The two-phonon $1^-$ candidate in $^{114}$Sn at 3452keV fits well into the Sn systematics in terms of the [(a)]{} $B(E1;0^+_1 \rightarrow 1^-)$ strength, [(b)]{} $R(E1)$ ratio and [(c)]{} energy systematics. As seen in all other Sn isotopes, its energy lies slightly below the sum energy of the constituent phonons and seems to be more sensitive to the evolution of the excitation energy of the $2^+_1$ state, i.e. a shallow maximum is observed, compare Fig.\[fig:e1\_Sn\][(c)]{}. Note that no other suitable candidate is observed in the relevant energy range. Furthermore, as in all other Sn isotopes, no $\gamma$ decay besides the ground-state decay is observed. It shall be mentioned that the comparably large uncertainty for the lifetime of the $^{114}$Sn candidate is caused by the large $F(\tau)$ value of 0.93(3) and its proximity to unity. The situation in $^{112}$Sn remains unsatisfying. Although the candidate is clearly identified, none of the measurements are found in agreement with any other measurement in terms of the $E1$ strength, see Fig.\[fig:e1\_Sn\][(a)]{}. The value determined by A. Kumar [et al.]{}, however, seems to be too large. It does not match the empirically determined range of two-phonon $B(E1)$ strengths and its proposed connection to the $B(E1;3^-_1 \rightarrow 2^+_1)$ value[@Piet99a], see Fig.\[fig:e1\_Sn\][(b)]{}. Since there were also ambiguities in the efficiency determination for the $^{112}$Sn$(\gamma,\gamma')$ experiment[@Pys04a], no decision in favor of any of the remaining experiments can be made. The $E1$ strength, see Fig.\[fig:e1\_Sn\][(a)]{}, seems to follow the evolution of the $B(E2;0^+_1 \rightarrow 2^+_1)$ value (grey-shaded area) rather than the evolution of the $B(E3;0^+_1 \rightarrow 3^-_1)$ value (light-blue shaded area), which might hint at a common origin of the strength. This should be investigated further using a stringent comparison to theory. Calculations in the framework of the RQTBA have already been performed for $^{112,116,120,124}$Sn[@Lit10a; @Lit13b] and new calculations for $^{114,118,122}$Sn were added in this work. Unfortunately, the RQTBA overestimates the experimental values by a factor of about 2. In Fig.\[fig:e1\_Sn\][(a)]{}, the theoretical results have been scaled with this factor and are shown as a red dashed line. Note, that, even though a two-phonon structure is predicted by the RQTBA, the origin of the strength evolution and especially the strength increase for $^{112}$Sn are presently not understood. One should mention that the RQTBA is a QCD-based self-consistent approach, which does not involve any adjustment of parameters besides the meson masses and meson couplings. These are fitted to global nuclear properties. In this theory, the two-phonon states are considered as tiny structures and an agreement within a factor of 2 is often considered sufficient. However, future advancements of the RQTBA, i.e. the implementation of higher-order correlations are expected to improve the agreement with experiment. Presently, the RQTBA is limited to phonon+2QP configurations. The excitation energy of the $1^-$ two-phonon state is also rather approximate in the RQTBA since it appears to be quite sensitive to the pairing strength. Still, in fair agreement with experiment the QOC $1^-$ is predicted between 2.66MeV ($^{116}$Sn) and 3.98MeV ($^{124}$Sn) in the stable Sn isotopes. The candidates in $^{112}$Sn and $^{114}$Sn are predicted at 3.85MeV and 3.70MeV, respectively.\ Interestingly, additional candidates for the $1^-$ QOC state might be observed around the expected energy, see Tables \[tab:qoc\_112sn\] and \[tab:qoc\_114sn\]. Unfortunately, a conclusive spin-parity assignment is not possible at the moment. A $J^{\pi} = 2^+$ assignment would be possible as well. Still, the additional candidates in $^{112}$Sn and $^{114}$Sn show similar decay properties, i.e. small $B(E1)$ values and $B(E2;(1^-) \rightarrow 3^-_1) \approx 6$W.u.. As will be discussed in the next part, such $B(E2)$ values are in fact expected for the members of the QOC quintuplet. ### The other quintuplet members ------------------ ----------- --------------- ----------- -------------- -------------- ------------------- ------------------- -- $E_x$ $J^{\pi}$ $J^{\pi}_f$ $E_f$ $E_{\gamma}$ $I_{\gamma}$ B(E1)$\downarrow$ B(E2)$\downarrow$ $\mathrm{[keV]}$ \[keV\] \[keV\] \[mW.u.\] \[W.u.\] 1256.5(2) $2^+_1$ $0^+_1$ 0 1256.5(2) 1 - 12.5(7)[^1] 2353.7(2) $3^-_1$ $2^+_1$ 1256.5(2) 1097.2(2) 1 1.13(8) - $2^+_1$ 1256.5(2) 2126.8(2) 0.85(2) 0.120(9) - $2^+_2$ 2150.5(3) 1232.9(2) 0.041(9) 0.030(7) - $(2^+,3,4^+)$ 2917.0(2) 466.5(2) 0.11(2) 1.5(2) - $2^+_1$ 1256.5(2) 2139.9(2) 0.057(13) 0.005(2) - $2^+_2$ 2150.5(2) 1246.1(2) 0.64(3) 0.30(9) - $3^-_1$ 2353.7(2) 1042.4(2) 0.27(5) - 9$^{+3}_{-7}$ $2^+_4$ 2720.6(2) 675.8(2) 0.039(9) 0.11(5) - 3433.4(2) $1^{(-)}$ $0^+_1$ 0 3433.4(2) 1 1.31(15) - $3^-_1$ 2353.7(2) 1144.2(2) 0.70(4) - 29(13) $4^+_2$ 2520.5(2) 977.1(2) 0.27(6) 0.39(18) - $4^+$ 2783.5(2) 714.7(3) $\leq 0.03$ $\leq 0.19$ - $2^+_1$ 1256.5(2) 2296.8(2) 0.83(3) 0.06(2) - $3^+_1$ 2755.2(3) 797.7(3) 0.17(3) 0.30(10) - $0^+_1$ 0 3827.1(2) 0.58(3) 0.040(5) - $2^+_1$ 1256.5(2) 2570.8(2) 0.29(5) 0.066(11) - $3^-_1$ 2353.7(2) 1473.0(7) 0.13(3) - 4.5(11) $0^+_1$ 0 3984.7(3) 0.79(2) 0.08(2) - $3^-_1$ 2353.7(2) 1630.0(3) 0.14(2) - 5(2) $2^+_3$ 2475.5(2) 1507.8(4) 0.07(2) 0.137(95) - ------------------ ----------- --------------- ----------- -------------- -------------- ------------------- ------------------- -- ------------------ ----------- ------------- ----------- --------------- -------------- ------------------- ------------------- -- $E_x$ $J^{\pi}$ $J^{\pi}_f$ $E_f$ $E_{\gamma}$ $I_{\gamma}$ B(E1)$\downarrow$ B(E2)$\downarrow$ $\mathrm{[keV]}$ \[keV\] \[keV\] \[mW.u.\] \[W.u.\] 1299.7(2) $2^+_1$ $0^+_1$ 0 1299.7(2) 1 - 11.1(7)[^2] 2274.5(2) $3^-_1$ $2^+_1$ 1299.7(2) 974.8(2) 1 0.65(8) - $4^+_1$ 2187.3(3) 627.4(2) 0.88(2) $\leq 0.77$ - $3^-_1$ 2274.5(2) 539.9(2) 0.12(3) - $\leq 38$ $2^+_1$ 1299.7(2) 1605.1(4) 0.026(5) 0.0030(14) - $4^+_1$ 2187.3(3) 717.3(2) 0.77(2) 0.7(3) - $3^+_1$ 2514.4(2) 390.2(2) 0.20(3) 1.6(7) - $4^+_2$ 2613.7(4) 290.3(4) 0.011(4) 0.21(12) - $2^+_1$ 1299.7(2) 1925.4(2) 0.920(14) 0.11(2) - $2^+_3$ 2453.8(2) 771.4(4) 0.019(7) 0.04(2) - $3^-$ 2904.9(3) 319.9(4) 0.061(13) - - $2^+_1$ 1299.7(2) 2097.6(2) 0.31(5) 0.06(2) - $2^+_2$ 2238.6(2) 1158.3(2) 0.13(2) 0.14(6) - $3^-_1$ 2274.5(2) 1122.0(4) 0.44(2) - $3^{+11}_{-3}$ $2^+_3$ 2453.8(2) 943.2(2) 0.12(2) 0.24(10) - 3452.1(2) $(1^{-})$ $0^+_1$ 0 3452.1(2) 1 1.6(7) - $2^+_1$ 1299.7(2) 2184.1(2) 0.671(13) 0.06(2) - $0^+_3$ 2155.9(2) 1327.7(3)[^3] 0.094(14) 0.037(11) - $3^-_1$ 2274.5(2) 1209.0(2) 0.235(14) - 5.2(13) $2^+_1$ 1299.7(2) 2214.4(2) 0.76(3) 0.14(6) - $4^+_1$ 2187.4(3) 1327.0(4)[^4] 0.07(2) 0.06(3) - $2^+_2$ 2238.6(2) 1275.0(3) 0.17(3) 0.17(8) - $2^+_1$ 1299.7(2) 2224.5(3) 0.55(3) 0.023(17) - $4^+_1$ 2187.3(3) 1158.3(2) 0.15(3) 0.028(22) - $3^-_1$ 2274.5(2) 1122.0(4) 0.13(2) - 1.2(10) $2^+_3$ 2453.8(2) 943.2(2) 0.17(3) 0.08(6) - 3610.2(4) $5^{(-)}$ $4^+_1$ 2187.3(3) 1422.9(3) 1 1.0(3) - $0^+_1$ 0 3650.1(3) 0.44(3) 0.012(4) - $2^+_1$ 1299.7(2) 2350.3(3) 0.11(2) 0.011(5) - $0^+_3$ 2155.9(2) 1493.7(3) 0.09(2) 0.04(2) - $3^-_1$ 2274.5(2) 1374.6(2) 0.36(6) - 6(2) ------------------ ----------- ------------- ----------- --------------- -------------- ------------------- ------------------- -- If the two-phonon interpretation is correct, a quintuplet of negative-parity states, i.e. $(2^+ \otimes 3^-)_{1^--5^-}$, should be observed close to the sum energy of the constituent-phonon states. In $^{112}$Sn this sum energy is 3.61MeV and in $^{114}$Sn it is 3.57MeV. Furthermore, these coupled states should approximately decay according to the properties of their constituent phonons: $$\begin{aligned} B(E2;(2^+_1 \otimes 3^-_1) \rightarrow 3^-_1) = B(E2;2^+_1 \rightarrow 0^+_1) \\ B(E3;(2^+_1 \otimes 3^-_1) \rightarrow 2^+_1) = B(E3;3^-_1 \rightarrow 0^+_1)\end{aligned}$$ For the case of $^{112}$Sn, candidates have already been proposed in Ref.[@Kum05a]. The possible candidates for $^{112,114}$Sn, which have been observed in this work, are shown in Tables \[tab:qoc\_112sn\] and \[tab:qoc\_114sn\]. Despite the $5^-$ state at 3.1MeV, which was discussed as a member of the multiplet in Ref.[@Kum05a], the $2^-$ and $3^-$ candidates at about 3.4MeV were also observed in our experiment. For the case of the tentatively assigned $2^-$ state at 3396.6(2)keV, the decay to the $3^-_1$ state was observed and the $B(E2)\downarrow$ agrees with the expectations. The rather small $E1$ transition rate to the $2^+_1$ state might hint at a non-negligible $E3$ or $M2$ contribution. However, assuming a pure $E3$ character of this transition results in an unphysically large value. Certainly, multipole mixing ratios should be determined.\ In general, the unnatural-parity states are only weakly excited in the present experiments, i.e. besides the $1^-$ multiplet candidates, mainly candidates for the $3^-$ and $5^-$ state have been observed. The observed $E2$ decay rate from the $5^-$ state at 3497.9(2)keV to the $3^-_1$ state also matches the expectations, while no such $\gamma$-decay branching could be observed for any $3^-$ candidate in $^{112}$Sn. The situation in $^{114}$Sn is reversed. Two $3^-$ states are observed close to the sum energy which decay to the $3^-_1$ state. However, only the $B(E2)$ value of the state at 3397.3(2)keV might allow a QOC interpretation within its comparably large uncertainties. Interestingly, the $B(E1;3^- \rightarrow 2^+_1)$ value is one order of magnitude smaller than the $B(E1;3^-_1 \rightarrow 2^+_1)$ value. This might raise the question whether enhanced $E1$ transitions are indeed expected for QOC candidates or if other mechanisms and structures have to be considered. The $B(E1;5^- \rightarrow 4^+_1)$ of the state at 3610.2(4)keV is as large as the $B(E1;3^-_1 \rightarrow 2^+_1)$ value, see Table \[tab:qoc\_114sn\].\ In this context, it is necessary to mention that for the case of $^{112}$Cd the identification of the $5^-$ multiplet candidates in terms of $E2$ transition rates to the $3^-_1$ state as well as in terms of energy arguments[@Garr99a] was certainly not sufficient. These states exhibited strong neutron single-particle character as was shown in a $(d,p)$ reaction leading to excited states of $^{112}$Cd[@Jami14a]. The $5^-$ state at 2373keV was interpreted to have a dominant $3s_{1/2} \otimes 1h_{11/2}$ configuration and to be a rotational-band member of the $3^-_1$ one-octupole phonon state which might its explain its enhanced $B(E2;5^- \rightarrow 3^-_1)$ value.\ To shed some light, we compiled the excitation energies of the lowest excited states in the odd-$A$ Sn isotopes, see Fig.\[fig:odd\_Sn\][(a)]{}. These states might in good approximation be identified as being the major fragments of the corresponding single-particle levels. In addition, we calculated the energy difference $\Delta E_{s.p.}$ between the $1h_{11/2}$ state and the $2d_{5/2}$ as well as the $1g_{7/2}$ state in Fig.\[fig:odd\_Sn\][(b)]{}, respectively. As can be seen, the energy of $3^-_1$ closely follows both energy differences while the $5^-_1$ state’s excitation energy evolves according to the energy of the $1h_{11/2}$ orbital, compare Fig.\[fig:odd\_Sn\][(c)]{} to the other two panels. Since the $3^-_1$ lies below the $5^-_1$ and the necessary quantities are partly known in $^{112-116}$Sn, the $B(E2;5^-_1 \rightarrow 3^-_1)$ can be at least estimated: $$\begin{aligned} &^{112}\mathrm{Sn}: B(E2;5^-_1 \rightarrow 3^-_1) \leq 8.2\,\mathrm{W.u.}\\ &^{114}\mathrm{Sn}: B(E2;5^-_1 \rightarrow 3^-_1) \leq 38\,\mathrm{W.u.}\\ &^{116}\mathrm{Sn}: B(E2;5^-_1 \rightarrow 3^-_1) = 2.45(12)\,\mathrm{W.u.}\end{aligned}$$ We see that assumming pure configurations of $(2d_{5/2})^{-1}(1h_{11/2})^1$ ($\Delta j = \Delta l = 3$) for the $3^-_1$ and $(3s_{1/2})^{-1}(1h_{11/2})^1$ ($\Delta j = \Delta l = 5$) for the $5^-_1$ state might also generate some $E2$ collectivity between the two levels, i.e. the transfer of a valence neutron from the $3s_{1/2}$ orbital to the $2d_{5/2}$ orbital ($\Delta j = \Delta l = 2$) or vice versa. The fact that the $5^-_1$ excitation energy saturates at approximately 2.2MeV in the more neutron-rich Sn isotopes where the ground state has a $(1h_{11/2})_{0^+}$ configuration further supports this hypothesis. In a nutshell, approximately 2MeV are needed to break a pair and the energy difference between the $1h_{11/2}$ and $3s_{1/2}$ orbital is about 200keV in the more neutron-rich stable Sn isotopes, see Fig.\[fig:odd\_Sn\][(a)*]{}. However, we want to stress that two things were shown in old shell-model calculations employing a finite-range force and the generalized-seniority scheme with $v \leq 4$, i.e. two broken pairs[@Bon85a]. First of all, configurations with $v > 2$ and a finite-range force are needed to reproduce the excitation energy and lifetime of the $5^-_1$ state as was shown for $^{112,116}$Sn. These admixtures are on the order of 20$\%$. Second, to account for the experimentally observed excitation energy of the $3^-_1$, 1p-1h configurations are needed which require excitations through the $^{100}$Sn inert core. These admixtures could be as large as 43$\%$ highlighting the collective nature of the $3^-_1$ state. More modern shell-model calculations which were, however, limited to a $^{100}$Sn inert core support these previous findings[@Guaz04a; @Guaz12a].\ $^{112}$Cd and $^{114}$Sn have both $N = 64$ and similar neutron components are expected to contribute. The $5^-_1$ state critically discussed in $^{112}$Cd might, thus, have the same structure as the $5^-_1$ state in $^{114}$Sn which is certainly not a member of the QOC quintuplet. Based on this discussion, we proposse the tenatively assigned $2^-$ state at 3396.6(2)keV and the $5^-$ state at 3497.9(2)keV as possible members of the quintuplet in $^{112}$Sn. For $^{114}$Sn, only the $3^-$ candidate at 3397.3(2)keV could be identified based on its $\gamma$-decay to the octupole vibrational $3^-_1$ state. We note, however, that the possible $5^-$ candidate at 3610.2(4)keV has only been weakly excited in our experiment. Assuming $B(E2;5^- \rightarrow 3^-_1) = B(E2;2^+_1 \rightarrow 0^+_1)$ leads to an estimated $I_{\gamma}$ of about 40$\%$ for $E_{\gamma} = 1335.7$keV. A strongly excited $3^-$ state at 3524.4(2)keV with a $\gamma$ transition of $E_{\gamma} = 1337.0(2)$keV prevented the detection of the $\gamma$-decay branch to the $3^-_1$ in the present $p\gamma$-coincidence data. No indications were found in our $\gamma\gamma$-coincidence data when applying a gate onto the $\gamma$-decay of the $3^-_1$ state. All other candidates named were cross-checked using the aforementioned $\gamma\gamma$-coincidence data. conclusion ========== We have performed two inelastic proton-scattering experiments at the Institute for Nuclear Physics of the University of Cologne to study excited states in the lightest stable tin isotopes $^{112,114}$Sn. Level lifetimes and $\gamma$-decay branching ratios were determined using the combined spectroscopy setup SONIC@HORUS to acquire $p\gamma$- and $\gamma\gamma$-coincidence data.\ In this publication, we have studied and identified the low-spin members of the proton 2p-2h intruder configuration in $^{112}$Sn and $^{114}$Sn. With respect to their $E2$ transitions strengths, these states are more similar to corresponding states in the 0p-4h Pd nuclei than to the 4p-0h states in the Xe nuclei. Our observations are supported by $sd$ IBM-2 mixing calculcations we performed for $^{114}$Sn. Systematic calculations along these lines will further help to understand shape coexistence and isospin symmetry in the vicinity of the Sn isotopes. Especially, measuring the lifetimes of the $4^+_{\mathrm{intr.}}$ and $6^+_{\mathrm{intr.}}$ states in $^{116}$Sn would be instructive. Then, a stringent comparison of the mixing between and the evolution of the two different configurations, which we sketched for $^{114}$Sn, would be possible.\ Since the two-phonon states, if present at all, already mix with the intruder configuration, the identification of possible three-phonon quadrupole states is even less straight-forward in $^{112,114}$Sn. We have, however, identified possible candidates which decay very similar to the candidates previously proposed in $^{124}$Sn. Still, the $B(E2)$ strengths of these states strongly deviates from the simple vibrational picture. Further systematic experimental and theoretical investigations are highly desirable.\ Besides the coupling of quadrupole phonons, we studied possible members of the QOC quintuplet and identified candidates. The new $J^{\pi} = 1^-$ candidate in $^{114}$Sn fits nicely into the systematics established for the previously studied nuclei. Therefore, it has been clearly shown that our new method provides the means to study such structures in nuclei with low abundance where the amount of target material needed to study these with other methods, ${\it e.g.}$, $(n,n'\gamma)$ or $(\gamma,\gamma')$, is hardly affordable. The situation in $^{112}$Sn, however, remains unsatisfying. No clear agreement is observed between the different measurements. It is desirable to remeasure $^{112}$Sn with the NRF technique to check the efficiency and photon flux ambiguities previously encountered[@Pys04a; @Pys06a]. Disturbingly, the expected $B(E2;1^- \rightarrow 3^-_1)$ has only been observed in a few nuclei up to now not including the Sn isotopes, see Ref.[@Der16a] and references therein. Possible $J^{\pi} = 1^-$ states have been observed close to the sum energy which would decay as expected from a QOC $1^-$ state. Firm spin-parity assignments and further investigations are needed. We have also reported candidates for the $2^-$, $3^-$ and $5^-$ members of the quintuplet making $^{112,114}$Sn the only two Sn nuclei where several members are identified based on their excitation energy and transition strengths. Therefore, future experiments using different experimental probes to identify the complete multiplet in the other Sn isotopes are highly desirable. Candidates have been reported in $^{116}$Sn[@Ram91a] and some negative-parity states at approximately the expected energies are already known in the other Sn isotopes[@ENSDF]. However, lifetime data is missing so far. As shown in this publication, the $(p,p'\gamma)$ DSA coincidence technique could be used to measure these lifetimes. Furthermore, the assignment of spin and parity will be possible with enhanced statistics as well as with the new and improved SONIC spectrometer based on $p\gamma$-angular correlations. We gratefully acknowledge the help of the accelerator staff at the IKP Cologne. We want to thank F. Iachello, J.Jolie, C.Petrache, A. Schreckling, and N.Warr for helpful and stimulating discussions as well as A.Blazhev and K.-O.Zell for the target preparation. We also want to thank H.-W.Becker for his support during the RBS measurements as well as the NNDC for a consistency check of our data prior to publication. E.L. acknowledges support from US-NSF grant PHY-1404343 and NSF CAREER grant $\#$1654379. This work was supported by the Deutsche Forschungsgemeinschaft under Contract ZI 510/7-1. [^1]: Ref.[@Jungcl11a] [^2]: Ref.[@Jungcl11a] [^3]: No clear assignment possible, see Table\[tab:114Sn\_tau\]. [^4]: No clear assignment possible, see Table\[tab:114Sn\_tau\].
--- abstract: 'We discuss the evidence for the presence of QCD saturation effects in the data collected in d+Au collisions at RHIC. In particular we focus our analysis on forward hadron yields and azimuthal correlations. Approaches alternative to the CGC description of these two observables are discussed in parallel.' address: - 'Institut de Physique Théorique, CEA/Saclay, 91191 Gif-sur-Yvette cedex, France' - 'Physics Department, Theory Unit, CERN, 1211 Genève 23, Switzerland' author: - 'Javier L. Albacete' - Cyrille Marquet title: 'Single and double inclusive particle production in d+Au collisions at RHIC, leading twist and beyond' --- Relativistic Heavy Ion Collisions ,Color Glass Condensate Introduction {#intro} ============ The large amount of experimental data collected at RHIC during the last decade over a wide kinematical range allows to explore novel QCD effects. Indeed RHIC measurements present a number of features which are well described within the Color Glass Condensate effective theory of QCD at high energies (see e.g. [@Gelis:2010nm; @Weigert:2005us] and references therein). The main physical ingredient in the CGC is the inclusion of unitarity effects through the proper consideration of both non-linear recombination effects in the quantum evolution of hadronic wave functions and multiple scatterings at the level of particle production. Such effects are expected to be relevant when partons with a small enough energy fraction $x$ are probed in the nuclear (or, in full generality, hadronic) wave function. The CGC formalism itself is only reliable at small $x$, in that kinematic regime gluon densities are large and gluon self-interactions become highly probable, thus taming, or saturating, further growth of the gluon occupation numbers. While the need for unitarity effects comprised in the CGC is, at a theoretical level, clear, the real challenge from a phenomenological point of view is to assess to what extent they are present in available data. Such is a difficult task, since different physical mechanisms concur in data, and also because the limit of asymptotically high energy in which the CGC formalism is developed may not be realized in current experiments. Notwithstanding these difficulties, we argue below that RHIC data [@Arsene:2004ux; @Adams:2006uz; @Braidot:2010zh] offer compelling evidence for the presence of saturation effects. Such claim is based on the successful simultaneous description of the suppression of particle production [@Albacete:2010bs] and azimuthal correlations [@Albacete:2010pg] at forward rapidities in d+Au collisions compared to p+p collisions, using the most up-to-date theoretical tools available in the CGC approach. We focus on forward particle production for the following reason: such processes are sensitive only to high-momentum partons inside one of the colliding hadron, which appears dilute, while mainly small-momentum partons inside the other dense hadron contribute to the scattering. Since the high$-x$ part of a proton wave function is well understood in perturbative QCD, forward particle production in high-energy d+Au (or generically p+A) collisions is ideal to investigate the small$-x$ part of the nucleus wave function. In the case of single-inclusive hadron production $pA\!\to\!hX$, denoting $p_{\perp}$ and $y$ the transverse momentum and rapidity of the final state particles, the partons that can contribute to the cross section have a fraction of longitudinal momentum bounded from below, by $x_p$ (for partons from the proton wave function) and $x_A$ (for partons from the nucleus wave function) given by $$x_p=x_F\ ,\hspace{0.5cm}x_A=x_F\ e^{-2y}\ , \label{kin1}$$ where we introduced the Feynman variable $x_F=\|p_{\perp}\|e^{y}/\sqrt{s_{NN}}$ with $\sqrt{s_{NN}}$ the collision energy per nucleon. With $\sqrt{s_{NN}}\gg\|p_{\perp}\|$ and forward rapidities $y\!>\!0,$ the process features $x_p\!\lesssim\!1$ and $x_A\!\ll\!1,$ meaning that the scattering involves on the proton side, quantum fluctuations well understood in QCD, and on the nucleus side, quantum fluctuations whose non-linear QCD dynamics can be studied. In the case of double-inclusive hadron production $pA\!\to\!h_1h_2X$, denoting $p_{1\perp},$ $p_{2\perp}$ and $y_1,$ $y_2$ the transverse momenta and rapidities of the final-state particles, the Feynman variables are $x_i=\|p_{i\perp}\|e^{y_i}/\sqrt{s_{NN}}$ and $x_p$ and $x_A$ read $$x_p=x_1+x_2\ ,\hspace{0.5cm} x_A=x_1\ e^{-2y_1}+x_2\ e^{-2y_2}\ . \label{kin2}$$ Therefore, only the forward-forward case is sensitive to values of $x$ as small as in the single particle production case: $x_p\!\lesssim\!1$ and $x_A\!\ll\!1$. The central-forward measurement does not probe such kinematics: moving one particle forward increases significantly the value of $x_p$ compared to the central-central case (for which $x_p=x_A=\|p_{\perp}\|/\sqrt{s_{NN}}$), but decreases $x_A$ only marginally. For this reason we shall refer to these two situations as mid-rapidity ones. The approximations made in CGC calculations apply best with forward-rapidity observables, at RHIC mid-rapidity ones are contaminated too much by large-$x$ physics to be treated at a quantitative level. Saturation-based approaches were the only ones to correctly predict the suppression of forward particle production [@Kharzeev:2003wz; @Albacete:2003iq] and the azimuthal decorrelation of forward hadron pairs [@Marquet:2007vb]. In the following, we also comment on alternative mechanisms that successfully describe mid-rapidity particle production in d+Au collisions. We note that, while some of these approaches are also able to describe, a posteriori, the suppression of forward yields, we are not aware of any formalism that can also describe the azimuthal decorrelation of forward hadron pairs. Nuclear modifications at mid-rapidity ===================================== In relativistic heavy-ion collisions, nuclear effects on single particle production are typically evaluated in terms of ratios of particle yields called nuclear modification factors: $$R^h_{pA}=\frac{dN^{pA\to hX}/dyd^2p_\perp}{N_{coll}\ dN^{pp\to hX}/dyd^2p_\perp}\ ,$$ where $N_{coll}$ is the number of nucleon-nucleon collisions in the p+A collision. If high-energy nuclear reactions were a mere incoherent superposition of nucleon-nucleon collisions, then the observed $R_{pA}$ would be equal to unity. However, RHIC measurements in d+Au collisions (or peripheral Au+Au collisions) [@Arsene:2004ux; @Adams:2006uz] exhibit two opposite regimes: at mid rapidities the nuclear modification factors exhibit an enhancement in particle production at intermediate momenta $\|p_\perp\|\sim 2\div 4$ GeV. In turn, a suppression at smaller momenta is observed. However, at forward rapidities the Cronin enhancement disappears, turning into an almost homogeneous suppression. ![RHIC experimental results for $R_{dAu}^{\pi^0}$ at $\eta=0$ compared with different calculations. From left to right: comparison with EPS09 parametrization [@Eskola:2009uj], a Glauber-Eikonal calculation [@Accardi:2003jh], and the CGC approach of [@Kharzeev:2004yx].[]{data-label="eta0"}](Phenix2007pi0.eps "fig:"){height="3.9cm"} ![RHIC experimental results for $R_{dAu}^{\pi^0}$ at $\eta=0$ compared with different calculations. From left to right: comparison with EPS09 parametrization [@Eskola:2009uj], a Glauber-Eikonal calculation [@Accardi:2003jh], and the CGC approach of [@Kharzeev:2004yx].[]{data-label="eta0"}](FIG3bPHENIXpi0.eps "fig:"){height="3.8cm"} ![RHIC experimental results for $R_{dAu}^{\pi^0}$ at $\eta=0$ compared with different calculations. From left to right: comparison with EPS09 parametrization [@Eskola:2009uj], a Glauber-Eikonal calculation [@Accardi:2003jh], and the CGC approach of [@Kharzeev:2004yx].[]{data-label="eta0"}](rda0.eps "fig:"){height="4.0cm"} Single-hadron production data at mid-rapidity have been successfully analyzed through different formalisms and techniques. Below we sketch an incomplete, but representative list of the variety in the theory spectrum: - [Leading-twist perturbation theory]{}. The assumption is that standard collinear factorization holds in nuclear reactions, meaning that highly-virtual partons in nuclei behave independently as they do in protons. For each parton species $i$ the nuclear parton distribution functions are taken to be proportional to that of a proton: $f_i^A(x,Q^2)=R_i^A(x,Q^2)\,f_i^N(x,Q^2)$. The proportionality factors $R_i^A(x,Q^2)$ are fitted, in part, to available d+Au data and in some cases, as in the EPS parametrization [@Eskola:2009uj], evolved according to DGLAP evolution. The resulting data description is displayed in a. - [Glauber-eikonal multiple scatterings]{}. This approach takes into account power corrections to the leading-twist approximation. It relies on a resummation of incoherent multiple scatterings. Typically, this results in a momentum broadening of the scattered parton which is responsible for the Cronin enhancement and, due to unitarity constraints, to a depletion of particle production at small transverse momenta, in agreement with the qualitative features of the data as can be seen in b. Performing the complete resummation including energy-momentum conservation is a challenging task. Sometimes, a detour of the strict calculation is taken by resorting to unintegrated parton distributions which include information about the intrinsic transverse momentum of the partons $k_0$, and the average transverse momentum gained during the interaction with the nucleus $\Delta k$: $f_i^A(x,Q^2)\rightarrow F_i^A(x,Q^2,<\!k_0^2\!>\!+\!<\!\Delta k^2\!>(\sqrt{s},b,p_\perp)$. While the intrinsic $k_0$ is adjusted in p+p collisions, the gained transverse momentum is let to depend on the collision energy, centrality and $p_\perp$ of the detected hadron. - [Color Glass Condensate]{}. The CGC approach resums all power corrections which are dominant in the high-energy/small-$x$ limit. It relies on two main assumptions: the scattering process is fully coherent and the propagation of the leading parton through the nucleus is eikonal, i.e. the momentum transfered during the collision is only transverse. Then, non-linearities or saturation effects can be taken into account either at the semiclassical level or, more accurately, including the quantum corrections embodied in the JIMWLK evolution equation. The work presented in c relies on a combination of both, with quantum corrections modeled according to analytical estimates [@Kharzeev:2004yx]. Note that in the CGC framework, the saturation scale $Q_s$ which characterizes the onset of non-linear effects in the nuclear wave function, also determines both the intrinsic transverse momentum and the amount of it gained during the collision. In the small-$x$ limit, it is not possible to distinguish saturation from multiple scatterings, such a separation would be frame dependent. Both become important when a large gluon density is reached, and including one without the other is not consistent. Simply by looking at Fig. 1, one concludes that the three different approaches above offer a comparably good description of the data, so it is difficult to extract any clean conclusion about the physical origin of the Cronin enhancement. This is probably due to the kinematic region probed in these measurements. For a hadron momentum of $\|p_\perp\|\sim 2$ GeV, one is sensitive to $(x_p\sim)x_A\sim 0.01\div0.1$. In this region different physical mechanisms concur, so neither of the physical assumptions underlying the approaches above is comletely fulfilled. Indeed, the coherence length, estimated to be $l_c\sim 1/(2m_N\,x_A)\sim 1 \div 10$ fm is, on average, smaller than the nuclear length, so the fully coherent treatment of the collision implicit in the CGC is not completely justified, and large-$x$ corrections are expected to be relevant. Moreover, both the coherent or incoherent treatments of the collision in [@Accardi:2003jh; @Kharzeev:2004yx] need to invoke the presence of an intrinsic scale, presumably of non-perturbative origin, of the order of 1 Gev in order to reproduce the data, whose dynamical origin is not evident at all. ![The coincidence probability at mid-rapidity as a function of $\Delta\phi$. RHIC data show that in the central-central case the away-side peak is similar in d+Au and p+p collisions. In the central-forward case, the Glauber-eikonal (left plot from [@Qiu:2004da]) and CGC (right plot from [@Kharzeev:2004bw]) calculations predict that the away-side peak is suppressed in d+Au compared to p+p collisions, in agreement with later data.[]{data-label="fwd-cent"}](fwd-cent-1.eps "fig:"){height="6.2cm"} ![The coincidence probability at mid-rapidity as a function of $\Delta\phi$. RHIC data show that in the central-central case the away-side peak is similar in d+Au and p+p collisions. In the central-forward case, the Glauber-eikonal (left plot from [@Qiu:2004da]) and CGC (right plot from [@Kharzeev:2004bw]) calculations predict that the away-side peak is suppressed in d+Au compared to p+p collisions, in agreement with later data.[]{data-label="fwd-cent"}](fwd-cent-2.eps "fig:"){height="6.3cm"} More insight is gained with di-hadron correlations. In particular, let us focus on the $\Delta\phi$ dependence of the double-inclusive hadron spectrum, where $\Delta\phi$ is the difference between the azimuthal angles of the measured particles $h_1$ and $h_2$. Nuclear effects on di-hadron correlations are typically evaluated in terms of the coincidence probability to, given a trigger particle in a certain momentum range, produce an associated particle in another momentum range. In a p+p or p+A collision, the coincidence probability is given by $CP(\Delta\phi)=N_{pair}(\Delta\phi)/N_{trig}$ with $$N_{pair}(\Delta\phi)=\int\limits_{y_i,\|p_{i\perp}\|}\frac{dN^{pA\to h_1 h_2 X}}{d^3p_1 d^3p_2}\ ,\quad N_{trig}=\int\limits_{y,\ p_\perp}\frac{dN^{pA\to hX}}{d^3p}\ . \label{kinint}$$ First measurements were performed at RHIC at mid-rapidity by the PHENIX and STAR collaborations [@Adams:2006uz; @Adler:2006hi]. In the central-central case, the coincidence probability features a near-side peak around $\Delta\phi=0,$ when both measured particles belong to the same mini-jet, and an away-side peak around $\Delta\phi=\pi,$ corresponding to hadrons produced back-to-back. In the central-forward case, there is naturally no near-side peak. Either in p+p or d+Au collisions, the sizeable width of the away-side peak cannot be described within the leading-twist collinear factorization framework. This indicates that, while it may not be obvious in single particle production, power corrections are important when $\|p_\perp\|\sim 2$ GeV. At such low transverse momenta, collinear factorization does not provide a global picture of particle production at RHIC, even at mid-rapidity. On the contrary, both the Glauber multiple scattering [@Qiu:2004da] and CGC [@Kharzeev:2004bw] approaches can qualitatively describe the data, including the depletion of the away-side peak in d+Au collisions when going from central-central to central-forward production. This is illustrated in . Such a depletion does not occur in p+p collisions, it is due to nuclear-enhanced power corrections, and therefore the p+A to p+p ratio of the integrated coincidence probabilites $$I_{pA}=\frac{\int d\phi\ CP_{pA}(\Delta\phi)}{\int d\phi\ CP_{pp}(\Delta\phi)}$$ is below unity. In , recent PHENIX data on $I_{dAu}$ are displayed as a function of centrality. At the moment, since $x_A$ is not that small, it is not clear whether the mechanism for this suppression is due to saturation effects rather than incoherent multiple scatterings. ![Central-forward preliminary $I_{dAu}$ data as a function of centrality [@Meredith:2009fp]. In central collisions, the integral of the coincidence probabilty is about half that in p+p collisions, reflecting the depletion of the away-side peak. Collinear factorization cannot reproduce this behavior, while both Glauber-eikonal and CGC calculations predicted it.[]{data-label="IdA"}](phenix-IdA2.eps){height="6cm"} Moving forward ============== As outlined in the introduction, data collected in the deuteron fragmentation region offer a cleaner opportunity to explore saturation effects. Let us first explain our implementation of the CGC framework. The CGC is equipped with a set of non-linear renormalization group equations to describe the evolution of hadronic wave functions towards small-$x$. In the large-$N_c$ limit, they reduce to the BK equation [@Balitsky:1995ub; @Kovchegov:1999yj]. The recent determination of running coupling corrections to the original leading-log equations [@Balitsky:2006wa; @Kovchegov:2006vj] has proven an essential step in promoting the BK equation to a phenomenological tool. Indeed, the running coupling BK equation (rcBK) has been employed to successfully describe inclusive structure functions in e+p scattering [@Albacete:2009fh] and also the energy and multiplicity dependence of total hadron multiplicities in Au+Au collisions at RHIC [@Albacete:2007sm]. The rcBK equation reads $$\begin{aligned} \frac{\mathcal{N}(r,Y)}{\partial\ln(1/x)}=\int d^2{\bf r_1}\ K^{{\rm run}}({\bf r},{\bf r_1},{\bf r_2}) \left[\mathcal{N}(r_1,Y)+\mathcal{N}(r_2,Y) \right.\nonumber\\\left.-\mathcal{N}(r,Y)-\mathcal{N}(r_1,Y)\,\mathcal{N}(r_2,Y)\right]\ , \label{bk1}\end{aligned}$$ where ${\bf r_2}={\bf r}-{\bf r_1}$ (we use the notation $v\equiv \|{\bf v}\|$ for two-dimensional vectors in and ). $\mathcal{N}(r,Y)$ is the dipole scattering amplitude in the fundamental representation, with $Y=\ln(x_0/x)$ the rapidity and $r$ the dipole transverse size. It turns out that the evolution kernel $$K^{{\rm run}}({\bf r},{\bf r_1},{\bf r_2})=\frac{N_c\,\alpha_s(r^2)}{2\pi^2} \left[\frac{1}{r_1^2}\left(\frac{\alpha_s(r_1^2)}{\alpha_s(r_2^2)}-1\right)+ \frac{r^2}{r_1^2\,r_2^2}+\frac{1}{r_2^2}\left(\frac{\alpha_s(r_2^2)}{\alpha_s(r_1^2)}-1\right) \right] \label{kbal}$$ proposed in [@Balitsky:2006wa] minimizes the role of higher order corrections, making it better suited for phenomenological applications. Detailed discussions about other prescriptions proposed to define the running coupling kernel, and about the numerical method to solve the rcBK equation can be found in [@Albacete:2007yr]. needs to be suplemented with initial conditions, which we choose to be of the McLerran-Venugopalan type: $$\mathcal{N}(r,Y\!=\!0)= 1-\exp\left[-\frac{r^2\,\bar{Q}_{s0}^2}{4}\,\ln\left(\frac{1}{\Lambda\,r}+e\right)\right]\ ,$$ where $\Lambda=0.241$ GeV. This introduces two free parameters: the value $x_0$ where the evolution starts and the initial saturation scale felt by quarks $\bar{Q}_{s0}$. Nuclear modification factors ---------------------------- According to Ref. [@Dumitru:2005gt], the differential cross section for forward hadron production in p+A collisions is given by $$\begin{aligned} \frac{dN^{pA\to hX}}{dy\,d^2p_\perp}=K\sum_{q}\int_{x_F}^1\,\frac{dz}{z^2}\ \left[x_1f_{q\,/\,p}(\tilde{x}_p,p_\perp^2)\ F\left(\tilde{x}_A,\frac{p_\perp}{z}\right)\ D_{h\,/\,q}(z,p_\perp^2)\right.\nonumber\\ +\left. x_1f_{g\,/\,p}(\tilde{x}_p,p_\perp^2)\ \tilde{F}\left(\tilde{x}_A,\frac{p_\perp}{z}\right)\,D_{h\,/\,g}(z,p_\perp^2)\right] \label{hyb}\ ,\end{aligned}$$ where the unintegrated gluon distributions $F$ and $\tilde{F}$ are related to the dipole scattering amplitude through Fourier transformations: $$\begin{aligned} F(x,k)=\int \frac{d^2{\bf r}}{(2\pi)^2}\ e^{-i{\bf k}\cdot{\bf r}}\left[1-\mathcal{N}(r,Y\!=\!\ln(x_0/x))\right]\ , \label{ugdfund}\\ \tilde{F}(x,k)=\int \frac{d^2{\bf r}}{(2\pi)^2}\ e^{-i{\bf k}\cdot{\bf r}}\left[1-\mathcal{N}(r,Y\!=\!\ln(x_0/x))\right]^2\ . \label{ugdadj}\end{aligned}$$ In principle $\tilde{F}$ is the Fourier transform of the dipole scattering amplitude in the adjoint representation [@Kovner:2001vi; @Kovchegov:2001sc; @Marquet:2004xa], we have used to large-$N_c$ limit in . In , $f_{i/p}$ and $D_{h/i}$ refer to the parton distribution function (pdf) of the incoming proton and to the final-state hadron fragmentation function respectively. Here we will use the CTEQ6 NLO pdf’s [@Pumplin:2002vw] and the DSS NLO fragmentation functions [@deFlorian:2007aj; @deFlorian:2007hc]. We have assumed that the factorization and fragmentation scales are both equal to the transverse momentum of the produced hadron. Note that the projectile in our calculations is actually a deuteron, and we obtain the neutron pdf from the proton one assuming isospin symmetry. For light hadron production discussed here, the difference between the rapidity and pseudo-rapidity $\eta$ of the produced hadron can be neglected, yielding the following kinematics: $\tilde{x}_p=x_F/z$ and $\tilde{x}_A=(x_F/z)\exp{(-2y)}$ with $x_F=\sqrt{m_h^2+p_\perp^2}/\sqrt{s_{NN}}\ \exp{(\eta)}\approx\|p_\perp\|/\sqrt{s_{NN}}\ \exp{(y)}$ introduced before. Due to parton fragmentation, the values of $\tilde{x}$’s actually probed are generically higher than $x_p$ and $x_A$ defined in . With this set up we reach a very good description [@Albacete:2010bs] of the forward negatively charged hadron and neutral pion yields measured by the BRAHMS [@Arsene:2004ux] and STAR [@Adams:2006uz] collaborations respectively, in p+p and minimum bias d+Au collisions. The parameters of the fit are $x_0=0.025\div 0.005$ (0.0075) and $\bar{Q}_{s0}^2=0.5\div0.4$ (0.2) GeV$^2$ for the initial nucleus (proton) wave function. Moreover, no $K$-factors are needed except for the most forward pions $\eta\!>\!3$, where we find that $K=0.3$ (0.4) is needed to describe the data. ![Negatively charged hadrons and neutral pions at forward rapidities at RHIC in p+p (left) and minimum bias d+Au (right) collisions, compared to our calculation [@Albacete:2010bs].[]{data-label="forward"}](spec_pp.eps "fig:"){height="4.5cm"} ![Negatively charged hadrons and neutral pions at forward rapidities at RHIC in p+p (left) and minimum bias d+Au (right) collisions, compared to our calculation [@Albacete:2010bs].[]{data-label="forward"}](spec_dAu.eps "fig:"){height="4.5cm"} Our results are shown in . By simply taking the ratios of the corresponding spectra, we get a very good description of the nuclear modification factors at forward rapidities, and this is shown in a. It should be noted that we use the same normalization as experimentalists do in their analysis of minimum bias d+Au collisions: $N_{coll}=7.2$. Physically, the observed suppression is due to the relative enhancement of non-linear terms in the small-$x$ evolution of the nuclear wave function with respect to that of a proton. After the data confirmed the CGC expectations, it has been argued that the observed suppression of particle production at forward rapidities is not an effect associated to the small values of $x_A$ probed in the nuclear wave function, but rather to energy-momentum conservation corrections relevant for $x_F\to 1$ [@Kopeliovich:2005ym; @Frankfurt:2007rn]. Such corrections are not present in the CGC, built upon the eikonal approximation (this may explain why a $K$-factor is needed to describe the suppression of very forward pions). The energy degradation of the projectile parton, neglected in the CGC, through either elastic scattering or induced gluon brehmstralung would be larger in a nucleus than in proton on account of the stronger color fields of the former, resulting in the relative suppression observed in data. A successful description of forward ratios based on the energy loss calculation in [@Kopeliovich:2005ym] is shown in b. Calculating forward di-hadron correlations in both frameworks could help pin down which is the correct picture. In the following, we show that the CGC calculations predicts correctly the azimuthal distribution. ![Nuclear modification factors at forward rapidities in minimum bias d+Au collisions in CGC [@Albacete:2010bs] (left) and energy-loss [@Kopeliovich:2005ym] (right) calculations. []{data-label="ratios"}](rdAubnl.eps "fig:"){height="5cm"} ![Nuclear modification factors at forward rapidities in minimum bias d+Au collisions in CGC [@Albacete:2010bs] (left) and energy-loss [@Kopeliovich:2005ym] (right) calculations. []{data-label="ratios"}](eta32.eps "fig:"){height="5cm"} Di-hadron azimuthal correlations -------------------------------- The kinematic range for forward particle detection at RHIC is such that $x_p\!\sim\!0.4$ and $x_A\!\sim\!10^{-3}.$ Therefore the dominant partonic subprocess is initiated by valence quarks in the proton and, at lowest order in $\alpha_s,$ the $pA\!\to\!h_1h_2X$ double-inclusive cross-section is obtained from the $qA\to qgX$ cross-section, the valence quark density in the proton $f_{q/p}$, and the appropriate hadron fragmentation functions $D_{h/q}$ and $D_{h/g}$: $$\begin{aligned} dN^{pA\to h_1 h_2 X}&=&\int_{x_1}^1 dz_1 \int_{x_2}^1 dz_2 \int_{\frac{x_1}{z_1}+\frac{x_2}{z_2}}^1 dx\ f_{q/p}(x,\mu^2)\nonumber\\&&\times\left[dN^{qA\to qgX}\left(xP,\frac{p_1}{z_1},\frac{p_2}{z_2}\right) D_{h_1/q}(z_1,\mu^2)D_{h_2/g}(z_2,\mu^2)+\right.\nonumber\\&&\left. dN^{qA\to qgX}\left(xP,\frac{p_2}{z_2},\frac{p_1}{z_1}\right)D_{h_1/g}(z_1,\mu^2)D_{h_2/q}(z_2,\mu^2)\right]\ . \label{collfact}\end{aligned}$$ We shall use CTEQ6 NLO quark distributions [@Pumplin:2002vw] and KKP NLO fragmentation functions [@Kniehl:2000fe]. The factorization and fragmentation scales are both chosen equal to the transverse momentum of the leading hadron, which we choose to denote hadron 1, $\mu=\|p_{1\perp}\|.$ We have assumed independent fragmentation of the two final-state hadrons, therefore cannot be used to compute the near-side peak around $\Delta\Phi=0$. Doing so would require the use of poorly-known di-hadron fragmentation functions, rather we will focus on the away-side peak around $\Delta\Phi=\pi$ where saturation effects are important. For the proton, one has $x_p<x<1$, and if $x_p$ would be smaller (this will be the case at the LHC), the gluon initiated processes $gA\to q\bar{q}X$ and $gA\to ggX$ should also be included in . For the nucleus, we shall see that the parton momentum fraction varies between $x_A$ and $e^{-2y_1}+e^{-2y_2}$. Therefore with large enough rapidities, only the small$-x$ part of the nuclear wave function is relevant when calculating the $qA\to qgX$ cross section, and that cross section cannot be factorized further: $dN^{qA\to qgX}\neq f_{g/A}\otimes dN^{qg\to qgX}$. Indeed, when probing the saturation regime, $dN^{qA\to qgX}$ is expected to be a non-linear function of the nuclear gluon distribution, which is itself, through evolution, a non-linear function of the gluon distribution at higher $x$. Using the CGC approach to describe the small$-x$ part of the nucleus wave function, the $qA\to qgX$ cross section was calculated in [@JalilianMarian:2004da; @Nikolaev:2005dd; @Baier:2005dv; @Marquet:2007vb]. It was found that the nucleus cannot be described by only the single-gluon distribution, a direct consequence of the fact that small-x gluons in the nuclear wave function behave coherently, and not individually. The $qA\!\to\!qgX$ cross section is instead expressed in terms of correlators of Wilson lines (which account for multiple scatterings), with up to a six-point correlator averaged over the CGC wave function. At the moment, it is not known how to practically evaluate the six-point function. In the large-$N_c$ limit, it factorizes into a dipole scattering amplitude times a trace of four Wilson lines, and the latter can in principle be obtained by solving an evolution equation written down in [@JalilianMarian:2004da]. However, this implies a significant amount of numerical work and also requires to introduce an unknown initial condition. Rather we shall follow the approach of [@Marquet:2007vb], and use an approximation that allows to express the six-point function in terms of the two-point function . The resulting cross section for the inclusive production of the quark-gluon system in the scattering of a quark with momentum $xP^+$ off the nucleus $A$ reads [@Marquet:2007vb]: $$\begin{aligned} \frac{dN^{qA\to qgX}}{d^3kd^3q}&=& \frac{\alpha_S C_F}{4\pi^2}\ \delta(xP^+\!-\!k^+\!-\!q^+)\ F(\tilde{x}_A,\Delta) \nonumber\\&&\times\sum_{\lambda\alpha\beta} \left\|I^{\lambda}_{\alpha\beta}(z,k_\perp\!-\!\Delta;{\tilde{x}_A})\!-\! \psi^{\lambda}_{\alpha\beta}(z,k_\perp\!-\!z\Delta)\right\|^2\ , \label{cs}\end{aligned}$$ where $q$ and $k$ are the momenta the quark and gluon respectively, and with $\Delta=k_\perp+q_\perp$ and $z=k^+/xP^+$. In this formula, $\tilde{x}_A$ denotes the longitudinal momentum fraction of the gluon in the nucleus, and $\tilde{x}_A=x_1\ e^{-2y_1}/z_1+x_2\ e^{-2y_2}/z_2>x_A$ when the cross section is plugged into . ![The $\Delta\phi$ distribution $(1/\sigma)d\sigma/d\Delta\phi$ computed in [@Marquet:2007vb] at fixed $y_1=3.5$, $y_2=2$ and $\|p_{1\perp}\|=5$ GeV. The azimuthal correlation is suppressed as $\|p_{2\perp}\|$ decreases, this is due to the onset of non-linearities in the nuclear wave function.[]{data-label="pred"}](cp-pred.eps){height="5cm"} The second line of features a $k_T$-factorization breaking term: $$I^{\lambda}_{\alpha\beta}(z,k_\perp;x) =\int d^2q_\perp \psi^{\lambda}_{\alpha\beta}(z,q_\perp) F(x,k_\perp\!-\!q_\perp)\ , \label{split}$$ and where $\psi^{\lambda}_{\alpha\beta}$ is the well-known amplitude for $q\!\to\!qg$ splitting ($\lambda,$ $\alpha$ and $\beta$ are polarization and helicity indices). While no additionnal information than the two-point function is needed to compute , since higher-point correlators needed in principle have been expressed in terms of $F(x,k_\perp)$, the cross section is still a non-linear function of that gluon distribution, invalidating $k_T-$factorization. The rather simple form of the $k_T-$factorization breaking term is due to the use of a Gaussian CGC color source distribution, and to the large$-N_c$ limit. Finally, the factor $\delta(xP^+\!-\!k^+\!-\!q^+)$ in is due to the fact that the eikonal approximation, valid at high energies, is used to compute the $qA\to qgX$ cross section. The delta function is a manifestation of the fact that in a high-energy hadronic collision, the momentum transfer is mainly transverse. Before comparing our predictions to the recent RHIC data, we first display in the predictions for the $\Delta\phi$ distributions made in [@Marquet:2007vb] when only the leading-order BK evolution was known. The different curves are obtained for fixed $y_1$, $y_2$ and $\|p_{1\perp}\|$, while $\|p_{2\perp}\|$ is varied. Although these results are only qualitative since for instance parton fragmentation was not included, the suppression of the $\Delta\phi$ distribution as $\|p_{2\perp}\|$ gets closer to the saturation scale was predicted. Then, the derivation of the rcBK equation made robust quantitative calculations possible. Indeed, after the parameters of the theory ($x_0$ and $\bar{Q}_{s0}$ for both the proton and the gold nucleus) were contrained by single-inclusive forward spectra, parameter-free predictions for the coincidence probability $CP(\Delta\phi)$ could be made [@Albacete:2010pg]. In , when computing $N_{pair}$ from and $N_{trig}$ from for d+Au and p+p collisions, the integration bounds for the rapidities are $2.4<y<4$, in order to compare with RHIC data [@Braidot:2010zh]. In addition, such a restriction does insure that only small-momentum partons are relevant in the nucleus wave function, as is assumed in our calculation. For the trigger (leading) particle $\|p_{1\perp}\|>2$ GeV and for the associated (sub-leading) hadron $1\ \mbox{GeV}<\|p_{2\perp}\|<\|p_{1\perp}\|.$ ![The coincidence probability as a function of $\Delta\phi$ for p+p (left) and central d+Au (right) collisions. These preliminary data from [@Braidot:2010zh] show a striking nuclear modification of di-hadron azimuthal correlations. The away-side peak, corresponding to hadrons emitted back-to-back, is prominent in p+p collisions but is absent in the central d+Au case. Such production of monojets was anticipated in the CGC as a signal of parton saturation.[]{data-label="starCP"}](star-cp-pp.eps "fig:"){height="5.9cm"} ![The coincidence probability as a function of $\Delta\phi$ for p+p (left) and central d+Au (right) collisions. These preliminary data from [@Braidot:2010zh] show a striking nuclear modification of di-hadron azimuthal correlations. The away-side peak, corresponding to hadrons emitted back-to-back, is prominent in p+p collisions but is absent in the central d+Au case. Such production of monojets was anticipated in the CGC as a signal of parton saturation.[]{data-label="starCP"}](star-cp-dau.eps "fig:"){height="5.9cm"} The recent data obtained by the STAR collaboration for the coincidence probability obtained with two neutral pions are displayed in for both p+p and central d+Au collisions. The nuclear modification of the di-pion azimuthal correlation is quite impressive, the prominent away-side peak in p+p collisions is absent in central d+Au collisions, in agreement with the behavior predicted in [@Marquet:2007vb]. In a, these data are compared with the rcBK calculations of [@Albacete:2010pg]. As mentioned before, the complete near-side peak is not computed, as does not apply around $\Delta\phi=0$. We see that the disapearence of the away-side peak in central d+Au collisions, compared to p+p collisions, is quantitatively consistent with the CGC calculations. The latter are only robust for the d+Au case, but the extrapolation to the p+p case is displayed in order to show that it is qualitatively consistent with the presence of the away-side peak in p+p, and also with the fact that the near-side peak is identical in the two cases, and is not sensitive to saturation physics. Note that since uncorrelated background has not been extracted from the data, the overall normalization of the data points has been adjusted by subtracting a constant shift, as indicated on the figure. ![The coincidence probability as a function of $\Delta\phi$. Left: for p+p and central d+Au collisions, preliminary data are compared with CGC predictions; the away-side peak in p+p is qualitatively described by the CGC calculation while its disapearence in central d+Au is quantitatively consistent with the prediction. Right: CGC predictions for different centralities of the d+Au collision; the near-side peak is independent of the centrality, while the away-side peak reappears as collisions are more and more peripheral.[]{data-label="cgcCP"}](CP_dAu_pp.eps "fig:"){height="4.5cm"} ![The coincidence probability as a function of $\Delta\phi$. Left: for p+p and central d+Au collisions, preliminary data are compared with CGC predictions; the away-side peak in p+p is qualitatively described by the CGC calculation while its disapearence in central d+Au is quantitatively consistent with the prediction. Right: CGC predictions for different centralities of the d+Au collision; the near-side peak is independent of the centrality, while the away-side peak reappears as collisions are more and more peripheral.[]{data-label="cgcCP"}](CP_cent.eps "fig:"){height="4.5cm"} To deal with the centrality dependence, we have identified the centrality averaged initial saturation scale $\bar{Q}^2_{s0}$, extracted from minimum-bias single-inclusive hadron production data, with the value of $Q^2_{s0}$ at $b=5.47$ fm, and used the Woods-Saxon distribution $T_A(b)$ to calculate the saturation scale at other centralities: $$Q^2_{s0}(b)=\frac{\bar{Q}_{s0}^2\ T_A(b)}{T_A(5.47\ \mbox{fm})}\ ,\quad\bar{Q}^2_{s0}=0.4\ \mbox{GeV}^2\ .$$ The value used in a in the central d+Au case is $Q^2_{s0}(0)\simeq 0.6$ GeV$^2$ at $x_0=0.02$. The corresponding saturation scale felt by gluons is about $1.2$ GeV$^2$ and of course it gets bigger with decreasing $x$. In b, we show the centrality dependence of the coincidence probability. Although it is difficult to trust our formalism all the way to peripheral collisions, we predict that the near-side peak does not change with centrality, and that the away-side peak reappears for less central collisions. This is consistent with the fact that peripheral d+Au collisions are p+p collisions. The fact that the away-side peak disappears from peripheral to central collisions shows that indeed this effect is correlated with the nuclear density. Moreover dihadron correlations at mid-rapidity, which are sensitive to larger values of $x_A$, feature an away-side peak whatever the centrality. The fact that for central collisions the away-side peak disappears from central to forward rapidities also indicates that the effect is correlated with the nuclear gluon density. In a similar way, we predict that for higher transverse momenta, the away-side peak will reappear. We are not aware of any descriptions of this phenomena that does not invoke saturation effects. We note that apart from our CGC calculation, a successful description based on the KLN saturation model was also recently proposed [@Tuchin:2009nf]. There, although different assumptions are used, the existence of the saturation scale is the crucial ingredient to successfully reproduce the data. While more differential measurements of the coincidence probability, as a function of transverse momentum or rapidity, will provide further quantitative tests of our CGC predictions, the piece of evidence we have discussed in this work strongly indicates that we have observed manifestations of the saturation regime of QCD at RHIC. Predictions for the LHC ======================= The huge leap forward in collision energy reached at the LHC allows for an exploration of small-$x$ effects already at mid-rapidity. There, both the target and projectile are probed at small values of $x$, and energy loss effects associated to large-$x_F$ effects are expected to be small. In we present our CGC predictions for the nuclear modification factor for negative charged hadrons in p+Pb collisions at two LHC energies. Our curves correspond to rapidities $y=2$ and larger. Technical difficulties related to the intrinsic asymmetry of the hybrid formalism used for particle production prevent us from calculating the ratios at mid-rapidity. However, the smooth rapidity dependence suggests that a large suppression $\sim 0.6$ is also expected at mid-rapidity in the LHC. It should be taken into account that the normalization taken to produce the curves in was $N_{coll}=3.6$. We also compare the $y=2$ and 4 curves with predictions obtained with the $k_T$-factorization formalism, in order to check the validity of that approach, and especially to test up to what value of $y$ it can be used. The $k_T$-factorization formula (see [@Albacete:2010bs]) is valid when the dominant contributions to the cross section come from small values of x, for both the projectile ($x_p\ll1$) and the target ($x_A\ll1$). For instance, it only includes gluonic degrees of freedom. This approach is clearly insufficient at very forward rapidities or large $p_\perp$, where valence quarks of the projectile are important ($x_p\to 1$). However, as can be seen in , both formalisms give comparable results, as the lines from $k_T$-factorization overlap with the uncertainty bands spanned by the results from the hybrid formalism. This seems to identify a kinematical window where both approximations are valid. ![CGC predictions for the nuclear modification factor in p+Pb collisions at two LHC energies and rapidities $y=2,4$ and 6.[]{data-label="LHC"}](r_pPb_hpm2.eps){height="6.2cm"} [38]{} natexlab\#1[\#1]{}\[2\][\#2]{} , , , , (). , , () . , et al., , () . , et al., , () . , , (). , , , () . , , (). , , , , () . , , , , , , () . , , () . , , , , () . , , , () . , , , , () . , , , () . , , , , () . , et al., , () . , , () . , , () . , , () . , , () . , , , () . , , , , , () . , , () . , , , () . , , , , () . , , , () . , , , () . , , () . , et al., , () . , , , , () . , , , , () . , , , , , , () . , , , () . , , , , () . , , , () . , , , , , () . , , , , , () . , , () .
--- abstract: 'We remark on the utility of an observational relation between the absorption column density in excess of the Galactic absorption column density, $\Delta N_{\rm H} = N_{\rm H, fit} - N_{\rm H, gal}$, and redshift, z, determined from all [55]{} -observed long bursts with spectroscopic redshifts as of 2006 December. The absorption column densities, $N_{\rm H, fit}$, are determined from powerlaw fits to the X-ray spectra with the absorption column density left as a free parameter. We find that higher excess absorption column densities with $\Delta N_{\rm H} > 2\times 10^{21}$ cm$^{-2}$ are only present in bursts with redshifts z$<$2. Low absorption column densities with $\Delta N_{\rm H} < 1\times 10^{21}$ cm$^{-2}$ appear preferentially in high-redshift bursts. Our interpretation is that this relation between redshift and excess column density is an observational effect resulting from the shift of the source rest-frame energy range below 1 keV out of the XRT observable energy range for high redshift bursts. We found a clear anti-correlation between $\Delta N_{\rm H}$ and z that can be used to estimate the range of the maximum redshift of an afterglow. A critical application of our finding is that rapid X-ray observations can be used to optimize the instrumentation used for ground-based optical/NIR follow-up observations. Ground-based spectroscopic redshift measurements of as many bursts as possible are crucial for GRB science.' author: - 'Dirk Grupe, John A. Nousek, Daniel E. vanden Berk, Peter W.A. Roming, David N. Burrows, Olivier Godet, Julian Osborne, Neil Gehrels' title: 'Redshift Filtering by [Swift]{} Apparent X-ray Column Density ' --- 2[[$\alpha^2$]{} ]{} Introduction ============ The mission [@gehrels04] has revolutionized the study of Gamma-Ray Burst (GRB) afterglows. The mission, which relies upon training sensitive X-ray and optical telescopes on new GRBs as rapidly as possible, has resulted in the accurate positioning of GRB afterglows on a timescale of minutes. Especially for long GRBs, this rapid localization has proven highly effective at identifying afterglows for study at other wavelengths. Already, in about 2 years of operation, has localized more than twice as many GRB afterglows than had been localized in the eight years preceding [@burrows06b]. However, alone cannot address all the important issues in GRB research. One of the most important GRB parameters is the redshift of a burst. Knowledge of the redshift is not only crucial for determining the luminosity and other physical parameters of the burst, but also permits optimization of ground-based observations. For example a determination of the redshift for high-redshift GRBs requires large telescopes with infra-red sensitivity, because the Lyman absorption edge gets shifted beyond the long wavelength end of the -UVOT sensitivity for redshifts above 5 [@roming06]. GRBs fade rapidly and only spectroscopy can provide reliable redshifts. Therefore, to search for redshifts as high as GRB050904 [z=6.29; @kawai05], or even to the unexplored $z\sim 7-10$ range, requires telescopes in the 8-10 m class making observations within the first night or two of the Swift discovery. As listed in this paper, about 6% of all -observed GRBs with spectroscopic redshift have redshift z$>$5. Observing time on such large, world-class telescopes is an extremely precious commodity. Currently the limited number of target of opportunity programs must triggered based on very limited information in order to spectroscopically observe the many discovered GRB afterglow. While nearly all promptly observed long GRBs can be localized by the X-ray Telescope [XRT; @burrows05], only about 34% are detected by the UV/Optical Telescope [UVOT; @roming05; @roming06]. Thus observers looking for high redshift bursts can filter the approximately 100 GRBs found by each year, by using the criterion: an afterglow is detected by the XRT, but not detected by the UVOT. Unfortunately this criterion alone only reduces the rate of high-z candidates by about 1/3, leaving about 67 GRBs to be observed per year. Moreover, many other factors, e.g. reddening, can result in suppression of afterglow emission in the UVOT sensitivity range (see Roming et al. 2006 for a discussion). Of course large telescope observers can wait to see if any of the smaller robotic telescopes can select candidates based on the broad-band photometric studies conducted by these telescopes. Often the cost of doing this is to lose several hours waiting for the results of these small telescope studies to be reduced and transmitted, and to be subject to the vagaries of weather and other observing constraints on the ground. Also GRB afterglows decay rapidly. Every hour of waiting reduces the chances of obtaining an optical and/or NIR spectrum of the afterglow and therefore significantly decreases the chances of measuring the redshift of the burst from a spectrum. To see whether Swift XRT data alone can help to provide very early information, we conducted a study of all [55]{} Swift long GRB afterglows with known redshifts by November 2006. On a timescale of one or two hours after the initial XRT detection of a new GRB, the XRT telemeters information to the ground from which many properties of the burst can be determined, including accurate positions, X-ray flux and X-ray spectral information. The Swift team has been routinely analyzing these data, and reporting them to the world via the GRB Coordinates Network [GCN; @barthelmy95]. It has been found that the typical afterglow can be fit with a simple power law model spectrum plus the effects of a variable amount of absorbing material [e.g. @stratta04; @campana06]. This absorbing material can be located either in our Galaxy, in the host galaxy of the GRB, or in intervening gas clouds. As shown on a theoretical basis by @ramirez02, extinction in GRBs is expected. Recently @prochaska06 presented the results of high-resolution optical spectroscopy of the interstellar medium of GRB host galaxies of GRB afterglows. For low redshift bursts, in the observer’s rest-frame, the effects of local, and intervening absorption appear at roughly similar energies, and thus the effects are intermixed. For high redshift bursts, the absorbing material in the host galaxy will incur a substantial redshift. The result is that the energy band of the Swift XRT is shifted to higher energies in the GRB rest frame, making it difficult to detect X-ray absorption. Without prior knowledge of the redshift, X-ray observations result in measurements of NH that are systematically low compared to the actual absorbing column in the host galaxy. Thus if we take the apparent absorption column density $N_{\rm H, fit}$, using the photo-electric cross-section as given by @morrison83, in the observer’s rest-frame, and subtract off the known absorption column density in our Galaxy [$N_{\rm H, gal}$ as given by @dic90], the residual $\Delta N_{\rm H} = N_{\rm H, intr} - N_{\rm H, gal}$ will reflect the redshifted column density either in the source or in the intervening line-of-sight. If this residual column density appears high, then it is very likely that the GRB is close, because distant GRB absorption effects are overwhelmed by the redshift effect. Of course if the column density is low, we cannot tell whether the GRB is near - with low intrinsic absorption - or far - with either low intrinsic absorption or redshifted high absorption. A similar method has also been proposed to estimate the redshifts of high-redshift quasars [@wang04]. We present the values for absorption measured in the first orbit of data for all [55]{} Swift GRBs with known spectroscopic redshifts (section 2). The X-ray data are typically available 1-2 hours after the detection of the burst, except for those few bursts for which observing constraints prevent from slewing immediately. The relation we are proposing is not a functional prediction (i.e. we do not suggest that GRB redshift is derivable from XRT absorption), but instead we argue that we can predict the maximum redshift of an afterglow on the basis of the excess absorption $\Delta N_{\rm H}$. In section 3 we present the results. Finally, in section 4 we present our conclusion. Throughout the paper spectral index [$\beta_{\rm X}$]{} is defined as $F_{\nu}(\nu)\propto\nu^{-\beta_{\rm X}}$. All errors are 1$\sigma$ unless stated otherwise. \[observe\] Observations and data reduction =========================================== Table\[grb\_list\] lists all [55]{} long GRBs with reported spectroscopic redshifts (up to 2006 November) which were observed by . We did not include short bursts since they most likely have different physical processes than the long bursts. Furthermore, short burst have typically been detected at relatively low redshifts [e.g. @berger06d], even though @levan06 suggested that the short GRB 060121 is possibly at z$>$4.5. We limit our sample to those bursts with spectroscopic redshifts only, because these are the most reliable redshift measurements. Other methods such as photometric redshifts are less certain because a drop-out in bluer filters can also be caused by strong dust reddening. The XRT data were reduced by the [xrtpipeline]{} software version 0.10.4 which is part of the HEASOFT version 6.1.1. For XRT Photon Counting mode data [PC; @hillj04], source photons were selected by [XSELECT]{} version 2.4 in a circular region with a radius of r=47$^{''}$ and the background photons were collected in a circular region close by with a radius r=137$^{''}$. For bright afterglows with PC mode count rates $>$ 1 count s$^{-1}$ the source photons were selected in an annulus that excludes the inner pixels in order to avoid the effects of pileup. For Windowed Timing mode [WT; @hillj04] data we extracted source and background photons in boxes with a length of 40 pixel each, except from very bright bursts like e.g. GRB 060729 [@grupe07] for which we applied the method as described in @romano06. For spectral fitting of the PC and WT mode data only events with grades 0-12 and 0-2, respectively, were included. The X-ray spectra were re-binned by [grppha]{} 3.0.0 having 20 photons per bin and analyzed by [XSPEC]{} version 12.3.0 [@arnaud96]. The auxiliary response files (arfs) were created by [xrtmkarf]{} using arfs version 008. We used the standard response matrix swxpc0to12\_20010101v008.rmf for the PC mode data and swxwt0to2\_20010101v008.rmf for the WT data. Note: These are standard reduction techniques as used in the first XRT refined analysis GCN circular from bursts. \[results\] Results =================== Table\[grb\_list\] lists the redshift, Galactic absorption column density $N_{\rm H, gal}$, $\Delta N_{\rm H} = N_{\rm H, fit} - N_{\rm H, gal}$, the intrinsic column density $N_{\rm H, intr}$ at the redshift of the burst, X-ray energy spectral slope [$\beta_{\rm X}$]{}, $\chi^2/\nu$ of the power law fit with the absorption column density as a free parameter ($N_{\rm H, free}$) and fixed to the Galactic value as given by @dic90, detection flag for the UVOT, and the reference for the spectroscopic redshift measurement. In cases where the free-fit absorption column density $N_{\rm H, fit}$ is within the errors consistent with the Galactic value, we set $\Delta N_{\rm H}$=0. The left panel of Figure\[z\_delta\_nh\] compares redshift versus $\Delta N_{\rm H}$. The dotted lines at z=2.30 and $\Delta N_{\rm H}=5.95\times 10^{20}$ cm$^{-2}$ are the medians in z and $\Delta N_{\rm H}$. These lines are used as cutoff lines to separate between low and high-redshift and low and high excess absorption groups in a $2\times2$ contingency table[^1]. The results of grouping these data by these cutoff lines are given in Table\[2x2tab\]. We can immediately see that bursts with high excess absorption column densities $\Delta N_{\rm H}$ will most likely be at low redshifts while afterglows with small $\Delta N_{\rm H}$ are most likely at high redshifts. From the $2\times2$ contingency table a probability of only P=0.0011 (using a two-tailed Fisher Exact Probability test) that this result is random can be calculated. Another statistical test to check whether the relation is purely random is the Spearman rank order test. A Spearman rank order test results in a correlation coefficient $r_s=-0.51$ with a Student’s T-test $T_s=-4.3$ and a probability P$< 10^{-4}$ of a random distribution. The right panel of Figure\[z\_delta\_nh\] displays the above results as $log (1+z)$ vs. $log(1+\Delta N_{\rm H})$. We use this diagram to conservatively draw a line along the envelope of the $log (1+z)$ vs. $log(1+\Delta N_{\rm H})$ distribution of the bursts, including the errors, in order to determine maximum redshift of a burst with respect to its excess absorption $\Delta N_{\rm H}$. This line is displayed as the dashed line in the right panel of Figure\[z\_delta\_nh\] and can be described as: $$log(1+z) < 1.3 - 0.5\times log(1+\Delta N_{\rm H})$$ with $\Delta N_{\rm H}$ in units of $10^{20}$ cm$^{-2}$. From this equation we can limit the maximum expected redshift based on the excess absorption. Note that this equation does not estimate the redshift of a burst. The equation only predicts the maximum redshift of a burst. As examples, the redshift of a burst with an excess absorption $\Delta N_{\rm H}=4\times 10^{21}$ cm$^{-2}$ is expected not to exceed z=2.1, $\Delta N_{\rm H}=2\times 10^{21}$ cm$^{-2}$ has z$<$3.4, and $\Delta N_{\rm H}=1\times 10^{21}$ has z$<$5. Also note that the number of bursts with redshifts z$>$2.3 (the median redshift) and no excess absorption detected, so the free-fit absorption column density $N_{\rm H.fit}$ is consistent with the Galactic value, is three times as high as the number of bursts with no excess absorption and a redshift z$<$2.3. In case we do not detect excess absorption in a burst, most-likely this will be a burst with a redshift z$>$2.3. Our findings of an observational relation between the excess absorption column density $\Delta N_{\rm H}$ and redshift is due purely to an observational artefact: we only detect any excess absorption in high-redshift bursts which have a very large intrinsic column density, as shown in Figure\[z\_nh\]. The dashed line at the lower boundary of the $N_{\rm H, intr}$ - z distribution displays the maximum redshift at which an intrinsic column density can be detected in the XRT energy window. \[discuss\] Discussion ====================== Our main result is that there is a clear anti-correlation between absorption column density in excess of the Galactic value $\Delta N_{\rm H} = N_{H, fit} - N_{\rm H, gal}$ and redshift. This relation can be used to limit the range of possible redshifts. GRBs for which early X-ray spectra are consistent with the Galactic absorption column density are most-likely at higher redshifts ($z>2$), although a few examples of low-redshift GRBs also fall into this category. However, GRBs with a significant excess column density of $\Delta N_{\rm H}>2 \times 10^{21}$ cm$^{-2}$ are exclusively at redshifts z$<$2.0. The strong correlation we find between redshift and intrinsic $N_{\rm H}$ is an observational artefact. We can only detect an intrinsic absorber if the absorption column density is large enough to affect the observable energy range. For a z=4 burst the low energy cutoff of the detector at 0.3 keV is at 1.5 keV in the rest frame of the burst. In order to detect any significant additional absorption in the observed energy window above the Galactic value the intrinsic (redshifted) absorption has to be in the order of at least $10^{22}$ cm$^{-2}$, as shown in Figure\[z\_nh\]. There are a few points one should be cautious about: 1) Our study is biased towards afterglows which have spectroscopic redshifts and are therefore detectable at optical and/or NIR wavelengths. As a result, we 2) may miss afterglows in galaxies seen edge-on. These afterglows would suffer from significant absorption columns on the order of several times $10^{23}$ cm$^{-2}$ as is commonly observed in e.g. Seyfert 2 galaxies. 3) Some of the early XRT WT mode data are not well-fitted by a single absorbed power law model, e.g. GRBs 060614 and 060729 [@mangano07; @grupe07 respectively]. These GRBs display dramatic changes in their X-ray spectra within minutes and the spectrum cannot be modelled by one single power law. In some cases the spectra require multi-component spectral models as it is the case for GRB 060729 [@grupe07]. The immediate application of our column density - redshift relation is to optimize ground-based optical and NIR follow up observations. We plan to include the redshift limit information in future XRT refined analysis GCNs. Photometric redshifts can also be calculated on the basis of UVOT data (Vanden Berk et al., in preparation). We tested our X-ray method with bursts that have UVOT photometric redshifts, but no spectroscopic redshifts and found that the results agree with each other. The advantage, however, of using the -XRT data instead of the UVOT data to estimate a maximum redshift is that -XRT data are typically processed faster at the NASA Data Center than the UVOT data. This is due to the larger data volume of the UVOT compared to the XRT data. Therefore a redshift prediction can be given faster on the basis of the XRT data than on the UVOT data. This will give ground-based observers at large telescopes a tool to decide which spectrograph to use - an optical spectrograph for the low-redshift bursts and a NIR spectrograph for the high-redshift bursts and will optimize the use of large ground-based telescopes. Note, however, that the purpose of this paper is not to discourage observers from obtaining spectra of bursts with predicted low redshifts. Each spectroscopic redshift - low or high redshift - is important, because it enables us to determine physical parameters of the burst such as the isotropic energy or the break times in the light curve. Therefore we encourage ground-based observers to continue to obtain spectra of afterglows whenever possible. The larger the number of bursts with spectroscopic redshifts the better our understanding of the physics of GRBs will be. We would like to thank all observers at ground-based optical telescopes for their effort to obtain redshifts of the afterglows. We would also like to thank Sergio Campana for discussion related to the XRT calibration, Cheryl Hurkett for sending us a draft of her paper on GRB 050505, and Abe Falcone for various discussions on the determination of the absorption column densities. In particular we want to thank Eric Feigelson for various discussions on statistics, and our referee Johan Fynbo for a fast and detailed referee’s report that significantly improved the paper. This research has made use of data obtained through the High Energy Astrophysics Science Archive Research Center Online Service, provided by the NASA/Goddard Space Flight Center. At Penn State we acknowledge support from the NASA Swift program through contract NAS5-00136. Arnaud, K. A., 1996, ASP Conf. 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Berger, E., & Ofek, E., 2005, GCN 4186 Still, M., et al., 2006, GCN 5226 Stratta, G., Fiore, F., Antonelli, L.A., Piro, L., & De Pasquale, M., 2004, , 608, 846 Thoene, C.C., et al., 2006a, GCN 5373 Thoene, C.C., Fynbo, J.P.U., Jakobsson, P., Vreeswijk, P.M., & Hjorth, J., 2006b, GCN 5812 Wang, J.X., Malhatra, S., Rhoads, J.E., & Norman, C.A., 2004, , 612, L109 Watson, D., et al., , 652, 1011 [llrrrccccl]{} 050126 & 1.29 & 5.28 & — & — & 1.01[$\pm$]{}0.18 & — & 16/15 & n & @berger05b\ 050315 & 1.94 & 4.34 & 8.55$^{+3.80}_{-3.48}$ & 0.66$^{+0.30}_{-0.27}$ & 1.42$^{+0.18}_{-0.16}$ & 43/45 & 61/46 & n & @berger05b\ 050318 & 1.44 & 2.79 & 4.23$^{+2.24}_{-2.00}$ & 0.16$^{+0.10}_{-0.09}$ & 1.03$^{+0.11}_{-0.10}$ & 73/74 & 86/75 & y & @berger05b\ 050319 & 3.24 & 1.13 & — & — & 1.10$^{+0.17}_{-0.15}$ & 27/36 & 31/37 & y & @jakobsson06\ 050401 & 2.90 & 4.84 & 11.96$^{+1.39}_{-1.33}$ & 1.95$^{+0.22}_{-0.24}$ & 1.09$^{+0.06}_{-0.05}$ & 296/258 & 614/259 & n & @watson06\ 050408 & 1.2357 & 1.74 & 28.36$^{+6.60}_{-5.97}$ & 1.40$^{+0.39}_{-0.33}$ & 1.31$^{+0.20}_{-0.19}$ & 39/36 & 120/37& y & @berger05b\ 050416A & 0.6535 & 2.07 & 26.28$^{+3.37}_{-3.64}$ & 0.61$^{+0.10}_{-0.09}$ & 1.20$^{+0.12}_{-0.11}$ & 65/84 & 320/85 & y & @jakobsson06c\ 050505 & 4.27 & 2.04 & 7.23$^{+2.45}_{-2.25}$ & 2.40$^{+0.80}_{-0.73}$ & 1.18$^{+0.12}_{-0.11}$ & 51/77 & 85/78 & n & @jakobsson06c\ 050525 & 0.606 & 9.10 & 9.81$^{+6.53}_{-5.96}$ & 0.21$^{+0.13}_{-0.15}$ & 1.08$^{+0.23}_{-0.21}$ & 23/27 & 31/28 & y & @jakobsson06c\ 050603 & 2.821 & 1.19 & — & — & 0.75[$\pm$]{}0.10 & 30/34 & 31/35 & y & @jakobsson06c\ 050730 & 3.967 & 3.06 & 3.97$^{+0.91}_{-0.87}$ & 1.24$^{+0.25}_{-0.27}$ & 0.71[$\pm$]{}0.04 & 177/174 & 243/175 & y & @jakobsson06c\ 050802 & 1.71 & 1.78 & 5.31$^{+2.66}_{-2.35}$ & 0.31$^{+0.17}_{-0.15}$ & 1.00$^{+0.12}_{-0.11}$ & 61/67 & 77/68 & y & @jakobsson06c\ 050803 & 0.422 & 5.63 & 17.19$^{+5.65}_{-5.02}$ & 2.65$^{+1.01}_{-0.83}$ & 1.05$^{+0.19}_{-0.17}$ & 57/59 & 99/60 & n & @bloom05\ 050820 & 2.612 & 4.71 & — & — & 0.00$^{+0.04}_{-0.04}$ & 87/100 & 87/101 & y & @jakobsson06c\ 050824 & 0.83 & 3.62 & — & — & 0.83$^{+0.24}_{-0.22}$ & 23/20 & 22/21 & y & @jakobsson06c\ 050826 & 0.297 & 21.70& 30.54$^{+15.6}_{-13.3}$ & 0.53$^{+0.30}_{-0.24}$ & 0.93$^{+0.26}_{-0.23}$ & 29/25 & 45/26 & n & @halpern06\ 050904 & 6.29 & 4.93 & 4.05$^{+1.16}_{-1.12}$ & 3.93$^{+1.10}_{-1.03}$ & 0.47[$\pm$]{}0.04 & 126/87 & 104/92 & n & @kawai05\ 050908 & 3.344 & 2.14 & — & — & 2.01$^{+0.41}_{-0.31}$ & 15/17 & 15/18 & y & @jakobsson06c\ 050922C & 2.198 & 3.43 & — & — & 0.99$^{+0.11}_{-0.10}$ & 22/25 & 22/26 & y & @piranomonte05\ 051016B & 0.9364 & 3.64 & 23.67$^{+6.77}_{-5.65}$ & 0.78$^{+0.26}_{-0.21}$ & 1.13$^{+0.21}_{-0.19}$ & 30/32 & 93/33 & y & @nardini06\ 051022 & 0.80 & 4.06 & 107.3$^{+18.8}_{-16.7}$ & 4.58$^{+0.76}_{-0.69}$ & 1.38$^{+0.22}_{-0.20}$ & 35/47 & 206/48 & n & @gal-yam05\ 051109A & 2.346 & 17.5 & — & — & 1.03[$\pm$]{}0.11 & 43/34 & 43/35 & y & @nardini06\ 051109B & 0.080 & 13.1 & 12.26$^{+11.6}_{-10.0}$ & 0.14$^{+0.14}_{-0.12}$ & 1.16$^{+0.33}_{-0.29}$ & 11/15 & 15/16 & n & @perley05\ 051111 & 1.55 & 5.02 & 25.77$^{+11.36}_{-10.01}$ & 1.85$^{+0.72}_{-0.85}$ & 1.77$^{+0.47}_{-0.41}$ & 16/12 & 36/13 & y & @nardini06\ 060115 & 3.53 & 12.60 & 5.00$^{+3.00}_{-3.25}$ & 1.29$^{+0.84}_{-0.74}$ & 0.92$^{+0.12}_{-0.11}$ & 78/85 & 86/86 & n & @delia06\ 060123 & 1.099 & 1.48 & — & — & 0.77$^{+0.26}_{-0.30}$ & 11/15 & 14/16 & n & @berger06a\ 060124 & 2.30 & 9.16 & 6.51[$\pm$]{}0.50 & 0.77[$\pm$]{}0.06 & 0.36[$\pm$]{}0.01 & 654/453 & 1215/454 & y & @mirabal06a\ 060206 & 4.05 & 0.94 & — & — & 1.25$^{+0.34}_{-0.29}$ & 9/11 & 13/12 & y & @fynbo06\ 060210 & 3.91 & 8.52 & 6.23$^{+1.03}_{-1.00}$ & 1.95$^{+0.29}_{-0.30}$ & 1.05[$\pm$]{}0.04 & 195/107 & 315/108 & n & @cucchiara06a\ 060218 & 0.033 & 11.00 & 39.01$^{+11.38}_{-8.73}$ & 0.40$^{+0.12}_{-0.10}$ & 3.13$^{+0.53}_{-0.38}$ & 17/26 & 90/27 & y & @mirabal06b\ 060223A & 4.41 & 5.93 & — & — & 0.84$^{+0.15}_{-0.13}$ & 15/14 & 16/15 & n & @berger06b\ 060418 & 1.49 & 9.27 & 14.27$^{+1.00}_{-0.99}$ & 0.82[$\pm$]{}0.06 & 1.22[$\pm$]{}0.04 & 218/174 & 1073/149 & y & @ellison06\ 060502A & 1.51 & 2.97 & 9.90$^{+4.56}_{-3.79}$ & 0.57$^{+0.21}_{-0.19}$ & 2.36$^{+0.46}_{-0.38}$ & 64/35 & 89/36 & y & @cucchiara06b\ 060510B & 4.90 & 3.78 & 9.61$^{+0.92}_{-0.89}$ & 4.55$^{+0.51}_{-0.48}$ & 0.38[$\pm$]{}0.03 & 194/183 & 238/183 & n & @price06\ 060512 & 0.4428 & 1.43 & 4.40[$\pm$]{}3.24 & 0.25[$\pm$]{}0.17 & 3.07[$\pm$]{}0.38 & 22/10 & 23/11 & y & @bloom06\ 060522 & 5.11 & 4.83 & — & — & 0.70$^{+0.19}_{-0.18}$ & 17/17 & 17/18 & y & @cenko06\ 060526 & 3.21 & 5.46 & 5.95$^{+0.81}_{-0.78}$ & 1.14$^{+0.16}_{-0.15}$ & 0.79[$\pm$]{}0.03 & 217/158 & 408/159 & y & @berger06c\ 060604 & 2.68 & 4.55 & 15.96$^{+1.72}_{-1.79}$ & 2.22$^{+0.25}_{-0.26}$ & 1.39[$\pm$]{}0.07 & 83/51 & 392/52 & y & @castro06\ 060605 & 3.711 & 5.11 & — & — & 1.03[$\pm$]{}0.09 & 31/35 & 31/36 & y & @still06\ 060607 & 2.937 & 2.67 & 2.87$^{+0.82}_{-0.79}$ & 0.50[$\pm$]{}0.13 & 0.79[$\pm$]{}0.04 & 200/188 & 240/189 & y & @ledoux06\ 060614 & 0.125 & 3.07 & 2.07[$\pm$]{}0.37 & 0.03[$\pm$]{}0.01 & 0.45[$\pm$]{}0.15 & 446/256 & 541/257 & y & @mangano07\ 060707 & 3.43 & 1.76 & — & — & 0.94[$\pm$]{}0.12 & 13/18 & 13/9 & y & @jakobsson06\ 060714 & 2.71 & 6.72 & 12.36$^{+1.59}_{-1.52}$ & 1.83$^{+0.05}_{-0.04}$ & 0.98$^{+0.06}_{-0.05}$ & 261/252 & 501/253 & y & @jakobsson06\ 060729 & 0.54 & 4.82 & 14.29$^{+0.09}_{-0.08}$ & 0.26[$\pm$]{}0.01 & 1.76[$\pm$]{}0.04 & 505/297 & 1817/298 & y & @grupe07\ 060904B & 0.703 & 12.10 & 28.80$^{+1.45}_{-1.40}$ & 0.78[$\pm$]{}0.05 & 1.16[$\pm$]{}0.04 & 540/425 & 2570/426 & y & @fugazza06\ 060906 & 3.685 & 9.66 & — & — & 1.51$^{+0.62}_{-0.51}$ & 12/10 & 16/11 & n & @jakobsson06\ 060908 & 2.43 & 2.73 & — & — & 1.30$^{+0.28}_{-0.25}$ & 29/25 & 31/26 & y & @rol06\ 060912 & 0.937 & 4.23 & 14.67$^{+7.25}_{-6.47}$ & 0.48$^{+0.26}_{-0.21}$ & 1.17$^{+0.25}_{-0.23}$ & 11/20 & 27/21 & y & @jakobsson06\ 060926 & 3.208 & 7.30 & — & — & 0.97$^{+0.33}_{-0.30}$ & 11/15 & 16/16 & y & @piranomonte06\ 060927 & 5.6 & 5.20 & — & — & 0.82$^{+0.30}_{-0.27}$ & 11/15 & 11/16 & n & @fynbo06b\ 061007 & 1.261 & 2.22 & 11.87[$\pm$]{}0.04 & 0.52[$\pm$]{}0.02 & 1.01[$\pm$]{}0.01 & 1072/587 & 4784/588 & y & @schady07\ 061110A & 0.757 & 4.87 & 16.15$^{+1.20}_{-1.56}$ & 0.97$^{+0.19}_{-0.18}$ & 2.05$^{+0.11}_{-0.10}$ & 242/197 & 579/198 & n & @fynbo06c\ 061110B & 3.44 & 4.83 & — & — & 1.04$^{+0.41}_{-0.37}$ & 4/3 & 4/4 & n & @thoene06b\ 061121 & 1.314 & 5.09 & 15.93$^{+2.50}_{-3.20}$ & 0.81$^{+0.22}_{-0.17}$ & 0.25[$\pm$]{}0.07 & 290/286 & 339/287 & y & @page07\ 061222B & 3.355 & 27.70 & 12.70$^{+5.86}_{-5.35}$ & 4.18$^{+1.81}_{-1.53}$ & 1.60$^{+0.20}_{-0.18}$ & 78/47 & 95/47 & n & @berger06d [lrr]{} $\Delta N_{\rm H}\geq 5.95 \times 10^{21}$ cm$^{-2}$ & 20 & 7\ $\Delta N_{\rm H}< 5.95 \times 10^{21}$ cm$^{-2}$ & 8 & 20 [^1]: We evaluate the statistical significance of our results by use of 2$\times$2 contingency tables. A 2$\times$2 contingency table is a statistical tool that compares a property of two groups, such as in the present example: low- and high-redshift bursts with or without significant additional absorption above the Galactic column density. The way this method works is the following: assume a number of low redshift bursts $n = l + k$, with $l$ of them having low and $k$ having high column densities in excess of the Galactic value. The ratio of low to high absorption column objects is then $l/k$. For the high-redshift bursts the numbers are $z = x + y$ with $x$ representing the number of bursts with low excess absorption column densities and $y$ representing high excess absorption column densities. If the ratio of low to high absorption bursts in high-redshift bursts is the same as among low-redshift bursts, the number of high-redshift bursts with high absorption column densities is $y = k\times x/l$. The 2x2 contingency table compares the number of objects in each cell to the expected number under this assumption, and can be used to calculate the probability that the deviation from the expected number of objects is just random.
--- abstract: \| We consider linear delay differential equations at the verge of Hopf instability, i.e. a pair of roots of the characteristic equation are on the imaginary axis of the complex plane and all other roots have negative real parts. When nonlinear and noise perturbations are present, we show that the error in approximating the dynamics of the delay system by certain two dimensional stochastic differential equation without delay is small (in an appropriately defined sense). Two cases are considered: (i) linear perturbations and multiplicative noise (ii) cubic perturbations and additive noise. The two-dimensional system without-delay is related to the projection of the delay equation onto the space spanned by the eigenfunctions corresponding to the imaginary roots of the characteristic equation. A part of this article is an attempt to relax the Lipschitz restriction imposed on the coefficients in [@LNNSN_SDDEavg] for additive noise case. Also, the multiplicative noise case is not considered in [@LNNSN_SDDEavg]. Examples without rigorous proofs are worked in [@LNPRLE]. address: \| University of Illinois\ Urbana, IL, USA. author: - Nishanth Lingala title: Approximation of delay differential equations at the verge of instability by equations without delay --- Introduction {#sec:intro} ============ Consider the stochastic delay differential equation (SDDE) $$\begin{aligned} \label{eq:motivsys_SDDE} dx(t)=\left(\mu x(t-1)+x^3(t)\right)dt + {\varepsilon}c_1x(t-1)dV_1(t) + {\varepsilon}^2 c_2dV_2(t),\end{aligned}$$ where $0<{\varepsilon}\ll 1$ and $V_1,V_2$ are Wiener processes. The above equation represents a noisy perturbation of the following deterministic system: $$\begin{aligned} \label{eq:motivsys} \dot{x}(t)=\mu x(t-1)+x^3(t).\end{aligned}$$ The linear system corresponding to is $$\begin{aligned} \label{eq:motivsys_linear} \dot{x}(t)=\mu x(t-1).\end{aligned}$$ Seeking a solution of the form $x(t)=e^{t\lambda}$ to the linear system, we find that $\lambda$ must satisfy the characteristic equation $\lambda - \mu e^{-\lambda}=0$. When $\mu \in (-\frac{\pi}{2},0)$, all roots of the characteristic equation have negative real parts (see corollary 3.3 on page 53 of [@Stepanbook]). When $\mu=-\frac{\pi}{2}$ a pair of roots $\pm i\frac{\pi}{2}$ are on the imaginary axis and all other roots have negative real parts. When $\mu<-\frac{\pi}{2}$ some of the roots have positive real part. Hence, the system is on the verge of instability at $\mu=-\frac{\pi}{2}$. Close to the verge of instability, the behaviour of the solution is oscillatory with amplitude increasing or decreasing depending on whether the root with the largest real part has positive real part or negative. To study close to the verge of instability, set $\mu=-\frac{\pi}{2}+{\varepsilon}^2\tilde{\mu}$ and zoom-in near $x=0$, i.e. write $y(t)={\varepsilon}^{-1}x(t)$ for $x$ governed by . We get $$\begin{aligned} \label{eq:exmpHpfyprc} dy(t)=-\frac{\pi}{2}y(t-1)dt+{\varepsilon}^2(y^3(t)+\tilde{\mu}y(t-1))dt+{\varepsilon}c_1y(t-1)dV_1(t) + {\varepsilon}c_2dV_2(t).\end{aligned}$$ The equations studied in this paper are of the above kind. Before stating the equations in more precise terms below, we describe briefly the motivation for studying such equations. Delay equations at the verge of instability arise, for example, in machining processes [@Gabor1], in the response of eye-pupil to incident light [@Long90], in human balancing [@Yao_delay_SDDE_stand]. In machining processes, the motion of the cutting tool can be described by a DDE—the tool cuts a work-piece placed on a rotating shaft and the delay is the time-period of the rotating shaft. For each rotation period there is certain rate of cutting below which the tool is stable and above which the tool breaks. The inhomogenities in the properties of the material being cut can be modeled by noise (see [@buck_kusk_noise_machine_tool]). The eye-pupil exhibits oscillations in response to incident light—however there is some delay in the response because neurons have finite processing speed. This phenomenon can be modeled using a DDE at the verge of instability [@Long90]. So, a study of the effect of noise perturbations on ‘DDE at the verge of instability’ is indeed useful. Now we describe the equations studied in this article in more precise terms. Let $\{y_t : t \geq 0\}$ be an ${\mathbb{R}}$-valued process governed by an SDDE. Let $r>0$ be the maximum of the delays involved in the drift and diffusion coefficients of the SDDE. To find the evolution at time $t$ of the process, we need to keep track of $y_s$ for $t-r\leq s \leq t$. For this purpose, let ${\mathcal{C}}:=C([-r,0];{\mathbb{R}})$ be the space of ${\mathbb{R}}$-valued continuous functions on $[-r,0]$, and equip ${\mathcal{C}}$ with sup norm: $$\|\|\eta\|\|:=\sup_{\theta \in [-r,0]}\|\eta(\theta)\|, \qquad \text{for } \eta\in {\mathcal{C}}.$$ Define the segment extractor ${\Pi}_t$ as follows: for $f\in C([-r,\infty);{\mathbb{R}})$, $$({\Pi}_tf)(\theta)=f(t+\theta), \qquad \theta \in [-r,0], \quad t\in[0,\infty).$$ Then, consider equation of the form: $$\begin{aligned} \label{eq:exmpHpfyprc_C1} dy(t)=L_0({\Pi}_t y)dt+{\varepsilon}^2G({\Pi}_ty)dt+{\varepsilon}L_1({\Pi}_ty)dV_1(t) + {\varepsilon}c_2dV_2(t), \qquad t\geq 0,\end{aligned}$$ where $L_0,L_1:{\mathcal{C}}\to {\mathbb{R}}$ are bounded linear operators, $G:{\mathcal{C}}\to {\mathbb{R}}$, and $V_i$ are Wiener processes. Of course, as an initial condition we specify ${\Pi}_0y=\xi \in {\mathcal{C}}$. If we choose the maximum delay $r=1$ and $L_0(\eta)=-\frac{\pi}{2}\eta(-1)$, $L_1(\eta)=c_1\eta(-1)$, $G(\eta)=\eta^3(0)+\tilde{\mu}\eta(-1)$, we see that represents . We make the following assumption on $L_0$ to reflect the Hopf-bifurcation scenario: \[ass:assumptondetsys\] We assume that the corresponding unperturbed DDE $$\begin{aligned} \label{eq:detDDE} \dot{x}(t)=L_0({\Pi}_t x)\end{aligned}$$ is critical, i.e. a pair of roots ($\pm i{\omega}_c$) of the characteristic equation $\lambda-L_0(e^{\lambda \,\cdot})=0$ are on the imaginary axis (critical eigenvalues) and all other roots have negative real parts (stable eigenvalues). Roughly speaking, under the above assumption, the solution of the unperturbed system is oscillatory with constant amplitude. However for the perturbed system , it can be shown that for the amplitude of oscillation of $y$ to change considerably, we need to wait for a time of order ${\varepsilon}^{-2}$. Hence we change the time scale, i.e. define $Y^{\varepsilon}(t)=y(t/{\varepsilon}^2)$. To be able to put the rescaled process $Y^{\varepsilon}$ in a form akin to we need to define the rescaled segment extractor ${{{\Pi}}^{{\varepsilon}}}_t$ as follows: for $f\in C([-{\varepsilon}^2r,\infty);{\mathbb{R}})$, $$({{{\Pi}}^{{\varepsilon}}}_tf)(\theta)=f(t+{\varepsilon}^2\theta), \qquad \theta \in [-r,0], \quad t\in[0,\infty).$$ Then, , with $Y^{\varepsilon}(t)=y(t/{\varepsilon}^2)$, can be written as $$\begin{aligned} \label{eq:exmpHpfyprc_C} dY^{\varepsilon}(t)={\varepsilon}^{-2}L_0({{{\Pi}}^{{\varepsilon}}}_t Y^{\varepsilon})dt+G({{{\Pi}}^{{\varepsilon}}}_t Y^{\varepsilon})dt+ L_1({{{\Pi}}^{{\varepsilon}}}_t Y^{\varepsilon})dW_1(t) + c_2dW_2(t),\end{aligned}$$ where $W_i(t)={\varepsilon}V_i(t/{\varepsilon}^2)$ are again Wiener processes. In this paper, we restrict to equations of the form: $$\begin{aligned} \label{eq:considerMAINadd} \begin{cases}d{X^{\varepsilon}}(t)&={\varepsilon}^{-2}L_0({{{\Pi}}^{{\varepsilon}}}_t {X^{\varepsilon}})dt+G({{{\Pi}}^{{\varepsilon}}}_t {X^{\varepsilon}})dt+ \sigma dW(t), \qquad t\in [0,T],\\ {X^{\varepsilon}}(t)&=\xi({\varepsilon}^{-2}t), \qquad t\in[-{\varepsilon}^2r,0], \quad \xi\in {\mathcal{C}},\\ G(\eta)&=\int_{-r}^0\eta(\theta)d\nu_1(\theta)+\int_{-r}^0\eta^3(\theta)d\nu_3(\theta), \end{cases}\end{aligned}$$ where for $i=1,3$, $\nu_i:[-r,0]\to {\mathbb{R}}$, are bounded functions continuous from the left on $(-r,0)$ and normalized with $\nu_i(0)=0$; and also equations of the form: $$\begin{aligned} \label{eq:considerMAINmul} \begin{cases}d{X^{\varepsilon}}(t)&={\varepsilon}^{-2}L_0({{{\Pi}}^{{\varepsilon}}}_t {X^{\varepsilon}})dt+G({{{\Pi}}^{{\varepsilon}}}_t {X^{\varepsilon}})dt+ L_1({{{\Pi}}^{{\varepsilon}}}_t {X^{\varepsilon}}) dW(t), \qquad t\in [0,T],\\ {X^{\varepsilon}}(t)&=\xi({\varepsilon}^{-2}t), \qquad t\in[-{\varepsilon}^2r,0], \quad \xi\in {\mathcal{C}},\\ &\|G(\eta_1)-G(\eta_2)\|\,\leq\,K_G\|\|\eta_1-\eta_2\|\|, \qquad G(0)=0. \end{cases}\end{aligned}$$ We refer to as the additive noise case and as the mulitplicative noise case. In both cases we assume that the initial condition $\xi$ is deterministic (not a random variable). Equations of the form $d{X^{\varepsilon}}(t)={\varepsilon}^{-2}L_0({{{\Pi}}^{{\varepsilon}}}_t {X^{\varepsilon}})dt+G({{{\Pi}}^{{\varepsilon}}}_t {X^{\varepsilon}})dt+ \sigma dW(t)$ were studied in [@LNNSN_SDDEavg] but the coefficient $G$ was assumed to be Lipschitz. A quantity ${\mathcal{H}}^{\varepsilon}$ was identified which, roughly speaking, gives the amplitude of oscillations of ${X^{\varepsilon}}$. It was shown that the distribution of ${\mathcal{H}}^{\varepsilon}$ converges weakly to the distribution of a process ${\mathcal{H}}^0$ governed by a stochastic differential equation (SDE) without delay. For small ${\varepsilon}$, this ${\mathcal{H}}^0$ gives good approximation for the dynamics of ${X^{\varepsilon}}$. The advantage is three fold: (i) equations without delay are easier to analyze, (ii) for numerical simulations, ${X^{\varepsilon}}$ requires storage of ${{{\Pi}}^{{\varepsilon}}}_t{X^{\varepsilon}}$ (the entire segment) whereas ${\mathcal{H}}^0$ requires just the storage of current value ${\mathcal{H}}^0_t$, (iii) numerical simulation of ${X^{\varepsilon}}$ requires very small time-step for integration because the drift coefficient is of the order ${\varepsilon}^{-2}$, whereas ${\mathcal{H}}^0$ does not require such a small time-step. In this article we relax the Lipschitz assumption on the coefficient $G$ for the additive noise case. Note that the presence of $\nu_3$ in makes $G$ non-Lipschitz. The process ${\mathcal{H}}^{\varepsilon}$ mentioned above encodes information only about the critical eigenspace (space spanned by the eigenvectors corresponding to the imaginary roots of the characteristic equation), and to obtain the convergence to ${\mathcal{H}}^0$ one needs to show that the projection of ${X^{\varepsilon}}$ onto stable eigenspace is small (details would be provided later). In [@LNNSN_SDDEavg] this was easy to show because of the Lipschitz condition on $G$. In this article we need to follow a different approach. The case of multiplicative noise is also considered here. However, for the multiplicative noise case the Lipschitz condition could not be relaxed. The presence of cubic nonlinearites causes the following problem: in trying to estimate a moment of certain order we encounter terms with higher order moments. [@LNPRLE] discusses the approaches in the literature towards SDDE at the verge of instability and shows the mistakes and shortcomings of those approaches (see section 1 and appendix A of [@LNPRLE]). Hence, here we refrain from mentioning these works again. Though here we discuss rigorously only ${\mathbb{R}}$-valued processes, the multi-dimensional processes are dealt with in [@LNPRLE] without proofs. An applications-oriented reader would benefit from [@LNPRLE] rather than this article. Before stating the goals of this paper, we give a brief overview of the unperturbed system , and the variation-of-constants formula relating the solutions of and with . The material in section \[subsec:unpertsys\_overview\] can be found in chapter 7 of [@Halebook] (see also [@Diekmanbook]). The unperturbed system {#subsec:unpertsys_overview} ----------------------- The solution of gives rise to the strongly continuous semigroup $T(t):{\mathcal{C}}\to {\mathcal{C}}, \, t\geq 0$, defined by $T(t){\Pi}_0x={\Pi}_t x$. The state space ${\mathcal{C}}$ can be split in the form ${\mathcal{C}}=P\oplus Q$ where $P=\text{span}_{{\mathbb{R}}}\{\Phi_1,\,\Phi_2\}$ where $$\Phi_1(\theta)=\cos({\omega}_c\theta), \qquad \Phi_2(\theta)=\sin({\omega}_c\theta), \qquad \theta\in[-r,0].$$ Write $\Phi=[\Phi_1,\,\Phi_2]$. Any $\eta \in P$ can be written as $\Phi z=z_1\Phi_1+z_2\Phi_2$ for $z\in {\mathbb{R}}^2$, i.e. $\Phi$ is a basis for the two-dimensional space $P$ and the $z$ are coordinates of $\eta \in P$ with respect to the basis $\Phi$. Let $\pi$ denote the projection of ${\mathcal{C}}$ onto $P$ along $Q$, i.e. $\pi:{\mathcal{C}}\to P$ with $\pi^2=\pi$ and $\pi(\eta)=0$ for $\eta\in Q$. The operator $\pi$ can be written down explicitly, but we would not need the explicit form. ### Behaviour of the solution on $P$ and $Q$ It is easy to see that ${\Pi}_tx=\cos(\omega_c(t+\cdot))$ is a solution to with the initial condition ${\Pi}_0x=\cos(\omega_c\cdot)$, and ${\Pi}_tx=\sin(\omega_c(t+\cdot))$ is a solution to with the initial condition ${\Pi}_0x=\sin(\omega_c\cdot)$. Using the identity $\cos(\omega_c(t+\cdot))=\cos(\omega_ct)\cos(\omega_c\cdot)-\sin(\omega_ct)\sin(\omega_c\cdot)$ and the linearity of $L_0$, it can be shown that $$\begin{aligned} \label{eq:TPhi_eq_PhieB} T(t)\Phi(\cdot)=\Phi(\cdot)e^{Bt}, \qquad B=\left[\begin{array}{cc}0 & \omega_c \\ -\omega_c & 0 \end{array}\right].\end{aligned}$$ There exists positive constants $\kappa$ and $K$ such that $$\begin{aligned} \label{eq:expdecesti} \|\|T(t)\phi\|\|\leq Ke^{-\kappa t}\|\|\phi\|\|, \qquad \quad \forall\, \phi \in Q. \end{aligned}$$ The above is a consequence of the fact that, except for the roots $\pm i{\omega}_c$ all other roots of the characteristic equation have negative real parts. Write the solution to as $${\Pi}_tx=\pi{\Pi}_tx+(1-\pi){\Pi}_tx=:\Phi z(t)+y_t$$ where $z$ is ${\mathbb{R}}^2$-valued and $y$ is ${\mathcal{C}}$ valued. Then we find that[^1] $z$ oscillate harmonically according to $\dot{z}(t)=Bz(t)$ and $\|\|y_t\|\|$ decays exponentially fast, i.e. $$\begin{aligned} \label{eq:expdecunpertsys} \|\|y_t\|\|\,\,\leq\,\,Ke^{-\kappa t}\|\|y_0\|\|.\end{aligned}$$ The variation-of-constants formula ---------------------------------- The solution of the perturbed systems or can be expressed in terms of the solution of with the initial condition ${\Pi}_0x={\mathbf{1}_{\{0\}}}$ where $${\mathbf{1}_{\{0\}}}(\theta)=\begin{cases} 1, \qquad \theta=0, \\ 0, \qquad \theta \in [-r,0).\end{cases}$$ However, note that ${\mathbf{1}_{\{0\}}}$ does not belong to ${\mathcal{C}}$ and so we need to extend the space ${\mathcal{C}}$ to accommodate the discontinuity. See p.192-193, 206-207 of [@SEAM_book] for the results pertaining to the extension. Let ${\hat{{\mathcal{C}}}}:=\hat{C}([-r,0];{\mathbb{R}})$ be the Banach space of all bounded measurable maps $[-r,0]\to {\mathbb{R}}$, given the sup norm. Solving the unperturbed system for initial conditions in ${\hat{{\mathcal{C}}}}$, we can extend the semigroup $T$ to one on ${\hat{{\mathcal{C}}}}$. Denote the extension by ${\hat{T}}$. Again ${\hat{{\mathcal{C}}}}$ splits in the form ${\hat{{\mathcal{C}}}}=P\oplus \hat{Q}$. The projection $\pi$ can be extended to ${\hat{{\mathcal{C}}}}$. The extension is denoted by ${\hat{\pi}}$. There exists a two component column vector ${\tilde{\Psi}}\in {\mathbb{R}}^2$ such that $$\begin{aligned} \label{eq:piIdeqPsiz} {\hat{\pi}}{\mathbf{1}_{\{0\}}}= \Phi {\tilde{\Psi}}= {\tilde{\Psi}}_1\Phi_1 + {\tilde{\Psi}}_2\Phi_2. \end{aligned}$$ Also, there exists positive constants $\kappa$ and $K$ such that $$\begin{aligned} \label{eq:expdecesti} \|\|{\hat{T}}(t)\phi\|\|\leq Ke^{-\kappa t}\|\|\phi\|\|, \qquad \quad \forall\, \phi \in \hat{Q}. \end{aligned}$$ ### Additive noise case The solution of can be represented as (see theorem IV.4.1 on page 200 in [@SEAM_book]) $$\begin{aligned} \label{eq:vocformstateW_full_addn} {{{\Pi}}^{{\varepsilon}}_{t}{X^{\varepsilon}}} \,\,=\,\,{\hat{T}}(t/{\varepsilon}^2){{{\Pi}}^{{\varepsilon}}_{0}{X^{\varepsilon}}}\,\,&+\,\,\int_0^t{\hat{T}}(\frac{t-u}{{\varepsilon}^2}){\mathbf{1}_{\{0\}}}G({{{\Pi}}^{{\varepsilon}}_{u}{X^{\varepsilon}}})du \,\,+\,\,\int_0^t{\hat{T}}(\frac{t-u}{{\varepsilon}^2}){\mathbf{1}_{\{0\}}}\, \sigma dW_u.\end{aligned}$$ The third term in the RHS of is an element in ${\mathcal{C}}$ and its value at $\theta \in [-r,0]$ is given by $\int_0^t\left({\hat{T}}(\frac{t-u}{{\varepsilon}^2}){\mathbf{1}_{\{0\}}}\right)(\theta)\, \sigma dW_u.$ Write $${{{\Pi}}^{{\varepsilon}}_{t}{X^{\varepsilon}}}=\Phi z^{\varepsilon}_t+y^{\varepsilon}_t.$$ Here $(y^{\varepsilon}_t)_{t\geq 0}$ is the ${\mathcal{C}}$-valued process $y^{\varepsilon}_t=(1-\pi){{{\Pi}}^{{\varepsilon}}_{t}{X^{\varepsilon}}}$ and $\Phi z_t = \pi{{{\Pi}}^{{\varepsilon}}_{t}{X^{\varepsilon}}}$. Note that $z$ is ${\mathbb{R}}^2$-valued process. Taking projection of onto the space $P$, and using the facts (i) $\pi{{{\Pi}}^{{\varepsilon}}_{t}{X^{\varepsilon}}}=\Phi z^{\varepsilon}_t$, (ii) ${\hat{T}}$ commutes with ${\hat{\pi}}$, (iii) ${\hat{\pi}}{\mathbf{1}_{\{0\}}}= \Phi {\tilde{\Psi}}$, (iv) ${\hat{T}}(t)\Phi z =\Phi e^{tB}z$, we get for $z^{\varepsilon}$ (see corollary IV.4.1.1 on page 207 in [@SEAM_book]) $$\begin{aligned} \label{eq:zproj_addn} dz^{\varepsilon}_t={\varepsilon}^{-2}Bz^{\varepsilon}_tdt+{\tilde{\Psi}}G(\Phi z^{\varepsilon}_t+y^{\varepsilon}_t)dt+{\tilde{\Psi}}\sigma dW_t, \qquad \Phi z^{\varepsilon}_0=\pi {{{\Pi}}^{{\varepsilon}}_{0}{X^{\varepsilon}}}.\end{aligned}$$ Using the fact that ${\hat{T}}$ commutes with ${\hat{\pi}}$, $y^{\varepsilon}_t$ satisfies $$\begin{aligned} \label{eq:vocformstateW_addn} y^{\varepsilon}_t \,\,=\,\,{\hat{T}}(t/{\varepsilon}^2)y^{\varepsilon}_0\,\,&+\,\,\int_0^t{\hat{T}}(\frac{t-u}{{\varepsilon}^2})(1-{\hat{\pi}}){\mathbf{1}_{\{0\}}}G(\Phi z^{\varepsilon}_u+y^{\varepsilon}_u)du \,\,+\,\,\int_0^t{\hat{T}}(\frac{t-u}{{\varepsilon}^2})(1-{\hat{\pi}}){\mathbf{1}_{\{0\}}}\, \sigma dW_u.\end{aligned}$$ ### Multiplicative noise case The solution of can be represented in a form analogous to with $\sigma$ replaced by $L_1({{{\Pi}}^{{\varepsilon}}_{u}{X^{\varepsilon}}})$ (see [@Reis_Emery_Ineq_voc_for_SDDE]): $$\begin{aligned} \label{eq:vocformstateW_full} {{{\Pi}}^{{\varepsilon}}_{t}{X^{\varepsilon}}} \,\,=\,\,{\hat{T}}(t/{\varepsilon}^2){{{\Pi}}^{{\varepsilon}}_{0}{X^{\varepsilon}}}\,\,&+\,\,\int_0^t{\hat{T}}(\frac{t-u}{{\varepsilon}^2}){\mathbf{1}_{\{0\}}}G({{{\Pi}}^{{\varepsilon}}_{u}{X^{\varepsilon}}})du \,\,+\,\,\int_0^t{\hat{T}}(\frac{t-u}{{\varepsilon}^2}){\mathbf{1}_{\{0\}}}\, L_1({{{\Pi}}^{{\varepsilon}}_{u}{X^{\varepsilon}}}) dW_u.\end{aligned}$$ For the projections onto $P$ and $Q$ we have: $$\begin{aligned} \label{eq:zproj_muln} dz^{\varepsilon}_t={\varepsilon}^{-2}Bz^{\varepsilon}_tdt+{\tilde{\Psi}}G(\Phi z^{\varepsilon}_t+y^{\varepsilon}_t)dt+{\tilde{\Psi}}L_1(\Phi z^{\varepsilon}_t+y^{\varepsilon}_t) dW, \qquad \Phi z^{\varepsilon}_0=\pi {{{\Pi}}^{{\varepsilon}}_{0}{X^{\varepsilon}}},\end{aligned}$$ $$\begin{aligned} \label{eq:vocformstateW} y^{\varepsilon}_t \,\,=\,\,{\hat{T}}(t/{\varepsilon}^2)y^{\varepsilon}_0\,\,&+\,\,\int_0^t{\hat{T}}(\frac{t-u}{{\varepsilon}^2})(1-{\hat{\pi}}){\mathbf{1}_{\{0\}}}G(\Phi z^{\varepsilon}_u+y^{\varepsilon}_u)du\\ \notag & \,\,+\,\,\int_0^t{\hat{T}}(\frac{t-u}{{\varepsilon}^2})(1-{\hat{\pi}}){\mathbf{1}_{\{0\}}}\, L_1(\Phi z^{\varepsilon}_u+y^{\varepsilon}_u) dW_u.\end{aligned}$$ Crucial role would be played in proofs by $$\begin{aligned} \label{eq:stabsoldefprob} {\gamma}(t):=({\hat{T}}(t)(1-{\hat{\pi}}){\mathbf{1}_{\{0\}}})(0).\end{aligned}$$ From we already know that $$\begin{aligned} \label{eq:stabsolexpdecesti} \|{\gamma}(t)\|\leq Ke^{-\kappa t}\|\|(1-{\hat{\pi}}){\mathbf{1}_{\{0\}}}\|\|, \qquad t\geq 0.\end{aligned}$$ Further, for $t> 0$ $$\begin{aligned} \label{eq:expdecayforhprime} \left\|\frac{d}{dt}{\gamma}(t)\right\|=\left\|L_0\bigg({\hat{T}}(t)(1-{\hat{\pi}}){\mathbf{1}_{\{0\}}}\bigg)\right\|\leq \|\|L_0\|\|_{op}\,\|\|{\hat{T}}(t)(1-{\hat{\pi}}){\mathbf{1}_{\{0\}}}\|\|\leq \|\|L_0\|\|_{op}\,\|\|(1-{\hat{\pi}}){\mathbf{1}_{\{0\}}}\|\|\,Ke^{-\kappa t}.\end{aligned}$$ Thus, both ${\gamma}$ and ${\gamma}'$ have exponential decay. Goal of this paper {#sec:subsec:goals} ------------------ Let ${X^{\varepsilon}}$ evolve according to either or . Write ${{{\Pi}}^{{\varepsilon}}_{t}{X^{\varepsilon}}}=\Phi z^{\varepsilon}_t+y^{\varepsilon}_t,$ and define $$\begin{aligned} \label{eq:ydecstbpdef} {\mathfrak{Y}}^{\varepsilon}_t:={\hat{T}}(t/{\varepsilon}^2)y^{\varepsilon}_0, \qquad {\mathscr{Y}}^{\varepsilon}_t:=y^{\varepsilon}_t-{\mathfrak{Y}}^{\varepsilon}_t.\end{aligned}$$ Note that ${\mathfrak{Y}}^{\varepsilon}_t$ depends only on the unperturbed system . Given the initial condition ${{{\Pi}}^{{\varepsilon}}}_0 {X^{\varepsilon}}$, ${\mathfrak{Y}}^{\varepsilon}_t$ is a deterministic quantity. Note that $\|\|{\mathfrak{Y}}^{\varepsilon}_t\|\|$ decays exponentially fast: $$\begin{aligned} \label{eq:ydecexpfastdec} \|\|{\mathfrak{Y}}^{\varepsilon}_t\|\|\,\,\leq\,\,Ke^{-\kappa t/{\varepsilon}^2}\|\|(1-\pi){{{\Pi}}^{{\varepsilon}}}_0 {X^{\varepsilon}}\|\|.\end{aligned}$$ ### Goal for the multiplicative noise case {#subsubsec:goalsformultipnoise} Roughly speaking, the goals are 1. Show that, until time $T>0$, ${\mathbb{E}}\sup_{t\in[0,T]}\|\|{\mathscr{Y}}^{{\varepsilon}}_t\|\|^n\,\,\xrightarrow{{\varepsilon}\to 0}0$, so that we can approximate $y^{\varepsilon}_t$ with the deterministic quantity ${\mathfrak{Y}}^{\varepsilon}_t$ 2. Consider the process $$\begin{aligned} \label{eq:zproj_muln_nodel} d{\bf z}^{\varepsilon}_t={\varepsilon}^{-2}B{\bf z}^{\varepsilon}_tdt+{\tilde{\Psi}}G(\Phi {\bf z}^{\varepsilon}_t)dt+{\tilde{\Psi}}L_1(\Phi {\bf z}^{\varepsilon}_t) dW, \qquad \Phi {\bf z}^{\varepsilon}_0=\pi {{{\Pi}}^{{\varepsilon}}_{0}{X^{\varepsilon}}}.\end{aligned}$$ Note that ${\bf z}^{\varepsilon}$ is two-dimensional process without delay totally ignoring $y^{\varepsilon}$. Show that $$\begin{aligned} \label{eq:goalsformultipnoise_z} {\mathbb{E}}\sup_{t\in [0,T]}\|\|{\bf z}^{\varepsilon}_t-z^{\varepsilon}_t\|\|_2^2 \xrightarrow{{\varepsilon}\to 0} 0\end{aligned}$$ where $\|\|\cdot\|\|_2$ is $\ell_2$ norm of vectors in ${\mathbb{R}}^2$. The above tasks justify the approximation of ${{{\Pi}}^{{\varepsilon}}_{t}{X^{\varepsilon}}}$ by $\Phi {\bf z}^{\varepsilon}_t + {\mathfrak{Y}}^{\varepsilon}_t$ for small ${\varepsilon}$. Note that ${\bf z}^{\varepsilon}$ is a two-dimensional process without delay and ${\mathfrak{Y}}^{\varepsilon}_t$ is an exponentially decaying deterministic process. For small ${\varepsilon}$ one could study this non-delay system instead of the original stochastic DDE . The advantage is that the 2-dimensional system without delay would be easier to analyze or simulate numerically. Further simplification can be obtained by studying the process $$\begin{aligned} \label{eq:hprcdefz2n} {\mathcal{H}}^{\varepsilon}_t:=\frac12\|\|z^{\varepsilon}_t\|\|_2^2. \end{aligned}$$ Roughly speaking[^2], $\sqrt{2{\mathcal{H}}^{\varepsilon}}$ is the amplitude of oscillations of ${X^{\varepsilon}}$. We will show that there is a constant $C$ such that $$\begin{aligned} \label{eq:hprcconvgreqzbound} {\mathbb{E}}\sup_{t\in [0,T]}\|\|{\bf z}^{\varepsilon}_t\|\|_2^2<C\end{aligned}$$ for all ${\varepsilon}$ smaller than some ${\varepsilon}_$. Using and it follows that $$\begin{aligned} \label{eq:hprcconvgmul} {\mathbb{E}}\sup_{t\in [0,T]}\|{\mathcal{H}}^{\varepsilon}_t-\frac12\|\|{\bf z}^{\varepsilon}_t\|\|_2^2\| \xrightarrow{{\varepsilon}\to 0} 0.\end{aligned}$$ One can use standard averaging techniques for stochastic differential equations (without delay) to show that the distribution of $\frac12\|\|{\bf z}^{\varepsilon}\|\|_2^2$ converges to the distribution of some one-dimensional process ${\mathcal{H}}^0$ without delay. By theorem 3.1 in [@Billingsley1999], the distribution of ${\mathcal{H}}^{\varepsilon}$ converges to the distribution of ${\mathcal{H}}^0$. For small ${\varepsilon}$, the distribution of ${\mathcal{H}}^0$ gives a good approximation to the distribution of ${\mathcal{H}}^{\varepsilon}$. The advantages of having ${\mathcal{H}}^0$ were mentioned in section \[sec:intro\]. ### Goal for the additive noise case {#subsubsec:goal4addnoisecase} The presence of cubic nonlinearites causes the following problem: in trying to estimate a moment of certain order we face the task of estimating terms with higher order moments. So the approach taken for does not work here. We take the following approach. Recall the projection operator $\pi:{\mathcal{C}}\to P$. Fix a constant $C_{{\mathfrak{e}}}>0$ and define the stopping time ${\mathfrak{e}^{\varepsilon}}=\inf\{t\geq 0:\|\|\pi {{{\Pi}}^{{\varepsilon}}}_t{X^{\varepsilon}}\|\|\geq C_{{\mathfrak{e}}}\}$. (Note that the stopping time depends on ${\varepsilon}$). - Show that for $t\in [0,T\wedge{\mathfrak{e}^{\varepsilon}}]$, $\|\|{\mathscr{Y}}^{\varepsilon}_t\|\|$ is small with high probability - Define a 2-dimensional process ${\bf z}^{\varepsilon}$ as $$\begin{aligned} d{\bf z}^{\varepsilon}_t={\varepsilon}^{-2}B{\bf z}^{\varepsilon}_tdt+{\tilde{\Psi}}G(\Phi {\bf z}^{\varepsilon}_t)dt+{\tilde{\Psi}}\sigma dW, \qquad \Phi {\bf z}^{\varepsilon}_0=\pi {{{\Pi}}^{{\varepsilon}}_{0}{X^{\varepsilon}}}.\end{aligned}$$ Note that ${\bf z}^{\varepsilon}$ is a 2 dimensional process without delay. Show that for $t\in [0,T\wedge{\mathfrak{e}^{\varepsilon}}]$, error in approximating $z^{\varepsilon}$ by ${\bf z}^{\varepsilon}$ is small with high probability - Using estimates on ${\bf z}^{\varepsilon}$ process, get rid of the stopping time and obtain approximation results until time $T$, by leveraging some arbitrarily small probability. The stopping time helps in arriving at a bound on the norm of stable-mode $(1-\pi){{{\Pi}}^{{\varepsilon}}}_t{X^{\varepsilon}}$ without worrying about what happens to the critical-mode $\pi{{{\Pi}}^{{\varepsilon}}}_t{X^{\varepsilon}}$. Examples illustrating the usefulness of the above approximation results are shown in sections \[subsec:mul\_examp\] and \[sec:example\]. For related work on stochastic partial differential equations see [@Blom_amp_eq]. However note that in [@Blom_amp_eq] the bifurcation scenario is different—analogous situation in the DDE framework would be if one root of characteristic equation is zero and all other roots have negative real parts. Multiplicative noise {#sec:mulnoi} ==================== In this section we consider with $T>0$ fixed. The constants here can depend on $T$. The first goal is to show that ${\mathbb{E}}\sup_{t\in [0,T]}\|\|{\mathscr{Y}}^{{\varepsilon}}_t\|\|^n\to 0$, which is the content of proposition \[prop:mul:stabnorm\]. For this purpose, we use the variation of constants formulas –. Recalling the definition of ${\mathscr{Y}}^{{\varepsilon}}$ from , to estimate ${\mathbb{E}}\sup_{t\in [0,T]}\|\|{\mathscr{Y}}^{{\varepsilon}}_t\|\|^n$, we need to estimate the last two terms on the RHS of . Roughly speaking, the integral in the last term of RHS of can be split as $\int_0^s=\int_0^{s-{\varepsilon}^{\delta}r}+\int_{s-{\varepsilon}^{\delta}r}^s$ with $0<\delta<2$. For $\int_0^{s-{\varepsilon}^{\delta}r}$ we can use exponential decay of ${\hat{T}}$ on $\hat{Q}$. For $\int_{s-{\varepsilon}^{\delta}r}^s$, making note that the length of the interval of integration is small ($r{\varepsilon}^\delta$), we need to be concerned with increments of Wiener process over small intervals, i.e. the modulus of continuity of the Wiener process. Lemma \[lem:mul:quest:supuntilTisconst\] is needed to be able to use the results of [@FischerNappo] on moments of modulus-of-continuity of Ito processes. Using results from [@FischerNappo], proposition \[prop:mul:supUpsT\] shows that the stochastic term in is small. Then, straight forward estimation yields proposition \[prop:mul:stabnorm\] which is the result that we need. \[lem:mul:quest:supuntilTisconst\] Fix $n\geq 0$. There exists constants $\mathfrak{C}>0$ and ${\varepsilon}_>0$ such that $\forall {\varepsilon}<{\varepsilon}_$, $$\begin{aligned} \label{eq:quest2} {\mathbb{E}}\sup_{t\in[0,T]}\|\|{{{\Pi}}^{{\varepsilon}}}_t{X^{\varepsilon}}\|\|^n<\mathfrak{C}.\end{aligned}$$ Proof is given in appendix \[apsec:prf:lem:mul:quest:supuntilTisconst\]. Note that, though one of the drift coefficients in is of the order ${\varepsilon}^{-2}$, the constant $\mathfrak{C}$ above does not depend on ${\varepsilon}$. Proof uses: (i) the variation-of-constants formula to exploit the exponential decay and on $\hat{Q}$, and oscillatory behaviour on $P$; (ii) Burkholder-Davis-Gundy inequality to estimate supremum of martingales by their quadratic-variation; and then (iii) Gronwall inequality. \[def:modcont\] Define the modulus of continuity for $f:[0,\infty)\to {\mathbb{R}}$ by $${\mathfrak{w}}(a,b;f)=\sup_{\substack{\|u-v\|\,\leq\, a \\ u,v\,\in\, [0,b]}}\|f(u)-f(v)\|.$$ Define $$\begin{aligned} \label{eq:UpsandZdef} \Upsilon^{\varepsilon}_s:=\sup_{\theta \in [-r,0]}\left\|\int_0^s\left({\hat{T}}(\frac{s-u}{{\varepsilon}^2})(1-{\hat{\pi}}){\mathbf{1}_{\{0\}}}\right)(\theta) dZ_u\right\|, \qquad Z_t:=\int_0^tL_1({{{\Pi}}^{{\varepsilon}}_{s}{X^{\varepsilon}}}) d{W}_s.\end{aligned}$$ Note that $Z$ dependens also on ${\varepsilon}$. \[prop:mul:supUpsT\] Fix $n\geq 1$. There exists constant $\hat{C}>0$ and a family of constants $\hat{{\varepsilon}}_{\delta}>0$ (indexed by $0<\delta<2$) such that, given $\delta \in (0,2)$ we have for ${\varepsilon}< \hat{{\varepsilon}}_{\delta}$ $$\begin{aligned} \label{prop:mul:supUpsT_statement_mully} {\mathbb{E}}\sup_{s\in [0,T]}(\Upsilon^{\varepsilon}_s)^n \,\,\,\leq\,\,\,\hat{C}\left({r{\varepsilon}^\delta}\ln\left(\frac{2T}{r{\varepsilon}^\delta}\right)\right)^{n/2} \,\,\xrightarrow{{\varepsilon}\to 0} 0. \end{aligned}$$ Proof is given in appendix \[apsec:prf:prop:mul:supUpsT\]. The essential idea of writing $\int_0^s=\int_0^{s-{\varepsilon}^{\delta}r}+\int_{s-{\varepsilon}^{\delta}r}^s$ and using [@FischerNappo] is mentioned earlier. Let ${\mathfrak{Y}}^{\varepsilon}_t$ and ${\mathscr{Y}}^{\varepsilon}_t$ be as defined in . \[prop:mul:stabnorm\] Fix $n\geq 1$. $\exists {\varepsilon}_>0$ such that $\forall\,{\varepsilon}<{\varepsilon}_$, $$\begin{aligned} \label{prop:mul:stabnorm_statement_b} {\mathbb{E}}\sup_{s\in[0,T]}\|\|{\mathscr{Y}}^{{\varepsilon}}_s\|\|^n\,\,\leq\,\, {\varepsilon}^{2n}2^{n-1}\left(\frac{K_GK}{\kappa}\right)^n\mathfrak{C}\,+\,2^{n-1}{\mathbb{E}}\sup_{s\in [0,T]}(\Upsilon^{{\varepsilon}}_s)^n\,\,\xrightarrow{{\varepsilon}\to 0}0,\end{aligned}$$ where $\mathfrak{C}$ is from lemma \[lem:mul:quest:supuntilTisconst\]. Proof given in appendix \[apsec:prf:prop:mul:stabnorm\]. Recall that when we write ${{{\Pi}}^{{\varepsilon}}_{t}{X^{\varepsilon}}}=\Phi z^{\varepsilon}_t+y^{\varepsilon}_t$, the ${\mathbb{R}}^2$-valued process $z^{\varepsilon}$ satisfies equation . Removing the fast rotation induced by $B$, i.e. writing ${\mathfrak{z}^{\varepsilon}}_t=e^{-tB/{\varepsilon}^2}z^{\varepsilon}_t$ we have $$d{\mathfrak{z}^{\varepsilon}}=e^{-tB/{\varepsilon}^2}{\tilde{\Psi}}G(\Phi e^{tB/{\varepsilon}^2}{\mathfrak{z}^{\varepsilon}}_t+y^{\varepsilon}_t)dt+e^{-tB/{\varepsilon}^2}{\tilde{\Psi}}L_1(\Phi e^{tB/{\varepsilon}^2} {\mathfrak{z}^{\varepsilon}}_t+y^{\varepsilon}_t) dW_t, \qquad {\mathfrak{z}^{\varepsilon}}_0=z^{\varepsilon}_0.$$ Let ${\widehat{\mathfrak{z}}^{\varepsilon}}$ be governed by $$d{\widehat{\mathfrak{z}}^{\varepsilon}}=e^{-tB/{\varepsilon}^2}{\tilde{\Psi}}G(\Phi e^{tB/{\varepsilon}^2}{\widehat{\mathfrak{z}}^{\varepsilon}}_t+{\mathfrak{Y}}^{\varepsilon}_t)dt+e^{-tB/{\varepsilon}^2}{\tilde{\Psi}}L_1(\Phi e^{tB/{\varepsilon}^2} {\widehat{\mathfrak{z}}^{\varepsilon}}_t+{\mathfrak{Y}}^{\varepsilon}_t) dW_t, \qquad {\widehat{\mathfrak{z}}^{\varepsilon}}_0={\mathfrak{z}^{\varepsilon}}_0.$$ i.e. we are totally ignoring $y$ part except for the effect of initial condition (note that ${\mathfrak{Y}}^{\varepsilon}_t={\hat{T}}(t/{\varepsilon}^2)y^{\varepsilon}_0$). As an intermediate step towards the end goal, we want to show that, until time $T$ the error in approximating ${\mathfrak{z}^{\varepsilon}}$ by ${\widehat{\mathfrak{z}}^{\varepsilon}}$ is small. For this purpose, define $$\alpha^{\varepsilon}_t=\frac12\|\|{\mathfrak{z}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|_2^2=\frac12(({\mathfrak{z}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t)_1^2+({\mathfrak{z}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t)_2^2).$$ Here $({\mathfrak{z}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t)_i$ denotes the $i^{th}$ component of the ${\mathbb{R}}^2$-valued vector ${\mathfrak{z}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t$. Let $$\begin{aligned} \label{eq:BigGammaDriftCoefDef} \Gamma_t=\sum_{i=1}^2({\mathfrak{z}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t)_i(e^{-tB/{\varepsilon}^2}{\tilde{\Psi}})_i.\end{aligned}$$ Then $\alpha^{\varepsilon}_t$ is governed by $$d\alpha^{\varepsilon}_t=\mathscr{B}_tdt+\Sigma_t dW_t, \qquad \alpha^{\varepsilon}_0=0,$$ where $$\begin{aligned} \mathscr{B}_t&=\Gamma_t\left(G(\Phi e^{tB/{\varepsilon}^2}{\mathfrak{z}^{\varepsilon}}_t+y^{\varepsilon}_t)-G(\Phi e^{tB/{\varepsilon}^2}{\widehat{\mathfrak{z}}^{\varepsilon}}_t+{\mathfrak{Y}}^{\varepsilon}_t)\right)\\ &\qquad +\frac12 \|\| {\tilde{\Psi}}\|\|_2^2\bigg(L_1(\Phi e^{tB/{\varepsilon}^2} ({\mathfrak{z}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t))\,\,+\,\,L_1(y^{\varepsilon}_t-{\mathfrak{Y}}^{\varepsilon}_t)\bigg)^2,\end{aligned}$$ and $$\Sigma_t = \Gamma_t\bigg(L_1(\Phi e^{tB/{\varepsilon}^2} ({\mathfrak{z}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t))\,\,+\,\,L_1(y^{\varepsilon}_t-{\mathfrak{Y}}^{\varepsilon}_t)\bigg).$$ The following lemma gives processes dominating $\mathscr{B}_t$ and $\Sigma_t$. These help in applying Gronwall inequality to arrive at proposition \[prop:mul:alphaissmall\]. \[lem:mul:aux4comparison\] Define $$\begin{aligned} \mathfrak{B}(\alpha,p)\,\,&:=\,\,C_{\mathscr{B}}(\alpha+p^2), \qquad C_{\mathscr{B}}=2\|\| {\tilde{\Psi}}\|\|_2^2\|\|L_1\|\|^2+3\|\| {\tilde{\Psi}}\|\|_2K_G,\\ \mathfrak{S}^2(\alpha,p)\,\,&:=\,\,C_{\Sigma}(\alpha^2+p^4), \qquad C_{\Sigma}=16\|\| {\tilde{\Psi}}\|\|_2^2\|\|L_1\|\|^2.\end{aligned}$$ Then $\|\mathscr{B}_t\|\leq \mathfrak{B}(\alpha^{\varepsilon}_t,\|\|{\mathscr{Y}}^{\varepsilon}_t\|\|)$ and $\Sigma_t^2\leq\mathfrak{S}^2(\alpha^{\varepsilon}_t,\|\|{\mathscr{Y}}^{\varepsilon}_t\|\|)$ for $t\geq0$. Proof given in appendix \[apsec:prf:lem:mul:aux4comparison\] \[prop:mul:alphaissmall\] Fix $\delta \in (0,2)$. There exists constants $C,\hat{{\varepsilon}}_\delta>0$ such that $\forall {\varepsilon}< \hat{{\varepsilon}}_\delta$ $$\begin{aligned} {\mathbb{E}}\sup_{s\in [0,T]}\left(\alpha^{\varepsilon}_s\right)^2\,\,\leq&\,\,C\left(r{\varepsilon}^\delta\ln(\frac{2T}{r{\varepsilon}^\delta})\right)^2\,\,\xrightarrow{{\varepsilon}\to 0} 0.\end{aligned}$$ Proof is given in appendix \[apsec:prf:prop:mul:alphaissmall\] and is by using lemma \[lem:mul:aux4comparison\], result , applying Gronwall pathwise (see [@StchGrwlScheu]) and Doob’s $L^p$ inequalities. As final step, consider the system $$\begin{aligned} \label{eq:defzedtproc} d\,{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t=e^{-tB/{\varepsilon}^2}{\tilde{\Psi}}G(\Phi e^{tB/{\varepsilon}^2}{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t)dt+e^{-tB/{\varepsilon}^2}{\tilde{\Psi}}L_1(\Phi e^{tB/{\varepsilon}^2} {\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t) dW, \qquad {\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_0=z^{\varepsilon}_0,\end{aligned}$$ i.e. we are totally ignoring the $Q$ part—even the effect ${\mathfrak{Y}}$ of the initial condition. Define $\beta^{\varepsilon}_t=\frac12\|\|{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|_2^2$. Using exactly the same technique as the one employed for $\alpha^{\varepsilon}$ and using the exponential decay of ${\mathfrak{Y}}^{\varepsilon}$, it is trivial to get the following result analogous to proposition \[prop:mul:alphaissmall\]. \[prop:mul:betaissmall\] There exists $C>0$ and ${\varepsilon}_>0$ such that $\forall {\varepsilon}<{\varepsilon}_$ $$\begin{aligned} {\mathbb{E}}\sup_{s\in [0,T]}\left(\beta^{\varepsilon}_s\right)^2\,\,\leq&\,\,C{\varepsilon}^2.\end{aligned}$$ Proof given in appendix \[apsec:prf:prop:mul:betaissmall\]. Combining propositions \[prop:mul:stabnorm\], \[prop:mul:alphaissmall\] and \[prop:mul:betaissmall\] we get \[prop:mul:finalthem\] Fix $\delta \in (0,2)$. There exists constants $C,\hat{{\varepsilon}}_\delta>0$ such that $\forall {\varepsilon}< \hat{{\varepsilon}}_\delta$ $$\begin{aligned} \label{eq:mul:finthem_zedhapprx} {\mathbb{E}}\sup_{t\in [0,T]}\big\|X^{\varepsilon}(t)-\big(\Phi(0)e^{tB/{\varepsilon}^2}{\widehat{\mathfrak{z}}^{\varepsilon}}_t+{\mathfrak{Y}}^{\varepsilon}_t(0)\big)\big\|^4\,\,\leq\,\,C\left(r{\varepsilon}^\delta\ln(\frac{2T}{r{\varepsilon}^\delta})\right)^2\,\,\xrightarrow{{\varepsilon}\to 0} 0.\end{aligned}$$ There exists constants $C>0$ and ${\varepsilon}_>0$ such that $\forall {\varepsilon}<{\varepsilon}_$ $$\begin{aligned} \label{eq:mul:finthem_zedtapprx} {\mathbb{E}}\sup_{t\in [0,T]}\big\|X^{\varepsilon}(t)-\big(\Phi(0)e^{tB/{\varepsilon}^2}{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t+{\mathfrak{Y}}^{\varepsilon}_t(0)\big)\big\|^4\,\,\leq\,\,C{\varepsilon}^2.\end{aligned}$$ Proof given in appendix \[apsec:prf:prop:mul:mainTh\] Note that both the approximating processes* ${\widehat{\mathfrak{z}}^{\varepsilon}}$ and ${\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}$ are processes without delay. However, ${\widehat{\mathfrak{z}}^{\varepsilon}}$ considers the effect of the initial condition $y^{\varepsilon}_0$, but ${\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}$ ignores it. Hence the approximation using ${\widehat{\mathfrak{z}}^{\varepsilon}}$ is better than the approximation . For example, choosing $\delta$ close to two in we can get the bound $O({\varepsilon}^{4-})$ whereas the bound in is $O({\varepsilon}^2)$. Now we revisit the goals stated in section \[subsubsec:goalsformultipnoise\]. Note that for ${\bf z}^{\varepsilon}_t$ defined in we have ${\bf z}^{\varepsilon}_t=e^{tB/{\varepsilon}^2}{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t$. Hence, ${\bf z}^{\varepsilon}_t-z^{\varepsilon}_t = e^{tB/{\varepsilon}^2}({\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t-{\mathfrak{z}^{\varepsilon}}_t)$. Using the results of this section and the fact that for any ${\mathbb{R}}^2$-vector $v$, $\|\|e^{tB/{\varepsilon}^2}v\|\|_2=\|\|v\|\|_2$, we can easily see that is satisfied. The condition is equivalent to the following condition . Lemma \[lem:mul:auxzbound\] is proved in appendix \[apsec:prf:lem:mul:auxzbound\]. \[lem:mul:auxzbound\] There exists constants $C$ and ${\varepsilon}_>0$ such that $\forall {\varepsilon}<{\varepsilon}_$ $$\begin{aligned} \label{eq:hprcconvgreqzbound_equiv} {\mathbb{E}}\sup_{t\in [0,T]}\|\|{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t\|\|_2^2\,\,<\,\,C.\end{aligned}$$ Hence, follows. We summarize the discussion in section \[subsubsec:goalsformultipnoise\] in the following theorem. Define ${\mathcal{H}}^{\varepsilon}_t:=\frac12\|\|z^{\varepsilon}_t\|\|_2^2$ where $z^{\varepsilon}$ are given by $\pi{{{\Pi}}^{{\varepsilon}}_{t}{X^{\varepsilon}}}=\Phi z^{\varepsilon}_t$. Let ${\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}$ be the two-dimensional process (without delay) defined in . Then $$\begin{aligned} \label{eq:hprcconvgmul_zedt} {\mathbb{E}}\sup_{t\in [0,T]}\|{\mathcal{H}}^{\varepsilon}_t-\frac12\|\|{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t\|\|_2^2\| \xrightarrow{{\varepsilon}\to 0} 0.\end{aligned}$$ If the process $\frac12\|\|{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}\|\|_2^2$ converges weakly to a process ${\mathcal{H}}^0$, then ${\mathcal{H}}^{\varepsilon}$ converges weakly to ${\mathcal{H}}^0$. Because ${\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}$ is a process without delay, weak convergence of $\frac12\|\|{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}\|\|_2^2$ can be dealt using standard averaging techniques for stochastic differential equations. Example {#subsec:mul_examp} ------- Consider with $G\equiv 0$. The corresponding ${\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}$ satisfies $$\begin{aligned} d{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}&=e^{-tB/{\varepsilon}^2}M e^{tB/{\varepsilon}^2} {\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t dW, \qquad M={\tilde{\Psi}}L_1\Phi,\end{aligned}$$ with ${\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_0$ such that $\Phi {\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_0=\pi\xi$. Let ${\mathcal{H}}^{\varepsilon}_t:=\frac12\|\|z^{\varepsilon}_t\|\|_2^2$ and $H^{\varepsilon}_t:=\frac12\|\|{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t\|\|_2^2$. Then applying Ito formula we have $$\begin{aligned} \label{eq:varsigeq} dH^{\varepsilon}_t=({\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t)^e^{-tB/{\varepsilon}^2}M e^{tB/{\varepsilon}^2} {\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t dW_t \,+\,\frac12\left(\left(e^{-tB/{\varepsilon}^2}M e^{tB/{\varepsilon}^2} {\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t\right)_1^2+\left(e^{-tB/{\varepsilon}^2}M e^{tB/{\varepsilon}^2} {\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t\right)_2^2\right)dt.\end{aligned}$$ Averaging out the fast oscillations of ${\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}$, it can be shown that[^3] as ${\varepsilon}\to 0$, the distribution of $H^{\varepsilon}$ converges weakly to the distribution of $$\begin{aligned} \label{eq:varsigeqavg} d{\mathcal{H}}^0_t\,\,=\,\,C_{1}{\mathcal{H}}^0_t\, dW_t \,\,+\,\, C_2 {\mathcal{H}}^0_tdt, \end{aligned}$$ where $2C_1^2=3(M_{11}^2+M_{22}^2)+(M_{12}+M_{21})^2+ 2M_{11}M_{22}$ and $C_2=\frac12(\sum_{i,j=1}^2M_{ij}^2)$. Using $M={\tilde{\Psi}}L_1\Phi$ we get $$\begin{aligned} \label{eq:varsigeqavg_fin} d{\mathcal{H}}^0_t\,\,=\,\sqrt{\left(\frac12\|\|{\tilde{\Psi}}\|\|_2^2\,\|\|L_1\Phi\|\|_2^2\,+\,(L_1\Phi\,{\tilde{\Psi}})^2\right)}\,{\mathcal{H}}^0_t\, dW_t \,\,+\,\, \frac12\|\|{\tilde{\Psi}}\|\|_2^2\,\|\|L_1\Phi\|\|_2^2\, {\mathcal{H}}^0_tdt. \end{aligned}$$ For solution can be written explicitly. For small ${\varepsilon}$, the distribution of ${\mathcal{H}}^0$ gives good approximation to the distribution of ${\mathcal{H}}^{\varepsilon}$. Note that, roughly speaking, $\sqrt{2{\mathcal{H}}^{\varepsilon}}$ is the amplitude of oscillations of ${X^{\varepsilon}}$. Hence ${\mathcal{H}}^0$ can be used to understand the dynamics of ${X^{\varepsilon}}$. The advantage is that ${\mathcal{H}}^0$ does not involve any delay and is one-dimensional, and hence easier to analyze and simulate numerically (see [@LNPRLE] for examples involving numerical simulations). Additive noise {#sec:addnoi} ============== In this section we consider with $T>0$ fixed. The constants here can depend on $T$. The strategy employed for in the previous section does not work for due to the problem of moment-closure, i.e. in trying to estimate a lower moment we end up with the task of estimating a higher moment (because of the cubic nonlinearity). For we employ the strategy stated in section \[subsubsec:goal4addnoisecase\]. Define $$\begin{aligned} \label{eq:UpsandZdef_add} \Upsilon^{\varepsilon}_s:=\sup_{\theta \in [-r,0]}\left\|\int_0^s\left({\hat{T}}(\frac{s-u}{{\varepsilon}^2})(1-{\hat{\pi}}){\mathbf{1}_{\{0\}}}\right)(\theta) dZ_u\right\|, \qquad Z_t:=\sigma W_t.\end{aligned}$$ \[prop:add:supUpsT\] Fix $n\geq 1$. There exists constant $\hat{C}>0$ and a family of constants $\hat{{\varepsilon}}_{\delta}>0$ (indexed by $0<\delta<2$) such that, given $\delta \in (0,2)$ we have for ${\varepsilon}< \hat{{\varepsilon}}_{\delta}$\ $$\begin{aligned} \label{prop:add:supUpsT_statement_mully} {\mathbb{E}}\sup_{s\in [0,T]}(\Upsilon^{\varepsilon}_s)^n \,\,\,\leq\,\,\,\hat{C}\left({r{\varepsilon}^\delta}\ln\left(\frac{2T}{r{\varepsilon}^\delta}\right)\right)^{n/2} \,\,\xrightarrow{{\varepsilon}\to 0} 0. \end{aligned}$$ Proof is same as that of proposition \[prop:mul:supUpsT\] with appropriate changes to account for $Z_t=\sigma W_t$; and we dont need anything analogous to lemma \[lem:mul:quest:supuntilTisconst\]. Fix a constant $C_{{\mathfrak{e}}}>0$ and define the stopping time $$\begin{aligned} \label{eq:stoptimedefy} {\mathfrak{e}^{\varepsilon}}=\inf\{t\geq 0:\|\|\pi {{{\Pi}}^{{\varepsilon}}}_t{X^{\varepsilon}}\|\|\geq C_{{\mathfrak{e}}}\}.\end{aligned}$$ The stopping time helps in arriving at a bound on the norm of stable-mode $(1-\pi){{{\Pi}}^{{\varepsilon}}}_t{X^{\varepsilon}}$ (until time $T\wedge {\mathfrak{e}^{\varepsilon}}$) without worrying about what happens to the critical-mode $\pi{{{\Pi}}^{{\varepsilon}}}_t{X^{\varepsilon}}$. Hence, as an intermediate step we establish results that hold until time $T\wedge {\mathfrak{e}^{\varepsilon}}$ and later get rid of the stopping time ${\mathfrak{e}^{\varepsilon}}$. Write ${{{\Pi}}^{{\varepsilon}}_{t}{X^{\varepsilon}}}=\Phi z^{\varepsilon}_t+y^{\varepsilon}_t$. Then $z^{\varepsilon}_t$ and $y^{\varepsilon}_t$ satisfy the variation-of-constants formula and . Define ${\mathfrak{Y}}^{\varepsilon}$ and ${\mathscr{Y}}^{\varepsilon}$ as in . \[prop:add:stabnorm\] Let $\hat{C}$ and $\hat{{\varepsilon}}_{\delta}$ be the same as in proposition \[prop:add:supUpsT\]. There exists a family of constants ${\varepsilon}_{a,C_{{\mathfrak{e}}}}>0$ such that, given $a\in [0,1)$ and $\delta \in (2a,2)$, we have for ${\varepsilon}< \min\{\hat{{\varepsilon}}_{\delta},\,{\varepsilon}_{a,C_{{\mathfrak{e}}}}\}$ $$\begin{aligned} \label{add:newprop:supboundGron_statement_Gcub_lin_a_add} {\mathbb{P}}\bigg[\sup_{s\in [0,T\wedge{\mathfrak{e}^{\varepsilon}}]}\|\|{\mathscr{Y}}^{\varepsilon}_s\|\| \,\,\leq\,\,{{8{\varepsilon}^a} } \bigg] \,\,\,\geq\,\,\,1-\hat{C}{\varepsilon}^{-a}\sqrt{{r{\varepsilon}^\delta}\ln\left(\frac{T}{r{\varepsilon}^\delta}\right)}.\end{aligned}$$ Here ${\varepsilon}_{a,C_{{\mathfrak{e}}}}$ is of the order ${\mathcal{O}}(\min\{C_{{\mathfrak{e}}}^{-3/(2-a)},C_{{\mathfrak{e}}}^{-3/2a}\})$ for large $C_{{\mathfrak{e}}}$. In we obtain a bound on $\|\|{\mathscr{Y}}^{\varepsilon}\|\|$ which does not depend on $C_{{\mathfrak{e}}}$ in spite of the cubic nonlinearity—hence the ${\varepsilon}$ should be made really small. Larger the $C_{{\mathfrak{e}}}$, smaller the ${\varepsilon}$ we need to consider. Proof is by straight forward application of exponential decay on $\hat{Q}$, Markov and Gronwall inequalities. Proof is given in appendix \[apsec:prf:prop:add:stabnorm\] Removing the fast rotation induced by $B$, i.e. writing ${\mathfrak{z}^{\varepsilon}}_t=e^{-tB/{\varepsilon}^2}z^{\varepsilon}_t$ we have $$d{\mathfrak{z}^{\varepsilon}}_t=e^{-tB/{\varepsilon}^2}{\tilde{\Psi}}G(\Phi e^{tB/{\varepsilon}^2}{\mathfrak{z}^{\varepsilon}}_t+y^{\varepsilon}_t)dt+e^{-tB/{\varepsilon}^2}{\tilde{\Psi}}\sigma dW_t, \qquad {\mathfrak{z}^{\varepsilon}}_0=z^{\varepsilon}_0.$$ Let ${\widehat{\mathfrak{z}}^{\varepsilon}}$ be governed by $$d{\widehat{\mathfrak{z}}^{\varepsilon}}_t=e^{-tB/{\varepsilon}^2}{\tilde{\Psi}}G(\Phi e^{tB/{\varepsilon}^2}{\widehat{\mathfrak{z}}^{\varepsilon}}_t+{\mathfrak{Y}}^{\varepsilon}_t)dt+e^{-tB/{\varepsilon}^2}{\tilde{\Psi}}\sigma dW_t, \qquad {\widehat{\mathfrak{z}}^{\varepsilon}}_0=z^{\varepsilon}_0,$$ i.e., in ${\widehat{\mathfrak{z}}^{\varepsilon}}$ we are totally ignoring $y$ part except for the effect of the initial condition (${\mathfrak{Y}}^{\varepsilon}_t={\hat{T}}(t/{\varepsilon}^2)y^{\varepsilon}_0$). Note that ${\widehat{\mathfrak{z}}^{\varepsilon}}$ is a process without delay. We want to show that until time $T\wedge{\mathfrak{e}^{\varepsilon}}$, error in approximating ${\mathfrak{z}^{\varepsilon}}$ by ${\widehat{\mathfrak{z}}^{\varepsilon}}$ is small. For this purpose, define $$\alpha^{\varepsilon}_t=\frac12\|\|{\mathfrak{z}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|_2^2=\frac12(({\mathfrak{z}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t)_1^2+({\mathfrak{z}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t)_2^2).$$ and let $\Gamma_t=\left(\sum_{i=1}^2({\mathfrak{z}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t)_i(e^{-tB/{\varepsilon}^2}{\tilde{\Psi}})_i\right)$. Then $\alpha^{\varepsilon}_t$ is governed by $$d\alpha^{\varepsilon}_t\,\,=\,\,\mathscr{B}_t\,dt, \qquad \alpha^{\varepsilon}_0=0,$$ where $$\begin{aligned} \mathscr{B}_t&=\Gamma_t\left(G(\Phi e^{tB/{\varepsilon}^2}{\mathfrak{z}^{\varepsilon}}_t+y_t)-G(\Phi e^{tB/{\varepsilon}^2}{\widehat{\mathfrak{z}}^{\varepsilon}}_t+{\mathfrak{Y}}^{\varepsilon}_t)\right).\end{aligned}$$ The following lemma gives a process dominating $\mathscr{B}_t$. This helps in applying Gronwall inequality to arrive at proposition \[prop:add:alhaissmall\]. \[lem:add:aux4comparison\] $\exists\, C>0$ (is of the order ${\mathcal{O}}(C_{{\mathfrak{e}}}^2)$ for large $C_{{\mathfrak{e}}}$) such that if $\mathfrak{B}$ is defined by $$\begin{aligned} \label{eq:def:mfBadd} \mathfrak{B}(\alpha,p)\,\,&:=\,\,C\,\sqrt{2\alpha}\,\sum_{j=1}^3(p^j+(\sqrt{2\alpha})^j),\end{aligned}$$ then $\|\mathscr{B}_t\|\leq \mathfrak{B}(\alpha_t,\|\|{\mathscr{Y}}^{\varepsilon}_t\|\|)$ for $t\in [0,T\wedge{\mathfrak{e}^{\varepsilon}}]$. Proof given in appendix \[apsec:prf:lem:add:aux4comparison\]. \[prop:add:alhaissmall\] Let $\hat{C}$ and $\hat{{\varepsilon}}_{\delta}$ be the same as in proposition \[prop:add:supUpsT\]. There exists two families of constants ${\varepsilon}_{a,C_{{\mathfrak{e}}},1}>0$, ${\varepsilon}_{a,C_{{\mathfrak{e}}},2}>0$ such that, given $a\in [0,1)$ and $\delta \in (2a,2)$, we have for ${\varepsilon}< \min\{\hat{{\varepsilon}}_{\delta},\,{\varepsilon}_{a,C_{{\mathfrak{e}}},1},\,{\varepsilon}_{a,C_{{\mathfrak{e}}},2}\}$ $$\begin{aligned} \label{eq:pepsdefalp} {\mathbb{P}}[\sup_{t\in [0,T\wedge {\mathfrak{e}^{\varepsilon}}]}\alpha^{\varepsilon}_t \,\,\leq\,\,{\varepsilon}^{a/2}]\,\,\geq \,\,1-\hat{C}{\varepsilon}^{-a}\sqrt{{r{\varepsilon}^\delta}\ln\left(\frac{T}{r{\varepsilon}^\delta}\right)}\,\,=:\,\,p_{{\varepsilon}} \xrightarrow{{\varepsilon}\to 0}1. \end{aligned}$$ Here ${\varepsilon}_{a,C_{{\mathfrak{e}}},1}$ is of the order ${\mathcal{O}}(\min\{C_{{\mathfrak{e}}}^{-3/(2-a)},C_{{\mathfrak{e}}}^{-3/2a}\})$ (these are from proposition \[prop:add:stabnorm\]) and ${\varepsilon}_{a,C_{{\mathfrak{e}}},2}$ is of the order ${\mathcal{O}}(\exp(-30C_{{\mathfrak{e}}}^2T/a))$ for large $C_{{\mathfrak{e}}}$. Proof is given in appendix \[apsec:prf:prop:add:alhaissmall\]. Finally, the stopping time ${\mathfrak{e}^{\varepsilon}}$ can be got rid as follows. Let ${\Omega}$ be the set of all realizations of the Brownian motion $W$ and ${\omega}\in {\Omega}$ denote one particular realization. \[def:propPTqDeff\] Given $T>0$ and $q>0$, we say that “${\widehat{\mathfrak{z}}^{\varepsilon}}$ system possesses the property $\mathscr{P}(T,q)$” if $\exists \,C_{{\mathfrak{e}}},\, {\varepsilon}_ \,>\,0$ such that $\forall {\varepsilon}<{\varepsilon}_$, we have ${\mathbb{P}}[E^{\varepsilon}]\geq1-q$ where $$\begin{aligned} \label{assonzedh_setEeps} E^{\varepsilon}:=\left\{{\omega}\,:\,\sup_{t\in [0,T]}\|\|\Phi e^{tB/{\varepsilon}^2}{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|\,<\,0.99C_{{\mathfrak{e}}}\right\}.\end{aligned}$$ \[prop:helpinprobconvg\] Fix $T>0$. Define $$H^{\varepsilon}:=\left\{{\omega}\,:\,\sup_{t\in[0,T]}\alpha^{{\varepsilon}}_t \,\geq\,{\varepsilon}^{a/2}\right\}, \qquad S^{\varepsilon}:=\left\{{\omega}\,:\,\sup_{t\in[0,T]}\|\|{\mathscr{Y}}^{\varepsilon}_t\|\| \,\geq\,8{\varepsilon}^{a}\right\},$$ for $a\in [0,1)$. Fix $q>0$ and assume ${\widehat{\mathfrak{z}}^{\varepsilon}}$ system possesses the property $\mathscr{P}(T,q)$. Then $\exists\,{\varepsilon}_q>0$ such that $\forall\,{\varepsilon}<{\varepsilon}_q$, $$\begin{aligned} {\mathbb{P}}[H^{\varepsilon}]\,\,<\,\,q+2(1-p_{{\varepsilon}}), \qquad {\mathbb{P}}[S^{\varepsilon}]\,\,<\,\,q+2(1-p_{{\varepsilon}}),\end{aligned}$$ where $p_{{\varepsilon}}\to 1$ as ${\varepsilon}\to 0$ and is given explicitly in . Proof is given in appendix \[apsec:prf:prop:add:helpinprobconvg\]. Note that we have extended our results on $[0,T\wedge {\mathfrak{e}^{\varepsilon}}]$ to $[0,T]$ by leveraging a small probability $q$, provided that ${\widehat{\mathfrak{z}}^{\varepsilon}}$ system possess property $\mathscr{P}(T,q)$. Now we discuss under what conditions does ${\widehat{\mathfrak{z}}^{\varepsilon}}$ system possesses the property $\mathscr{P}(T,q)$ for arbitrary $q>0$. Fix $T>0$. In general one cannot expect $\mathscr{P}(T,q)$ to hold for arbitrary $q>0$—for example, if the cubic nonlinearities have a destabilizing effect then there is a non-zero probability that trajectories blow-up in finite time. Similar situation arises in stochastic partial differential equations—see remark 5.2 in [@Blom_amp_eq]. When cubic nonlinearities have stabilizing effect, it is reasonable to expect $\mathscr{P}(T,q)$ to hold for arbitrary $q>0$ (see proposition \[lem:add:propertyPTq\_stable\] below). The following two propositions help in checking if the property $\mathscr{P}(T,q)$ is satisfied. Proofs of them are similar in nature to the proof of Theorem 5.1 in [@Blom_amp_eq]. [@Blom_amp_eq] deals with stochastic partial differential equations and the instability scenario there is different—analogous situation in delay equations case would be that “one root of the characteristic equation is zero, and all other roots have negative real parts”. For the scenario that we are considering in this paper, one pair of roots lie on the imaginary axis, and so there are oscillations in the system and the proofs requires a bit more work than that in [@Blom_amp_eq]. Proposition \[lem:add:propertyPTq\] does not assume anything about the nature of the nonlinearity $G$—consequently its result is weak. Proposition \[lem:add:propertyPTq\_stable\] assumes that the nonlinearity is stabilizing and concludes that ${\widehat{\mathfrak{z}}^{\varepsilon}}$ possesses the property $\mathscr{P}(T,q)$ for any $q>0$. \[lem:add:propertyPTq\] Fix $q>0$. Then $\exists T_q>0$ such that the ${\widehat{\mathfrak{z}}^{\varepsilon}}$ system possesses the property $\mathscr{P}(T,q)$ for $T\in [0,T_q]$. Proof is given in appendix \[appsec:lem:add:propertyPTq\]. \[lem:add:propertyPTq\_stable\] Fix $T>0$. Assume the cubic nonlinearity of $G$ is stabilizing, i.e., $\exists C_G>0$ such that $$\begin{aligned} \label{lem:add:propertyPTq_stable_condition} \frac{{\omega}_c}{2\pi}\int_0^{2\pi/{\omega}_c}\left((e^{tB}z)^{\tilde{\Psi}}\int_{-r}^0(\Phi(\theta)e^{tB}z)^3d\nu_3(\theta)\right)dt \,\,<\,\,-C_G\|\|z\|\|_2^4, \qquad \forall z\in {\mathbb{R}}^2.\end{aligned}$$ Then the ${\widehat{\mathfrak{z}}^{\varepsilon}}$ system possesses the property $\mathscr{P}(T,q)$ for arbitrary $q>0$. Proof is given in appendix \[appsec:lem:add:propertyPTq\_stableNony\]. Now consider the system $$d\,{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t=e^{-tB/{\varepsilon}^2}{\tilde{\Psi}}G(\Phi e^{tB/{\varepsilon}^2}{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t)dt+e^{-tB/{\varepsilon}^2}{\tilde{\Psi}}L_1(\Phi e^{tB/{\varepsilon}^2} {\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t) dW, \qquad {\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_0=z^{\varepsilon}_0,$$ i.e. we are totally ignoring the $Q$ part—even the effect ${\mathfrak{Y}}$ of the initial condition. Define $$\beta^{\varepsilon}_t=\frac12\|\|{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|_2^2.$$ \[prop:betaepsSmallAddNoiseFin\] Assume the cubic nonlinearity is such that is satisfied, i.e. nonlienarity is stabilizing. Fix $T>0$. Given any $q>0$, $\exists\,C>0$ and ${\varepsilon}_{\circ}>0$ such that $\forall {\varepsilon}<{\varepsilon}_{\circ}$ $${\mathbb{P}}\left[\sup_{t\in [0,T]}\beta^{\varepsilon}_t \geq C{\varepsilon}^4\right]\leq q.$$ Proof is in appendix \[sec:proofofprop\_betaepsSmallAddNoiseFin\]. Comibining theorem \[prop:helpinprobconvg\] and proposition \[prop:betaepsSmallAddNoiseFin\] we get the following result. \[thm:cubnonstab\_probconvg\_apprx\] Assume the cubic nonlinearity is such that is satisfied, i.e. nonlinearity is stabilizing. Fix any $a\in [0,1)$. For any $q>0$, $\exists {\varepsilon}_q>0$ such that $\forall {\varepsilon}<{\varepsilon}_q$ $$\begin{aligned} {\mathbb{P}}\bigg[\sup_{s\in [0,T]}\big\|{X^{\varepsilon}}_s- &\left(\Phi(0)e^{sB/{\varepsilon}^2}{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_s+{\mathfrak{Y}}^{\varepsilon}_s\right)\big\|> 6{\varepsilon}^{a/4}\bigg] \,< \, 3q + 4(1-p_{\varepsilon})\end{aligned}$$ where $p_{{\varepsilon}}\to 1$ as ${\varepsilon}\to 0$ and is given explicitly in . Using ${X^{\varepsilon}}_s = \Phi(0)e^{sB/{\varepsilon}^2}{\mathfrak{z}^{\varepsilon}}_s+{\mathscr{Y}}^{\varepsilon}_s + {\mathfrak{Y}}^{\varepsilon}_s$ the above probability is bounded by $$\begin{aligned} {\mathbb{P}}\bigg[\sup_{s\in [0,T]}& \big\|\Phi(0)e^{sB/{\varepsilon}^2}({\mathfrak{z}^{\varepsilon}}_s-{\widehat{\mathfrak{z}}^{\varepsilon}}_s)\big\| +\big\|\Phi(0)e^{sB/{\varepsilon}^2}({\widehat{\mathfrak{z}}^{\varepsilon}}_s-{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_s)\big\|+\|{\mathscr{Y}}^{\varepsilon}_s(0)\|>6{\varepsilon}^{a/4}\bigg] \\ &\leq \,{\mathbb{P}}\left[\sup_{s\in [0,T]}\sqrt{2\alpha^{\varepsilon}_t}>2{\varepsilon}^{a/4}\right]+{\mathbb{P}}\left[\sup_{s\in [0,T]}\sqrt{2\beta^{\varepsilon}_t}>2{\varepsilon}^{a/4}\right]+{\mathbb{P}}\left[\sup_{s\in [0,T]}\|\|{\mathscr{Y}}^{\varepsilon}_s\|\|>2{\varepsilon}^{a/4}\right] \\ & < q+2(1-p_{{\varepsilon}}) \,\,+\,\,q\,\,+\,\, q+2(1-p_{{\varepsilon}}) .\qquad \quad (\text{for } {\varepsilon}\text{ sufficiently small.})\end{aligned}$$ Note that ${\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}$ is a 2-dimensional system without delay and ${\mathfrak{Y}}^{\varepsilon}$ is a deterministic process that has exponential decay. The above theorem shows that, for small enough ${\varepsilon}$, the delay system ${X^{\varepsilon}}$ can be approximated by the ${\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}$ system without delay, with probability close to 1. When the nonlinearity is stabilizing, using standard averaging techniques for equations without delay (see for example [@NavamSowers]), it can be shown that the distribution of ${\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}$ converges as ${\varepsilon}\to 0$ to the distribution of a 2-dimensional process ${\pmb{\widetilde{\mathfrak{z}}}^0}$. Theorem \[prop:helpinprobconvg\] and propositions \[prop:betaepsSmallAddNoiseFin\] show that $\sup_{t\in [0,T]}\beta^{\varepsilon}_t$ and $\sup_{t\in [0,T]}\alpha^{\varepsilon}_t$ converge to zero in probability. Hence, by theorem 3.1 in [@Billingsley1999], the distribution of ${\mathfrak{z}^{\varepsilon}}$ converges as ${\varepsilon}\to 0$ to the distribution of ${\pmb{\widetilde{\mathfrak{z}}}^0}$. Also, the distribution of ${\mathcal{H}}^{\varepsilon}$ process, where ${\mathcal{H}}^{\varepsilon}_t:=\frac12\|\|z^{\varepsilon}_t\|\|_2^2=\frac12\|\|{\mathfrak{z}^{\varepsilon}}_t\|\|_2^2$, converges as ${\varepsilon}\to 0$ to the distribution of ${\mathcal{H}}^0$, where ${\mathcal{H}}^0$ is the weak-limit as ${\varepsilon}\to 0$ of the process $H^{\varepsilon}_t=\frac12\|\|{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t\|\|_2^2$. Great simplification can be obtained when $\frac12\|\|{\mathfrak{z}^{\varepsilon}}\|\|_2^2$ can be used to approximate the required quantities. For example, consider the exit time $$\begin{aligned} \tau^{\varepsilon}&:=\inf \{t\geq 0: \|{X^{\varepsilon}}_t\|\geq \sqrt{2H^}\}\end{aligned}$$ where $H^$ is fixed and is such that $\sqrt{2H^}\gg \|\|(I-\pi){{{\Pi}}^{{\varepsilon}}}_0 {X^{\varepsilon}}\|\|$. Noting that $\Phi(0)e^{tB/{\varepsilon}^2}{\mathfrak{z}^{\varepsilon}}_{t}=({\mathfrak{z}^{\varepsilon}}_t)_1\cos({\omega}t/{\varepsilon}^2)+({\mathfrak{z}^{\varepsilon}}_t)_2\sin({\omega}t/{\varepsilon}^2)$; because of the fast oscillations of ${\mathfrak{z}^{\varepsilon}}$ and fast decay of ${\mathfrak{Y}}^{\varepsilon}$ and smallness of ${\mathscr{Y}}^{\varepsilon}$, the exit time $\tau^{\varepsilon}$ would be very close to the exit time ${\tau'}^{\varepsilon}$ where $$\begin{aligned} {\tau'}^{\varepsilon}&:=\inf \{t\geq 0: \sqrt{({\mathfrak{z}^{\varepsilon}}_t)_1^2+({\mathfrak{z}^{\varepsilon}}_t)_2^2}\geq \sqrt{2H^}\}.\end{aligned}$$ To approximate the distribution of $\tau^{\varepsilon}$, one can study $H^{\varepsilon}_t:=\frac12\|\|{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t\|\|_2^2$ and consider the distribution of $$\begin{aligned} \tau^{{\varepsilon},\hbar}&:=\inf \{t\geq 0: H^{\varepsilon}_t\geq H^\}.\end{aligned}$$ The distribution of $\tau^{{\varepsilon},\hbar}$ would be close to that of $\tau^{\varepsilon}$. Since ${\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t$ does not involve any delay, standard averaging techniques can be used to show that the distribution of $H^{\varepsilon}$ converges as ${\varepsilon}\to 0$ to the distribution of a specific 1-dimensional process ${\mathcal{H}}^0$ (without delay). Then the exit times $$\begin{aligned} \tau^{\hbar}&:=\inf \{t\geq 0: {\mathcal{H}}^0_t\geq H^\}\end{aligned}$$ would closely approximate $\tau^{\varepsilon}$. The advantages in doing so are: (i) ${\mathcal{H}}^0$ is a process without delay and hence easier to simulate (ii) numerical simulation of ${\mathcal{H}}^0$ can be done with a much coarser numerical mesh than that required for ${X^{\varepsilon}}$. Example {#sec:example} ------- Consider the following equation: $$\begin{aligned} \label{eq:examplesys_quadadded} d{X^{\varepsilon}}(t)={\varepsilon}^{-2}L_0({{{\Pi}}^{{\varepsilon}}}_t {X^{\varepsilon}})dt + G({{{\Pi}}^{{\varepsilon}}}_t{X^{\varepsilon}})dt + \sigma dW, \qquad \quad {{{\Pi}}^{{\varepsilon}}}_0{X^{\varepsilon}}=\xi,\end{aligned}$$ where $L_0\eta=-\frac{\pi}{2}\eta(-1)$ and $G(\eta)=\gamma_c\eta^3(-1)$. The characteristic equation $\lambda+\frac{\pi}{2}e^{-\lambda}=0$ has countably infinite roots on the complex plane. The roots with the largest real part are $\pm i\frac{\pi}{2}$. Hence $L_0$ satisfies the assumption \[ass:assumptondetsys\]. The basis $\Phi$ for $P$ and the vector ${\tilde{\Psi}}$ can be evaluated as $$\Phi(\theta)=[\cos(\frac{\pi}{2}\theta)\,\,\,\sin(\frac{\pi}{2}\theta)], \qquad \quad {\tilde{\Psi}}=\frac{2}{(1+(\pi/2)^2)}\left[\begin{array}{c}1 \\ \pi/2 \end{array}\right].$$ The corresponding ${\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}$ satisfies $$\begin{aligned} d{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}\,=\,\gamma_c e^{-tB/{\varepsilon}^2}{\tilde{\Psi}}\left(\Phi(-1)e^{tB/{\varepsilon}^2}{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t\right)^3dt + e^{-tB/{\varepsilon}^2}{\tilde{\Psi}}\sigma dW, \qquad \quad \Phi {\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_0=\pi\xi.\end{aligned}$$ Let $H^{\varepsilon}_t=\frac12(({\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_1)^2+({\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_2)^2)_t$. Applying Ito formula we have $$\begin{aligned} dH^{\varepsilon}_t\,&=\,\gamma_c(e^{-tB/{\varepsilon}^2}{\tilde{\Psi}})^\,{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}(\Phi(-1)e^{tB/{\varepsilon}^2}{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}})^3\,dt + (e^{-tB/{\varepsilon}^2}{\tilde{\Psi}})^\,{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}\sigma dW \\ & \qquad \qquad + \frac12\sigma^2\left((e^{-tB/{\varepsilon}^2}{\tilde{\Psi}})_1^2+(e^{-tB/{\varepsilon}^2}{\tilde{\Psi}})_2^2\right)dt.\end{aligned}$$ Averaging the fast oscillations we get that the probability distribution of $H^{\varepsilon}$ converges as ${\varepsilon}\to 0$ to the probability distribution of $$\begin{aligned} \label{eq:examplesysavgd_quadadded} d{\mathcal{H}}^0_t=\left(-\frac{3\gamma_c\pi/2}{1+(\pi/2)^2}({\mathcal{H}}^0_t)^2 + \frac{2\sigma^2}{1+(\pi/2)^2}\right)dt + \frac{2}{\sqrt{1+(\pi/2)^2}}\sqrt{{\mathcal{H}}^0_t}\,\sigma dW.\end{aligned}$$ Now we illustrate our results employing numerical simulations. Draw a random sample of $N_{samp}$ particles with initial $H$ values $\{h_i\}_{i=1}^{Nsamp}$. Simulate them according to for $0\leq t \leq T_{end}$. Simulate for $0 \leq t \leq T_{end}$ using initial trajectories $\{\sqrt{2h_i} \cos({\omega}_c\cdot)\}_{i=1}^{Nsamp}$. Let $\tau^{\varepsilon}:=\inf \{t\geq 0: \|{X^{\varepsilon}}_t\|\geq \sqrt{2H^}\}$ and $\tau^h:=\inf \{t\geq 0: H^0(t)\geq H^\}$ We can check whether the following pairs are close. 1. the distribution of $\frac12((z^{\varepsilon}_{T_{end}})_1^2+(z^{\varepsilon}_{T_{end}})_2^2)$ from and the distribution of ${\mathcal{H}}^0_{T_{end}}$ from , 2. distribution of $\tau^{\varepsilon}$ and the distribution of $\tau^h$. We took ${\varepsilon}=0.025$, $H^=1.5$, $T_{end}=2$, $N_{samp}=4000$, and $\sqrt{2\{h_i\}_{i=1}^{Nsamp}}=1.2$. Figures \[fig:tcdf\] and \[fig:taucdf\] answer the above questions. Two cases are considered with $\sigma=1$ fixed: $\gamma_c=1$ and $\gamma_c=0$. ![cdf of $\tau^{\varepsilon}$ (dashed line) and cdf of $\tau^h$ (solid line). The cdf value at $\tau^{\varepsilon}=2$ indicates the fraction of particles whose modulus exceeded $\sqrt{2H^}$ before the time $T=2$.[]{data-label="fig:taucdf"}](revtransdens_pcdAm) ![cdf of $\tau^{\varepsilon}$ (dashed line) and cdf of $\tau^h$ (solid line). The cdf value at $\tau^{\varepsilon}=2$ indicates the fraction of particles whose modulus exceeded $\sqrt{2H^}$ before the time $T=2$.[]{data-label="fig:taucdf"}](revexittimes_pcdAm) More examples (oscillators with cubic nonlinearity) are discussed in[^4] [@LNPRLE]. [0]{} N. Lingala, and N. Sri Namachchivaya. [Nonlinear and additive white noise perturbations of linear delay differential equations at the verge of instability: an averaging approach]{}, Stochastics and Dynamics, 16(4), 2016. <http://dx.doi.org/10.1142/S0219493716500131> N. Lingala, and N. Sri Namachchivaya. [Perturbations of linear delay differential equations at the verge of instability]{}. <http://arxiv.org/abs/1403.3029> G. Stepan. Retarded dynamical systems: stability and characteristic functions.* Longman Scientific & Technical, 1989. G. Stepan. Modelling nonlinear regenerative effects in metal cutting. Phil. Trans. R. Soc. Lond. A, 359:0 739 – 757, 2001. <http://dx.doi.org/10.1098/rsta.2000.0753>. A. Longtin, J.G. Milton, J.E. Bos, and M.C. Mackey. Noise and critical behavior of the pupil light reflex at oscillation onset. Phys. Rev. A, 41:0 6992–7005, Jun 1990. URL <http://link.aps.org/doi/10.1103/PhysRevA.41.6992>. W. Yao, P. Yu, and C. Essex. Delayed stochastic differential model for quiet standing. Phys. Rev. E, 63:0 021902, Jan 2001. URL <http://link.aps.org/doi/10.1103/PhysRevE.63.021902>. E. Buckwar, R. Kuske, B. L’Esperance, and T. Soo. Noise-sensitivity in machine tool vibrations. International Journal of Bifurcation and Chaos, 16(08):2407–2416, 2006. URL <http://www.worldscientific.com/doi/abs/10.1142/S021812740601615X>. J. K. Hale and S. M. Verduyn Lunel, [Introduction to functional differential equations]{}, ([Springer Verlag]{}, 1993). O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel, and H. O. Walther, [[Delay equations]{}]{}, (Springer Verlag, 1995). S-E. A. Mohammed, [[Stochastic functional differential equations]{}]{}, (Pitman, 1984). M. Reiś, M. Riedle and O. van Gaans, [On Émery’s Inequality and a Variation-of-Constants Formula]{}, Stochastic Analysis and Applications, 25:353-379, 2007. <http://dx.doi.org/10.1080/07362990601139586> P. Billingsley, Convergence of probability measures, John Wiley & Sons, $2^{nd}$ edition, 1999. D. Blomker, [Amplitude equations for locally cubic nonautonomous nonlinearities]{}, SIAM Journal on Applied Dynamical Systems, 2:464-486, 2003. <http://epubs.siam.org/doi/abs/10.1137/S1111111103421355> M. Fischer and G. Nappo. [On the Moments of the Modulus of Continuity of Ito Processes,]{} Stochastic Analysis and Applications, 28:103-122, 2009. M. Scheutzow, [A stochastic Gronwall lemma]{}. <http://arxiv.org/abs/1304.5424> N. Sri Namachchivaya and R. Sowers, [Rigorous stochastic averaging at a center]{}, Meccanica, 37:85-114, 2002. <http://dx.doi.org/10.1023/A%3A1019614613583> D. Revuz, and M. Yor. Continuous martingales and Brownian motion, Springer, 1999. D. Bainov, and P. Simeonov. Integral inequalities and applications, Kluwer Academic, 1992. J.K. Hale. Ordinary differential equations, Krieger Publishing Company, 2nd edition, 1980. Proofs of results in section \[sec:mulnoi\] =========================================== Proof of Lemma \[lem:mul:quest:supuntilTisconst\] {#apsec:prf:lem:mul:quest:supuntilTisconst} ------------------------------------------------- For $\eta \in {\mathcal{C}}$, by $\eta(\theta)$ we mean $\eta$ evaluated at $\theta \in [-r,0]$. Let ${X^{\varepsilon}}_{q,t}:=((1-\pi){{{\Pi}}^{{\varepsilon}}_{t}{X^{\varepsilon}}})(0)$, and ${X^{\varepsilon}}_{p,t}:=(\pi{{{\Pi}}^{{\varepsilon}}_{t}{X^{\varepsilon}}})(0).$ And for the unperturbed system , let $x_{q,t}:=((1-\pi){\Pi}_tx)(0)$, and $x_{p,t}:=(\pi{\Pi}_tx)(0)$ with the initial condition ${\Pi}_0x={{{\Pi}}^{{\varepsilon}}_{0}{X^{\varepsilon}}}=\xi$. Let $$Z_t:=\int_0^tL_1({{{\Pi}}^{{\varepsilon}}_{s}{X^{\varepsilon}}}) d{W}_s.$$ In , ${\gamma}$ was defined. Let $${\chi}(t)=({\hat{T}}(t){\hat{\pi}}{\mathbf{1}_{\{0\}}})(0).$$ Using ${\hat{\pi}}{\mathbf{1}_{\{0\}}}=\Phi {\tilde{\Psi}}$ from and ${\hat{T}}(t)\Phi=\Phi e^{tB}$, we get ${\chi}(t)=\Phi(0)e^{tB}{\tilde{\Psi}}$. Using the variation-of-constants formula - we have for $t\geq 0$ $$\begin{aligned} {X^{\varepsilon}}_t={X^{\varepsilon}}_{p,t}+{X^{\varepsilon}}_{q,t}=x_{p,t/{\varepsilon}^2}+x_{q,t/{\varepsilon}^2}+D_{p,t}+D_{q,t}+A_{p,t}+A_{q,t}\end{aligned}$$ with $$\begin{aligned} D_{q,t}&:=\int_0^t{\gamma}\left(\frac{t-s}{{\varepsilon}^2}\right)G({{{\Pi}}^{{\varepsilon}}_{s}{X^{\varepsilon}}}) ds, \qquad A_{q,t}:=\int_0^t{\gamma}\left(\frac{t-s}{{\varepsilon}^2}\right)dZ_s, \\ D_{p,t}&:=\int_0^t{\chi}\left(\frac{t-s}{{\varepsilon}^2}\right)G({{{\Pi}}^{{\varepsilon}}_{s}{X^{\varepsilon}}}) ds, \qquad A_{p,t}:=\int_0^t{\chi}\left(\frac{t-s}{{\varepsilon}^2}\right)dZ_s.\end{aligned}$$ For any process $M$, we define $M^_t:=\sup_{0\leq s\leq t}\|M_s\|$. Now, what we mean by $D^_{q,t}$, $A^_{q,t}$ and $x^_{q,t}$ etc is clear. Also define $${\mathfrak{X}^{\varepsilon}}_{t}:=\sup_{s\in[0,t]}\|{X^{\varepsilon}}_{s}\|^n.$$ We then have, $$\begin{aligned} \label{eq:fischNapLemm_to_collect} 2^{-5(n-1)}{\mathbb{E}}\,{\mathfrak{X}^{\varepsilon}}_{t}\,\,\quad \leq &\,\,\quad {\mathbb{E}}\,\|x^_{p,t/{\varepsilon}^2}\|^n\,\,+\,\,{\mathbb{E}}\,\|x^_{q,t/{\varepsilon}^2}\|^n\,\, \\ &\qquad +\,\,{\mathbb{E}}\,\|D^_{p,t}\|^n\,\,+\,\,{\mathbb{E}}\,\|D^_{q,t}\|^n+\,\,{\mathbb{E}}\,\|A^_{p,t}\|^n+\,\,{\mathbb{E}}\,\|A^_{q,t}\|^n. \notag\end{aligned}$$ First we focus on the terms involving the process $A$. Using integration by parts we have $$\begin{aligned} A_{q,s}={\gamma}(0)Z_s+\int_0^s{\varepsilon}^{-2}{\gamma}'\left(\frac{s-u}{{\varepsilon}^2}\right)Z_u\,du.\end{aligned}$$ Using Minkowski inequality, $$\begin{aligned} \label{FmultGcubReq_eq:Dtuseful} {\mathbb{E}}\,\|A^_{q,t}\|^n\,\,&\leq\,\,2^{n-1}\|{\gamma}(0)\|^n{\mathbb{E}}\sup_{s\in [0,t]}\|Z_s\|^n\,\,+\,\,2^{n-1}{\mathbb{E}}\sup_{s\in [0,t]}\left\|\int_0^s{\varepsilon}^{-2}{\gamma}'\left(\frac{s-u}{{\varepsilon}^2}\right)Z_u\,du\right\|^n.\end{aligned}$$ The second term on the RHS of is bounded above (using the exponential decay ) by $$\begin{aligned} 2^{n-1}{\mathbb{E}}\sup_{s\in [0,t]}\left\|\int_0^s{\varepsilon}^{-2}\left\|{\gamma}'\left(\frac{s-u}{{\varepsilon}^2}\right)\right\|\,\left\|Z_u\right\|\,du\right\|^n\,\,&\leq\,\,2^{n-1}{\mathbb{E}}\sup_{s\in [0,t]}\left\|\int_0^s{\varepsilon}^{-2}\widetilde{K}\|\|L_0\|\|e^{-\kappa(s-u)/{\varepsilon}^2}\,\left\|Z_u\right\|\,du\right\|^n\\ &\leq\,\,2^{n-1}(\widetilde{K}\|\|L_0\|\|/\kappa)^n{\mathbb{E}}\sup_{s\in [0,t]}\|Z_s\|^n,\end{aligned}$$ where $\widetilde{K}=K\|\|(1-{\hat{\pi}}){\mathbf{1}_{\{0\}}}\|\|$. Hence, [using Burkholder-Davis-Gundy inequality]{} and Holder inequality $$\begin{aligned} {\mathbb{E}}\,\|A^_{q,t}\|^n\,\,&\leq\,\,2^{n-1}\left(\|{\gamma}(0)\|^n+(\widetilde{K}\|\|L_0\|\|/\kappa)^n\right){\mathbb{E}}\sup_{s\in [0,t]}\|Z_s\|^n \\ &\leq\,\,2^{n-1}\left(\|{\gamma}(0)\|^n+(\widetilde{K}\|\|L_0\|\|/\kappa)^n\right)C_m{\mathbb{E}}\left(\int_0^tL_1^2({{{\Pi}}^{{\varepsilon}}_{u}{X^{\varepsilon}}})du\right)^{n/2} \\ &\leq\,\,2^{n-1}\left(\|{\gamma}(0)\|^n+(\widetilde{K}\|\|L_0\|\|/\kappa)^n\right)C_{m}t^{\frac{n-2}{2}}\|\|L_1\|\|^n\left(\int_0^t{\mathbb{E}}{\mathfrak{X}^{\varepsilon}}_u du\,+{\mathbb{E}}\|\|\xi\|\|^n(t\wedge {\varepsilon}^2r)\right)\\ &=\,\,C_{m,L}t^{\frac{n-2}{2}}\left(\int_0^t{\mathbb{E}}{\mathfrak{X}^{\varepsilon}}_u du\,+{\mathbb{E}}\|\|\xi\|\|^n(t\wedge {\varepsilon}^2r)\right)\end{aligned}$$ where $C_{m,L}=2^{n-1}\left(\|{\gamma}(0)\|^n+(\widetilde{K}\|\|L_0\|\|/\kappa)^n\right)C_{m}\|\|L_1\|\|^n$ and $t\wedge {\varepsilon}^2r$ means $\min\{t,{\varepsilon}^2r\}$. Now we focus on $A_{p,t}$. $$\begin{aligned} A_{p,t}\,\,&=\,\, \int_0^t{\chi}\left(\frac{t-s}{{\varepsilon}^2}\right)\,dZ_s \,\,=\,\,\int_0^t\left(\Phi(0)e^{(t-s)B/{\varepsilon}^2}{\tilde{\Psi}}\right)\,dZ_s\,\,=\,\,\Phi(0)e^{tB/{\varepsilon}^2}\left(\int_0^te^{-sB/{\varepsilon}^2}dZ_s\right){\tilde{\Psi}}.\end{aligned}$$ Let $M^c_t=\int_0^t\cos({\omega}_cs/{\varepsilon}^2)dZ_s$ and $M^s_t=\int_0^t\sin({\omega}_cs/{\varepsilon}^2)dZ_s$. Then $$\begin{aligned} {\mathbb{E}}\|A^_{p,t}\|^n\,\,&\leq\,\, (\|\|\Phi(0)\|\|_2\|\|{\tilde{\Psi}}\|\|_2)^n{\mathbb{E}}(M^{c,}_t+M^{s,}_t)^n.\end{aligned}$$ Using BDG and Holder inequalities, $$\begin{aligned} {\mathbb{E}}\|A^_{p,t}\|^n\,\,&\leq\,\, 2^{n-1}(\|\|\Phi(0)\|\|_2\|\|{\tilde{\Psi}}\|\|_2)^nCt^{\frac{n-2}{2}}\|\|L_1\|\|^n\left(\int_0^t{\mathbb{E}}{\mathfrak{X}^{\varepsilon}}_u du\,+{\mathbb{E}}\|\|\xi\|\|^n(t\wedge {\varepsilon}^2r)\right).\end{aligned}$$ Now we focus on the process $D$. Using exponential decay of ${\gamma}$ we have $$\begin{aligned} \|D_{q,t}\|\,\,&\leq\,\, \int_0^t\left\|{\gamma}\left(\frac{t-s}{{\varepsilon}^2}\right)\right\|\,\|G({{{\Pi}}^{{\varepsilon}}_{s}{X^{\varepsilon}}})\| ds \\ &\leq\,\,\sup_{s\in[0,t]}\|G({{{\Pi}}^{{\varepsilon}}_{s}{X^{\varepsilon}}})\|\int_0^t\widetilde{K}e^{-\kappa(t-s)/{\varepsilon}^2}ds\,\,\leq\,\,{\varepsilon}^2(\widetilde{K}/\kappa)\sup_{s\in[0,t]}\|G({{{\Pi}}^{{\varepsilon}}_{s}{X^{\varepsilon}}})\|.\end{aligned}$$ Hence, using the Lipschitz condition $\|G(\eta)\|\leq K_G\|\|\eta\|\|$ we have, $$\begin{aligned} {\mathbb{E}}\,\|D^_{q,t}\|^n\,\,\leq\,\,{\varepsilon}^{2n}(\widetilde{K}K_G/\kappa)^n({\mathbb{E}}{\mathfrak{X}^{\varepsilon}}_{t}+{\mathbb{E}}\|\|\xi\|\|^n).\end{aligned}$$ Now, $$\begin{aligned} \|D_{p,t}\|\,\,&\leq\,\, \int_0^t\left\|{\chi}\left(\frac{t-s}{{\varepsilon}^2}\right)\right\|\,\|G({{{\Pi}}^{{\varepsilon}}_{s}{X^{\varepsilon}}})\| ds \,\,=\,\,\int_0^t\left\|\Phi(0)e^{(t-s)B/{\varepsilon}^2}{\tilde{\Psi}}\right\|\,\|G({{{\Pi}}^{{\varepsilon}}_{s}{X^{\varepsilon}}})\| ds\\ &\leq\,\,\|\|\Phi(0)\|\|_2\|\|{\tilde{\Psi}}\|\|_2\int_0^t\|G({{{\Pi}}^{{\varepsilon}}_{s}{X^{\varepsilon}}})\| ds \,\,\leq\,\,\|\|\Phi(0)\|\|_2\|\|{\tilde{\Psi}}\|\|_2K_G\int_0^t\|\|{{{\Pi}}^{{\varepsilon}}_{s}{X^{\varepsilon}}}\|\| ds.\end{aligned}$$ Hence, using BDG and Holder inequalities, $$\begin{aligned} {\mathbb{E}}\,\|D^_{p,t}\|^n\,\,\leq\,\,(\|\|\Phi(0)\|\|_2\|\|{\tilde{\Psi}}\|\|_2K_G)^nt^{n-1}\left(\int_0^t{\mathbb{E}}{\mathfrak{X}^{\varepsilon}}_{s} ds+{\mathbb{E}}\|\|\xi\|\|^n(t\wedge{\varepsilon}^2r)\right).\end{aligned}$$ Now, we focus on the deterministic terms. Because of our assumption on $L_0$, there exists $C_{L_0}>0$ such that $x^_{q,t/{\varepsilon}^2} \leq \sqrt{C_{L_0}}\|\|(1-\pi)\xi\|\|e^{-\kappa t/{\varepsilon}^2}$ and $x^_{p,t/{\varepsilon}^2}\leq \sqrt{C_{L_0}}\|\|\pi\xi\|\|$. Collecting all the above results in , we have for $n>2$, $$\begin{aligned} (2^{-5(n-1)}-{\varepsilon}^{2n}(\widetilde{K}K_G/\kappa)^n){\mathbb{E}}\,{\mathfrak{X}^{\varepsilon}}_{t}\,\,\leq\,\,C_1+C_2\int_0^t{\mathbb{E}}{\mathfrak{X}^{\varepsilon}}_{s} ds,\end{aligned}$$ where $$\begin{aligned} C_1&=C_{L_0}^{n/2}({\mathbb{E}}\|\|\pi\xi\|\|^n+{\mathbb{E}}\|\|(1-\pi)\xi\|\|^n)+{\varepsilon}^{2n}(\widetilde{K}K_G/\kappa)^n{\mathbb{E}}\|\|\xi\|\|^n\\ & \qquad \quad + (\|\|\Phi(0)\|\|_2\|\|{\tilde{\Psi}}\|\|_2)^n(T^{n-1}K_G^n+2^{n-1}\|\|L_1\|\|^nCT^{(n-2)/2}){\mathbb{E}}\|\|\xi\|\|^n{\varepsilon}^2r,\end{aligned}$$ $$C_2=(\|\|\Phi(0)\|\|_2\|\|{\tilde{\Psi}}\|\|_2K_G)^nT^{n-1}+C_{m,L}T^{(n-2)/2}+2^{n-1}\|\|\Phi(0)\|\|_2^n\|\|{\tilde{\Psi}}\|\|_2^nCT^{\frac{n-2}{2}}\|\|L_1\|\|^n.$$ The initial condition $\xi$ is assumed to be deterministic and hence $C_1$ can be written as $C_1=C_{L_0}^{n/2}(\|\|\pi\xi\|\|^n+\|\|(1-\pi)\xi\|\|^n)+{\varepsilon}^{2n}(KK_G/\kappa)^n\|\|\xi\|\|^n.$ Applying Gronwall inequality we have ${\mathbb{E}}\,{\mathfrak{X}^{\varepsilon}}_{T}\,\,\leq\,\,\frac{2C_1}{2^{-5(n-1)}}\exp\left(\frac{2C_2}{2^{-5(n-1)}}T\right)$ for small enough ${\varepsilon}$. For $0\leq n\leq 2$ we can use ${\mathbb{E}}\sup_{t\in [0,T]}\|{X^{\varepsilon}}_t\|^n \,\leq\, 1+ {\mathbb{E}}\sup_{t\in [0,T]}\|{X^{\varepsilon}}_t\|^3.$ Proof of Proposition \[prop:mul:supUpsT\] {#apsec:prf:prop:mul:supUpsT} ----------------------------------------- Recall the ${\gamma}$ defined in . We have $$\begin{aligned} \left({\hat{T}}(\frac{s-u}{{\varepsilon}^2})(1-{\hat{\pi}}){\mathbf{1}_{\{0\}}}\right)(\theta)=\begin{cases}{\gamma}\left(\frac{s+{\varepsilon}^2\theta -u}{{\varepsilon}^2}\right), \qquad s+{\varepsilon}^2\theta-u\geq 0\\ {\hat{\pi}}{\mathbf{1}_{\{0\}}}\left(\frac{s+{\varepsilon}^2\theta -u}{{\varepsilon}^2}\right), \qquad s+{\varepsilon}^2\theta-u< 0. \end{cases}\end{aligned}$$ Note that $$\begin{aligned} \label{add:newlem:supbound:eq:2_split1} \sup_{s\in [0,T]}\Upsilon^{{\varepsilon}}_s \,\,&\leq\,\,\sup_{s\in [0,T]}\sup_{\theta\in [-r,0]} \left\|\int_0^{(s+{\varepsilon}^2\theta)\vee 0}{\gamma}\left(\frac{s+{\varepsilon}^2\theta -u}{{\varepsilon}^2}\right) dZ_u\right\| \\ \notag & \qquad + \sup_{s\in [0,T]}\sup_{\theta\in [-r,0]} \left\|\int_{(s+{\varepsilon}^2\theta)\vee 0}^s{\hat{\pi}}{\mathbf{1}_{\{0\}}}\left(\frac{s+{\varepsilon}^2\theta -u}{{\varepsilon}^2}\right)dZ_u\right\| \qquad =: \mathscr{J}_1 + \mathscr{J}_2.\end{aligned}$$ In the above $t\vee s$ means $\max \{t,\,s\}$. For $\mathscr{J}_1$ we have (with $\delta \in (0,2)$) $$\begin{aligned} \mathscr{J}_1 \,\,&=\,\,\sup_{t\in [0,T]} \left\|\int_0^{t}{\gamma}\left(\frac{t -u}{{\varepsilon}^2}\right)dZ_u\right\| \\ &\leq \,\,\sup_{t\in [r {\varepsilon}^\delta,T]} \left\|\int_0^{t-r {\varepsilon}^\delta}{\gamma}\left(\frac{t -u}{{\varepsilon}^2}\right)dZ_u\right\| + \sup_{t\in [0,T]} \left\|\int_{(t-r {\varepsilon}^\delta)\vee 0}^t{\gamma}\left(\frac{t -u}{{\varepsilon}^2}\right)dZ_u\right\| \qquad =: \mathscr{J}_{1a} + \mathscr{J}_{1b}.\end{aligned}$$ Using integration by parts and exponential decay of ${\gamma}$ and ${\gamma}'$ (see –) in $\mathscr{J}_{1a}$ we have $$\begin{aligned} \mathscr{J}_{1a} \,\,\,\leq &\,\,\,\sup_{t\in [r {\varepsilon}^\delta,T]} \|{\gamma}(r {\varepsilon}^{\delta-2}) Z_{t-r {\varepsilon}^{\delta}}\| + \sup_{t\in [r {\varepsilon}^\delta,T]} \frac{1}{{\varepsilon}^2} \int_0^{t-r{\varepsilon}^\delta}\left\|{\gamma}'\left(\frac{t -u}{{\varepsilon}^2}\right)\right\|\,\|Z_u\|\,du \\ \leq & \,\,\, \widetilde{K} e^{-\kappa r {\varepsilon}^{\delta-2}}\sup_{t\in [0,T]}\|Z_t\|\,\,+\,\, \sup_{t\in [r {\varepsilon}^\delta,T]} \frac{1}{{\varepsilon}^2}\|\|L_0\|\|\widetilde{K}\int_0^{t-r{\varepsilon}^\delta}e^{-\kappa (t-u)/{\varepsilon}^2}\,\|Z_u\|\,du \\ \leq & \,\,\, \widetilde{K} \left(1+\frac{\|\|L_0\|\|}{\kappa}\right)e^{-\kappa r {\varepsilon}^{\delta-2}}\sup_{t\in [0,T]}\|Z_t\|,\end{aligned}$$ where $\widetilde{K}=K\|\|(1-\pi){\mathbf{1}_{\{0\}}}\|\|$. For $\mathscr{J}_{1b}$ we use $$\begin{aligned} {\gamma}\left(\frac{t -u}{{\varepsilon}^2}\right)={\gamma}\left(\frac{t -((t-r{\varepsilon}^{\delta})\vee 0)}{{\varepsilon}^2}\right)-\frac{1}{{\varepsilon}^2}\int_{(t-r{\varepsilon}^{\delta})\vee 0}^u {\gamma}'\left(\frac{t -\tau}{{\varepsilon}^2}\right)d\tau, \qquad \text{ for } u\in [(t-r{\varepsilon}^{\delta})\vee 0,t].\end{aligned}$$ Now, using the definition \[def:modcont\] of modulus of continuity, we have $$\begin{aligned} \mathscr{J}_{1b}\,\,&\leq\,\, \sup_{t\in[0,r{\varepsilon}^\delta]}\|{\gamma}(t {\varepsilon}^{-2})\|\,\|Z_t\| \,+\, \sup_{t\in[r{\varepsilon}^\delta,T]}\|{\gamma}(r{\varepsilon}^{\delta-2})\|\,\|Z_t-Z_{t-r{\varepsilon}^\delta}\| \\ & \qquad \qquad + \sup_{t\in [0,T]}\frac{1}{{\varepsilon}^2}\left\|\int_{(t-r {\varepsilon}^\delta)\vee 0}^t\left(\int_{(t-r {\varepsilon}^\delta)\vee 0}^u{\gamma}'\left(\frac{t -\tau}{{\varepsilon}^2}\right)d\tau\right) dZ_u\right\| \\ &\leq 2\widetilde{K} {\mathfrak{w}}(r{\varepsilon}^{\delta},T;Z) + \sup_{t\in [0,T]}\frac{1}{{\varepsilon}^2}\left\|\int_{(t-r {\varepsilon}^\delta)\vee 0}^t\left(\int_{\tau}^t dZ_u\right) {\gamma}'\left(\frac{t -\tau}{{\varepsilon}^2}\right)d\tau\right\| \\ & \leq 2\widetilde{K} {\mathfrak{w}}(r{\varepsilon}^{\delta},T;Z) + \sup_{t\in [0,T]}\frac{1}{{\varepsilon}^2}\int_{(t-r {\varepsilon}^\delta)\vee 0}^t \|Z_t-Z_{\tau}\| \left\|{\gamma}'\left(\frac{t -\tau}{{\varepsilon}^2}\right)\right\|d\tau \\ & \leq 2\widetilde{K} {\mathfrak{w}}(r{\varepsilon}^{\delta},T;Z) + {\mathfrak{w}}(r{\varepsilon}^{\delta},T;Z) \sup_{t\in [0,T]}\frac{1}{{\varepsilon}^2}\int_{(t-r {\varepsilon}^\delta)\vee 0}^t K\|\|L_0\|\|e^{-\kappa(t-\tau)/{\varepsilon}^2}d\tau \\ & \leq 2\widetilde{K} \left(1+\frac{\|\|L_0\|\|}{2\kappa}\right){\mathfrak{w}}(r{\varepsilon}^\delta,T;Z).\end{aligned}$$ For $\mathscr{J}_2$ we make use of the following facts: $$\begin{aligned} {\hat{\pi}}{\mathbf{1}_{\{0\}}}(v-u)&={\tilde{\Psi}}_1 \cos({\omega}_c(v-u))+{\tilde{\Psi}}_2\sin({\omega}_c(v-u)) \\ &=({\tilde{\Psi}}_1\cos {\omega}_cv+{\tilde{\Psi}}_2\sin {\omega}_cv)\cos {\omega}_cu + ({\tilde{\Psi}}_1\sin {\omega}_cv-{\tilde{\Psi}}_2\cos {\omega}_cv)\sin {\omega}_cu, \end{aligned}$$ and $\|{\tilde{\Psi}}_1\cos {\omega}_cv+{\tilde{\Psi}}_2\sin {\omega}_cv\|\leq \sqrt{{\tilde{\Psi}}_1^2+{\tilde{\Psi}}_2^2}=\|\|{\tilde{\Psi}}\|\|_2$. Using these it is easy to see that the $$\begin{aligned} \label{add:newlem:supbound:eq:2_split2_bound} \mathscr{J}_2 \,\,\leq \,\, \|\|{\tilde{\Psi}}\|\|_2\,\sup_{s\in [0,T]}\sup_{\theta\in [-r,0]} \left(\left\|M^{c,{\varepsilon}}_{s}-M^{c,{\varepsilon}}_{(s+{\varepsilon}^2\theta)\vee 0}\right\| + \left\|M^{s,{\varepsilon}}_{s}-M^{s,{\varepsilon}}_{(s+{\varepsilon}^2\theta)\vee 0}\right\|\right)\end{aligned}$$ where $$\begin{aligned} M^{c,{\varepsilon}}_t=\int_0^t \cos({\omega}_cu/{\varepsilon}^2)dZ_u, \qquad M^{s,{\varepsilon}}_t=\int_0^t \sin({\omega}_cu/{\varepsilon}^2)dZ_u.\end{aligned}$$ Using the definition \[def:modcont\] of modulus of continuity, we have $$\begin{aligned} \mathscr{J}_2 \,\,\leq \,\,\|\|{\tilde{\Psi}}\|\|_2 \,\left( \,{\mathfrak{w}}({\varepsilon}^2r,T;M^{c,{\varepsilon}}) \,+\, {\mathfrak{w}}({\varepsilon}^2r,T;M^{s,{\varepsilon}}) \,\right).\end{aligned}$$ Collecting all the above estimates in we have $$\begin{aligned} \sup_{s\in [0,T]}\Upsilon^{{\varepsilon}}_s \,\,&\leq\,\,\widetilde{K} \left(1+\frac{\|\|L_0\|\|}{\kappa}\right)e^{-\kappa r {\varepsilon}^{\delta-2}}\sup_{t\in [0,T]}\|Z_t\|\\ & \qquad +\,\,2\widetilde{K} \left(1+\frac{\|\|L_0\|\|}{2\kappa}\right){\mathfrak{w}}(r{\varepsilon}^\delta,T;Z)\\ & \qquad +\,\,\|\|{\tilde{\Psi}}\|\|_2 \,\left( \,{\mathfrak{w}}({\varepsilon}^2r,T;M^{c,{\varepsilon}}) \,+\, {\mathfrak{w}}({\varepsilon}^2r,T;M^{s,{\varepsilon}}) \,\right).\end{aligned}$$ Now we take expectations. Using Burkholder-Davis-Gundy inequality and lemma \[lem:mul:quest:supuntilTisconst\], we have for $n \geq 1$, $${\mathbb{E}}\sup_{t\in [0,T]}\|Z_t\|^n\,\leq \,C{\mathbb{E}}{\langle}Z{\rangle}_{T}^{n/2}\,\leq\,C{\mathbb{E}}\left(T \sup_{t\in[0,T]}\|L_1({{{\Pi}}^{{\varepsilon}}}_t{X^{\varepsilon}})\|^2\right)^{n/2}\leq CT^{n/2}\mathfrak{C}.$$ Using the Theorem 1 in section 3 of [@FischerNappo] and lemma \[lem:mul:quest:supuntilTisconst\], we get that there exists constants $C_{{\mathfrak{w}}}$, $C_{{\mathfrak{w}}}^c$ and $C_{{\mathfrak{w}}}^s$ such that, ${\mathbb{E}}\,{\mathfrak{w}}^n(r {\varepsilon}^\delta,T;Z) \leq C_{{\mathfrak{w}}}\left({r{\varepsilon}^\delta}\ln\left(\frac{2T}{r{\varepsilon}^\delta}\right)\right)^{n/2}$, ${\mathbb{E}}\,{\mathfrak{w}}^n( {\varepsilon}^2r,T;M^{c,{\varepsilon}}) \leq C_{{\mathfrak{w}}}^c\left({{\varepsilon}^2r}\ln\left(\frac{2T}{{\varepsilon}^2r}\right)\right)^{n/2}$ and ${\mathbb{E}}\,{\mathfrak{w}}^n( {\varepsilon}^2r,T;M^{s,{\varepsilon}}) \leq C_{{\mathfrak{w}}}^s\left({{\varepsilon}^2r}\ln\left(\frac{2T}{{\varepsilon}^2r}\right)\right)^{n/2}$. Collecting all, we have $$\begin{aligned} 2^{-5(n-1)}{\mathbb{E}}\sup_{s\in [0,T]}(\Upsilon^{{\varepsilon}}_s)^n \,\,&\leq\,\,\widetilde{K}^n\left(1+\frac{\|\|L_0\|\|}{\kappa}\right)^nCT^{n/2}\mathfrak{C}\,e^{-n\kappa r {\varepsilon}^{\delta-2}}\\ & \qquad +\left(2\widetilde{K} \left(1+\frac{\|\|L_0\|\|}{2\kappa}\right)\right)^n C_{{\mathfrak{w}}}\left({{\varepsilon}^\delta r}\ln\left(\frac{2T}{{\varepsilon}^\delta r}\right)\right)^{n/2} \\ & \qquad +\|\|{\tilde{\Psi}}\|\|_2^{n}(C_{{\mathfrak{w}}}^c+C_{{\mathfrak{w}}}^s)\left({{\varepsilon}^2r}\ln\left(\frac{2T}{{\varepsilon}^2r}\right)\right)^{n/2}.\end{aligned}$$ As ${\varepsilon}\to 0$, the 2nd term on the RHS dominates and hence we have . Proof of Proposition \[prop:mul:stabnorm\] {#apsec:prf:prop:mul:stabnorm} ------------------------------------------ Using the variation of constants formula , we have $$\begin{aligned} \label{locap03_eq:vocnonlinstable} \|\|{\mathscr{Y}}^{{\varepsilon}}_s\|\|\,\,\,\leq &\,\,\|\|\int_0^s{\hat{T}}(\frac{s-u}{{\varepsilon}^2})(1-{\hat{\pi}}){\mathbf{1}_{\{0\}}}G({{{\Pi}}^{{\varepsilon}}_{u}{X^{\varepsilon}}})du \|\| \,\,+\,\,\Upsilon^{{\varepsilon}}_s.\end{aligned}$$ Using the exponential decay we have $$\begin{aligned} \|\|\int_0^s & {\hat{T}}(\frac{s-u}{{\varepsilon}^2})(1-{\hat{\pi}}){\mathbf{1}_{\{0\}}}G({{{\Pi}}^{{\varepsilon}}_{u}{X^{\varepsilon}}})du \|\| \,\, \leq \,\,\,\int_0^s\|\|{\hat{T}}(\frac{s-u}{{\varepsilon}^2})(1-{\hat{\pi}}){\mathbf{1}_{\{0\}}}\|\|\,\|G({{{\Pi}}^{{\varepsilon}}_{u}{X^{\varepsilon}}})\|\,du \\ \notag & \leq \,\,\, K_G\sup_{u\in [0,s]}\|\|{{{\Pi}}^{{\varepsilon}}_{u}{X^{\varepsilon}}}\|\| \int_0^s \widetilde{K}e^{-\kappa(s-u)/{\varepsilon}^2}\,du\,\,\,\leq \,\,\,({\varepsilon}^2K_G\widetilde{K}/\kappa)\sup_{u\in [0,s]}\|\|{{{\Pi}}^{{\varepsilon}}_{u}{X^{\varepsilon}}}\|\|,\end{aligned}$$ where $\widetilde{K}=K\|\|(1-{\hat{\pi}}){\mathbf{1}_{\{0\}}}\|\|$. Hence for $s\in [0,T]$ $$\begin{aligned} \|\|{\mathscr{Y}}^{\varepsilon}_s\|\|\,\,&\leq\,\,({\varepsilon}^2K_G\widetilde{K}/\kappa)\sup_{u\in [0,s]}\|\|{{{\Pi}}^{{\varepsilon}}_{u}{X^{\varepsilon}}}\|\|+\,\,\Upsilon^{{\varepsilon}}_s. \end{aligned}$$ Hence $$\begin{aligned} \sup_{s\in [0,t]}\|\|{\mathscr{Y}}^{\varepsilon}_s\|\|\,\,&\leq\,\,({\varepsilon}^2K_G\widetilde{K}/\kappa)\sup_{s\in [0,t]}\|\|{{{\Pi}}^{{\varepsilon}}_{s}{X^{\varepsilon}}}\|\|+\,\,\sup_{s\in [0,t]}\Upsilon^{{\varepsilon}}_s.\end{aligned}$$ Raise to power $n$, take expectation and apply lemma \[lem:mul:quest:supuntilTisconst\] for the first term on the RHS and proposition \[prop:mul:supUpsT\] for the second term to get . Proof of lemma \[lem:mul:aux4comparison\] {#apsec:prf:lem:mul:aux4comparison} ----------------------------------------- For any ${\mathbb{R}}^2$-vector $v$, and $\theta\in [-r,0]$, we have $\Phi(\theta) e^{tB/{\varepsilon}^2}v=v_1\cos(({\omega}_ct/{\varepsilon}^2)+\theta)+v_2\sin(({\omega}_ct/{\varepsilon}^2)+\theta)$. Hence $$\begin{aligned} \label{eq:normPhietBepsqv} \|\|\Phi e^{tB/{\varepsilon}^2}v\|\|=\sup_{\theta\in [-r,0]}\|\Phi(\theta) e^{tB/{\varepsilon}^2}v\| \,\leq\,\sqrt{v_1^2+v_2^2}.\end{aligned}$$ Using Lipshitz condition on $G$, and then using $y^{\varepsilon}_t-{\mathfrak{Y}}^{\varepsilon}_t={\mathscr{Y}}^{\varepsilon}_t$ and , we get $$\begin{aligned} \label{eq:auxHelpGDiff_boundDrifCoef} \left\|G(\Phi e^{tB/{\varepsilon}^2}{\mathfrak{z}^{\varepsilon}}_t+y^{\varepsilon}_t)-G(\Phi e^{tB/{\varepsilon}^2}{\widehat{\mathfrak{z}}^{\varepsilon}}_t+{\mathfrak{Y}}^{\varepsilon}_t)\right\|\,\,\leq\,\,K_G(\|\|{\mathscr{Y}}^{\varepsilon}_t\|\|+\sqrt{2\alpha^{\varepsilon}_t}).\end{aligned}$$ Using the definition of $\Gamma_t$ we have $$\begin{aligned} \|\Gamma_t\|\,\,&=\,\,\big\|\big({\tilde{\Psi}}_1({\mathfrak{z}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t)_1+{\tilde{\Psi}}_2({\mathfrak{z}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t)_2\big)\cos({\omega}_ct/{\varepsilon}^2)+\big({\tilde{\Psi}}_1({\mathfrak{z}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t)_2-{\tilde{\Psi}}_2({\mathfrak{z}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t)_1\big)\sin({\omega}_ct/{\varepsilon}^2)\big\| \\ &\leq \,\,\sqrt{({\tilde{\Psi}}_1({\mathfrak{z}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t)_1+{\tilde{\Psi}}_2({\mathfrak{z}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t)_2)^2+({\tilde{\Psi}}_1({\mathfrak{z}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t)_2-{\tilde{\Psi}}_2({\mathfrak{z}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t)_1)^2} \\ &= \,\,\|\|{\tilde{\Psi}}\|\|_2\,\sqrt{2\alpha^{\varepsilon}_t}.\end{aligned}$$ Using the above inequality and in the definition of $\mathscr{B}_t$ we get $$\begin{aligned} \|\mathscr{B}_t\|\,\,&\leq\,\,\|\|{\tilde{\Psi}}\|\|_2\sqrt{2\alpha^{\varepsilon}_t}\,K_G(\|\|{\mathscr{Y}}^{\varepsilon}_t\|\|+\sqrt{2\alpha^{\varepsilon}_t})\,\,+\,\,\frac12\|\|{\tilde{\Psi}}\|\|_2^2\|\|L_1\|\|^2(\sqrt{2\alpha^{\varepsilon}_t}+\|\|{\mathscr{Y}}^{\varepsilon}_t\|\|)^2 \\ &\leq\,\,\|\|{\tilde{\Psi}}\|\|_2K_G(\frac{\|\|{\mathscr{Y}}^{\varepsilon}_t\|\|^2+2\alpha^{\varepsilon}_t}{2}+2\alpha^{\varepsilon}_t)\,\,+\,\,\|\|{\tilde{\Psi}}\|\|_2^2\|\|L_1\|\|^2(2\alpha^{\varepsilon}_t+\|\|{\mathscr{Y}}^{\varepsilon}_t\|\|^2) \\ &\leq\,\,C_{\mathscr{B}}(\alpha^{\varepsilon}_t+\|\|{\mathscr{Y}}^{\varepsilon}_t\|\|^2).\end{aligned}$$ Using $\|\Gamma_t\| \leq \|\|{\tilde{\Psi}}\|\|_2\,\sqrt{2\alpha^{\varepsilon}_t}$ in the definition of $\Sigma_t$ we get $$\begin{aligned} \Sigma_t^2\,\,&\leq\,\,\Gamma_t^2\|\|L_1\|\|^2(\sqrt{2\alpha^{\varepsilon}_t}+\|\|{\mathscr{Y}}^{\varepsilon}_t\|\|)^2\\ &\leq\,\,\|\|{\tilde{\Psi}}\|\|_2^2\|\|L_1\|\|^22\alpha^{\varepsilon}_t(\sqrt{2\alpha^{\varepsilon}_t}+\|\|{\mathscr{Y}}^{\varepsilon}_t\|\|)^2\,\,\leq\,\,16\|\|{\tilde{\Psi}}\|\|_2^2\|\|L_1\|\|^2((\alpha^{\varepsilon}_t)^2+\|\|{\mathscr{Y}}^{\varepsilon}_t\|\|^4).\end{aligned}$$ Proof of Proposition \[prop:mul:alphaissmall\] {#apsec:prf:prop:mul:alphaissmall} ---------------------------------------------- Using lemma \[lem:mul:aux4comparison\] we have that $$d\alpha^{\varepsilon}_t\,\leq\,C_{\mathscr{B}}(\alpha^{\varepsilon}_t+\|\|{\mathscr{Y}}^{\varepsilon}_t\|\|^2)dt+\Sigma_tdW_t.$$ Let $$\begin{aligned} H_t:=C_{\mathscr{B}}\int_0^t\|\|{\mathscr{Y}}^{\varepsilon}_s\|\|^2ds, \qquad M_t:=\int_0^t\Sigma_sdW_s, \qquad L_t:=\int_0^te^{-C_{\mathscr{B}}s}dM_s.\end{aligned}$$ Then, $$\begin{aligned} \label{eq:alphaepsIneq} \alpha^{\varepsilon}_t\,\leq\,\int_0^tC_{\mathscr{B}}\alpha^{\varepsilon}_sds\,+\,H_t\,+\,M_t.\end{aligned}$$ Applying Gronwall inequality pathwise, we get, $$\begin{aligned} \label{eq:Gronpathw} \alpha^{\varepsilon}_te^{-C_{\mathscr{B}}t}\,\leq\,\,(H_t\,+\,M_t)e^{-C_{\mathscr{B}}t}\,+\,\int_0^t(H_s+M_s)C_{\mathscr{B}}e^{-C_{\mathscr{B}}s}ds.\end{aligned}$$ Using integration by parts we get $$\begin{aligned} \int_0^tH_sC_{\mathscr{B}}e^{-C_{\mathscr{B}}s}ds\,\,&=\,\,-H_te^{-C_{\mathscr{B}}t}+\int_0^te^{-C_{\mathscr{B}}s}dH_s \\ &\leq\,\,-H_te^{-C_{\mathscr{B}}t}+\int_0^tdH_s\quad=\quad-H_te^{-C_{\mathscr{B}}t}+H_t.\end{aligned}$$ Using integration by parts we get $\int_0^tM_sC_{\mathscr{B}}e^{-C_{\mathscr{B}}s}ds\,=\,-M_te^{-C_{\mathscr{B}}t}+L_t.$ Using these results in we get $$\begin{aligned} 0\,\,\leq\,\,\alpha^{\varepsilon}_te^{-C_{\mathscr{B}}t}\,\,\leq\,\,L_t+H_t.\end{aligned}$$ Note that $L$ is a martingale. We have $$\begin{aligned} {\mathbb{E}}\sup_{s\in [0,t]}\left(\alpha^{\varepsilon}_se^{-C_{\mathscr{B}}s}\right)^2\,\,\leq\,\, {\mathbb{E}}\sup_{s\in [0,t]} (L_s+H_s)^2 \,\,&\leq\,\, 2{\mathbb{E}}\sup_{s\in [0,t]} L_s^2 + 2{\mathbb{E}}\sup_{s\in [0,t]} H_s^2 \\ &\leq\,\, 8{\mathbb{E}}L_t^2 + 2{\mathbb{E}}H_t^2\end{aligned}$$ where in the last step we have used Doob’s $L^p$ inequality (Theorem 2.1.7 in [@RevuzYor]) and the fact that $H$ is non-decreasing. Now, using BDG inequality $$\begin{aligned} {\mathbb{E}}L_{t}^2\,\,&=\,\,{\mathbb{E}}\int_0^{t}e^{-2C_{\mathscr{B}}s}\Sigma_s^2ds\,\,\leq\,\,C_{\Sigma}{\mathbb{E}}\int_0^{t}e^{-2C_{\mathscr{B}}s}((\alpha^{\varepsilon}_s)^2+\|\|{\mathscr{Y}}^{\varepsilon}_s\|\|^4)ds,\\ &\leq\,\,C_{\Sigma}\int_0^{t}{\mathbb{E}}\sup_{u\in [0,s]}\left(\alpha^{\varepsilon}_ue^{-C_{\mathscr{B}}u}\right)^2ds\,\,+\,\,C_{\Sigma}\int_0^{t}{\mathbb{E}}\|\|{\mathscr{Y}}^{\varepsilon}_s\|\|^4ds.\end{aligned}$$ Using Holder inequality we have $$2{\mathbb{E}}H_t^2\,\,=\,\,2{\mathbb{E}}\left(C_{\mathscr{B}}\int_0^t\|\|{\mathscr{Y}}^{\varepsilon}_s\|\|^2ds\right)^2\,\,\leq\,\,2C_{\mathscr{B}}^2t\int_0^t{\mathbb{E}}\|\|{\mathscr{Y}}^{\varepsilon}_s\|\|^4ds.$$ Hence, $$\begin{aligned} {\mathbb{E}}\sup_{s\in [0,t]}\left(\alpha^{\varepsilon}_se^{-C_{\mathscr{B}}s}\right)^2\,\,\leq\,\,8C_{\Sigma}\int_0^{t}{\mathbb{E}}\sup_{u\in [0,s]}\left(\alpha^{\varepsilon}_ue^{-C_{\mathscr{B}}u}\right)^2ds\,\,+\,\,(8C_{\Sigma}+2C_{\mathscr{B}}^2t)\int_0^{t}{\mathbb{E}}\|\|{\mathscr{Y}}^{\varepsilon}_s\|\|^4ds.\end{aligned}$$ Using Gronwall and then we have $$\begin{aligned} {\mathbb{E}}\sup_{s\in [0,T]}\left(\alpha^{\varepsilon}_se^{-C_{\mathscr{B}}s}\right)^2\,\,\leq&\,\,(8C_{\Sigma}+2C_{\mathscr{B}}^2T)T\,2^6\left({\varepsilon}^{8}2^{3}\left(\frac{K_GK}{\kappa}\right)^4\mathfrak{C}^4\,+\,2^{3}{\mathbb{E}}\sup_{s\in [0,T]}(\Upsilon^{{\varepsilon}}_s)^4\right)e^{8C_{\Sigma}\,T}\\ \leq&\,\,C\left(r{\varepsilon}^\delta\ln(\frac{2T}{r{\varepsilon}^\delta})\right)^2, \qquad \text{for small enough } {\varepsilon}. \end{aligned}$$ Hence $$\begin{aligned} {\mathbb{E}}\sup_{s\in [0,T]}\left(\alpha^{\varepsilon}_s\right)^2\,\, \leq\,\,Ce^{2C_{\mathscr{B}}T}\left(r{\varepsilon}^\delta\ln(\frac{2T}{r{\varepsilon}^\delta})\right)^2, \qquad \text{for small enough } {\varepsilon}. \end{aligned}$$ Proof of Proposition \[prop:mul:betaissmall\] {#apsec:prf:prop:mul:betaissmall} --------------------------------------------- Following exactly the same technique as for $\alpha^{\varepsilon}$, we arrive at $$\begin{aligned} {\mathbb{E}}\sup_{s\in [0,t]}\left(\beta_se^{-C_{\mathscr{B}}s}\right)^2\,\,\leq\,\,8C_{\Sigma}\int_0^{t}{\mathbb{E}}\sup_{u\in [0,s]}\left(\beta_ue^{-C_{\mathscr{B}}u}\right)^2ds\,\,+\,\,(8C_{\Sigma}+2C_{\mathscr{B}}^2t)\int_0^{t}{\mathbb{E}}\|\|{\mathfrak{Y}}^{\varepsilon}_s\|\|^4ds.\end{aligned}$$ Using the exponential decay we have that $\int_0^{t}{\mathbb{E}}\|\|{\mathfrak{Y}}^{\varepsilon}_s\|\|^4ds \,\leq\,\widetilde{K}^4\int_0^te^{-4\kappa s/{\varepsilon}^2}ds \,\leq\,{\varepsilon}^2(\widetilde{K}^4/4\kappa)$ where $\widetilde{K}=K\|\|(I-{\hat{\pi}}){\mathbf{1}_{\{0\}}}\|\|$. Using Gronwall inequality we have $$\begin{aligned} {\mathbb{E}}\sup_{s\in [0,T]}\left(\beta_se^{-C_{\mathscr{B}}s}\right)^2\,\,&\leq\,\,{\varepsilon}^2(8C_{\Sigma}+2C_{\mathscr{B}}^2T)T (\widetilde{K}^4/4\kappa) e^{8C_{\Sigma}\,T}.\end{aligned}$$ Proof of theorem \[prop:mul:finalthem\] {#apsec:prf:prop:mul:mainTh} --------------------------------------- Using $X^{\varepsilon}(t)=\Phi(0)e^{tB/{\varepsilon}^2}{\mathfrak{z}^{\varepsilon}}_t+y^{\varepsilon}_t(0)$ and Minkowski inequality in and then using we have $$\begin{aligned} \big\|X^{\varepsilon}(t)-\big(\Phi(0)e^{tB/{\varepsilon}^2}{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t+{\mathfrak{Y}}^{\varepsilon}_t(0)\big)\big\|^4 \,&\leq\, 8\left(\|\|\Phi e^{tB/{\varepsilon}^2}({\mathfrak{z}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t)\|\|^4+\|\|\Phi e^{tB/{\varepsilon}^2}({\widehat{\mathfrak{z}}^{\varepsilon}}_t-{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t)\|\|^4+\|\|y^{\varepsilon}_t-{\mathfrak{Y}}^{\varepsilon}_t\|\|^4\right) \\ &\leq\, 8\left(\|\|{\mathfrak{z}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|_2^4+\|\|{\widehat{\mathfrak{z}}^{\varepsilon}}_t-{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t\|\|_2^4+\|\|{\mathscr{Y}}^{\varepsilon}_t\|\|^4\right) \\ &\leq\, 8\left(4(\alpha^{\varepsilon}_t)^2+4(\beta^{\varepsilon}_t)^2+\|\|{\mathscr{Y}}^{\varepsilon}_t\|\|^4\right).\end{aligned}$$ Combining propositions \[prop:mul:stabnorm\], \[prop:mul:alphaissmall\] and \[prop:mul:betaissmall\] and realizing that $\left(r{\varepsilon}^\delta\ln(\frac{2T}{r{\varepsilon}^\delta})\right)^2 \,\ll\,{\varepsilon}^2$ for small enough ${\varepsilon}$ when $\delta \in (1,2)$, we get . Similar is the proof for . Proof of lemma \[lem:mul:auxzbound\] {#apsec:prf:lem:mul:auxzbound} ------------------------------------ Define $\zeta^{\varepsilon}_t=\frac12\|\|{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t\|\|_2^2$. Using Ito formula we have $d\zeta^{\varepsilon}_t=\widetilde{\mathscr{B}}_tdt+\widetilde{\Sigma}_tdW_t$ where $$\begin{aligned} \widetilde{\mathscr{B}}_t\,\,=\,\,\widetilde{\Gamma}_tG(\Phi e^{tB/{\varepsilon}^2}{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t)\,+\, \frac12\|\|e^{-tB/{\varepsilon}^2}{\tilde{\Psi}}\|\|_2^2 \big(L_1(\Phi e^{tB/{\varepsilon}^2}{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t)\big)^2, \qquad \quad \widetilde{\Sigma}_t=\widetilde{\Gamma}_tL_1(\Phi e^{tB/{\varepsilon}^2}{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t),\end{aligned}$$ and $\widetilde{\Gamma}_t=\sum_{i=1}^2({\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t)_i(e^{-tB/{\varepsilon}^2}{\tilde{\Psi}})_i$. Using similar technique as in proof of lemma \[lem:mul:aux4comparison\] it can be shown that $\|\widetilde{\mathscr{B}}_t\| \leq C_{\widetilde{\mathscr{B}}}\zeta^{\varepsilon}_t$ and $\widetilde{\Sigma}_t^2\leq C_{\widetilde{\Sigma}}(\zeta^{\varepsilon}_t)^2$ where $C_{\widetilde{\mathscr{B}}}=2\|\|{\tilde{\Psi}}\|\|_2K_G+\|\|{\tilde{\Psi}}\|\|_2^2\|\|L_1\|\|^2$ and $C_{\widetilde{\Sigma}}=4\|\|{\tilde{\Psi}}\|\|_2^2\|\|L_1\|\|^2$. Hence we have $$\begin{aligned} \zeta^{\varepsilon}_t \,\,\leq\,\, \int_0^tC_{\widetilde{\mathscr{B}}}\zeta^{\varepsilon}_sds + \widetilde{H}_t + \widetilde{M}_t, \qquad \widetilde{H}_t:=\zeta^{\varepsilon}_0, \quad \widetilde{M}_t:=\int_0^t \widetilde{\Sigma}_sdW_s,\end{aligned}$$ which is analogous to . Following the same technique as in section \[apsec:prf:prop:mul:alphaissmall\] we get $$\begin{aligned} {\mathbb{E}}\sup_{s\in [0,T]}\left(\zeta^{\varepsilon}_s\right)^2\,\, \leq\,\,{\mathbb{E}}(\zeta^{\varepsilon}_0)^2 e^{(2C_{\widetilde{\mathscr{B}}}+8C_{\widetilde{\Sigma}})T}.\end{aligned}$$ Proofs of results in section \[sec:addnoi\] =========================================== Proof of proposition \[prop:add:stabnorm\] {#apsec:prf:prop:add:stabnorm} ------------------------------------------ Using the variation of constants formula and definition , we have $$\begin{aligned} \label{locadd:eq:vocnonlinstable} \|\|{\mathscr{Y}}^{{\varepsilon}}_s\|\|\,\,\,\leq &\,\,\,\|\|\int_0^s{\hat{T}}(\frac{s-u}{{\varepsilon}^2})(1-{\hat{\pi}}){\mathbf{1}_{\{0\}}}G({{{\Pi}}^{{\varepsilon}}}_u {X^{\varepsilon}})du \|\| \,\,+\,\,\Upsilon^{{\varepsilon}}_s.\end{aligned}$$ For $G$ defined in we have $$\begin{aligned} \label{eq:boundonGforTstopt} \|G(\eta)\| &\leq \int \|\pi\eta\| \|d\nu_1\|+\int \|(1-\pi)\eta\| \|d\nu_1\|+ \sum_{j=0}^3\binom{3}{j}\int_{-r}^0\|\pi\eta\|^{3-j}\|(1-\pi)\eta\|^j\|d\nu_3\|.\end{aligned}$$ For $s\in [0,T\wedge{\mathfrak{e}^{\varepsilon}}]$ we have that $\|\|\pi{{{\Pi}}^{{\varepsilon}}}_s{X^{\varepsilon}}\|\|\leq C_{{\mathfrak{e}}}$. Using this fact and $\|\|(1-\pi){{{\Pi}}^{{\varepsilon}}}_s {X^{\varepsilon}}\|\|\leq \|\|{\mathscr{Y}}^{\varepsilon}_s\|\|+\|\|{\mathfrak{Y}}^{\varepsilon}_s\|\|$ in , and using inequalities $q \leq 1+q^3$, $q^2\leq 1+q^3$ for $q>0$; we have for $s\in [0,T\wedge{\mathfrak{e}^{\varepsilon}}]$ $$\|G({{{\Pi}}^{{\varepsilon}}}_s{X^{\varepsilon}})\|\leq C(1+\|\|{\mathscr{Y}}^{\varepsilon}_s\|\|^3+\|\|{\mathfrak{Y}}^{\varepsilon}_s\|\|^3).$$ This $C$ is of the order of $C_{{\mathfrak{e}}}^3$ for large $C_{{\mathfrak{e}}}$. Now, using the above inequality and the exponential decays and we have $$\begin{aligned} \|\|\int_0^s & {\hat{T}}(\frac{s-u}{{\varepsilon}^2})(1-{\hat{\pi}}){\mathbf{1}_{\{0\}}}G({{{\Pi}}^{{\varepsilon}}_{u}{X^{\varepsilon}}})du \|\| \,\, \leq \,\,\,\int_0^s\|\|{\hat{T}}(\frac{s-u}{{\varepsilon}^2})(1-{\hat{\pi}}){\mathbf{1}_{\{0\}}}\|\|\,\|G({{{\Pi}}^{{\varepsilon}}_{u}{X^{\varepsilon}}})\|\,du \\ \notag & \leq \,\,\, C\int_0^s e^{-\kappa(s-u)/{\varepsilon}^2}(1+\|\|{\mathscr{Y}}^{\varepsilon}_u\|\|^3+\|\|{\mathfrak{Y}}^{\varepsilon}_u\|\|^3)\,du \,\,\,\,\\ &\leq \,\,\,(C{\varepsilon}^2/\kappa)(1+K^3\|\|(1-\pi){{{\Pi}}^{{\varepsilon}}}_0{X^{\varepsilon}}\|\|^3/2)\,\,+\,\,C\int_0^s e^{-\kappa(s-u)/{\varepsilon}^2}\,\|\|{\mathscr{Y}}^{\varepsilon}_u\|\|^3\,du.\end{aligned}$$ Plugging the above inequality in we have for $s\in [0,T\wedge {\mathfrak{e}^{\varepsilon}}]$ $$\begin{aligned} \|\|{\mathscr{Y}}^{\varepsilon}_s\|\|-\left({C}{\varepsilon}^2\,\,+\,\,{C}\int_0^s e^{-\kappa(s-u)/{\varepsilon}^2}\,\|\|{\mathscr{Y}}^{\varepsilon}_u\|\|^3\,du\right)\,\,\leq\,\,\Upsilon^{{\varepsilon}}_s,\end{aligned}$$ where $C$ above is of the order of $C_{{\mathfrak{e}}}^3$ for large $C_{{\mathfrak{e}}}$. For the RHS of the above inequality we use Markov inequality, i.e. $$\begin{aligned} {\mathbb{P}}\left[\sup_{s\in [0,T\wedge{\mathfrak{e}^{\varepsilon}}]}\Upsilon^{{\varepsilon}}_s\,\geq\,{\varepsilon}^a\right] \,\,\leq\,\,{\varepsilon}^{-a}{\mathbb{E}}\left[\sup_{s\in [0,T]}\Upsilon^{{\varepsilon}}_s\right]\end{aligned}$$ and then proposition \[prop:add:supUpsT\]. Then we have the following statement: Fix $a\in [0,1)$. For $\delta \in (2a,2)$, there exists constants $\hat{C}>0$ (independent of $\delta$ and $a$) and ${\varepsilon}_{\delta}>0$, such that for ${\varepsilon}< {\varepsilon}_{\delta}$ $$\begin{aligned} {\mathbb{P}}\bigg[\forall s\in [0,T\wedge{\mathfrak{e}^{\varepsilon}}],\quad \|\|{\mathscr{Y}}^{\varepsilon}_s\|\| \,\,\leq\,\,& C{\varepsilon}^2\,\,+\,\,C\int_0^s e^{-\kappa(s-u)/{\varepsilon}^2}\,\|\|{\mathscr{Y}}^{\varepsilon}_u\|\|^3\,du+ 2{\varepsilon}^a \bigg] \\ & \geq\,\,\,1-\hat{C}{\varepsilon}^{-a}\sqrt{{r{\varepsilon}^\delta}\ln\left(\frac{T}{r{\varepsilon}^\delta}\right)}. \end{aligned}$$ Using Gronwall kind of inequality (Theorem 2.4.8 in [@Bainov]) we have that LHS of above inequality is bounded above by $$\begin{aligned} {\mathbb{P}}\bigg[\forall s\in [0,T\wedge{\mathfrak{e}^{\varepsilon}}],\quad \|\|{\mathscr{Y}}^{\varepsilon}_s\|\| \,\,\leq\,\,\frac{C{\varepsilon}^2+2{\varepsilon}^a }{\sqrt{1-2\int_0^s\left(C{\varepsilon}^2+2{\varepsilon}^a \right)^2Cdu}} \bigg] \end{aligned}$$ which is bounded above by (for small enough ${\varepsilon}$, i.e. ${\varepsilon}\ll (2/C)^{1/(2-a)}$) $$\begin{aligned} {\mathbb{P}}\bigg[\forall s\in [0,T\wedge{\mathfrak{e}^{\varepsilon}}],\quad \|\|{\mathscr{Y}}^{\varepsilon}_s\|\| \,\,\leq\,\,\frac{4{\varepsilon}^a }{\sqrt{1-2CT(4{\varepsilon}^a)^2}} \bigg]. \end{aligned}$$ which is bounded above by (for small enough ${\varepsilon}$, i.e. ${\varepsilon}\ll (1/C)^{1/2a}$) $$\begin{aligned} {\mathbb{P}}\bigg[\forall s\in [0,T\wedge{\mathfrak{e}^{\varepsilon}}],\quad \|\|{\mathscr{Y}}^{\varepsilon}_s\|\| \,\,\leq\,\,{8{\varepsilon}^a } \bigg]. \end{aligned}$$ Hence follows. Proof of lemma \[lem:add:aux4comparison\] {#apsec:prf:lem:add:aux4comparison} ----------------------------------------- Recall that $G(\eta)=\int_{-r}^0\eta(\theta)d\nu_1(\theta)+\int_{-r}^0\eta^3(\theta)d\nu_3(\theta)$. For brevity, let ${\mathbf{e}}$ denote $e^{tB/{\varepsilon}^2}$. Now, $$\begin{aligned} \bigg\|\int_{-r}^0&(\Phi {\mathbf{e}}{\mathfrak{z}^{\varepsilon}}_t+y^{\varepsilon}_t)^3d\nu_3 - \int_{-r}^0(\Phi {\mathbf{e}}{\widehat{\mathfrak{z}}^{\varepsilon}}_t+{\mathfrak{Y}}^{\varepsilon}_t)^3d\nu_3 \bigg\|\\ &=\,\left\|\int_{-r}^0(\Phi {\mathbf{e}}{\mathfrak{z}^{\varepsilon}}_t+{\mathfrak{Y}}^{\varepsilon}_t+{\mathscr{Y}}^{\varepsilon}_t)^3d\nu_3 - \int_{-r}^0(\Phi {\mathbf{e}}{\mathfrak{z}^{\varepsilon}}_t+\Phi {\mathbf{e}}({\widehat{\mathfrak{z}}^{\varepsilon}}_t-{\mathfrak{z}^{\varepsilon}}_t)+{\mathfrak{Y}}^{\varepsilon}_t)^3d\nu_3 \right\| \\ &\leq \sum_{j=1}^3\binom{3}{j}\left\|\int_{-r}^0(\Phi {\mathbf{e}}{\mathfrak{z}^{\varepsilon}}_t)^{3-j}(({\mathfrak{Y}}^{\varepsilon}_t+{\mathscr{Y}}^{\varepsilon}_t)^j-(\Phi {\mathbf{e}}({\widehat{\mathfrak{z}}^{\varepsilon}}_t-{\mathfrak{z}^{\varepsilon}}_t)+{\mathfrak{Y}}^{\varepsilon}_t)^j)d\nu_3 \right\| \\ & \leq \left(3\sum_{j=0}^2\|\|\Phi {\mathbf{e}}{\mathfrak{z}^{\varepsilon}}_t\|\|^j\right)\left(3\sum_{j=0}^2\|\|{\mathfrak{Y}}^{\varepsilon}_t\|\|^j\right) \left(\int\|d\nu_1\|+\|d\nu_3\|\right)\sum_{j=1}^3(\|\|{\mathscr{Y}}^{\varepsilon}_t\|\|^j+\|\|\Phi {\mathbf{e}}({\widehat{\mathfrak{z}}^{\varepsilon}}_t-{\mathfrak{z}^{\varepsilon}}_t)\|\|^j).\end{aligned}$$ Note that for $t\in [0,T\wedge{\mathfrak{e}^{\varepsilon}}]$, ${\mathfrak{z}^{\varepsilon}}$ is bounded. Also, due to the exponential decay we have that $\|\|{\mathfrak{Y}}^{\varepsilon}_t\|\| \,<\,K\|\|(I-\pi){{{\Pi}}^{{\varepsilon}}_{0}{X^{\varepsilon}}}\|\|$. Hence we have, $$\begin{aligned} \bigg\|\int_{-r}^0(\Phi {\mathbf{e}}{\mathfrak{z}^{\varepsilon}}_t+y^{\varepsilon}_t)^3d\nu_3 - \int_{-r}^0(\Phi {\mathbf{e}}{\widehat{\mathfrak{z}}^{\varepsilon}}_t+{\mathfrak{Y}}^{\varepsilon}_t)^3d\nu_3 \bigg\| \,\, \leq \,\,C\sum_{j=1}^3(\|\|{\mathscr{Y}}^{\varepsilon}_t\|\|^j+(\sqrt{2\alpha^{\varepsilon}_t})^j),\end{aligned}$$ where $C$ in the above inequality is of the order of $C_{{\mathfrak{e}}}^2$ for large $C_{{\mathfrak{e}}}$. Similarly, $$\begin{aligned} \left\|\int_{-r}^0(\Phi {\mathbf{e}}{\mathfrak{z}^{\varepsilon}}_t+y^{\varepsilon}_t)d\nu_1 - \int_{-r}^0(\Phi {\mathbf{e}}{\widehat{\mathfrak{z}}^{\varepsilon}}_t+{\mathfrak{Y}}^{\varepsilon}_t)d\nu_1 \right\|\,&\leq\,C(\|\|{\mathscr{Y}}^{\varepsilon}_t\|\|+(\sqrt{2\alpha^{\varepsilon}_t})).\end{aligned}$$ Combining, we get that for $t\in [0,T\wedge{\mathfrak{e}^{\varepsilon}}]$, $$\|\Gamma_t\|\left\|G(\Phi e^{tB/{\varepsilon}^2}{\mathfrak{z}^{\varepsilon}}_t+y^{\varepsilon}_t)-G(\Phi e^{tB/{\varepsilon}^2}{\widehat{\mathfrak{z}}^{\varepsilon}}_t)\right\|\,\,\leq\,\,\|\Gamma_t\|C\sum_{j=1}^3(\|\|{\mathscr{Y}}^{\varepsilon}_t\|\|^j+(\sqrt{2\alpha^{\varepsilon}_t})^j).$$ We have shown $\|\Gamma_t\|\leq \|\|{\tilde{\Psi}}\|\|_2\sqrt{2\alpha^{\varepsilon}_t}$ in section \[apsec:prf:lem:mul:aux4comparison\]. Hence, if we define $\mathfrak{B}$ by , then we have, $\|\mathscr{B}_t\|\leq \mathfrak{B}(\alpha_t,\|\|{\mathscr{Y}}^{\varepsilon}_t\|\|)$ for $t\in[0,T\wedge {\mathfrak{e}^{\varepsilon}}]$. Proof of proposition \[prop:add:alhaissmall\] {#apsec:prf:prop:add:alhaissmall} --------------------------------------------- Using lemma \[lem:add:aux4comparison\] and $\sqrt{2\alpha}(\sqrt{2\alpha})^2\leq 4\alpha(1+\alpha)$ we have $$d\alpha^{\varepsilon}_t \,\,\leq\,\,(6C\alpha^{\varepsilon}_t+8C(\alpha^{\varepsilon}_t)^2)dt + C\sqrt{2\alpha^{\varepsilon}_t}(\sum_{j=1}^3\|\|{\mathscr{Y}}^{\varepsilon}_t\|\|^j)dt, \qquad t\in [0,T\wedge{\mathfrak{e}^{\varepsilon}}], \qquad \alpha^{\varepsilon}_0=0,$$ where $C$ is from lemma \[lem:add:aux4comparison\]. This $C$ is of the order ${\mathcal{O}}(C_{{\mathfrak{e}}}^2)$ for large $C_{{\mathfrak{e}}}$. Let ${\varepsilon}_{a,C_{{\mathfrak{e}}}}$ be as in proposition \[prop:add:stabnorm\] and define ${\varepsilon}_{a,C_{{\mathfrak{e}}},1}=\min\{1,{\varepsilon}_{a,C_{{\mathfrak{e}}}}\}$. Then we have, for ${\varepsilon}< \min\{\hat{{\varepsilon}}_{\delta},\,{\varepsilon}_{a,C_{{\mathfrak{e}}},1}\}$, with probability atleast $p_{{\varepsilon}}:=1-\hat{C}{\varepsilon}^{-a}\sqrt{{r{\varepsilon}^\delta}\ln\left(\frac{T}{r{\varepsilon}^\delta}\right)},$ $$\sum_{j=1}^3\|\|{\mathscr{Y}}^{\varepsilon}_t\|\|^j \,\,\leq\,\,24{\varepsilon}^a, \qquad \forall t\in [0,T\wedge{\mathfrak{e}^{\varepsilon}}].$$ (we have used that for ${\varepsilon}\leq 1$, ${\varepsilon}^{3a}\leq {\varepsilon}^a$.) Let $\mathfrak{s}:=\inf\{t\geq 0\,:\,\alpha^{\varepsilon}_t\geq 1\}$. Using $\sqrt{2\alpha}\leq 2(1+\alpha)$, and $\alpha^2<\alpha$ when $\alpha<1$; we have for $t\in [0,T\wedge{\mathfrak{e}^{\varepsilon}}\wedge \mathfrak{s}]$ $$\begin{aligned} \frac1C d\alpha^{\varepsilon}_t \,\,\leq\,\,\alpha^{\varepsilon}_t\left(6+8+48{\varepsilon}^a\right)dt\,+\,48{\varepsilon}^adt\end{aligned}$$ Using Gronwall we get for $t\in [0,T\wedge{\mathfrak{e}^{\varepsilon}}\wedge \mathfrak{s}]$ $$\begin{aligned} \alpha^{\varepsilon}_t\,\,\leq\,\,\frac{48{\varepsilon}^a}{14+48{\varepsilon}^a}(e^{(14+48{\varepsilon}^a)Ct}-1).\end{aligned}$$ Define ${\varepsilon}_{a,C_{{\mathfrak{e}}},2}=\min\{48^{-1/a},\,(14e^{-15CT}/48)^{2/a}\}$. Then for ${\varepsilon}< \min\{\hat{{\varepsilon}}_{\delta},\,{\varepsilon}_{a,C_{{\mathfrak{e}}},1},\,{\varepsilon}_{a,C_{{\mathfrak{e}}},2}\}$ we have $\alpha^{\varepsilon}_t \leq {\varepsilon}^{a/2}$ for $t\in [0,T\wedge{\mathfrak{e}^{\varepsilon}}\wedge \mathfrak{s}]$. But, since ${\varepsilon}^{a/2}<1$ we have $\mathfrak{s}> T\wedge{\mathfrak{e}^{\varepsilon}}$ and hence $\alpha^{\varepsilon}_t \leq {\varepsilon}^{a/2}$ for $t\in [0,T\wedge{\mathfrak{e}^{\varepsilon}}]$. Note that $C$ is of the order ${\mathcal{O}}(C_{{\mathfrak{e}}}^2)$ and hence ${\varepsilon}_{a,C_{{\mathfrak{e}}},2}$ is of the order ${\mathcal{O}}(e^{-30C_{{\mathfrak{e}}}^2T/a})$. Proof of theorem \[prop:helpinprobconvg\] {#apsec:prf:prop:add:helpinprobconvg} ----------------------------------------- Since the ${\widehat{\mathfrak{z}}^{\varepsilon}}$ system possesses the property $\mathscr{P}(T,q)$, $\,\,\exists\, C_{{\mathfrak{e}}},\,{\varepsilon}_>0$ such that $\forall {\varepsilon}<{\varepsilon}_$, we have ${\mathbb{P}}[E^{\varepsilon}]\geq 1-q$ where $E^{\varepsilon}$ is given in . The stopping time ${\mathfrak{e}^{\varepsilon}}$ was defined at . Using $$\begin{aligned} {\Omega}\,\,&=\,\,E^{\varepsilon}\,\cup\, ({\Omega}\setminus E^{\varepsilon}) \\ &=\,\,(E^{\varepsilon}\cap \{{\mathfrak{e}^{\varepsilon}}\leq T\}) \,\cup\, (E^{\varepsilon}\cap \{{\mathfrak{e}^{\varepsilon}}> T\}) \,\cup\, ({\Omega}\setminus E^{\varepsilon}), \end{aligned}$$ we have $$\begin{aligned} \label{eq:auxlevprob2remstopt_add} H^{\varepsilon}\,\,&=\,\,(E^{\varepsilon}\cap \{{\mathfrak{e}^{\varepsilon}}\leq T\}\cap H^{\varepsilon}) \,\cup\, (E^{\varepsilon}\cap \{{\mathfrak{e}^{\varepsilon}}> T\}\cap H^{\varepsilon}) \,\cup\, (H^{\varepsilon}\cap ({\Omega}\setminus E^{\varepsilon}) ).\end{aligned}$$ Now we deal with the first term on the RHS of . Note that $$\begin{aligned} \sup_{t\in[0,T\wedge{\mathfrak{e}^{\varepsilon}}]}\sqrt{2\alpha^{{\varepsilon}}_t} \,\,\geq\,\sup_{t\in[0,T\wedge{\mathfrak{e}^{\varepsilon}}]}\|\|\Phi e^{tB/{\varepsilon}^2}({\mathfrak{z}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t)\|\|\,\,\geq\,\sup_{t\in[0,T\wedge{\mathfrak{e}^{\varepsilon}}]}\|\|\Phi e^{tB/{\varepsilon}^2}{\mathfrak{z}^{\varepsilon}}_t\|\|-\sup_{t\in[0,T\wedge{\mathfrak{e}^{\varepsilon}}]}\|\|\Phi e^{tB/{\varepsilon}^2}{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|.\end{aligned}$$ In $E^{\varepsilon}$ we have $\sup_{t\in[0,T\wedge{\mathfrak{e}^{\varepsilon}}]}\|\|\Phi e^{tB/{\varepsilon}^2}{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|<0.99C_{{\mathfrak{e}}}$, and in $\{{\mathfrak{e}^{\varepsilon}}\leq T\}$ we have $\sup_{t\in[0,T\wedge{\mathfrak{e}^{\varepsilon}}]}\|\|\Phi e^{tB/{\varepsilon}^2}{\mathfrak{z}^{\varepsilon}}_t\|\|\geq C_{{\mathfrak{e}}}$. Hence, in $E^{\varepsilon}\cap \{{\mathfrak{e}^{\varepsilon}}\leq T\}$ we have that $\sup_{t\in[0,T\wedge{\mathfrak{e}^{\varepsilon}}]}\sqrt{2\alpha^{{\varepsilon}}_t}>0.01C_{{\mathfrak{e}}}$. Hence $E^{\varepsilon}\cap \{{\mathfrak{e}^{\varepsilon}}\leq T\} \,\subset\,J^{\varepsilon}$ where $J^{\varepsilon}:=\left\{{\omega}\,:\,\sup_{t\in[0,T\wedge{\mathfrak{e}^{\varepsilon}}]}\alpha^{{\varepsilon}}_t \,\geq\,\frac12(0.01C_{{\mathfrak{e}}})^2\right\}$. By proposition \[prop:add:alhaissmall\], $\exists \,{\varepsilon}_1$ such that $\forall {\varepsilon}<{\varepsilon}_1$, ${\mathbb{P}}[J^{\varepsilon}]<1-p_{{\varepsilon}}$. Now we deal with the second term on the RHS of . Note that $\{{\mathfrak{e}^{\varepsilon}}> T\}\cap H^{\varepsilon}\,\subset\,\widetilde{J}^{\varepsilon}$ where $$\widetilde{J}^{\varepsilon}:=\left\{{\omega}\,:\,\sup_{t\in[0,T\wedge{\mathfrak{e}^{\varepsilon}}]}\alpha^{{\varepsilon}}_t \,\geq\,{\varepsilon}^{a/2}\right\}.$$ By proposition \[prop:add:alhaissmall\], $\exists \,{\varepsilon}_2$ such that $\forall {\varepsilon}<{\varepsilon}_2$, ${\mathbb{P}}[\widetilde{J}^{\varepsilon}]<1-p_{{\varepsilon}}$. And $\forall {\varepsilon}<{\varepsilon}_$ ${\mathbb{P}}[{\Omega}\setminus E^{\varepsilon}]<q$. Combining we have, when ${\varepsilon}<\min\{{\varepsilon}_1,{\varepsilon}_2,{\varepsilon}_\}=:{\varepsilon}_q$, $${\mathbb{P}}[H^{\varepsilon}]<q+2(1-p_{{\varepsilon}}).$$ Note that is true with $H^{\varepsilon}$ replaced by $S^{\varepsilon}$. Using that $$\{{\mathfrak{e}^{\varepsilon}}> T\}\cap S^{\varepsilon}\,\subset\,\{{\omega}:\sup_{t\in [0,T\wedge{\mathfrak{e}^{\varepsilon}}]}\|\|{\mathscr{Y}}^{\varepsilon}_t\|\|\geq 8{\varepsilon}^a\}$$ and also that the probability of the latter set is bounded above by $1-p_{{\varepsilon}}$ we get the desired result. Proof of proposition \[lem:add:propertyPTq\] {#appsec:lem:add:propertyPTq} -------------------------------------------- We have $$\begin{aligned} \label{eq:lem:add:propertyPTq:eq1} {\widehat{\mathfrak{z}}^{\varepsilon}}_t={\widehat{\mathfrak{z}}^{\varepsilon}}_0+\int_0^te^{-sB/{\varepsilon}^2}{\tilde{\Psi}}G(\Phi e^{sB/{\varepsilon}^2}{\widehat{\mathfrak{z}}^{\varepsilon}}_s+{\mathfrak{Y}}^{\varepsilon}_s)ds+{\mathfrak{w}}_t, \qquad {\mathfrak{w}}_t:=\int_0^te^{-sB/{\varepsilon}^2}{\tilde{\Psi}}\sigma dW_s.\end{aligned}$$ To keep things simple, we prove assuming $\|\|{\mathfrak{Y}}^{\varepsilon}_0\|\|=0$ (which ensures that $\|\|{\mathfrak{Y}}^{\varepsilon}_t\|\|=0$ for all $t\geq 0$). Using $\int_0^t \|\|{\mathfrak{Y}}^{\varepsilon}_s\|\|^nds \leq {\varepsilon}^2(K/nk)\|\|{\mathfrak{Y}}^{\varepsilon}_0\|\|^n$ (because of exponential decay ), it is easy to see that the following ideas work even if we assume that $\|\|{\mathfrak{Y}}^{\varepsilon}_0\|\|\neq 0$ (we assume the initial condition is deterministic). We will make use of the inequality[^5] that for ${\mathbb{R}}^2$ vector $v$, $$\begin{aligned} \label{eq:Phievleqv1} \|\|\Phi e^{tB/{\varepsilon}^2}v\|\| \leq \|\|v\|\|_1\end{aligned}$$ where $\|\|\cdot\|\|_1$ indicates the 1-norm. Using the structure of $G$ specified at in we have (with some $K_G>0$) $$\begin{aligned} \label{eq:lem:add:propertyPTq:auxeq1p5} \|\|{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|_1\leq\|\|{\widehat{\mathfrak{z}}^{\varepsilon}}_0\|\|_1+\|\|{\tilde{\Psi}}\|\|_1\int_0^tK_G (\|\|{\widehat{\mathfrak{z}}^{\varepsilon}}_s\|\|_1+\|\|{\widehat{\mathfrak{z}}^{\varepsilon}}_s\|\|_1^3)ds+\|\|{\mathfrak{w}}_t\|\|_1.\end{aligned}$$ Because the initial condition is deterministic, we have a $C_0>0$ such that $\|\|{\widehat{\mathfrak{z}}^{\varepsilon}}_0\|\|_1<C_0$. For any $C_a>4C_0$, define $T_{C_a}:=\left(2(1+C_a^2)K_G\|\|{\tilde{\Psi}}\|\|_1\right)^{-1}.$ Suppose that $\sup_{t\in [0,T]}\|\|{\mathfrak{w}}_t\|\|_1<C_a/4$. If $T\leq T_{C_a}$, as long as $\|\|{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|_1<C_a$, we have (using ) for $t\in [0,T]$ $$\begin{aligned} \|\|{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|_1\,\,&<\,\, C_0+K_G\|\|{\tilde{\Psi}}\|\|_1(C_a+C_a^3)T+\frac14C_a \\ &< \,\,\frac14C_a +K_G\|\|{\tilde{\Psi}}\|\|_1(C_a+C_a^3)T_{C_a}+\frac14C_a \,\,=\,\,C_a.\end{aligned}$$ This means that, if $C_a>4C_0$ and $T\leq T_{C_a}$, then we have $\sup_{t\in[0,T]}\|\|{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|_1<C_a$ provided $\sup_{t\in [0,T]}\|\|{\mathfrak{w}}_t\|\|_1<C_a/4$. Hence, for $C_a>4C_0$ and $T\leq T_{C_a}$, $$\begin{aligned} \label{eq:lem:add:propertyPTq:auxeq2} {\mathbb{P}}\left[\sup_{t\in[0,T]}\|\|{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|_1 \geq C_a\right]\,\,\leq\,\,{\mathbb{P}}\left[\sup_{t\in[0,T]}\|\|{\mathfrak{w}}_t\|\|_1 \geq C_a/4\right].\end{aligned}$$ Using Markov inequality and Burkholder-Davis-Gundy inequality we have $$\begin{aligned} {\mathbb{P}}[\sup_{t\in[0,T]}\|\|{\mathfrak{w}}_t\|\|_1\,\geq\,C_a/4]\,\,&\leq\,\,\frac{{\mathbb{E}}\sup_{t\in [0,T]}\|\|{\mathfrak{w}}_t\|\|_1}{C_a/4}\\ &\leq\,\, \frac{\sum_{j=1}^2C_{bdg}{\mathbb{E}}\sqrt{\int_0^T (e^{-sB/{\varepsilon}^2}{\tilde{\Psi}}\sigma)^2ds}}{C_a/4} \,\,\leq\,\,\frac{8\|\sigma\|\,\|\|{\tilde{\Psi}}\|\|_2\,\sqrt{T}C_{bdg}}{C_a}.\end{aligned}$$ Using the above inequality in we have for $C_a>4C_0$ and $T \leq T_{C_a}$ $$\begin{aligned} \label{eq:lem:add:propertyPTq:auxeq3} {\mathbb{P}}\left[\sup_{t\in[0,T]}\|\|{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|_1 \geq C_a\right]\,\,\leq\,\,\frac{8\|\sigma\|\,\|\|{\tilde{\Psi}}\|\|_2C_{bdg}}{C_{a}\sqrt{2K_G\|\|{\tilde{\Psi}}\|\|_1(1+C_a^2)}}\,\,=:\,\,f(C_a).\end{aligned}$$ Given $q>0$, let $C_{a,q}>4C_0$ be such that $f(C_a)<q$, $\forall C_a \geq C_{a,q}$. Such a $C_{a,q}$ exists because $f$ is monotonically decreasing in $C_a$. Set $T_q=T_{C_{a,q}}$. Choose $C_{{\mathfrak{e}}}>C_{a,q}/0.99$. Let $$\begin{aligned} \label{eq:def:ETqwidetilde} \widetilde{E}^{\varepsilon}:= \left\{{\omega}\,:\,\sup_{t\in [0,T_q]}\|\|\Phi e^{tB/{\varepsilon}^2}{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|\,<\,0.99C_{{\mathfrak{e}}}\right\}.\end{aligned}$$ Now, using and $$\begin{aligned} {\mathbb{P}}[{\Omega}\setminus \widetilde{E}^{\varepsilon}]\,\,& = {\mathbb{P}}\left[\sup_{t\in [0,T_q]}\|\|\Phi e^{tB/{\varepsilon}^2}{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|\,\geq\,0.99C_{{\mathfrak{e}}}\right]\\ &\leq {\mathbb{P}}\left[\sup_{t\in [0,T_q]}\|\|{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|_1\,\geq\,0.99C_{{\mathfrak{e}}}\right]\,\, \leq\,\,{\mathbb{P}}\left[\sup_{t\in[0,T_q]}\|\|{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|_1\geq C_{a,q}\right]\,\,\leq \,\,f(C_{a,q})\,\,<\,\,q.\end{aligned}$$ Hence ${\mathbb{P}}[\widetilde{E}^{\varepsilon}] \geq 1-q$. But, for $T \leq T_q$ the set $E^{\varepsilon}$ defined in contains $\widetilde{E}^{\varepsilon}$ and hence we have that for $T\in [0,T_q]$, ${\mathbb{P}}[E^{\varepsilon}]\,\,\geq\,\,1-q.$ Hence possesses the property $\mathscr{P}(T,q)$ for $T\in [0,T_q]$. Proof of proposition \[lem:add:propertyPTq\_stable\] {#appsec:lem:add:propertyPTq_stableNony} ---------------------------------------------------- To keep things simple, we prove assuming $\|\|{\mathfrak{Y}}^{\varepsilon}_0\|\|=0$ (which ensures that $\|\|{\mathfrak{Y}}^{\varepsilon}_t\|\|=0$ for all $t\geq 0$). Using $\int_0^t \|\|{\mathfrak{Y}}^{\varepsilon}_s\|\|^nds \leq {\varepsilon}^2(K/nk)\|\|{\mathfrak{Y}}^{\varepsilon}_0\|\|^n$ (because of exponential decay ), it is easy to see that the following ideas work even if we assume that $\|\|{\mathfrak{Y}}^{\varepsilon}_0\|\|\neq 0$. For simplicity of notation we write $G=G_1+G_3$ where $G_1$ is the linear part and $G_3$ is the cubic part. We have $$\begin{aligned} {\widehat{\mathfrak{z}}^{\varepsilon}}_t={\widehat{\mathfrak{z}}^{\varepsilon}}_0+\int_0^te^{-sB/{\varepsilon}^2}{\tilde{\Psi}}G(\Phi e^{sB/{\varepsilon}^2}{\widehat{\mathfrak{z}}^{\varepsilon}}_s)ds+{\mathfrak{w}}_t, \qquad {\mathfrak{w}}_t:=\int_0^te^{-sB/{\varepsilon}^2}{\tilde{\Psi}}\sigma dW_s.\end{aligned}$$ Writing ${\mathfrak{y}}_t={\widehat{\mathfrak{z}}^{\varepsilon}}_t-{\mathfrak{w}}_t$, we have $$\begin{aligned} \label{eq:howzpwevolves} \dot{{\mathfrak{y}}}_t=e^{-tB/{\varepsilon}^2}{\tilde{\Psi}}G(\Phi e^{tB/{\varepsilon}^2}({\mathfrak{y}}_t+{\mathfrak{w}}_t))\end{aligned}$$ from which we can write (using that the transpose of $e^{tB/{\varepsilon}^2}$ is $e^{-tB/{\varepsilon}^2}$) $$\begin{aligned} \frac12\frac{d}{dt}\|\|{\mathfrak{y}}_t\|\|_2^2\,&=\,(e^{tB/{\varepsilon}^2}{\mathfrak{y}}_t)^{\tilde{\Psi}}G(\Phi e^{tB/{\varepsilon}^2}({\mathfrak{y}}_t+{\mathfrak{w}}_t)) \\ &=\,\,(e^{tB/{\varepsilon}^2}{\mathfrak{y}}_t)^{\tilde{\Psi}}G(\Phi e^{tB/{\varepsilon}^2}{\mathfrak{y}}_t)\,\,+\,\,(e^{tB/{\varepsilon}^2}{\mathfrak{y}}_t)^{\tilde{\Psi}}\left(G(\Phi e^{tB/{\varepsilon}^2}({\mathfrak{y}}_t+{\mathfrak{w}}_t))-G(\Phi e^{tB/{\varepsilon}^2}{\mathfrak{y}}_t)\right).\end{aligned}$$ Using $G=G_1+G_3$, and the Lipschitz condition on the linear part $\|G_1(\eta_1)-G_1(\eta_2)\|\leq K_G\|\|\eta_1-\eta_2\|\|$, and that $\|\|\Phi e^{tB/{\varepsilon}^2}{\mathfrak{w}}_t\|\|\leq \|\|{\mathfrak{w}}_t\|\|_1$, and $\|(e^{tB/{\varepsilon}^2}{\mathfrak{y}}_t)^{\tilde{\Psi}}\|\leq \|\|{\tilde{\Psi}}\|\|_2\|\|{\mathfrak{y}}_t\|\|_2$, we have $$\begin{aligned} \label{eq:rateofevolnormzpw2} \frac12\frac{d}{dt}\|\|{\mathfrak{y}}_t\|\|_2^2\,\,\,&\leq\,\,\,K_G\|\|{\tilde{\Psi}}\|\|_2\|\|{\mathfrak{y}}_t\|\|_2^2+ (e^{tB/{\varepsilon}^2}{\mathfrak{y}}_t)^{\tilde{\Psi}}G_3(\Phi e^{tB/{\varepsilon}^2}{\mathfrak{y}}_t) \\ \notag & \qquad \qquad +\,\,\|\|{\tilde{\Psi}}\|\|_2\|\|{\mathfrak{y}}_t\|\|_2K_G\|\|{\mathfrak{w}}_t\|\|_1\,\,+\,\,\sum_{j=1}^3c_j\|\|{\mathfrak{y}}_t\|\|_2^{4-j}\|\|{\mathfrak{w}}_t\|\|_1^j,\end{aligned}$$ for some constants $c_j>0$. Define the time averaging operator ${\mathbb{T}}$ as follows: For a periodic function $f:{\mathbb{R}}\to {\mathbb{R}}$ with period $2\pi/{\omega}_c$, the action of ${\mathbb{T}}$ is given by ${\mathbb{T}}(f)=\frac{1}{2\pi/{\omega}_c}\int_0^{2\pi/{\omega}_c}f(s)ds$. Note that the condition means that $$\begin{aligned} \label{eq:TavgG3conditionMeaning} {\mathbb{T}}\left((e^{\cdot B}z)^{\tilde{\Psi}}G_3(\Phi e^{\cdot B}z)\right) \,<\,-C_G\|\|z\|\|_2^4.\end{aligned}$$ Define $$\begin{aligned} \label{eq:def:G3tildefeq} \tilde{G}_3(z,t):=(e^{t B}z)^{\tilde{\Psi}}G_3(\Phi e^{t B}z)-{\mathbb{T}}((e^{\cdot B}z)^{\tilde{\Psi}}G_3(\Phi e^{\cdot B}z)).\end{aligned}$$ Then, using and in we have $$\begin{aligned} \frac12\frac{d}{dt}\|\|{\mathfrak{y}}_t\|\|_2^2\,\,\,&\leq\,\,K_G\|\|{\tilde{\Psi}}\|\|_2\|\|{\mathfrak{y}}_t\|\|_2^2 \,\,-\,\,C_G\|\|{\mathfrak{y}}\|\|_2^4 \,\,+\,\, \tilde{G}_3({\mathfrak{y}}_t,t/{\varepsilon}^2) \\ & \qquad \qquad +\,\,\|\|{\tilde{\Psi}}\|\|_2\|\|{\mathfrak{y}}_t\|\|_2K_G\|\|{\mathfrak{w}}_t\|\|_1\,\,+\,\,\sum_{j=1}^3c_j\|\|{\mathfrak{y}}_t\|\|_2^{4-j}\|\|{\mathfrak{w}}_t\|\|_1^j.\end{aligned}$$ Using Young’s inequality we have for some $C_Y>0$ $$\begin{aligned} \label{eq:afterYoungEq} \frac12\frac{d}{dt}\|\|{\mathfrak{y}}\|\|_2^2\,\,&<\,\,-\frac12C_G\|\|{\mathfrak{y}}\|\|_2^4 \,\,+\,\,C_Y\|\|{\mathfrak{w}}_t\|\|_1^4\,\,+C_Y \,\,+\,\,\tilde{G}_3({\mathfrak{y}}_t,t/{\varepsilon}^2).\end{aligned}$$ Assume $$\begin{aligned} \label{eq:ass:supmfwLER} \sup_{t\in[0,T]}\|\|{\mathfrak{w}}_t\|\|_1^4<R,\end{aligned}$$ and let $\widetilde{C}=C_Y(1+R)$. Then $$\begin{aligned} \label{eq:lem:add:propertyPTq_stable_aux} \frac12\frac{d}{dt}\|\|{\mathfrak{y}}\|\|_2^2\,\,&<\,\,-\frac12C_G\|\|{\mathfrak{y}}\|\|_2^4 \,\,+\,\,\widetilde{C}\,\,+\,\,\tilde{G}_3({\mathfrak{y}}_t,t/{\varepsilon}^2).\end{aligned}$$ Using comparison principle (theorem 6.1 on page 31 of [@HaleODEbook]), we have that $\|\|{\mathfrak{y}}_t\|\|_2^2\leq {\mathfrak{v}}_t$ where ${\mathfrak{v}}$ is governed by $$\begin{aligned} \label{eq:lem:add:propertyPTq_stable_aux_vsup} \frac{d}{dt}{\mathfrak{v}}_t\,\,&=\,\,-C_G{\mathfrak{v}}_t+C_G({\mathfrak{v}}_t-{\mathfrak{v}}_t^2) \,\,+\,\,2\widetilde{C}\,\,+\,\,2\tilde{G}_3({\mathfrak{y}}_t,t/{\varepsilon}^2), \qquad {\mathfrak{v}}_0=\|\|{\mathfrak{y}}_0\|\|_2^2.\end{aligned}$$ Using variation-of-constants formula, the fact that ${\mathfrak{v}}-{\mathfrak{v}}^2<1$, and integration-by-parts, we find that $$\begin{aligned} \label{eq:lem:add:propertyPTq_stable_aux_vsup_upbound} {\mathfrak{v}}_t\,\,&<\,\,{\mathfrak{v}}_0e^{-C_Gt}+\frac{2\widetilde{C}+C_G}{C_G}(1-e^{-C_Gt})+2\int_0^t\tilde{G}_3({\mathfrak{y}}_s,s/{\varepsilon}^2)ds \\ & \qquad \qquad -C_G\int_0^te^{-C_G(t-s)}\left(2\int_0^s\tilde{G}_3({\mathfrak{y}}_u,u/{\varepsilon}^2)du\right)ds. \notag\end{aligned}$$ Now we try to obtain some bounds on the last two terms of the above inequality. Using the structure of ${G}_3$ (defined in ) and $\tilde{G}_3$ (defined in ) and that ${\mathbb{T}}(\tilde{G}_3(z,\cdot))=0$, it is easy to see that $\tilde{G}_3$ can be expressed as $$\tilde{G}_3({\mathfrak{y}},t)=\sum_{j=1}^4(\alpha_j\cos(j{\omega}_ct)+\beta_j\sin(j{\omega}_ct))$$ where $\alpha_j$ and $\beta_j$ are fourth order polynomials in the components of ${\mathfrak{y}}$. Define $${\mathfrak{g}}(z,t):=2\int_0^t\tilde{G}_3(z,s)ds.$$ Using the structure of $\tilde{G}_3$ it is easy to see that (note that $\tilde{G}_3$ is mean zero and periodic as a function of its second argument) there exists $C_{{\mathfrak{g}}}>0$ such that $$\|{\mathfrak{g}}({\mathfrak{y}},t)\|\leq C_{{\mathfrak{g}}}(1+\|\|{\mathfrak{y}}\|\|_2^4), \qquad \qquad \|\|\frac{\partial {\mathfrak{g}}}{\partial {\mathfrak{y}}}({\mathfrak{y}},t)\|\|_2\leq C_{{\mathfrak{g}}}(1+\|\|{\mathfrak{y}}\|\|_2^3).$$ Also, from , it is easy to see that $\exists C_>0$ such that $\|\|\dot{{\mathfrak{y}}}\|\|_2\leq C_(1+\|\|{\mathfrak{y}}+{\mathfrak{w}}\|\|_2^3)$. Since $$\begin{aligned} {\varepsilon}^2{\mathfrak{g}}({\mathfrak{y}}_t,t/{\varepsilon}^2)-{\varepsilon}^2{\mathfrak{g}}({\mathfrak{y}}_0,0)-{\varepsilon}^2\int_0^t\frac{\partial {\mathfrak{g}}}{\partial {\mathfrak{y}}}({\mathfrak{y}}_s,s/{\varepsilon}^2)\dot{{\mathfrak{y}}}_sds = 2\int_0^t\tilde{G}_3({\mathfrak{y}}_s,s/{\varepsilon}^2)ds,\end{aligned}$$ and ${\mathfrak{g}}({\mathfrak{y}},0)=0$, we have $$\begin{aligned} \left\|2\int_0^t\tilde{G}_3({\mathfrak{y}}_s,s/{\varepsilon}^2)ds\right\| \,\,&\leq\,\,{\varepsilon}^2C_{{\mathfrak{g}}}(1+\|\|{\mathfrak{y}}_t\|\|_2^4)+{\varepsilon}^2C_{{\mathfrak{g}}}C_\int_0^t(1+\|\|{\mathfrak{y}}_s\|\|_2^3)(1+\|\|{\mathfrak{y}}_s+{\mathfrak{w}}_s\|\|_2^3)ds \\ &\leq\,\,{\varepsilon}^2C_{{\mathfrak{g}}}(1+\|\|{\mathfrak{y}}_t\|\|_2^4)+{\varepsilon}^211C_{{\mathfrak{g}}}C_\int_0^t(1+\|\|{\mathfrak{y}}_s\|\|_2^6+\|\|{\mathfrak{w}}_s\|\|_1^6)ds.\end{aligned}$$ Let $$\begin{aligned} \label{eq:auxeqtauepsdef_lem:add:propertyPTq_stable} \tau^{\varepsilon}:= \inf \{t\geq 0\,:\,\|\|{\mathfrak{y}}_t\|\|_2 \geq \frac{1}{{\varepsilon}^{1/6}}\}.\end{aligned}$$ Then, for $t \leq \min\{\tau^{\varepsilon}, T\}$ we have $$\begin{aligned} \left\|2\int_0^t\tilde{G}_3({\mathfrak{y}}_s,s/{\varepsilon}^2)ds\right\| \,\,&\leq \,\,{\varepsilon}^2C_{{\mathfrak{g}}}+{\varepsilon}^{4/3}C_{{\mathfrak{g}}} + {\varepsilon}^211C_{{\mathfrak{g}}}C_\int_0^t(1+{\varepsilon}^{-1}+R^{3/2})ds. \end{aligned}$$ When ${\varepsilon}<1$, we have (from the above inequality) that for $t \leq \tau^{\varepsilon}\wedge T$ $$\begin{aligned} \left\|2\int_0^t\tilde{G}_3({\mathfrak{y}}_s,s/{\varepsilon}^2)ds\right\| \,\,&\leq\,\,{\varepsilon}^{4/3}2C_{{\mathfrak{g}}}\,\,+\,\,{\varepsilon}\widehat{C}t \label{eq:lem:add:propertyPTq_stable_aux_intg_aftbou_simp}\end{aligned}$$ where $\widehat{C}=22C_{{\mathfrak{g}}}C_(1+R^{3/2})$. Using in , we have[^6] for ${\varepsilon}<1$ and $t \leq \tau^{\varepsilon}\wedge T$ $$\begin{aligned} \label{eq:lem:add:propertyPTq_stable_aux_vsup_upbound_up} \|\|{\mathfrak{y}}_t\|\|_2^2\,\,\leq\,\,{\mathfrak{v}}_t\,\,&<\,\,{\mathfrak{v}}_0e^{-C_Gt}+\frac{2\widetilde{C}+C_G}{C_G}(1-e^{-C_Gt})+{\varepsilon}2\widehat{C}t+{\varepsilon}^{4/3}4C_{{\mathfrak{g}}} \\ \notag &<\,\,\max\{\|\|{\mathfrak{y}}_0\|\|_2^2,\,\frac{2\widetilde{C}+C_G}{C_G}\}+{\varepsilon}2(\widehat{C}T+2C_{{\mathfrak{g}}}).\end{aligned}$$ Hence, for ${\varepsilon}<1$ and $t \leq \tau^{\varepsilon}\wedge T$ $$\begin{aligned} \|\|{\mathfrak{y}}_t\|\|_2\,\,<\,\,\|\|{\mathfrak{y}}_0\|\|_2+1+\sqrt{\frac{2\widetilde{C}}{C_G}}+\sqrt{{\varepsilon}}\sqrt{2\widehat{C}T}+\sqrt{{\varepsilon}}\sqrt{4C_{{\mathfrak{g}}}}. \end{aligned}$$ Using $\widehat{C}=22C_{{\mathfrak{g}}}C_(1+R^{3/2})$ and that $\widetilde{C}=C_Y(1+R)$ we find that $$\begin{aligned} \|\|{\mathfrak{y}}_t\|\|_2\,\,&<\,\,\|\|{\mathfrak{y}}_0\|\|_2+(1+\sqrt{{\varepsilon}}\sqrt{4C_{{\mathfrak{g}}}})+\sqrt{\frac{2C_Y}{C_G}}\sqrt{1+R}+\sqrt{{\varepsilon}}\sqrt{44C_{{\mathfrak{g}}}C_T}(1+R^{3/4}). \end{aligned}$$ Note that $\exists \,{\varepsilon}_{(2)}$ such that $\forall {\varepsilon}< {\varepsilon}_{(2)}$ we have $\sqrt{{\varepsilon}}\sqrt{4C_{{\mathfrak{g}}}}<1$. Also, $\exists \,{\varepsilon}_{(3)}$ such that $\forall {\varepsilon}< {\varepsilon}_{(3)}$ we have $\sqrt{{\varepsilon}}\sqrt{\frac{44C_{{\mathfrak{g}}}C_T}{2C_Y/C_G}}<1$. Hence, for ${\varepsilon}<\min\{1,{\varepsilon}_{(2)},{\varepsilon}_{(3)}\}=:{\varepsilon}_{(4)}$ and $t \leq \tau^{\varepsilon}\wedge T$ we have[^7] $$\begin{aligned} \label{eq:lem:add:propertyPTq_stable_aux_zpw_upbound_upup} \|\|{\mathfrak{y}}_t\|\|_2\,\,<\,\,\|\|{\mathfrak{y}}_0\|\|_2+2+6\sqrt{\frac{2C_Y}{C_G}}\sqrt{1+R^{6/4}}.\end{aligned}$$ Hence, for ${\varepsilon}<{\varepsilon}_{(4)}$ if $\tau^{\varepsilon}\geq T$ we have (using $\|\|{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|_2\leq \|\|{\mathfrak{y}}_t\|\|_2+\|\|{\mathfrak{w}}_t\|\|_1$) $$\begin{aligned} \sup_{t\in[0,T]}\|\|{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|_2\,\,&<\,\, \|\|{\widehat{\mathfrak{z}}^{\varepsilon}}_0\|\|_2+2+6\sqrt{\frac{2C_Y}{C_G}}\sqrt{1+R^{6/4}} + R^{1/4}\\ &<\,\,\|\|{\widehat{\mathfrak{z}}^{\varepsilon}}_0\|\|_2+2+6\left(\frac{1}{6}+\sqrt{\frac{2C_Y}{C_G}}\right)\sqrt{1+R^{6/4}} \\ & =: \quad \|\|{\widehat{\mathfrak{z}}^{\varepsilon}}_0\|\|_2+2+C_{YG}\sqrt{1+R^{6/4}} \end{aligned}$$ Because the initial condition is deterministic $\exists C_0>0$ such that $\|\|{\widehat{\mathfrak{z}}^{\varepsilon}}_0\|\|_2<C_0$. Hence $$\begin{aligned} \label{eq:proof:aux:howmuchboundonz} \sup_{t\in[0,T]}\|\|{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|_2\,\,&<\,\, C_0 +2+C_{YG}\sqrt{1+R^{6/4}}. \end{aligned}$$ Define $C_R$ by $$C_R \overset{{\tt def}}{=} C_0 + 2+C_{YG}\sqrt{1+R^{6/4}}.$$ For ${\varepsilon}<{\varepsilon}_{(4)}$ and $t \leq \tau^{\varepsilon}\wedge T$, we have from that $\|\|{\mathfrak{y}}_t\|\|_2 < C_R$. So, if we define ${\varepsilon}_R:=(1/C_{R})^6$, then for ${\varepsilon}< \min\{ {\varepsilon}_{(4)},{\varepsilon}_R\}$ we have that $\|\|{\mathfrak{y}}_t\|\|_2 < \frac{1}{{\varepsilon}^{1/6}}$ and hence $\tau^{\varepsilon}>T$. Hence, from we have that for ${\varepsilon}< \min\{ {\varepsilon}_{(4)},{\varepsilon}_R\}$ $$\begin{aligned} \label{eq:proof:aux:howmuchboundonz_nice} \sup_{t\in[0,T]}\|\|{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|_2\,\,&<\,\, C_R. \end{aligned}$$ Recalling the definition of $R$ from we have for ${\varepsilon}< \min\{ {\varepsilon}_{(4)},{\varepsilon}_R\}$ $$\begin{aligned} \label{eq:proof:aux:auxauxnice} {\mathbb{P}}\left[\sup_{t\in[0,T]}\|\|{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|_2\,\geq\, C_R\right]\,\,&\leq\,\,{\mathbb{P}}\left[\sup_{s\in[0,T]}\|\|{\mathfrak{w}}_t\|\|_1\,\geq \, \left( \left( \frac{C_R-C_0-2}{C_{YG}} \right)^2-1 \right)^{1/6} \right].\end{aligned}$$ Lets estimate the RHS of the above equation. Using the definition of ${\mathfrak{w}}$ and then Markov and Burkholder-Davis-Gundy inequalities we have that $$\begin{aligned} {\mathbb{P}}\left[\sup_{s\in[0,T]}\|\|{\mathfrak{w}}_t\|\|_1\,\geq \, \rho \right] \,\,\,&\leq\,\,\,\sum_{j=1}^2{\mathbb{P}}\left[\sup_{s\in[0,T]}\left\|\int_0^t(e^{-sB/{\varepsilon}^2}{\tilde{\Psi}})_j \sigma dW_s\right\|\,\geq \, \frac12\rho \right] \\ &\leq\,\,\,\frac{2}{\rho}\sum_{j=1}^2{\mathbb{E}}\left[\sup_{s\in[0,T]}\left\|\int_0^t(e^{-sB/{\varepsilon}^2}{\tilde{\Psi}})_j \sigma dW_s\right\|\right] \\ &\leq\,\,\,\frac{2C_{bdg}}{\rho}\sum_{j=1}^2{\mathbb{E}}\sqrt{\int_0^t(e^{-sB/{\varepsilon}^2}{\tilde{\Psi}})_j^2 \sigma^2 ds} \quad \leq \quad \frac{2\sqrt{2}C_{bdg}}{\rho}\|\|{\tilde{\Psi}}\|\|_2\sigma \sqrt{T} \end{aligned}$$ Hence, from we have that for ${\varepsilon}< \min\{ {\varepsilon}_{(4)},{\varepsilon}_R\}$ $$\begin{aligned} \label{eq:proof:aux:auxauxnice2} {\mathbb{P}}\left[\sup_{t\in[0,T]}\|\|{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|_2\,\geq\, C_R\right]\quad &\leq\quad \frac{2\sqrt{2}C_{bdg}\|\|{\tilde{\Psi}}\|\|_2\sigma \sqrt{T}}{\left( \left( \frac{C_R-C_0-2}{C_{YG}} \right)^2-1 \right)^{1/6}} \quad \overset{{\tt def}}{=}: \quad f(C_R).\end{aligned}$$ Let ${C}_{R,q}>C_0+2+C_{YG}$ be such that $f(C_R)<q$ for $C_R>C_{R,q}$. Such a ${C}_{R,q}$ exists because $f$ is monotonically decreasing in $C_R$ (for $C_R>C_0+2+C_{YG}$). Choose $C_{{\mathfrak{e}}}>C_{R,q}/0.99$ and ${\varepsilon}_=\min\{{\varepsilon}_{(4)},(C_{R,q})^{-6}\}$. Let $E^{\varepsilon}$ be as defined in . Now, using for ${\varepsilon}< {\varepsilon}_$ $$\begin{aligned} {\mathbb{P}}[{\Omega}\setminus E^{\varepsilon}]\,\,& = {\mathbb{P}}\left[\sup_{t\in [0,T]}\|\|\Phi e^{tB/{\varepsilon}^2}{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|\,\geq\,0.99C_{{\mathfrak{e}}}\right]\\ &\leq {\mathbb{P}}\left[\sup_{t\in [0,T]}\|\|{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|_2\,\geq\,0.99C_{{\mathfrak{e}}}\right]\,\, \leq\,\,{\mathbb{P}}\left[\sup_{t\in[0,T]}\|\|{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|_2\geq C_{R,q}\right]\,\,\leq \,\,f(C_{R,q})\,\,<\,\,q.\end{aligned}$$ Hence ${\mathbb{P}}[E^{\varepsilon}]\,\,\geq\,\,1-q,$ and so possesses the property $\mathscr{P}(T,q)$. As mentioned above, for any $q>0$ it is possible to select ${\varepsilon}_$ and $C_{R,q}$ such that $f(C_{R,q})<q$ and hence ${\widehat{\mathfrak{z}}^{\varepsilon}}$ possesses the property $\mathscr{P}(T,q)$ for arbitrary $q>0$. Proof of proposition \[prop:betaepsSmallAddNoiseFin\] {#sec:proofofprop_betaepsSmallAddNoiseFin} ----------------------------------------------------- Using Ito formula, $\beta^{\varepsilon}_t$ satisfies $$d\beta^{\varepsilon}_t={\mathcal{B}}_tdt, \qquad \beta^{\varepsilon}_0=0,$$ where $${\mathcal{B}}_t=\Gamma_t\left(G(\Phi e^{tB/{\varepsilon}^2}{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t)-G(\Phi e^{tB/{\varepsilon}^2}{\widehat{\mathfrak{z}}^{\varepsilon}}_t+{\mathfrak{Y}}_t)\right), \qquad \Gamma_t=\sum_{i=1}^{2}\left(e^{-tB/{\varepsilon}^2}{\tilde{\Psi}}\right)_i\left({\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t\right)_i.$$ Using the structure of $e^{-tB}$ and ${\tilde{\Psi}}$, we have $\|\Gamma_t\|\leq \sqrt{{\tilde{\Psi}}^{\tilde{\Psi}}}\sqrt{2\beta^{\varepsilon}_t}$. Writing ${\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t$ as ${\widehat{\mathfrak{z}}^{\varepsilon}}_t+({\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}_t-{\widehat{\mathfrak{z}}^{\varepsilon}}_t)$ and expanding $G$ in ${\mathcal{B}}_t$ we get $$\begin{aligned} \|{\mathcal{B}}_t\|\,\leq\,C\sqrt{\beta^{\varepsilon}_t}\left(\sqrt{\beta^{\varepsilon}_t}+\|\|{\mathfrak{Y}}^{\varepsilon}_t\|\| + \sum_{j=1}^{3}\|\|\Phi e^{tB/{\varepsilon}^2}{\widehat{\mathfrak{z}}^{\varepsilon}}_t\|\|^{3-j}\left((\sqrt{\beta^{\varepsilon}_t})^j+\|\|{\mathfrak{Y}}^{\varepsilon}_t\|\|^j\right)\right)\end{aligned}$$ Because the nonlinearity is such that is satisfied, by lemma \[lem:add:propertyPTq\_stable\], ${\widehat{\mathfrak{z}}^{\varepsilon}}$ possesses property $\mathscr{P}(T,q)$ for abitrary $q>0$. Hence, it is possible to select $C_{{\mathfrak{e}}}>0$ so that $\exists\,{\varepsilon}_>0$ such that $\forall {\varepsilon}<{\varepsilon}_$, we have ${\mathbb{P}}[E^{\varepsilon}]\geq1-q$ where $E^{\varepsilon}$ is defined in . So, with probability at least $1-q$ we have $$\begin{aligned} \|{\mathcal{B}}_t\|\,\leq\,C\sqrt{\beta^{\varepsilon}_t}\left((1+C_{{\mathfrak{e}}}^2)\sqrt{\beta^{\varepsilon}_t} + C_{{\mathfrak{e}}}(\sqrt{\beta^{\varepsilon}_t})^2+(\sqrt{\beta^{\varepsilon}_t})^3 +(1+C_{{\mathfrak{e}}}^2)\sum_{j=1}^3\|\|{\mathfrak{Y}}^{\varepsilon}_t\|\|^j \right).\end{aligned}$$ Let $\mathscr{C}:=\|\|(1-\pi){{{\Pi}}^{{\varepsilon}}_{0}{X^{\varepsilon}}}\|\|$. As long as $\sqrt{\beta^{\varepsilon}_t}<1$ we have $$\begin{aligned} \|{\mathcal{B}}_t\|\,\leq\,C\sqrt{\beta^{\varepsilon}_t}\left((3+2C_{{\mathfrak{e}}}^2)\sqrt{\beta^{\varepsilon}_t}+(1+C_{{\mathfrak{e}}}^2)\left(\sum_{j=1}^3\mathscr{C}^j\right)e^{-kt/{\varepsilon}^2} \right).\end{aligned}$$ Hence, as long as $\sqrt{\beta^{\varepsilon}_t}<1$ $$\begin{aligned} d\sqrt{\beta^{\varepsilon}_t}\,\leq\,C(3+2C_{{\mathfrak{e}}}^2)\sqrt{\beta^{\varepsilon}_t}+C(1+C_{{\mathfrak{e}}}^2)\left(\sum_{j=1}^3\mathscr{C}^j\right)e^{-kt/{\varepsilon}^2}.\end{aligned}$$ Using Gronwall, we get (as long as $\sqrt{\beta^{\varepsilon}_t}<1$) $$\begin{aligned} \sqrt{\beta^{\varepsilon}_t} \,&\leq\, \frac{{\varepsilon}^2 C(1+C_{{\mathfrak{e}}}^2)(\sum_{j=1}^3 \mathscr{C}^j)}{k+{\varepsilon}^2C(3+2C_{{\mathfrak{e}}}^2)} \left(e^{C(3+2C_{{\mathfrak{e}}}^2)t}-e^{-kt/{\varepsilon}^2}\right) \\ & < \frac{{\varepsilon}^2 C(1+C_{{\mathfrak{e}}}^2)(\sum_{j=1}^3 \mathscr{C}^j)}{k} e^{C(3+2C_{{\mathfrak{e}}}^2)T}\end{aligned}$$ Choose ${\varepsilon}_{}$ small enough so that the above expression is less than one. Set ${\varepsilon}_{\circ}=\min\{{\varepsilon}_{},{\varepsilon}_{*}\}$. Then, we have $\forall {\varepsilon}<{\varepsilon}_{\circ}$, $\sup_{t\in [0,T]}\beta^{\varepsilon}_t < C{\varepsilon}^4$ with probability at least $1-q$. $\hfill \qed$ [^1]: Multiply by $z(0)$ and realize (using the fact $T$ commutes with $\pi$) that $T(t)\Phi(\cdot)z(0)=T(t)\pi {\Pi}_0x=\pi T(t){\Pi}_0x=\pi {\Pi}_tx=\Phi z(t)$ to get that $\Phi z(t)=\Phi e^{Bt}z(0)$ from which $\dot{z}=Bz$ follows. [^2]: Writing ${{{\Pi}}^{{\varepsilon}}_{t}{X^{\varepsilon}}}=\Phi z(t)+y_t$ and showing $y$ is small, we can write ${X^{\varepsilon}}(t)={{{\Pi}}^{{\varepsilon}}_{t}{X^{\varepsilon}}}(0)\approx \Phi(0)z_t = (z_t)_1$. Since the dynamics of $z$ is small perturbation of a predominant oscillation according $\dot{z}=Bz$, the approximate amplitude of $(z)_1$ is $\sqrt{(z)_1^2+(z)_2^2}=\|\|z\|\|_2=\sqrt{2{\mathcal{H}}}$. [^3]: If $\frac12\|\|{\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}\|\|_2^2={\mathcal{H}}^0$ then $$\lim_{\mathfrak{T}\to 0}\frac{1}{\mathfrak{T}}\int_0^{\mathfrak{T}}\frac12\|\|e^{-tB/{\varepsilon}^2}M e^{tB/{\varepsilon}^2} {\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}\|\|_2^2 \,dt\,=\, C_2{\mathcal{H}}^0 \quad \text{ and }\quad \lim_{\mathfrak{T}\to 0}\frac{1}{\mathfrak{T}}\int_0^{\mathfrak{T}}\left(({\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}})^e^{-tB/{\varepsilon}^2}M e^{tB/{\varepsilon}^2} {\pmb{\widetilde{\mathfrak{z}}}^{\varepsilon}}\right)^2dt \,=\, (C_1{\mathcal{H}}^0)^2.$$ [^4]: [@LNPRLE] employs complex coordinates and so the form of answers would differ from this paper. However the numerical values would be same. For example, in this paper we write an element in $P$ as $z_1\cos({\omega}_c\cdot)+z_2\sin({\omega}_c\cdot)$ with $z_i \in {\mathbb{R}}$. But in [@LNPRLE] we write $z_1e^{i{\omega}_c\cdot}+z_2e^{-i{\omega}_c\cdot}$ with $z_1$ and $z_2$ being complex conjugates. For multidimensional systems as treated in [@LNPRLE] this complex coordinates is more convenient. [^5]: Note that $\|\|\Phi e^{tB/{\varepsilon}^2}v\|\|\,=\,\sup_{\theta\in [-r,0]}\|v_1\cos(\theta+ {\omega}_ct/{\varepsilon}^2)+v_2\sin(\theta + {\omega}_ct/{\varepsilon}^2)\| \,\leq\,\|v_1\|+\|v_2\|.$ [^6]: For the last term in RHS of we have used that $$\|C_G\int_0^te^{-C_G(t-s)}f(s)ds\| \quad \leq \quad (\sup_{s\in [0,t]}\|f(s)\|)\,C_G\int_0^te^{-C_G(t-s)}ds\quad \leq\quad (\sup_{s\in [0,t]}\|f(s)\|).$$ [^7]: We use $\sqrt{1+R}+(1+R^{3/4})\,\,\, <\,\,\, 6\sqrt{1+R^{6/4}}$.
--- abstract: 'A partial matrix over a field $\mathbb{F}$ is a matrix whose entries are either an element of $\mathbb{F}$ or an indeterminate and with each indeterminate only appearing once. A completion is an assignment of values in $\mathbb{F}$ to all indeterminates. Given a partial matrix, through elementary row operations and column permutation it can be decomposed into a block matrix of the form $\left[\begin{smallmatrix}{\bf W} & * & * \\ 0 & {\bf S} & * \\ 0 & 0 & {\bf T} \end{smallmatrix}\right]$ where ${\bf W}$ is wide (has more columns than rows), ${\bf S}$ is square, ${\bf T}$ is tall (has more rows than columns), and these three blocks have at least one completion with full rank. And importantly, each one of the blocks ${\bf W}$, ${\bf S}$ and ${\bf T}$ is unique up to elementary row operations and column permutation whenever ${\bf S}$ is required to be as large as possible. When this is the case $\left[\begin{smallmatrix}{\bf W} & * & * \\ 0 & {\bf S} & * \\ 0 & 0 & {\bf T} \end{smallmatrix}\right]$ will be called a WST-decomposition. With this decomposition it is trivial to compute maximum rank of a completion of the original partial matrix: $\operatorname{rows}({\bf W})+\operatorname{rows}({\bf S})+\operatorname{cols}({\bf T})$. In fact we introduce the WST-decomposition for a broader class of matrices: the ACI-matrices.' author: - \| Alberto Borobia, Roberto Canogar\ Dpto. Matemáticas, Universidad Nacional de Educación a Distancia (UNED), 28040 Madrid, Spain\ e-mail: $aborobia@mat.uned.es$, $rcanogar@mat.uned.es$ title: 'The WST-decomposition for partial matrices [^1] [^2] [^3]' --- Introduction ============ Preliminaries ------------- The ACI-matrices were introduced in 2010 by Brualdi, Huang and Zhan [@MR2680270] as a generalization of [partial matrices]{} (matrices whose entries are either a constant or an indeterminate and with each indeterminate only appearing once). Let $\mathbb{F}[x_1, \ldots , x_k]$ denote the set of polynomials in the indeterminates $x_1, \ldots , x_k$ with coefficients on a field $\mathbb{F}$. A matrix over $\mathbb{F}[x_1, \ldots, x_k]$ is an Affine Column Independent matrix or [ACI-matrix]{} if its entries are polynomials of degree at most one and no indeterminate appears in two different columns. A completion of an ACI-matrix $A$ is an assignment of values in $\mathbb{F}$ to all indeterminates so that it gives a constant matrix in $\mathbb{F}$. All definitions and most of the results work for any field $\mathbb{F}$, so we will usually omit in what field we are working on. The [Rank]{} of an ACI-matrix $M$ is the set of all possible ranks of completions of $M$. The [maxRank]{} of $M$ is the maximum rank for a completion of $M$. The [minRank]{} of $M$ is the minimum rank for a completion of $M$. We say that $M$ is [constantRank]{} if $\operatorname{maxRank}(M)=\operatorname{minRank}(M)$. The Rank of partial matrices has a substantial literature (see Section 1 of [@MR2680270]). The constantRank partial matrices were studied in [@McTigueQuinlan3], and the constantRank ACI-matrices were studied in [@BoCa1; @BoCa2; @BoCa3; @MR2680270; @MR2775784]. Multiplying an ACI-matrix by a constant square matrix on the left produces an ACI-matrix of the same size. If, in addition, the constant matrix is nonsingular then the new ACI-matrix will share the same Rank, minRank and maxRank with the old one. The same happens if we permute the columns of an ACI-matrix. Since we are concerned with the Rank of ACI-matrices, the following definition makes sense. Let $M$ be an $m\times n$ ACI-matrix. For any nonsingular constant matrix $R$ of order $m$ and for any permutation matrix $Q$ of order $n$, the ACI-matrix $RMQ$ is said to be [equivalent]{} to $M$. We represent this equivalence by $M \sim RMQ$. In this work we are interested in the study of the Rank of a given ACI-matrix $M$. In order to do it we will consider the equivalence class of $M$ so that we can find a representative in the class with an easier structure that, for example, reveal directly its maxRank. This easier structure will be the WST-decomposition of $M$ as we will see in Section \[WST\]. It is important to point out that this definition of equivalence can not be applied to partial matrices since $RMQ$ will not be necessarily a partial matrix, it will be an ACI-matrix. Basic definitions ----------------- The relation between the number of rows and the number of columns of ACI-matrices will play an important role, that is why we introduce the following terminology: \[\] Let $M$ be an ACI-matrix. - [rows($M$)]{} denotes the number of rows of $M$. - [cols($M$)]{} denotes the number of columns of $M$. - $M$ is wide if $\operatorname{cols}(M)> \operatorname{rows}(M)$. - $M$ is tall if $\operatorname{rows}(M)> \operatorname{cols}(M)$. - $M$ is square if $\operatorname{rows}(M)= \operatorname{cols}(M)$. For technical reasons we will consider as ACI-matrices the ones without rows or/and without columns, namely: (i) the [wide degenerate]{} ACI-matrix $0\times q$ with $q>0$; (ii) the [tall degenerate]{} ACI-matrix $p\times 0$ with $p>0$; and (iii) the square degenerate or void ACI-matrix $0\times 0$. A constant matrix $M$ is full row rank if $\operatorname{rank}(M)=\operatorname{rows}(M)$, is full column rank if $\operatorname{rank}(M)=\operatorname{cols}(M)$, and is full rank if $\operatorname{rank}(M)=\min\{\operatorname{rows}(M),\operatorname{cols}(M)\}$. We will adapt this common terminology to the maxRank of ACI-matrices. \[mfr\] The ACI-matrix $M$ is - [Full Row maxRank]{} or [FRmR]{} if $\operatorname{maxRank}(M)=\operatorname{rows}(M)$. - [Full Column maxRank]{} or [FCmR]{} if $\operatorname{maxRank}(M)=\operatorname{cols}(M)$. - [Full maxRank]{} or [FmR]{} if $\operatorname{maxRank}(M)=\min \{\operatorname{rows}(M), \operatorname{cols}(M)\}$ or, equivalently, if $M$ has a completion with full rank. Just to emphasize: $(i)$ FRmR is wide or square FmR; $(ii)$ FCmR is tall or square FmR; $(iii)$ FmR is FRmR or FCmR or both. Again, for technical reasons we will consider a tall degenerate to be FRmR, a wide degenerate to be FCmR, and the void to be FRmR and FCmR. The next proposition shows how to build new FmR ACI-matrices from known FmR ACI-matrices. Its proof is straightforward. \[FRmR-FCmR\] Let $M$ be an ACI-matrix. 1. If $M$ is FRmR and $M \sim M'$ then $M'$ is FRmR. 2. If $M$ is FCmR and $M \sim M'$ then $M'$ is FCmR. 3. \[dmp7\] If $M=\left[\begin{smallmatrix} A & B \\ 0 & C \end{smallmatrix}\right]$ where $A$ and $C$ are FRmR then $M$ is FRmR. 4. If $M=\left[\begin{smallmatrix} A & B \\ 0 & C \end{smallmatrix}\right]$ where $A$ and $C$ are FCmR then $M$ is FCmR. 5. If $M=\left[\begin{smallmatrix} A & B \\ 0 & C \end{smallmatrix}\right]$ where $A$ and $C$ are square FmR then $M$ is square FmR. The main Theorem ---------------- The first important result in ACI-matrices appeared in the work where they were introduced. (see [@MR2680270 Theorem 3]) \[SubmatrizCerosFirst\] Let $M$ be an $m \times n$ ACI-matrix and let $\rho$ be an integer such that $0\leq \rho < \min \{m, n\}$. The following two statements are equivalent: (i) $\operatorname{maxRank}(M)\leq \rho$. (ii) For some positive integers $r$ and $s$ with $\rho=(m-r)+(n-s)$ there exist a nonsingular constant matrix $R$ and a permutation matrix $Q$ such that $RMQ= \left[\begin{smallmatrix}A & B \\ 0 & C \end{smallmatrix}\right]$ where $0$ is an $r\times s$ submatrix with all its entries equal to zero. It is important to note that the values of $r$ and $s$ in Theorem \[SubmatrizCerosFirst\] are not unique in general, and neither are $A$ and $C$ (see the example below). The inspiration of the present work has been to generalize Theorem \[SubmatrizCerosFirst\] to find an analogous decomposition which is unique in some sense. In our main theorem, Theorem \[existance&uniqueness\], we will show that any ACI-matrix is equivalent to an ACI-matrix $$\left[\begin{array}{ccc} {\bf W} & * & * \\ 0 & {\bf S} & * \\ 0 & 0 & {\bf T} \\ \end{array}\right]$$ where ${\bf W}$ is wide FRmR or void, ${\bf S}$ is square FmR or void and ${\bf T}$ is tall FCmR or void. And importantly, each one of the ACI-matrices ${\bf W}$, ${\bf S}$ and ${\bf T}$ are unique up to equivalence whenever ${\bf S}$ is required to be as large as possible. When this is the case the ACI-matrix $\left[\begin{smallmatrix}{\bf W} & * & * \\ 0 & {\bf S} & * \\ 0 & 0 & {\bf T} \end{smallmatrix}\right]$ is called a [WST-decomposition]{}. This decomposition even works for FmR matrices (note that Theorem \[SubmatrizCerosFirst\] did not), but then at least one of the blocks ${\bf W}$ or ${\bf T}$ become void or degenerate ACI-submatrices. The WST-decomposition will allow us to restrict the study of some properties of ACI-matrices (and for that matter partial matrices) to the case of FmR ACI-matrices. For instance, in this work we will be focused on the maxRank and if we know a WST-decomposition of an ACI-matrix it will be trivial to compute its maxRank: $\operatorname{rows}({\bf W})+\operatorname{rows}({\bf S})+\operatorname{cols}({\bf T})$. So we might ask how to find the WST-decomposition in practice. A work that explains an algorithm that computes efficiently the WST-decomposition is in preparation. The maxRank for partial matrices was treated in [@CJRW] where the authors provide an procedure to compute it. Our algorithm will permit us to compute the maxRank for the broader class of ACI-matrices. [Example:]{} Below we present a $5\times 5$ ACI-matrix (it is actually a partial matrix) with maxRank 4, and with three different block partitions that verify the condition (ii) of Theorem \[SubmatrizCerosFirst\]: [$$\begin{aligned} \label{example4x4} M=\left[\begin{array}{cc\|ccc} 1 & x_1 & y_1 & z_1 & 1\\ \hline 0 & 0 & y_2 & z_2 & t_1 \\ 0 & 0 & 0 & z_3 & t_2\\ 0 & 0 & 0 & 0 & t_3\\ 0 & 0 & 0 & 0 & 1 \end{array}\right] =\left[\begin{array}{ccc\|cc} 1 & x_1 & y_1 & z_1 & 1\\ 0 & 0 & y_2 & z_2 & t_1 \\ \hline 0 & 0 & 0 & z_3 & t_2\\ 0 & 0 & 0 & 0 & t_3\\ 0 & 0 & 0 & 0 & 1 \end{array}\right] = \left[\begin{array}{cccc\|c} 1 & x_1 & y_1 & z_1 & 1\\ 0 & 0 & y_2 & z_2 & t_1 \\ 0 & 0 & 0 & z_3 & t_2\\ \hline 0 & 0 & 0 & 0 & t_3\\ 0 & 0 & 0 & 0 & 1 \end{array}\right].\end{aligned}$$]{} For $M$ a valid WST-decomposition is: $$\begin{aligned} {\small\left[\begin{array}{cc\|cc\|c} 1 & x_1 & * & * & \\ \hline 0 & 0 & y_2 & z_2 & \\ 0 & 0 & 0 & z_3 & \\ \hline 0 & 0 & 0 & 0 & t_3\\ 0 & 0 & 0 & 0 & 1 \end{array}\right]} \text{ where } {\bf W}= \begin{bmatrix}1 & x_1\end{bmatrix},\ {\bf S}= \begin{bmatrix} y_2 & z_2 \\ 0 & z_3\end{bmatrix},\ {\bf T}=\begin{bmatrix}t_3 \\ 1\end{bmatrix}.\end{aligned}$$ The property of ${\bf S}$ being as big as possible is required for the uniqueness of ${\bf W}$, ${\bf S}$ and ${\bf T}$ up to equivalence. Because decompositions like the following meet all the other requirements [$$\begin{aligned} \left[\begin{array}{cc\|c\|cc} 1 & x_1 & & * & \\ \hline 0 & 0 & y_2 & & * \\ \hline 0 & 0 & 0 & z_3 & t_2\\ 0 & 0 & 0 & 0 & t_3\\ 0 & 0 & 0 & 0 & 1 \end{array}\right].\end{aligned}$$]{} Zero blocks =========== Given an ACI-matrix, we will be interested in finding equivalent ACI-matrices which have a lot of zeros. A submatrix with all its entries equal to 0 will be referred as a [zero block]{}. A measure associated to the size of a zero block that we will frequently use is its number of rows plus its number of columns. \[defBigZero\] Let $\left[\begin{smallmatrix} A & B \\ 0 & C \end{smallmatrix}\right]$ be an $m\times n$ ACI-matrix where the zero block $0$ is of size $r\times s$. - The zero block in $\left[\begin{smallmatrix} A & B \\ 0 & C \end{smallmatrix}\right]$ is Big when $r+s>\max\{ m, n\}$. - The zero block in $\left[\begin{smallmatrix} A & B \\ 0 & C \end{smallmatrix}\right]$ is Medium when $r+s=\max\{ m, n\}$. Note that a Medium zero block measures one less than the smallest Big zero block. Again, for technical reasons we include the possibility for a Medium zero block to be degenerate. In our next result we provide equivalent and more intuitive definitions for Big and for Medium zero blocks. \[corBig\] For an ACI-matrix $M=\left[\begin{smallmatrix} A & B \\ 0 & C \end{smallmatrix}\right]$ with a zero block the following properties are satisfied: (i) The zero block is Big if and only if $A$ is wide and $C$ is tall. (ii) The zero block is Medium if and only if either (1) $M$ is tall, $A$ is square and $C$ is tall; (2) $M$ is square, $A$ and $C$ are square; or (3) $M$ is wide, $A$ is wide and $C$ is square. Let $M=\left[\begin{smallmatrix} A & B \\ 0 & C \end{smallmatrix}\right]$ be an $m\times n$ ACI-matrix with a $r\times s$ zero block, that is, $$\begin{aligned} \label{MDecomp} M =\left[\begin{array}{cc} A & B \\ \smash{\underset{s}{\underbrace{0}}} & \smash{\underset{n-s}{\underbrace{C}}} \end{array}\right] \hspace{-2mm}\begin{array}{ll} \} \ m-r \\ \} \ r \end{array} \end{aligned}$$ (i) The zero block of $M$ is Big if and only if $$\begin{aligned} r+s>\max\{ m, n \} \Leftrightarrow \begin{cases} s>m -r \\ r>n -s \end{cases} \Leftrightarrow \begin{cases} \operatorname{cols}(A)>\operatorname{rows}(A)\\ \operatorname{rows}(C)>\operatorname{cols}(C) \end{cases} \Leftrightarrow \begin{cases} A \text{ is wide}\\ C \text{ is tall} \end{cases}\end{aligned}$$ (ii) The zero block of $M$ is Medium if and only if $$\begin{aligned} r+s=\max\{ m, n \} \Leftrightarrow \begin{cases} \text{if } M \text{ tall }\;\;\;\; \begin{cases} s=m -r \\ r> n -s \end{cases} \Leftrightarrow \begin{cases} \operatorname{cols}(A)=\operatorname{rows}(A)\\ \operatorname{rows}(C)>\operatorname{cols}(C) \end{cases} \Leftrightarrow \begin{cases} A \text{ square}\\ C \text{ tall} \end{cases} \vspace{3mm} \\ \text{if } M \text{ square} \begin{cases} s=m -r \\ r= n -s \end{cases} \Leftrightarrow \begin{cases} \operatorname{cols}(A)=\operatorname{rows}(A)\\ \operatorname{rows}(C)=\operatorname{cols}(C) \end{cases} \Leftrightarrow \begin{cases} A \text{ square}\\ C \text{ square} \end{cases} \vspace{3mm} \\ \text{if } M \text{ wide }\;\; \begin{cases} s> m -r \\ r=n -s \end{cases} \Leftrightarrow \begin{cases} \operatorname{cols}(A)> \operatorname{rows}(A)\\ \operatorname{rows}(C)=\operatorname{cols}(C) \end{cases} \Leftrightarrow \begin{cases} A \text{ wide}\\ C \text{ square} \end{cases} \end{cases}.\end{aligned}$$ Let us see a consequence when an ACI-matrix has a Big zero block. \[Big->nofFullMaxRank\] An ACI-matrix with a Big zero block is not FmR. Let $M=\left[\begin{smallmatrix} A & B \\ 0 & C \end{smallmatrix}\right]$ be as in (\[MDecomp\]). If the zero block is Big then $$\operatorname{maxRank}\left[\begin{smallmatrix} A & B \\ 0 & C \end{smallmatrix}\right] \leq \operatorname{rows}(A) + \operatorname{cols}(C) = (m-r)+(n-s)=(m+n)-(r+s)<\min\{m,n\}$$ and therefore $M$ is not FmR. If a Big or Medium zero block is present in an ACI-matrix it makes it trivial to compute the maxRank when the diagonal blocks are FmR. \[Mrank+Mrank=Mrank\] If the ACI-matrix $\left[\begin{smallmatrix} A & B \\ 0 & C \end{smallmatrix}\right]$ has a Big or Medium zero block then the following conditions are equivalent: (i) $A$ is FRmR and $C$ is FCmR. (ii) $\operatorname{maxRank}\left[\begin{smallmatrix} A & B \\ 0 & C \end{smallmatrix}\right]=\operatorname{rows}(A)+\operatorname{cols}(C)$. Since 0 is a Big or Medium zero block of $\left[\begin{smallmatrix} A & B \\ 0 & C \end{smallmatrix}\right]$ then $A$ is wide or square and $C$ is tall or square. $(i) \Rightarrow (ii)$ : Let $\left[\begin{smallmatrix} \widehat{A} & \widehat{B} \\ 0 & \widehat{C}\end{smallmatrix}\right]$ be a completion of $\left[\begin{smallmatrix} A & B \\ 0 & C \end{smallmatrix}\right]$ such that $\widehat{A}$ and $\widehat{C}$ are full rank. Then we have $$\begin{aligned} \operatorname{maxRank}\left[\begin{smallmatrix} A & B \\ 0 & C \end{smallmatrix}\right] \leq \operatorname{rows}(A) + \operatorname{cols}(C) = \operatorname{rank}(\widehat{A})+\operatorname{rank}(\widehat{C}) \leq \operatorname{rank}\left[\begin{smallmatrix} \widehat{A} & \widehat{B} \\ 0 & \widehat{C} \end{smallmatrix}\right] \leq \operatorname{maxRank}\left[\begin{smallmatrix} A & B \\ 0 & C \end{smallmatrix}\right]\end{aligned}$$ and the result follows. $(ii) \Rightarrow (i)$ : We divide the proof in two parts: (a) Note that $\operatorname{maxRank}\left[\begin{smallmatrix}B\\ C \end{smallmatrix}\right] \leq \operatorname{cols}(C)$ and $\operatorname{maxRank}(A)\leq \operatorname{rows}(A)$, so $$\operatorname{maxRank}\left[\begin{smallmatrix} A & B \\ 0 & C \end{smallmatrix}\right] \leq \operatorname{maxRank}\left[\begin{smallmatrix} A \\ 0 \end{smallmatrix}\right]+ \operatorname{maxRank}\left[\begin{smallmatrix}B\\ C \end{smallmatrix}\right] \leq \operatorname{rows}(A) + \operatorname{cols}(C).$$ Since $ \operatorname{maxRank}\left[\begin{smallmatrix} A & B \\ 0 & C \end{smallmatrix}\right] = \operatorname{rows}(A) + \operatorname{cols}(C) $ then $\operatorname{maxRank}(A)=\operatorname{rows}(A)$. So $A$ is FRmR. (b) Note that $\operatorname{maxRank}\left[\begin{smallmatrix} A & B \end{smallmatrix}\right] \leq \operatorname{rows}(A)$ and $\operatorname{maxRank}(C) \leq \operatorname{cols}(C)$, so $$\operatorname{maxRank}\left[\begin{smallmatrix} A & B \\ 0 & C \end{smallmatrix}\right] \leq \operatorname{maxRank}\left[\begin{smallmatrix} A & B \end{smallmatrix}\right]+ \operatorname{maxRank}\left[\begin{smallmatrix}0 & C \end{smallmatrix}\right] \leq \operatorname{rows}(A) + \operatorname{cols}(C).$$ Since $ \operatorname{maxRank}\left[\begin{smallmatrix} A & B \\ 0 & C \end{smallmatrix}\right] = \operatorname{rows}(A) + \operatorname{cols}(C) $ then $\operatorname{maxRank}(C)=\operatorname{cols}(C)$. So $C$ is FCmR. Factor and semifactor sets ========================== Frequently, we will need to permute the columns of an ACI-matrix so that a certain set $F$ of columns appear as the first $\#F$ columns. This will be achieved by right multiplying the ACI-matrix by the appropriate permutation matrix. Let $F=\{f_1,\ldots, f_s\}\subset \{1,\ldots, n\}$ and let $\overline{F}=\{1,\ldots, n\}-\{f_1,\ldots, f_s\}=\{g_1, \ldots,g_{n-s}\}$. We define the permutation $\sigma_F$ of $\{1,\ldots,n\}$ by $$\begin{cases} \sigma_F(f_i) =i & \text{ for all } i \in \{1,\ldots,s\} \\ \sigma_F(g_j)=s+j & \text{ for all } j \in \{1,\ldots,n-s\} \end{cases} .$$ Note that $ \sigma_F(F)= \{1,\ldots, s\}$ and $\sigma_F(\overline{F})= \{s+1,\ldots, n\} $. Finally, ${Q_{F}}$ denotes the permutation matrix of order $n$ such that for each $k=1,\ldots,n$ the $k-$th column of any $m \times n$ ACI-matrix $M$ is equal to the $\sigma_F(k)-$th column of $M Q_F$. We now introduce two concepts associated to ACI-matrices: factor and semifactor sets. It will be crucial in this work to determine when an ACI-matrix has factor sets or has semifactor sets, and also to determine the relation between all of its factor sets or between all of its semifactor sets. Let $M$ be an $m\times n$ ACI-matrix. The set $F\subseteq \{1, \ldots, n\}$ is a [factor set]{} of $M$ if there exists a nonsingular matrix $R$ of order $m$ such that $$\label{defDiv27} R M Q_{F}=\left[\begin{array}{cc} \smash{\overset{\sigma_F(F)}{\overbrace{A}}} & B\\ 0 & C \end{array}\right]$$ where the zero block is Big, $A$ is (wide) FRmR and $C$ is (tall) FCmR. We will say that $RMQ_F$ is an $F$-decomposition of $M$. Note that in (\[defDiv27\]) $A$ is wide and $C$ is tall since the zero block is Big. For completeness we put in Table \[tabla1\] all cases that are possible in (\[defDiv27\]) for $R M Q_{F}$ taking into account when degenerate ACI-submatrices appear. $A$ wide non-degenerate and FRmR $A$ wide degenerate ---------------------------------- ---------------------------------------------------------------------------------- ------------------------------------------------------------------------------ $C$ tall non-degenerate and FCmR $\left[\begin{smallmatrix} A & B \\ 0 & C \end{smallmatrix}\right]$ $\left[\begin{smallmatrix} 0 & C \end{smallmatrix}\right]$ $C$ tall degenerate $\left[\begin{smallmatrix} A \\0 \end{smallmatrix}\right]$$F=\{1, \ldots, n\}$ $\left[\begin{smallmatrix} 0 \end{smallmatrix}\right]$ $F=\{1, \ldots, n\}$ : \[tabla1\]$\protect R M Q_{F}$ for factor sets. Now we introduce the concept of semifactor sets, where the Medium zero blocks take the same role as the Big zero blocks for factor sets. \[semifactorSetDefinition\] Let $M$ be an $m\times n$ ACI-matrix. The set $F\subseteq \{1, \ldots, n\}$ is a [semifactor set]{} of $M$ if there exists a nonsingular $R$ of order $m$ such that $$\label{defDiv28} R M Q_{F}=\left[\begin{array}{cc} \smash{\overset{\sigma_F(F)}{\overbrace{A}}} & B\\ 0 & C \end{array}\right]$$ where the zero block is Medium, $A$ is (wide or square) FRmR and $C$ is (tall or square) FCmR. Then $RMQ_F$ is called an $F$-semidecomposition of $M$. Note that in (\[defDiv28\]) $A$ or/and $C$ are square since the zero block is Medium. For completeness we put in Table \[tabla2\] all cases that are possible in (\[defDiv28\]) for $R M Q_{F}$ taking into account when degenerate ACI-submatrices appear. $A$ wide/square non-degenerate and FRmR $A$ wide degenerate $A$ void ----------------------------------------- ----------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------- $C$ tall/square non-degenerate and FCmR Case 1: $\left[\begin{smallmatrix} A & B \\ 0 & C \end{smallmatrix}\right]$ Case 2: $\left[\begin{smallmatrix} 0 & C \end{smallmatrix}\right]$ $C$ square since 0 is Medium Case 3: $\left[\begin{smallmatrix} C \end{smallmatrix}\right]$ Tall degenerate zero block $F=\emptyset$ $C$ tall degenerate Case 4: $\left[\begin{smallmatrix} A \\0 \end{smallmatrix}\right]$$A$ square since 0 is Medium $F=\{1, \ldots, n\}$ $\left[\begin{smallmatrix} 0 \end{smallmatrix}\right]$ NOT possible since 0 is Big Degenerate $C$ void Case 5: $\left[\begin{smallmatrix} A \end{smallmatrix}\right]$ Wide degenerate zero block $F=\{1, \ldots, n\}$ Degenerate Degenerate : \[tabla2\]$\protect R M Q_{F}$ for semifactor sets. Table \[tabla2\] shows that FmR ACI-matrices have at least one semifactor set. Note that if $M$ is FmR then $M$ is wide FRmR or square FmR or tall FCmR. Now if $M$ is wide/square FRmR then $\{1, \ldots, n\}$ is a semifactor set since we can always take $M=A$ with $C$ void and the Medium zero block being wide degenerate (Case 5 in Table \[tabla2\]); and if $M$ is tall/square FCmR then $\emptyset$ is a semifactor set since we can always take $M= C$ with $A$ void and the Medium zero block being tall degenerate (Case 3 in Table \[tabla2\]). In the next result we characterize when an ACI-matrix has a factor or a semifactor set. As we will see, the part corresponding to factor sets is quite related with Theorem \[SubmatrizCerosFirst\]. \[maxRank<->noFactorSet\] The following assertions about factor and semifactor sets are true: 1. An ACI-matrix has a factor set if and only if it is not FmR. 2. An ACI-matrix has a semifactor set if and only if it is FmR. Let $M$ be an $m\times n$ ACI-matrix. $(i) \Rightarrow$ : If $M$ has a factor set $F$ then there exists a nonsingular $R$ such that $$R M Q_{F}=\left[\begin{array}{cc} \smash{\overset{\sigma_F(F)}{\overbrace{A}}} & B\\ 0 & C \end{array}\right]$$ where the zero block is Big. So $R M Q_{F}$ is not FmR (Proposition \[Big->nofFullMaxRank\]) and thus $M$ is not FmR. $\quad \Leftarrow$ : If $M$ is not FmR then $\operatorname{maxRank}(M)<\min\{m, n\}$ and, by Theorem \[SubmatrizCerosFirst\], for some positive integers $r$ and $s$ with $\operatorname{maxRank}(M)=(m-r)+(n-s)$ there exist a nonsingular $R$ and a permutation $Q$ such that $RMQ=\left[\begin{smallmatrix} A & B \\ 0 & C \end{smallmatrix}\right]$ where 0 is an $r\times s$ zero block. Note that $$\min\{m,n\} > \operatorname{maxRank}(M)=(m-r)+(n-s) \ \Rightarrow \ r+s > m+n-\min\{m,n\} = \max\{m,n\}$$ so the zero block of $\left[\begin{smallmatrix} A & B \\ 0 & C \end{smallmatrix}\right]$ is Big. On the other hand $$\begin{aligned} \operatorname{maxRank}\begin{bmatrix} A & B \\ 0 & C \end{bmatrix}=\operatorname{maxRank}(M)= (m-r)+(n-s) = \operatorname{rows}(A)+\operatorname{cols}(C)\end{aligned}$$ which implies (see Theorem \[Mrank+Mrank=Mrank\]) that $A$ is FRmR and $C$ is FCmR. And so, the existence of a factor set for $M$ is proved. $(ii) \Rightarrow$ : If $M$ has a semifactor set $F$ then there exists a nonsingular $R$ such that $$\label{defDiv29} R M Q_{F}=\left[\begin{array}{cc} \smash{\overset{\sigma_F(F)}{\overbrace{A}}} & B\\ 0 & C \end{array}\right]$$ where the zero block is Medium, $A$ is FRmR and $B$ is FCmR. By Proposition \[corBig\] we know that $A$ or/and $C$ are square, and so both are FRmR or both are FCmR. This implies (see Proposition \[FRmR-FCmR\]) that $R M Q_{F}$ is FRmR or FCmR. So $R M Q_{F}$ is FmR, and then $M$ is FmR. $\quad \Leftarrow$ : If $M$ is FmR then, as we have seen just after Table \[tabla2\], $M$ has at least one semifactor set. Linear independent rows ======================= Let $M$ be an ACI-matrix. Remember that each column of $M$ has its own indeterminates. Suppose that the first column of $A$ has intedeterminates $x_1, x_2, \ldots, x_i$; that the second column $y_1, y_2, \ldots, y_j$; and so on. Now, let us represent the vector space where the entries of the first column lie by $\mathbb{F}+\mathbb{F}x_1+\ldots+\mathbb{F}x_i$, and for the second column $\mathbb{F}+\mathbb{F}y_1+\ldots+\mathbb{F}y_j$, and so on. All these sets are vector spaces over $\mathbb{F}$. And the row vectors of $M$ are in the vector space $$\begin{aligned} \label{vectorSpaceACIrows} \big(\mathbb{F}+\mathbb{F}x_1+\ldots+\mathbb{F}x_i\big)\times \big(\mathbb{F}+\mathbb{F}y_1+\ldots+\mathbb{F}y_j\big)\times \ldots\end{aligned}$$ From now on when we talk about linear independence or linear dependence of the rows of an ACI-matrix $M$, we are talking about the vector space given in (\[vectorSpaceACIrows\]). The next Proposition and Remark expose the relation of ACI-matrices with linear independent rows and ACI-matrices which are FRmR. It is important to keep in mind this relation. \[SC->lir\] An FRmR ACI-matrix has linear independent rows. Suppose that $M$ is an $m\times n$ ACI-matrix with linear dependent rows. Then any completion of $M$ is a constant matrix with linear dependent rows whose rank is less than $m$. So $\operatorname{maxRank}(M)<\operatorname{rows}(M)$ and so M is not FRmR. The linear independence of the rows of an ACI-matrix does not imply that it is FRmR. For example, over any field the ACI-matrix $$\begin{aligned} \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & x \\ 1 & 1 & 1 & y \end{bmatrix}\end{aligned}$$ has linear independent rows and maxRank equal to 2. So it is not FRmR. The next result will be relevant in the proof of a key result: Lemma \[RLItoFRC\]. In this somewhat long statement the condition that we want to emphasize is that $A_1$ as well as $A_2$ have linear independent rows. This condition will reappear in Lemma \[RLItoFRC\], and actually it will be a central theme of many proofs of our work. \[TwoDecompDivisor2\] Consider two ACI-matrices of size $m \times n$ given by $$M_{1}=\begin{bmatrix} A_{1} & B_{1} \\ 0 & C_{1} \end{bmatrix} \quad \text{and} \quad M_{2}=\begin{bmatrix} A_{2} & B_{2} \\ 0 & C_{2} \end{bmatrix}.$$ where $A_{1}$ and $A_{2}$ have the same number $n_1$ of columns. Let $R$ be a nonsingular constant matrix of order $m$ and let $Q$ and $Q'$ be permutation matrices of orders $n_1$ and $(n-n_1)$ respectively, such that $$\begin{bmatrix} A_{2} & B_{2} \\ 0 & C_{2} \end{bmatrix}=R\begin{bmatrix} A_{1} & B_{1} \\ 0 & C_{1} \end{bmatrix} \begin{bmatrix} Q & 0 \\ 0 & Q' \end{bmatrix}.$$ If $A_{1}$ as well as $A_{2}$ have linearly independent rows then $A_{1} \sim A_{2}$ and $C_{1} \sim C_{2}$. By hypothesis $A_1$ and $A_2$ have the same number $n_1$ of columns, but nothing is said about the number of rows. Nevertheless, as $ R \left[\begin{smallmatrix} A_{1} \\ 0 \end{smallmatrix}\right] Q= \left[\begin{smallmatrix} A_{2} \\ 0 \end{smallmatrix}\right]$ where $A_{1}$ and $A_{2}$ have linearly independent rows then $A_{1}$ and $A_{2}$ also have the same number $m_1$ of rows. Therefore $A_{1}$ and $A_{2}$ have the same size $m_1\times n_1$. Thus $C_{1}$ and $C_{2}$ also have the same size $(m-m_1)\times (n-n_1)$. Writing $$R=\begin{bmatrix}S& T \\ U & V \end{bmatrix}$$ as a block matrix where $S$ is $m_1\times m_1$ and $V$ is $(m-m_1)\times (m-m_1)$ we have $$\label{blockEquiv} \begin{bmatrix} A_{2} & B_{2} \\ 0 & C_{2} \end{bmatrix}= \begin{bmatrix} S& T \\ U & V \end{bmatrix} \begin{bmatrix} A_{1} & B_{1} \\ 0 & C_{1} \end{bmatrix} \begin{bmatrix} Q & 0 \\ 0 & Q' \end{bmatrix} = \begin{bmatrix} * & * \\ U A_{1} Q & * \end{bmatrix}.$$ Since $UA_1Q=0$, $A_{1}$ has linearly independent rows, and $Q$ is a permutation then $U=0$. So $$\label{blockEquiv2} \begin{bmatrix} A_{2} & B_{2} \\ 0 & C_{2} \end{bmatrix}= \begin{bmatrix} S& T \\ 0 & V \end{bmatrix} \begin{bmatrix} A_{1} & B_{1} \\ 0 & C_{1} \end{bmatrix} \begin{bmatrix} Q & 0 \\ 0 & Q' \end{bmatrix} = \begin{bmatrix} {\bf S}A_{1} Q & * \\ 0 & V C_{1} Q' \end{bmatrix}.$$ As $S$ and $V$ are nonsingular then $A_{1} \sim A_{2}$ and $C_{1}\sim C_{2}$. Let $M$ be an ACI-matrix. Imagine that we want to find out if $F$ is a factor or a semifactor set of $M$. It makes sense to try to find an equivalent ACI-matrix with as many zero rows as possible in the ACI-submatrix formed by the columns indexed by $F$. An efficient way to do this is the procedure of a sweep from bottom to top that we are going to introduce now. Let $M$ be an $m\times n$ ACI-matrix and let $F\subseteq \{1,\ldots,n\}$. A sweep from bottom to top in $M$ is a procedure that transforms $M$ into an equivalent ACI-matrix that has all its nonzero rows linearly independent. It consists of $m-1$ steps and step $i$ is the following: step i: : If it is possible, make the ($m-i$)-th row equal to the zero row by adding linear combinations of rows $m-i+1,\dots,m$ below it. A sweep from bottom to top in $M$ with respect to the columns of $F$ is a procedure as the previous one but only requiring to do zeros in the entries allocated in the columns corresponding to $F$. For the field or reals consider the ACI-matrix $$M=\begin{bmatrix} x+2 & 1 & z \\ x+1 & 8y & 3z-5 \\ x & 4y & z-2 \\ 1 & 4y & 2z-3 \end{bmatrix}.$$ If we do a sweep from bottom to top in $M$ then $$\begin{bmatrix}[ccc] x+2 & 1 & z \\ x+1 & 8y & 3z-5 \\ x & 4y & z-2 \\ 1 & 4y & 2z-3 \end{bmatrix} \stackrel{\text{step 2}}{\longrightarrow} \begin{bmatrix} x+2 & 1 & z \\ 0 & 0 &0 \\ x & 4y & z-2 \\ 1 & 4y & 2z-3 \end{bmatrix}$$ and we have finished with an equivalent ACI-matrix whose nonzero rows are linearly independent. If we do a sweep from bottom to top in $M$ with respect to $F=\{ 2 \}$ then $$\begin{bmatrix}[c\|c\|c] x+2 & 1 & z \\ x+1 & 8y & 3z-5 \\ x & 4y & z-2 \\ 1 & 4y & 2z-3 \end{bmatrix} \stackrel{\text{step 1}}{\longrightarrow} \begin{bmatrix}[c\|c\|c] x+2 & 1 & z \\ x+1 & 8y & 3z-5 \\ x-1 & 0 & -z+1 \\ 1 & 4y & 2z-3 \end{bmatrix} \stackrel{\text{step 2}}{\longrightarrow} \begin{bmatrix}[c\|c\|c] x+2 & 1 & z \\ x-1 & 0 &-z+1 \\ x-1 & 0 & -z+1 \\ 1 & 4y & 2z-t \end{bmatrix}$$ and we have finished with an equivalent ACI-matrix whose nonzero rows in the second column are linearly independent. Note that the linear combinations employed to make zeros in the second column of $M$ were extended to the entire rows of $M$. The definition of factor (resp. semifactor) set just requires one decomposition to exist. If we know somehow that $F$ is a factor (resp. semifactor) set of $M$ and we perform a sweep from bottom to top in $M$ with respect to $F$, we might ask the following question: Do we always arrive, up to permutation of rows and columns, to an $F$-decomposition (resp. $F$-semidecomposition)? The answer is yes, as we will see in the next result where we assume that the sweep from bottom to top with respect to $F$ has already occurred. Then we only need to permute rows and columns to leave a zero block in the bottom left part. \[RLItoFRC\] Let $M$ be an $m\times n$ ACI-matrix. Suppose that $F$ is a factor (resp. semifactor) set of $M$ and $P$ is a permutation matrix of order $m$ such that $$\label{NAQD5} P M Q_F = \left[\begin{array}{cc} \smash{\overset{\sigma_F(F)}{\overbrace{A}}} & B \\ 0 & C \end{array}\right]$$ with $A$ having linearly independent rows. Then (\[NAQD5\]) is an $F$-decomposition (resp. $F$-semidecomposition). That is: the zero block is Big (resp. Medium), $A$ is FRmR and $C$ is FCmR. Note that $PMQ_F$ is obtained from $M$ by permuting its rows by $P$ and its columns by $Q_F$. As $F$ is a factor (resp. semifactor) set of $M$ then there exists a nonsingular $R$ such that $$\label{taqs} R M Q_F= \begin{bmatrix} \smash{\overset{\sigma_F(F)}{\overbrace{A'}}} & B' \\ 0 & C' \end{bmatrix}$$ is an $F$-decomposition (resp. $F$-semidecomposition). Therefore $A'$ is FRmR and so, by Proposition \[SC->lir\], it has all its rows linearly independent. Note that $$\label{} \begin{bmatrix} A' & B' \\ 0 & C' \end{bmatrix}=R M Q_F= R\left( P^{-1} \left[\begin{array}{cc} A & B \\ 0 & C \end{array}\right] Q_F^{-1} \right)Q_F= R P^{-1}\left[\begin{array}{cc} A & B \\ 0 & C \end{array}\right] .$$ From Lemma \[TwoDecompDivisor2\] we conclude that $A\sim A'$ and $C\sim C'$. As $A'$ and $C'$ are FmR then $A$ and $C$ are also FmR. And since the zero block of (\[taqs\]) is Big (resp. Medium) then the zero block of (\[NAQD5\]) will also be Big (resp. Medium). So $PMQ_F$ is an $F$-decomposition (resp. $F$-semidecomposition). The union and intersection of factors and of semifactor sets ============================================================ Most of the heavy lifting of the main result is done in this section. In the following three results we will study the relative position of two factor sets or of two semifactor sets of an ACI-matrix. It is important to recall (see Proposition \[maxRank<->noFactorSet\]) that an ACI-matrix has a factor set if and only if it is not FmR, and that an ACI-matrix has a semifactor set if and only if it is FmR. \[constant-nonFullRank.2factorSetsCantBeDisjoint\] Two factor sets of an ACI-matrix can not be disjoint. Suppose $F_1$ and $F_2$ are two disjoint factor sets of an $m\times n$ ACI-matrix $M$. As the empty set is not a factor set of any ACI-matrix then, up to permutation of columns, we can assume that $F_1=\{1,\ldots,h\}$ and $F_2=\{h+1,\ldots,k\}$ with $1\leq h < k \leq n$. And let $U=\{1,\ldots, n\}\setminus (F_1\cup F_2)$. First we do a sweep from bottom to top in $M$ with respect to $F_1$. After reordering the rows we obtain $$\begin{aligned} M' =\left[\begin{array}{c\|cc} \smash{\overset{F_1}{\overbrace{A}}} & \smash{\overset{F_2}{\overbrace{B}}} & \smash{\overset{U}{\overbrace{C}}} \\ \cline{1-3} 0 & D & E \end{array}\right] \hspace{-2mm}\begin{array}{rr} \} \ r \\ \} \ t \end{array} \end{aligned}$$ where $A$ has linearly independent rows. As $M\sim M'$ and $M'$ is obtained from $M$ without permuting columns, then $F_1$ and $F_2$ are factor sets of $M'$. Now we do a sweep from bottom to top in $M'$ with respecto to $F_2$. After reordering the first $r$ rows and the last $t$ rows we obtain: $$\begin{aligned} M''=\left[\begin{array}{c\|cc} \smash{\overset{F_1}{\overbrace{A''}}} & \smash{\overset{F_2}{\overbrace{0}}} & \smash{\overset{U}{\overbrace{C''}}} \\ A' & B' & C' \\ \cline{1-3} 0 & D' & E' \\ 0 & 0 & E'' \end{array}\right] \hspace{-2mm} \begin{array}{rr} \} \ r_1\\ \} \ r_2 \\ \} \ t_1 \\ \} \ t_2 \end{array} \end{aligned}$$ with $r_1+r_2=r$, $t_1+t_2=t$, and where $\left[\begin{smallmatrix} B' \\ D' \end{smallmatrix}\right]$ has linearly independent rows. As $A\sim \left[\begin{smallmatrix} A'' \\ A' \end{smallmatrix}\right]$ then also $\left[\begin{smallmatrix} A'' \\ A' \end{smallmatrix}\right]$ has linear independent rows. As $M'\sim M''$ and $M''$ is obtained from $M'$ without permuting columns then $F_1$ and $F_2$ are factor sets of $M''$. Now let us deduce some inequalities that will be key to our analysis. On one hand we have a zero block corresponding to the factor set $F_1$, it is formed by the two zeros of the first block column of $M''$. As $\left[\begin{smallmatrix} A'' \\ A' \end{smallmatrix}\right]$ has linear independent rows then this zero block must be Big (see Lemma \[RLItoFRC\]). So $$\begin{aligned} \label{m,n<t_1+t_2+F_1} t_1+t_2+\#F_1 > \max\{ m,n \}.\end{aligned}$$ On the other hand we have a zero block corresponding to the factor set $F_2$, it is formed by the two zeros of the second block column of $M''$. As $\left[\begin{smallmatrix} B' \\ D' \end{smallmatrix}\right]$ has linearly independent rows then this zero block must be Big (see Lemma \[RLItoFRC\]). So $$\begin{aligned} \label{m,n<s_2+t_1+F_2} r_1+t_2+\#F_2> \max\{ m,n \}.\end{aligned}$$ Finally, the culprit of the contradiction will be the $(t_1+t_2)\times ( \# F_2+ \# U)$ ACI-submatrix of $M''$ $$\begin{aligned} \left[\begin{array}{cc} D' & E' \\ 0 & E'' \end{array}\right],\end{aligned}$$ since we will prove that it is FmR and not FmR at the same time: 1. $\left[\begin{smallmatrix} D' & E' \\ 0 & E'' \end{smallmatrix}\right]$ is FmR because $F_1$ is a factor set of $M''$ and $\left[\begin{smallmatrix} A'' \\ A' \end{smallmatrix}\right]$ has linearly independent rows (see Lemma \[RLItoFRC\]). 2. The zero block of $\left[\begin{smallmatrix} D' & E' \\ 0 & E'' \end{smallmatrix}\right]$ is Big if $t_2+\#F_2> \max\{\operatorname{rows}\left[\begin{smallmatrix} D' & E' \\ 0 & E'' \end{smallmatrix}\right], \operatorname{cols}\left[\begin{smallmatrix} D' & E' \\ 0 & E'' \end{smallmatrix}\right]\}$. Let us see that this inequality is true: 1. $t_2+\#F_2>t_1+t_2$. From (\[m,n<s\_2+t\_1+F\_2\]) it follows that $$\begin{aligned} r_1+t_2+\#F_2 >m=r_1+r_2+t_1+t_2\end{aligned}$$ which implies the required inequality. 2. $t_2+\#F_2 > \# F_2+ \# U $. From (\[m,n<t\_1+t\_2+F\_1\]) it follows that: $$\begin{aligned} \label{parte1-raz-L41} t_1+t_2+\#F_1 >m= r_1+r_2+t_1+t_2 \quad \Longrightarrow \quad \#F_1>r_1.\end{aligned}$$ From (\[m,n<s\_2+t\_1+F\_2\]) it follows that: $$\begin{aligned} \label{parte2-raz-L41} r_1+t_2+\#F_2>n= \#F_1+\#F_2+\#U.\end{aligned}$$ From (\[parte1-raz-L41\]) and (\[parte2-raz-L41\]) we obtain the required inequality. So $\left[\begin{smallmatrix} D' & E' \\ 0 & E'' \end{smallmatrix}\right]$ is not FmR because its zero block is Big (see Proposition \[Big->nofFullMaxRank\]). Although the proof of the next Lemma\[SemifactorSet-intersection\] is very similar to the proof of Lemma \[constant-nonFullRank.2factorSetsCantBeDisjoint\] we will include it for clarity because the differences are subtle. \[SemifactorSet-intersection\] Two semifactor sets of a wide ACI-matrix can not be disjoint. Suppose $F_1$ and $F_2$ are two disjoint semifactor sets of a wide ACI-matrix $M$ of size $m\times n$. The first half of the proof of Lemma \[constant-nonFullRank.2factorSetsCantBeDisjoint\] is the same as this one with the difference that in this case we have two semifactor sets and so the zero blocks that will appear will be Medium instead of Big. So, after the two sweeps from bottom to top in $M$ and reordering the rows we obtain: $$\begin{aligned} M''=\left[\begin{array}{c\|cc} \smash{\overset{F_1}{\overbrace{A''}}} & \smash{\overset{F_2}{\overbrace{0}}} & \smash{\overset{U}{\overbrace{C''}}} \\ A' & B' & C' \\ \cline{1-3} 0 & D' & E' \\ 0 & 0 & E'' \end{array}\right] \hspace{-2mm} \begin{array}{rr} \} \ r_1\\ \} \ r_2 \\ \} \ t_1 \\ \} \ t_2 \end{array} \end{aligned}$$ Now let us deduce some equalities that will be key to our analysis. On one hand we have a zero block corresponding to the semifactor set $F_1$, it is formed by the two zeros of the first block column of $M''$. As $\left[\begin{smallmatrix} A'' \\ A' \end{smallmatrix}\right]$ has linear independent rows then this zero block must be Medium (see Lemma \[RLItoFRC\]). So $$\begin{aligned} \label{m,n=t_1+t_2+F_1} t_1+t_2+\#F_1 = \max\{ m,n \}=n.\end{aligned}$$ On the other hand we have a zero block corresponding to the semifactor set $F_2$, it is formed by the two zeros of the second block column of $M''$. As $\left[\begin{smallmatrix} B' \\ D' \end{smallmatrix}\right]$ has linearly independent rows then this zero block must be Medium (see Lemma \[RLItoFRC\]). So $$\begin{aligned} \label{m,n=s_2+t_1+F_2} r_1+t_2+\#F_2= \max\{ m,n \}=n.\end{aligned}$$ Again the culprit of the contradiction will be the $(t_1+t_2)\times ( \# F_2+ \# U)$ ACI-submatrix of $M''$ $$\begin{aligned} \left[\begin{array}{cc} D' & E' \\ 0 & E'' \end{array}\right], \end{aligned}$$ since we will prove that it is FmR and not FmR at the same time: 1. $\left[\begin{smallmatrix} D' & E' \\ 0 & E'' \end{smallmatrix}\right]$ is FmR because $F_1$ is a semifactor set of $M''$ and $\left[\begin{smallmatrix} A'' \\ A' \end{smallmatrix}\right]$ has linearly independent rows (see Lemma \[RLItoFRC\]). 2. The zero block of $\left[\begin{smallmatrix} D' & E' \\ 0 & E'' \end{smallmatrix}\right]$ is Big if $t_2+\#F_2> \max\{\operatorname{rows}\left[\begin{smallmatrix} D' & E' \\ 0 & E'' \end{smallmatrix}\right], \operatorname{cols}\left[\begin{smallmatrix} D' & E' \\ 0 & E'' \end{smallmatrix}\right]\}$. Let us see that this inequality is true: 1. $t_2+\#F_2>t_1+t_2$. From (\[m,n=s\_2+t\_1+F\_2\]) it follows that $$\begin{aligned} r_1+t_2+\#F_2 =n>m=r_1+r_2+t_1+t_2\end{aligned}$$ which implies the required inequality. 2. $t_2+\#F_2 > \# F_2+ \# U $. From (\[m,n=t\_1+t\_2+F\_1\]) it follows that: $$\begin{aligned} \label{parte1-raz-L42} t_1+t_2+\#F_1 =n>m= r_1+r_2+t_1+t_2 \quad \Longrightarrow \quad \#F_1>r_1.\end{aligned}$$ From (\[m,n=s\_2+t\_1+F\_2\]) it follows that: $$\begin{aligned} \label{parte2-raz-L42} r_1+t_2+\#F_2=n= \#F_1+\#F_2+\#U.\end{aligned}$$ From (\[parte1-raz-L42\]) and (\[parte2-raz-L42\]) we obtain the required inequality. So $\left[\begin{smallmatrix} D' & E' \\ 0 & E'' \end{smallmatrix}\right]$ is not FmR because its zero block is Big (see Proposition \[Big->nofFullMaxRank\]). \[SemifactorsInTall\] Two semifactor sets of a tall or square ACI-matrix can be disjoint or not disjoint. We provide an example for each case. For tall ACI-matrices $$\begin{array}{lcl} \vspace{-2mm} \ \ \overset{F_1}{\overbrace{\phantom{x}}} \ \ \overset{F_2}{\overbrace{\phantom{x}}} & & \ \ \ \overset{F_1}{\overbrace{\phantom{xxxxxx}}} \\ \vspace{-3mm} \begin{bmatrix} \ \ 1 \ \ & \ \ 0 \ \ & \ \ x \ \ \\ \ \ 0 \ \ & \ \ 1 \ \ & \ \ y \ \ \\ \ \ 0 \ \ & \ \ 0 \ \ & \ \ 1 \ \ \\ \ \ 0 \ \ & \ \ 0 \ \ & \ \ 1 \ \ \\ \end{bmatrix} & \text{and} & \begin{bmatrix} \ \ 1 \ \ & \ \ 0 \ \ & \ \ 0 \ \ \\ \ \ x \ \ & \ \ 1 \ \ & \ \ y \ \ \\ \ \ 0 \ \ & \ \ 0 \ \ & \ \ 1 \ \ \\ \ \ 0 \ \ & \ \ 0 \ \ & \ \ 0 \ \ \\ \end{bmatrix} \\ & & \phantom{xxxxxx} \underset{F_2}{\underbrace{\phantom{xxxxxx}}} \end{array}$$ and for square ACI-matrices it is enough to delete the last row on each one. In what follows our main objective will be to prove that the intersection and the union of two factor (resp. semifactor) sets is a factor (resp. semifactor) set. That is what Theorem \[cor6\] below says. Note that Lemmas \[constant-nonFullRank.2factorSetsCantBeDisjoint\], \[SemifactorSet-intersection\] and \[SemifactorsInTall\] conclude that two factor or two semifactor sets always overlap, except in the case of some semifactor sets of tall or square ACI-matrices. So in order to achieve our objective we will first study this exceptional case of disjoint semifactor sets (Theorem \[2semifactor-disjointcase\]), and then the generic case of overlapping factor or semifactor sets (Theorem \[2non-overlappingCupAndCapFS\]). \[2semifactor-disjointcase\] The intersection and the union of two disjoint semifactor sets of a tall or square ACI-matrix are semifactor sets. Let $F_1$ and $F_2$ be two disjoint semifactor sets of a tall or square ACI-matrix $M$ of size $m\times n$. Tall and square FmR ACI-matrices are the only ACI-matrices for which the empty set is a semifactor set (see Table \[tabla2\]). Then $F_1\cap F_2=\emptyset$ is a semifactor set of $M$. The proof for $F_1\cup F_2$ starts again like the proof of Lemma \[constant-nonFullRank.2factorSetsCantBeDisjoint\]. So after the two sweeps from bottom to top in $M$ and reordering the rows we obtain: $$\begin{aligned} \label{MatrixWellOrderedQWZ11} M''=\left[\begin{array}{c\|cc} \smash{\overset{F_1}{\overbrace{A''}}} & \smash{\overset{F_2}{\overbrace{0}}} & C'' \\ A' & B' & C' \\ \cline{1-3} 0 & D' & E' \\ 0 & 0 & E'' \end{array}\right] \hspace{-2mm} \begin{array}{rr} \} \ r_1\\ \} \ r_2 \\ \} \ t_1 \\ \} \ t_2 \end{array} \end{aligned}$$ Focusing on the $F_2$ semifactor set, by Proposition \[corBig\] we know that $\left[\begin{smallmatrix} B' \\ D' \end{smallmatrix}\right]$ is square and so $$\begin{aligned} \label{t1<F2} \#F_2 \geq t_1.\end{aligned}$$ Focusing on the $F_1$ semifactor set, the zero block formed by the two zeros of the first column of $M''$ is Medium. So according to Definition \[defBigZero\]: $$\begin{aligned} \label{m,n<t_1+t_2+F_1-2} t_1+t_2+\#F_1=\max\{ m,n \}= m.\end{aligned}$$ Let $Z$ be the zero block composed by the two zero blocks of the last row of $M''$. From (\[t1<F2\]) and (\[m,n<t\_1+t\_2+F\_1-2\]) $$\begin{aligned} \label{maxM,N<t_2+F_1+F_2} t_2+\#F_1+\#F_2\geq \max\{ m,n \}= m\end{aligned}$$ and so $Z$ is either a Big (if inequality is strict) or a Medium (if there is equality) zero block in $M''$. It can not be Big, otherwise $M''$ would not be FmR (Proposition \[Big->nofFullMaxRank\]) and this contradicts that an ACI-matrix has a semifactor set if and only if it is FmR (Proposition \[maxRank<->noFactorSet\]). So $Z$ is a Medium zero block in $M''$. So now we know that there must be equality in (\[maxM,N<t\_2+F\_1+F\_2\]), which in turn means that there is equality in (\[t1<F2\]): $\#F_2 = t_1$. So $D'$ is square, which implies that $B'$ is wide degenerate and therefore $r_2=0$. So (\[MatrixWellOrderedQWZ11\]) is simplified into: $$\begin{aligned} M''=\left[\begin{array}{c\|cc} \smash{\overset{F_1}{\overbrace{A''}}} & \smash{\overset{F_2}{\overbrace{0}}} & C'' \\ \cline{1-3} 0 & D' & E' \\ 0 & 0 & E'' \end{array}\right] \hspace{-2mm} \begin{array}{rr} \} \ r_1\\ \} \ t_1 \\ \} \ t_2 \end{array} . \\[-3.5mm] \nonumber\end{aligned}$$ To prove that $F_1\cup F_2$ is a semifactor set, apart from $Z$ being a Medium zero block, we still need to prove that: i. $\left[\begin{smallmatrix} A'' & 0 \\ 0 & D' \end{smallmatrix}\right]$ is FRmR. Since $F_1$ is a semifactor set we know that $\left[\begin{smallmatrix} A'' \\ A' \end{smallmatrix}\right]= \left[\begin{smallmatrix}A''\end{smallmatrix}\right]$ is FRmR, and since $F_2$ is a semifactor set we know that $\left[\begin{smallmatrix} B' \\ D' \end{smallmatrix}\right]= \left[\begin{smallmatrix}D'\end{smallmatrix}\right]$ is FRmR. Now we apply Proposition \[FRmR-FCmR\] to deduce this item. ii. $E''$ is FCmR. Since $F_1$ is a semifactor set we know that $\left[\begin{smallmatrix} D' & E' \\ 0 & E'' \end{smallmatrix}\right]$ is FCmR. This together with $D'$ being square implies that $E''$ is FCmR. For the proof of our next theorem it will be convenient to introduce the notion of complementary of an ACI-submatrix. Let $A$ be an ACI-submatrix of an ACI-matrix $M$, the complementary $\overline{A}$ of $A$ in $M$ is obtained by deleting all the rows and columns of $M$ that are involved in $A$. \[2non-overlappingCupAndCapFS\] The intersection and the union of two overlapping factor (resp. semifactor) sets of an ACI-matrix are factor (resp. semifactor) sets. Let $F_1$ and $F_2$ be two overlapping factor (resp. semifactor) sets of an ACI-matrix $M$. If $F_1 \subset F_2$ or $F_2 \subset F_1$ the result is trivial. So, without loss of generality we can assume that $F_1=\{1,\ldots,k\}$ and $F_2=\{h+1,\ldots,l\}$ with $1\leq h < k < l$. We do a sweep from bottom to top in $M$ with respect to $F_1$ and after reordering the rows we obtain $$\begin{aligned} \label{M'} & \phantom{XXXx} \overset{F_1}{\overbrace{\phantom{XXXx}}} \nonumber \\[-5mm] & M' =\left[\begin{array}{cc\|cc} {A} & B & C & D\\ \cline{1-4} 0 & 0 & E & F \end{array}\right] \hspace{-2mm}\begin{array}{rr} \} \ r \\ \} \ t \end{array} \\[-5mm] & \phantom{XXXXXl} \underset{F_2}{\underbrace{\phantom{XXXx}}} \nonumber \end{aligned}$$ where $\left[\begin{smallmatrix} A & B \end{smallmatrix}\right]$ has linearly independent rows and $\left[\begin{smallmatrix} E & F \end{smallmatrix}\right] $ is FCmR (see Lemma \[RLItoFRC\]). Now we do a sweep from bottom to top in $M'$ with respect to $F_2$ and after reordering the first $r$ rows and the last $t$ rows we obtain $$\begin{aligned} & \phantom{XXXX} \overset{F_1}{\overbrace{\phantom{XXXX}}} \nonumber \\[-5mm] & M''=\left[\begin{array}{cc\|cc} A'' & 0 & 0 & D'' \\ A' & B' & C' & D' \\ \cline{1-4} 0 & 0 & E' & F' \\ 0 & 0 & 0 & F'' \end{array}\right] \hspace{-2mm} \begin{array}{rr} \} \ r_1\\ \} \ r_2 \\ \} \ t_1 \\ \} \ t_2 \end{array} \\[-5mm] & \phantom{XXXXXXl} \underset{F_2}{\underbrace{\phantom{XXXx}}} \nonumber \end{aligned}$$ with $r_1+r_2=r$, $t_1+t_2=t$, $r_1, r_2, t_1, t_2\geq 0$, and where $\left[\begin{smallmatrix} A'' & 0 \\ A' & B' \end{smallmatrix}\right]$ and $\left[\begin{smallmatrix} B' & C' \\ 0 & E' \end{smallmatrix}\right]$ have linearly independent rows. Note that the fourth column of $M''$ could be tall degenerate with size $(r_1+r_2+t_1+t_2)\times 0$ while the other three columns will never be tall degenerate since $1\leq h < k < l$. Let us see that $t_1=0$ is impossible: if this was the case then the third row of $M''$ would be wide degenerate and since $F_1$ is a factor set then $\left[\begin{smallmatrix} 0 & F'' \end{smallmatrix}\right]$ should be FCmR, but this is impossible since it has columns full of zeros. So, from now on $t_1>0$. The value $r_2$ might be positive or zero and the arguments we will provide hold for both. Four possibilities appear depending on the values of $r_1$ and $t_2$. - Suppose $r_1>0$ and $t_2>0$. i. \[iStep\] Since $F_2$ is a factor (resp. semifactor) set and $\left[\begin{smallmatrix} B' & C' \\ 0 & E \end{smallmatrix}\right]$ has linearly independent rows, then (see Lemma \[RLItoFRC\]) its complementary $ \left[\begin{smallmatrix} A'' & D'' \\ 0 & F'' \end{smallmatrix}\right]$ in $M''$ is FCmR. In particular $A''$ is FCmR. ii. \[ABstep\] Since $F_1$ is a factor (resp. semifactor) set and $\left[\begin{smallmatrix} A'' & 0 \\ A' & B' \end{smallmatrix}\right]$ has linear independent rows, then (see Lemma \[RLItoFRC\]) $\left[\begin{smallmatrix} A'' & 0 \\ A' & B' \end{smallmatrix}\right]$ is FRmR. In particular $A''$ is FRmR. iii. \[iiiStep\] Since $A''$ is FCmR and FRmR at the same time then $A''$ is a square FmR. iv. \[4Step\] Since $F_1$ is a factor (resp. semifactor) set and $\left[\begin{smallmatrix} A'' & 0 \\ A' & B' \end{smallmatrix}\right]$ has linear independent rows, then (see Lemma \[RLItoFRC\]) its complementary $\left[\begin{smallmatrix} E' & F' \\ 0 & F'' \end{smallmatrix}\right]$ in $M''$ is FCmR. In particular $E'$ is FCmR. v. \[5Step\] Since $F_2$ is a factor (resp. semifactor) set and $\left[\begin{smallmatrix} B' & C' \\ 0 & E' \end{smallmatrix}\right]$ has linear independent rows then (see Lemma \[RLItoFRC\]) $\left[\begin{smallmatrix} B' & C' \\ 0 & E' \end{smallmatrix}\right]$ is FRmR. In particular $E'$ is FRmR. vi. \[viStep\] Since $E'$ is FCmR and FRmR at the same time then $E'$ is a square FmR. vii. \[viiStep\] Since $\left[\begin{smallmatrix} E' & F' \\ 0 & F'' \end{smallmatrix}\right]$ is FCmR and $E'$ is square then $F''$ is FCmR. viii. \[ivStep\] Since $\left[\begin{smallmatrix} A'' & 0 \\ A' & B' \end{smallmatrix}\right]$ is FRmR and $A''$ is square then $B'$ is FRmR. ix. \[xiStep\] The complementary matrix of $F''$ in $M''$ is $\overline{F''}= \left[\begin{smallmatrix} A'' & 0 & 0 \\ A' & B' & C' \\ 0 & 0 & E' \end{smallmatrix}\right]\sim \left[\begin{smallmatrix} B' & A' & C' \\ 0 & A'' & 0 \\ 0 & 0 & E' \end{smallmatrix}\right]$. Since $B'$, $A''$ and $E'$ are FRmR (\[ivStep\], \[ABstep\] and \[5Step\]) then Proposition \[FRmR-FCmR\] implies that $\overline{F''}$ is also FRmR. x. \[viiiStep\] The complementary matrix of $B'$ in $M''$ is $ \overline{B'}=\left[\begin{smallmatrix} A'' & 0 & D'' \\ 0 & E' & F' \\ 0 & 0 & F'' \end{smallmatrix}\right]$. Since $A''$, $E'$ and $F''$ are FCmR (\[iStep\], \[4Step\] and \[viiStep\]) then Proposition \[FRmR-FCmR\] implies that $\overline{B'}$ is FCmR. xi. \[ixStep\] Consider the zero block $Z$ obtained by joining together the three zero blocks in the second column of $M''$, and consider the zero block $Z_1$ corresponding to the factor (resp. semifactor) set $F_1$. Note that $Z$ has size $(r_1+t_1+t_2)\times \#(F_1\cap F_2)$, and that $Z_1$ has size $(t_1+t_2)\times \#F_1$. The number $\operatorname{rows}+\operatorname{cols}$ for $Z$ and for $Z_1$ is equal since $A''$ is square (\[iiiStep\]). As $Z_1$ is Big (resp. Medium) by hypothesis and the number $\operatorname{rows}+\operatorname{cols}$ is what determines if a zero block is Big (resp. Medium), then $Z$ is Big (resp. Medium). xii. \[xiiStep\] Consider the zero block $Z'$ obtained by joining together the three zero blocks in the last row of $M''$, and consider again the zero block $Z_1$ corresponding to the factor (resp. semifactor) set $F_1$. Note that $Z'$ has size $t_2 \times (\#F_1 + \#(F_2\setminus F_1))$ and that $Z_1$ has size $(t_1+t_2)\times \#F_1$. The number $\operatorname{rows}+\operatorname{cols}$ for $Z'$ and $Z_1$ is equal since $E'$ is square (\[viStep\]). As $Z_1$ is Big (resp. Medium) by hypothesis then $Z'$ is Big (resp. Medium). xiii. $F_1 \cap F_2$ is a factor (resp. semifactor) set of $M''$ since $Z$ is a Big (resp. Medium) zero block (\[ixStep\]), $B'$ is FRmR (\[ivStep\]) and $\overline{B'}$ is FCmR (\[viiiStep\]). xiv. $F_1 \cup F_2$ is a factor (resp. semifactor) set of $M''$ since $Z'$ is a Big (resp. Medium) zero block (\[xiiStep\]), $F''$ is FCmR (\[viiStep\]) and $\overline{F''}$ is FRmR (\[xiStep\]). - Suppose $r_1>0$ and $t_2=0$. Then $$\begin{aligned} & \phantom{XXXX} \overset{F_1}{\overbrace{\phantom{XXXX}}} \nonumber \\[-5mm] & M''=\left[\begin{array}{cc\|cc} A'' & 0 & 0 & D'' \\ A' & B' & C' & D' \\ \cline{1-4} 0 & 0 & E & F \\ \end{array}\right] \hspace{-2mm} \begin{array}{l} \} \ r_1\\ \} \ r_2 \\ \} \ t=t_1 \\ \end{array} \\[-5mm] & \phantom{XXXXXXl} \underset{F_2}{\underbrace{\phantom{XXXx}}} \nonumber \end{aligned}$$ i. \[1Step2\] Since $F_2$ is a factor (resp. semifactor) set and $\left[\begin{smallmatrix} B' & C' \\ 0 & E \end{smallmatrix}\right]$ has linearly independent rows, then (see Lemma \[RLItoFRC\]) its complementary $ \left[\begin{smallmatrix} A'' & D'' \end{smallmatrix}\right]$ in $M''$ is FCmR. In particular $A''$ is FCmR. ii. \[2Step2\] Since $F_1$ is a factor (resp. semifactor) set and $\left[\begin{smallmatrix} A'' & 0 \\ A' & B' \end{smallmatrix}\right]$ has linear independent rows, then (see Lemma \[RLItoFRC\]) $\left[\begin{smallmatrix} A'' & 0 \\ A' & B' \end{smallmatrix}\right]$ is FRmR. In particular $A''$ is FRmR. iii. \[3Step2\] Since $A''$ is FCmR and FRmR at the same time then $A''$ is a square FmR. iv. \[4Step2\] Since $F_1$ is a factor (resp. semifactor) set and $\left[\begin{smallmatrix} A'' & 0 \\ A' & B' \end{smallmatrix}\right]$ has linear independent rows, then (see Lemma \[RLItoFRC\]) its complementary $\left[\begin{smallmatrix} E & F \end{smallmatrix}\right]$ in $M''$ is FCmR. In particular $E$ is FCmR. v. \[5Step2\] Since $F_2$ is a factor (resp. semifactor) set and $\left[\begin{smallmatrix} B' & C' \\ 0 & E \end{smallmatrix}\right]$ has linear independent rows then (see Lemma \[RLItoFRC\]) $\left[\begin{smallmatrix} B' & C' \\ 0 & E \end{smallmatrix}\right]$ is FRmR. In particular $E$ is FRmR. vi. \[6Step2\] Since $E$ is FCmR and FRmR at the same time then $E$ is a square FmR. vii. \[7Step2\] Since $\left[\begin{smallmatrix} A'' & 0 \\ A' & B' \end{smallmatrix}\right]$ is FRmR and $A''$ is square FmR then $B'$ is FRmR. viii. \[8Step2\] In this step the arguments diverge significantly depending on $F_1, F_2$ being factor or semifactor sets, so we consider the cases separately: a. $F_1$ and $F_2$ are factor sets. Since $F_1$ is a factor set then the two zero blocks on the last row of $M''$ compose a Big zero block. This implies (see Proposition \[corBig\]) that $\left[\begin{smallmatrix} E & F \end{smallmatrix}\right]$ is tall, which is imposible since $E$ is square (\[6Step2\]). b. $F_1$ and $F_2$ are semifactor sets. Since $F_1$ is a semifactor set then the two zero blocks on the last row of $M''$ compose a Medium zero block. This implies (see Proposition \[corBig\]) that $\left[\begin{smallmatrix} E & F \end{smallmatrix}\right]$ is tall or square, and since $E$ is square (\[6Step2\]) this forces $F$ to be tall degenerate with size $t_1\times 0$. So $$\begin{aligned} & \phantom{XXXX} \overset{F_1}{\overbrace{\phantom{XXXX}}} \nonumber \\[-5mm] & M''=\left[\begin{array}{cc\|cc} A'' & 0 & 0 \\ A' & B' & C' \\ \cline{1-4} 0 & 0 & E \\ \end{array}\right] \hspace{-2mm} \begin{array}{ll} \} \ r_1\\ \} \ r_2 \\ \} \ t=t_1 \\ \end{array} \\[-5mm] & \phantom{XXXXXXl} \underset{F_2}{\underbrace{\phantom{XXXx}}} \nonumber \end{aligned}$$ - In $M''$ the complementary of $B'$ is $ \overline{B'}=\left[\begin{smallmatrix} A'' & 0 \\ 0 & E \end{smallmatrix}\right]$. Since $A''$ and $E$ are square FmR (\[3Step2\] and \[6Step2\]) then Proposition \[FRmR-FCmR\] says that $\overline{B'}$ is square FmR. - Consider the zero block $Z$ obtained by joining together the zero blocks below and above $B'$. By Proposition \[corBig\] $Z$ is Medium since $B'$ is wide or square (\[7Step2\]) and $\overline{B'}$ is square. - As $A''$ is square (\[3Step2\]), $B'$ is wide or square (\[7Step2\]) and $E$ is square (\[6Step2\]) then $M''$ is wide or square. As $M''$ has semifactor sets then $M''$ is FmR (see Proposition \[maxRank<->noFactorSet\]). Moreover, $F_1\cup F_2$ span all columns of $M''$. Recall the discussion after Definition \[semifactorSetDefinition\] where it was explained that in a wide or square FRmR ACI-matrix the set $\{1,\ldots,n\}$ is a semifactor set. So $F_1\cup F_2$ is a semifactor set. - $F_1 \cap F_2$ is a semifactor set of $M''$ since $Z$ is a Medium zero block, $B'$ is FRmR (\[7Step2\]) and $\overline{B'}$ is FCmR (it is square FmR). - Suppose $r_1=0$ and $t_2>0$. Then $$\begin{aligned} & \phantom{XXXx} \overset{F_1}{\overbrace{\phantom{XXXx}}} \nonumber \\[-5mm] & M''=\left[\begin{array}{cc\|cc} A & B & C & D \\ \cline{1-4} 0 & 0 & E' & F' \\ 0 & 0 & 0 & F'' \end{array}\right] \hspace{-2mm} \begin{array}{l} \} \ r=r_2 \\ \} \ t_1 \\ \} \ t_2 \end{array} \\[-5mm] & \phantom{XXXXxXl} \underset{F_2}{\underbrace{\phantom{XXxx}}} \nonumber \end{aligned}$$ As $F_2$ is a factor (resp. semifactor) set of $M''$ and $\left[\begin{smallmatrix} B & C \\ 0 & E' \end{smallmatrix}\right]$ has linearly independent rows, then its complementary in $M''$ $\left[\begin{smallmatrix} 0 & F''\end{smallmatrix}\right]$ is FCmR (see Lemma \[RLItoFRC\]). Which is impossible because FCmR ACI-matrices can not have columns full of zeros. - Suppose $r_1=0$ and $t_2=0$. Then $M''=M'$ (see (\[M’\])). As $\left[\begin{smallmatrix} B & C \\ 0 & E \end{smallmatrix}\right]$ has linearly independent rows then $F_2$ is not a factor (resp. semifactor) set of $M''$. Contradiction. As we explained before, Theorem \[2semifactor-disjointcase\] together with Theorem \[2non-overlappingCupAndCapFS\] add up to the following result. \[cor6\] The intersection and the union of two factor (resp. semifactor) sets of an ACI-matrix are factor (resp. semifactor) sets. The WST-decomposition for ACI-matrices {#WST} ====================================== Note that the set of factor sets of a non FmR ACI-matrix is a partial order set where the order is given by set inclusion. Indeed, Theorem \[cor6\] tells us that this set is a lattice. Since it is a finite lattice then it is bounded. So there is a factor set that is the maximum or [top factor set]{}: the union of all factor sets which we will denote $F_{\top}$. And there is another factor set that is the minimum or [bottom factor set]{}: the intersection of all factor sets which we will denote $F_{\bot}$. The previous paragraph is also valid when we substitute factor sets of a non FmR ACI-matrix by semifactor sets of a FmR ACI-matrix. \[existance&uniqueness\] For any ACI-matrix $M$ there exists a nonsingular $R$ and a permutation matrix $Q$ such that $M$ can be decomposed as follows: $$\begin{aligned} \label{ACI-decompositionABC} RMQ= \left[\begin{array}{ccc} \bf{W} & * & * \\ 0 & \bf{S} & * \\ 0 & 0 & \bf{T} \\ \end{array}\right] \end{aligned}$$ where ${\bf W}$ is a wide FmR or void, ${\bf S}$ is square FmR or void, and ${\bf T}$ is a tall FmR or void. Moreover, the ACI-matrices ${\bf W}$, ${\bf S}$ and ${\bf T}$ in decomposition (\[ACI-decompositionABC\]) are unique up to equivalence if we impose that ${\bf S}$ is as large as possible for such a decomposition. Recall that when $M$ is not FmR (resp. $M$ is FmR) then it has at least one factor (resp. semifactor) set. Then this factor (resp. semifactor) set provides a decomposition of type (\[ACI-decompositionABC\]) where ${\bf S}$ is void. In this way the existence is solved in a trivial way, but we want to be more demanding and give the decompositions where the ACI-submatrix ${\bf S}$ is as large as possible because these decompositions will lead to uniqueness. How do we find such decompositions? Suppose we are given a decomposition as in (\[ACI-decompositionABC\]) where ${\bf W}$ is a wide FmR or void, ${\bf S}$ is square FmR or void, and ${\bf T}$ is a tall FmR or void. Let $F_1$ be the set of columns corresponding to ${\bf W}$, and $F_2$ be the set of columns corresponding to $\left[\begin{smallmatrix} {\bf W}& * \\ 0 & {\bf S} \end{smallmatrix}\right]$. It is easy to check that the $2\times 2$ block partition $\scriptsize\left[\begin{array}{c\|cc} \bf{W} & * & * \\ \hline 0 & \bf{S} & * \\ 0 & 0 & \bf{T} \\ \end{array}\right]$ is an $F_1$-decomposition, and $\scriptsize\left[\begin{array}{cc\|c} \bf{W} & * & * \\ 0 & \bf{S} & * \\ \hline 0 & 0 & \bf{T} \\ \end{array}\right]$ is an $F_2$-decomposition. Note that the order of $\bf{S}$ corresponds to the difference $\#F_2-\#F_1$. If we take $F_1=F_{\bot}$ and $F_2=F_\top$ then we will see (Existance) that we obtain a decomposition of type (\[ACI-decompositionABC\]). Moreover, as $F_\top$ is the union of all factor (resp. semifactor) sets it is the largest and is unique, and as $F_\bot$ is the intersection of all factor (resp. semifactor) sets it is the smallest and is unique. Since we are taking the extreme sizes, then we will obtain the largest possible order for ${\bf S}$. Existence. : Let $M$ be an $m\times n$ ACI-matrix. We will make a systematic analysis to be sure that nothing wrong happens even when some of the ACI-submatrices become degenerate or void: $M$ is FmR. : We divide the proof into three cases: $M$ is tall. : As we saw after Definition \[semifactorSetDefinition\] the empty set is a semifactor set, so $F_\bot=\emptyset$. Without loss of generality we can assume that $F_\top=\{1,\ldots,k\} $ with $0\leq k \leq n$. Three subcases are possible: (a) $\emptyset=F_\bot=F_{\top}$. Take ${\bf W}$ void, ${\bf S}$ void, and ${\bf T}=M$ . (b) $\emptyset=F_\bot \subsetneq F_{\top} \subsetneq \{1,\ldots,n\}$. We do a sweep from bottom to top in $M$ with respect to $F_\top$ and after reordering the rows we obtain $$\begin{aligned} & \phantom{XXX} \overset{F_\top}{\overbrace{\phantom{XX}}} \\[-5mm] & M\sim \left[\begin{array}{cc} A & B \\ 0 & C\\ \end{array}\right] \hspace{-2mm} \nonumber \end{aligned}$$ where $A$ is square with linearly independent rows and $C$ is tall (see Proposition \[corBig\]). And from Lemma \[RLItoFRC\] $A$ is FmR and $C$ is FCmR. Take ${\bf W}$ is void, ${\bf S}=A$ and ${\bf T}=C$. (c) $\emptyset=F_\bot \subset F_{\top} = \{1,\ldots,n\}$. We do a sweep from bottom to top in $M$ with respect to $F_\top$ and after reordering the rows we obtain $M\sim \left[\begin{smallmatrix} A \\ 0 \end{smallmatrix}\right]$ were $A$ is square (see Proposition \[corBig\]) and has linearly independent rows. And from Lemma \[RLItoFRC\]) $A$ is FmR. Take ${\bf W}$ void, ${\bf S}=A$ and ${\bf T}$ tall degenerate. $M$ is wide. : As we saw after Definition \[semifactorSetDefinition\] the set $\{1,\ldots,n\}$ is a semifactor set, so $F_\top=\{1,\ldots,n\}$. Without loss of generality we can assume that $F_\bot=\{1,\ldots,k\} $ with $1\leq k \leq n$. Note that $F_{\bot}=\emptyset$ is not possible since it will not generate a Medium zero block. Three subcases are possible: (a) $\emptyset \neq F_\bot = F_{\top} = \{1,\ldots,n\}$. Take ${\bf W}=M$, ${\bf S}$ void and ${\bf T}$ void. (b) $\emptyset \neq F_\bot \subsetneq F_{\top} = \{1,\ldots,n\}$ and all the entries of the columns corresponding to $F_{\bot}$ are equal to zero. Then $M=\left[\begin{smallmatrix} 0 & C\end{smallmatrix}\right]$ where $C$ is square (see Proposition \[corBig\]) and FmR. Take ${\bf W}$ wide degenerate, ${\bf S}=C$ and ${\bf T}$ void. (c) $ \emptyset \neq F_\bot \subsetneq F_{\top} = \{1,\ldots,n\}$ and not all the entries of the columns corresponding to $F_{\bot}$ are equal to zero. We do a sweep from bottom to top in $M$ with respect to $F_\bot$ and after reordering the rows we obtain $$\begin{aligned} & M\sim \left[\begin{array}{cc} A& B \\ 0 & C\\ \end{array}\right] \hspace{-2mm} \\[-4mm] & \phantom{XXx} \underset{\sigma_{F_\bot}(F_\bot)}{\underbrace{\phantom{Xx}}} \nonumber \end{aligned}$$ were $A$ is wide with linearly independent rows and $C$ is square (see Proposition \[corBig\]). And from Lemma \[RLItoFRC\] $A$ is FRmR and $C$ is FmR. Take ${\bf W}=A$, ${\bf S}=C$ and ${\bf T}$ void. $M$ is square. : Take ${\bf W}$ void, ${\bf S}=M$ and ${\bf T}$ void. $M$ is not FmR. : Then $M$ has factor sets. Note that $F_{\bot}=\emptyset$ is not possible since it will not generate a Big zero block. Without loss of generality we can assume that $$F_\bot=\{1,\ldots,h\} \quad \text{and} \quad F_\top=\{1,\ldots,k\}$$ with $0< h \leq k \leq n$. We consider four cases: (1) $\mathbf{0< h < k < n}$. We distinguish two possibilities: (a) Not all the entries of the columns corresponding to $F_{\bot}$ are equal to zero. We do a sweep from bottom to top in $M$ with respect to $F_\top$ and after reordering the rows we obtain $$\begin{aligned} & \phantom{XXXXXX} \overset{F_\top}{\overbrace{\phantom{XXXx}}} \nonumber \\[-4mm] & M\sim M'=\left[\begin{array}{cc\|c} A & B & C \\ \cline{1-3} 0 & 0 & D \\ \end{array}\right] \hspace{-2mm} \\[-3.5mm] & \phantom{XXXXXX} \underset{F_\bot}{\underbrace{\phantom{XX}}} \nonumber \end{aligned}$$ where $\left[\begin{smallmatrix} A & B\end{smallmatrix}\right]$ has linearly independent rows. By Lemma \[RLItoFRC\] $\left[\begin{smallmatrix} A & B\end{smallmatrix}\right]$ is FRmR and $D$ is FCmR. As $M'$ is obtained from $M$ without permuting columns, then $F_\bot$ and $F_\top$ are factor sets of $M'$. Now we do a sweep from bottom to top in $M'$ with respect to $F_\bot$ and after reordering the rows we obtain $$\begin{aligned} \label{M''} & \phantom{XXXXXXX} \overset{F_\top}{\overbrace{\phantom{XXXx}}} \nonumber \\[-4mm] & M' \sim M''=\left[\begin{array}{cc\|c} A'' & B'' & C'' \\ 0 & B' & C' \\ \cline{1-3} 0 & 0 & D \\ \end{array}\right] \hspace{-2mm} \\[-3.5mm] & \phantom{XXXXXXX} \underset{F_\bot}{\underbrace{\phantom{XX}}} \nonumber \end{aligned}$$ where $A''$ has linearly independent rows. By Lemma \[RLItoFRC\] $A''$ is FRmR and $\left[\begin{smallmatrix} B' & C' \\ 0 & D \\ \end{smallmatrix}\right]$ is FCmR. And thus $B'$ is FCmR. On the other hand, as $\left[\begin{smallmatrix} A & B \end{smallmatrix}\right]$ is FRmR and $\left[\begin{smallmatrix} A & B \end{smallmatrix}\right] \sim \left[\begin{smallmatrix} A'' & B'' \\ 0 & B' \end{smallmatrix}\right]$ then $\left[\begin{smallmatrix} A'' & B'' \\ 0 & B' \end{smallmatrix}\right]$ is also FRmR. And thus $B'$ is FRmR. Since $B'$ is FRmR and FCmR then it must be square FmR. Take ${\bf W}=A''$, ${\bf S}=B'$ and ${\bf T}=D$. (b) All the entries of the columns corresponding to $F_{\bot}$ are equal to zero. Then we proceed as in case (1)(a) although now it is not necessary to perform the second sweep. We finish with $M'=\left[\begin{smallmatrix} 0 & B & C \\ 0 & 0 & D \end{smallmatrix}\right]$. Take ${\bf W}$ wide degenerate, ${\bf S}=B$ and ${\bf T}=D$. (2) $\mathbf{0< h < k = n}$. We distinguish two possibilities: (a) Not all the entries of the columns corresponding to $F_{\bot}$ are equal to zero. The argument is the same as in the case (1)(a) although in this case the last column does not appear as $F_{\top}=\{1,\ldots, n\}$. So $M'=\left[\begin{smallmatrix} A & B \\ 0 & 0 \end{smallmatrix}\right]$ and $M''=\left[\begin{smallmatrix} A'' & B'' \\ 0 & B' \\ 0 & 0 \end{smallmatrix}\right]$. Take ${\bf W}=A''$, ${\bf S}=B'$ and ${\bf T}$ tall degenerate. (b) All the entries of the columns corresponding to $F_{\bot}$ are equal to zero. Then we proceed as in case (2)(a) although now it is not necessary to perform the second sweep. So we finish with $M'=\left[\begin{smallmatrix} 0 & B \\ 0 & 0 \end{smallmatrix}\right]$. Take ${\bf W}$ wide degenerate, ${\bf S}=B$, and ${\bf T}$ tall degenerate. (3) $\mathbf{0< h = k < n}$. We distinguish two possibilities: (a) Not all the entries of the columns corresponding to $F_{\bot}$ are equal to zero. The argument is the same as (1)(a) although in this case the second column does not appear as $F_{\bot}=F_{\top}$. Then only one sweep is necessary. So $M'=\left[\begin{smallmatrix} A & C \\ 0 & D \end{smallmatrix}\right]$. Take ${\bf W}=A$, ${\bf S}$ void and ${\bf T}=D$. (b) All the entries of the columns corresponding to $F_{\bot}$ are equal to zero. Then $M=\left[\begin{smallmatrix} 0 & D\end{smallmatrix}\right]$. Take ${\bf W}$ wide degenerate, ${\bf S}$ void, and ${\bf T}=D$. (4) $\mathbf{0< h = k = n}$. We distinguish two possibilities: (a) Not all the entries of the columns corresponding to $F_{\bot}$ are equal to zero. The argument is the same as (3)(a) although in this case the last column does not appear as $F_{\top}=\{1,\ldots, n\}$. Then only one sweep is necessary. So $M'=\left[\begin{smallmatrix} A \\ 0 \end{smallmatrix}\right]$. Take ${\bf W}=A$, ${\bf S}$ void and ${\bf T}$ tall degenerate. (b) All the entries of the columns corresponding to $F_{\bot}$ are equal to zero. Then $M=\left[\begin{smallmatrix} 0 \end{smallmatrix}\right]$. Take ${\bf W}$ wide degenerate, ${\bf S}$ void, and ${\bf T}$ tall degenerate. Uniqueness up to equivalence of $\bf{W}$, $\bf{S}$ and $\bf{T}$. : We will do the FmR case and the non FmR case together. Actually, we will do all subcases that were studied in the Existence part together, since at this point to adapt the general argument to the different subcases should be straightforward (for example, some of the submatrices $P$, $P'$ or $P''$ involved in (\[wsca\]) can be void). So assume that we have two different decompositions $$\begin{aligned} & \phantom{XXXXXX} \overset{F_\top}{\overbrace{\phantom{XXXXx}}} \hspace{53mm} \overset{F_\top}{\overbrace{\phantom{XXXxx}}} \\[-5mm] & R_1MQ_1=\left[\begin{array}{ccc} \mathbf{W}_1 & * & * \\ 0 & \mathbf{S}_1 & * \\ 0 & 0 & \mathbf{T}_1 \end{array}\right] \quad \text{ and }\quad R_2MQ_2=\left[\begin{array}{ccc} \mathbf{W}_2 & * & * \\ 0 & \mathbf{S}_2 & * \\ 0 & 0 & \mathbf{T}_2 \\ \end{array}\right]\hspace{-2mm} \\[-4mm] & \phantom{XXXXXX} \underset{F_\bot}{\underbrace{\phantom{XX}}} \hspace{63mm} \underset{F_\bot}{\underbrace{\phantom{XX}}} \end{aligned}$$ where $R_1$ and $R_2$ are nonsingular matrices, $Q_1$ and $Q_2$ are permutation matrices, $\mathbf{W}_1$ and $\mathbf{W}_2$ are wide FmR or void, $\mathbf{S}_1$ and $\mathbf{S}_2$ are square FmR or void, and $\mathbf{T}_1$ and $\mathbf{T}_2$ are tall FmR or void. Then $$\begin{matrix} \hspace{-20mm} \overset{F_\top}{\overbrace{\phantom{XXXxx}}} \phantom{XXx} \phantom{XXXXXXs} \overset{F_\top}{\overbrace{\phantom{XXXxx}}} \\[-4mm] \left[\begin{array}{ccc} \mathbf{W}_1 & * & * \\ 0 & \mathbf{S}_1 & * \\ 0 & 0 & \mathbf{T}_1 \end{array}\right] = R_1 R_2^{-1}\left[\begin{array}{ccc} \mathbf{W}_2 & * & * \\ 0 & \mathbf{S}_2 & * \\ 0 & 0 & \mathbf{T}_2 \\ \end{array}\right] Q_2^{-1} Q_1. \hspace{-2mm} & \\[-3.5mm] \hspace{-30mm} \underset{F_\bot}{\underbrace{\phantom{XX}}} \phantom{XXXXx} \phantom{XXXXXXx} \underset{F_\bot}{\underbrace{\phantom{XX}}} \end{matrix}$$ Note that the three groups of columns $F_\bot$, $F_\top \setminus F_\bot$ and $\{1,\ldots, n\}\setminus F_\top$ do not change of position. But the columns of each group might get permuted so there are three permutation matrices ($P$ of order $\#F_{\bot}$, $P'$ of order $\#F_{\top}-\#F_{\bot}$, and $P''$ of order $n-\#F_{\top}$) such that $Q_2^{-1} Q_1=\left[\begin{smallmatrix} P & 0 & 0 \\ 0 & P' & 0 \\ 0 & 0 & P'' \end{smallmatrix}\right]$. So $$\begin{aligned} \label{wsca} \left[\begin{array}{c\|cc} \mathbf{W}_1 & * & * \\ \cline{1-3} 0 & \mathbf{S}_1 & * \\ 0 & 0 & \mathbf{T}_1 \end{array}\right] = R_1 R_2^{-1} \left[\begin{array}{c\|cc} \mathbf{W}_2 & * & * \\ \cline{1-3} 0 & \mathbf{S}_2 & * \\ 0 & 0 & \mathbf{T}_2 \\ \end{array}\right] \left[\begin{array}{c\|cc} P & 0 & 0 \\ \cline{1-3} 0 & P' & 0 \\ 0 & 0 & P'' \end{array}\right]\end{aligned}$$ where the lines define $2\times 2$ block ACI-matrices: we consider $\left[\begin{smallmatrix} \mathbf{S}_1 & * \\ 0 & \mathbf{T}_1 \end{smallmatrix}\right]$, $\left[\begin{smallmatrix} \mathbf{S}_2 & * \\ 0 & \mathbf{T}_2 \end{smallmatrix}\right]$ and $\left[\begin{smallmatrix} P' & 0 \\ 0 & P'' \end{smallmatrix}\right]$ as just one block. Since $\mathbf{W}_1$ and $\mathbf{W}_2$ are wide FRmR then they have linear independent rows and Lemma \[TwoDecompDivisor2\] implies that $$\begin{aligned} \label{a1anda2Etc} \mathbf{W}_1\sim \mathbf{W}_2 \quad \text{and} \quad \begin{bmatrix} \mathbf{S}_1 & * \\ 0 & \mathbf{T}_1 \end{bmatrix} \sim \begin{bmatrix} \mathbf{S}_2 & * \\ 0 & \mathbf{T}_2 \end{bmatrix}.\end{aligned}$$ In the proof of Lemma \[TwoDecompDivisor2\] we saw that $R_1 R_2^{-1}=\left[\begin{smallmatrix} R & * \\ 0 & R' \end{smallmatrix}\right]$ where $R$ is nonsingular of order $\operatorname{rows}(\mathbf{W}_2)$ and $R'$ is nonsingular of order $\operatorname{rows}(\mathbf{S}_2)+\operatorname{rows}(\mathbf{T}_2)$ with $$\begin{bmatrix} \mathbf{S}_1 & * \\ 0 & \mathbf{T}_1 \end{bmatrix} = R' \begin{bmatrix} \mathbf{S}_2 & * \\ 0 & \mathbf{T}_2 \end{bmatrix} \begin{bmatrix} P' & 0 \\ 0 & P'' \end{bmatrix}.$$ Since $\mathbf{S}_1$ and $\mathbf{S}_2$ are square FmR then they have linear independent rows and Lemma \[TwoDecompDivisor2\] implies that $\mathbf{S}_1\sim \mathbf{S}_2$ and $\mathbf{T}_1\sim \mathbf{T}_2$. Which finishs the uniqueness part. Since the decomposition of Theorem \[existance&uniqueness\] involve a Wide (or void) $\bf{W}$, a Square (or void) $\bf{S}$ and a Tall (or void) $\bf{T}$, we will denote this decomposition the WST-decomposition for ACI-matrices whenever $\bf{S}$ is as large as possible. ACI-matrices of constantRank ============================ As a application of the WST-decomposition we will refine the main theorem of [@MR2775784 Theorem 5] by Huang and Zhan, which is a characterization of constantRank ACI-matrices. The version of the theorem that will be given below makes the degenerate cases much more explicit than the original one. The same version of the theorem was already used in our work on ACI-matrices [@BoCa2] as “Theorem 2.4 (detailed version)”. ([@MR2775784 Theorem 5], see also [@BoCa2 Theorem 2.4 (detailed version)]) \[HZCharacterization-1\] Let $M$ be a $m\times n$ ACI-matrix of constantRank $\rho$ with $1 \leq \rho \leq \min \{m, n\}$ over a field $\mathbb{F}$ with $\|\mathbb{F}\|\geq \max\{m,n+1\}$. Depending on $m$, $n$ and $\rho$ we have the following possibilities: (i) $\rho=m<n$ if and only if $ M \sim \left[\begin{smallmatrix} \vspace{-3mm}1 & & * &\|& \\ \vspace{-1mm} &\ddots & &\| &\\ 0 & & 1 &\|& \end{smallmatrix}\right]$. (ii) $\rho=m=n$ if and only if $M \sim \left[\begin{smallmatrix} \vspace{-2mm} 1 & & \\ & \ddots & \\ 0 & & 1 \end{smallmatrix}\right].$ (iii) $\rho=n<m$ if and only if $ M \sim \left[\begin{smallmatrix} \vspace{-1mm} & * & \\ \cline{1-3} \vspace{-2mm} 1 & & \\ &\ddots & \\ 0 & & 1 \end{smallmatrix}\right]$. (iv) $1\leq \rho<\min\{m,n\}$ if and only if for some positive integers $r$ and $s$ with $r+s=m+n-\rho$ $$\label{HZtype} M \sim {\footnotesize \left[\begin{array}{cccc\|ccc} 1 & & \multicolumn{1}{c\|}{} & \multicolumn{1}{c\|}{\multirow{3}{}{\Large $\ \ $}} & \multicolumn{3}{c}{\multirow{3}{}{\Large $$}} \\ & \ddots & \multicolumn{1}{c\|}{} & \\ 0 & & \multicolumn{1}{c\|}{1} & & & & \\ \hline \multicolumn{4}{c\|}{\multirow{4}{}{\Large$0_{r\times s}$}} & \multicolumn{3}{c}{\multirow{1}{}{\Large $$}}\\ &&&&&& \\ \cline{5-7} & & & & 1 & & \\ & & & & & \ddots & \\ & & & & 0 & & 1 \\ \end{array}\right]}$$ where the upper blocks do not appear if $r=m$ and the right blocks do not appear if $s=n$. Finally, we present the refinement of Theorem \[HZCharacterization-1\] that we were talking about. \[maincorollary\] Let $M$ be an $m\times n$ ACI-matrix $M$ over a field $\mathbb{F}$ such that $\|\mathbb{F}\|\geq \max\{m,n+1\}$. Then $M$ is constantRank if and only if there exist a nonsingular $R$ and a permutation $Q$ such that $$\begin{aligned} \label{RefinamentTheorem71} RMQ= {\footnotesize\begin{bmatrix} \begin{array}{\|ccc\|c\|}\cline{1-4} 1 & & * & \\ &\ddots & & * \\ 0 & & 1 & \\ \cline{1-4} \end{array} & * & * \\ 0 & \hspace{-2.4ex}\begin{array}{\|ccc\|}\cline{1-3} 1 & & * \\ & \ddots & \\ 0 & & 1 \\ \cline{1-3} \end{array} & * \\ 0 & 0 & \hspace{-2.4ex} \begin{array}{\|ccc\|}\cline{1-3} & * & \\ \cline{1-3} 1 & & \\ & \ddots & \\ 0 & & 1 \\ \cline{1-3} \end{array} \end{bmatrix}}.\end{aligned}$$ where instead of the ACI-submatrix $\left[\begin{smallmatrix} \vspace{-3mm}1 & & * &\|& \\ \vspace{-1mm} &\ddots & &\| &\\ 0 & & 1 &\|& \\ \end{smallmatrix}\right]$ there could be a wide degenerate or a void, instead of $\left[\begin{smallmatrix} \vspace{-2mm}1 & & \\ &\ddots & \\ 0 & & 1 \\ \end{smallmatrix}\right]$ there could be a void, and instead of $\left[\begin{smallmatrix} \vspace{-1mm} & * & \\ \cline{1-3} \vspace{-2mm} 1 & & \\ &\ddots & \\ 0 & & 1 \\ \end{smallmatrix}\right]$ there could be a tall degenerate or a void. If we impose that $\left[\begin{smallmatrix} \vspace{-2mm}1 & & \\ &\ddots & \\ 0 & & 1 \\ \end{smallmatrix}\right]$ is as large as possible then the three blocks are unique up to equivalence. The sufficiency is obvious. So we proceed with the necessity. Let $R'$ be a nonsingular constant matrix and $Q'$ a permutation matrix such that $$\begin{aligned} R'MQ'= \begin{bmatrix} {\bf W}& * & * \\ 0 & {\bf S}& * \\ 0 & 0 & {\bf T} \end{bmatrix}\end{aligned}$$ is a WST-decomposition of $M$ where: ${\bf W}$ is wide FRmR or void, ${\bf S}$ is square FmR or void, ${\bf T}$ is tall FCmR or void, and ${\bf S}$ is as large as possible. Therefore $$\operatorname{maxRank}({\bf W})=\operatorname{rows}({\bf W}),\quad \operatorname{maxRank}({\bf S})=\operatorname{rows}({\bf S})\quad \text{and} \quad \operatorname{maxRank}({\bf T})=\operatorname{cols}({\bf T}).$$ Moreover, as $M$ is constantRank then $R'MQ'$ is also constantRank and then $$\begin{aligned} \label{minRank=maxRankEc1} \operatorname{minRank}\left[\begin{smallmatrix}{\bf W}& * & \\ 0 & {\bf S}& \\ 0 & 0 & {\bf T}\end{smallmatrix}\right]=\operatorname{maxRank}\left[\begin{smallmatrix}{\bf W}& * & \\ 0 & {\bf S}& \\ 0 & 0 & {\bf T}\end{smallmatrix}\right] =\operatorname{rows}({\bf W})+\operatorname{rows}({\bf S})+\operatorname{cols}({\bf T}).\end{aligned}$$ We will prove that $M$ is equivalent to an ACI-matrix as in (\[RefinamentTheorem71\]) in three steps. We will assume that ${\bf W}, {\bf S}$ and ${\bf T}$ are not void nor degenerate, otherwise the proof simplifies. (i) First we will prove that ${\bf W}$, ${\bf S}$ and ${\bf T}$ are full constantRank, that is: $$\begin{aligned} \operatorname{minRank}({\bf W}) &=\operatorname{maxRank}({\bf W}), \label{minRank=maxRankW}\\ \operatorname{minRank}({\bf S}) &=\operatorname{maxRank}({\bf S}) \text{ and }\label{minRank=maxRankEc-1} \\ \operatorname{minRank}({\bf T}) &=\operatorname{maxRank}({\bf T}). \label{minRank=maxRankEc0}\end{aligned}$$ Suppose that $\operatorname{minRank}({\bf T})<\operatorname{maxRank}({\bf T})$. Let $\left[\begin{smallmatrix}\widehat{{\bf W}} & * & * \\ 0 & \widehat{{\bf S}} & * \\ 0 & 0 & \widehat{{\bf T}}\end{smallmatrix}\right]$ be a completion of $\left[\begin{smallmatrix}{\bf W}& * & \\ 0 & {\bf S}& \\ 0 & 0 & {\bf T}\end{smallmatrix}\right]$ such that $\operatorname{rank}(\widehat{{\bf T}})<\operatorname{maxRank}({\bf T})=\operatorname{cols}({\bf T})$. Then $$\begin{aligned} \label{minRank=maxRankEc2}\nonumber \operatorname{minRank}\left[\begin{smallmatrix}{\bf W}& * & \\ 0 & {\bf S}& \\ 0 & 0 & {\bf T}\end{smallmatrix}\right] &\leq \operatorname{rank}\left[\begin{smallmatrix}\widehat{{\bf W}} & * & * \\ 0 & \widehat{{\bf S}} & * \\ 0 & 0 & \widehat{{\bf T}}\end{smallmatrix}\right]\leq\operatorname{rows}(\widehat{{\bf W}})+\operatorname{rows}(\widehat{{\bf S}})+\operatorname{rank}(\widehat{{\bf T}})\\ &<\operatorname{rows}({\bf W})+\operatorname{rows}({\bf S})+\operatorname{cols}({\bf T}).\end{aligned}$$ As (\[minRank=maxRankEc1\]) and (\[minRank=maxRankEc2\]) are contradictory then (\[minRank=maxRankEc0\]) is true. Similar arguments prove (\[minRank=maxRankW\]) and (\[minRank=maxRankEc-1\]). (ii) Now, we apply Theorem \[HZCharacterization-1\] $(i)$ to ${\bf W}$ to obtain a nonsingular $R_1$ and a permutation $Q_1$ such that $R_1 {\bf W}Q_1= \left[\begin{smallmatrix} \vspace{-3mm}1 & & * &\|& \\ \vspace{-1mm} &\ddots & &\| &\\ 0 & & 1 &\|& \\ \end{smallmatrix}\right]$. We apply Theorem \[HZCharacterization-1\] $(ii)$ to ${\bf S}$ to obtain a nonsingular $R_2$ and a permutation $Q_2$ such that $R_2 {\bf S}Q_2= \left[\begin{smallmatrix} \vspace{-2mm}1 & & \\ &\ddots & \\ 0 & & 1 \\ \end{smallmatrix}\right]$. And we apply Theorem \[HZCharacterization-1\] $(iii)$ to ${\bf T}$ to obtain a nonsingular $R_3$ and a permutation $Q_3$ such that $R_3 {\bf T}Q_3= \left[\begin{smallmatrix} \vspace{-1mm} & * & \\ \cline{1-3} \vspace{-2mm} 1 & & \\ &\ddots & \\ 0 & & 1 \\ \end{smallmatrix}\right]$. (iii) Finally, if $R:=\left[\begin{smallmatrix} R_1 & 0 & 0\\ 0 &R_2 & 0 \\ 0 & 0 & R_3 \end{smallmatrix}\right]R'$ and $Q=Q' \left[\begin{smallmatrix} Q_1 & 0 & 0\\ 0 &Q_2 & 0 \\ 0 & 0 & Q_3 \end{smallmatrix}\right]$ then we obtain the desired result since $$\begin{aligned} \small RMQ &= \begin{bmatrix} R_1 & 0 & 0\\ 0 &R_2 & 0 \\ 0 & 0 & R_3 \end{bmatrix}R' M Q' \begin{bmatrix} Q_1 & 0 & 0\\ 0 &Q_2 & 0 \\ 0 & 0 & Q_3 \end{bmatrix}= \begin{bmatrix} R_1 & 0 & 0\\ 0 &R_2 & 0 \\ 0 & 0 & R_3 \end{bmatrix}\begin{bmatrix} {\bf W}& & * \\ 0 & {\bf S}& * \\ 0 & 0 & {\bf T} \end{bmatrix} \begin{bmatrix} Q_1 & 0 & 0\\ 0 &Q_2 & 0 \\ 0 & 0 & Q_3 \end{bmatrix}\\ & = \begin{bmatrix} R_1 {\bf W}Q_1 & * & * \\ 0 & R_2 {\bf S}Q_2 & * \\ 0 & 0 & R_3 {\bf T}Q_3 \end{bmatrix}\end{aligned}$$ is an ACI-matrix of type (\[RefinamentTheorem71\]) where $R_2 {\bf S}Q_2$ is as large as possible. The characterization of constantRank ACI-matrices of Theorem \[HZCharacterization-1\] has a caveat: there is a restriction on the number of elements of the field that can not be avoided. In [@BoCa2 Theorem 2.5.] we extended the characterization without any restriction on the field. It is possible to apply the WST-decomposition to refine this extension analogously. [1]{} A. Borobia, R. Canogar, Nonsingular ACI-matrices over integral domains. , 436:4311–4316, 2012. A. Borobia, R. Canogar, Characterization of full rank ACI-matrices over fields. , 439:3752–3762, 2013. A. Borobia, R. Canogar, ACI-matrices of constant rank over arbitrary fields. , 527:232–259, 2017. R. Brualdi, Z. Huang, X. Zhan, Singular, nonsingular, and bounded rank completions of ACI-matrices. , 433:1452–1462, 2010. N. Cohen, C.R. Johnson, L. Rodman, H. Woerdeman, Ranks of completions of partial matrices. , 40:165–185,1989. Z. Huang, X. Zhan, ACI-matrices all of whose completions have the same rank, , 434:1956–1967, 2011. J. McTigue, R. Quinlan, Partial matrices of constant rank. , 446:177–191, 2014. [^1]: [Keywords:]{} Partial matrix; ACI-matrix; Completion problem; Rank; Matrix decomposition. [^2]: [Mathematics subject classification:]{} 15A83 [^3]: Supported by the Spanish Ministerio y Tecnología MTM2017-90682-REDT.
--- abstract: 'Ni coarsening in Ni-yttria stabilized zirconia (YSZ) solid oxide fuel cell anodes is considered a major reason for anode degradation. We present a predictive, quantative modeling framework based on the phase-field approach to systematically examine coarsening kinetics in such anodes. The initial structures for simulations are experimentally acquired functional layers of anodes. Sample size effects and error analysis of contact angles are examined. Three phase boundary (TPB) lengths and Ni surface areas are quantatively identified on the basis of the active, dead-end, and isolated phase clusters throughout coarsening. Tortuosity evolution of the pores is also investigated. We find that phase clusters with larger characteristic length evolve slower than those with smaller length scales. As a result, coarsening has small positive effects on transport, and impacts less on the active Ni surface area than the total counter part. TPBs, however, are found to be sensitive to local morphological features and are only indirectly correlated to the evolution kinetics of the Ni phase.' address: - 'Department of Materials Science and Engineering, University of Michigan, 2300 Hayward St., Ann Arbor, MI 48109, USA' - 'Department of Materials Science and Engineering, Northwestern University, 2220 Campus Drive, Evanston, IL 60201, USA' author: - 'Hsun-Yi Chen' - 'Hui-Chia Yu' - 'J. Scott Cronin' - 'James R. Wilson' - 'Scott A. Barnett' - Katsuyo Thornton bibliography: - 'SOFCanode.bib' title: 'Ni coarsening in the three-phase solid oxide fuel cell anode – a phase-field simulation study' --- solid oxide fuel cell ,coarsening ,phase-field model ,three-phase boundary ,nickel ,yttria-stabilized zirconia Introduction ============ Nickel coarsening in SOFC anodes -------------------------------- Solid oxide fuel cells (SOFCs) are one of the most promising clean energy conversion devices for stationary applications because of their low pollutant emissions, high efficiency, and ability to operate using various hydrocarbon fuels. The need for precious-metal catalysts is eliminated in SOFCs because the reaction kinetics is enhanced at their operating temperatures, which are between 500 and 1000$^\circ$C [@Atkinson]. However, high operating temperatures also lead to disadvantages such as slow startup, high fabrication costs, and rapid component degradation [@SMHaile]. For SOFCs to be commercially viable for stationary applications, their lifetime must meet a minimum of $\sim$50,000 hours. This goal has not yet been achieved. Understanding SOFC degradation mechanisms is therefore crucial to improving their durability. The degradation mechanisms of SOFCs have been reviewed in a few articles [@Yokokawa08; @TuJPS04]. SOFC degradation is usually evaluated in terms of cell power or voltage decrease or area-specific resistance (ASR) increase, which can be determined by the electrochemical impedance spectroscopy (EIS). These electrochemical methods allow the cell degradation to be monitored in situ. However, even though a few studies [@Hagen1; @Hagen2] have attempted to address SOFC degradation, it is very difficult to unambiguously characterize the impacts of individual mechanisms using these techniques. Many non-electrochemical methods have been used to identify and monitor the degradation and failure mechanisms of SOFCs in situ or pre-/post-operation (for a review, see Ref. [@Malzbender09]), including X-ray tomography and X-ray diffraction. Among these degradation mechanisms, the microstructural change in a Ni-based cermet anode is one of the least understood, because experiments that can provide detailed time-dependent, three-dimensional (3D) structural information are difficult. The anode of SOFCs is usually made of a composite material of complex morphology that facilitates electrochemical reactions, which require simultaeous transport of fuels, ions, and electrons. The most commonly used anode material up to date is a porous cermet comprised of Ni and YSZ. Electrochemical reactions at the SOFC anode mainly occur at the vicinity of TPBs, where the pore, Ni, and YSZ phases are in contact [@Mogensen]. Although the electrochemical reaction mechanisms are not very well understood, the length of TPBs is considered one of the most important geometrical parameters that dictates the resistance in a SOFC anode [@SSunde]. The anode is thus designed to have a complex microstructure and increase the TPB lengths and the simultaneous transport of three different species. However, this intricate anode microstructure is typically not stable. The agglomeration and coarsening of nickel particles have been considered the major mechanisms responsible for microstructural change in SOFC anodes [@Simwonis]. Ni coarsening in the SOFC anode is a capillarity-driven phenomenon. Regions with high curvatures have higher chemical potentials than those with lower curvatures in accordance with the Gibbs-Thomson effect. Materials will therefore be transported from higher- to lower-curvature regions when mobility is sufficiently large for the time scale of interest. This phenomenon increases the resistance in SOFCs and results in cell degradation. Long-term coarsening experiments have been conducted for thousands of hours to study Ni coarsening in Ni-YSZ anodes. Simwonis et al. [@Simwonis] measured a $33\%$ decrease in electrical conductivity after 4000-hour exposure of a Ni-YSZ anode to a H$_2$ environment at 1000$^\circ$C. They also found a $26\%$ average Ni particle-size increase via analysis of micrographs of cross-sections. Thyden et al. [@Thyden08; @Hagen1] performed an aging experiment of a SOFC over $17,500$ hours. The cell was operated at 850$^\circ$C with an initial current density of 1 A/cm$^2$. Optical microscopy, field emission-scanning electron microscopy (FE-SEM), SEM-charge contrast (SEM-CC), focused ion beam (FIB)-SEM, and EIS measurements were utilized to analyze the microstructural evolution in the Ni-YSZ anode. Their results suggested that an increase in the H$_2$O concentration can promote Ni particle coarsening and lead to conductivity loss within the Ni-YSZ cermet. Tanasini et al. [@Tanasini09] also conducted coarsening experiments for a single SOFC operated at 850$^\circ$C with humidified H$_2$. Although cell degradation is often attributed to cathodic processes (e.g., Cr poisoning and cathode-electrolyte interface formation reaction), they reported that the major cell performance reduction stems from the anode degradation due to coarsening. They also found that the cell potential drop and Ni particle size increase reach a plateau after $\sim 1000$ hours of operation. Despite these experimental efforts, a quantitative correlation between microstructure and coarsening still awaits explication. Even though the performance of SOFCs can be largely affected by microstructural changes, only a few models have been proposed to study the effects of coarsening on the performance of electrodes [@Ioselevich; @IoselLehnert; @Vaben]. Due to the inability of acquiring 3D microstructural information for Ni-YSZ anodes, these models were mostly based on empirically fitted parameters or simplified microstructures. The use of oversimplified microstructures and empirical parameters without extensive validations can be problematic because it can lead to incorrect conclusions. Recently, we have demonstrated that the microstructure of SOFC electrodes can be three-dimensionally reconstructed using dual-beam FIB-SEM [@James]. The data can be acquired to produce a wealth of information, including the tortuosity of individual phases and the TPB density. While FIB-SEM allows the reconstruction of anode microstructure in three dimensions, this technique damages the materials. Thus, no evolution information can be obtained for the specimen after the procedure. Modeling offers the advantage of allowing systematic analyses of coarsening effects on the performance of SOFC anodes. The phenomenon of anode coarsening can be described as a free-boundary problem in the sharp-interface modeling framework; however, explicit tracking of the evolving phase boundaries is highly impracticable in three dimensions. For a 3D anode with a complex microstructure, it is therefore advantageous to use a diffuse interface approach, such as a phase-field modeling, to model the microstructural evolution. Therefore, we have developed a diffuse-interface modeling framework, based on the phase-field model and the smoothed boundary method (SBM), in conjunction with FIB-SEM experiments to quantitatively investigate Ni coarsening. The phase field model {#PFM} --------------------- The fundamental basis of a phase-field approach is to define field variables, or order parameters (OPs), that distinguish the phases in a multi-phase system. OPs have a constant value in each bulk phase, while interfaces are represented by finite regions in which the OPs smoothly vary from one bulk value to another. Since the boundary information is embedded in OPs, explicit tracking of moving boundaries is no longer necessary. This leads to a computational advantage in modeling multi-phase systems with multiple interfaces, especially in three dimensions. The phase-field model, one of the phase-field approachs, can also be considered as a type of diffuse interface model, which describes interfaces using a finite thickness. The inclusion of OPs into the free energy density is generally attributed to Landau and Ginzburg for their work related to the modeling of superconductivity in the 1950s [@Landau]. Later, in order to describe the interfacial energy of an inhomogeneous system, Cahn and Hilliard [@CahnHilliard] proposed to describe the free energy of a system by its OPs and their spatial derivatives. The concept of describing the evolution of an interface between two phases differing in composition with a Ginzburg-Landau-type functional was introduced by Langer [@JSLanger]. Allen and Cahn [@AllenCahn] developed the theory for the motion of a coherent anti-phase boundary. Formal asymptotic analyses have been used to show that a variety of phase-field models (PFMs) recover the corresponding sharp interface models when the width of the interfaces approaches zero [@WBM; @GCaginalp]. PFMs have been successfully utilized to describe phase transition in two-phase systems (see, e.g., [@WBM]), dendrite growth (see, e.g., [@Karma97]), and Ostwald ripening (see, e.g., [@Warren96]). However, in three-phase systems, a PFM with a generic free energy functional may introduce an artificial third-phase contribution at a two-phase boundary, if no special treatment is applied [@Nestler; @Folch]. This effect can compromise the study of cermet anode coarsening since the emergence of a phantom phase at a two-phase boundary can contribute to extra TPBs. This foreign-phase creation will lead to an erroneous estimation of the electrochemically active sites. Nestler [et al.]{} [@Nestler] proposed two types of remedies for this problem in a multi-component liquid-solid system. In each of these approaches, the potentials penalize equal contributions from each of the OPs to reduce the third phase appearance. However, the modified potentials may not eliminate the PFM simulation artifact for a simple three-phase conserved-field system undergoing coarsening. To address this issue, Folch [et al.]{} [@Folch] developed a specific minimal model for a three-phase system that ensures that there is no third phase invasion at a pure two-phase boundary. This model can also accommodate systems with unequal surface tensions by adding tunable saddle-lifting terms. However, the computations become too expensive due to the steeper free-energy landscape needed for large-scale simulations. The PFM handles the interface implicitly because the interface information is embedded in OPs. Immiscibility and interfacial energy are naturally incorporated in the PFM via the bulk free energy and the gradient energy penalty over the diffuse-interfacial region. The interfacial energy ratios among different interfaces can be specified in the PFM using proper parameterization of the free energy functional. Therefore, we developed a first phase-field model, named Model A, within the context of a multiphase PFM, to model the Ni coarsening in three-phase anodes [@hsunyi]. As discussed later, Model A allows a small mobility of YSZ, which significantly affects the evolution. Therefore, we developed an alternative model based on the smoothed boundary method (SBM), which is briefly described in the following section. The smoothed boundary method {#SBM} ---------------------------- Complex geometries are abundant in naturally and man-made objects. To study physical processes or phenomena occurring within such objects, the numerical solution of partial differential equations (PDEs) with prescribed boundary conditions is necessary. A standard scheme requires triangulation of the complex shapes, followed by solving of the PDEs using the finite element method (FEM). However, automatic generation of proper meshes for complex 3D domains is challenging. In addition, in many cases the complex geometries can evolve during a physical process, thus demanding a dynamic remeshing of the evolved domain. To address these difficulties, one can alternatively embed complex geometries within a larger, simpler domain (such as a cube). The PDEs can then be solved with regularly shaped meshes on the extended domain, provided that the original boundary conditions can be properly applied. Methods employing such a concept include composite FEM [@Hackbusch97], extended FEM [@Duarte2000], the immersed interface method [@Gong08], the immersed boundary method [@Mittal05], and the cut-cell method [@Ji06]. The smoothed boundary method differs from the above mentioned methods in that it represents the complex geometries with a phase-field-like function, which veries smoothly across the domain boundary. Thus, the sharp boundaries of the complex geometry $\partial \Omega$ are instead described by a thin interface of a finite width. This diffuse-domain approach was utilized to study the diffusion of chemoattractant inside a cell with a no-flux boundary condition (BC) at the cell surface [@Kockelkoren03]. Spectral methods were later coupled with the SBM to model electrical wave propagation in cardiac tissues with a no-flux BC [@BuenoOrovio05; @BuenoOrovio06]. In Ref. [@hcy09], the SBM was extended to solve PDEs in complex geometries with Dirichlet, Neumann, and Robin BCs. An alternative but similar approach was independently developed, [@XLi09]. In the SBM, an auxiliary variable, a domain parameter $\psi$, is introduced to identify the domain of interest $\Omega$ in which the PDEs are solved. The domain parameter $\psi$ usually has a value of $1$ within $\Omega$, $0$ exterior to $\Omega$, and $0< \psi <1$ in $\partial \Omega$. Since the complex geometry is embedded in a regularly shaped, expanded domain, the exterior of the domain of interest is included only to facilitate computation; any numerical solutions obtained in the external region are devoid of meaningful information. The initial construction of the domain parameter $\psi$ can be achieved by solving a phase-field equation or by transforming the distance function of the domain structure with the hyperbolic tangent. It is generally believed that the YSZ phase has a very low mobility in an operating SOFC anode; that is, YSZ serves as the supporting structure in which Ni coarsens. We therefore propose a model, named Model B, utilizing the SBM for Ni coarsening simulations. In Model B, we assume that YSZ is stationary. It is therefore treated as the geometry within which the Ni and pore phases evolve. We also assume that the triple junctions in the Ni-YSZ anode possess the contact angles deduced from Young’s equation for a locally flat surface. The dynamics of this system can thus be modeled by a single OP Cahn-Hilliard equation with two complementary BCs implemented with the SBM: the contact-angle BC at triple junctions and the no-flux BC at YSZ interfaces. Methods ======= In this section, we elaborate the model framework that we previously published in a short communication for Ni coarsening [@hsunyi]. The non-dimensionalization, asymptotic analysis, and numerical methods are discussed in detail. Moreover, the methodology we used to identify the TPBs and distinguish the percolated or isolated phases is discussed. An error analysis is performed to examine how the ratios between the characteristic length and the interfacial thickness affect the contact angles at triple junctions in our model. Model Formulation ----------------- We have proposed two PFMs for coarsening simulations in three-phase SOFC cermet anodes. Our models are based on a free energy funtional that is computationally inexpensive and can circumvent the phantom phase issue associated with some other solutions. The approximations we made in both of our models are: (1) the volume (mass) of each phase in the Ni-YSZ anode is conserved so that the Cahn-Hilliard dynamics is applicable; (2) material properties required for modeling are estimated at $1000^\circ$C; (3) Ni surface free energy is assumed to be isotropic; (4) Ni surface diffusivity is estimated by the mean value of multiple crystal orientations; (5) surface diffusion dominates. In our PFMs, the Cahn-Hilliard evolution equations can be considered as a type of diffusion equation, where the material transport is driven by the chemical potential. This potential can be formulated by the design of a free energy functional that is based on the effective driving forces for material transport. At ordinary operating temperatures, the three phases comprise the Ni-YSZ anode can be assumed immiscible; therefore, lattice misfit as well as elastic energy contribution toward the diffusion are negligible. In addition, external forces such as gravity are not considered. We thus formulate our free energy functional as the simplest Ginzburg-Landau type functional $ F = \int\limits_{V} f dV$, where the free-energy density $f$ simply depends on the OPs and their derivatives. The two PFMs we have proposed differ in their treatment of the YSZ phase. If we consider that the YSZ phase can transport with its mobility, the Ni-YSZ anode is then a system of three mobile phases (Ni, YSZ and Pore). This system requires two evolution equations to resolve the kinetics. We call this model [Model A]{}. In [Model B]{}, we consider the YSZ phase to be completely immobile, which is justified because its mobility is orders of magnitude smaller than that of the Ni phase. Therefore, only two phases, Ni and pore, are allowed to evolve in between of the YSZ matrix. To incorporate the YSZ phase as the internal boundary in the computational domain, we have developed a model utilizing the smoothed boundary method. ### Model A In Model A, the OPs are vectorized as $\overrightarrow{\phi} = (\phi_{1},\phi_{2},\phi_{3})$ for generalization. Each vector component represents the volume fraction of the corresponding phase so that an additional constraint $\phi_{1}+\phi_{2}+\phi_{3}=1$ applies; thus, only two evolution equations are needed. The governing equation sets can be written as $$\begin{array}{l} \displaystyle \frac{\partial \phi_{1}}{\partial t}=\triangledown \cdot M(\overrightarrow{\phi}) \triangledown \frac{\delta F}{\delta \phi_{1}},\\ \displaystyle \\ \displaystyle \frac{\partial \phi_{2}}{\partial t}=\triangledown \cdot M(\overrightarrow{\phi}) \triangledown \frac{\delta F}{\delta \phi_{2}}, \end{array} \label{eq:eqCH2}$$ where $M(\overrightarrow{\phi})$ is the surface mobility. A standard bulk free energy for a three-phase system should contain three local minima, each of which represents a bulk-phase value. We select the three local minima as $\Phi_{a} = (1,0,0), \Phi_{b} = (0,1,0), \Phi_{c} = (0,0,1)$. However, in the three-phase Cahn-Hilliard dynamics, a foreign phase can be introduced at a two-phase interface. Two remedies are that one can use the procedure developed in [@Folch] to generate a specific free energy, or one can use an interpolation function similar to that described in [@Moelans2010] (the latter method may only reduce the amount of the third phase contribution). Since the three phases in the Ni-YSZ anodes are immiscible, the generation of a foreign phase at a two-phase boundary is a problematic artifact that leads to extra TPB sites that do not physically exist. Our approach to resolve this issue is to select a free energy functional that can limit a foreign-phase appearance and can also incorporate an unequal surface tension at a reasonable computational expense. In the Cahn-Hilliard dynamics, the excess free energy at the interfaces, i.e., the interfacial energy, is determined by gradient energy coefficients and the bulk free energy at intermediate values of OPs. According to Young’s relation, the contact angles among the different phases are determined by the interfacial energies of the intersecting interfaces. For instance, if phases 1, 2, and 3 are in contact at a triple junction, the contact angle formed in phase 1 is related to the interfacial energy of the interface 2-3 relative to that of the interfaces 1-3 and 1-2. It is, however, easier to understand the interfacial energy as a factor reflecting the affinity of one phase to another. A lower interfacial energy suggests a stronger bonding of the two phases that are in contact. This tendency appears to have profound effects on the long-term microstructural evolution as demonstrated in our simulations [@hsunyi]. Applying the commonly used gradient and bulk free energy densities for multiple-OP systems, our free energy functional can be written as $$F = \int \limits_V dV \left ( \sum_{i=1}^{3} k_{i} \phi_{i}^{2} (1-\phi_{i})^{2} -\sum_{i,j=1,i>j}^{3} \alpha_{ij}^{2}\nabla \phi_{i} \nabla \phi_{j} \right ) . \label{eq:FEF}$$ The specific interfacial energy is calculated from the equilibrium solution at a planar interface. To do so, the free energy functional must be minimized subject to a constraint that the sum of the order parameters equals unity. This is carried out by taking the variational derivative of the functional using the method of Lagrange multipliers. The resulting equation is given by $$\left ( \frac{\delta F}{\delta \phi_{i}} \right )_{\phi_{1}+\phi_{2}+\phi_{3}=1}= \frac{\delta F}{\delta \phi_{i}} - \left( \frac{1}{3} \right ) \sum_j \frac{\delta F}{\delta \phi_{j}}, \label{eq:DfDphi}$$ where the variational derivatives at the right hand side of the equation are taken as if all $\phi_i$’s were independent [@Folch]. If we consider an interface between phase $i$ and phase $j$ without the existence of the third phase, the interfacial free energy can be calculated from the integration of the free energy density over the interface as $$\gamma_{ij} = 2 \alpha_{ij} \sqrt{(k_{i}+k_{j})} \int_{0}^{1} \phi_{i} (1-\phi_{i}) d\phi_{i} = \frac{\alpha_{ij}}{3} \sqrt{(k_{i}+k_{j})}. \label{eq:SurfE}$$ Similarly, the interfacial width $\delta_{ij}$ can also be found as $$\delta_{ij} = \frac{\alpha_{ij}}{ \sqrt{(k_{i}+k_{j})}}. \label{eq:SurfW}$$ After multiplying Eq. \[eq:SurfE\] by Eq. \[eq:SurfW\] and rearranging the result, we find $$3 \delta_{ij} \gamma_{ij} = \alpha_{ij}^2. \label{eq:SurfEW}$$ By choosing $\delta_{12}=\delta_{23}=\delta_{13}$, the relationships between the gradient energy coefficients and the interfacial energy in this three-phase system is simplified as $$\frac{\gamma_{12}}{\alpha_{12}^{2}} = \frac{\gamma_{23}}{\alpha_{23}^{2}} = \frac{\gamma_{13}}{\alpha_{13}^{2}}. \label{eq:SFEGEC}$$ The values of the surface free energies can be set to those found in the literature or be left as a parameter if data are missing or uncertain. The interfacial width is a computational parameter that must be chosen to ensure thin-interface limit for accuracy while providing a numerically well-resolved interface. Once the interfacial width $\delta_{ij}$ and interfacial energies $\gamma_{ij}$ are set, the gradient energy coefficients $\alpha_{ij}$ can be calculated from Eq. \[eq:SurfEW\], and the bulk energy coefficients $k_i$ can be calculated from Eq. \[eq:SurfW\]. Surface diffusion is generally considered to be the main mechanism of transport for the capillarity-driven microstructural evolution in these SOFC anodes [@Vaben]. We have confirmed the validity of this assumption based on an analysis of order of magnitudes. Since the OPs with integer values represent the bulk phases in the PFMs, the surface mobility function in a one-OP, two-phase case is commonly formulated as $M(\phi) = \phi^2 (1-\phi)^2$; such a formulation guarantees a non-zero mobility at two-phase boundaries only. In Model A, the surface mobility function is given by: $$M(\phi_{1},\phi_{2},\phi_{3}) =\sum_{i,j=1,i>j}^{3} M_{ij} \textstyle \prod_{C_{l}C_{h}}(\phi_{i}) \prod_{C_{l}C_{h}}(\phi_{j}) \big ( \phi_{i} \phi_{j} (1-\phi_{i})(1-\phi_{j}) \big ), \label{eq:mobF}$$ where we introduce a boxcar function $\prod_{C_{l}C_{h}}(\phi_i)$ to avoid excess mobility resulting from the appearance of a small amount (less than 3% in fraction) of a foreign phase at a two-phase boundary. We choose $C_{l}=0.05$ as the lower cutoff OP value in the mobility function, and $C_{h}=1-C_{l}$ as the upper cutoff value and resulting from the complementary value of the lower cutoff. All three mobility prefactors have positive values, and thus, materials flow from high to low chemical potentials. ### Model B In this model, we assume that YSZ is stationary and that the Ni phase evolves by diffusion along the Ni-pore interfaces. The dynamics of this system can thus be described by a single-OP Cahn-Hilliard equation with one contact-angle BC at triple junctions, and one no-flux BC at YSZ interfaces, where the OP distinguishes the Ni and pore phases. The treatment of a contact-angle BC in a non-conserved (Allen-Cahn) PFM was proposed by Warren [et al.]{} [@Warren] to model heterogeneous nucleation. Following [@Warren], we developed Model B using the SBM for a Ni-YSZ anode coarsening with conserved dynamics. This SBM approach is an entirely diffuse-interface treatment that implicitly handles complex geometries (while Warren et al. applied delta function [@Warren]). In the following derivation, we demonstrate that both no-flux and contact-angle BCs can be coupled with a Cahn-Hilliard equation to give a single evolution equation using the SBM framework. For one-OP Cahn-Hilliard dynamics within a SBM framework, the evolution equation can be written as $$\frac{\partial \phi}{\partial t} = \nabla \cdot M(\phi,\psi) \nabla \mu. \label{eq:CHev}$$ We consider the free energy functional as being: $$\mathcal{F} = \int_V dV [\frac{\epsilon^{2}}{2} \left \| \nabla \phi \right \vert^{2} + f(\phi)], \label{eq:Teng}$$ where $f(\phi)$ is a generic double-well function. The chemical potential $\mu$ is by definition the variational derivative of free energy $\mathcal{F}$ with respect to the order parameter $\phi$, [i.e.]{}, $\mu = \delta \mathcal{F} /\delta \phi = \partial f /\partial \phi - \epsilon^2 \nabla^{2} \phi$. We introduce in the SBM a domain parameter $\psi$ to incorporate general BCs on internal boundaries. In this case, the domain parameter $\psi$ distinguishes regions having YSZ ($\psi=0$) and other phases ($\psi=1$), as well as the YSZ interfaces ($0<\psi<1$). Multiplying Eq. by $\psi$ and letting $\vec J = M \nabla \mu$ the formulation becomes $$\frac{\partial ( \psi \phi)}{\partial t} = \psi \nabla \cdot \vec J = \nabla \cdot (\psi \vec J) - \vec J \cdot \nabla \psi. \label{eq:CHevM}$$ The aforementioned no-flux BC, $\nabla \mu \cdot \nabla \psi = 0$, should be applied to the internal boundaries (YSZ interfaces) to ensure mass conservation. This no-flux BC eliminates the last term of Eq. and the equation becomes $$\frac{\partial ( \psi \phi)}{\partial t} = \nabla \cdot (\psi \vec J) = \nabla \cdot (\psi M \nabla \mu). \label{eq:CHm1}$$ Like an ordinary OP in a phase-field approach, $\psi$ varies continuously across the interface; thus, the unit interface normal $\vec n$ of the YSZ interface can be described as a function of the gradient of $\psi$, i.e., $\vec {n} = \nabla \psi / \left \| \nabla \psi \right \vert$. By assuming that the effects of the YSZ on the Ni phase are only of short range (immiscible) and $\phi=1$ represents the bulk Ni phase, the contact angle $\theta$ at triple junctions can be formulated as $$\vec n \cdot \frac{\nabla \phi}{\left \| \nabla \phi \right \vert} = \frac{\nabla \psi}{\left \| \nabla \psi \right \vert} \cdot \frac{\nabla \phi}{\left \| \nabla \phi \right \vert} = -\cos \theta. \label{eq:AngBC}$$ The negative sign comes from the convention that $\nabla \psi$ points into YSZ and $\nabla \phi$ points out of Ni. The mechanical equilibrium at the triple junction corresponds to an extremum of the free energy, i.e., $\delta \mathcal{F} = 0$. We can use the planar solution of the thermodynamic equilibrium condition within the interface to find a useful equality, $\left \| \nabla \phi \right \vert = \sqrt{2 f} / \epsilon$, which can be substituted into Eq. to derive the SBM contact-angle BC as $$\nabla \psi \cdot \nabla \phi = - \left \| \nabla \psi \right \vert \cos \theta \frac{\sqrt{2f}}{\epsilon}. \label{eq:AngBCF}$$ This contact-angle BC results in energy only near triple junctions, rather than in the bulk volume, to achieve the force balance at the triple junction that is dictated by Young’s equation for a flat surface. The final evolution equation derived from the SBM with no-flux and contact-angle BCs is: $$\frac{\partial ( \psi \phi)}{\partial t} = \nabla \cdot \bigg \{ \psi M \nabla \bigg [ f_{\phi} - \frac{\epsilon^{2}}{\psi} \bigg ( \nabla \cdot ( \psi \nabla \phi ) + \frac{\left \| \nabla \psi \right \vert \sqrt{2f}}{\epsilon} \cos \theta \bigg ) \bigg ] \bigg \}. \label{eq:CHSBMF}$$ Again, the gradient energy coefficient $\epsilon$, the contact angle $\theta$, and the bulk energy coefficients in $f(\phi)$ are determined according to the interfacial energies of the Ni-YSZ cermet and the selected interfacial widths. The mobility function in Model B is formulated as $$M(\phi,\psi) = M_{Ni-Pore} \textstyle \prod_{C_{l}C_{h}}(\phi) \big ( \phi^2 (1-\phi^2) \big ) g(\psi), \label{eq:SBMMob}$$ where $g(\psi) = \psi^6 (10\psi^2-15\psi+6)$ is introduced to control the mobility at and near triple junctions. This one-sided interpolation function $g(\psi)$ transitions smoothly from 1 to the order of $0.01$ as the domain parameter varies from 1 to $0.5$. In other words, this choice of the mobility function ensures an immobile YSZ phase by limiting the mobilities at YSZ interfaces. Subsequently, the mobility near a triple junction decreases from the Ni-pore value to a value that is $~10^{-6}$ smaller as $\psi$ varies from 1 to about $0.1$. Nondimensionalization and Asymptotic Analysis --------------------------------------------- Having the appropriate values of the mobility prefactors is essential in simulating coarsening kinetics. In order to quantitatively correlate simulation results with physical phenomena, asymptotic analyses are required in PFMs because of their diffuse interface nature. Using an asymptotic analysis, we determined the relationship between diffusivity and mobility and the characteristic simulation time scale. Unlike some of the previous studies, which determined model parameters by fitting coarsening experimental results, our model is a predictive model that is free of fitting parameters when the material properties, such as surface diffusivities, are accurately known. In the Ni-YSZ anode of SOFCs, coarsening proceeds mostly via surface diffusion. The anisotropic effect of crystal facets on surface diffusion is lumped into one ensemble diffusivity in our models. In this case, the normal velocity $V_{n}$ of the interface $\Gamma$ incurred by surface diffusion, in the dimensional sharp-interface form, can be represented as $$V_{n} = \frac{\gamma_{s} D_{s} \delta_{s}}{k_{B} T N_{v}} \nabla_{s}^2 \kappa_{c} = \frac{\gamma_{s} D_{s} \delta_{s}}{k_{B} T N_{v}} \frac{\partial^{2} \kappa_{c}}{\partial s^{2}}, \label{eq:SHP}$$ , where $\gamma_{s}$ is the surface energy, $D_{s}$ is the surface diffusion coefficient, $\delta_{s}$ is the interfacial thickness, $N_{v}$ is the atomic number density per volume, $\kappa_{c}$ is the local curvature, $\nabla_{s}^2$ is the surface Laplacian and $\partial / \partial s$ is the gradient operator along the interface [@Balluffi2005]. To link the corresponding diffuse-interface model to the aforementioned sharp-interface model, a pure two-phase boundary in this three-phase system is considered. For both Model A and Model B, the dimensional Cahn-Hilliard evolution equation at a two-phase boundary can be re-arranged as $$\frac{\partial \phi}{\partial t} = \triangledown \cdot ( M(\phi) \triangledown \mu ), \label{eqDCH}$$ $$\mu = \frac{\partial f(\phi)}{\partial \phi} - \epsilon^{2} \triangledown^{2} \phi, \label{eqDmu}$$ , where $\phi$ is the OP distinguishing two phases, $f(\phi) = W \phi^{2}(1-\phi)^{2}/4$ is the energy density, and $M(\phi) = 6M_{s} \phi^2(1-\phi)^2$ is the surface mobility. Eq. \[eqDCH\] and Eq. \[eqDmu\] possess the physical steady-state solution for a planar interface, i.e., $\phi(x)=[1-\tanh(x/2 \delta)]/2$, with the interface thickness $\delta = \epsilon \sqrt{(2/W)}$, and the interfacial energy $\gamma = \epsilon \sqrt{(W/72)}$. Following the derivation in [@SteveWise], a new set of variables for a non-dimensionalization of the governing equations is utilized. We assume that $\tau = L^{4}/D$, wherein $L$ is the characteristic length scale of the sample and $\tau$ is the characteristic time scale for surface diffusion, and consider that other nondimensional quantities (denoted by overbars) are defined by $$\begin{aligned} && \bar{M}_{s} = \frac{36 M_{s} \gamma \delta}{D},~~ \bar{\delta} = \frac{\delta}{L},~~ \bar x = \frac{x}{L},~~\bar t = \frac{t}{\tau} \nonumber \\ && \bar{M}(\phi) = \phi^{2}(1-\phi)^{2},~~ \bar{f}(\phi) = \frac{1}{2} \phi^{2}(1-\phi)^{2}. \label{eq:NDParm}\end{aligned}$$ Eqs. \[eqDCH\] and \[eqDmu\] can be written using the non-dimensional parameters to derive the nondimensional evolution equation as $$\frac{\partial \phi}{\partial \bar t} =\frac{1}{\bar \delta^{2}} \triangledown \bar M(\phi) \triangledown \bar \mu \label{eq:NDCH3}$$ $$\bar \mu = \frac{\partial \bar f(\phi)}{\partial \phi} - \bar \delta^{2} \triangledown^{2}\phi, \label{eq:NDmu}$$ The asymptotic analysis derivation is similar to [@DongHee] and is detailed in the appendix. Comparing the dimension-restored equation derived from the asymptotic analysis with the corresponding sharp interface equation, Eq. \[eq:SHP\], we find $$\frac{D \bar{M_{s}}}{36} = M_{s} \delta \gamma = \frac{\gamma_{s} D_{s} \delta_{s}}{k_{B} T N_{v}}, \label{eq:MS}$$ , which indicates that $M_{s}=D_{s} \delta_{s}/k_{B}TN_{v} \delta$, provided the choice of $\gamma = \gamma_{s}$; that is, we find that the surface mobility is proportional to surface diffusivity. Choosing $\bar{M}_{s} = 36$, the model variable $D$ is connected to the surface diffusivity $D_{s}$ as follows: $$D = \frac{\gamma_{s} D_{s} \delta_{s}}{k_{B} T N_{v}}. \label{eq:DDs}$$ The time scale that links the simulation time to the physical time is acquired from Eq. \[eq:DDs\] as $$\tau = \frac{L^{4} k_{B} T N_{v}}{\gamma_{s} D_{s} \delta_{s}}. \label{eq:Tau2}$$ Numerical methods ----------------- One of the challenges in modeling Ni-YSZ anode coarsening is the fact that solving Cahn-Hilliard equations with an explicit time iteration scheme is too expensive, especially for a large scale simulations in three dimensions. Thus, we solve the nondimensional evolution equations, Eqs. \[eq:NDCH3\] and \[eq:NDmu\], with the algorithm based on splitting the fourth-order Cahn-Hilliard equation into two second-order equations and soloving for the OPs and chemical potential simultaneously [@SteveWise]. We use the central-differencing method for the spatial discretization and the Crank-Nicholson algorithm for the time discretization. This semi-implicit scheme significantly reduces the stiffness of the numerical integration, which allows much larger time step size. To solve the nonlinear finite-difference equations, Newton’s method is utilized for the nonlinear terms. A pointwise Gauss-Sidel relaxation scheme and a red-black checkerboard iteration scheme are used together to accerelate the convergence rate as well to facilitate the parallelization. In most cases, we find this solver is over 100 times faster in comparison to the explicit scheme. However, attention must be paid to the choice of the time stepping size. We find in some cases, when an overly large time step is used in our solver, the simulation results are incorrect even if the numerical scheme is stable. When solving the coupled governing equations in Model A, the solver is parallelized with Message Passing Interface (MPI) library to take advantage of the multiple processors. In our MPI code, the domain is decomposed equally in size in each axial direction, if possible, to achieve load balance. For instance, if 64 CPUs are allocated to the computation, the entire domain is decomposed into 4 by 4 by 4 sub-domains; that is, each sub-domain has a domain size of 1/64 of the original domain in volume or, more explicitly, 1/4 of the length of the original domain in each axis (assuming there are no residuals). The solver for Model B is parallelized with both the MPI and openMP libraries. Since only one Cahn-Hilliard equation is solved, it is much more numerically stable and efficient compare to the Model A solver. Error analysis: contact angles and the interfacial width -------------------------------------------------------- The phase-field model is known to smooth out microstructures with length scales below the diffuse interface thickness. This artificial smoothing process introduces errors during the early stages of a simulation, but has no negative effects on the analysis of long-term coarsening kinetics. In contrast, microstructural evolution kinetics can only be accurately resolved in PFMs when the length scale of the microstructure is larger than certain multiples of the interfacial thickness. In other words, there is a critical ratio between the microstructural length scale and the interfacial thickness that is required to obtain a sufficient agreement with the sharp-interface limit. For example, if a system contains particles with typical radii a few times smaller than the interface thickness, the simulations of its evolution will incur a large error. The commonly recognized critical ratio is of about 10. However, in large scale 3D simulations, to resolve a complicated system with microstructural features of different length scales, achieving this critical value for all features is impracticable because it would require a very high resolution. In practice, we use a smaller ratio, especially when there are smaller features among a range of feature sizes within the microstructure, and quantify the errors introduced by the selected value. Because the coarsening kinetics of Ni-YSZ anodes has been found to depend strongly upon the Ni-YSZ contact angle, an error analysis is performed via the investigation of the contact angle of Ni on the YSZ phase in two dimensions (2D). The two-dimensional domain is initialized with a bottom that is half occupied by YSZ and a top half that is equally divided into the Ni and pore phases with a $90^\circ$ contact angle. The remainder of the domain is filled with the pore phase. The domain size is designed to be large enough so that no-flux BCs have negligible effects on the contact angles at the triple junction. The system is evolved with our Model B to its steady state. The final contact angle $\theta_C$ is calculated from the average value of the dot product of the normal, $$\frac{\nabla \psi}{\left \| \nabla \psi \right \vert} \cdot \frac{\nabla \phi}{\left \| \nabla \phi \right \vert} = -\cos \theta_C, \label{eq:ctAng}$$ over the region where the domain parameter $\psi$ and the order parameter $\phi$ are both between 0.1 and 0.9, indicating the TPB region. Two contact angles are studied: $120^\circ$ and $93^\circ$. The $120^\circ$ case is selected as a reference case because the cosine function is away from the extrema or inflection point at this contact angle. The $93^\circ$ case corresponds to a physical contact angle of Ni on the YSZ phase that is based on our selected interfacial energies. As the characteristic length of the system, we choose the domain size, which is varied from 10 points to 100 points in each direction, while the interfacial width is held at 4 grid points. Therefore, the ratio of the domain size to the interfacial width varies from 2.5 to 10. For the $10\times 10$ domain, the interfacial region (either the domain or order parameters are between 0.1 and 0.9) occupies about half of the domain. As the domain size increases, the fraction becomes very small. The contact angles normalized to the set value versus the ratios of the domain size to the interfacial width are plotted in Fig. \[fig:ErrAng\]. In the case of a $120^\circ$ contact angle, we find that the angle deviate from the sharp-interface value by less than $0.2^\circ$ when the ratio is larger than $5$. Even at a ratio of $2.5$, the deviation is below $1$ degree or $1\%$. In the case of a $93^\circ$ contact angle, the contact angle differs from the sharp-interface value by less than $0.5^\circ$, or $0.5 \%$. ![Normalized angle plotted with the ratio of the domain size to the interfacial width. The error decreases with an increasing ratio of the domain size to the interfacial width. \[fig:ErrAng\]](ErrAna_ContAng.eps){width="10.37cm" height="7.5cm"} The interfacial width utilized in coarsening simulations is approximately $0.13~\mu$m. Based on our error analysis results, the contact angles of the Ni particles in the active and dead-end categories with a length scale larger than $0.32~\mu$m can fairly accurately be modeled by using our Model B. In the isolated network, a length scale larger than $0.64~\mu$m is needed for a fair accuracy. For the $(4~\mu$m$)^3$ domain, only $3\%$ in volume of the Ni phase belongs to clusters outside the accurate range. In addition, based on Young’s equation for a flat surface, a $0.5^\circ$ difference in contact angle of the Ni phase corresponds to a $1.1\%$ difference in Ni-YSZ interfacial energy. In other words, this model error has very limited effects on the coarsening kinetics of the Ni-YSZ anode. Parameterization of the models {#ModelPara} ============================== The aforementioned asymptotic analysis provides the mathematical ground for correlating simulations on the coarsening phenomenon with experimental results. However, to quantitatively model the physical process, the material-specific parameters in the governing equations must be specified based on the material properties. Most of these parameters can be found in the literature. Mobility prefactors ------------------- As demonstrated in our asymptotic analysis, the mobility prefactor corresponding to the Ni-pore interface should be proportional to the surface self-diffusivity of Ni, which is usually anisotropic or depends on the crystallographic orientation. The Ni diffusivities based on the field ion microscope (FIM) measurements range from $10^{-13}~$m$^{2}$s$^{-1}$ in (110) to $10^{-9}~$m$^{2}$s$^{-1}$ in (331) at 1273 K [@Seebauer; @Tung; @Fu]; these values are extrapolated at high temperatures and could be inaccurate, because FIM can be conducted only at low temperature regions ($T < 0.2 T_{m}$, $T_m$ is the melting temperature). On the other hand, the surface smoothing method (SSm) measures mass-transfer diffusion at high temperatures ($T > 0.7 T_{m}$), which is generally averaged over crystallographic orientations [@Seebauer]. The latter method provides a more appropriate value for the diffusivity ($\sim 10^{-10}~$m$^{2}$s$^{-1}$) during Ni coarsening at the modeled temperature (1000$^\circ$C). In addition, the anisotropy of the Ni surface diffusivity was found to be relatively small in high temperature regions ($0.82~T_{m}< T < T_{m}$) in [@Maiya]. We thus consider an ensemble value $\sim 10^{-10}~$m$^{2}$s$^{-1}$ for the Ni surface diffusivity in our model. Cation diffusion in oxides is related to various parameters, such as valence, atomic radius, impurity, and oxygen activity. In Ref. [@Kilo], it is reported that the bulk diffusivity of yttrium in YSZ (containing 10 to 32 mol% Y$_{2}$O$_{3}$) is slightly larger than that of zirconium in YSZ, and that the difference is less than an order of magnitude. The surface diffusion of Zr in YSZ is calculated from the measurements of the surface area reduction in powder compacts during sintering in Ref. [@Mayo], and the diffusivity at 1000 $^\circ$C is found to be $\sim 10^{-16}~$m$^{2}$s$^{-1}$. We thus consider a surface diffusivity of YSZ of $10^{-16}~$m$^{2}$s$^{-1}$ in our simulations. The diffusion mechanisms at metal-ceramic interfaces are less understood than those on metal or ceramic surfaces. The cohesiveness of the Ni-YSZ interface, which depends on the process used to fabricate the porous cermets, determines the interfacial structure and affects the effective interfacial diffusivity. A series of diffusion bonding experiments indicated that the metal-ceramic interface does not act as an efficient vacancy sink or mass transport path [@Derby]. These findings suggest that the diffusivity at the Ni-YSZ interface should be much smaller than that of the Ni surface and the YSZ surface. In addition, the exact value of the diffusivity at the Ni-YSZ interface is not as important as that of Ni because the redistribution of Ni is observed to be the dominant morphological change during anode coarsening. The diffusivity of the Ni-YSZ interface is considered to be approximately $\sim10^{-20}~$m$^{2}$s$^{-1}$. Using these values, the mobility ratios among the three materials are set at $M_{\mathrm{NiP}}:M_{\mathrm{YP}}:M_{\mathrm{NiY}}=1:10^{-6}:10^{-10} $, where the subscript Ni represents nickel, Y the YSZ and P the pore phase. For non-dimensionalization, a mobility scale of $10^{-10}~$m$^{2}$s$^{-1}$ is used. Bulk and gradient energy coefficients ------------------------------------- In a PFM, the balance between the bulk and the gradient energy terms in the free energy functional determines the thickness of the interface. The ratio of the interfacial thickness to the characteristic length of the system and the ratio of the domain size to the characteristic length are crucial for the validity of a PFM. It is known that an interface that is too thin can cause an unphysical pinning or halting of coarsening, whereas an interface that is too thick can lead to unphysical dissolution of particles [@NingMa2006]. Therefore, our task is to select bulk and gradient energy coefficients that result in an appropriate interfacial energy, while keeping the interfacial thickness sufficiently large so that the computation is feasible. In a multi-phase system, using an unequal interfacial thickness among different interfaces without a special treatment can cause erroneous evolution kinetics in a PFM simulation because the interfacial thickness changes the ratio of the simulation time scale to the physical time scale. Therefore, a single value for the interfacial thickness of all interfaces is set. For example, we set $\delta_{ij} = \Delta x$, which gives 4 points in the interfacial region to avoid the aforementioned pinning while optimizing the computational efficiency ($\Delta x$ depends on the selected length scale in a non-dimensionalization). The interfacial energies are obtained from existing experimental or computational studies. These choices are then used in Eqs. \[eq:SurfW\] and \[eq:SurfEW\] to determine the bulk and gradient energy coefficients. The three interfaces that exist in the Ni-YSZ anode are the Ni surface (Ni-pore interface), the YSZ surface (YSZ-pore interface), and the Ni-YSZ interface. The Ni surface free energy has been widely investigated and the values reported are in good agreement. At 1000$^\circ$C, the surface energy of $\sim 1.9~$Jm$^{-2}$ for Ni has been measured [@Mantzouris05]. A 8YSZ surface (8 mol % Y$_{2}$O$_{3}$) has been reported in Ref. [@Tsoga]. It was studied at 1300$\sim$1600$^\circ$C using a multiphase equilibrium technique. The YSZ surface free energy $\gamma_{\mathrm{YSZ}}$ was found to decrease linearly from 1.26 to 1.13 Jm$^{-2}$ in the range of temperatures studied. By extrapolating this data, one obtains $\gamma_{\mathrm{YP}} \sim 1.4~$Jm$^{-2}$ at 1000$^\circ$C. This value is used to parameterize our model. In Ref. [@Ballabio], ab initio calculations have been reported with $\gamma_{\mathrm{YP}}$ ranging from 1.04 to 1.75 Jm$^{-2}$ at $T = 0$, depending on the crystallographic orientation. The free energy of a heterogeneous Ni-YSZ interface depends on the interfacial structure, which is dependant on the fabrication process. Nikolopoulos [et al.]{} experimentally measured the non-reactive contact angle between molten Ni and 8YSZ at 1500$^\circ$C and found a value of 117$^\circ$, which suggests an interfacial energy of 1.95 Jm$^{-2}$ [@Nikolopoulos96]. Although this wetting experiment was not conducted at normal SOFC operating temperatures, the result implies a poor wettability at the Ni-YSZ interface. Several ab initio calculations on bond formation at Ni-YSZ interfaces have been reported [@Christensen]. A strong bonding between Ni-Zr and Ni-O was found at the Ni$(100)/$ZrO$_{2}(100)$ polar interfaces, and an interfacial tension of $\sigma_{(100)}=1.04~$Jm$^{-2}$ was reported. Another local minimum appeared at the ZrO$_{2}/$Ni$(111)$ interface where $\sigma_{(111)}=1.80~$ Jm$^{-2}$ [@Beltran]. However, Ni-YSZ interfaces formed subsequent to sintering are multifaceted. Despite the lack of understanding of these interfaces, we know the range of reasonable interfacial energies at Ni-YSZ interfaces. In our simulations, $\gamma_{\mathrm{NiY}}=1.50~$Jm$^{-2}$ is used. In view of the above discussion, we assume that the gradient energy coefficient ratios are $\gamma_{\mathrm{NiP}} : \alpha_{\mathrm{NiY}}^2 : \alpha_{\mathrm{YP}}^2 = 1.9:1.5:1.4$ based on a surface energy scale of 1 Jm$^{-2}$, which leads to the bulk energy coefficient ratios of $k_{\mathrm{Ni}} : k_{\mathrm{Y}} : k_{\mathrm{P}} = 1:0.5:0.9$. 3D data analysis {#PconTPBact} ================ In three dimensions, a boundary at which three phases coincide has only one degree of freedom in space, i.e., the TPB is a line in the Ni-YSZ anode where the Ni, YSZ, and pore phases are all in contact. However, to identify the TPBs based on the microstructure reconstructed by FIB-SEM, some data processing is necessary since the microstructural data are represented by a 3D matrix. In this matrix, each phase is represented by regions in which voxel values are equal to a pre-determined constant. There are several methods to determine the TPB length in 3D volumetric data. For example, in Ref. [@JWilson], the TPBs are identified as the edges where three different voxels, each of which belongs to a different phase, are in contact. With some geometric corrections, the TPB length can be acquired with fair accuracy, but is limited by the resolution. In our work, we adopt a thinning, or skeletonization, algorithm to determine the TPB regions, and count the voxels. In PFM simulations, the interfacial region is commonly identified as the zone over which an OP varies from 0.1 to 0.9. Because the interfaces in PFMs span multiple grid points, one can identify the diffuse TPB regions as voxels in which all three OPs are between 0.1 and 0.9. For Model B, the threshold values are slightly different due to the additional pre- and post- treatment of the data (Ref. [@HsunYi11]). In order to recover the one-dimensional nature of TPBs from this data, we use a thinning algorithm developed in Ref. [@Palagyi], which reduces the diffuse TPB regions to the corresponding skeleton of chains of voxels. The key feature of this algorithm is that it preserves the topological characteristics of the original image/voxelated data (i.e., it does not allow the pinching or connection of regions). The final TPB length is calculated via a multiplication of the physical grid size and the total number of TPB voxels after skeletonization. The used procedures may either overestimate or underestimate the TPB lengths depending on the angle at which the TPB lies within the grid; for an isotropic distribution of lines, the over/underestimation has been determined to be less than $13\%$. Because we have nearly isotropic distribution (i.e., anisotropy that only results from statistical variations), the error should be relatively consistent throughout the coarsening process and therefore not impact our investigation of the evolution of the TPB length. Not all of the TPBs are active; active TPBs must be simultaneously in contact with the three active phases that facilitate simultaneous transport of fuel, electrons, and oxygen ions. Since the electrochemical reactions occur only at or near these active TPBs, it is these active TPBs that contributes to the anode performance. Before identifying the active TPBs, we must first identify the active phases. Physically, a Ni phase cluster is active only if it connects the TPB sites to the current collector; a YSZ cluster is active only if it connects the TPBs to the electrolyte; and a pore cluster is active only if it connects the TPBs to the gas flow channels. However, the determination of such connectivities is feasible only if an anode structure that spans from the electrolyte to the current collector and to the gas channel is available. Our reconstructed microstructure, in contrast, constitutes only a portion of the entire span. As a compromise, we follow the procedures described in Ref. [@JWilson], which categorize each bulk region into active, dead-end, or isolated clusters. Active clusters are clusters that are connected to at least two sides of the sample domain boundaries, while isolated clusters are those that are not connected to any of the sides and are therefore electrochemically inactive. Dead-end clusters are defined as those that are in contact with only one side of the sample domain. In Ref. [@JWilson], a pre-smoothing procedure is used before this calculation to avoid artifacts resulting from image processing. Because PFM simulations naturally smooth microstructures over the lengh scale of the interfacial thickness, the pre-smoothing procedure is not necessary in our analysis. Note that in this procedure, a voxel is assumed to belong to a cluster if at least one of the 6 nearest neighbors is in the cluster. The active TPBs are then identified as those simultaneously in contact with the active networks of all three different phases. The TPB is marked as inactive as long as one of the phases adjacent to a TPB voxel is isolated. The remainder of the TPBs are considered unknown in terms of activity. Results and Discussion ====================== Phase connectivity and TPB activity {#phaseCon} ----------------------------------- TPB length has been recognized as one of the most important geometric parameter in a three-phase SOFC anode that influences the electrochemical performance. However, how the coarsening of microstructures affects the TPB activity has not been fully explored. As previously mentioned, TPBs are active only if they are simultaneously connected to conducting ionic, electronic, and gaseous transport pathways (phases). Therefore, the connectivity evolution of a phase induced by coarsening can alter the activity of TPBs and the effective conductivity of that phase. In addition, the evolution of the anode microstructure results in changes in the amounts and distributions of TPBs. In turn, coarsening has significant impacts on the overall performance of the electrode. By simulating coarsening with our models, a series of microstructures are acquired at various stage of evolution. Using the methods described in section \[PconTPBact\], we can analyze the evolution of the active TPB length based on the evolving TPB distribution and phase connectivity during coarsening. ![Initial Ni-YSZ anode microstructure for a set of larger simulations. The dimensions of the microstructure are 4 $\mu$m $\times$ 4 $\mu$m $\times$ 4 $\mu$m. The Ni, pore, and YSZ phases are represented in green, blue, and semi-transparent, respectively (with volume fractions 23.8%, 18.7%, and 55.4%, respectively). The initial TPB density is $\sim$5.2$~\mu$m$^{-2}$. \[fig:InitMS\]](NiPoreYSZ_126cu_init.eps){width="9.cm" height="9.cm"} The initial microstructure of our simulation is based on an FIB-SEM reconstructed SOFC anode of dimensions 9.73 $\mu$m $\times$ 8.35 $\mu$m $\times$ 11.2 $\mu$m [@James]. We selected a 4.0 $\mu$m $\times$ 4.0 $\mu$m $\times$ 4.0 $\mu$m portion of the specimen (Fig. \[fig:InitMS\]), resolved by a domain of a 126$\times$126$\times$126 computational grid. The governing evolution equations are solved using the finite-difference algorithm implemented using Open-MP. No-flux boundary conditions are imposed on the computational domain boundaries to reflect the assumption that materials that comprise the SOFC anode are conserved. In the simulated sample, the volume fractions are 23.8%, 18.7%, and 57.5% for the Ni, pore, and YSZ phases, respectively. Each phase region is categorized as an active, isolated, or dead-end cluster according to the procedure described in Sec. \[PconTPBact\]. By examining the YSZ phase with the aforementioned nearest-neighbor scheme, we find that, in the initial sample, the entire YSZ phase is fully percolated and active within the volume. Using the same procedure, we find that there are 89.1% active, 6.2% isolated, and 4.7% dead-end clusters within the Ni phase, and 94.7% active, 1.9% isolated, and 3.4% dead-end clusters in the pore phase. Because our 3D microstructure data are represented as a set of voxels with values corresponding to various phases, this categorization procedure seems similar to the problem that consists of identifying the percolating clusters in a finite system on the basis of a simple cubic network with a coordination number of 6. According to the percolation theory, the site percolation of a phase in a simple cubic network is achieved when the volume fraction of that phase is above a threshold value of 0.3116. The YSZ phase in our sample is fully percolated and active because its volume fraction is much higher than this threshold value. However, it is surprising that nearly $90\%$ of the volume of the Ni and pore phases are active while their volume fractions are much lower than 0.3116. This finding may be attributed to two reasons. First, the functional layer of our Ni-YSZ anode sample is fabricated by sintering a 50/50wt$\%$ NiO-YSZ mixture and exposing it to humidified H$_2$ to reduce NiO to Ni. This specific fabrication process results in highly percolated pore and Ni phases, even at volume fractions below 0.3116. Second, in our characterization algorithm, a cluster is considered active if it connects any two domain boundaries of the sample volume. Therefore, a cluster with a length scale smaller than the sample dimensions can be active as long as the cluster connects two neighboring domain boundaries of the sample. In terms of identifying the percolated clusters, this criterion is less stringent than the percolation theory that requires that the cluster is percolated over an infinite volume. ![The pore volume change in three different categories over 500 hours of coarsening. The change in each category is relative to its initial value. \[fig:Pore\_Vol\_1000hr\]](Pore_Catego_126cu.eps){width="11.06cm" height="7.6cm"} A crucial question arises is how these phase clusters evolve with coarsening. The coarsening dynamics are greatly simplified in Model B because the Ni surface is the only mobile interface in the system and the YSZ phase is immobile. While coarsening progresses, the system reduces its total energy by reducing mobile regions with high curvatures (which possess smaller length scales and larger surface to volume ratios). Because the Ni and pore phases share the same Ni-pore interface, the mobile surfaces are the same for both phases. The driving force for material transport thus depends on the curvature gradient, which is inversely linked to the characteristic length scale that is the inverse of the interfacial area per unit volume of the phase ($S_V^{-1}$). For example, the characteristic length scale of the Ni structure is defined by the volume of Ni divided by the total interfacial area of Ni. Note that this definition of the characteristic length scale is well suited for analyzing multiphase composites with unequal volume fractions because it takes into account the effects of volume on the typical size of a phase domain. For example, if the volume fraction is large, then we expect that the typical size of the domain be larger even if the surface area is the same (this is the case in a two-phase system where that phase is the majority phase). Using this definition, an average Ni length scale 36% larger than the average pore length scale is found in our sample. Inactive pore regions should thus evolve faster and possibly merge with active regions. According to the simulation results, during the first 500-hr simulation, the Ni volume fractions in each category and the mean Ni length scale remain roughly constant. In contrast, as shown in Fig. \[fig:Pore\_Vol\_1000hr\], the dead-end and isolated pore volumes decrease significantly, which confirms the faster evolution of regions associated with smaller length scales during coarsening. Interestingly, the active-pore volume increases in our simulation due to the fact that the smaller isolated and dead-end clusters merge into the larger pore clusters, that are most likely in the active cluster category. One interesting observation is that even though clusters with higher curvatures possess higher free energy than those with lower curvatures, they may be trapped in a local equilibrium state because the transport pathway is confined to the Ni-pore interface. This kinetic constraint explains why the isolated and dead-end clusters in the pore phase did not suffer major loss of volume after 500 hours of coarsening. ![TPB density change in each category over 500 hours of coarsening. The active TPB length reduction is $\sim$89$\%$ of the total TPB reduction after 500 hours. \[fig:TPB\_OT\]](TPB_OldCatego_126cu.eps){width="11.06cm" height="7.6cm"} The evolution of TPBs is correlated with the evolution of the phases that comprise the SOFC anode. The material transport among different clusters of each constituting phases not only changes the distribution of the TPBs but also dictates their activity. As shown in Fig. \[fig:TPB\_OT\], the reduction trend of the active TPB density agrees well with the total TPB density. The overall TPB density decreased by 20.2% after a 500-hour coarsening, while the active and isolated TPB density decreased by 31.4% and 22.3%, respectively. In contrast, the unknown TPB density slightly increased by 1.6%. Most of the TPB density reduction comes from the 31.4% decrease in the active TPBs. At the early stage of coarsening, the rapid reduction of active TPBs is due to the local coarsening of high-curvature microstructural features. The small increase in the unknown TPBs observed in the early stage of evolution stems from coalescence of inactive Ni domains, which occurs rarely. Since the evolution of TPBs is sensitive to local microstructural features, it is only indirectly correlated to the coarsening kinetics. To summarize, our findings indicate that the coarsening of a Ni-YSZ anode consumes the mobile phase clusters with high curvatures. The isolated and dead-end pore clusters are thus the first clusters that evolve, leading to coarsening of the pore phase, which is indicated by the change of $S_{V(pore)}^{-1} = 0.1105~\mu$m to $S_{V(pore)}^{-1} = 0.1154~\mu$m, i.e., a $4.4\%$ increase over 500 hours. While the Ni phase has the same mobile area as the pore phase, its mean characteristic length scale, which is defined by $S_{V(Ni)}^{-1}$, is significantly larger than that of the pore phase and results in a slower evolution. In addition, the Ni-pore area reduction is found to be balanced by the increase of the Ni-YSZ interfacial area. The coarsening has thus no significant effect on the mean length scale of Ni, which varies from $S_{V(Ni)}^{-1} = 0.1711~\mu$m to $S_{V(Ni)}^{-1} = 0.1719~\mu$m (which corresponds to a $0.5\%$ change during the course of our simulation. In turn, the coarsening leads to a significant reduction in TPBs and the active TPB sites decreases in a very similar fashion as the overall TPBs. However, the TPB evolution is sensitive to microstructural details and is only weakly linked to the evolution of bulk phases during coarsening. Tortuosity of pores ------------------- Tortuosity represents the geometric aspect of the transport property of a phase that comprises a composite. Therefore, the tortuosity of the pore phase plays a crucial role in the generation of electricity because the fuels need to be transported through the pores to reach the reaction sites. The tortuosity factors of the pore phase are evaluated in three orthogonal directions over a 500-hr coarsening period. As shown in Fig. \[fig:TortuRedRate\], although the absolute values differ, the tortuosity factors decrease moderately in all directions during coarsening. This trend is consistent with the fact that the active pore volume increases slightly during coarsening (see Sec. \[phaseCon\], Fig. \[fig:Pore\_Vol\_1000hr\]). These results indicate a coalescence of pore clusters and a lack of breakup of active clusters, which lead to somewhat enhanced transport properties. ![Tortuosity factors in three orthogonal directions of the pore phase over 500 hours of coarsening. \[fig:TortuRedRate\]](Tortuosity_xyz_126cu.eps){width="11.06cm" height="7.6cm"} The active Ni surface --------------------- One advantage of SOFCs is their ability to operate with hydrocarbon fuels as well as hydrogen. This is because hydrocarbon fuels can be internally reformed to hydrogen with the catalytic effect of Ni at the typical operating temperatures of SOFCs. This reforming rate is highly related to the active Ni surface area. Depending on the fuel, the reforming kinetics can be very different and involve multiple elementary reaction steps; however, the reforming reactions of all types of fuels involve gas species and electron transfer, regardless of the detailed reforming mechanisms. Therefore, the active Ni surface for hydrocarbon reforming must reside at the interface of active pore clusters and active Ni clusters. To evaluate the catalytic performance of the Ni phase in SOFC anodes, an important question to ask is how coarsening affects the active Ni surface area. The evolution of an active Ni surface area is dictated by the evolution of active clusters of Ni and pore phases. The active Ni surface is identified as the interfaces between active Ni and active pore clusters. As shown in Fig. \[fig:NiSARedRate\], the active Ni surface decrease by 9% after 500-hr coarsening, while the total Ni-pore interface decrease by 13.6%. Our finding indicates that coarsening reduces more inactive Ni surfaces than those in contact with the active clusters of Ni and pore. This is due to the fact that smaller clusters, which have smaller length scales and thus have larger driving force toward coarsening, coarsen faster than the larger ones. ![Ni surface area reduction over 500 hours of coarsening. \[fig:NiSARedRate\]](NiSurfA_ActA_126cu.eps){width="11.06cm" height="7.6cm"} Size effects and evolution kinetics {#SizeEff} ----------------------------------- We next simulate coarsening in Ni-YSZ anode samples of different sizes, namely, (3.2 $\mu$m)$^3$, (4.0 $\mu$m)$^3$, and (4.8 $\mu$m)$^3$, to examine the size effects. Each of these samples is acquired from a portion of the original 914.55 $\mu$m$^3$ sample. Specifically, the sample of dimensions (3.2 $\mu$m)$^3$ is cropped from the sample of dimensions (4.8 $\mu$m)$^3$, while the sample of dimensions (4.0 $\mu$m)$^3$ belongs to a different portion of the original sample. ![Reduction of the Ni surface areas for three different sample sizes over 300 hours of coarsening. \[fig:Size\_NiSurfA\]](NiSurfA_101_126_151_comp.eps){width="11.06cm" height="7.6cm"} ![Reduction of the Ni surface area per Ni volume for three different sample sizes over 300 hours of coarsening. \[fig:Size\_NiS\_v\]](S_v_101_126_151.eps){width="11.06cm" height="7.6cm"} The Ni surface areas and the TPB lengths are compared over the first 300 hours of the coarsening period. The evolution kinetics can be best interpreted from the evolution of the Ni surface area in our Model B simulations. As shown in Figs. \[fig:Size\_NiSurfA\] and \[fig:Size\_NiS\_v\], the Ni surface area reduction rate of the (4.0 $\mu$m)$^3$ and the (4.8 $\mu$m)$^3$ specimen are very similiar, while the reduction rate of the (3.2 $\mu$m)$^3$ specimen deviates from the other two significantly even though the specimen is a portion of the (4.8 $\mu$m)$^3$ sample. In addition, some reduction steps appear in the (3.2 $\mu$m)$^3$ curve whereas the other two curves are relatively smooth. This suggests that the two larger sample sizes may be sufficient to eliminate the boundary effects from the simulations, while smaller volumes would likely suffer from them and may not contain enough particles or statistics for coarsening simulations. As shown in Fig. \[fig:Size\_TPB\], the change of the TPB densities of the three specimens are compared over 300-hr coarsening. Although the behavior of the TPBs is only indirectly related to the evolution kinetics, we find similar trend in the TPB evolution among the three specimens as those found in the Ni surface area. Although the TPB density reduction kinetics are different, after 300-hour coarsening, the difference of the TPB densities between the two larger samples is less than 3%. Since TPB evolution is more sensitive to microstructures, this difference may serve as the indication of the local variation due to microsctructural details. Thus, in order to determine whether the two larger sample volumes are sufficient to represent the microstructure of the anode, we would need to simulate a larger sample volume. ![TPB densities for three differnt sample sizes over 500 hours of coarsening. The initial TPB densities of the (3.2 $\mu$m)$^3$, (4 $\mu$m)$^3$, and the (4.8 $\mu$m)$^3$ samples are 5.45 $\mu$m$^{-2}$, 5.22 $\mu$m$^{-2}$, and 5.25 $\mu$m$^{-2}$, respectively. \[fig:Size\_TPB\]](TPB_101_126_151_comp.eps){width="11.06cm" height="7.6cm"} Although rapid reductions of TPBs and the Ni surface area at early stages of coarsening is a result of the large thermodynamic driving force for mass transport, which mostly results from regions with high curvatures, the overly steep slopes in Figs. \[fig:Size\_NiSurfA\] and \[fig:Size\_TPB\] deserve further investigations. To examine the length scales involved in the microstructures, the mean curvature and the interfacial shape distribution (ISD) of the Ni interfaces in the (4 $\mu$m)$^3$ specimen are plotted in Fig. \[fig:UnderResolve\] (a) and (b), respectively. Due to the diffuse interface nature of our models and the model parameters we utilized in our simulations (based on the resolution that made simulations feasible), very small microstructural features with absolute principal curvature values larger than 0.17 (which accounts for approximately $30\%$ of the Ni interfacial area) are not fully resolved. As shown in Fig. \[fig:UnderResolve\] (c) and (d), if those numerically under-resolved regions are mobile, they will be smoothed out very rapidly, leading to coarsening that is faster than expected. ![Numerical smoothing of Ni microstructures in Model B. The circled particle is numerically smoothed after a very short time of coarsening. (a) mean-curvature ($H = (\kappa_1+\kappa_2)/2$) plot of Ni interfaces; (b) ISD diagram of Ni interfaces; (c) the Ni microstructure before coarsening; (d) Ni microstructure after coarsening for 3 minutes. (Figures (a) and (b) were plotted by Chal Park in Thornton’s group.) \[fig:UnderResolve\]](UnderResolve_MCL.eps){width="10.8cm" height="10.8cm"} This numerical smoothing due to under-resolution contributes in part to the rapid evolution in the early stage of coarsening simulations and leads to an overestimation of the TPB reduction rate. However, these mobile, high-curvature regions inherently possess very large thermodynamic driving force and will coarsen sooner or later. Therefore, even though the rapid evolution at the early stage of simulations may be caused by the insufficient resolution of microstructures, the stabilized value of TPB density after further coarsening remains unaffected by it. That is, the prediction of the stabilization of TPBs, as well as its predicted value, remains robust, even though the early kinetics may be overestimated. Since the microstructures of Ni-YSZ anodes are found to stabilize after coarsening for a period of time, the final or stablized TPB density rather than the short-term evolution kinetics is of importance for the operation of SOFCs. Conclusion ========== We developed two models to study the coarsening kinetics in the Ni-YSZ anode. An asymptotic analysis was conducted to link our simulation results to the physical system. The size effects were studied and an error analysis was performed to validate our models. For the model parameters selected, no obvious boundary effects were observed when the simulated domain was larger than (4 $\mu$m)$^3$, even though the sample size was still insufficient to be statistically representative of the entire microstructure at this volume. In addition, the use of the selected mesh resolution had very minor effects on the quasi-equilibrium contact angles and evolution kinetics. While the short-term evolution kinetics may be affected by the insufficient resolution of microstructures, the amount of TPBs and other properties after long-term coarsening can be identified by our simulations. Although our models contain approximations and the simulation results may be affected by uncertainties in the material properties, a reasonable agreement could be found for the TPB length reduction between coarsening experiments and simulation results using Model B. Unlike Model B, Model A overestimates the TPB reduction due to the evolution of the YSZ structure that is induced by an excess mobility at triple junctions built into the model. Our simulations show that the major portion of TPB reduction occurs during the early stages of coarsening and that stability of TPBs is observed provided that YSZ is nearly immobile. The evolution of active TPBs, active Ni surface areas, and tortuosity of the pore phase are investigated in our simulations. We found that coarsening has a smaller impact on the active Ni surface areas than on their total respective amounts or their inactive counterparts. In addition, the tortuosity of the pore phase was found to decrease slowly in all three orthogonal directions during coarsening. These phenomena are due to the fact that percolated phase clusters that dictate the active parameters and transport have typically larger characteristic length scales, $S_V^{-1}$, than isolated/dead-end phase clusters. Smaller clusters experience larger driving forces for coarsening, while larger clusters may grow at the expense of these regions if transport can be facilitated between them. Active Ni surface areas are also less reduced than their dead/isolated counterparts, and the transport property of the pore phase is slightly enhanced. In contrast, the reduction of active TPBs are found to account for most of the total TPBs reduction, which suggests that the TPB evolution is sensitive to microstructural details and is indirectly related to the evolution of bulk phases during coarsening. The proposed coarsening models provide insights into directing the design of anode microstructures. The model framework is general and can be applied to many other three-phase coarsening systems. Further experiments can help validate and improve these models. Appendix ======== Asymptotic analysis ------------------- The asymptotic analysis that was used is similar to that described in [@DongHee]. Briefly, we first determine the outer solution. Substituting equation \[eq:NDmu\] into \[eq:NDCH3\], dropping the overbars, and replacing the variables $\delta$ with $\zeta$, $\mu$ with $\mu_{out}(\overrightarrow{r},t,\zeta)$, and $\phi$ with $\phi_{out}(\overrightarrow{r},t,\zeta)$, we obtain the non-dimensional evolution equations $$\zeta^{2} \frac{\partial \phi_{out}}{\partial t} = \triangledown \cdot \left [ M(\phi_{out}) \triangledown \left ( \frac{\partial f}{\partial \phi_{out}} - \zeta^{2} \triangledown^{2} \phi_{out} \right ) \right ] \label{eq:NDCH4}$$ $$\mu_{out} = \frac{\partial f}{\partial \phi_{out}} - \zeta^{2} \triangledown^{2} \phi_{out} \label{eq:MuOut}$$ ### Outer expansion An outer expansion is performed by expanding the field $\phi_{out}$ and $\mu_{out}$ in our model in powers of $\zeta$ as follows: $$\phi_{out}(\overrightarrow{r},t,\zeta) = \phi_{out}^{(0)}(\overrightarrow{r},t) + \phi_{out}^{(1)}(\overrightarrow{r},t) \zeta + \phi_{out}^{(2)}(\overrightarrow{r},t) \zeta^{2} + \cdots \label{eq:Cexp}$$ $$\mu_{out}(\overrightarrow{r},t,\zeta) = \mu_{out}^{(0)}(\overrightarrow{r},t) + \mu_{out}^{(1)}(\overrightarrow{r},t) \zeta + \mu_{out}^{(2)}(\overrightarrow{r},t) \zeta^{2} + \cdots \label{eq:Muexp}$$ Substituting Eq. \[eq:Cexp\] into Eq. \[eq:NDCH4\], we obtain to the zeroth order in $\zeta$ $$\triangledown \cdot \left [M(\phi_{out}^{(0)}) \triangledown \left (\frac{\partial f}{\partial \phi_{out}^{(0)}} \right ) \right ] = 0. \label{eq:Outexp}$$ Eq. \[eq:Outexp\] is satisfied with the solution of $$\begin{aligned} \phi_{out}^{(0)}(\overrightarrow{r})=1~~~~~~ \forall \overrightarrow{r} \in \Omega_{+} \nonumber \\ \phi_{out}^{(0)}(\overrightarrow{r})=0~~~~~~ \forall \overrightarrow{r} \in \Omega_{-}, \label{eq:OutSolu}\end{aligned}$$ where $\Omega_{\pm}$ are the two bulk phase regions. This solution asserts that far from the interface, we have one of the equilibrium phases. By substituting Eqs. \[eq:Cexp\] and \[eq:Muexp\] into Eq. \[eq:MuOut\] and collecting the zeroth order of $\zeta$, we find $$\mu_{out}^{(0)} = \frac{\partial f}{\partial \phi_{out}^{(0)}}. \label{eq:MuC0}$$ Substituting Eq. \[eq:OutSolu\] into Eq. \[eq:MuC0\] yields $$\mu_{out}^{(0)}(\overrightarrow{r} \in \Omega_{\pm}) = 0. \label{eq:MuC0solu}$$ ### Inner expansion To denote the inner solution, we replace the variable $\delta$ with $\zeta$, $\mu$ with $\mu_{in}$ and $\phi$ with $\phi_{in}$. To facilitate the inner expansion, we introduce a moving coordinate system (M.C.S.): one coordinate $r$ is parallel to $\triangledown \phi$, while the other coordinate is the arclength $s$ along the interface (which is located at $r = 0$). The operators close to the interface ($r \approx 0$) are given by: $$\triangledown = \overrightarrow{e}_{r} \frac{\partial}{\partial r} + \overrightarrow{e}_{s} \frac{\partial}{\partial s},~~~~ \triangledown^{2} = \frac{\partial^{2}}{\partial r^{2}} + \kappa_{c} \frac{\partial}{\partial r} + \frac{\partial^{2}}{\partial s^{2}}, \label{eq:Opt}$$ where $\kappa_c$ is the local curvature. Introducing a stretched variable $z=r/\zeta$, $$\triangledown = \overrightarrow{e}_{r} \frac{1}{\delta} \frac{\partial}{\partial z} + \overrightarrow{e}_{s} \frac{\partial}{\partial s},~~~~ \triangledown^{2} = \frac{1}{\delta^{2}}\frac{\partial^{2}}{\partial z^{2}} + \kappa_{c} \frac{1}{\delta} \frac{\partial}{\partial z} + \frac{\partial^{2}}{\partial s^{2}}. \label{eq:StOpt1}$$ $$\frac{\partial}{\partial t} = \left ( \frac{D}{D t} \right ) - V_{n} \frac{1}{\delta} \frac{\partial}{\partial z} - V_{s} \frac{\partial}{\partial s}, \label{eq:StOpt2}$$ where $V_{n}$ and $V_{s}$ are the normal and tangential velocities, and $D/Dt$ represents the material derivative, which is the derivative taken based on the M.C.S. Multiplying Eq. \[eq:NDCH4\] by $\zeta^{2}$ and rewriting it in terms of a new field variable $\phi$ in the M.C.S. yields $$-V_{n} \frac{\partial \phi}{\partial z} \zeta^{3} + \left (\frac{D \phi}{D t} - V_{s} \frac{\partial \phi}{\partial s} \right ) \zeta^{4} = \frac{\partial}{\partial z} \left ( M \frac{\partial \mu}{\partial z} \right ) + \zeta \kappa_{c} M \frac{\partial \mu}{\partial z} + \zeta^{2} \frac{\partial}{\partial s} \left (M \frac{\partial \mu}{\partial s} \right ). \label{eq:InExpGov}$$ Expanding the fields in the power of $\zeta$ gives: $$\phi_{in}(z,s,t,\zeta) = \phi_{in}^{(0)}(z,s,t) + \phi_{in}^{(1)}(z,s,t) \zeta + \phi_{in}^{(2)}(z,s,t) \zeta^{2} + ... \label{eq:PhiExp}$$ $$\mu_{in} = \mu_{in}^{(0)} + \mu_{in}^{(1)} \zeta + \mu_{in}^{(2)} \zeta^{2} + ... \label{eq:MuExp}$$ Substituting Eq. \[eq:PhiExp\] and \[eq:MuExp\] into Eq. \[eq:InExpGov\] and collecting the zeroth order of $\zeta$ leads to: $$\frac{\partial}{\partial z} \left [ M \left (\phi_{in}^{(0)} \right ) \frac{\partial{\mu_{in}^{(0)}}}{\partial z} \right ] = 0. \label{eq:InExp0}$$ Integrating Eq. \[eq:InExp0\] with respect to $z$ yields $$M (\phi_{in}^{(0)} ) \frac{\partial{\mu_{in}^{(0)}}}{\partial z} = g_{0}(s,t). \label{eq:IntIn0}$$ Taking the limit $z \rightarrow \pm \infty$ and matching with Eq. \[eq:MuC0solu\], we obtain $$M (\phi_{in}^{(0)} ) \frac{\partial{\mu_{in}^{(0)}}}{\partial z} = 0. \label{eq:MatchIn0}$$ Because $\phi_{in}^{(0)}$ should vary smoothly from 1 to 0 as $z$ transitions from $+ \infty$ to $- \infty$, $M(\phi_{in}^{(0)} )$ is expected to be nonzero. We thus have $$\frac{\partial{\mu_{in}^{(0)}}}{\partial z} = 0 \quad \mathrm{or} \quad \mu_{in}^{(0)}(z,s,t) = g_{1}(s,t). \label{eq:MuIn0od}$$ Matching Eq. \[eq:MuIn0od\] with the zeroth order in outer field, the profile of $\phi_{in}^{0}$ along the $z$-direction is governed by $$\mu_{in}^{(0)} = 0 = \frac{\partial f}{\partial \phi_{in}^{(0)}} - \frac{\partial^{2} \phi_{in}^{(0)}}{\partial z^{2}},$$ which leads to the following relationship that is useful in change of veriables: $$\frac{\partial \phi_{in}^{(0)}}{\partial z} = \sqrt{2 f(\phi_{in}^{(0)})}. \label{eq:ChVar}$$ Collecting the first order of $\zeta$ in Eq. \[eq:InExpGov\] gives: $$\frac{\partial}{\partial z} \left (\frac{\partial \mu_{in}^{(1)}}{\partial z} \right ) + \kappa_{c} \frac{\partial \mu_{in}^{(0)}}{\partial z} = 0,$$ which in turn gives $\mu_{in}^{(1)} = \mu_{in}^{(1)}(s)$, or $\mu_{in}^{(1)}$ is independent of $z$. Similarly, collecting the second order of $\zeta$ in Eq. \[eq:InExpGov\], we find that $\mu_{in}^{(2)} = \mu_{in}^{(2)}(s)$, or $\mu_{in}^{(2)}$ is independent of $z$. Finally, by collecting the third order of $\zeta$ in Eq. \[eq:InExpGov\] and considering that $\mu_{in}^{(1)}$ and $\mu_{in}^{(2)}$ are independent of $z$, we have: $$-V_{n} \frac{\partial \phi_{in}^{(0)}}{\partial z} = \frac{\partial}{\partial z} \left [M(\phi_{in}^{(0)}) \frac{\partial \mu_{in}^{(3)}}{\partial z} \right ] + \frac{\partial}{\partial s} \left [M(\phi_{in}^{(0)}) \frac{\partial \mu_{in}^{(1)}}{\partial s} \right ]. \label{eq:InO3}$$ Integrating Eq. \[eq:InO3\] from $z = -\infty$ to $z = \infty$ gives: $$V_{n} = - \frac{\partial}{\partial s} \left [\int\limits_{-\infty}^{\infty}M(\phi_{in}^{(0)}) dz \right ] \frac{\partial \mu_{in}^{(1)}}{\partial s} = - I \cdot \frac{\partial^{2} \mu_{in}^{(1)}}{\partial s^{2}}, \label{eq:InIntO3}$$ where we assume a constant $I = \int\limits_{-\infty}^{\infty} M(\phi_{in}^{(0)}) dz$ as it is independent of $s$. Also, substituting Eq. \[eq:PhiExp\] and \[eq:MuExp\] into Eq. \[eq:NDmu\] and collecting the first order terms of $\zeta$ leads to: $$\mu_{in}^{(1)} = \frac{\partial^{2} f}{\partial \phi^{2}} \phi_{in}^{(1)} - \frac{\partial^{2} \phi_{in}^{(1)}}{\partial z^{2}} - \kappa_{c} \frac{\partial \phi_{in}^{(0)}}{\partial z}. \label{eq:Che0}$$ Multiplying Eq. \[eq:Che0\] by $\partial \phi_{in}^{(0)}/\partial z$ and integrating over z from $-\infty$ to $\infty$ gives: $$\mu_{in}^{(1)} = - \kappa_{c} \int\limits_{-\infty}^{\infty} \left (\frac{\partial \phi_{in}^{(0)}}{\partial z} \right )^{2} dz. \label{eq:Che1}$$ Performing a change of variables in Eq. \[eq:Che1\] using the relationship given in Eq. \[eq:ChVar\] leads to: $$\mu_{in}^{(1)} = - \kappa_{c} \int\limits_{0}^{1} \sqrt{2 f(\phi_{in}^{(0)})} d\phi = - \kappa_{c} \cdot J, \label{eq:Che2}$$ , wherein we assume a constant $J = \int\limits_{0}^{1} \sqrt{2 f(\phi_{in}^{(0)})} d\phi_{in}^{(0)}$. Combining Eq. \[eq:Che2\] and Eq. \[eq:InIntO3\] results in: $$V_{n} = I \cdot J \cdot \frac{\partial^{2} \kappa_{c}}{\partial s^{2}} \label{eq:Vn1}$$ Performing the integration of $I$ and $J$ (dropping the superscripts and subscripts for convenience) leads to: $$J = \int\limits_{0}^{1} \phi (1-\phi) d\phi = \frac{1}{6}.$$ $$I = \int\limits_{-\infty}^{\infty} M(\phi) dz = \int\limits_{0}^{1} \frac{M(\phi)}{\sqrt{2f}} d \phi= \bar{M}_{s} \int\limits_{0}^{1} \phi (1-\phi) d\phi = \frac{\bar{M}_{s}}{6}.$$ Substituting $I$ and $J$ into Eq. \[eq:Vn1\] and restoring the dimensions gives: $$V_{n} \frac{L^{3}}{D} = - \frac{\bar{M_{s}} L^{3}}{36} \frac{\partial^{2} \kappa_{c}}{\partial s^{2}}. \label{eq:Vn2}$$
--- abstract: 'We consider the problem of distinguishing between a set of arbitrary quantum states in a setting in which the time available to perform the measurement is limited. We provide simple upper bounds on how well we can perform state discrimination in a given time as a function of either the average energy or the range of energies available during the measurement. We exhibit a specific strategy that nearly attains this bound. Finally, we consider several applications of our result. First, we obtain a time-dependent Tsirelson’s bound that limits the extent of the Bell inequality violation that can be in principle be demonstrated in a given time $t$. Second, we obtain a Margolus-Levitin type bound when considering the special case of distinguishing orthogonal pure states.' author: - 'Andrew C. Doherty' - Stephanie Wehner title: 'A time-dependent Tsirelson’s bound from limits on the rate of information gain in quantum systems' --- Introduction ============ Entropic measures tell us how much information a quantum register $E$ contains about some classical register $X$ in principle. But just how quickly does this information become available to us? In this little note, we derive bounds on the amount of information available after a given time $t$. As expected, our bounds depend on the resources we have available in the form of the available energy. Throughout this paper, we will choose to measure information in terms of the [min-entropy]{}, which is the relevant quantity when we consider single-shot experiments and quantum cryptography. As we will explain in detail below, this measure is directly related [@renato:operational] to the probability of success in state discrimination [@helstrom:detection; @holevo; @belavkin:optimal; @yuen:maxState; @surveyDiscr]. As a result, we focus on bounding the probability of success in distinguishing states $\{\rho_x\}_{x \in {\mathcal{X}}}$ where we are given $\rho_x$ with probability $p_x$. Let $P_{\rm guess}(X\|E)_{H,t}$ denote this success probability after time $t$ when using a particular Hamiltonian $H$ in the measurement process. After providing a more careful discussion of the measurement process, we show the following results. Results ------- A bound for two states: We first consider the case of only two input states $\rho_0,\rho_1$, for which it is easy to compute the optimal success probability if we have unlimited time (or resources) available [@helstrom:detection]. We first provide a general bound in terms of the spectrum of the Hamiltonian (Corollary \[cor:traceDist\]). For the special case of of two equiprobable states ($p_0 = p_1 = 1/2$), this bound simply reads $$\begin{aligned} \label{eq:twoStateSummary} P_{\rm guess}(X\|E)_{H,t} \leq \frac{1}{2} + \frac{\gamma t \\|H\\|_\infty D(\rho_0,\rho_1)}{2\hbar}\ ,\end{aligned}$$ where $D(\rho_0,\rho_1)$ is the trace distance between the two states, and $\gamma$ is a small constant. This bound is directly related to our ability to distinguish two inputs states given an unlimited amount of time, where the best measurements gives us [@helstrom:detection] $$\begin{aligned} P_{\rm guess}(X\|E) = \frac{1}{2} + \frac{D(\rho_0,\rho_1)}{2}\ .\end{aligned}$$ We proceed to show that our bound is nearly tight up to a constant factor (Theorem \[thm:attaining\]) by providing an explicit measurement strategy. Finally, we prove a bound in terms of the average energies of the input states (Theorem \[thm:avgEnergyTwoStates\]). However, this bound does not compare as easily to the case of unlimited time. A bound for many input states: When considering the case of an arbtirary number of input states $\rho_0,\ldots,\rho_{N-1}$ it is difficult to compute the maximum success probability even in the case of unlimited time. In particular, no general closed form expression is known – only for the case of single qubit encodings does there exist a way to construct the optimal measurements geometrically [@barbara:qubit]. In general, we can only approximate the optimal measurements numerically [@yuen:maxState; @eldar:sdpDetector; @eldar:sdp; @werner:iterate; @jezek:iterate; @jezek:iterate2; @tyson:oldIterate; @tyson:newIterate], or resort to bounds on the success probability [@belavkin:optimal; @belavkin:radio; @mochon:pgm; @wootters:pgm; @tyson:pgm; @tyson:estimates; @deepthi:pi]. As such, it becomes harder to relate $P_{\rm guess}(X\|E)_{H,t}$ to case of unlimited time. We hence provide a general bound in terms of the average energies alone. In particular, we show (Theorem \[thm:manyStates\]) that $$\begin{aligned} P_{\rm guess}(X\|E) \leq p_{{ {x_{\max}} }} + \sum_{x=0}^{N-1} p_x {\mathop{\mathrm{tr}}\nolimits}\left(H\rho_x\right)\ ,\end{aligned}$$ where ${ {x_{\max}} }$ is the smallest $x \in \{0,\ldots,N-1\}$ such that $p_{{ {x_{\max}} }} \geq p_x$ for all $x$. Applications: Finally, we discuss two applications of our bound. The first is to the study of Bell inequalities [@bell]. Typically, we care about determining the maximum quantum violation of such inequalities. In contrast, we ask what is the maximum violation that can be achieved in a fixed amount of time. When considering such inequalities as games between two players Alice and Bob (see Section \[sec:game\]), the ”amount” of quantum violation is determined by the probability $p_{\rm win}$ that the players win the game maximized over all states and measurements. For the CHSH inequality [@chsh], we have that classically $$\begin{aligned} p_{\rm win} \leq \frac{3}{4}\ \end{aligned}$$ for any strategy of Alice and Bob. However in quantum mechanics there exists a strategy that achieves $$\begin{aligned} p_{\rm win} = \frac{1}{2} + \frac{1}{2\sqrt{2}}\ ,\end{aligned}$$ which is optimal [@tsirel:original]. Here, we show (Corollary \[cor:tsirel\]) that if we demand answers from Alice and Bob after time $t$ $$\begin{aligned} p_{\rm win} \leq \frac{3}{4} + \frac{\gamma t \\|H\\|_\infty}{\hbar\sqrt{2}}\ ,\end{aligned}$$ where $H$ is Bob’s Hamiltonian involved in the measurement process, and $\gamma$ is a small constant. We will also see that to achieve Tsirelson’s bound, Alice and Bob need time at least $$\begin{aligned} t \geq \frac{\hbar}{\gamma \\|H\\|_\infty}\ .\end{aligned}$$ Our bounds tell us that there does indeed a fundamental time that is needed to establish non-local correlations of a certain strength. We will discuss these bounds in detail in Section \[sec:tsirel\]. As a second application, we use our bound to obtain a form of the Margolus-Levitin theorem [@mlTheorem] which provides us with a lower bound on how much time it takes to transform a pure state into an orthogonal state. Since the Margolus-Levitin theorem provides a bound on the speed of evolution, it clearly provides a bound on the minimum amount of time that is required to obtain the optimal (time unlimited) success probability for state discrimination. Yet, note that we are interested in bounding $P_{\rm guess}(X\|E)_{H,t}$ even for shorter periods of time. We will discuss the relation of our work and the Margolus-Levitin theorem in detail in Section \[sec:ML\]. Related work ------------ Next to the Margolus-Levitin theorem [@mlTheorem], our work is related to several bounds [@levitin1; @pendry] on how fast information can be transmitted in principle given energy constraints (see [@bekenstein:survey] for a survey of results). These bounds generally consider the von Neumann entropy as a measure of information and are concerned with determining the capacity for sending information as a function of energy. That is, they consider how fast we could convey information in the best possible way. In contrast, we consider the case of arbitrary encodings $\rho_x$, which may not be optimal to transmit classical information. In fact, even in the case of unlimited time the probability that we can reconstruct $x$ from $\rho_x$ could be very small. Our setting also differs in the sense that we focus solely on extracting classical information into a classical register in a sense that we will make precise below. Our work is also related to several previous papers [@lazyStates; @gogolin:mthesis; @hayashi:purity] that study the rate of change in entropies of a system that is in contact with an environment. Again, our work is a somewhat different flavor since we are interested in extracting classical information, and our bounds furthermore involve average energies, rather than the largest energy $\\|H\\|_\infty$ of the (interaction) Hamiltonian $H$ alone. Gaining classical information ============================= Quantifying information ----------------------- Let us now consider more formally what we mean by gaining classical information encoded in a quantum system. Imagine that there is some finite set ${\mathcal{X}}$ of possible classical symbols to be encoded. For any symbol $x \in {\mathcal{X}}$, we thereby use $\rho_x \in {\mathcal{B}}({\mathcal{H}_{\rm enc}})$ to denote its encoding into a quantum state on the system ${\mathcal{H}_{\rm enc}}$. We also refer to ${\mathcal{H}_{\rm enc}}$ as the encoding space. Our a priori ignorance about the classical information $x$ is captured by the probability distribution $p_x$ according to which the encoding space is prepared in the state $\rho_x$. Throughout, we quantify how much information we have about $x$ given access to the encoding space ${\mathcal{H}_{\rm enc}}$ in terms of the min-entropy [@renato:operational] $$\begin{aligned} {\ensuremath{ {\rm H}}_{\infty}}(X\|E) := - \log P_{\rm guess}(X\|E)\ ,\end{aligned}$$ where $$\begin{aligned} P_{\rm guess}(X\|E) := \sup_{\substack{\forall x M_x \geq 0\\ \sum_{x \in {\mathcal{X}}} M_x = {\mathbb{I}}}} \sum_{x \in {\mathcal{X}}} p_x {\mathop{\mathrm{tr}}\nolimits}\left(M_x \rho_x\right)\ ,\end{aligned}$$ is the probability that we guess $x$, maximized over all possible measurements on the encoding space. Finding the optimal measurement is known as state discrimination and can be done using semidefinite programming [@yuen:maxState; @eldar:sdpDetector]. The min-entropy accurately measures information in a cryptographic setting [@renato:diss], and for single shot experiments. This is in contrast to the von Neumann entropy which is concerned with the asymptotic case of a large number of identical experiments. The min-entropy and the von Neumann entropy can be arbitrarily different, as is easily seen by considering the example where the encoding is trivial, that is, $\rho_x = \rho_{x'}$ for all $x$ and $x'$. The strategy that maximizes the guessing probability $P_{\rm guess}(X\|E)$ is then simply given by outputting the most likely symbol, i.e., ${\ensuremath{ {\rm H}}_{\infty}}(X\|E) = - \log \max_x p_x$ [^1], and the conditional von Neumann entropy obeys ${\ensuremath{ {\rm H}}}(X\|E) = {\ensuremath{ {\rm H}}}(X) = - \sum_x p_x \log p_x$. Consider now $\Sigma = \{0,1\}^n$ to be the set of bitstrings of length $n$ and suppose the all ’0’ string occurs with probability $p_{0^n} = 1/2$, and with probability $1/2$ any of the remaining strings occurs with equal probability. Clearly, we have ${\ensuremath{ {\rm H}}_{\infty}}(X\|E) = 1$, whereas ${\ensuremath{ {\rm H}}}(X) \approx n/2$. That is, the von Neumann entropy can be very large, even if there is one symbol that occurs with extremely high probability. We will remark on the rate of information extraction from a quantum system in terms of the von Neumann entropy later on, but focus on the single shot case given by the min-entropy, or equivalently the probability of error in state discrimination. Producing a classical output ---------------------------- To determine how quickly we can acquire classical information, we first need to specify what it means to output classical information from a measurement. Here, we model this process with the help of an additional ‘classical’ ancilla system ${\mathcal{H}_{\rm anc}}$ that contains the output. A classical system is associated with a fixed basis, which without loss of generality we take to be the computational basis. Preparation and measurement of a classical system can only be done in this basis, which intuitively corresponds to the idea of storing classical information: The ancilla can be prepared in any state of the fixed basis, and is subsequently measured in this basis after time $t$. The information contained in this register captures the notion of a classical probability distribution over the basis elements. We model the process of state discrimination as follows. The problem is to discriminate between $N$ states $\rho_x$ on the encoding space ${\mathcal{H}_{\rm enc}}$, where $N$ is the number of possible classical symbols. At the beginning of the experiment the ancilla system is initialized to the symbol occurring with the largest probabily ${{\| {\rm x_{\max}}\rangle}}$ where $$\begin{aligned} x_{\max} := {\mathop{\mathrm{argmax}}\nolimits}_x p_x\ .\end{aligned}$$ This initial condition captures the distinguisher’s apriori knowledge: recall without access to the quantum register ${\ensuremath{ {\rm H}}_{\infty}}(X\|E) = - \log p_{{ {x_{\max}} }}$. If the there are multiple classical symbols with the same value ${p_{{ {x_{\max}} }}}$, we take the smallest one in lexicographic order. We will discuss the choice of initial state in detail below. The ancilla system has total dimension $d_{{\mathcal{H}_{\rm anc}}} = N$ and the other directions correspond to the classical symbols $x$. The experimenter implements a unitary $U$ on ${\mathcal{H}_{\rm enc}}\otimes {\mathcal{H}_{\rm anc}}$ during a specified time $t$. At this point the ancilla system is passed to a referee who will decide whether information has been gathered successfully by measuring ${\mathcal{H}_{\rm anc}}$ in the computational basis, using measurement operators $$\begin{aligned} P_x := {\|{x}\rangle\langle{x}\|}\ ,\end{aligned}$$ where the subscript $x$ denotes the corresponding classical output. Hence the success probability of correctly identifying the state $\rho_x$ using this procedure when the ancilla was initially in the state ${{\| {\rm x_{\max}}\rangle}}$ is given by $$\begin{aligned} \label{eq:succProb} {\mathop{\mathrm{tr}}\nolimits}\left( ({\mathbb{I}}_{{\mathcal{H}_{\rm enc}}} \otimes P_x) U (\rho_x \otimes {{\|{ {\rm x_{\max}}}\rangle\langle{ {\rm x_{\max}}}\|}}\|)U^\dagger\right)\ .\end{aligned}$$ See Figure \[fig:ancillaUse\] for a schematic depiction of this process. Note that the ancilla is measured by the referee at no time cost. This is a natural assumption in our setting where we imagine that the final information is extracted by a referee who is not limited by any energy constraints. Such a referee naturally arises in, for example, the setting of Bell inequalities which we consider later. We will from now on assume that measurements producing classical outcomes are always performed this way. ![Our protocol for distinguishing quantum states in finite time. First, the encoding register is placed into an encoding $\rho_x$ of the classical symbol $x$ chosen with probability $p_x$. The ancilla is intialized in the state ${{\| {\rm x_{\max}}\rangle}}$. Second, we can perform a unitary interaction $U = \exp(-i Ht/\hbar)$ for time $t$ between the encoding and the ancilla register. Finally, the ancilla register is measured by the referee in the computational basis to determine a guess $x'$ for $x$. If $x'=x$, then we successfully recovered the classical information. In the setting of Bell inequalities considered later on, the ancilla register is simply the message returned to the referee.[]{data-label="fig:ancillaUse"}](ancillaUse.pdf) To bound how much min-entropy we have after time $t$, our goal is to place bounds on the success probability in terms of the unitary $$\begin{aligned} U = \exp\left(-\frac{iHt}{\hbar}\right)\ ,\end{aligned}$$ that is, in terms of the interaction Hamiltonian $$\begin{aligned} H = \sum_n E_n {\|{E_n}\rangle\langle{E_n}\|}\end{aligned}$$ and the time $t$. Throughout, we will assume that $H \geq 0$ and that the lowest energy level is in fact $E_0 = 0$. Any other Hamiltonian differs from such an $H$ by a term proportional to the identity, which does not contribute to the speed of information gain. We explictly chose not to use the common convention $\hbar=1$ to make it easier to draw comparisons to the Margolus-Levitin theorem [@mlTheorem] later on. Before turning to our actual bounds, let us first introduce some additional notation which we will refer to throughout the paper. We will use $$\begin{aligned} \tilde{\rho}_x := \rho_x \otimes {{\|{ {\rm x_{\max}}}\rangle\langle{ {\rm x_{\max}}}\|}}\ ,\end{aligned}$$ to denote the combined state consisting of the input state $\rho_x$ on the encoding space, and the initial state of the ancilla ${{\|{ {\rm x_{\max}}}\rangle\langle{ {\rm x_{\max}}}\|}}$. We also write $$\begin{aligned} R := U - I = \sum_n (\exp(-i E_n t/\hbar) - 1) {\|{E_n}\rangle\langle{E_n}\|}\ .\end{aligned}$$ Furthermore, it will be convenient to rewrite the success probability in terms of measurement operators $$\begin{aligned} \label{eq:measurementOperators} M_x := U^\dagger({\mathbb{I}}\otimes P_x)U = {\mathbb{I}}\otimes P_x + W_x\ ,\end{aligned}$$ as ${\mathop{\mathrm{tr}}\nolimits}(M_x \tilde{\rho}_x)$, where $$\begin{aligned} \label{eq:Wdef} W_x = W_{x}^{1} + W_{x}^{2}\ ,\end{aligned}$$ and $$\begin{aligned} W_{x}^{1} &:= ({\mathbb{I}}\otimes P_x)R + R^\dagger ({\mathbb{I}}\otimes P_x)\ ,\\ W_{x}^{2} &:= R^\dagger ({\mathbb{I}}\otimes P_x) R\ .\end{aligned}$$ The average success probability for a particular Hamiltonian $H$ and time $t$ can now be written as $$\begin{aligned} P_{\rm guess}(X\|E)_{H,t} := \sum_{x \in {\mathcal{X}}} p_x {\mathop{\mathrm{tr}}\nolimits}\left(M_x\tilde{\rho}_x\right)\ .\end{aligned}$$ Time vs. information gain ========================= We are now ready to derive our bounds. For simplicity, we will outline how this can be done for the case of two equiprobable states, and merely state our general result. Precise statements as well as a detailed derivation can be found in the appendix. An upper bound to $P_{\rm guess}(X\|E)$ -------------------------------------- We now first derive an upper bound to the guessing probability. For the case of two equiprobable states (i.e., $N= 2$ and $p_x = 1/2$ for all $x \in {\mathcal{X}}$, such bounds are easy to obtain when we allow unlimited time (or energy). In particular, it is well known that in this case the success probability is given by [@helstrom:detection] $$\begin{aligned} \label{eq:distStandard} P_{\rm guess}(X\|E) := \frac{1}{2} + \frac{D(\rho_0,\rho_1)}{2}\ ,\end{aligned}$$ where $D(\rho_0,\rho_1) = \frac{1}{2} \\|\rho_0 - \rho_1\\|_1$ is the trace distance of the two states. Let us now consider what happens in our time limited scenario for a particular interaction Hamiltonian $H$. First of all, recall that for two equiprobable states, the ancilla is initialized to the smallest value $$\begin{aligned} {{\| {\rm x_{\max}}\rangle}}= {\|0\rangle}\ .\end{aligned}$$ For two states, the success probability $P_{\rm succ}$ averaged over the choice of input state, using the measurement given by operators $M_1$ and $M_0 = {\mathbb{I}}- M_1$ from , can now be expressed as $$\begin{aligned} &P_{\rm succ}(X\|E)_{H,t} \\ &= \frac{1}{2}\left[ {\mathop{\mathrm{tr}}\nolimits}\left(M_0 \tilde{\rho}_0\right) + {\mathop{\mathrm{tr}}\nolimits}\left(M_1 \tilde{\rho}_1\right)\right]\nonumber\\ &=\frac{1}{2}\left[1 + {\mathop{\mathrm{tr}}\nolimits}\left(M_1(\tilde{\rho}_1 - \tilde{\rho}_0)\right)\right]\\ &=\frac{1}{2}\left[1 + {\mathop{\mathrm{tr}}\nolimits}\left(\rho_1 - \rho_0\right){\mathop{\mathrm{tr}}\nolimits}\left(P_1 {{\|{ {\rm x_{\max}}}\rangle\langle{ {\rm x_{\max}}}\|}}\right) + \nonumber \right.\\ &\qquad \left.{\mathop{\mathrm{tr}}\nolimits}\left(W_1(\tilde{\rho}_1 - \tilde{\rho}_0)\right)\right]\\ &=\frac{1}{2} + \frac{{\mathop{\mathrm{tr}}\nolimits}\left(W_1(\tilde{\rho}_1 - \tilde{\rho}_0)\right)}{2}\ ,\label{eq:finalPSUCC}\end{aligned}$$ where the fourth equality follows immediately from the fact that $P_1 {{\|{ {\rm x_{\max}}}\rangle\langle{ {\rm x_{\max}}}\|}}= {{\|{ {\rm x_{\max}}}\rangle\langle{ {\rm x_{\max}}}\|}}P_1 = 0$. Let us now upper bound the term involving $W_1$. Again using that $P_1 {{\|{ {\rm x_{\max}}}\rangle\langle{ {\rm x_{\max}}}\|}}= {{\|{ {\rm x_{\max}}}\rangle\langle{ {\rm x_{\max}}}\|}}P_1 = 0$, we have $$\begin{aligned} {\mathop{\mathrm{tr}}\nolimits}\left(W_{1}^{1}(\tilde{\rho}_1 - \tilde{\rho}_0)\right) = 0\ .\end{aligned}$$ Define $\tilde{A} := \tilde{\rho}_1 - \tilde{\rho}_0$, and consider its diagonalization $\tilde{A} = \sum_j \lambda_j {\|{u_j}\rangle\langle{u_j}\|}$. Let $\tilde{A}^+ := \sum_{j,\lambda_j \geq 0} \lambda_j {\|{u_j}\rangle\langle{u_j}\|}$ and $\tilde{A}^- := \tilde{A} - \tilde{A}^+$. Using the fact that $R \cdot R^\dagger$ is a positive map [@bathia:posBook] and $0 \leq {\mathbb{I}}\otimes P_x \leq {\mathbb{I}}$, we can now bound the term $$\begin{aligned} {\mathop{\mathrm{tr}}\nolimits}\left(W_{1}^{2} \tilde{A}\right) &\leq {\mathop{\mathrm{tr}}\nolimits}\left(R \tilde{A}^+ R^\dagger\right)\\ &\leq 2 \sum_n (1 - \cos(t E_n/\hbar)) {\langleE_n\|} \tilde{A}^+ {\|E_n\rangle}. \end{aligned}$$ Substituting back into our original bound (\[eq:finalPSUCC\]) gives us $$P_{\rm succ}(X\|E)_{H,t}\leq \frac{1}{2} +\sum_n (1 - \cos(t E_n/\hbar)) {\langleE_n\|} \tilde{A}^+ {\|E_n\rangle}\ . \label{eq:protoBound}$$ This is the basic inequality that we can use, along with some restriction on the allowed energies $E_n$, to bound the success probability for state discrimination in time $t$. In the rest of the paper we will apply this in two main settings, bounded maximum energy, and bounded average energy. ### A bound in terms of the maximum energy From , we can immediately obtain a bound on the success probability for state discrimination in terms of the maximum energy $\\|H\\|_\infty$ of the coupling Hamiltonian $H$. ($\\|H\\|_\infty$ is just the largest eigenvalue of $H$.) This bound is attractive since it is simple to derive and has the appealing feature that it involves the trace distance between the two states, and is thus directly related to the probability that we distinguish the two states given an unlimited amount of time. However, there are many systems of physical interest where the maximum energy of system states is effectively unbounded. Even though we may without loss of generality assume that the spectrum is bounded for a particular set of input states (see appendix), this bound is nevertheless quite unsatisfying in these situations since it can be very weak. For this reason, we use the fundamental inequality in the next section to derive a bound on the success probability that depends only on the average energy. Note that since ${\mathop{\mathrm{tr}}\nolimits}(\tilde{A}^+) = D(\tilde{\rho}_0,\tilde{\rho}_1) = D(\rho_0,\rho_1)$ we immediately obtain that the success probability obeys $$\begin{aligned} &P_{\rm succ}(X\|E)_{H,t} \\ &\leq \frac{1}{2} + (1 - \cos(t C_{\max}/\hbar)) D(\rho_0,\rho_1)\ ,\nonumber\end{aligned}$$ where $C_{\max} = {\mathop{\mathrm{argmax}}\nolimits}_{E_n} (1 - \cos(t E_n/\hbar))$. If $t E_{n}/\hbar \leq 1$ for all $n$, then this upper bound simply reads $$\begin{aligned} \label{eq:spectrumBound} &P_{\rm succ}(X\|E)_{H,t} \\ &\leq \frac{1}{2} + (1 - \cos(t \\|H\\|_\infty/\hbar)) D(\rho_0,\rho_1)\ ,\nonumber\end{aligned}$$ which will be useful for comparison below. For larger values of $t E_n/\hbar$ it is easy to see that $$\begin{aligned} \label{eq:simpleTrace} P_{\rm succ}(X\|E)_{H,t} \leq \frac{1}{2} + \frac{\gamma t \\|H\\|_\infty D(\rho_0,\rho_1)}{2 \hbar}\ ,\end{aligned}$$ where $$\begin{aligned} \gamma := \left\{ \begin{array}{ll}5/\pi & \mbox{if } 1 < t E_n/\hbar < 4\ ,\\ 3/\pi & \mbox{otherwise}\ . \end{array}\right.\end{aligned}$$ ### A bound in terms of the average energy A sometimes more satisfying bound can be obtained in terms of the average energy. Note that we can upper bound as $$\begin{aligned} \frac{1}{2} + \frac{1}{2}\frac{\gamma t}{\hbar} \sum_n E_n {\langleE_n\|} \tilde{A}^+ {\|E_n\rangle}\ ,\end{aligned}$$ and hence we may use the fact that $$\begin{aligned} \tilde{A}^+ &= \frac{1}{2}(\tilde{A}^+ - \tilde{A}^-) + \frac{1}{2}(\tilde{A}^+ + \tilde{A}^-)\\ &\frac{1}{2}(\tilde{\rho}_1 - \tilde{\rho}_0) + \frac{1}{2}\|\tilde{\rho}_1 - \tilde{\rho}_0\|\ ,\end{aligned}$$ to obtain $$\begin{aligned} &P_{\rm succ}(X\|E)_{H,t}\\ &\leq \frac{1}{2} + \frac{ \gamma t \left({\mathop{\mathrm{tr}}\nolimits}(H\|\tilde{\rho}_1 - \tilde{\rho}_0\|) + {\mathop{\mathrm{tr}}\nolimits}(H\tilde{\rho}_1) - {\mathop{\mathrm{tr}}\nolimits}(H\tilde{\rho}_0)\right)} {4 \hbar} \ .\nonumber\end{aligned}$$ Now, the asymmetry between the labels $0$ and $1$ is inessential. The bound is true if we swap the two state labels, as may be seen by repeating the above derivation swapping the role of the two state labels. Averaging these two bounds we find the following symmetric bound $$\begin{aligned} \label{eq:avg2States} P_{\rm succ}(X\|E)_{H,t} \leq \frac{1}{2} + \frac{ \gamma t {\mathop{\mathrm{tr}}\nolimits}(H\|\tilde{\rho}_1 - \tilde{\rho}_0\|)} {4 \hbar} \ .\end{aligned}$$ This bound should be compared with the bound in terms of the maximum energy in which the trace distance appears. The quantity on the right hand side of is loosely an energy-weighted trace distance. Whereas this bound is certainly stronger for a particular choice of $H$, it does not any longer bear an obvious quantitative relation to the Helstrom bound in terms of the trace distance. In deriving we have made use of the knowledge of the optimal measurements for distinguishing a pair of states. This is no longer possible in more complicated cases, even where unlimited time is allowed [@surveyDiscr]. We can weaken the bound somewhat, using the fact that $\rho,H\geq 0$ to obtain a bound explicitly in terms of the average energy as follows $$\begin{aligned} \label{eq:avg2States2} &P_{\rm succ}(X\|E)_{H,t}\\ &\leq \frac{1}{2} + \frac{ \gamma t {\mathop{\mathrm{tr}}\nolimits}[H(\tilde{\rho}_0 + \tilde{\rho}_0)]} {4 \hbar} \ .\nonumber\end{aligned}$$ So we see that the average energy of the joint system and ancilla place a bound on the success probability of state discrimination, as claimed. This bound may be generalized easily to the case of more than two classical symbols and an aribtrary distribution $\{p_x\}_x$. We show in the appendix that Suppose $H \geq 0$. Then the probability of distinguishing $\rho_0,\ldots,\rho_{N-1}$ given with probabilities $p_0,\ldots,p_{N-1}$ using the Hamiltonian $H$ obeys $$\begin{aligned} &P_{\rm succ}(X\|E)_{H,t}\\ &\leq {p_{{ {x_{\max}} }}}+\frac{\hat{\gamma}t}{\hbar} \sum_{x=0}^{N-1} p_x {\mathop{\mathrm{tr}}\nolimits}\left(H\tilde{\rho}_x\right)\ ,\nonumber \end{aligned}$$ where $$\begin{aligned} \hat{\gamma} := \left\{ \begin{array}{ll}5/\pi & \mbox{if } \forall E_n, 1 < t E_n/\hbar < 4\ ,\\ 3/\pi & \mbox{otherwise}\ . \end{array}\right. \end{aligned}$$ Note that the term $\sum_{x} p_x {\mathop{\mathrm{tr}}\nolimits}\left(H\tilde{\rho}_x\right)$ is the energy of the encoding and ancilla register averaged over the choice of input symbols. A lower bound on $P_{\rm guess}(X\|E)$ ------------------------------------- We now exhibit a specific measurement strategy for two equiprobable states, which attains our upper bound up to a constant factor. We again focus on the case of two possible input states, as for the general setting there is no analytic procedure of obtaining the optimal measurements even in the setting of unlimited time. Our construction for two states will make explicit use of this optimal measurement. Let $A = \rho_1 - \rho_0$. It is well known [@helstrom:detection] that the optimal distinguishing measurement in the time unlimited case without the use of an ancilla is given by $\{\Pi_{A^+},\Pi_{A^-}\}$, where $\Pi_{A^+}$ and $\Pi_{A^-}$ are projectors on the positive and negative eigenspace of $A$ respectively. To construct our Hamiltonian $H$, let us diagonalize $A = \sum_j \lambda_j {\|{u_j}\rangle\langle{u_j}\|}$, and define $A^+ := \sum_{j,\lambda_j \geq 0} \lambda_j {\|{u_j}\rangle\langle{u_j}\|}$ and $A^- := A^+-A$. Consider the operator $$\begin{aligned} \hat{H} &:= \Pi_{A^-} \otimes {\mathbb{I}}+\Pi_{A^+} \otimes ({\|0\rangle\langle1\|} + {\|1\rangle\langle0\|})\ .\end{aligned}$$ Clearly, $\hat{H}$ is Hermitian and unitary, and hence has eigenvalues $\pm 1$. In fact, $\hat{H}$ is the unitary we would use to achieve the optimum distinguishing probability if we were unconcerned with time. We now define a Hamiltonian $H$ $$\begin{aligned} \label{eq:achieveH} H := E_{\rm max} (\hat{H} + {\mathbb{I}})/2\ .\end{aligned}$$ For comparison with our upper bound of $H$ obeys the condition $H \geq 0$ and has largest eigenvalue equal to $E_{\rm max} = \\|H\\|_\infty$. A simple calculation provided in the appendix shows that for our choice of $H$ we have $$\begin{aligned} P_{\rm succ}(X\|E)_{H,t} = \frac{1}{2} + \frac{1}{4}(1 - \cos(t \\|H\\|_\infty/\hbar)) D(\rho_0,\rho_1)\ \end{aligned}$$ which gives a lower bound to $P_{\rm succ}(X\|E)$ maximized over all possible $H$ in time $t$. This bound matches the upper bound of up to a factor of $1/4$. Note that $\hat{H}$ effectively implements a variant of the controlled-NOT (c-NOT) operation on the encoding space and the ancilla. For more than two inputs states, one could construct a similar $\hat{H}$ implementing a controlled addition mod $N$ on the ancilla, as long as the optimum distinguishing measurement in the case of unlimited time is a projective measurement on the encoding space. This would give a similar relation between time and the original probability of distinguishing the given states. However, it is known that there do exist choices of encodings $\rho_x$ such that the optimum measurement is not projective, and hence we omit this restricted form of generalization. Applications ============ Let us now consider several applications of our simple bound. Minimum distinguishing time and the Margolus-Levitin theorem {#sec:ML} ------------------------------------------------------------ The first application we are interested in, is a return to our initial question: Just how quickly can we acquire information? That is, what is the minimum time needed to extract classical information encoded in a quantum system? Note that with the Hamiltonian $H$ in the lower bound for two equiprobale states, there does indeed exist a way to optimally distinguish the two states in time $t = \hbar \pi/\\|H\\|_\infty$. However, since there is a small gap to our upper bound it would be an open question, whether it is possible to achieve the same in an even shorter amount of time. ### Minimum time Yet, note that our upper bounds on $P_{\rm succ}(X\|E)_{H,t}$ can also be understood as lower bounds on the time required to optimally distinguish the given states, retrieving the maximum amount of information from the encoding. Let us first consider our most general bound for large ${\mathcal{X}}$. We have that if we can distinguish optimally in time $t_{\rm distinguish}$ our upper bound must be at least as large as the optimum $P_{\rm guess}(X\|E)$. That is, $$\begin{aligned} {p_{{ {x_{\max}} }}}+ \frac{\gamma t_{\rm distinguish}}{\hbar}\sum_{x=0}^{N-1} p_x {\mathop{\mathrm{tr}}\nolimits}\left(H\tilde{\rho}_x\right) \geq P_{\rm guess}(X\|E)\ ,\end{aligned}$$ and hence $$\begin{aligned} \label{eq:minTime} t_{\rm distinguish} \geq \frac{(P_{\rm guess}(X\|E) - {p_{{ {x_{\max}} }}}) \hbar}{\gamma \sum_{x=0}^{N-1} p_x {\mathop{\mathrm{tr}}\nolimits}\left(H\tilde{\rho}_x\right)}\ .\end{aligned}$$ ### Margolus-Levitin theorem Let us now consider the special case where two equiprobable input encodings are perfectly distinguishable. That is, $\rho_0 = {\|{0}\rangle\langle{0}\|}$ and $\rho_1 = {\|{1}\rangle\langle{1}\|}$. Our task is now quite simple: We merely wish to turn the state ${\|1\rangle}{\|0\rangle}$ of the encoding and ancilla system to the state ${\|1\rangle}{\|1\rangle}$, that is, we wish to transform one vector into its orthogonal. Note that given unlimited time (or energy) we can succeed perfectly at this task and hence $P_{\rm guess}(X\|E) = 1$. From we thus have $$\begin{aligned} t_{\rm distinguish} &\geq \frac{\hbar}{2\gamma {\mathop{\mathrm{tr}}\nolimits}\left(H\tilde{\rho}_1\right)}\ .\end{aligned}$$ Our bound can hence also be understood as putting a limit on the time that it takes to turn a state vector to its orthogonal (on the ancilla), given some additional resource (the encoding register). A bound on the minimum time that it takes to turn a vector into its orthogonal is indeed known as the Margolus-Levitin theorem [@mlTheorem]. In particular, their bound applied to our situation involving both the encoding and the ancilla register gives $$\begin{aligned} \label{eq:ML} t_{\rm ML} & \geq \frac{\hbar \pi}{2 {\mathop{\mathrm{tr}}\nolimits}\left(H\tilde{\rho}_1\right)}\ .\end{aligned}$$ Such a bound had previously only been derived from the time-energy uncertainty principle where instead of the average energy, we have the energy spread, i.e, the difference in the largest and smallest eigenvalue of the Hamiltonian (see [@lloyd:margolus] for a review of history). The Margolus-Levitin theorem has been used to place bounds on the fundamental speed of computation [@lloyd:margolus], and was even slightly improved for some special cases [@mlImprovement]. Note however that for the Hamiltonian constructed in we have ${\mathop{\mathrm{tr}}\nolimits}\left(H\tilde{\rho}_1\right) = E_{\rm max}/2$ and hence the bound provided by Margolus-Levitin is in fact tight as we know that lets us achieve the optimum success probability in time $t = \hbar \pi/E_{\rm max}$. This shows that it is our upper, rather than our lower bound that can be improved. Since we have $\gamma = 3/\pi$ or $\gamma = 5/\pi$ depending on the parameters, our bound is slightly worse than the Margolus-Levitin bound which stems from our somewhat crude bound on $(1 - \cos(t E_n/\hbar))$. Note, however, that our bound considers a more specialized situation, namely turning the ancilla to its orthogonal given the encoding, but in turn applies to any kind of input states. That we obtain a Margolus-Levitin type theorem as a side effect of our analysis is not very surprising: Clearly, the speed of dynamical evolution places a bound on how quickly we can transfer information from one system into the other. In turn however, note that a bound on how quickly transformation can be transferred does translate into bounds on the speed of evolution as well and one can think of the speed of dynamical evolution when applied to a computation [@lloyd:margolus] as being limited by how quickly one can transfer the necessary information required for the subsequent stage of computation. Time-dependent Tsirelson-bound {#sec:tsirel} ------------------------------ As another example on how our bound can be used we will derive a time-dependent Tsirelson’s bound [@tsirel:original] for the Bell inequality [@bell] known as the CHSH inequality [@chsh]. ### CHSH as a game {#sec:game} We briefly describe the CHSH inequality in its more modern form as a game involving two distant players, Alice and Bob. A detailed account of this formulation and how it allows us to recover the original form of the CHSH inequality can for example be found in [@steph:diss]. In the CHSH game, we imagine that we pose a question $y \in \01$ to Alice and a question $z \in \01$ to Bob, chosen uniformly at random, i.e., $p(y) = p(z) = 1/2$. These questions can be identified with the choice of measurement setting in the usual formulation. Alice and Bob now return answers $a \in \01$ and $b \in \01$ respectively, where we say that Alice and Bob win the game if and only if $$\begin{aligned} \label{eq:chshCondition} y \cdot z = a + b \mod 2\ .\end{aligned}$$ Alice and Bob may thereby agree on any strategy beforehand, but they can no longer communicate once the game starts. In the quantum setting, this strategy corresponds to a choice of shared state and measurements, and in an experiment the no-signaling assumption is employed to enforce their inability to communicate. Clearly, one may write the probability that Alice and Bob win for a particular strategy as $$\begin{aligned} p_{\rm win} = \frac{1}{4} \sum_{y,z \in \01} \sum_{\substack{a,b\\a + b = y \cdot z}} \Pr[a,b\|y,z]\ ,\end{aligned}$$ where $\Pr[a,b\|y,z]$ denotes the probability that Alice and Bob return answers $a$ and $b$ given questions $y$ and $z$. For any classical strategy, $p_{\rm win} \leq 3/4$ but quantumly there exist a strategy that achieves $p_{\rm win} = 1/2 + 1/(2\sqrt{2}) \approx 0.853$. This is in fact optimal, since Tsirelson has shown [@tsirel:original; @tsirel:separated] that for any quantum strategy $$\begin{aligned} \label{eq:tsirelBound} p_{\rm win} \leq \frac{1}{2} + \frac{1}{2\sqrt{2}}\ .\end{aligned}$$ ### Strategies and state discrimination {#sec:tsirelNotation} For our purposes, it will be convenient to employ a simple observation about what Bob has to do in order to produce the right answer in the game, which was described in more detail in [@steph:diss]. Let $\rho_{y,a}$ denote the state of Bob’s system conditioned on the fact that Alice received question $y$ and has given answer $a$. Note that Bob’s system will be placed in this state with probability $p(y,a) = p(a\|y)/2$. For $z=0$, the rules of the game state that Alice and Bob win if and only if Bob returns the same answer as Alice, that is, $b = a$. In other words, Bob would like to determine, which of the following two states he is given $$\begin{aligned} \sigma_0^{z=0} &:= \left(q^{z=0,0}_0\rho_{0,0} + q^{z=0,0}_1 \rho_{1,0}\right)\ ,\\ \sigma_1^{z=0} &:= \left(q^{z=0,1}_0\rho_{0,1} + q^{z=0,1}_1\rho_{1,1}\right)\ ,\end{aligned}$$ where $$\begin{aligned} q^{z=0,0}_y &= p(0\|y)/(p(0\|0) + p(0\|1))\ ,\\ q^{z=0,1}_y &= p(1\|y)/(p(1\|0) + p(1\|1))\ ,\end{aligned}$$ and the probability of $\sigma_x^{z=0}$ is given by $p_x^{z=0} = (p(x\|0) + p(x\|1))/2$. That is, Bob would simply try to extract classical information stored in quantum states, which is exactly the setting that our bound applies to. Producing a classical outcome on the ancilla system is very natural in this setting as we can imagine that when giving his answer Bob simply returns his ancilla to a referee who decides whether Alice and Bob win [^2]. Similarly, if $z=1$ Bob would like to determine which of the following two states he is given $$\begin{aligned} \sigma_0^{z=1} &:= \left(q^{z=1,0}_0\rho_{0,0} + q^{z=1,0}_1 \rho_{1,1}\right)\ ,\\ \sigma_1^{z=1} &:= \left(q^{z=1,1}_0\rho_{0,1} + q^{z=1,1}_1\rho_{1,0}\right)\ ,\end{aligned}$$ where $$\begin{aligned} q^{z=1,0}_{y=0} &= p(0\|0)/(p(0\|0) + p(1\|1))\ ,\\ q^{z=1,0}_{y=1} &= p(1\|1)/(p(0\|0) + p(1\|1))\ ,\\ q^{z=1,1}_{y=0} &= p(1\|0)/(p(1\|0) + p(0\|1))\ ,\\ q^{z=1,1}_{y=1} &= p(0\|1)/(p(1\|0) + p(0\|1))\ ,\end{aligned}$$ the probability of $\sigma_0^{z=1}$ is $p_0^{z=1} = (p(0\|0) + p(1\|1))/2$, and the probability of $\sigma_1^{z=1}$ is $p_1^{z=1} = (p(1\|0) + p(0\|1))/2$. The probability that Alice and Bob win the game for a particular strategy can now be expressed as $$\begin{aligned} p_{\rm win} = \frac{1}{2}\sum_{z \in \01} P_{\rm guess}(X^z\|E^z)\ ,\end{aligned}$$ where we write $P_{\rm guess}(X^z\|E^z)$ for Bob’s success probability in solving the state discrimination problems described above for $z \in \01$. From this perspective, Tsirelson’s bound provides us with an upper bound on how well we can solve these two problems on average. ### A time limited game In the usual setting of this game, Alice and Bob are essentially given an unlimited amount of time and energy to produce their answers. But how well can they do given only a limited amount of energy and time? Here, we consider a time-limited version of the CHSH game, in which Alice and Bob are given a fixed time $t$ to produce their answers. If no answers are given at time $t$, we automatically rule that Alice and Bob loose. Our goal will be to derive a time-dependent version of . For simplicity, we will thereby assume that Alice has an essentially unlimited amount of energy at her disposal and only Bob will be restricted in some fashion. Given the perspective that Bob has to solve a state discrimination problem to produce the right answer as explained above, it is clear that we can use our general bound to address this setting. The use of an ancilla register is very natural, as we can view it as the message system holding Bob’s answer that is returned to the referee. In the usual scenario, Alice and Bob can choose which state to share at the start of the game as part of their strategy. Note, however, that we cannot allow arbitrary starting states to begin with, as we want to put a limit on the energy that Bob has at his disposal. For simplicity, however, we will make the sole assumption that Bob’s Hamiltonian is bounded as $\\|H\\|_\infty$. In the appendix, we will derive a general time dependent Tsirelson bound from this assumption where we will need our generalization of the time bound for two input states to the case of non-uniform input distributions. Here, we will focus on the essential idea that underlies this bound which already becomes apparent if we consider a slightly simpler scenario in which Alice’s marginal distributions are uniform ($p(a\|y) = 1/2$ for all $y$). This scenario is well motivated if we imagine that there is a source supplying Alice and Bob with the maximally entangled state which lies outside of their control, and their strategy is restricted to their choice of two-outcome observables. In this case, Alice’s outcome distribution will either be deterministic or uniform. In the deterministic case, Alice essentially plays a classical strategy. To obtain a quantum advantage in the case of unlimited time, Alice’s outcome distributions will be uniform, and we will hence focus on this case. To obtain a time-dependent Tsirelson bound, we now employ our simple bound involving the original trace distance of the two states that we wish to discriminate . We have by Tsirelson’s bound that $$\begin{aligned} \frac{1}{2}\sum_{z \in \01} P_{\rm guess}(X^z\|E^z) \leq \frac{1}{2} + \frac{1}{2\sqrt{2}}\ ,\end{aligned}$$ and hence by and the fact that $p(a\|y) = 1/2$ $$\begin{aligned} \frac{1}{2}\sum_{z \in \01} D(\sigma_0^z,\sigma_1^z) \leq \frac{1}{\sqrt{2}}\ ,\end{aligned}$$ otherwise there would exist a better strategy for Alice and Bob at long times. So we have from that $$\begin{aligned} p_{\rm win} \leq \frac{1}{2} + \frac{\gamma t \\|H\\|_\infty}{2\sqrt{2}\hbar}\ .\end{aligned}$$ In particular, this means that if we allow only a limited amount of energy by Bob (e.g., by demanding that $\\|H\\|_\infty = 1$), then Bob needs time at least $$\begin{aligned} \label{eq:mintime} t \geq \frac{\hbar}{\gamma \\|H\\|_\infty}\ \end{aligned}$$ to achieve the optimum quantum violation of CHSH. Note that to achieve the optimum quantum violation, Alice’s marginals will in fact be uniform, and hence this is indeed the minimum time required. Clearly, for small time frames, it would be better for Alice and Bob to play a classical strategy in which Bob can just return the ancilla ${\|0\rangle}$ ”as is” to the referee. The tradeoff betweeen the classical and quantum strategies in our setting can be captured when considering arbitrary distributions, which we will address in the appendix. In particular, we will show that Let Bob’s Hamiltonian be scaled such that $H \geq 0$. Then the maximum success probability of winning the CHSH game for Alice and Bob in time $t$ obeys $$\begin{aligned} \label{eq:timeTsirel} p_{\rm win}^t \leq \frac{3}{4} + \frac{\gamma t \\|H\\|_\infty}{\sqrt{2} \hbar}\ , \end{aligned}$$ where $$\begin{aligned} \gamma := \left\{ \begin{array}{ll}5/\pi & \mbox{if } 1 < t E_n/\hbar < 4 \ ,\\ 3/\pi & \mbox{otherwise} \ . \end{array}\right. \end{aligned}$$ We could also derive a more general bound in terms of Bob’s average energy using . However, such a bound does not compare easily to the original Tsirelon’s bound. Of course, the minimum time is extremely small, and irrelevant for any practical tests of CHSH. Indeed, it is not our intention to question the validity of present CHSH experiments or suggest any loopholes caused by an insufficient distance for Alice and Bob compared to the time it takes them to achieve Tsirelson’s bound. Instead, we provided the present analysis as an illustrative example of how our bound applies. We would like to point out that tells us that the strength of non-local correlation is indeed a function of time. Furthermore, tells us that there exists a fundamental time required to establish maximally strong quantum correlations. Finally, we note that one can also interpret in another way: Let’s suppose that we were to fix a time $t$ and observe that Alice and Bob tend to win the game with probability at least $q$. We can now rewrite to obtain a lower bound on $\\|H\\|_\infty$. That is we can conclude that Bob had a certain energy at his disposal, and the strength of non-local correlations in this setting provides us with a form of ”energy witness” for Bob. This also holds for the most general case discussed in the appendix. Discussion ========== Choice of initial state ----------------------- We obtained a series of simple bounds on how well we can recover classical information stored in a quantum system within a certain timeframe. Let us now first consider what role the choice of initial state of the ancilla played in our bounds. During our discussions we assumed that the ancilla started out in the classical state corresponding to the most likely symbol ${ {x_{\max}} }$. This reflects the fact that the distinguisher does have full knowledge not only about the states $\rho_x$ themselves, but also about the distribution $p_x$. In particular, this means that without touching the quantum register, he can always achieve a success probability of ${p_{{ {x_{\max}} }}}$ by outputting ${ {x_{\max}} }$. Clearly, we could have chosen any other classical symbol as our starting point, and our bounds can easily be adapted accordingly. This holds even for an arbitrary pure state of the ancilla. Yet, such a choice does not reflect the distinguisher’s apriori knowledge. Another option would be to let the ancilla start out in a special blank state, which intuitively corresponds to an outcome of “don’t know”. and is orthogonal to any other outputs. It is straightforward to apply our methods to obtain a similar bound for this case. Yet, note that using a blank ancilla state is conceptually rather different since it means that we essentially neglect the apriori knowledge that a distinguisher has available. Input size ---------- Our bound is especially useful, if we are merely concerned with the probability of success that can be achieved withing a certain time $t$ in principle, using any physically allowed operation $H$. This is indeed interesting when we consider the problem of Bell inequalities where we wanted to obtain a bound on how well Alice and Bob can violated CHSH within a given time frame, when they can choose any Hamiltonian they like subject to energy constraints alone. In particular, we would like to emphasize that the time required to acquire classical information in our setting is not limited by the size of the alphabet ${\mathcal{X}}$, but merely by the choice of encodings. In practise, however, there are much more stringent constraints on how quickly information can be transferred that depend on the geometry of the ancilla, leading to additional constraints on the interaction Hamiltonian $H$. For example, it could be that $H$ can consist only of two qubit interactions, and interactions between the encoding system and the ancilla are limited to their boundary. In this case, the size of the alphabet ${\mathcal{X}}$ clearly does matter, and stronger bounds therefore should depend strongly on the exact form of $H$. We note that some bounds on time scales for particular Hamiltonians $H$ do follow from the decoherence and thermodynamics literature [@gogolin:mthesis; @lazyStates] for pure state encodings, yet since such bounds typically involve $\\|H_{\rm int}\\|_\infty$, where $H_{\rm int}$ is the interacting part of $H$ they offer little advantage in our setting. To see how such bounds are related to ours is most easily seen when considering the conditional von Neumann entropy ${\ensuremath{ {\rm H}}}(X\|E)$. Note that if all $\rho_x$ are pure the overall cqq-state $\rho_{XEA}$ [^3] is pure as well. Hence, ${\ensuremath{ {\rm H}}}(X\|E) = {\ensuremath{ {\rm H}}}(XE) - {\ensuremath{ {\rm H}}}(E) = {\ensuremath{ {\rm H}}}(A) - {\ensuremath{ {\rm H}}}(E)$. To determine how ${\ensuremath{ {\rm H}}}(X\|E)$ can change with time we would thus like to determine how the entropy of the reduced systems $A$ and $E$ evolves with time which has been studied for the von Neumann entropy in the decoherence literature where an upper bound for the rate of change in entropy was obtained in terms of $\\|H_{\rm int}\\|_\infty$ [@lazyStates]. Similar considerations can be made for other entropies [@adrian:mthesis]. It is an interesting open question to obtain good bounds on such quantities for arbitrary $H$ that take more of their structure into account. Open questions -------------- Clearly, this is not the only interesting open question. Closely related is the question of how much time is required to demonstrate non-local correlations if Alice and Bob are yet more restricted. Again, this could take the form of physical constraints on the ancilla, or be considered in the framework of circuit complexity where one cares about the number of two qubit interactions, i.e., gates, that they have to apply. The example of CHSH is too small for such constraints to make a difference, but do play an important role when considering more complicated inequalities. Furthermore, it would be nice to see if the slight gap between our bound and the Margolus-Levitin theorem can be closed completely using a more stringent analysis for the case of orthogonal encodings $\rho_x$. In particular, this means that one would rederive the exact form of the Margolus-Levitin theorem from the rate of information transfer alone. SW thanks Oscar Dahlsten, Artur Ekert, Christian Gogolin, Peter Janotta, Jonathan Oppenheim, Renato Renner, Thomas Vidick and CQT’s ”non-local club” for interesting comments and discussions. SW would also like to thank Artur Ekert and Jonathan Oppenheim for useful pointers to existing literature [@mlTheorem; @bekenstein:survey]. SW was supported by the National Research Foundation (Singapore), and the Ministry of Education (Singapore). ACD acknowledges support through the ARC Centre of Excellence in Engineered Quantum Systems (EQuS), project number CE110001013. [37]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , , , ** (). , , (). , , (), . , , (). , , , (). , , (). , , (). , , , , (). , , (). , , (). , , , , (). , , , , (). (), . , , (). , , (). , , (). , , (). , , (). , , (). , , (). , , (). , , , , , (). , , (). , , (). , , (). , (). , , (). , , , , , (). , Ph.D. thesis (), . , , , , (). , Ph.D. thesis, (), . , (, ). , **, (). , , (). , Ph.D. thesis, (), . , **, (). (), . Basic observations ================== In this appendix, we provide the technical details of our claims. To this end, we first establish two simple lemmas from which we later derive all our results. Since we will use these in everything that follows we consider the generalized problem where we wish to distinguish $N$ states $\rho_0,\ldots,\rho_{N-1}$. The first lemma will be used to bound the success probabilies using measurement operators $M_x = {\mathbb{I}}\otimes P_x + W_x$ where the label $x \in \{0,\ldots,N-1\}$ corresponds to one of the $N$ states we wish to idenitfy. \[lem:basicBound\] For any Hermitian operator $A \in {\mathcal{B}}({\mathcal{H}_{\rm in}})$ with diagonalization $A = \sum_j \lambda_j {\|{u_j}\rangle\langle{u_j}\|}$, and any $x\in\{0,\ldots,N-1\}$ the operator $\tilde{A} := A \otimes {{\|{ {\rm x_{\max}}}\rangle\langle{ {\rm x_{\max}}}\|}}$ satisfies $$\begin{aligned} {\mathop{\mathrm{tr}}\nolimits}\left(W_x\tilde{A}\right) \leq 2 \sum_n (1 - \cos(t E_n/\hbar)) {\langleE_n\|}\tilde{A}_x{\|E_n\rangle}\ , \end{aligned}$$ where $$\begin{aligned} \tilde{A}_x &:= \left\{ \begin{array}{ll} \|A\| \otimes {{\|{ {\rm x_{\max}}}\rangle\langle{ {\rm x_{\max}}}\|}}& {\rm for\ } x = { {x_{\max}} }\ ,\\ A^+ \otimes {{\|{ {\rm x_{\max}}}\rangle\langle{ {\rm x_{\max}}}\|}}& {\rm otherwise}\ , \end{array}\right. \\ A^+ &:= \sum_{j,\lambda_j \geq 0} \lambda_j {\|{u_j}\rangle\langle{u_j}\|}\ .\end{aligned}$$ Using the definition of $W_x$ from we evaluate the terms involving $W_x^1$ and $W_x^2$ separately. Let us now first bound the term involving $W_x^1$. For $x\neq { {x_{\max}} }$ we have that $$\begin{aligned} &{\mathop{\mathrm{tr}}\nolimits}\left(W_x^1 \tilde{A}\right)\nonumber\\ &\qquad={\mathop{\mathrm{tr}}\nolimits}\left(({\mathbb{I}}\otimes P_x)R \tilde{A}\right) + {\mathop{\mathrm{tr}}\nolimits}\left(R^\dagger ({\mathbb{I}}\otimes P_x)\tilde{A}\right)\\ &\qquad=0\ , \end{aligned}$$ where we used the linearity and cyclicity of the trace, as well as the fact that $P_x {{\|{ {\rm x_{\max}}}\rangle\langle{ {\rm x_{\max}}}\|}}= {{\|{ {\rm x_{\max}}}\rangle\langle{ {\rm x_{\max}}}\|}}P_x = 0$ for all $x \neq { {x_{\max}} }$. Let $A^- := \sum_{j,\lambda_j < 0} {\|{u_j}\rangle\langle{u_j}\|}$, and define $\tilde{A}^+ := A^+ \otimes {{\|{ {\rm x_{\max}}}\rangle\langle{ {\rm x_{\max}}}\|}}$ and $\tilde{A}^- := A^- \otimes {{\|{ {\rm x_{\max}}}\rangle\langle{ {\rm x_{\max}}}\|}}$. Note that $\tilde{A} = \tilde{A}^+ - \tilde{A}^-$. For $x={ {x_{\max}} }$ we can now use the fact that $$\begin{aligned} \label{eq:RDef} R &= U - I = \sum_n (\exp(-itE_n/\hbar) - 1) {\|{E_n}\rangle\langle{E_n}\|}\ , \end{aligned}$$ to write $$\begin{aligned} &{\mathop{\mathrm{tr}}\nolimits}\left(W_{ {x_{\max}} }^1 \tilde{A}\right)\\ &= {\mathop{\mathrm{tr}}\nolimits}\left(({\mathbb{I}}\otimes P_{ {x_{\max}} })R \tilde{A}\right) + {\mathop{\mathrm{tr}}\nolimits}\left(R^\dagger ({\mathbb{I}}\otimes P_{ {x_{\max}} })\tilde{A}\right)\nonumber\\ &={\mathop{\mathrm{tr}}\nolimits}\left( (R + R^\dagger)\tilde{A}\right)\\ &= \sum_{n=0}^\infty (\exp(-itE_n/\hbar) +\exp(itE_n/\hbar) - 2)\nonumber\\ &\qquad\qquad\qquad {\langleE_n\|}\tilde{A}{\|E_n\rangle}\\ &=2 \sum_{n=0}^{\infty} (\cos(t E_n/\hbar) - 1) {\langleE_n\|}\tilde{A}{\|E_n\rangle}\ .\\ &=2 \sum_{n=0}^{\infty} (\cos(t E_n/\hbar) - 1) {\langleE_n\|}(\tilde{A}^+-\tilde{A}^-){\|E_n\rangle}\\ &\leq 2 \sum_{n=0}^{\infty} (1 - \cos(t E_n/\hbar)) {\langleE_n\|}\tilde{A}^-{\|E_n\rangle}\ , \end{aligned}$$ where the fourth equality follows from Euler’s formula, and the first inequality from the fact that $\cos(t E_n/\hbar) - 1 \leq 0$ and $\tilde{A}^+,\tilde{A}^- \geq 0$. It remains to bound the term involving $W_x^2$. First of all, since $\Lambda(X) = R X R^\dagger$ is a positive map [@bathia:posBook], and $\tilde{A}^+,\tilde{A}^- \geq 0$, we have that $$\begin{aligned} \label{eq:posTerms} R\tilde{A}^+R^\dagger &\geq 0\ ,\\ R\tilde{A}^-R^\dagger &\geq 0\ . \end{aligned}$$ Note that for any $X,Z \geq 0$, we have ${\mathop{\mathrm{tr}}\nolimits}(XZ) \geq 0$, and hence ${\mathop{\mathrm{tr}}\nolimits}\left( ({\mathbb{I}}\oplus P_x) R\tilde{A}^-R^\dagger\right) \geq 0$. Second, note that $RR^\dagger = R^\dagger R$ and we have $$\begin{aligned} \label{eq:RR} &RR^\dagger = \\ &= \sum_{n=0}^\infty \left(2 - \exp(i t E_n/\hbar) - \exp(-i t E_n/\hbar)\right)\nonumber\\ &\qquad\qquad\qquad{\|{E_n}\rangle\langle{E_n}\|} \\ &= 2 \sum_n (1 - \cos(t E_n/\hbar)) {\|{E_n}\rangle\langle{E_n}\|}\ , \end{aligned}$$ where the second equality follows by applying Euler’s formula. We thus have $$\begin{aligned} &{\mathop{\mathrm{tr}}\nolimits}\left(W_x^2 \tilde{A}\right)\\ &= {\mathop{\mathrm{tr}}\nolimits}\left(({\mathbb{I}}\otimes P_x)R \tilde{A}^+ R^\dagger\right) - {\mathop{\mathrm{tr}}\nolimits}\left(({\mathbb{I}}\otimes P_x)R \tilde{A}^- R^\dagger\right)\nonumber\\ &\leq {\mathop{\mathrm{tr}}\nolimits}\left(R^\dagger R \tilde{A}^+\right)\\ &= 2 \sum_n (1 - \cos(t E_n/\hbar)) {\langleE_n\|}\tilde{A}^+{\|E_n\rangle} \end{aligned}$$ where the first inequality follows from , the fact that $0 \leq {\mathbb{I}}\otimes P_x \leq {\mathbb{I}}$ and the cyclicity of the trace, and the last equality from . Putting everything together, ${\mathop{\mathrm{tr}}\nolimits}(W_x\tilde{A}) = {\mathop{\mathrm{tr}}\nolimits}(W_x^1\tilde{A}) + {\mathop{\mathrm{tr}}\nolimits}(W_x^2\tilde{A})$, we obtain the claimed result. We will also make repeated use of the following bound. Note that whereas the bound applies to a very large range of values $E_n \geq 0$, we will later be particularly interested in the case of $t E_n/\hbar < 1$. Indeed the bound below is a great overestimate if $t E_n/\hbar > 2\pi$, as $2(1-\cos(k)) \leq \gamma (k - 2 \pi {\lfloor{k/(2\pi)}\rfloor})$. \[lem:gammaBound\] Let $E_n \geq 0$. Then $2(1-\cos(t E_n/\hbar)) \leq \gamma t E_n/\hbar$ where $$\begin{aligned} \gamma := \left\{ \begin{array}{ll}5/\pi & \mbox{if } 1 < t E_n/\hbar < 4\ ,\\ 3/\pi & \mbox{otherwise}\ . \end{array}\right. \end{aligned}$$ A bound for two states ====================== We now first consider the case were we are given just two states, $\rho_0$ and $\rho_1$. Here, we will consider the most general problem where $p_0$ and $p_1$ can be arbitrary. A bound in terms of the trace distance -------------------------------------- First of all, note that even for a general distribution $\{p_x\}_x$ the problem of distinguishing two states is easy to analyze [@helstrom:detection]. In particular, we have that in the time-unlimited case for measurement operators acting directly on the encoding space $$\begin{aligned} \label{eq:unequalProb} P_{\rm guess}(X\|E) &= \max_{M_0,M_1} p_0 {\mathop{\mathrm{tr}}\nolimits}(M_0 \rho_0) + p_1 {\mathop{\mathrm{tr}}\nolimits}(M_1 \rho_1)\\ &= p_0 + \max_{M_1} {\mathop{\mathrm{tr}}\nolimits}\left(M_1(p_1 \rho_1 - p_0 \rho_0)\right)\\ &= p_0 + \Delta(p_1 \rho_1, p_0 \rho_0)\ ,\end{aligned}$$ where $\Delta(p_1 \rho_1, p_0 \rho_0)$ is given by $$\begin{aligned} \Delta(p_1 \rho_1,p_0 \rho_0) &= \max_{0 \leq P \leq {\mathbb{I}}} {\mathop{\mathrm{tr}}\nolimits}\left(PA\right)\\ &={\mathop{\mathrm{tr}}\nolimits}(A^+)\ , \label{eq:DeltaTrace}\end{aligned}$$ where $A := p_1 \rho_1 - p_0 \rho_0$ with diagonalization $A = \sum_j \lambda_j {\|{u_j}\rangle\langle{u_j}\|}$ and $A^+ = \sum_{j,\lambda_j \geq 0}{\|{u_j}\rangle\langle{u_j}\|}$ (Note that $\Delta$ is not symmetric here and hence formally does not form a distance measure.) Similarly, we have $$\begin{aligned} \Delta(p_0\rho_0,p_1\rho_1) = {\mathop{\mathrm{tr}}\nolimits}(A-)\ .\end{aligned}$$ Note that for the time unlimited case, we could have equivalently expressed the success probability as $$\begin{aligned} P_{\rm guess}(X\|E) = p_1 + \Delta(p_0\rho_0,p_1\rho_1)\ .\end{aligned}$$ It will also be useful to note that for $\tilde{\rho}_x = \rho_x \otimes {{\|{ {\rm x_{\max}}}\rangle\langle{ {\rm x_{\max}}}\|}}$, $$\begin{aligned} \label{eq:ancillaDoesntChangeDelta} \Delta(p_1 \rho_1,p_0 \rho_0) = \Delta(p_1 \tilde{\rho}_1,p_0 \tilde{\rho}_0)\ .\end{aligned}$$ Before stating our bound, let us introduce some additional notation. For two states, define $$\begin{aligned} {x_{\min}}:= 1 - { {x_{\max}} }\ .\end{aligned}$$ We now first relate the problem of discriminating the two states in time $t$ to the original success probability. \[lem:proto2\] The probability of distinguishing $\rho_0$ and $\rho_1$ given with probabilities $p_0$ and $p_1$ using the Hamiltonian $H = \sum_n E_n {\|{E_n}\rangle\langle{E_n}\|} \geq 0$ is bounded by $$\begin{aligned} &P_{\rm succ}(X\|E)_{H,t} \leq {p_{{ {x_{\max}} }}}+ \\ &\qquad 2(1 - \cos(t C_{\rm max}/\hbar)) \Delta({p_{{x_{\min}}}}{\rho_{{x_{\min}}}},{p_{{ {x_{\max}} }}}{\rho_{{ {x_{\max}} }}})\ ,\nonumber \end{aligned}$$ where $C_{\rm max} = {\mathop{\mathrm{argmax}}\nolimits}_{E_n} (1 - \cos(t E_n/\hbar))$. Using , and the fact that ${P_{{x_{\min}}}}{{\| {\rm x_{\max}}\rangle}}= 0$ we may bound the success probability as $$\begin{aligned} \label{eq:rewriteProb} &P_{\rm succ}(X\|E)_{H,t} \leq {p_{{ {x_{\max}} }}}+\\ &\qquad {\mathop{\mathrm{tr}}\nolimits}\left({W_{{x_{\min}}}}({p_{{x_{\min}}}}\tilde{\rho}_{{x_{\min}}} - {p_{{ {x_{\max}} }}}\tilde{\rho}_{{ {x_{\max}} }})\right)\ .\nonumber \end{aligned}$$ Applying Lemma \[lem:basicBound\] for $A = {p_{{x_{\min}}}}{\rho_{{x_{\min}}}}- {p_{{ {x_{\max}} }}}{\rho_{{ {x_{\max}} }}}$ we have that $$\begin{aligned} &{\mathop{\mathrm{tr}}\nolimits}\left({W_{{x_{\min}}}}(A\otimes {{\|{ {\rm x_{\max}}}\rangle\langle{ {\rm x_{\max}}}\|}})\right)\\ &\leq 2 \sum_n (1 - \cos(t E_n/\hbar)) {\langleE_n\|}A^+ \otimes {{\|{ {\rm x_{\max}}}\rangle\langle{ {\rm x_{\max}}}\|}}{\|E_n\rangle}\ .\nonumber \end{aligned}$$ Hence from and we have $$\begin{aligned} &{\mathop{\mathrm{tr}}\nolimits}\left({W_{{x_{\min}}}}({p_{{x_{\min}}}}\tilde{\rho}_{{x_{\min}}} - {p_{{ {x_{\max}} }}}\tilde{\rho}_{{ {x_{\max}} }})\right)\\ &\leq 2(1-\cos(t C_{\rm max}/\hbar)) \Delta({p_{{x_{\min}}}}{\rho_{{x_{\min}}}},{p_{{ {x_{\max}} }}}{\rho_{{ {x_{\max}} }}})\ . \nonumber \end{aligned}$$ Our claim now follows by plugging this bound into . With the help of Lemma \[lem:gammaBound\] one may now also use the fact that $\forall E_n, E_n \leq \\|H\\|_\infty$ to obtain a very simple bound in terms of the spectrum of the Hamiltonian. \[cor:traceDist\] The probability of distinguishing $\rho_0$ and $\rho_1$ given with probabilities $p_0$ and $p_1$ using the Hamiltonian $H = \sum_n E_n {\|{E_n}\rangle\langle{E_n}\|} \geq 0$ is bounded by $$\begin{aligned} &P_{\rm succ}(X\|E)_{H,t} \leq {p_{{ {x_{\max}} }}}+\\ &\qquad \frac{\gamma t \\|H\\|_\infty \Delta({p_{{x_{\min}}}}{\rho_{{x_{\min}}}}, {p_{{ {x_{\max}} }}}{\rho_{{ {x_{\max}} }}})}{\hbar}\ .\nonumber \end{aligned}$$ A bound in terms of the average energy -------------------------------------- Inspecting the proof above with Lemma \[lem:gammaBound\] in mind, it is indeed easy to see that we can also obtain a bound in terms of average energies. We first derive a somewhat stronger bound for two equiprobable states that actually depends on the “average energy” of a function of both states. \[thm:avgEnergyTwoStates\] The probability of distinguishing $\rho_0$ and $\rho_1$ given with probabilities $p_0$ and $p_1$ using the Hamiltonian $H = \sum_n E_n {\|{E_n}\rangle\langle{E_n}\|} \geq 0$ in time $t$ is bounded by $$\begin{aligned} &P_{\rm succ}(X\|E)_{H,t} \leq {p_{{ {x_{\max}} }}}+\\ &\qquad\frac{\gamma t}{2\hbar} \left[{\mathop{\mathrm{tr}}\nolimits}(H\|{p_{{x_{\min}}}}\tilde{\rho}_{{x_{\min}}}- {p_{{ {x_{\max}} }}}\tilde{\rho}_{{ {x_{\max}} }}\|)\right. + \nonumber\\ &\qquad \left. {p_{{x_{\min}}}}{\mathop{\mathrm{tr}}\nolimits}(H\tilde{\rho}_{{x_{\min}}}) - {p_{{ {x_{\max}} }}}{\mathop{\mathrm{tr}}\nolimits}(H\tilde{\rho}_{{ {x_{\max}} }})\right]\ ,\nonumber \end{aligned}$$ where $$\begin{aligned} \gamma := \left\{ \begin{array}{ll}5/\pi & \mbox{if } 1 < t E_n/\hbar < 4 \ ,\\ 3/\pi & \mbox{otherwise} \ . \end{array}\right. \end{aligned}$$ Recall that by applying Lemma \[lem:basicBound\] with $A = {p_{{x_{\min}}}}{\rho_{{x_{\min}}}}- {p_{{ {x_{\max}} }}}{\rho_{{ {x_{\max}} }}}$ we have that $$\begin{aligned} &{\mathop{\mathrm{tr}}\nolimits}\left({W_{{x_{\min}}}}({p_{{x_{\min}}}}\tilde{\rho}_{{x_{\min}}} - {p_{{ {x_{\max}} }}}\tilde{\rho}_{{ {x_{\max}} }})\right) \\ &\leq 2 \sum_n (1 - \cos(t E_n/\hbar)) {\langleE_n\|}A^+ \otimes {{\|{ {\rm x_{\max}}}\rangle\langle{ {\rm x_{\max}}}\|}}{\|E_n\rangle}\ .\nonumber \end{aligned}$$ We may now use Lemma \[lem:gammaBound\] to obtain $$\begin{aligned} &{\mathop{\mathrm{tr}}\nolimits}\left({W_{{x_{\min}}}}({p_{{x_{\min}}}}\tilde{\rho}_{{x_{\min}}} - {p_{{ {x_{\max}} }}}\tilde{\rho}_{{ {x_{\max}} }})\right) \\ &\leq \frac{\gamma t}{\hbar} \sum_n E_n {\langleE_n\|}A^+ \otimes {{\|{ {\rm x_{\max}}}\rangle\langle{ {\rm x_{\max}}}\|}}{\|E_n\rangle}\nonumber\\ &= \frac{\gamma t {\mathop{\mathrm{tr}}\nolimits}(H\tilde{A}^+)}{\hbar}\ . \end{aligned}$$ Our claim now follows by noting that $$\begin{aligned} \tilde{A}^+ &= \frac{1}{2}\left(\tilde{A}^+ + \tilde{A}^{-}\right) + \frac{1}{2}\left(\tilde{A}^+ - \tilde{A}^-\right)\\ &=\frac{1}{2}\left\|{p_{{x_{\min}}}}\tilde{\rho}_{{x_{\min}}} - {p_{{ {x_{\max}} }}}\tilde{\rho}_{{ {x_{\max}} }}\right\| +\\ &\qquad\frac{1}{2}\left({p_{{x_{\min}}}}\tilde{\rho}_{{x_{\min}}} - {p_{{ {x_{\max}} }}}\tilde{\rho}_{{ {x_{\max}} }}\right)\ .\nonumber \end{aligned}$$ A bound for many input states ============================= Finally, we derive a bound for the most general case of distinguishing states $\rho_0,\ldots,\rho_{N-1}$ where we are given $\rho_x$ with probability $p_x$. \[thm:manyStates\] Suppose $H \geq 0$. Then the probability of distinguishing $\rho_0,\ldots,\rho_{N-1}$ given with probabilities $p_0,\ldots,p_{N-1}$ obeys $$\begin{aligned} P_{\rm succ}(X\|E)_{H,t} &\leq {p_{{ {x_{\max}} }}}+\frac{\hat{\gamma}t}{\hbar} \sum_{x=0}^{N-1} p_x {\mathop{\mathrm{tr}}\nolimits}\left(H\tilde{\rho}_x\right)\ , \end{aligned}$$ where $$\begin{aligned} \hat{\gamma} := \left\{ \begin{array}{ll}5/\pi & \mbox{if } \forall E_n, 1 < t E_n/\hbar < 4\\ 3/\pi & \mbox{otherwise} \end{array}\right. \ . \end{aligned}$$ Note that the success probability for a particular interaction $H$ is now given by $$\begin{aligned} \label{eq:pSUCCGeneral} &P_{\rm succ}(X\|E)_{H,t} = \sum_{x=0}^{N-1} p_x {\mathop{\mathrm{tr}}\nolimits}\left(M_x \tilde{\rho}_x\right)\ ,\end{aligned}$$ where $$\begin{aligned} {\mathop{\mathrm{tr}}\nolimits}\left(M_x \tilde{\rho}_x\right)= {\mathop{\mathrm{tr}}\nolimits}\left( ( {\mathbb{I}}\otimes P_x) \tilde{\rho}_x\right) + {\mathop{\mathrm{tr}}\nolimits}\left(W_x \tilde{\rho}_x\right)\ .\end{aligned}$$ Let us now first consider the case of $x = { {x_{\max}} }$. We have that $$\begin{aligned} {\mathop{\mathrm{tr}}\nolimits}\left(({\mathbb{I}}\otimes P_x)\tilde{\rho}_x\right) = {\mathop{\mathrm{tr}}\nolimits}(\rho_x) = 1\ .\end{aligned}$$ Using Lemma \[lem:basicBound\] with $A = \rho_x$ we hence have that $$\begin{aligned} &{\mathop{\mathrm{tr}}\nolimits}\left(M_x \tilde{\rho}_x\right) \leq\\ &\qquad 1 + 2 \sum_n (1-\cos(t E_n/\hbar)) {\langleE_n\|}\tilde{\rho}_x{\|E_n\rangle}\nonumber\ .\end{aligned}$$ We now turn to the case of $x \neq { {x_{\max}} }$. Since $P_x{{\|{ {\rm x_{\max}}}\rangle\langle{ {\rm x_{\max}}}\|}}= {{\|{ {\rm x_{\max}}}\rangle\langle{ {\rm x_{\max}}}\|}}P_x = 0$ and $\tilde{\rho}_x = \rho_x \otimes {{\|{ {\rm x_{\max}}}\rangle\langle{ {\rm x_{\max}}}\|}}$ we have $({\mathbb{I}}\otimes P_x)\tilde{\rho}_x = 0$ for all $x \neq { {x_{\max}} }$. Again by applying \[lem:basicBound\] with $A = \rho_x$ we obtain from $\rho_x \geq 0$ that $$\begin{aligned} {\mathop{\mathrm{tr}}\nolimits}\left(W_x \tilde{\rho}_x\right) \leq 2 \sum_n (1-\cos(t E_n/\hbar)) {\langleE_n\|}\tilde{\rho}_x{\|E_n\rangle}\ . \end{aligned}$$ Our claim now follows by using Lemma \[lem:gammaBound\] to obtain $2(1 - \cos(t E_n/\hbar)) \leq \gamma t E_n/\hbar$ for $E_n \geq 0$. Attaining the bound =================== We now exhibit a Hamiltonian that achieves our upper bound for the success probability of distinguishing two states given with apriori equal probability. \[thm:attaining\] Suppose we are given $\rho_0$ and $\rho_1$ with apriori equal probability. Let $E_{\max} \geq 0$. Then there exist a Hamiltonian $H$ with $\\|H\\|_\infty = E_{\max}$ that in time $t$ achieves success probability $$\begin{aligned} P_{\rm succ}(X\|E)_{H,t} = \frac{1}{2} + \frac{1}{4}(1-\cos(t\\|H\\|_\infty /\hbar)) D(\rho_0,\rho_1)\ . \end{aligned}$$ In particular, we can distinguish the two states perfectly in time $t = \hbar \pi/E_{\max}$. Let $A = \rho_1 - \rho_0$. We can diagonalize $A = \sum_j \lambda_j {\|{u_j}\rangle\langle{u_j}\|}$, and define $A^+ := \sum_{j,\lambda_j \geq 0} \lambda_j {\|{u_j}\rangle\langle{u_j}\|}$ and $A^- := A - A^+$. Consider the operator $$\begin{aligned} \hat{H} &:= \Pi_{A^-} \otimes {\mathbb{I}}+\Pi_{A^+} \otimes ({\|0\rangle\langle1\|} + {\|1\rangle\langle0\|})\ ,\end{aligned}$$ where $\Pi_{A^+}$ and $\Pi_{A^-}$ are projectors on the support of $A^+$ and $A^-$ respectively. Clearly, $\hat{H}$ is Hermitian and unitary, and hence has eigenvalues $\pm 1$. We now define the Hamiltonian $H$ $$\begin{aligned} H := E_{\max} (\hat{H} + {\mathbb{I}})/2\ .\end{aligned}$$ Since any term in the Hamiltonian proportional to the identity does not affect the dynamics, we may replace the time evolution operator $\exp(-itH/\hbar)$ with $$\begin{aligned} U = \exp(-i t\\|H\\|_\infty\hat{H}/2\hbar) = \sum_{n=0}^{\infty}\frac{1}{n!} \left(\frac{-i t\\|H\\|_\infty\hat{H}}{2\hbar}\right)^n\ .\end{aligned}$$ For our choice of $\hat{H}$ we have that $$\begin{aligned} (\hat{H})^n &=\left\{ \begin{array}{ll} {\mathbb{I}}&n \mbox{ even}\ ,\\ \hat{H} &n \mbox{ odd}\ .\end{array}\right. \end{aligned}$$ and as a result the Taylor expansion for $U$ gives $$\begin{aligned} U&=(\cos(t\\|H\\|_\infty/2\hbar){\mathbb{I}}- i \sin(t\\|H\\|_\infty/2\hbar)\hat{H})\ .\end{aligned}$$ Using this unitary in our state discrimination problem, we obtain $$\begin{aligned} &P_{\rm succ}(X\|E)_{H,t}\\ &= \frac{1}{2} + \frac{1}{2} {\mathop{\mathrm{tr}}\nolimits}\left(U^\dagger ({\mathbb{I}}\otimes P_1) U(\tilde{\rho}_1 - \tilde{\rho}_0)\right)\nonumber\\ &=\frac{1}{2} + \frac{\sin^2(t\\|H\\|_\infty/2\hbar)}{2} {\mathop{\mathrm{tr}}\nolimits}\left(\hat{H}({\mathbb{I}}\otimes P_1)\hat{H}(\tilde{\rho}_1 - \tilde{\rho}_0)\right)\ .\end{aligned}$$ It remains to evaluate the last term. First of all, note that $$\begin{aligned} \hat{H}({\mathbb{I}}\otimes P_1)\hat{H} &= (\Pi_{A^+} \otimes {\|{0}\rangle\langle{0}\|} + \Pi_{A^-} \otimes {\|{1}\rangle\langle{1}\|})\ .\end{aligned}$$ Since $\tilde{\rho}_1 - \tilde{\rho}_0 = (A^+ - A^-) \otimes {\|{0}\rangle\langle{0}\|}$ and $\Pi_{A^+}\Pi_{A^-} = \Pi_{A^-}\Pi_{A^+} = 0$, we thus have $$\begin{aligned} {\mathop{\mathrm{tr}}\nolimits}\left(\hat{H}({\mathbb{I}}\otimes P_1)\hat{H}(\tilde{\rho}_1 - \tilde{\rho}_0)\right) &={\mathop{\mathrm{tr}}\nolimits}(A^+)\\ &= D(\rho_0,\rho_1)\ .\end{aligned}$$ The claim follows by an application of the double angle formula. Constraining the eigenvalues of $H$ =================================== For completeness, we now remind ourselves why in many settings it is not unreasonable to assume that $\\|H\\|_\infty$ is indeed bounded. Note that when dealing with fixed input states $\rho_0,\ldots,\rho_{N-1}$, we can without loss of generality assume that the Hamiltonian $H$ that leads to the optimal success probability possible within a certain time $t$ is limited to the energy eigenspace sufficient to contain the support of the inputs states and the standard ancilla state. This holds even in an approximate sense. To see this, consider how the state on the encoding register and the ancilla evolve in time $$\begin{aligned} \tilde{\rho}_0(t) := U(t) \tilde{\rho}_0 U(t)^\dagger\ ,\\ \tilde{\rho}_1(t) := U(t) \tilde{\rho}_1 U(t)^\dagger\ .\end{aligned}$$ Choose an error parameter ${\varepsilon}$ and define ${\Pi_{\delta}^{{\varepsilon}}}$ to be the lowest rank operator such that $[{\Pi_{\delta}^{{\varepsilon}}},H] = 0$ and $$\begin{aligned} \label{eq:epsClose} \frac{1}{2}\\|\rho_0(0) - {\Pi_{\delta}^{{\varepsilon}}}\rho_0(0) {\Pi_{\delta}^{{\varepsilon}}}\\|_1 &\leq {\varepsilon}\ ,\\ \frac{1}{2}\\|\rho_1(0) - {\Pi_{\delta}^{{\varepsilon}}}\rho_1(0) {\Pi_{\delta}^{{\varepsilon}}}\\|_1 &\leq {\varepsilon}\ .\end{aligned}$$ Since $[{\Pi_{\delta}^{{\varepsilon}}},H] = 0$ and the L1-norm is unitarily invariant, we can immediately conclude that still holds when we replace $\rho_0(0)$ and $\rho_1(0)$ with any subsequent states $\rho_0(t)$ and $\rho_1(t)$. However, this tells us that we can approximate $U$ with a unitary $\hat{U}$ as $$\begin{aligned} \hat{H} &:= {\Pi_{\delta}^{{\varepsilon}}}H {\Pi_{\delta}^{{\varepsilon}}}\ ,\\ \hat{U}(t) &:= \exp(- i \hat{H} t/\hbar)\ ,\end{aligned}$$ without affecting any of the output states, except with a chosen error ${\varepsilon}$. By the definition of the L1-norm we have for any Hermitian operator $A$ that $$\begin{aligned} \frac{1}{2}\\|A\\|_1 = \sup_{-{\mathbb{I}}\leq P \leq {\mathbb{I}}}{\mathop{\mathrm{tr}}\nolimits}(PA)\ ,\end{aligned}$$ and hence implies that using the unitary $\hat{H}$ in place of any original $H$ leads to a change in success probability in the state discrimination problem of at most $2 {\varepsilon}$. A general time-dependent Tsirelson’s bound ========================================== Let us now consider a more general version of our time-dependent Tsirelson’s bound in which we drop the assumption that the source emits a particular state, and that Alice makes a two-outcome projective measurement. The only assumption we will make now is that Bob’s Hamiltonian is bounded $\\|H\\|_\infty = E_{\max}$. For our proof, we will need the more general version of the two state discrimination problem in which the two states are not necessarily given with equal probabilities (see Corollary \[cor:traceDist\]). Again, let us first briefly consider the time unlimited case, where $M_0$ and $M_1$ are just measurements on a single system. Recall that we could express the success probability of distinguishing $\rho_0$ and $\rho_1$ given with probabilities $p_0$ and $p_1$ respectively as $$\begin{aligned} P_{\rm guess}(X\|E) = p_0 + \Delta(p_1 \rho_1, p_0 \rho_0)\ .\end{aligned}$$ At first glance, this expression appears a bit assymetric - after all, what should be so special about $p_0$? Note, however, that by replacing $M_1 = {\mathbb{I}}- M_0$ in we could also have expressed the success probability as $$\begin{aligned} P_{\rm guess}(X\|E) = p_1 + \Delta(p_0 \rho_0, p_1 \rho_1)\ . \nonumber\end{aligned}$$ In particular, it will be convenient to note that we could have also written the success probability as the average of these two terms $$\begin{aligned} P_{\rm guess}(X\|E) = \frac{1}{2}\left(1 + \sum_{x\in\01}\Delta(p_{\bar{x}} \rho_{\bar{x}}, p_x \rho_x)\right)\ .\end{aligned}$$ Let us now return to the time limited case, involving an interaction of the encoding and ancilla system, followed by a measurement on the ancilla. Recall that we have from Corollary \[cor:traceDist\] that $$\begin{aligned} \label{eq:timeLimitedSym} &P_{\rm guess}(X\|E)_{H,t}\\ &\leq p_{{ {x_{\max}} }} + \left(\frac{t \gamma \\|H\\|_\infty}{\hbar}\right) \Delta({p_{{x_{\min}}}}{\rho_{{x_{\min}}}},{p_{{ {x_{\max}} }}}{\rho_{{ {x_{\max}} }}})\ . \nonumber\end{aligned}$$ Note that in the time limited case we cannot simply average – the proof of Corollary \[cor:traceDist\] yields a different bound had we placed $p_{{x_{\min}}}$ in front (a small calculation shows that it will again single out ${p_{{ {x_{\max}} }}}$). We are now ready to show our general bound, where we will use the notation developed in Section \[sec:tsirelNotation\]. Let Bob’s Hamiltonian be scaled such that $H \geq 0$. Then the maximum success probability of winning the CHSH game for Alice and Bob in time $t$ obeys $$\begin{aligned} p_{\rm win}^t \leq \frac{1}{2}\left(\sum_{z\in\01} {p_{{ {x_{\max}} }}}^z\right) + \frac{\gamma t \\|H\\|_\infty}{\sqrt{2} \hbar}\ , \end{aligned}$$ where $$\begin{aligned} \gamma := \left\{ \begin{array}{ll}5/\pi & \mbox{if } 1 < t E_n/\hbar < 4 \ ,\\ 3/\pi & \mbox{otherwise} \ . \end{array}\right. \end{aligned}$$ We have from Tsirelson’s bound [@tsirel:original] that for any strategy of Alice and Bob no matter how much time or energy they may have available $$\begin{aligned} p_{\rm win} &= \frac{1}{2}\sum_{z \in \01} P_{\rm guess}(X^z\|E^z)\\ &= \frac{1}{4}\sum_{z} \left(1 + \sum_{x \in \01} \Delta(p_{\bar{x}}^z \sigma_{\bar{x}}^z, p_x^z \sigma_x^z)\right)\\ &\leq \frac{1}{2} + \frac{1}{2\sqrt{2}}\ .\end{aligned}$$ Rearranging terms gives us $$\begin{aligned} \frac{1}{4}\sum_{z,x\in \01} \Delta(p_{\bar{x}}^z \sigma_{\bar{x}}^z, p_x^z \sigma_x^z) &\leq \frac{1}{2} + \frac{1}{2\sqrt{2}} - \frac{1}{2}\\ & = \frac{1}{2\sqrt{2}}\ .\end{aligned}$$ Since $\Delta(\cdot,\cdot) \geq 0$, this means that $$\begin{aligned} \frac{1}{2}\sum_{z} \Delta({p_{{x_{\min}}}}^z \sigma_{{x_{\min}}}^z, {p_{{ {x_{\max}} }}}^z \sigma_{{ {x_{\max}} }}^z) \leq \frac{1}{\sqrt{2}}\ .\end{aligned}$$ Our claim for the time limited case now follows by plugging this bound into $$\begin{aligned} p_{\rm win}^t &= \frac{1}{2}\sum_{z \in \01} P_{\rm guess}(X^z\|E^z)_{H,t} \\ &\leq \frac{1}{2}\left(\sum_{z \in \01} {p_{{ {x_{\max}} }}}^z\right) +\\ & \qquad\frac{C}{2} \sum_{z,x \in \01} \Delta({p_{{x_{\min}}}}^z \sigma_{{x_{\min}}}^z, {p_{{ {x_{\max}} }}}^z \sigma_{{ {x_{\max}} }}^z)\ ,\nonumber\\ &\leq \frac{1}{2}\left(\sum_{z \in \01} {p_{{ {x_{\max}} }}}^z\right) + \frac{C}{\sqrt{2}}\end{aligned}$$ where we have used the shorthand $$\begin{aligned} C := \frac{t \gamma \\|H\\|_\infty}{\hbar}\ .\end{aligned}$$ At first glance, this bound may seem somewhat strange as it involves a potentially unknown term $\sum_x {p_{{ {x_{\max}} }}}^z$. Note, however, that this term is determined by the distributions over the states that Bob should distinguish. It is this distribution, that determines the classical bound for CHSH and hence $$\begin{aligned} \frac{1}{2}\sum_z {p_{{ {x_{\max}} }}}^z \leq \frac{3}{4}\ .\end{aligned}$$ Note that since the ancilla is initialized to the most likely $x$ in each case, there exists a strategy for Alice and Bob with which they can play optimally classically in no time at all. Yet, since there is generally an interplay between the choice of distributions and the state Alice can create we derived the bound in its more general form above. \[cor:tsirel\] Let Bob’s Hamiltonian be scaled such that $H \geq 0$. Then the maximum success probability of winning the CHSH game for Alice and Bob in time $t$ obeys $$\begin{aligned} p_{\rm win}^t \leq \frac{3}{4} + \frac{\gamma t \\|H\\|_\infty}{\sqrt{2} \hbar}\ , \end{aligned}$$ where $$\begin{aligned} \gamma := \left\{ \begin{array}{ll}5/\pi & \mbox{if } 1 < t E_n/\hbar < 4 \ ,\\ 3/\pi & \mbox{otherwise} \ . \end{array}\right. \end{aligned}$$ [^1]: In analogy to the von Neumann entropy, ${\ensuremath{ {\rm H}}_{\infty}}(X\|E) = {\ensuremath{ {\rm H}}_{\infty}}(X)$ since ${\ensuremath{ {\rm H}}_{\infty}}(X) := - \log \max_x p_x$. [^2]: Note that our assumption that the referee is unrestricted contrasts with the view of computational complexity in which such games play a role in interactive proof systems. There, Alice and Bob are all-powerful, but the referee has limited time at his disposal to decide the outcome of the game. We would like to emphasize that our aim here is entirely different since we are merely interested in the strength of correlations between Alice and Bob that can be obtained within a certain time frame. [^3]: A cqq-state is a classical-quantum-classical state, here classical on the source registers $X$, quantum on the encoding system $E$ and, before the referee’s measurement, quantum on the ancilla $A$.
--- abstract: 'In recent years, Bayes filter methods in the labeled random finite set formulation have become increasingly powerful in the multi-target tracking domain. One of the latest outcomes is the Generalized Labeled Multi-Bernoulli (GLMB) filter which allows for stable cardinality and target state estimation as well as target identification in a unified framework. In contrast to the initial context of the GLMB filter, this paper makes use of it in the Track-Before-Detect (TBD) scheme and thus, avoids information loss due to thresholding and other data preprocessing steps. This paper provides a TBD GLMB filter design under the separable likelihood assumption that can be applied to real world scenarios and data in the automotive radar context. Its applicability to real sensor data is demonstrated in an exemplary scenario. To the best of the authors’ knowledge, the GLMB filter is applied to real radar data in a TBD framework for the first time.' address: - 'University of Stuttgart, Germany, (e-mail: david.meister@ist.uni-stuttgart.de)' - 'TU Darmstadt, Germany, (e-mail: holder@fzd.tu-darmstadt.de)' - 'TU Darmstadt, Germany, (e-mail: winner@fzd.tu-darmstadt.de)' author: - David Meister - 'Martin F. Holder' - Hermann Winner bibliography: - 'references.bib' title: 'A Track-Before-Detect Approach to Multi-Target Tracking on Automotive Radar Sensor Data' --- Bayesian methods, Particle filtering, Estimation and filtering, Sensor integration and perception, Random finite sets, Multi-object tracking, GLMB, Track-before-detect Introduction ============ Radar sensors are widely used for object detection in the automotive domain due to their ability to simultaneously measure range and relative velocity as well as their robustness against adverse weather conditions. The measurements of a radar sensor are initially available as a multidimensional spectrum in the dimensions distance, relative radial velocity and azimuth angle. In a typical radar signal processing chain, an (adaptive) threshold is applied to detect individual targets. Subsequently, a tracking process relates targets at different time steps. On the contrary, in the so-called Track-Before-Detect (TBD) framework, objects are estimated directly from the spectral data without any thresholding. This eliminates potential information loss and has been widely discussed in the literature, e.g. by [@Boers.2003], [@Vo.2010], or [@Papi.2013]. In general, multi-target tracking is an estimation problem where targets and their states are to be identified individually. Multi-target Bayes estimators in the Random Finite Set (RFS) formulation, introduced by [@Mahler.2003], [@Vo.2010], [@Vo.2013b], and others, allow for the simultaneous estimation of the number of objects and their states. The GLMB filter developed by [@Vo.2013b] is one of the latest RFS multi-target tracking approaches and according to [@Vo.2015], the first exact closed form solution to the multi-target Bayes recursion that allows for an unbiased estimation of target number and states as well as the unique identification of each target. [@Papi.2013] apply the TBD idea to the GLMB filter framework from a theoretical perspective. This paper extends their work to make it applicable to real radar sensor data. Multi-Target Tracking with the GLMB filter ========================================== This section introduces the GLMB filter recursion equations and the necessary labeled RFS theory. Let $${\ensuremath{1_S \!\left( X\right)}} = \begin{cases} 1, & X \subseteq S \\ 0, & \text{otherwise} \end{cases}, \quad {\ensuremath{\delta_S \!\left( X\right)}} = \begin{cases} 1, & X = S \\ 0, & \text{otherwise} \end{cases}$$ be the inclusion function and the generalization of the Dirac delta function, respectively. GLMB Random Finite Set ---------------------- We introduce the $\delta$-GLMB RFS distribution as the multi-object analogue of a probability density function (PDF) for the set of labeled target state vectors ${\ensuremath{\tilde{X}}}$: $${\ensuremath{\tilde{\pi}}}({\ensuremath{\tilde{X}}}) = \Delta({\ensuremath{\tilde{X}}}) \sum_{\left( I, \xi \right) \in {\ensuremath{\mathcal{F} \!\left( \mathbb{L}\right)}}\times\Xi} \omega^{(I, \xi)} \delta_{I}(\mathcal{L}({\ensuremath{\tilde{X}}})) \left[ p^{(\xi)} \right]^{{\ensuremath{\tilde{X}}}}, \label{eq:d_glmb_pdf_def}$$ where ${\ensuremath{\mathcal{F} \!\left( \mathbb{L}\right)}}$ and $\Xi$ denote the space of all finite subsets of the label space $\mathbb{L}$ and a discrete index set, $p^{(\xi)}({\ensuremath{\tilde{\bm{x}}}})$ refers to the distribution of a state vector ${\ensuremath{\tilde{\bm{x}}}}$ in ${\ensuremath{\tilde{X}}}$, $\omega^{(I, \xi)}$ represents a non-negative weight and a multi-target exponential is defined as $h^{{\ensuremath{\tilde{X}}}} = \prod_{{\ensuremath{\tilde{\bm{x}}}} \in {\ensuremath{\tilde{X}}}} {\ensuremath{h \!\left( {\ensuremath{\tilde{\bm{x}}}}\right)}}$ with $h^{\emptyset} = 1$. Furthermore, the distinct label indicator $\Delta({\ensuremath{\tilde{X}}}) = \delta_{\lvert {\ensuremath{\tilde{X}}} \rvert}(\lvert \mathcal{L}({\ensuremath{\tilde{X}}}) \rvert)$ compares the cardinality of the RFS ${\ensuremath{\tilde{X}}}$ with the one of its labels $\mathcal{L}({\ensuremath{\tilde{X}}}) = \lbrace {\ensuremath{\mathcal{L} \!\left( {\ensuremath{\tilde{\bm{x}}}}\right)}} : {\ensuremath{\tilde{\bm{x}}}} \in {\ensuremath{\tilde{X}}} \rbrace$ and therefore, indicates if ${\ensuremath{\tilde{X}}}$ has distinct labels. Let ${\ensuremath{\mathcal{L} \!\left( {\ensuremath{\tilde{\bm{x}}}}\right)}} = {\ensuremath{\mathcal{L} \!\left( \left( \bm{x}, l \right)\right)}} = l$ return the label $l$ of ${\ensuremath{\tilde{\bm{x}}}}$. Essentially, the labeled multi-object PDF is described by a weighted mixture of multi-target exponentials $[ p^{(\xi)} ]^{{\ensuremath{\tilde{X}}}}$. For more details, see [@Vo.2013b]. GLMB Filter Framework {#sec:GLMB_filter} --------------------- This subsection presents the GLMB filter recursion in the joint prediction and update version according to [@Vo.2017], slightly modified in favor of the TBD use case according to [@Papi.2015] as well as to accommodate the birth model presented in Section \[sec:abm\]. Let ${\ensuremath{\left\langle f, g \right\rangle}} = \int {\ensuremath{f \!\left( x\right)}} {\ensuremath{g \!\left( x\right)}} {\ensuremath{\, {\mathrm{d}}}}x$ abbreviate the standard inner product. The joint prediction and update directly leading from the previous GLMB posterior density $$\begin{gathered} {\ensuremath{\tilde{\pi}}}_{k-1}({\ensuremath{\tilde{X}}}_{k-1} {\ensuremath{\!\mid\!}}\bm{z}_{1:k-1}) \propto \Delta({\ensuremath{\tilde{X}}}_{k-1}) \\ \cdot \sum_{I_{k-1} \in {\ensuremath{\mathcal{F} \!\left( \mathbb{L}_{0:k-1}\right)}}} \omega_{k-1}^{(I_{k-1})} \delta_{I_{k-1}}(\mathcal{L}({\ensuremath{\tilde{X}}}_{k-1})) \left[ p_{k-1}^{(I_{k-1})} \right]^{{\ensuremath{\tilde{X}}}_{k-1}},\end{gathered}$$ to the one at the current time step can be expressed as $$\begin{gathered} {\ensuremath{\tilde{\pi}}}_{k}({\ensuremath{\tilde{X}}}_{k} {\ensuremath{\!\mid\!}}\bm{z}_{1:k}) \propto \Delta({\ensuremath{\tilde{X}}}_{k}) \\ \cdot \sum_{\substack{I_{k-1} \in {\ensuremath{\mathcal{F} \!\left( \mathbb{L}_{0:k-1}\right)}},\\ I_{k} \in {\ensuremath{\mathcal{F} \!\left( \mathbb{L}_{0:k}\right)}}}} \omega_{k-1}^{(I_{k-1})} \omega_{\bm{z}_{k}}^{(I_{k-1}, I_{k})} \delta_{I_{k}}(\mathcal{L}({\ensuremath{\tilde{X}}}_{k})) \left[ p_{k}^{(I_{k})} \right]^{{\ensuremath{\tilde{X}}}_{k}}, \label{eq:glmb_jup_post}\end{gathered}$$ where the $\delta$-GLMB form introduced in has been used and $\bm{z}_{1:k}$ abbreviates the set of measurements received from the first to the $k$-th time step. Likewise, $\mathbb{L}_{0:k}$ refers to the label space built from the start of the recursion to the $k$-th time step. Moreover, the following definitions apply: $$\begin{aligned} \omega_{\bm{z}_{k}}^{(I_{k-1}, I_{k})} =& \left[ 1-\bar{p}_{{\mathrm{S}}}^{(I_{k-1})} \right]^{I_{k-1}-I_{k}} \left[ \bar{p}_{{\mathrm{S}}}^{(I_{k-1})} \right]^{I_{k-1} \cap I_{k}} \nonumber\\ \cdot~& \left[ 1-\mathfrak{r}_{{\mathrm{B}},k} \right]^{{\ensuremath{I_{{\mathrm{B}}} \!\left( I_{k-1}\right)}} - I_{k}} \mathfrak{r}_{{\mathrm{B}},k}^{ {\ensuremath{I_{{\mathrm{B}}} \!\left( I_{k-1}\right)}} \cap I_{k}} \left[ \bar{\psi}_{\bm{z}_{k}}^{(I_{k})} \right]^{I_{k}}, \\ {\ensuremath{\bar{p}_{{\mathrm{S}}}^{(I_{k-1})} \!\left( l\right)}} =& {\ensuremath{\left\langle {\ensuremath{p_{k-1}^{(I_{k-1})} \!\left( \cdot, l\right)}}, {\ensuremath{p_{{\mathrm{S}}} \!\left( \cdot, l\right)}} \right\rangle}}, \\ {\ensuremath{\bar{\psi}_{\bm{z}_{k}}^{(I_{k})} \!\left( l\right)}} =& {\ensuremath{\left\langle {\ensuremath{p_{{\ifthenelse{\equal{1}{0}}{\ensuremath{k \mid k}}{\ensuremath{k \mid k-1}}}}^{(I_{k})} \!\left( \cdot, l\right)}}, {\ensuremath{\psi_{\bm{z}_{k}} \!\left( \cdot, l\right)}} \right\rangle}},\\ {\ensuremath{p_{{\ifthenelse{\equal{1}{0}}{\ensuremath{k \mid k}}{\ensuremath{k \mid k-1}}}}^{(I_{k})} \!\left( \bm{x}_{k}, l\right)}} =& {\ensuremath{1_{\mathbb{L}_{0:k-1}} \!\left( l\right)}} {\ensuremath{p_{{\mathrm{S}}}^{(I_{k})} \!\left( \bm{x}_k, l\right)}} + {\ensuremath{1_{\mathbb{L}_{k}} \!\left( l\right)}} {\ensuremath{p_{{\mathrm{B}}, k} \!\left( \bm{x}_{k}, l\right)}}, \\ {\ensuremath{p_{{\mathrm{S}}}^{(I_{k})} \!\left( \bm{x}_k, l\right)}} =& \frac{{\ensuremath{\left\langle {\ensuremath{p_{{\mathrm{S}}} \!\left( \cdot, l\right)}} {\ensuremath{f_{{\ifthenelse{\equal{1}{0}}{\ensuremath{k \mid k}}{\ensuremath{k \mid k-1}}}} \!\left( {\ensuremath{\tilde{\bm{x}}}}_{k} {\ensuremath{\!\mid\!}}\cdot, l\right)}}, {\ensuremath{p_{k-1}^{(I_{k-1})} \!\left( \cdot, l\right)}} \right\rangle}}}{ {\ensuremath{\bar{p}_{{\mathrm{S}}}^{(I_{k-1})} \!\left( l\right)}}},\\ {\ensuremath{p_{k}^{(I_{k})} \!\left( \bm{x}_{k}, l\right)}} =& \frac{{\ensuremath{p_{{\ifthenelse{\equal{1}{0}}{\ensuremath{k \mid k}}{\ensuremath{k \mid k-1}}}}^{(I_{k})} \!\left( \bm{x}_{k}, l\right)}} {\ensuremath{\psi_{\bm{z}_{k}} \!\left( \bm{x}_{k}, l\right)}}}{ {\ensuremath{\bar{\psi}_{\bm{z}_{k}}^{(I_{k})} \!\left( l\right)}}},\end{aligned}$$ with $I_i - I_j = \left(I_i \cup I_j\right) \backslash I_j$ and ${\ensuremath{f_{{\ifthenelse{\equal{1}{0}}{\ensuremath{k \mid k}}{\ensuremath{k \mid k-1}}}} \!\left( {\ensuremath{\tilde{\bm{x}}}}_{k} {\ensuremath{\!\mid\!}}{\ensuremath{\tilde{\bm{x}}}}_{k-1}\right)}}$ referring to the Markov transition density imposed by the transition model presented in Section \[sec:tm\]. The set $I_k$ contains all labels of a potential target constellation at time step $k$, also referred to as hypothesis, while ${\ensuremath{I_{{\mathrm{B}}} \!\left( I_{k-1}\right)}}$ denotes the label set of potential new born or appearing targets if $I_{k-1}$ is assumed as the previous target constellation. The motivation for changing the formulation by [@Vo.2017] is to allow for target birth taking into account the non-overlapping illumination region assumption leading to separable likelihoods required to derive the measurement model in Section \[sec:mm\]. Moreover, $p_{{\mathrm{S}}}({\ensuremath{\tilde{\bm{x}}}}_k)$ quantifies the survival probability of a target, i.e. the prior probability of continued existence of a target after one time step. Assuming a Labelled Multi-Bernoulli RFS for describing the birth or target appearing process, $\mathfrak{r}_{{\mathrm{B}}}$ and $p_{{\mathrm{B}}}({\ensuremath{\tilde{\bm{x}}}})$ can be interpreted as birth probability and state distribution of a new born target. For the TBD measurement model presented in Section \[sec:mm\], the measurement likelihood contribution of a present target can be written as $${\ensuremath{\psi_{\bm{z}_{k}} \!\left( {\ensuremath{\tilde{\bm{x}}}}\right)}} = \prod_{\iota \in {\ensuremath{C \!\left( {\ensuremath{\tilde{\bm{x}}}}\right)}}} \ell^{(\iota)}(z^{(\iota)} {\ensuremath{\!\mid\!}}{\ensuremath{\tilde{\bm{x}}}}) \label{eq:meas_lik_tgt_contr}$$ where $\ell^{(\iota)}(z^{(\iota)} \| {\ensuremath{\tilde{\bm{x}}}})$ is called likelihood ratio. What is more, $C({\ensuremath{\tilde{\bm{x}}}})$ refers to the illumination region of a specific target representing a set of measurement vector elements $\{z^{(\iota)}\}_{\iota \in {\ensuremath{C \!\left( {\ensuremath{\tilde{\bm{x}}}}\right)}}}$ influenced by the target’s presence. Gibbs sampling is utilized as the truncation method as suggested by [@Vo.2017]. Radar Sensor ============ The Fast Chirp Modulation radar sensor used in the experiments is widely employed in the automotive field. It combines frequency and pulse modulation by transmitting a sequence of linearly increasing frequency ramps (chirps). By analyzing phase and frequency shifts as well as time delays of incoming waves reflected by surfaces in the field of view, range, radial relative speed and azimuth angle of targets can be computed. The intensity of the received wave depends on the absorption and reflection properties of the target surface, the distance to the target and multi-path propagation mitigation effects. By performing a three-dimensional Fast Fourier Transform, the individual wave properties of multiple reflections can be isolated. [@Wintermantel.2009] describes the aforementioned process yielding the data structure referred to as radar cube in detail. The measurement space is discretized along the dimensions range, radial relative speed and azimuth angle leading to multiple measurement cells. They are assigned an intensity proportional to the power of the wave potentially received from a target at the respective location in the measurement space. By stacking these intensity values in a fixed order, the measurement vector $\bm{z}_k$ is obtained. Experimental Setup and Models ============================= This section presents the transition and measurement model as well as the employed adaptive birth model and track merging strategy. A more detailed discussion on all subsections is provided by [@Meister.2019]. Transition Model {#sec:tm} ---------------- In multi-target tracking literature, it is common to assume a constant velocity transition model as presented by [@Vo.2014] or [@Papi.2015]. With the state vector $\bm{x}_{k} = [x_{k}, \dot{x}_{k}, y_{k}, \dot{y}_{k}, \vartheta_{k}]^T$ consisting in planar position and velocity as well as a constant fifth state to be defined later on, the target transition equation can be expressed as $$\bm{x}_k = A \bm{x}_{k-1} + \bm{v}_{k-1}, \quad \bm{v}_{k-1} \sim \mathcal{N}(\bm{0},Q)$$ where $$\begin{aligned} A &= \mathrm{diag}(\bar{A}, \bar{A}, 1), & Q &= \mathrm{diag}(\sigma_{\ddot{x}}^2 \bar{Q}, \sigma_{\ddot{y}}^2 \bar{Q}, \sigma_{\dot{\vartheta}}^2 \Delta t^2) \\ \bar{A} &= \begin{bmatrix} 1 & \Delta t \\ 0 & 1 \end{bmatrix}, & \bar{Q} &= \begin{bmatrix} \frac{\Delta t^4}{4} & \frac{\Delta t^3}{2} \\ \frac{\Delta t^3}{2} & \Delta t^2 \end{bmatrix}\end{aligned}$$ and $\Delta t$ refers to the time step size. The process noise $\bm{v}_{k-1}$ can therefore be interpreted as being caused by a random acceleration between two time steps. Please note that the model can only give reasonable predictions if the rotational speed of the ego vehicle is negligible. Otherwise, the latter would need to be estimated, e.g. by an ego motion estimator, to compensate for the relative target state transition due to the sensor rotation. Thus, ego vehicle steering is avoided in the scenario in Section \[sec:results\]. TBD Measurement Model {#sec:mm} --------------------- An overview on multi-target likelihoods in the TBD framework is provided by [@Lepoutre.2016]. In this paper, we restrict the implementation to the Swerling 1 case with squared modulus measurements provided by the radar sensor. When assuming separable likelihoods due to non-overlapping illumination regions and neglecting the spatial coherence of the measurements, the likelihood ratio in for the Swerling 1 case and target $i$ can be derived to be $$\ell(z_{k}^{(\iota)} {\ensuremath{\!\mid\!}}{\ensuremath{\tilde{\bm{x}}}}_{k,i}) = \frac{1}{1 + {\ensuremath{\varsigma^{(\iota)} \!\left( {\ensuremath{\tilde{\bm{x}}}}_{k,i}\right)}}} \exp \left( \frac{z_{k}^{(\iota)} {\ensuremath{\varsigma^{(\iota)} \!\left( {\ensuremath{\tilde{\bm{x}}}}_{k,i}\right)}}}{ {\ensuremath{\mu_{z}^{(\iota)} \!\left( {\ensuremath{\tilde{\bm{x}}}}_{k,i}\right)}}} \right) \label{eq:sw1_lik_ratio}$$ where $ {\ensuremath{\varsigma^{(\iota)} \!\left( {\ensuremath{\tilde{\bm{x}}}}_{k,i}\right)}} = \frac{\sigma_{\rho_i}^2}{\sigma_{w}^2} \left\lvert {\ensuremath{h^{(\iota)} \!\left( {\ensuremath{\tilde{\bm{x}}}}_{k,i}\right)}} \right\rvert^2 $ and $ {\ensuremath{\mu_{z}^{(\iota)} \!\left( {\ensuremath{\tilde{\bm{x}}}}_{k,i}\right)}} = 2\sigma_{w}^2 + 2\sigma_{\rho_i}^2 \left\lvert {\ensuremath{h^{(\iota)} \!\left( {\ensuremath{\tilde{\bm{x}}}}_{k,i}\right)}} \right\rvert^2. \label{eq:sw1_exp_meas} $ The function ${\ensuremath{h^{(\iota)} \!\left( {\ensuremath{\tilde{\bm{x}}}}_{k,i}\right)}}$ returns the value of a point-spread function in measurement cell $\iota$ and $\sigma_{w}^2$ quantifies the measurement noise variance. In contrast to the suggestion by [@Lepoutre.2016], the following correspondence is established: $$\vartheta_{k,i} = \sigma_{\rho_i}^2 \cdot \frac{{\ensuremath{G \!\left( {\ensuremath{\tilde{\bm{x}}}}_{k,i}\right)}}^2}{{\ensuremath{r \!\left( {\ensuremath{\tilde{\bm{x}}}}_{k,i}\right)}}^4}, \label{eq:sw1_fifth_state}$$ where corrections according to the radar range equation (see [@Richards.2014]) by the antenna gain ${\ensuremath{G \!\left( {\ensuremath{\tilde{\bm{x}}}}_{k,i}\right)}}$ and the range ${\ensuremath{r \!\left( {\ensuremath{\tilde{\bm{x}}}}_{k,i}\right)}}$ are incorporated. Adaptive Birth Model {#sec:abm} -------------------- Since the non-overlapping target illumination region assumption is crucial to the performance of the filter, a birth model is designed that incorporates present targets and their illumination regions in the birth process. The idea behind the birth model utilized in this paper is to initialize targets in regions with significant measurements while ensuring that new born targets do not overlap with existing ones in their illumination regions. The latter are assumed to be of equal ellipsoidal size in the measurement space to achieve simple equations to check for overlapping illumination regions. For details, see [@Meister.2019]. Track Merging Strategy ---------------------- The GLMB filter under the non-overlapping illumination region assumption requires a track merging strategy. [@Vo.2010] and [@Mahler.2014] provide helpful insights into adequate methods for other RFS approaches. For the reason of simplicity, an either-or-relationship is established between overlapping targets in the Gibbs sampling process here: only one of the targets can survive the update step. This is achieved by modifying the Gibbs cost matrix rows of overlapping targets adequately as soon as one of them is sampled to exist, i.e. the other targets’ death probabilities are artificially set to 1. Further research could establish more sophisticated merging strategies for the TBD GLMB filter including a combination of elimination regions and track merging as suggested by [@Suzuki.2018]. Experimental Results {#sec:results} ==================== A Sequential Monte Carlo approximation according to [@Vo.2014] was utilized to obtain the following results. Scenario Description and Filter Tuning -------------------------------------- The examined scenario incorporates the ego vehicle with the radar sensor mounted at the front and two passenger cars as targets moving in line on a straight road. The vehicles accelerate to the speeds shown in Fig. \[fig:dyn\_scheme\]. The light grey area represents the sensor field of view. This scenario allows to analyze the filter behavior in a dynamic environment including target occlusion effects while maintaining the transition model assumption of no ego vehicle steering. ![Scenario illustration[]{data-label="fig:dyn_scheme"}](sc_dyn_scheme){width="8.7cm"} The survival probability $p_{{\mathrm{S}}}$ is set to $0.99$. The algorithm considers a maximum of 200 hypotheses and utilizes 15000 particles to represent a target state distribution. The initial birth existence probability $\mathfrak{r}_{{\mathrm{B}}}$ is kept at 0.3 and the illumination region ellipsoid radii are set to the doubled sensor resolution cell size. Birth targets are initialized uniformly around the centroid of a measurement cell with a significant intensity above $z_{th} = 10^{-5}$. The measurement noise $\sigma_{w}^2$ is quantified as $2 \cdot 10^{-6}$. A new measurement is approximately received every 70 ms. The process noise variances are $\sigma_{\ddot{x}}^2 = \sigma_{\ddot{y}}^2 = \left(\frac{5}{3}\right)^2~{\mathrm{m}}^2{\mathrm{s}}^{-4}$ and $\sigma_{\dot{\vartheta}}^2 = 10^{-3}$. Filter Behavior Analysis ------------------------ The estimated target positions and velocities over time are shown in Fig. \[fig:dyn\_pos\_matched\] and Fig. \[fig:dyn\_vel\_matched\]. They are depicted along with the estimates of the sensor internal estimation algorithm considered as a ground truth reference. The coloring of the estimates refers to the unique target identities. Target estimates at the borders of the field of view have been filtered out due to the fact that they originate from a data ambiguity imposed by the sensor design that was not incorporated in the implemented measurement model. ![Estimated target positions $x$ and $y$ over time $k$[]{data-label="fig:dyn_pos_matched"}](sc_dynamic_REAL_Sw1_20_09_2019_at_10_38_17_noIt_573_tStart_470_truth_0_matched_pos_over_k.pdf){width="9cm"} ![Estimated target velocities $\dot{x}$ and $\dot{y}$ over time $k$[]{data-label="fig:dyn_vel_matched"}](sc_dynamic_REAL_Sw1_20_09_2019_at_10_38_17_noIt_573_tStart_470_truth_0_matched_vel_over_k.pdf){width="9cm"} It can be observed that the implemented TBD GLMB filter estimates the target positions and velocities reasonably: The lateral distance $y$ to the ego vehicle center stays close to zero at all times. Small deviations from zero in the lateral direction can be explained by the physical size of the target indicated in yellow. It is worth highlighting that the filter tracks the centers of reflection and not the target centers. Moreover, the longitudinal distance $x$ follows the sensor internal estimates shown as black lines very accurately. Due to multi-path propagation, the sensor can also detect the occluded distant vehicle. Nonetheless, the filter looses track of the distant vehicle approximately from time step 370 onward. This is mainly caused by signal mitigation effects for an occluded distant vehicle. Further tuning of the assumed noise floor might allow for better tracking results like for the sensor internal estimator. Similar observations can be made for the velocity estimates depicted in Fig. \[fig:dyn\_vel\_matched\]. The longitudinal velocity estimates follow the radar sensor estimates consistently. The higher volatility in the targets’ lateral speed is induced by the missing measurement information regarding the velocity in that direction. Moreover, sudden shifts in the reflection center location, potentially induced by slight target orientation variations, result in significant velocity changes. Lastly, targets are labelled quite persistently which indicates that the TBD GLMB filter is capable of keeping track of target identities in an adequate fashion. Conclusion ========== This paper provides a proof of concept for the implemented TBD GLMB filter with an automotive radar sensor. A real data experiment has been performed that yields desirable tracking performance. A more detailed analysis, also considering more scenarios and elaborating on the made assumptions in depth, is provided by [@Meister.2019]. Further research could focus on more sophisticated track merging strategies or the assessment of different measurement models. What is more, including weakly reflecting targets in the experiments is expected to allow for a closer evaluation of the magnitude of the mentioned information loss. Designing a real-time capable TBD GLMB filter algorithm remains an open challenge as well.
--- abstract: 'In this paper we study a two-photon time-dependent Jaynes-Cummings model interacting with a Kerr-like medium. We assumed that the electromagnetic field is in different states such as coherent, squeezed vacuum and pair coherent, and that the atom is initially in the excited state. We studied the temporal evolution of the population of the excited level, and the second order coherence function. The results obtained show that this system has some similarities with the two-mode Stark system. We analize two photon entanglement for different initial conditions.' address: - \| Departamento de Física, Facultad de Ciencias Exactas y Naturales,\ Universidad de Buenos Aires, (1428) Buenos Aires, Argentina - ':' author: - 'Guido Berlin and J. Aliaga' title: 'Quantum dynamical properties of a two-photon non-linear Jaynes-Cummings model' --- [PACS numbers]{}: 42.50.-p, 42.50.Dv.\ [Corresponding author]{}: Jorge Aliaga. e-mail: jaliaga@df.uba.ar\ Departamento de Física, Facultad de Ciencias Exactas y Naturales,\ Universidad de Buenos Aires, Pabellón I, (1428) Buenos Aires, Argentina\ FAX: +54 - 11 - 4576 - 3357 Introduction ============ One of the most fundamental models in quantum optics is the interaction of a single two-level atom with a single quantized mode of radiation, described by the Jaynes-Cummings Hamiltonian [@Jaynes63; @Shore93]. Despite being simple enough to be analytically soluble in the rotating-wave approximation, this model is able to describe the quantum-mechanical aspects of the interaction between light and matter. It has led to nontrivial predictions, such as the existence of “collapses" and “revivals" in the atomic excitation [@Eberly80], and has also allowed a deeper understanding of the dynamical entangling and disentangling of the atom-field system in the course of time [@Phoenix88]. The interest in the Jaynes-Cummings model (JCM) is mainly due to the fact that some of its predictions are nowadays accessible to experimental verification [@Exper]. A JCM interaction can be experimentally realized in cavity-QED setups and, as an effective interaction, in laser-cooled trapped ions. During the last decade many theoretical and experimental efforts have been done in order to study two-photon processes involving atoms inside a cavity, stimulated by the experimental realization of a two-photon cascade micromaser [@Brune] Two-photon processes are also an efficient way of generating squeezed states of the electromagnetic field. It has also been established that two-photon degenerate atoms inside a Kerr-like medium can generate squeezing amplification [@Squeez_Amp]. The recent discovery of new materials with very high Kerr coupling [@H_Kerr] opened the possibility for the implementation of new experiments that generates entangled states [@Vitali] that can be used for a perfect Bell-state discrimination. A system composed by a three level atom in $\Xi$, $\Lambda$, and $V$ configurations interacting with two modes of the electromagnetic field was proposed and studied more than ten years ago by Yoo and Eberly [@Yoo]. Following these guidelines, Gou [@Gou1] investigated the $\Xi$ configuration when the intermediate level can be adiabatically eliminated [@Alsing]. This approximation turns the original bilinear photon-level interaction into a three-linear one (usually called “non-linear non-degenerate two-photon” interaction). This model has been broadly used in order to study the time evolution of the atomic and photon operators, the second order coherence function, the one and two-mode squeezing, the atomic-dipole squeezing and the emission spectra [@Gou1; @Joshi; @Gerry1; @Abdel; @Ashraf1; @Ashraf2; @Iwasawa]. Usually, the two-level system has been considered initially in the excited state and the two-photons have been chosen initially in two independent coherent states [@Gou1; @Abdel; @Ashraf1; @Iwasawa], two mode squeezed states [@Gou1], pair-coherent states [@Joshi], correlated SU(1,1) coherent states [@Gerry1], etc. The $\Lambda$ configuration, also called the Raman coupled model, when the intermediate level can be adiabatically eliminated was studied by Abdalla, Ahmed and Obada [@Abdalla], and independently by Gerry and Eberly [@Gerry2]. There have been investigations of the atomic inversion, the appearance of antibunched light, the violations of the Cauchy-Schwartz inequality, population trapping, and squeezing [@Abdalla; @Gerry2; @Cardimona; @Gerry3; @Deb1]. Some similarities in the Rabi frequencies of both configurations for some special conditions of the parameters were reported [@Gerry1], and the connection between these configurations was deeply studied in Ref. [@JC2]. It was reported that for the $\Xi$ [@Gou2; @Souza] and $\Lambda$ [@Nasreen; @Li] models, when doing the adiabatic approximation, the appearance of Stark shifts must be taken into account. In Ref. [@JC2] the Stark shifts were neglected in order to separate their contribution to the nontrivial dynamics studied. The intensity-dependent Stark shifts caused by off-resonant levels [@Joshi2] were modeled by Moya-Cessa, Bu$\check{{\rm z}}$ek and Knight [@Moya2] using a JCM with an intensity-dependent shift of the two-level transition. Recently, some similarities and differences between the models used in order to describe a cavity filled with a Kerr-like medium, modeled by an anharmonic oscillator, (first analyzed in detail by Jex and Bu$\check{{\rm z}}$ek [@Buzek] and discussed in many articles thereafter [@Werner; @Ho; @Joshi2; @Gruver2]) and Stark effects have been studied [@Moya; @Kerr-Stark]. In this paper we study a two-photon time-dependent Jaynes-Cummings model interacting with a nonlinearity that can be, for instance, a Kerr-like medium. The problem is solved using a technique based on obtaining those observables dynamically related [@JC2], and then solving their temporal evolution. We assumed that the electromagnetic field is in different states such as coherent, squeezed vacuum and pair coherent, and that the atom is initially in the excited state. We studied the temporal evolution of the population of the excited level, and the second order coherence function. The results obtained show that this system has some similarities with the two-mode Stark system. The system ========== We study an effective two-level atom [@JC2], whose levels $\|g>$ and $\|e>$, with energies $E_1$ and $E_2$, interact with two modes of electromagnetic radiation of frequencies $\omega_1$ and $\omega_2$ inside a non-linear Kerr-like medium. In two-photon processes there are more than two levels involved, but it is possible to neglect them if we assume that $\omega_1+\omega_2 \approx E_1 - E_2$ ($\hbar=1$) and we consider that the transition frequencies between $\|e>$, $\|g>$ and the intermediate levels are different from the frequencies of the field. So, the JCM in the rotating wave approximation, ($\Xi$ configuration), reads [@JC2]: $$\begin{aligned} \label{ham2m} \hat H = \sum_{i=1}^{2} E_{i} \hat b_{i}^{\dagger} \hat b_{i} + \sum_{i=1}^{2} \omega_{i} \hat a_{i}^{\dagger} \hat a_{i} + T(t)(\gamma \hat a_{1} \hat a_{2} \hat b_{1} \hat b_{2}^{\dagger} + \gamma^{} \hat b_{2} \hat b_{1}^{\dagger} \hat a_{2}^{\dagger} \hat a_{1}^{\dagger})\; , \end{aligned}$$ where $a_{i}$, $a_{i}^{\dagger}$, $b_{i}^{\dagger}$ and $b_{i}$, $i = 1,2$, are creation and annihilation bosonic and fermionic operators, respectively, and $\gamma$ is the coupling constant between the atomic levels and the fields. The Kerr medium can be modeled by an anharmonic oscillator coupled to the field [@Buzek; @Werner; @Ho; @Joshi2; @Gruver2; @Imoto]. Using the adiabatic approximation ($\omega_{medium}\ll \omega_{field}$) the non-linear medium can be represented by a non-linear power of the field. Thus, the total Hamiltonian reads: $$\begin{aligned} \label{hamkerr} \hat{H} &=& {\sum^2_{i=1}} E_i \hat{b}^{\dagger}_i \hat{b}_i + {\sum_{i=1}^2} \omega_i \hat{a}^{\dagger}_i \hat{a}_i + T(t)(\gamma \hat{a}_1 \hat{a}_2 \hat{b}_1 \hat{b}^{\dagger}_2 + \gamma^{\ast} \hat{b}_2 \hat{b}^{\dagger}_1 \hat{a}^{\dagger}_1 \hat{a}^{\dagger}_2)\nonumber\\ && + \chi_1 (\hat{a}^{\dagger}_1)^2 (\hat{a}_1)^2 + \chi_2 (\hat{a}^{\dagger}_2)^2 (\hat{a}_2)^2 +2{\sqrt{\chi_1 \chi_2}}\hat{a}^{\dagger}_2 \hat{a}^{\dagger}_1 \hat{a}_1 \hat{a}_2 \end{aligned}$$ It is important to notice that three non-linear terms appear due to the presence of the non-linear Kerr media. Two of them are similar to the ones appearing in the case of one mode while the third one is a bilinear connection between the modes. This bilinear interaction is the one proposed in Refs. [@H_Kerr; @Vitali] as a way of generating cat’s states. The dynamical evolution of the mean values of the operators can be obtained by using the Ehrenfest equation, $${d \langle \hat O_{j}\rangle_{t} \over{d t}} = -\sum_{i} g_{ij} \langle \hat O_{i} \rangle\; ,$$ where $g_{ij}$ are the structure constants of a semi-Lie Algebra closed under commutation with the Hamiltonian [@JC2], i.e. $$\left[ \hat{H}\left( t \right), \hat{O}_i \right] = i \hbar \sum_i g_{ji} \left( t \right) \hat{O}_j\; .\nonumber$$ The operators defined via the previous equation are called [relevant operators]{} (RO)[@Gruver3]. The relevant operators for the two-modes Jaynes-Cummings Hamiltonian read \[RO\] $$\begin{aligned} \hat{N}^{n,m}_1 &\equiv& (\hat{a}^{\dagger}_1)^n (\hat{a}^{\dagger}_2)^m \hat{b}^{\dagger}_1 \hat{b}_1 (\hat{a}_2)^m (\hat{a}_1)^n \; \\ \hat{N}^{n,m}_2 &\equiv& (\hat{a}^{\dagger}_1)^n (\hat{a}^{\dagger}_2)^m \hat{b}^{\dagger}_2 \hat{b}_2 (\hat{a}_2)^m (\hat{a}_1)^n \; ,\\ \hat{\Delta}^{n,m}_1 &\equiv& (\hat{a}^{\dagger}_1)^n (\hat{a}^{\dagger}_2)^m \hat{a}^{\dagger}_1 \hat{a}_1 (\hat{a}_2)^m (\hat{a}_1)^n \; , \\ \hat{\Delta}^{n,m}_2 &\equiv& (\hat{a}^{\dagger}_1)^n (\hat{a}^{\dagger}_2)^m \hat{a}^{\dagger}_2 \hat{a}_2 (\hat{a}_2)^m (\hat{a}_1)^n \; ,\\ \hat{I}^{n,m} &\equiv& (\hat{a}^{\dagger}_1)^n (\hat{a}^{\dagger}_2)^m (\gamma \hat{a}_1 \hat{a}_2 \hat{b}_1 \hat{b}^{\dagger}_2 + \gamma^{\ast} \hat{a}^{\dagger}_1 \hat{a}^{\dagger}_2 \hat{b}^{\dagger}_1 \hat{b}_2) (\hat{a}_2)^m (\hat{a}_1)^n \; ,\\ \hat{F}^{n,m} &\equiv& (\hat{a}^{\dagger}_1)^n (\hat{a}^{\dagger}_2)^m i(\gamma \hat{a}_1 \hat{a}_2 \hat{b}_1 \hat{b}^{\dagger}_2 - \gamma^{\ast} \hat{a}^{\dagger}_1 \hat{a}^{\dagger}_2 \hat{b}^{\dagger}_1 \hat{b}_2) (\hat{a}_2)^m (\hat{a}_1)^n\; ,\\ \hat{N}^{n,m}_{2,1} &\equiv& (\hat{a}^{\dagger}_1)^n (\hat{a}^{\dagger}_2)^m \hat{b}^{\dagger}_2 \hat{b}_2 \hat{b}^{\dagger}_1 \hat{b}_1 (\hat{a}_2)^m (\hat{a}_1)^n \; .\end{aligned}$$ Notice that $\hat{I}^{n,m}$ and $\hat{F}^{n,m}$ are the operators that have the information about the correlation between the modes and the atom levels (i.e. the entanglement). Using the Ehrenfest theorem, the evolution equations of the relevant operators are \[Set1\] $$\begin{aligned} {d \left< \hat{N}^{n,m}_1 \right> \over dt} &=& T(t) \hat{F}^{n,m} + nT(t) \hat{F}^{n-1,m} + mT(t) \hat{F}^{n,m-1} \nonumber\\ &&+ nmT(t) \hat{F}^{n-1,m-1}\; ,\\ {d \left< \hat{N}^{n,m}_2 \right> \over dt} &=& -T(t) \hat{F}^{n,m}\; ,\\ {d \left< \hat{\Delta}^{n,m}_1 \right> \over dt} &=& (n+1)T(t) \hat{F}^{n,m} + mT(t) \hat{F}^{n,m-1} \nonumber\\ &&+ mT(t) \hat{F}^{n+1,m-1} + nmT(t) \hat{F}^{n,m-1} \; ,\\ {d \left< \hat{\Delta}^{n,m}_2 \right> \over dt} &=& (m+1)T(t) \hat{F}^{n,m} + nT(t) \hat{F}^{n-1,m} \nonumber\\ &&+ nT(t) \hat{F}^{n-1,m+1}+ nmT(t) \hat{F}^{n-1,m}\; ,\\ {d \left< \hat{N}^{n,m}_{2,1} \right> \over dt} &=& 0\; ,\\ {d \left< \hat{I}^{n,m} \right> \over dt} &=& \left[ \alpha - 2 \left(n \chi_1 + m\chi_2 +(m+n+1){\sqrt{\chi_1 \chi_2}} \right) \right] \hat{F}^{n,m} \nonumber\\ &&-2 \left( \chi_1 + {\sqrt{\chi_1 \chi_2}} \right) \hat{F}^{n+1,m} -2 \left( \chi_2 + {\sqrt{\chi_1 \chi_2}}\right) \hat{F}^{n,m+1}\; ,\\ {d \left< \hat{F}^{n,m} \right> \over dt} &=& -\left[ \alpha - 2 \left( n \chi_1 + m\chi_2 + \left(m+n+1 \right){\sqrt{\chi_1 \chi_2}} \right) \right] \hat{I}^{n,m} \nonumber\\ &&+2 \left( \chi_1 + {\sqrt{\chi_1 \chi_2}} \right) \hat{I}^{n+1,m} + 2 \left( \chi_2 +{\sqrt{\chi_1 \chi_2}} \right) \hat{I}^{n,m+1} \nonumber\\ &&+2\|\gamma\|^2 T(t)[ (n+1)(m+1) \hat{N}^{n,m}_2 - (n+1)(m+1) \hat{N}^{n,m}_{2,1} \nonumber\\ &&+ (n+1) \hat{N}^{n,m+1}_2-(n+1) \hat{N}^{n,m+1}_{2,1} +(m+1) \hat{N}^{n+1,m}_2 \nonumber\\ &&- (m+1) \hat{N}^{n+1,m}_{2,1}- \hat{N}^{n+1,m+1}_1 +\hat{N}^{n+1,m+1}_2 ]\; .\end{aligned}$$ where $\alpha = E_2 - E_1 -\omega_1 - \omega_2$. So, using Eqs. (\[Set1\]) it is possible to evaluate the temporal evolution of the RO. These operators close a Lie Algebra. Thus, any extension of the two-photon JCM built up adding terms proportional to RO will have similar evolution equations. This fact will be used in the following section in order to study the two-photon JCM with Stark shifts. Equations (\[Set1\]) can be numerically solved for any temporal dependence of the atomic-field interaction, $T(t)$. For the time-independent case, Eqs. (\[Set1\]) can be analytically solved using the series expansion in terms of commutators with the Hamiltonian [@JC2; @Gruver2; @Gruver3] $$\begin{aligned} \label{sum} \left< \hat{O} \right>_t = \left< \hat{O} \right>_0 + \sum_{n\geq 0} {1 \over {n!}} \left({t \over {i \hbar}}\right)^n \left< \left[ \dots \left[\hat{O},\hat{H}\right], \dots \dots,\hat{H} \right] \right>_0\; .\end{aligned}$$ From Eqs. (\[Set1\]) we can notice that the evolution equations for all the operators different from $\hat{F}^{n,m}$ depend on $\hat{F}^{n,m}$. So, the double commutator $\left[\hat{H},\left[\hat{H},\hat{F}^{n,m}\right] \right]$ plays a central role. This double commutator reads $$\begin{aligned} \label{dobconm} \left[\hat{H},\left[\hat{H},\hat{F}^{n,m}\right] \right] &=& \beta^2_{n,m}\hat{F}^{n,m} + \left(\beta^2_{n+1,m} - \beta^2_{n,m} \right) \hat{F}^{n+1,m}+\nonumber \\ && \left(\beta^2_{n,m+1} - \beta^2_{n,m} \right)\hat{F}^{n,m+1} + \left[ 8 \epsilon_1 \epsilon_2 + 4 \|\gamma\|^2 \right] \hat{F}^{n+1,m+1} +\nonumber \\ &&4 \epsilon^2_1 \hat{F}^{n+2,m} + 4 \epsilon^2_2 \hat{F}^{n,m+2}\; ,\end{aligned}$$ where the generalized Rabi frequency $ \beta^2_{n,m}=\left[\alpha - 2\left(n \chi_1 + m \chi_2 + (n+m+1) {\sqrt{\chi_1 \chi_2}}\right) \right]^2 + 4 \|\gamma\|^2 \left(n+1\right) \left(m+1 \right)$ and $\epsilon_i=\chi_i + {\sqrt{\chi_1 \chi_2}}$. It is important to notice that the term proportional to ${\sqrt{\chi_1 \chi_2}}$ appearing in $\beta^2_{n,m}$ and $\epsilon_i$ is due to the bilinear connection between modes. Equation (\[dobconm\]) generates [paths]{}, or different ways of dynamically connecting the RO, representing quantum correlations [@JC2; @Gruver2; @Kerr-Stark; @Gruver3]. From Eq. (\[dobconm\]) it can be seen that those terms proportional to $\epsilon_i$ are due to the presence of the Kerr medium. Notice that taking into account the bilinear connection between modes doubles the value of $\epsilon_i$ in the $\chi_1 = \chi_2$ case usually studied. Making use of Eq. (\[dobconm\]) we can obtain the temporal evolution of the population of the excited level ($\langle \hat N_{2}^{0,0} \rangle_{t}$) and the second order intermodes coherence function (defined as $g^2_{12}(t)={{\langle \hat{a}^\dagger_1 \hat{a}_1 \hat{a}^\dagger_2 \hat{a}_2 \rangle} \over {\langle \hat{a}^\dagger_1 \hat{a}_1 \rangle \langle \hat{a}^\dagger_2 \hat{a}_2 \rangle } } -1$), for different states of the field, (See appendix). This is depicted in Figs. \[fig1\]-\[fig3\]. In all cases, we assume the atom initially in the excited state ($\langle \hat N_{1}^{0,0} \rangle(0)=0$, $\langle \hat N_{2}^{0,0} \rangle(0)=1$). The phenomenological third-oder nonlinear susceptibility for the Kerr media $\chi=\chi_1=\chi_2$ takes the values $0$, $0.5$ and $1$, which, in principle, could be obtained in the ultra-high susceptibility Kerr cells recently studied [@H_Kerr]. We also assume that there are initially 10 photons in each mode ($\|\alpha_i\|^2=10)$. We analyze the influence of the non-linear medium in the temporal evolution of the excited level and we find inhibited decay as it can be seen from Figs. \[fig1\]-b,\[fig2\]-b,\[fig3\]-b. In the Pair Coherent state case (Fig. \[fig3\]-b), we find that the revivals are of such a kind that the atom recovers its initial population in each revival, and this effect is increased as we make more important the coupling with the non-linear medium. Another interesting feature is that in the squeezed vacuum state (Fig. \[fig2\]-b) we see that the revivals are regular and sharp, and become more periodic as we increase the strength of the coupling with the Kerr medium. We can see from Figs. \[fig1\]-a and \[fig3\]-a, that the two-photon JCM presents antibunching, and that this effect is attenuated by a stronger interaction with the non-linear medium. In the particular case of the squeezed vacuum, we notice that the field recovers its initial intermodes coherence when the excited level population reaches a maximum. Finally, we want to mention that $\Lambda$ configuration can be studied by using the canonical transformation $\hat{\tilde{a}}_1 = \hat{a}^\dagger_1$ and redefining the RO following Ref. [@JC2]. Stark effect ============ The Stark effect is related to changes in the atomic energy levels due to virtual transitions from levels out of resonance. The effective Hamiltonian taking into account the Stark effect reads $$\begin{aligned} \label{hamstark2m} \hat H &=& \sum_{i=1}^{2} E_{i} \hat b_{i}^{\dagger} \hat b_{i} + \sum_{i=1}^{2} \omega_{s_i} \hat a_{i}^{\dagger} \hat a_{i} + T(t)(\gamma \hat a_{1} \hat a_{2} \hat b_{1} \hat b_{2}^{\dagger} + \gamma^{} \hat b_{2} \hat b_{1}^{\dagger} \hat a_{2}^{\dagger} \hat a_{1}^{\dagger})\nonumber\\ && + \hat a_{1}^{\dagger} \hat a_{1}\left(\eta_1 \hat b_{1}^{\dagger} \hat b_{1}+ \eta_2 \hat b_{2}^{\dagger} \hat b_{2} \right)+ \hat a_{2}^{\dagger} \hat a_{2}\left(\eta_1 \hat b_{1}^{\dagger} \hat b_{1}+ \eta_2 \hat b_{2}^{\dagger} \hat b_{2} \right)\; .\end{aligned}$$ Both the Kerr Hamiltonian and the Stark Hamiltonian can be written in terms of RO (for the one-mode case see Ref. [@Kerr-Stark]. Using the same technique is is easy to see that if $$\begin{aligned} \omega_1 + \sqrt{\chi_1\chi_2} &=& \omega_{s_1} + \eta_1 \nonumber \\ \omega_2 + \sqrt{\chi_1\chi_2} &=& \omega_{s_2} + \eta_2 \nonumber \\ \eta_1 - \eta_2 &=& 2 \epsilon_1 = 2 \epsilon_2 \; ,\end{aligned}$$ the evolution equations for the relevant operators will be the same. In this case, for the RO here studied, both systems will have the same temporal evolution [@Kerr-Stark]. This will not be the case if we study the temporal evolution of other R.O. like the electromagnetic field $(\hat{a} + \hat{a}^{\dagger})$. Conclusion ========== In the present paper we analyzed the non-degenerate two-photon JCM in the presence of a Kerr media. We have identified those relevant operators that are interrelated with the dynamics of the system. We find that the dynamical evolution of the relevant operators for this problem and for the model that takes into account Stark shifts are the same for these RO and for some values of the constants. We analyze the influence of the non-linear medium in the temporal evolution of the excited level and we find inhibited decay. In the Pair Coherent state case, we observe that the revivals are of such a kind that the modes of the field are disentangled, and this effect is increased as we make more important the coupling with the non-linear medium. We also notice that in the squeezed vacuum state the revivals are regular and sharp, and that the modes of the field are entangled. We also see that this system presents antibunching, and that this effect is attenuated by a stronger interaction with the non-linear medium. Thus, this system can be used in order to obtain very different two photon entanglement situations, depending on the initial state of the field that is considered. This fact can be very useful in experimental realizations. acknowledgments {#acknowledgments .unnumbered} =============== The authors acknowledge support from Consejo Nacional de Investigaciones Científicas y Técnicas (Conicet), Argentina. Exact expressions for the time evolution of the Relevant Operators {#exact-expressions-for-the-time-evolution-of-the-relevant-operators .unnumbered} ================================================================== In the time independent case, making use of equations (\[dobconm\],\[sum\]) we can obtain an exact expression for all the Relevant Operators. For instance $\langle \hat N_{1}^{0,0} \rangle(t)$ reads: $$\begin{aligned} \label{n1dete} \langle \hat N_{1}^{0,0} \rangle(t)=\langle \hat N_{1}^{0,0} \rangle(0) &-& \sum_{j,k=0}^{\infty} \frac{S_{j,k}}{\beta_{j,k}} \sum_{n=j}^{\infty} \sum_{m=k}^{\infty} a^{n,m}_{j,k} \langle \hat F^{n,m} \rangle_{0} \nonumber\\ &-& \sum_{j,k=0}^{\infty} \frac{C_{j,k}}{\beta_{j,k}^{2}} \sum_{n=j}^{\infty} \sum_{m=k}^{\infty} a^{n,m}_{j,k} [ 2 \epsilon_1 + 2 \epsilon_2 - [ \alpha - 2(n \epsilon_1 + m \epsilon_2 \nonumber \\ &+& \sqrt{\chi_1 \chi_2}) ]] \langle \hat I^{n,m} \rangle_{0} + 2 \|\gamma\|^2 [\langle \hat N_{2}^{n+1,m+1} \rangle - \langle \hat N_{1}^{n+1,m+1} \rangle] \nonumber \\ &+& 2 \|\gamma\|^2 (n+1)(m+1)[\langle \hat N_{2}^{n,m} \rangle - \langle \hat N_{2,1}^{n,m} \rangle] \nonumber \\ &+& 2 \|\gamma\|^2 (n+1)[\langle \hat N_{2}^{n,m+1} \rangle - \langle \hat N_{2,1}^{n,m+1} \rangle]\; ,\end{aligned}$$ where $S_{j,k}=sin(\beta_{j,k}t)$, $C_{j,k} =cos(\beta_{j,k}t) - 1$, and $a^{n,m}_{j,k} = (-1)^{n+m+j+k+1}/ [(n-j)! j! (m-k)! k!]$.\ We can evaluate this expression for different initial conditions for the population of the atomic levels, and the state of the field. If we assume that the field is initially in a coherent state, $$\begin{aligned} \|\alpha_{1} \alpha_{2}>=\sum_{i=0}^{\infty}P_{n_1}P_{n_2}\|n_1 n_2>\; ,\end{aligned}$$ where $$\begin{aligned} P_{n_{i}}={{e^{\|\alpha\|^2}{\|\alpha\|^2}^{n_{i}}}\over{n_{i}!}}\; ,\end{aligned}$$ then the expression (\[n1dete\]) takes the form: $$\label{N1coherente} \langle \hat N_{1}^{0,0} \rangle_{t} = 1 + 2\|\gamma\|^{2} \sum_{j,k=0}^{\infty} \frac{1}{j! k!} \frac{C_{j,k}}{\beta_{j,k}^{2}} \langle \hat \Delta_{1}^{0,0} \rangle_{0}^{j} \langle \hat \Delta_{2}^{0,0} \rangle_{0}^{k} e^{-\langle \hat \Delta_{1}^{0,0} \rangle_{0}} e^{-\langle \hat \Delta_{2}^{0,0} \rangle_{0}}\; ,$$ where $\Delta_{i}^{0,0}$ ($i=1,2$) is the number of photons present in each mode of the field. The same treatment can be applied to the case of having the field in the Squeezed Vacuum state [@walls], which can be written in the following way, $$\begin{aligned} \label{sv} \|r, \phi>=(cosh (r))^{-1} \sum_n (e^{i\phi}tanh (r))^{n} \|n, n>\; .\end{aligned}$$ In this case the expression (\[n1dete\]) reads, $$\begin{aligned} \label{n1sqv} \langle \hat N_{1}^{0,0} \rangle_{t} = 1 + \frac{2\|\gamma\|^{2}}{cosh^2(r)} \sum_{k=0}^{\infty} (k+1)^2 tanh^2(r) \frac{C_{k,k}}{\beta^2_{k,k}} \; .\end{aligned}$$ Finally, we assume the field is in the Pair Coherent state [@agarwal], $$\|\xi,q>=N_q \sum_{n=0}^{\infty}{{\xi^n}\over{[n!(n+q)!)]^{1 \over 2}}} \|n+q,n>\;,$$ where $N_q$ is a normalization constant. In this case we have: $$\begin{aligned} \label{n1pcs} \langle \hat N_{1}^{0,0} \rangle_{t} = 1 + 2\|\gamma\|^2 N_q \sum_{k=0}^{\infty}(k+1)(k+q+1) \frac{\|\xi\|^{2k}}{k!(k+q)!} C_{k,k}\end{aligned}$$ [99]{} E.T. Jaynes and F.W. Cummings, Proc. IEEE [51]{}, 89 (1963). B.W. Shore and P.L. 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Yamamoto, Phys. Rev. A [32]{}, (1985) 2287. J.L. Gruver, J Aliaga, Hilda A. Cerdeira y A. N. Proto, Phys Rev A [50]{}, (1994) 5274 D. F. Walls, G. J. Milburn, [Quantum Optics]{}, (Springer-Verlag, Berlin, 1994). G.S. Agarwal, J. Opt. Soc. Am. B 5, 1940 (1988) ![Figure 1. $\langle \hat N_{2}^{0,0} \rangle(t)$ and $g^2_{12}(t)$ for coherent state field, $\langle \hat N_{2}^{0,0} \rangle(0)=1$, $\langle \hat N_{1}^{0,0} \rangle(0)=0$ and $\|\alpha_i\|^2=10)$.[]{data-label="fig1"}](portcoh.ps){width="18cm" height="16cm"} ![Figure 2. Same as figure 1 but field in squeezed vacuum.[]{data-label="fig2"}](portsqv.ps){width="18cm" height="16cm"} ![Figure 3. Same as figure 1 but field in pair coherent field.[]{data-label="fig3"}](portpcs.ps){width="18cm" height="16cm"}
--- abstract: 'Linear control theory is used to develop an improved localized control scheme for spatially extended chaotic systems, which is applied to a Coupled Map Lattice as an example. The optimal arrangement of the control sites is shown to depend on the symmetry properties of the system, while their minimal density depends on the strength of noise in the system. The method is shown to work in any region of parameter space and requires a significantly smaller number of controllers compared to the method proposed earlier by Qu and Hu [@gang]. A nonlinear generalization of the method for a 1-d lattice is also presented.' address: \| Condensed Matter Physics 114-36 and Neural Systems Program 139-74\ California Institute of Technology, Pasadena CA 91125 author: - 'R. O. Grigoriev, M. C. Cross and H. G. Schuster' title: Pinning Control of Spatiotemporal Chaos --- psfig Controlling chaos in high-dimensional systems [@ding] and spatiotemporal chaos especially is a very important problem with numerous applications to turbulence [@katz], instabilities in plasma [@pentek], multi-mode lasers [@colet] and reaction-diffusion systems [@crowley]. The present letter represents an effort to develop a general control algorithm for spatiotemporally chaotic systems using the methodology of linear control theory, which already proved to be fruitful [@romeiras]. Clarifying a number of issues will have direct bearing on this. For instance, it is not clear how many parameters are required for successful control. If the control is applied locally, what is the minimal density of controllers and how should they be arranged to obtain optimal performance? What are the limitations of the linear control scheme and how can they be overcome? Consider the Coupled Map Lattice (CML), originally introduced by Kaneko [@kaneko]: $$\begin{aligned} \label{eq_cml} z_i^{t+1}&=&F(z_{i-1}^t,z_i^t,z_{i+1}^t)\cr &=&f((1-2\epsilon)z_i^t+\epsilon(z_{i-1}^t+z_{i+1}^t)), \end{aligned}$$ with $i=1,2,\cdots,L$ and periodic boundary conditions, i.e. $z_{i+L}^t=z_i^t$ imposed. We also assume that the local map $f(z,a)$ is a nonlinear function with parameter $a$, such that $f(z^,a^)=z^$. To be specific, we choose $$\label{eq_loc_map} f(z)=az(1-z),$$ but emphasize, that all the major results hold independent of this choice. This CML has a homogeneous steady state $z^=1-{1 \over a}$, which is unstable for $a>3.0$ and our goal is to stabilize it using a minimal number of controllers. The first attempt in this direction was undertaken by Hu and Qu [@gang]. The authors tried to stabilize the homogeneous state by controlling an array of $M$ [periodically]{} placed pinning sites $\{i_1,\cdots,i_M\}$ with appropriately chosen control $u_m^t$ $$\label{eq_cml_ctrl} z_i^{t+1}=F(z_{i-1}^t,z_i^t,z_{i+1}^t)+\sum_{m=1}^M\delta(i-i_m)u_m^t.$$ This however required a very dense array with distance between controllers $L_p=L/M\le 3$ in the physically interesting interval of parameters $3.57<a<4.0$. The reason for this is the spatial periodicity of the pinnings. Since the system is spatially uniform, its eigenmodes are just Fourier modes and the pinning sites do not affect the modes whose nodes happen to lie at the pinnings, i.e. modes with periods equal to $2L_p$, $2L_p/2$, $2L_p/3$, etc, provided those are integer. The control scheme worked only when [all]{} such modes were [stable]{}. It is however not necessary to destroy the periodicity completely to achieve control: that would complicate the analysis unnecessarily. Instead we [add]{} one more pinning site between each of the existing ones. Not all positions are good, but some do solve the problem — previously uncontrollable modes become controllable. In order to understand how the pinnings should be placed and see whether we achieve improved performance by introducing additional controllers, we have to use a few results of the linear control theory [@dorato]. We will start with linearizing eq. (\[eq\_cml\_ctrl\]) about the homogeneous steady state ${\bf z}^t=(z^,\cdots,z^)$ in both the state vector and control to obtain the following standard equation $$\label{eq_gen_lin} {\bf x}^{t+1}=A{\bf x}^t+B{\bf u}^t,$$ where we denoted the displacement ${\bf x}={\bf z}-{\bf z}^$. If we define $\alpha=\partial f(x^,a^) / \partial x$, then the $L\times L$ Jacobian $A$ is given by $$\label{eq_matr_a} A = \alpha \left( \matrix{ 1-2\epsilon & \epsilon & 0 & \cdots &\epsilon \cr \epsilon & 1-2\epsilon & \epsilon & \cdots & 0 \cr 0 & \epsilon & 1-2\epsilon & \cdots & 0 \cr \vdots & \vdots & \vdots & \ddots & \vdots\cr \epsilon & 0 & 0 & \cdots & 1-2\epsilon \cr } \right)$$ and the $L\times M$ control matrix $B_{ij}=\delta(j-m)\delta(i-i_m)$ depends on how we place the pinning sites. If we use synchronous linear feedback ${\bf u}^t=-K{\bf x}^t$, equation (\[eq\_gen\_lin\]) becomes $${\bf x}^{t+1}=(A-BK){\bf x}^t,$$ and the solution ${\bf x}=0$ can be made stable by a suitable choice of the feedback gain matrix $K$, if the [controllability]{} condition ${\rm rank}(C)=L$ is satisfied. The controllability matrix $C$ is defined via $$\label{eq_ctblm} C=(B\ AB\ \cdots\ A^{L-1}B).$$ One can easily verify that the matrix $B$ calculated for a periodic array of pinning sites does not satisfy the controllability condition and therefore the homogeneous steady state is not controllable. It can be stabilized if the weaker [stabilizability]{} condition is satisfied, i.e. all uncontrollable modes are stable. However this imposes excessive restrictions on the pinning density. The condition for stabilizability can be obtained from the spectrum of eigenvalues of the matrix (\[eq\_matr\_a\]) $$\label{eq_spectrum} \gamma_i=\alpha(1-2\epsilon(1-\cos(k_i)),$$ where $k_1=0$, $k_i=k_{i+1}=\pi i/L$ for $i=2,4,6,\cdots$ and, for $L$-even, $K_L=\pi$ and $\alpha=2-a$. Specifically, we need $\|(a-2)(1-2\epsilon(1-\cos(\pi j/L_p))\|<1$ for all $j=1,\cdots,L-2$, such that $L_p/j$ is integer. Using this criterion one can obtain the relation between the minimum coupling, the distance between controllers and parameter $a$ of the local chaotic map for a stabilizable system. For instance, $j=1$ yields $$\label{eq_min_coup} \epsilon={a-3 \over 2(a-2)(1-\cos(\pi/L_p))}.$$ The results are presented in fig. \[fig\_coupling\]. It can be easily verified that they coincide with the numerically obtained results of Hu and Qu. It is possible however to extend the limits of the control scheme quite substantially by making the system [controllable]{} as opposed to [stabilizable]{}. This is easily achieved by choosing a different matrix $B$, i.e. placing the pinning sites differently. Doing so will enable us to control the system [anywhere]{} in the parameter space at the same time using a [smaller]{} density of controllers. First one has to determine the dimensionality of the matrix $B$, in other words determine the minimal number of parameters required to control the CML (\[eq\_cml\]) of an arbitrary length. It can be shown [@self] that the minimal number of parameters required to control a system with degenerate Jacobian is equal to the greatest multiplicity of its eigenvalues. Since the system under consideration has parity symmetry, the eigenvalues (\[eq\_spectrum\]) of its Jacobian are in fact doubly degenerate, so the minimal number of control parameters yielding a controllable system in our case is two, meaning at least two pinning sites are required. One can easily verify that the controllability condition for an $L\times 2$ matrix $$B_{ij}=\delta(j-1)\delta(i-i_1)+\delta(j-2)\delta(i-i_2)$$ is indeed satisfied for a number of arrangements $\{i_1,i_2\}$. The restrictions on the mutual arrangement of the controllers are again given by the condition of controllability: $L$ should not be a multiple of $\|i_2-i_1\|$, otherwise the mode with the period $2\|i_2-i_1\|$ becomes uncontrollable. The next step in the algorithm is to determine the feedback gain $K$. Pole placement techniques based on Ackermann’s method [@barreto] are inapplicable to the problem of controlling spatially extended systems because they are [numerically unstable]{} [@kautsky] and break down rapidly for problems of order greater than 10. Instead we use the method of the linear-quadratic (LQ) control theory [@dorato], applicable to the unstable periodic trajectories as well as fixed points. This method is not only numerically stable, but also allows one to [optimize]{} the control algorithm to increase convergence speed, and at the same time minimize the strength of control. As we will see below, decreasing control enlarges the basin of attraction, which has very important consequences for the time to achieve control (capture the chaotic trajectory). The optimal solution is obtained by minimizing the cost functional $$\label{eq_functional} V({\bf x}^0)=\sum_{n=0}^{\infty} ({{\bf x}^t}^{\dagger}Q{\bf x}^t+{{\bf u}^t}^{\dagger}R{\bf u}^t),$$ where $Q$ and $R$ are the weight matrices that can be chosen as any positive-definite square matrices. The minimum of (\[eq\_functional\]) is reached when $$\label{eq_feedback} K=(R+B^{\dagger}PB)^{-1}B^{\dagger}PA,$$ where $P$ is the solution to the discrete-time algebraic Ricatti equation $$\label{eq_ricatti} P=(Q+A^{\dagger}PA)-A^{\dagger}P^{\dagger}B(R+B^{\dagger}PB)^{-1} B^{\dagger}PA.$$ Numerical simulations show that the CML (\[eq\_cml\],\[eq\_loc\_map\]) can indeed by stabilized by this linear control scheme in a wide range of parameters $a$ and $\epsilon$. The solution for $K$ is presented in figure \[fig\_feedback\] for $a=4.0$, $\epsilon=0.33$ and $L=8$ with $Q=I_{8\times 8}$ and $R=I_{2\times 2}$. The steady homogeneous state $z^=0.75$ has 3 unstable and 5 stable directions and we use 2 pinning sites to control it. The contribution $-K_{mi}x^n_i$ from the sites $i$ far away from the pinning site $i_m$ is larger, as one would expect: since the feedback is applied indirectly through coupling to the neighbors, the perturbation introduced by the controllers decays with increasing distance to the pinning sites. Noise limits our ability to locally control arbitrarily large systems with local interactions. We will use a simple illustrative approach to see the effect of noise on the control scheme. The rank of the matrix is given by the number of its nonzero singular values. The singular values of the controllability matrix (\[eq\_ctblm\]) scale roughly as $s_l\sim \|\gamma_1\|^l$, where $\gamma_1$ is the largest eigenvalue of the Jacobian $A$ $$\|\gamma_1\|=e^{\lambda_{max}}=\cases{ \|\alpha\|, & $\epsilon<0.5$,\cr \|\alpha(4\epsilon-1)\|, & $\epsilon>0.5$.}$$ Assuming that there is an uncertainty in the calculations (due to the uncertainty in the state vector, parameter vector or just numerical roundoff errors) of relative magnitude $\sigma$, we can say that the rank of the controllability matrix can be reliably determined to be equal to the length of the lattice if $s_0/s_L>\sigma$. This gives us the theoretical bounds on the size of the controllable system in the presence of noise: $$\label{eq_rank_bnd} L_{max}^{(1)}=-{\log(\sigma)\over\lambda_{max}}.$$ On the other hand, the perturbation $\delta x_i$ introduced by the controller $i$ affects the dynamics of the remote site $j$ after propagating a distance $\Delta=\|i-j\|$ in time $\tau=\Delta$, decaying by a factor of $\epsilon$ per iteration, while the noise at site $j$ increases roughly by a factor of $\gamma_1$ per iteration. We therefore need $\delta x_i\epsilon^\Delta> \sigma\|\gamma_1\|^\tau$. Since the maximum distance $\Delta$ to the closest controller is $L/2$ and $\delta x_i\sim 1$, we get another bound, complementing (\[eq\_rank\_bnd\]) $$\label{eq_coup_bnd} L_{max}^{(2)}={2\log(\sigma)\over\log(\epsilon)-\lambda_{max}}.$$ Similar constraints were obtained by Aranson et. al for the lattices with asymmetric coupling (cf. equation (15) of the ref. [@aranson]). The maximal length of the system, that can be stabilized by the LQ method with two pinning sites placed next to each other is obtained numerically by choosing the initial condition very close to the fixed point ($\|{\bf x}^0\|\ll\|\gamma_1\|^{-L/2}$) and letting the system evolve under control (\[eq\_feedback\]) calculated for $Q=I_{L\times L}$ and $R=I_{2\times 2}$. This length is quite large even in the presence of noise (fig. \[fig\_length\]) and agrees with the theoretical bounds (\[eq\_rank\_bnd\], \[eq\_coup\_bnd\]) rather well for such a crude estimate. The problem of controlling a large 1-dimensional system with the length $L>L_{max}(\sigma)$ exceeding the maximum allowed for a given noise level can be easily reduced to the problem of controlling a number of smaller systems with the length $L_p<L_{max}(\sigma)$. We partition the entire lattice $\{z^t_1,\cdots,z^t_L\}$ into $M=L/L_p$ subdomains $\{z^t_{(m-1)L_p+1}, \cdots,z^t_{mL_p}\}$, and control it with an array of pinning sites $i_{m1}=(m-1)L_p+1$, $i_{m2}=mL_p$, $m=1,\cdots\,M$ positioned periodically at the boundaries of these subdomains. The stabilization can be achieved by choosing $$\begin{aligned} \label{eq_nonlin} &u^t_{i_{m1}}&=F(z_{i_{m2}}^t,z_{i_{m1}}^t,z_{i_{m1}+1}^t)- F(z_{i_{m1}-1}^t,z_{i_{m1}}^t,z_{i_{m1}+1}^t)\cr &+&\prod_{i=1}^{L_p}\theta(\delta x_i-\|x^t_{(m-1)L_p+i}\|) \sum_{i=1}^{L_p}K_{1i}x^t_{(m-1)L_p+i}\cr &u^t_{i_{m2}}&=F(z_{i_{m2}-1}^t,z_{i_{m2}}^t,z_{i_{m1}}^t)- F(z_{i_{m2}-1}^t,z_{i_{m2}}^t,z_{i_{m2}+1}^t)\cr &+&\prod_{i=1}^{L_p}\theta(\delta x_i-\|x^t_{(m-1)L_p+i}\|) \sum_{i=1}^{L_p}K_{2i}x^t_{(m-1)L_p+i}, \end{aligned}$$ where $\theta(x)$ is a step-function. This arrangement effectively carries two functions. We use control (\[eq\_nonlin\]) to (nonlinearly) decouple the subdomains, simultaneously imposing periodic boundary condition for each subdomain (the first two terms) to make the system controllable. Then we stabilize each subdomain asynchronously by applying a linear (in deviation $x_i^t=z_i^t-z^$) feedback (the last term), inside the neighborhood of the fixed point determined by $\delta x_i$. The linear approximation (\[eq\_gen\_lin\]) is only valid if $$\delta x_i\ll \|K_{mi}\|^{-1},\qquad m=1,\cdots,M$$ and therefore strong feedback significantly decreases the size of the capture region, which makes the capture time vary large. Minimizing the capture time can be achieved by minimizing the feedback strength using the LQ method (\[eq\_feedback\],\[eq\_ricatti\]). We demonstrate this approach by stabilizing the homogeneous stationary state of the CML defined by equation (\[eq\_cml\],\[eq\_loc\_map\]) with $a=4.0$, $\epsilon=0.33$. $L=128$ sites were divided into $M=16$ subdomains of length $L_p=8$, each controlled by two pinning sites. The results presented in fig. \[fig\_multiple\] show the evolution of the system from the initial condition chosen to be a collection of random numbers in the interval $[0,1]$. Eqs. (\[eq\_rank\_bnd\],\[eq\_coup\_bnd\]) now give the minimal density of pinning sites that yields the controllable fixed point solution. It is indeed seen to be much lower than that given by (\[eq\_min\_coup\]), e.g. $2/L_p=1/20$ ($1/17$ from the numerics, see fig. \[fig\_length\]) as opposed to $1/L_p=1/2$ for the choice $a=4.0$, $\epsilon=0.4$ and the precision of calculations given by $\sigma=10^{-14}$. Although the resulting control scheme becomes nonlinear (and therefore requires full knowledge of the evolution equations), it has the additional benefit, that the capture time is determined by the length $L_p\ll L$ and is typically many orders of magnitude smaller than that obtained for the linear control scheme (obtained by linearizing (\[eq\_nonlin\])), which only requires the Jacobian to be known. In fact our computational resources were insufficient to observe even a single capture for $L>40$ with the linearized control. Generalizing this nonlinear approach to higher-dimensional systems remains a challenge. To summarize, we have shown that the restrictions on the minimal density of periodically placed single pinning sites obtained by Qu and Hu [@gang] as a result of numerical simulations can in fact be obtained analytically from the stabilizability condition. The efficiency of the control scheme can be improved significantly if one uses double pinnings instead of single ones. The homogeneous steady state becomes controllable for any values of the control parameters and the minimal density of pinning sites is reduced substantially. It is shown that the maximal distance between the pinnings depends on the strength of noise in the system and can be estimated analytically. The appropriately chosen (using the LQ technique) feedback can decrease the capture time for the chaotic trajectory by enlarging the capture region. The introduction of nonlinearity into the control scheme can decrease this time even more significantly by effectively decoupling the large lattice into a number of smaller subdomains. The authors thank Prof. J.C. Doyle for many fruitful discussions. This work was partially supported by the NSF through grant no. DMR-9013984. H.G.S. thanks C. Koch for the kind hospitality extended to him at Caltech and the Volkswagen Foundation for financial support. G. Hu and Z. Qu, Phys. Rev. Lett. [72]{} (1994) 68. M. Ding et al., Phys. Rev. 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Sompolinsky, Phys. Rev. Lett. [68*]{} (1992) 3495.
ibvs2.sty 2000[EX Hya]{}[4U 1249-28]{}[12 52 24.40 -29 14 56.7]{} Recent satellite observations demonstrate that the phase of maximum flux of the 67 min spin modulation of the white dwarf in the cataclysmic variable EX Hya is drifting away from the optical quadratic ephemeris of Hellier & Sproats (1992, hereafter HS92). Relative to that ephemeris, the peak of the spin-phase extreme ultraviolet (EUV) flux modulation measured with the [Extreme Ultraviolet Explorer]{} ([EUVE]{}) was $\phi_{67}=0.040\pm 0.002$ in 1994 May (Mauche 1999) and $\phi_{67}=0.115\pm 0.001$ in 2000 May (Belle et al. 2002). Similarly, the peak of the spin-phase X-ray flux modulation measured with the [Chandra X-ray Observatory]{} was $\phi_{67}\approx 0.1$ in 2000 May (Hoogerwerf, Brickhouse, & Mauche 2004) and $\phi_{67}\approx 0.2$ in 2007 May (Luna, Brickhouse, & Mauche 2008). Because the discrepancy between the observed $O$ and calculated $C$ phases of the spin-phase flux modulation of EX Hya is now approaching a significant fraction of a spin cycle, we have undertaken the task of updating the ephemeris. Toward that end, we have have combined the optical data of Vogt, Krzeminski, & Sterken (1980, hereafter VKS80), Gilliland (1982), Sterken et al. (1983), Hill & Watson (1984), Jablonski & Busko (1985), Bond & Freeth (1988), HS92, Walker & Allen (2000), and Belle et al. (2005) with the optical, EUV, and X-ray data listed in Table 1. The first set of optical data in Table 1 was obtained by CS at the European Southern Observatory, La Silla, Chile using the Danish 1.5-m telescope and the DFOSC CCD camera. Differential $V$-band magnitudes were obtained by aperture photometry extracted from flat-fielded and bias-corrected CCD frames. The second set of optical data in Table 1 was obtained by Beuermann & Reinsch (2008, hereafter BR08) and is included here to clear up an ambiguity in the units of the timings in their Table 3, which are labeled as HJD, described as BJD, and treated as BJD(TT), whereas they are in fact BJD(UT); this change affects all the $O-C$ values in their table. Other than the [EXOSAT]{}, [Ginga]{}, and BR08 data, which have been taken from the given references, all other times of spin maximum in the table have been derived by us from the various datasets. In the processes, we have corrected an error in the (spin [and]{} orbit) phases of the [ASCA]{} data published by Ishida, Mukai, & Osborne (1994) and the [RXTE]{} data published by Mukai et al. (1998). We note that our result for the second [EUVE]{} observation agrees within the errors with the result derived independently by Belle et al. (2002). Table 1 lists the observed times of spin maximum in Barycentric Julian Date, the corresponding cycle number $E$ derived from the HS92 quadratic ephemeris, and the $O-C$ residuals in days relative to the VKS80 linear ephemeris, the HS92 quadratic ephemeris, and our cubic ephemeris (eqn. 1). The task of combining optical, EUV, and X-ray data into a single ephemeris presents a number of challenges. First, the published times of optical flux maximum typically do not include error estimates. Second, the times of flux maximum are typically determined in different manners in the optical and higher-energy wavebands. In the optical, the [times]{} of the flux maxima are typically estimated directly from the light curves, whereas in the EUV and X-ray wavebands, where the event rates are often fairly low, the events are typically phase-folded to produce a mean light curve, from which the [phase offset]{} relative to the assumed ephemeris is calculated from an analytic (typically, sine) fit to the mean light curve. From this, the effective time of flux maximum is derived, typically referenced to the start or mid-point of the observation. This approach is capable of producing very high signal-to-noise ratio light curves and hence error values on the fit parameters, particularly the times of flux maxima, that are formally very small. Given these complications, we have taken a multi-step approach to calculate a revised spin ephemeris for EX Hya. First, we fit the optical data to a quadratic ephemeris without weights, producing the ephemeris constants listed in the first entry of Table 2. The standard deviation of this fit is 0.00360 days or 0.077 cycles (which, if used as a uniform error on the data, produces the same fit with a reduced $\chi^2 =1$). Second, we fit the EUV and X-ray data to a quadratic ephemeris accounting for the errors listed in Table 1, producing the ephemeris constants listed in the second entry of Table 2. The two results, optical on one hand and EUV and X-ray on the other, are consistent within the errors and are as well close to (but different from) the optical quadratic ephemeris constants of HS92. Next, we fit the combined data sets, using 0.00360 days for the error on the optical data and the errors listed in Table 1 for the errors on the EUV and X-ray data, producing the ephemeris constants listed in the third entry of Table 2. The ephemeris constants are now significantly different from those of the previous fits, although it is apparent that the fit is not ideal ($\chi^2$ per degree of freedom (dof) $=651.2/431 =1.51$), in part because the ephemeris rolls over too rapidly at early times. To remedy this deficiency, we fit the combined data sets to a cubic ephemeris, producing the ephemeris constants listed in the fourth entry of Table 2. The fit is now somewhat improved ($\chi^2/{\rm dof} =638.5/430 = 1.48$), the fit parameters are closer to those of the earlier quadratic fits, the ephemeris is close to that of HS92 through 1991 January (230[,]{}000 cycles; Fig. 1[a]{}), and it reproduces well all of the available EUV and X-ray data (Fig. 1[c]{}). Finally, by setting a lower limit of 0.02 cycles or 0.00093 days on the size of the timing errors on the EUV and X-ray data, the reduced $\chi^2$ of the fit is reduced to a very reasonable $\chi^2/{\rm dof} =471.0/430 = 1.10$. Based on these results, we recommend that the following cubic ephemeris be used for recent past and future timings of the flux maxima of the spin modulation of the white dwarf in EX Hya: $$T_{\rm max} = 2437699.8917(6) + 0.046546484(9) \, E - 7.3(4) \times 10^{-13}\, E^2 + 2.2(6) \times 10^{-19}\, E^3.$$ [Acknowledgements:]{} The ESO La Silla optical data used in the work were obtained with the Danish 1.5-m telescope, which is operated by the Astronomical Observatory, Niels Bohr Institute, Copenhagen University, Denmark. We thank K. Beuermann for clearing up the ambiguity in the optical timings of BR08 and for his rapid, positive, and helpful referee’s report. This research has made use of data obtained from the High Energy Astrophysics Science Archive Research Center (HEASARC), provided by NASA’s Goddard Space Flight Center. Support for this work was provided in part by NASA through [Chandra]{} Award Number GO7-8026X issued by the [Chandra]{} X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of NASA under contract NAS8-03060. NB acknowledges support from NASA contract NAS8-03060 to the [Chandra]{} X-ray Observatory Center. This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. Belle, K.E., et al., 2005, [AJ]{}, [129]{}, 1985 Belle, K.E., Howell, S.B., Sirk, M.M., & Huber, M.E., 2002, [ApJ]{}, [577]{}, 359 Beuermann, K., & Reinsch, K., 2008, [A&A]{}, [480]{}, 199 (BR08) Bond, I.A., & Freeth, R.V., 1988, [MNRAS]{}, [232]{}, 753 Córdova, F.A., Mason, K.O., & Kahn, S.M., 1985, [MNRAS]{}, [212]{}, 447 Gilliland, R.L., 1982, [ApJ]{}, [258]{}, 576 Hellier, C., & Sproats, L.N., 1992, [IBVS]{}, No. 3724 (HS92) Hill, K.M., & Watson, R.D., 1984, [Proc. ASA]{}, [5]{}, 532 Hoogerwerf, R., Brickhouse, N.S., & Mauche, C.W., 2004, [ApJ]{}, [610]{}, 411 Ishida, M., Mukai, K., & Osborne, J.P., 1994, [PASJ]{}, [46]{}, L81 Jablonski, F., & Busko, I.C., 1985, [MNRAS]{}, [214]{}, 219 Luna, G., Brickhouse, N., & Mauche, C., 2008, [HEAD]{}, [10]{}, \#13.09 Mauche, C.W., 1999, [ApJ]{}, [520]{}, 822 Mukai, K., Ishida, M., Osborne, J., Rosen, S., & Stavroyiannopoulos, D., 1998, in Wild Stars in the Old West, ed. S. Howell, E. Kuulkers, and C. Woodward (San Francisco: ASP), p. 554 Rosen, S.R., Mason, K.O., & Córdova, F.A., 1988, [MNRAS]{}, [231]{}, 549 Rosen, S.R., Mason, K.O., Mukai, K., & Williams, O.R., 1991, [MNRAS]{}, [249]{}, 417 Sterken, C., et al., 1983, [A&A]{}, [118]{}, 325 Vogt, N., Krzeminski, W., & Sterken, C., 1980, [A&A]{}, [85]{}, 106 (VKS80) Walker, W.S.G., & Allen, W.H., 2000, [Southern Skies]{}, [39]{}, 29 Table 1. Times and cycles of spin maxima and $O-C$ residuals. -------------------------------------------------- -------- ---------- ------------ ------------ ---------- $\rm BJD(TT)-2400000$ Cycle VKS80 HS92 Eqn. 1 Ref.$^1$ $45546.4450{\phantom{0}}\pm 0.0010{\phantom{0}}$ 168575 $-0.013$ $-0.00090$ $-0.00039$ 1 $46261.3044{\phantom{0}}\pm 0.0026{\phantom{0}}$ 183933 $-0.014$ $+0.00160$ $+0.00181$ 2 $46261.3471{\phantom{0}}\pm 0.0025{\phantom{0}}$ 183934 $-0.017$ $-0.00225$ $-0.00204$ 2 $46261.3928{\phantom{0}}\pm 0.0021{\phantom{0}}$ 183935 $-0.018$ $-0.00310$ $-0.00289$ 2 $46261.4450{\phantom{0}}\pm 0.0014{\phantom{0}}$ 183936 $-0.013$ $+0.00256$ $+0.00277$ 2 $46261.4905{\phantom{0}}\pm 0.0017{\phantom{0}}$ 183937 $-0.014$ $+0.00151$ $+0.00172$ 2 $46261.5353{\phantom{0}}\pm 0.0029{\phantom{0}}$ 183938 $-0.015$ $-0.00024$ $-0.00003$ 2 $46261.5789{\phantom{0}}\pm 0.0019{\phantom{0}}$ 183939 $-0.018$ $-0.00318$ $-0.00297$ 2 $46261.6239{\phantom{0}}\pm 0.0015{\phantom{0}}$ 183940 $-0.020$ $-0.00473$ $-0.00452$ 2 $46261.6730{\phantom{0}}\pm 0.0018{\phantom{0}}$ 183941 $-0.017$ $-0.00217$ $-0.00196$ 2 $46261.7218{\phantom{0}}\pm 0.0022{\phantom{0}}$ 183942 $-0.015$ $+0.00008$ $+0.00029$ 2 $46261.7636{\phantom{0}}\pm 0.0014{\phantom{0}}$ 183943 $-0.020$ $-0.00467$ $-0.00446$ 2 $46261.8148{\phantom{0}}\pm 0.0016{\phantom{0}}$ 183944 $-0.015$ $-0.00001$ $+0.00020$ 2 $46262.3707{\phantom{0}}\pm 0.0018{\phantom{0}}$ 183956 $-0.018$ $-0.00267$ $-0.00246$ 2 $46262.4227{\phantom{0}}\pm 0.0017{\phantom{0}}$ 183957 $-0.012$ $+0.00279$ $+0.00300$ 2 $46262.4668{\phantom{0}}\pm 0.0014{\phantom{0}}$ 183958 $-0.015$ $+0.00034$ $+0.00055$ 2 $46262.5130{\phantom{0}}\pm 0.0016{\phantom{0}}$ 183959 $-0.015$ $-0.00001$ $+0.00020$ 2 $46262.5552{\phantom{0}}\pm 0.0020{\phantom{0}}$ 183960 $-0.020$ $-0.00435$ $-0.00414$ 2 $47328.79044 \pm 0.00154 $ 206867 $-0.024$ $-0.00279$ $-0.00319$ 3 $47328.88757 \pm 0.00322 $ 206869 $-0.020$ $+0.00125$ $+0.00084$ 3 $47328.98132 \pm 0.00253 $ 206871 $-0.019$ $+0.00190$ $+0.00150$ 3 $47329.02481 \pm 0.00155 $ 206872 $-0.022$ $-0.00115$ $-0.00156$ 3 $47329.16375 \pm 0.00097 $ 206875 $-0.023$ $-0.00185$ $-0.00225$ 3 $47329.30569 \pm 0.00149 $ 206878 $-0.021$ $+0.00045$ $+0.00005$ 3 $49185.47425 \pm 0.00023 $ 246756 $-0.031$ $+0.00182$ $-0.00014$ 4 $49502.17402 \pm 0.00010 $ 253560 $-0.033$ $+0.00186$ $-0.00043$ 5 $50193.99031 \pm 0.00019 $ 268423 $-0.037$ $+0.00358$ $+0.00051$ 6 $51683.27876 \pm 0.00010 $ 300419 $-0.049$ $+0.00447$ $-0.00067$ 7 $51687.51537 \pm 0.00005 $ 300510 $-0.048$ $+0.00539$ $+0.00025$ 5 $52364.8102 $ 315061 $-0.050$ $+0.00908$ $+0.00283$ 8 $52364.8608 $ 315062 $-0.046$ $+0.01314$ $+0.00688$ 8 $52366.7276 $ 315102 $-0.041$ $+0.01810$ $+0.01184$ 8 $52366.7759 $ 315103 $-0.040$ $+0.01985$ $+0.01359$ 8 $53027.7678 $ 329304 $-0.054$ $ 0.01206$ $+0.00462$ 9 $53027.8117 $ 329305 $-0.056$ $ 0.00942$ $+0.00197$ 9 $53030.8389 $ 329370 $-0.055$ $ 0.01113$ $+0.00368$ 9 $54235.21476 \pm 0.00007 $ 355245 $-0.068$ $+0.01024$ $+0.00035$ 7 $54237.96000 \pm 0.00011 $ 355304 $-0.069$ $+0.00927$ $-0.00063$ 7 $54240.38080 \pm 0.00006 $ 355356 $-0.069$ $+0.00968$ $-0.00022$ 7 $54243.03427 \pm 0.00007 $ 355413 $-0.068$ $+0.01004$ $+0.00013$ 7 $54301.68235 \pm 0.00045 $ 356673 $-0.069$ $+0.01023$ $+0.00019$ 10 -------------------------------------------------- -------- ---------- ------------ ------------ ---------- $^1$References: 1: [EXOSAT]{} (Córdova, Mason, & Kahn 1985), 2: [EXOSAT]{} (Rosen, Mason, & Córdova 1988), 3: [Ginga]{} (Rosen et al. 1991), 4: [ASCA]{} (Sequence 20020000), 5: [EUVE]{} (Program IDs 93-067 and 99-009), 6: [RXTE]{} (ObsIDs 10032-01-01 through 10032-01-12), 7: [Chandra]{} (ObsIDs 1706 and 7449–7452), 8: optical (ESO La Silla), 9: optical (Beuermann & Reinsch 2008), 10: [Suzaku]{} (ObsID 402001010). Table 2. Spin ephemeris constants: $T_{\rm max}=\sum C_nE^n$. Data Included to 0.9in[$C_0-2400000$]{} to 0.9in[$C_1$]{} to 0.9in[$C_2$]{} to 0.9in[$C_3$]{} -------------------------------------- --------------------------- ------------------- --------------------------- --------------------------- to 1.5in[Opticalto 0.5em[.]{}]{} 37699.89157 $ +0.046546478$ $ -6.25\times 10^{-13}$ to 0.9in[$\cdots$]{} $\pm 0.00054$ $\pm 0.000000007$ $\pm 0.22\times 10^{-13}$ to 1.5in[EUV & X-rayto 0.5em[.]{}]{} 37699.88930 $ +0.046546477$ $ -6.19\times 10^{-13}$ to 0.9in[$\cdots$]{} $\pm 0.00165$ $\pm 0.000000011$ $\pm 0.17\times 10^{-13}$ to 1.5in[Allto 0.5em[.]{}]{} 37699.89300 $ +0.046546454$ $ -5.85\times 10^{-13}$ to 0.9in[$\cdots$]{} $\pm 0.00041$ $\pm 0.000000003$ $\pm 0.05\times 10^{-13}$ to 1.5in[Allto 0.5em[.]{}]{} 37699.89165 $ +0.046546484$ $ -7.34\times 10^{-13}$ $ +2.16\times 10^{-19}$ $\pm 0.00056$ $\pm 0.000000009$ $\pm 0.42\times 10^{-13}$ $\pm 0.61\times 10^{-19}$
--- author: - \| \ Helmholtz-Institut Jena, Fröbelstieg 3, D-07743 Jena, Germany &\ Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1,\ D-07743 Jena, Germany\ E-mail: title: The Symmetries of the Three Heavy Quark Bound State and the Singlet Static Energy --- Introduction ============ We have studied the symmetries of the three heavy-quark system under exchange of the heavy-quark fields and their implications for the various matching coefficients, i.e. the potentials, of potential non-relativistic QCD (pNRQCD) for the three heavy-quark system. Moreover, we have calculated the ultrasoft corrections of order $\alpha_{\rm s}^4\ln\alpha_{\rm s}$ to the singlet static energy and of order $\alpha_{\rm s}^4\ln\mu$ to the singlet static potential of a three heavy-quark bound state. Whereas this has been achieved for the case of $Q\bar Q$ systems more than ten years ago [@Brambilla:1999qa], the result for three heavy-quark systems are new [@wir]. Three heavy-quark composite fields {#sec:QQQ} ================================== Quarks transform under the fundamental representation, $3$, of the (color) gauge group ${\rm SU}(3)_c$. According to $$3\otimes3\otimes3=1\oplus8\oplus8\oplus10\,, \label{prodinirreps}$$ the three heavy-quark product state, $Q_i({\bf x}_1,t)Q_j({\bf x}_2,t)Q_k({\bf x}_3,t)$ ($i,j,k=1,2,3$ denote color indices), can thus be decomposed into a singlet, two different octets and a decuplet with respect to ${\rm SU}(3)_c$ gauge transformations at a common point $\bf R$. Employing equal-time straight Wilson strings, $$\begin{aligned} \phi({\bf y},{\bf x},t)= {\cal P}\exp\left\{ig\int_0^1{\rm d}s\ ({\bf y}-{\bf x})\cdot{\bf A}({\bf x}+({\bf y}-{\bf x})s,t)\right\},\end{aligned}$$ where ${\bf A}={\bf A}^a\lambda^a/2$ is the color gauge field, $\lambda^a$ are the Gell-Mann matrices, and ${\cal P}$ denotes path ordering of the color matrices, we write $$\begin{gathered} Q_{i}({\bf x}_1,t)Q_{j}({\bf x}_2,t)Q_{k}({\bf x}_3,t) =\phi_{ii'}({\bf x}_1,{\bf R},t)\phi_{jj'}({\bf x}_2,{\bf R},t)\phi_{kk'}({\bf x}_3,{\bf R},t) \\ \times\biggr\{S({\bf x}_1,{\bf x}_2,{\bf x}_3,t){\underline {\bf S}}_{i'j'k'} +\sum_{a=1}^8O^{Aa}({\bf x}_1,{\bf x}_2,{\bf x}_3,t){\underline {\bf O}}^{Aa}_{i'j'k'}\\+\sum_{a=1}^8O^{Sa}({\bf x}_1,{\bf x}_2,{\bf x}_3,t){\underline {\bf O}}^{Sa}_{i'j'k'} +\sum_{\delta=1}^{10}\Delta^{\delta}({\bf x}_1,{\bf x}_2,{\bf x}_3,t){\underline {\pmb \Delta}}^{\delta}_{i'j'k'}\biggl\}, \label{eq:dec}\end{gathered}$$ where ${\underline {\bf S}}_{ijk}$, ${\underline {\bf O}}^{Aa}_{ijk}$, ${\underline {\bf O}}^{Sa}_{ijk}$ and ${\underline {\pmb \Delta}}^{\delta}_{ijk}$ are orthogonal and normalized color tensors that satisfy the relations $$\begin{aligned} &{\underline {\bf S}}_{ijk}{\underline {\bf S}}_{ijk}=1\,, \quad {\underline {\bf O}}^{Aa}_{ijk}{\underline {\bf O}}^{Ab}_{ijk} =\delta^{ab}\,, \quad {\underline {\bf O}}^{Sa}_{ijk}{\underline {\bf O}}^{Sb}_{ijk} =\delta^{ab}\,, \quad {\underline {\pmb \Delta}}^{\delta}_{ijk}{\underline {\pmb \Delta}}^{\delta'}_{ijk}=\delta^{\delta\delta'}\,, \nonumber\\ &{\underline {\bf S}}_{ijk}{\underline {\bf O}}^{Aa}_{ijk}={\underline {\bf S}}_{ijk}{\underline {\bf O}}^{Sa}_{ijk} ={\underline {\bf S}}_{ijk}{\underline {\pmb \Delta}}^{\delta}_{ijk} ={\underline {\bf O}}^{Aa}_{ijk}{\underline {\bf O}}^{Sb}_{ijk} ={\underline {\bf O}}^{Aa}_{ijk}{\underline {\pmb \Delta}}^{\delta}_{ijk} ={\underline {\bf O}}^{Sa}_{ijk}{\underline {\pmb \Delta}}^{\delta}_{ijk}=0\,, \label{OrthoN}\end{aligned}$$ with $a,b\in \{1, \ldots, 8\}$, and $\delta,\delta'\in \{1, \ldots, 10\}$ [@Brambilla:2005yk]. For simplicity, we have omitted an explicit reference to ${\bf R}$ in the argument of the singlet, $S$, two octets, $O^A$ and $O^S$, and decuplet, $\Delta$, fields. Besides the time coordinate $t$, we rather list the position coordinates (${\bf x}_1,{\bf x}_2,{\bf x}_3$) of the heavy-quark fields in the order (from left to right) of their appearance on the right-hand side of Eq. (\[eq:dec\]). The same convention is used for the color indices ($i,j,k$). The heavy fields fulfill equal-time anticommutation relations, $$\begin{aligned} \{Q_{i}({\bf x},t),Q_{j}({\bf y},t)\}=0. \label{vert}\end{aligned}$$ Thus, it is natural to ask for the implications of Eq. for the singlet, octet and decuplet fields in Eq. . In order to do this, we stick to a specific (matrix) representation of the color tensors, namely [@Brambilla:2005yk] $${\underline {\bf S}}_{ijk}=\frac{1}{\sqrt{6}}\epsilon_{ijk}\,, \quad {\underline {\bf O}}^{Aa}_{ijk}=\frac{1}{2}\epsilon_{ijl}\lambda^a_{kl}\,, \quad {\underline {\bf O}}^{Sa}_{ijk}=\frac{1}{2\sqrt{3}}\left(\epsilon_{jkl}\lambda^a_{il}+\epsilon_{ikl}\lambda^a_{jl}\right)\,,$$ and $$\begin{aligned} {\underline {\pmb \Delta}}^{1}_{111}&={\underline {\pmb \Delta}}^{4}_{222}={\underline {\pmb \Delta}}^{10}_{333}=1\,, \quad\quad {\underline {\pmb \Delta}}^{6}_{\{123\}}=\frac{1}{\sqrt{6}}\,, \nonumber\\ {\underline {\pmb \Delta}}^{2}_{\{112\}}&={\underline {\pmb \Delta}}^{3}_{\{122\}}={\underline {\pmb \Delta}}^{5}_{\{113\}} ={\underline {\pmb \Delta}}^{7}_{\{223\}}={\underline {\pmb \Delta}}^{8}_{\{133\}}={\underline {\pmb \Delta}}^{9}_{\{233\}} =\frac{1}{\sqrt{3}}\,, \label{Delta}\end{aligned}$$ where the symbol $\{ijk\}$ denotes all permutations of the indices $ijk$; all components not listed explicitly in Eq. (\[Delta\]) are zero. Moreover, note that as all six possible orderings of the heavy-quark fields can be obtained by applying the following two independent operations on the product state $Q_i({\bf x}_1,t)Q_j({\bf x}_2,t)Q_k({\bf x}_3,t)$, $$(i):\quad Q_i({\bf x}_1,t)\leftrightarrow Q_j({\bf x}_2,t)\,, \quad (ii):\quad Q_i({\bf x}_1,t)\leftrightarrow Q_k({\bf x}_3,t)\,,$$ it suffices to determine the transformation behavior of $S$, $O^{Aa}$, $O^{Sa}$ and $\Delta$ under $(i)$ and $(ii)$. We find $$\begin{aligned} (i):\quad \begin{cases} S({\bf x}_1,{\bf x}_2,{\bf x}_3,t) \hspace{-3mm} &= \phantom{-}S({\bf x}_2,{\bf x}_1,{\bf x}_3,t) \\ \Delta^{\delta}({\bf x}_1,{\bf x}_2,{\bf x}_3,t) \hspace{-3mm} &= -\Delta^{\delta}({\bf x}_2,{\bf x}_1,{\bf x}_3,t) \\ O^{Aa}({\bf x}_1,{\bf x}_2,{\bf x}_3,t) \hspace{-3mm} &= \phantom{-}O^{Aa}({\bf x}_2,{\bf x}_1,{\bf x}_3,t) \\ O^{Sa}({\bf x}_1,{\bf x}_2,{\bf x}_3,t) \hspace{-3mm} &= -O^{Sa}({\bf x}_2,{\bf x}_1,{\bf x}_3,t) \label{block1} \end{cases} \,,\end{aligned}$$ and $$\begin{aligned} (ii):\quad \begin{cases} S({\bf x}_1,{\bf x}_2,{\bf x}_3,t) \hspace{-3mm} &= \phantom{-}S({\bf x}_3,{\bf x}_2,{\bf x}_1,t)\\ \Delta^{\delta}({\bf x}_1,{\bf x}_2,{\bf x}_3,t) \hspace{-3mm} &= -\Delta^{\delta}({\bf x}_3,{\bf x}_2,{\bf x}_1,t)\\ O^{Aa}({\bf x}_1,{\bf x}_2,{\bf x}_3,t) \hspace{-3mm} &= -\cos(\tfrac{\pi}{3})O^{Aa}({\bf x}_3,{\bf x}_2,{\bf x}_1,t) + \sin(\tfrac{\pi}{3})O^{Sa}({\bf x}_3,{\bf x}_2,{\bf x}_1,t)\\ O^{Sa}({\bf x}_1,{\bf x}_2,{\bf x}_3,t) \hspace{-3mm} &= \phantom{-}\sin(\tfrac{\pi}{3})O^{Aa}({\bf x}_3,{\bf x}_2,{\bf x}_1,t) + \cos(\tfrac{\pi}{3})O^{Sa}({\bf x}_3,{\bf x}_2,{\bf x}_1,t) \label{block2} \end{cases} \,. \end{aligned}$$ The pNRQCD Lagrangian for the three heavy-quark system ====================================================== As pNRQCD for the three heavy-quark system is usually formulated in terms of singlet, octet and decuplet fields, the symmetry relations and have some immediate consequences for the form of the pNRQCD Lagrangian, which is organized as a double expansion in $1/m$ and in the relative coordinates ${\bf r}_i$ ($i=1,2,3$). To zeroth order in both expansions, it reads [@Brambilla:2005yk] $$\begin{aligned} {\cal L}_{\rm pNRQCD}^{(0,0)}&=&\int{\rm d}^3\!\rho\,{\rm d}^3\!\lambda\;\Bigl\{ S^{\dag}\left[i\partial_0-V^s\right]S+\Delta^{\dag}\left[iD_0-V^{d}\right]\Delta+O^{A\dag}\left[iD_0-V^o_A\right]O^A \nonumber\\ &&\hspace{1.7cm}+O^{S\dag}\left[iD_0-V^o_S\right]O^S +O^{A\dag}\left[-V_{AS}^o\right]O^S+O^{S\dag}\left[-V_{AS}^o\right]O^A\Bigr\} \nonumber\\ &&+\sum_{l}\bar{q}^{\,l}i\slashed{D}q^l-\frac{1}{4}F^a_{\mu\nu}F^{a\mu\nu}\,. \label{LpNRQCD1}\end{aligned}$$ Equation describes at zeroth order in the multipole expansion the propagation of light quarks and ultrasoft gluons as well as the temporal evolution of static quarks. Higher order terms, not displayed explicitly here, account for corrections due to finite heavy-quark masses and interactions between heavy-quarks and ultrasoft gluons. As there is no preferred ordering of the heavy-quarks, Eq. has to be invariant under different orderings of the heavy-quark fields. The potentials $V$ can be expressed in terms of the relative vectors $${\bf r}_1={\bf x}_1-{\bf x}_2\,, \qquad {\bf r}_2={\bf x}_1-{\bf x}_3\,, \qquad {\bf r}_3={\bf x}_2-{\bf x}_3\,. \label{rx123}$$ For the ordering of the heavy-quarks as in Eq. , i.e. $S\equiv S({\bf x}_1,{\bf x}_2,{\bf x}_3,t)$, $O^A\equiv O^A({\bf x}_1,{\bf x}_2,{\bf x}_3,t)$, $O^S\equiv O^S({\bf x}_1,{\bf x}_2,{\bf x}_3,t)$ and $\Delta\equiv \Delta({\bf x}_1,{\bf x}_2,{\bf x}_3,t)$, they are defined as $V\equiv V({\bf r}_1,{\bf r}_2,{\bf r}_3)$. Given the symmetry relations for the singlet, octet and decuplet fields, Eqs. and , it is straightforward to also derive corresponding symmetry relations for the potentials in Eq. . The singlet and decuplet potentials remain invariant under $(i)$ and $(ii)$, whereas the octet potentials transform as $$\begin{aligned} (i):\quad \begin{cases} V^o_A({\bf r}_1,{\bf r}_2,{\bf r}_3) \hspace{-3mm} &= \phantom{-} V^o_A(-{\bf r}_1,{\bf r}_3,{\bf r}_2) \\ V^o_S({\bf r}_1,{\bf r}_2,{\bf r}_3) \hspace{-3mm} &= \phantom{-} V^o_S(-{\bf r}_1,{\bf r}_3,{\bf r}_2) \\ V_{AS}^o({\bf r}_1,{\bf r}_2,{\bf r}_3) \hspace{-3mm} &= - V_{AS}^o(-{\bf r}_1,{\bf r}_3,{\bf r}_2) \label{Vrel1} \end{cases}\,, \end{aligned}$$ and $$\begin{aligned} (ii):\quad \begin{pmatrix} V^o_A({\bf r}_1,{\bf r}_2,{\bf r}_3) \\ V^o_S({\bf r}_1,{\bf r}_2,{\bf r}_3) \\ V_{AS}^o({\bf r}_1,{\bf r}_2,{\bf r}_3) \end{pmatrix} = \begin{pmatrix} \cos^2(\tfrac{\pi}{3}) & \sin^2(\tfrac{\pi}{3}) & -\sin(\tfrac{2\pi}{3}) \\ \sin^2(\tfrac{\pi}{3}) & \cos^2(\tfrac{\pi}{3}) & +\sin(\tfrac{2\pi}{3}) \\ \frac{1}{2}\sin(\tfrac{2\pi}{3}) & -\frac{1}{2}\sin(\tfrac{2\pi}{3}) & -\cos(\tfrac{2\pi}{3}) \end{pmatrix} \begin{pmatrix} V^o_A(-{\bf r}_3,-{\bf r}_2,-{\bf r}_1) \\ V^o_S(-{\bf r}_3,-{\bf r}_2,-{\bf r}_1) \\ V_{AS}^o(-{\bf r}_3,-{\bf r}_2,-{\bf r}_1) \end{pmatrix}. \label{Vrel2}\end{aligned}$$ Similar relations hold, e.g. for the interaction vertices in the pNRQCD Lagrangian appearing at higher order in the multipole expansion in the relative vectors ${\bf r}_i$ ($i=1,2,3$). The symmetry properties of the three heavy-quark system under exchange of the heavy-quark fields thus have a deep impact on the structure of the pNRQCD Lagrangian. They in particular induce relations between different matching coefficients in the effective theory and thereby constrain their form. The singlet static energy up to order $\alpha_s^4\ln\alpha_s$ ============================================================= Besides studying the symmetries of the three heavy-quark system, we have used the effective field theory framework of pNRQCD to determine the leading ultrasoft contribution to the singlet static energy, which is of $\alpha_s^4\ln\alpha_s$, and to the singlet static potential, which is of order $\alpha_s^4\ln\mu$. Here, the symmetry relations and have served as an important check of the obtained result. Adding the newly determined leading ultrasoft corrections to the singlet static potential $V^s$, known analytically at next-to-next-to-leading order (NNLO) [@Brambilla:2009cd], the singlet static energy is now known up to order $\alpha_{\rm s}^4\ln\alpha_{\rm s}$ and reads $$\begin{aligned} E^s({\bf r}_1,{\bf r}_2,{\bf r}_3) &=& V^s_{\rm NNLO}({\bf r}_1,{\bf r}_2,{\bf r}_3) \nonumber\\ - \frac{\alpha_{\rm s}^4}{3\pi}\ln\alpha_{\rm s} && \hspace{-6mm} \left[ \left({\bf r}_1^2+\frac{({\bf r}_2+{\bf r}_3)^2}{3}\right) \left(\frac{1}{\|{\bf r}_1\|^2} + \frac{1}{\|{\bf r}_2\|^2} + \frac{1}{\|{\bf r}_3\|^2} -\frac{1}{4}\frac{\|{\bf r}_1\| + \|{\bf r}_2\| + \|{\bf r}_3\|}{\|{\bf r}_1\|\|{\bf r}_2\|\|{\bf r}_3\|}\right) \right. \nonumber\\ && \hspace{3.4cm} \times \left(\frac{1}{\|{\bf r}_1\|} + \frac{1}{\|{\bf r}_2\|} + \frac{1}{\|{\bf r}_3\|} \right) \nonumber\\ && \hspace{-6mm} + \left({\bf r}_1^2-\frac{({\bf r}_2+{\bf r}_3)^2}{3}\right) \left(\frac{1}{\|{\bf r}_1\|^2} + \frac{1}{\|{\bf r}_2\|^2} + \frac{1}{\|{\bf r}_3\|^2} +\frac{5}{4}\frac{\|{\bf r}_1\| + \|{\bf r}_2\| + \|{\bf r}_3\|}{\|{\bf r}_1\|\|{\bf r}_2\|\|{\bf r}_3\|}\right) \nonumber\\ && \hspace{3.4cm} \times \left(\frac{1}{\|{\bf r}_1\|} - \frac{1}{2\|{\bf r}_2\|} - \frac{1}{2\|{\bf r}_3\|} \right) \nonumber\\ && \hspace{-6mm} + {\bf r}_1\cdot({\bf r}_2+{\bf r}_3) \left(\frac{1}{\|{\bf r}_1\|^2} + \frac{1}{\|{\bf r}_2\|^2} + \frac{1}{\|{\bf r}_3\|^2} +\frac{5}{4}\frac{\|{\bf r}_1\| + \|{\bf r}_2\| + \|{\bf r}_3\|}{\|{\bf r}_1\|\|{\bf r}_2\|\|{\bf r}_3\|}\right) \nonumber\\ && \hspace{3.4cm} \left. \times \left(\frac{1}{\|{\bf r}_2\|} - \frac{1}{\|{\bf r}_3\|} \right) \right] \,. \label{eq:E0full}\end{aligned}$$ In contrast to the static energy, the singlet static potential explicitly depends on the factorization scale $\mu$ separating soft from ultrasoft contributions. It is now known up to order $\alpha_s^4\ln\mu$, where the quantity $\ln\mu$ is in general referred to as an ultrasoft logarithm. In a minimal subtraction scheme, it is given by $$\begin{aligned} V^s({\bf r}_1,{\bf r}_2,{\bf r}_3;\mu) &=& V^s_{\rm NNLO}({\bf r}_1,{\bf r}_2,{\bf r}_3) \nonumber\\ - \frac{\alpha_{\rm s}^4}{3\pi}\ln\mu && \hspace{-6mm} \left[ \left({\bf r}_1^2+\frac{({\bf r}_2+{\bf r}_3)^2}{3}\right) \left(\frac{1}{\|{\bf r}_1\|^2} + \frac{1}{\|{\bf r}_2\|^2} + \frac{1}{\|{\bf r}_3\|^2} -\frac{1}{4}\frac{\|{\bf r}_1\| + \|{\bf r}_2\| + \|{\bf r}_3\|}{\|{\bf r}_1\|\|{\bf r}_2\|\|{\bf r}_3\|}\right) \right. \nonumber\\ && \hspace{3.4cm} \times \left(\frac{1}{\|{\bf r}_1\|} + \frac{1}{\|{\bf r}_2\|} + \frac{1}{\|{\bf r}_3\|} \right) \nonumber\\ && \hspace{-6mm} + \left({\bf r}_1^2-\frac{({\bf r}_2+{\bf r}_3)^2}{3}\right) \left(\frac{1}{\|{\bf r}_1\|^2} + \frac{1}{\|{\bf r}_2\|^2} + \frac{1}{\|{\bf r}_3\|^2} +\frac{5}{4}\frac{\|{\bf r}_1\| + \|{\bf r}_2\| + \|{\bf r}_3\|}{\|{\bf r}_1\|\|{\bf r}_2\|\|{\bf r}_3\|}\right) \nonumber\\ && \hspace{3.4cm} \times \left(\frac{1}{\|{\bf r}_1\|} - \frac{1}{2\|{\bf r}_2\|} - \frac{1}{2\|{\bf r}_3\|} \right) \nonumber\\ && \hspace{-6mm} + {\bf r}_1\cdot({\bf r}_2+{\bf r}_3) \left(\frac{1}{\|{\bf r}_1\|^2} + \frac{1}{\|{\bf r}_2\|^2} + \frac{1}{\|{\bf r}_3\|^2} +\frac{5}{4}\frac{\|{\bf r}_1\| + \|{\bf r}_2\| + \|{\bf r}_3\|}{\|{\bf r}_1\|\|{\bf r}_2\|\|{\bf r}_3\|}\right) \nonumber\\ && \hspace{3.4cm} \left. \times \left(\frac{1}{\|{\bf r}_2\|} - \frac{1}{\|{\bf r}_3\|} \right) \right] \,. \label{Vs3loop}\end{aligned}$$ Specializing to an equilateral geometry, characterized by the single length scale $r=\|{\bf r}_1\|=\|{\bf r}_2\|=\|{\bf r}_3\|$, we have moreover managed to resum the leading ultrasoft logarithms that start appearing in the static potential at NNNLO to all orders by solving the corresponding renormalization group equations. For the singlet static potential this results in $$V^s(r;\mu) = V^s_{\rm NNLO}(r)-9\frac{\alpha_{\rm s}^3(1/r)}{\beta_0r} \ln\frac{\alpha_{\rm s}(1/r)}{\alpha_{\rm s}(\mu)}\,, \label{vsr}$$ where $\beta_0 = 11 -2/3n_l$, with $n_l$ the number of light-quark flavors. Equation provides the complete expression of the singlet static potential at next-to-next-to-leading-logarithmic (NNLL) accuracy in an equilateral geometry. Corresponding results for the octet and decuplet potentials can be found in [@wir]. Acknowledgments {#acknowledgments .unnumbered} --------------- The author gratefully acknowledges collaboration with N. Brambilla and A. Vairo on the presented topics, as well as financial support from the FAZIT foundation and from the DFG cluster of excellence “Origin and structure of the universe” (www.universe-cluster.de). [99]{} N. Brambilla, A. Pineda, J. Soto and A. Vairo, “The Infrared behavior of the static potential in perturbative QCD,” Phys. Rev. D [60]{}, 091502 (1999) \[hep-ph/9903355\]. N. Brambilla, F. Karbstein and A. Vairo, “Symmetries of the Three Heavy-Quark System and the Color-Singlet Static Energy at NNLL,” arXiv:1301.3013 \[hep-ph\]. N. Brambilla, A. Vairo and T. Rösch, “Effective field theory Lagrangians for baryons with two and three heavy quarks,” Phys. Rev. D [72]{}, 034021 (2005) \[hep-ph/0506065\]. N. Brambilla, J. Ghiglieri and A. Vairo, “The Three-quark static potential in perturbation theory,” Phys. Rev. D [81]{}, 054031 (2010) \[arXiv:0911.3541 \[hep-ph\]\].
--- author: - \| Damien Challet\ \ Laboratoire de Mathématiques et Informatique\ pour la Complexité et les Systèmes\ CentraleSupélec, Université Paris-Saclay, France\ Encelade Capital SA, Lausanne, Switzerland\ [damien.challet@centralesupelec.fr](damien.challet@centralesupelec.fr) bibliography: - 'biblio.bib' title: 'Regrets, learning and wisdom' --- Introduction ============ A good starting point to rebuild an economic theory is to use agent-based models that include learning, interaction and networks [@kirman2; @bouchaud2008economics; @bouchaud2009unfortunate; @battiston2016complexity]. This framework is a natural meeting point for Economics and Physics (and Psychology and Biology and Computer Science and ...), which already hints that Econophysics is only part of the solution. Statistical Physics’s strength comes from its familiarity with collective phenomena. Aggregating the non-linear actions of many interacting individuals leads to remarkable global phenomena and great mathematical simplifications [@MPV; @privman2005nonequilibrium]. Whether the outcome is optimal for the agents or the system is of central importance. This contribution argues that learning and optimality may occur at various levels, may be either implicit or explicit, and that Econophysics would be wise to incorporate some more ideas from Neuroscience, Computer Science and Experimental Psychology. Let me start with a few generic remarks about the difference of approaches to data analysis and modelling in Econophysics and Economics or Finance. Interdisciplinary communication =============================== The grass in other fields seems not only greener but also disconcerting at times. This, of course, works both ways. Statistics ---------- In their 10-year old worries about Econophysics [@gallegati2006worrying], Gallegati, Keen, Lux and Ormerod pinpointed the general disregard of econophysicists for Statistics, and rightly so. The spontaneous reaction of physicists is/was to assume to have collected enough data to dispense with statistical tests and tables. The situation has much changed in this respect. Confidence intervals of estimates are not unheard of nowadays. Physicists not only use some statistical tools, but even propose new statistics [@tumminello2011identification; @malevergne2011testing; @challet2015sharperatio; @gualdi2016statistically]. Star(t)ling fits ---------------- The abundance of linear fits in Economics and Finance papers often puzzles physicists. Let me discuss what model fitting implies generically. Measuring something is equivalent to projecting a system into a sub-space. A good example is that of a picture taken by a camera: it is projection of a 3+1-dimensional world into a 2-dimensional world. In addition, the position of the camera is also of crucial importance. Figure \[fig\_starl\] shows of a flock of bird. For centuries, people have puzzled about the 3-dimensional structure of such flocks, before it was realized that these clouds were dynamical two-dimensional objects, i.e., ribbons [@cavagna2010scale]. Had anyone been able to ask a bird what the shape was like from the inside, the shape of starlings’ flocks would be have been known a long time ago. In short, placing oneself in the right space is a necessary condition for meaningful fits. ![Starlings flocking and a line. Original image source: Wikipedia.org\[fig\_starl\]](Starling_murmuration.pdf) Taking linear regressions against a few well-known factors is the same thing as taking a picture from a limited number of popular standpoints. Say that one wishes to analyse the performance of a collection of hedge funds. A questionable approach is to project their performance in the space spanned by the few Fama-French factors [@famafrench1993] and then argue that hedge funds trade on such and such factors. This is inherently incomplete and very unlikely to yield real understanding of what drives hedge funds performance. What one really needs to do is to reverse-engineer the performance of hedge funds. This requires to place one self in a space that encompasses all likely used trading strategies, or equivalently, to define factors as the returns of a variety of these strategies. Reference [@weber2013hedge] is mostly successful in replicating the returns of several thousand funds, except two: the feeder fund of Madoff, which lived in a fantasy world, and Capital Fund Management Stratus Fund because the chosen strategy space did not include the strategies of that particular fund, which resulted in an meaningless projection. Fitting any kind of model, however more sophisticated, to data is a projection. For example, calibrating a simple agent-based model [@kirman1991epidemics] to financial data based on a moment method is a double projection, i.e., a double dimensionality reduction: from the data to moments and from the model to moments [@alfarano2006estimation]. Model and data meet in a third space. This may yield an incomplete fit, which therefore may not be more efficient than other approaches. This leads us to microscopic models. Lurking Ising models ==================== Initially, econophysicists tried to apply the models they were most familiar with to financial or economic situations, which sometimes was hair-raising, even for some physicists. There is indeed no reason why financial markets should be exactly equivalent to a gas of electrons, even if some random power-law exponent seems right. This was deeply worrying. Much has changed since then. The case of the Ising model is less controversial, if only because it is equivalent to Schelling’s model [@schelling1971dynamic] and because it is easy to see why it is quite likely to appear in discrete-choice agent-based models. It also illustrates the variety of what a classical spin may describe in other contexts.[^1] The simplest idea of course is to map the two possible values of a classical spin to two opposite decisions, which seems natural for investors [@CB; @iori1999avalanche; @bornholdt2001expectation]. Another possibility is to map two alternative possibilities to the two spin values: Ref. [@bouchaudopinion] proposes a test for the presence of social imitation in the choice between two alternative possibilities. The key point is that this test is based on exact results from mean-field random-field Ising model and consists in non-linear relationships between two quantities, all of which would be impossible to guess [a priori]{}. In other words, analytically tractable aggregation provides much more than moments of intellectual satisfaction. Yet, all the above models are built as Ising models from scratch: actions are directly mapped to classical spins. Assuming instead that the binary choice is which strategy to use also leads to disordered spin models: agents with very limited possible actions in a complex world are able to optimise a global quantity, which can be written as a mean-field spin Hamiltonian [@MGbook] where the disorder comes from agents’ heterogeneity. When binary choices are involved in interacting agent models, it is hard to avoid Ising models. Learning ======== Microscopic learning -------------------- Whereas Logit learning is often found in Econophysics literature, surprisingly few other types of learning have been investigated. There lies much potential to establish bridges with other disciplines. First, computer scientists have also applied learning to finance [@cover1991universal]. More generically, in a Markovian context, Q-learning rests on the assumption that the system may be classified in a finite number of states by the agents and converges towards the optimal policy [@watkins1992q]. Computer scientists duly applied this scheme to the Minority Game, for example, defining a state as either which strategy was used, or the previous correct decision. Although this is not discussed in their papers, such dynamics seem to converge to a Nash equilibrium (e.g. [@andrecut2001q]). Methods from Statistical Physics are without any doubt able to tackle this kind of learning scheme. Another central question is what to learn. The current consensus in Neuroscience is that we learn to regret what we did not do and that a kind of Q-learning describes well how the brain works [@montague2006imaging]. In the context of financial markets, this reinforces bubbles and crashes [@lohrenz2007fictive]. Indeed, when the price of an asset has an apparent trend, investors that do own any shares of the said asset regret not to have invested earlier, which triggers their investment. Reversely, when they are invested in an asset whose price begins to fall, they regret not to have closed their positions earlier. Finally, Econophysicists have incorporated remarkably few well-known behavioural biases in their models, as they often assume that agents are risk-neutral. True, there are many reasons why one should use the exponential or logarithmic utility functions with about a ton of salt. However, it is surprising that even Prospect Theory [@kahnemanprospect] is nowhere to be seen in our agent-based models. By contrast, in their beautiful paper [@barberis2001prospect], Barberis [et al.]{} further simplify Prospect Theory with linear approximations, keeping the crucial feature that losses are about twice as painful as gains for agents, and a reference point which is a moving average of past wealth. Prospect Theory hence requires to distinguish between gains and losses in the agent payoff equation updates, which inevitably leads to additional complications. Adding a pinch of Prospect theory in our agent-based models is doable. For example, asymmetric gains and losses can be included in the Minority Game and turn out to be another cause for the emergence of large fluctuations [@MGbettin]. There is little doubt that De Dominicis generating functionals [@dedominicis] can accomodate Prospect Theory in more complex agent-based models. Systemic learning ================= The Darwinian force in financial markets that makes them adaptive systems has long been noted [@farmerforce; @zhangecology; @lo2004adaptive]. It does not imply however that the agents themselves are adaptive (e.g. that speculative funds calibrate their strategies in real-time). Indeed, at a global level (and at long time scales), the relative importance of a single agent or of a sub-population of agents may evolve because of competitive processes (for food, wealth, price predictability, etc.): fitness selection is an indirect form of global learning (see e.g. [@galla2009minority]). As a result, even economic systems with zero-intelligence agents undergo a global learning process provided that some kind of selection is performed, as reflected by replicator equations in Evolutionary Game Theory [@weibull1995evolutionary]. This does not systematically result in a measurable global optimality. There are cases, however, where a single global quantity is fairly well optimised, sometimes for obvious reasons (e.g. financial predictability). This is the Vox Populi [@galton1907vox], or Widsom of Crowds [@wisdomcrowds] effect: the aggregation of inconsistent opinions may lead to consistent aggregate estimates. Although well-known historical examples are really about estimating a single outcome such as the position of a lost craft or the weight of some object, the power of aggregation extends to much more generic situations. Ensemble learning such as Random Forests [@breiman2001random] apply this idea to classification and regression problems. The latter kind of problem implies that an ensemble of imperfect learners may correctly learn functional relationships, a much more difficult task. The implication for Economics is that the many textbook noiseless “laws”, for example between price and excess demand, may be valid at an aggregate level. In other words, the clearly too simplistic economic intuition exposed in standard textbooks are in fact quite noisy relationships,. In passing the noise may be due in part to the heterogeneity of economic entities. A most striking illustration of how to extend these “theories” comes from a work on Marseilles Fish Market [@hardle1995nonclassical]: one of its figures plots the price paid for a type of fish as a function of the quantity sold for many transactions. A cloud of point emerges. Only when local averages are taken does emerge a relationship similar to those predicted by usual economic theories. More recently, collective portfolio optimisation in the presence of a complex transaction cost structure was found in brokerage data [@Lachapelle2010]. The conditions under which collective learning may occur are still unclear and provide a nice challenge for the years to come [@lorenz2011social; @celis2016sequential] and certainly one which Physicists can contribute to. Finding more examples of wisdom of the crowds will also be part of the fun. Beyond the average behavior, the origin and role of heterogeneity in the dynamics of these systems are complementary research topics. Conclusion ========== The question is not if Economics can become a physical science, but how to make it a science. 20 years of multidisciplinary research have convinced me that it will not be enough to sprinkle economic theory with a few more mechanistic ingredients and to take a slightly less axiomatic approach. A famous speech by Jean-Claude Trichet called for help from a wide range of hard and soft sciences [@Trichet2010]. Only the combination of Biology, Experimental Psychology, Computer Science and Physics is likely to make a difference. It is hard to disagree with this point of view [@kirman2; @bouchaud2008economics; @bouchaud2009unfortunate; @sornette2014physics; @battiston2016complexity]. Using common tools and concepts will certainly help achieving better cooperation. Agent-based models, learning and networks certainly qualify as common grounds. A good example of cooperation between economists and physicists is the European CRISIS project, which lead to substantial scientific cross-fertilization. More sophisticated tools of Statistical Physics such as generating functionals are nowadays used by mathematical economists, which is great news, but, expectedly, only by the most mathematically minded, very much as in Physics. Mean-field games [@lasry2007mean] will also contribute to establish bridges between Physics and Economics (see e.g. [@swiecicki2016schrodinger]). With regards to whether less open-minded economists will accept the resulting new economic thinking, there are many reasons to be optimistic. Each economic crisis is an Economics crisis [@kirman2], and leads to more realistic models. For example, the 2008 crisis has triggered much interest in real networks and self-excited processes (e.g. [@lando2010correlation]). Let us build it and they will use it. [^1]: For a review of the Ising model in Econophysics, see [@sornette2014physics] and references therein.
--- abstract: 'I give a new integral representation for the degree five (standard) $L$-function for automorphic representations of $\GSp(4)$ that is a refinement of integral representation of Piatetski-Shapiro and Rallis. The new integral representation unfolds to produce the Bessel model for $\GSp(4)$ which is a unique model. The local unramified calculation uses an explicit formula for the Bessel model and differs completely from Piatetski-Shapiro and Rallis.' address: 'Departement of Mathematics, 14 MacLean Hall, Iowa City, Iowa 52242-1419' author: - Daniel File title: 'On the degree five $L$-function for $\GSp(4)$' --- [^1] Introduction ============ In 1978 Andrianov and Kalinin established an integral representation for the degree $2n+1$ standard $L$-function of a Siegel modular form of genus $n$ [@andrianovkalinin1978]. Their integral involves a theta function and a Siegel Eisenstein series. The integral representation allowed them to prove the meromorphic continuation of the $L$-function, and in the case when the Siegel modular form has level $1$ they established a functional equation and determined the locations of possible poles. Piatetski-Shapiro and Rallis became interested in the construction of Andrianov and Kalinin because it seems to produce Euler products without using any uniqueness property. Previous examples of integral representations used either a unique model such as the Whittaker model, or the uniqueness of the invariant bilinear form between an irreducible representation and its contragradient. It is known that an automorphic representation of $\Sp_4$ (or $\GSp_4$) associated to a Siegel modular form does not have a Whittaker model. Piatetski-Shapiro and Rallis adapted the integral representation of Andrianov and Kalinin to the setting of automorphic representations and were able to obtain Euler products [@piatetski-shapirorallis1988]; however, the factorization is not the result of a unique model that would explain the local-global structure of Andrianov and Kalinin. They considered the integral $$\int \limits_{\Sp_{2n}(F) \backslash \Sp_{2n}(\mathbb{A})} \phi(g) \theta_T(g) E(s,g) \, dg$$ where $E(s,g)$ is an Eisenstein series induced from a character of the Siegel parabolic subgroup, $\phi$ is a cuspidal automorphic form, $T$ is a $n$-by-$n$ symmetric matrix determining an $n$ dimensional orthogonal space, and $\theta_T(g)$ is the theta kernel for the dual reductive pair $\Sp_{2n} \times \O(V_T)$. Upon unfolding their integral produces the expansion of $\phi$ along the abelian unipotent radical $N$ of the Siegel parabolic subgroup. They refer to the terms in this expansion as Fourier coefficients in analogy with the Siegel modular case. The Fourier coefficients are defined as $$\phi_T(g) = \int \limits_{N(F)\backslash N(\mathbb{A})} \phi(ng) \, \psi_T(n) \, dn.$$ Here, $T$ is associated to a character $\psi_T$ of $N(F)\backslash N(\mathbb{A})$. These functions $\phi_T$ do not give a unique model for the automorphic representation to which $\phi$ belongs. The corresponding statement for a finite place $v$ of $F$ is that for a character $\psi_v$ of $N(F_v)$ the inequality $$dim_\mathbb{C} \mathrm{Hom}_{N(F_v)}(\pi_v , \psi_v) \leq 1$$ does not hold for all irreducible admissible representation $\pi_v$ of $\Sp_{2n}(F_v)$. However, Piatetski-Shapiro and Rallis show that their local integral is independent of the choice of Fourier coefficient when $v$ is a finite place and the local representation $\pi_v$ is spherical. Specifically, they show that for any $\ell_T \in \mathrm{Hom}_{N(F_v)}(\pi_v , \psi_v)$ the integral $$\int \limits_{Mat_n(\mathcal{O}_v) \cap \GL_n(F_v)} \ell_T \left( \begin{bmatrix} g & \\ & ^{t} g ^{-1} \end{bmatrix} v_0 \right) \|\det(g)\|_v^{s-1/2} \, dg= d_v(s) L(\pi_v, \frac{2s+1}{2}) \ell_T(v_0)$$ where $v_0$ is the spherical vector for $\pi_v$, $\mathcal{O}_v$ is the ring of integers, and $d_v(s)$ is a product of local $\zeta$-factors. At the remaining “bad” places the integral does not factor, and there is no local integral to compute. However, they showed that the integral over the remaining places is a meromorphic function of $s$. In this paper I present a new integral representation for the degree five $L$-function for GSp$_4$ which is a refinement of the work of Piatetski-Shapiro and Rallis. Instead of working with the full theta kernel, the construction in this paper uses a theta integral for $\GSp_4 \times \GSO_2$. This difference has the striking effect of producing the Bessel model for $\GSp_4$ and the uniqueness that Piatetski-Shapior and Rallis expected. Therefore, this integral factors as an Euler product over all places. I compute the local unramified integral when the local representation is spherical using the formula due to Sugano [@sugano1985]. In some instances an integral representation of an $L$-function can be used to prove algebraicity of special values of that $L$-function (up to certain expected transcendental factors). Harris [@harris1981], Sturm [@sturm1981], Bocherer [@bocherer1985], and Panchishkin [@panchishkin1991] applied the integral representation of Andrianov and Kalinin to prove algebraicity of special values of the standard $L$-function of certain Siegel modular forms. Shimura [@shimura2000] also used an integral representation to prove algebraicity of these special values for many Siegel modular forms including forms for every congruence subgroup of $\Sp_{2n}$ over a totally real number field. Furusawa [@furusawa1993] gave an integral representation for the degree eight $L$-function for $\GSp_4 \times \GL_2$. Furusawa’s integral representation unfolds to give the Bessel model for $\GSp_4$ times the Whittaker model for $\GL_2$, and he uses Sugano’s formula for the spherical Bessel model to compute the unramified local integral. Let $\Phi$ be a genus $2$ Siegel eigen cusp form of weight $\ell$, and let $\pi=\otimes_v \pi_v$ be the automorphic representation for $\GSp_4$ associated to it. Let $\Psi$ be an elliptic (genus 1) eigen cusp form of weight $\ell$, and let $\tau=\otimes_v \tau_v$ be the associated representation for $\GL_2$. As an application of his integral representation Furusawa proved an algebraicity result for special values of the degree eight $L$-function $L(s, \Phi \times \Psi)$ provided that for all finite places $v$ both $\pi_v$ and $\tau_v$ are spherical. This condition is satisfied when $\Phi$ and $\Psi$ are modular forms for the full modular groups $\Sp_4(\mathbb{Z})$ and $\SL_2(\mathbb{Z})$, respectively. A recent result of Saha [@saha2009] includes the explicit computation of Bessel functions of local representations that are Steinberg. This allowed Saha to extend the special value result of Furusawa to the case when $\pi_p$ is Steinberg at some prime $p$. Pitale and Schmidt [@pitaleschmidt2009] considered the local integral of Furusawa for a large class of representations $\tau_p$ and as an application extended the algebraicity result of Furusawa further. In principle one could explicitly compute the local integral given in this paper using the formula of Saha at a place where the local representation is Steinberg. Considering the algebraicity results of Harris [@harris1981], Sturm [@sturm1981], Bocherer [@bocherer1985], and Panchishkin [@panchishkin1991] that involve the integral of Andrianov and Kalinin, and the explicit computations for Bessel models due to Sugano [@sugano1985], Furusawa [@furusawa1993], and Saha [@saha2009], it would be interesting to see if the integral representation of this paper can be used to obtain any new algebraicity results. This is a question I intend to address in a later work. Summary of Results ================== Let $\pi$ be an automorphic representation of GSp$_4(\mathbb{A})$, $\phi \in V_\pi$, $\nu$ an automorphic character on GSO$_2(\mathbb{A})$ the similitude orthogonal group that preserves the symmetric form determined by the symmetric matrix $T$, $\theta_\varphi(\nu^{-1})$ the theta lift of $\nu^{-1}$ to GSp$_4$ with respect to a Schwartz-Bruhat function $\varphi$, and $E(s, f, g)$ a Siegel Eisenstein series for a section $f(s, -) \in \text{Ind}_{P(\mathbb{A})}^{G(\mathbb{A})}( \delta_P ^{1/3(s-1/2)})$. Consider the global integral $$I(s;f, \phi, T, \nu, \varphi)=I(s):=\int \limits_{Z_\mathbb{A} GSp_4(F) \backslash GSp_4(\mathbb{A})} E(s, f, g) \phi(g) \theta_\varphi(\nu^{-1})(g) \, dg.$$ Section \[global\] contains the proof that $I(s)$ has an Euler product expansion $$I(s)=\int \limits_{N(\mathbb{A}_\infty) \backslash G_1(\mathbb{A}_\infty)} f(s,g) \phi^{T, \nu}(g) \omega(g, 1) \varphi(1_2) \, dg \cdot \prod \limits_{v < \infty} I_v(s)$$ where integrals $I_v(s)$ are defined to be $$I_v(s)=\int \limits_{N(F_v) \backslash G_1(F_v)} f_v(s, g_v) \, \phi_v ^{T, \nu}(g_v) \, \omega_v(g_v, 1) \varphi_v(1_2) \, dg_v.$$ The function $\phi_v^{T,\nu}$ belongs to the Bessel model of $\pi_v$. Section \[unramifiedchapter\] includes the proof that under certain conditions that hold for all but a finite number of places $v$, there is a normalization $I_v^(s)=\zeta_v(s+1)\zeta_v(2s) \, I_v(s)$ such that $$I_v^(s)=L(s, \pi_v \otimes \chi_T)$$ where $\chi_T$ is a quadratic character associated to the matrix $T$. Section \[ramified\] deals with the finite places that are not covered in Section \[unramifiedchapter\]. For these places there is a choice of data so that $I_v(s)=1$. Section \[archimedean\] deals with the archimedean places and shows that there choice of data to control the analytic properties of $I_v(s)$. Combining these analyses give the following theorem. \[maintheorem\] Let $\pi$ be a cuspidal automorphic representation of GSp$_4(\mathbb{A})$, and $\phi \in V_\pi$. Let $T$ and $\nu$ be such that $\phi^{T,\nu}\neq0$. There exists a choice of section $f(s, -) \in \text{Ind}_{P(\mathbb{A})}^{G(\mathbb{A})}( \delta_P ^{1/3(s-1/2)})$, and some $\varphi=\otimes_v \varphi_v \in \mathcal{S}( \mathbb{X}(\mathbb{A}))$ such that the normalized integral $$I^(s;f, \phi, T, \nu, \varphi)= d(s) \cdot L^{S}(s, \pi \otimes \chi_{T,v})$$ where $S$ is a finite set of bad places including all the archimedean places. Furthermore, for any complex number $s_0$, there is a choice of data so that $d(s)$ is holomorphic at $s_0$, and $d(s_0) \neq 0$. Notation ======== Let $F$ be a number field, and let $\mathbb{A}=\mathbb{A}_F$ be its ring of adeles. For a place $v$ of $F$ denote by $F_v$ the completion of $F$ at $v$. For a non-archimedean place $v$ let $\mathcal{O}_v$ be the ring of integers of $F_v$, and let $\mathfrak{p}_v$ be its maximal ideal. Let $q_v=[ \mathcal{O}_v : \mathfrak{p}_v ]$. Let $\varpi_v$ be a choice of uniformizer for $\mathfrak{p}_v$, and let $\| \cdot \|_v$ be the absolute value on $F_v$, normalized so that $\|\varpi_v \|_v=q_v^{-1}$. For a finite set of places $S$, let $\mathbb{A}^S= {\prod \limits_{v \notin S}}^\prime F_v$, and $\mathbb{A}_S={\prod \limits_{v \in S}} F_v$. In particular, $\mathbb{A}_\infty = {\prod \limits_{v \| \infty}} F_v$, and $\mathbb{A}_{\text{fin}}={\prod \limits_{v < \infty}}^\prime F_v$. Denote by $\text{Mat}_n$ the variety of $n \times n$ matrices defined over $F$. $\text{Sym}_n$ is the variety of symmetric $n \times n$ matrices defined over $F$. Let $G= \mathrm{GSp}_4=\{g \in GL_4 \big\| \ ^tg J g= \lambda_G(g) J\}$ where $$\begin{aligned} J=\begin{bmatrix} & & 1 & \\ & & & 1\\ -1 & & &\\ & -1 & &\end{bmatrix}.\end{aligned}$$ Fix a maximal compact subgroup $K$ of $G(\mathbb{A})$ such that $K=\prod_{v} K_v$ where $K_v$ is a maximal compact subgroup of $G(F_v)$, and at all but finitely many finite places $K_v =G(\mathcal{O}_v)$. According to [@moeglinwaldspurger1995]\[I.1.4]{} the subgroups $K_v$ can be chosen so that for every standard parabolic subgroup $P$, $G(\mathbb{A})=P(\mathbb{A})K$, and $M(\mathbb{A}) \cap K$ is a maximal compact subgroup of $M(\mathbb{A})$. Orthogonal Similitude Groups {#orthog} ============================ A matrix $T \in \text{Sym}_2(F)$ with $\det(T) \neq 0$ determines a non-degenerate symmetric bilinear form $( \ , \ )_T$ on an $V_T=F^2$: $$(v_1 , v_2)_T:= {^t}v_1 T v_2.$$ The orthogonal group associated to this form (and matrix $T$) is $$O(V_T)=\{h\in GL_2 \big\| {^t}h T h= T\}.$$ Similarly, the similitude group $GO(V_T) = \{h\in GL_n \big\| {^t}h T h=\lambda_T(h) T\}$, and $GSO(V_T)$ is defined to be the Zariski connected component of $GO(V_T)$. Note that since $dim(V_T)=2$, and $h\in GSO(V_T)$, then $\lambda(h) = \det(h)$. Let $\chi_T$ be the quadratic character associated to $V_T$. If $E/F$ is the discriminant field of $V_T$, i.e. $E=F \left(\sqrt{-\det(T)}\right)$, then $$\chi_T : F^\times \backslash \mathbb{A}^\times \rightarrow \mathbb{C}$$ is the idele class character associated to $E$ by class field theory. It has the property that $\chi_T=\otimes \chi_{T,v}$ where $\chi_{T,v}(a)= ( a , - \det(T) )_v$, and $(\, , \, )_v$ denotes the local Hilbert symbol [@soudry1988]\[$\S$ 0.3]{}. Consequently, each $\chi_{T.v} \circ N_{E_v/F_v} \equiv 1$ where $N_{E_v / F_v}$ is the norm map [@serre1973]\[Chapter III, Proposition 1]{}. Note that $N_{E_v / F_v}=\det=\lambda_T$. The Siegel Parabolic Subgroup ----------------------------- Let $P=MN$ be the Siegel parabolic subgroup of $G$, i.e. $P$ stabilizes a maximal isotropic subspace $X=\mathrm{span}_F \{e_1, e_2\}$ where $e_i$ is the ith standard basis vector. Then $P$ has Levi factor $M \cong \mathrm{GL}_1 \times \mathrm{GL}_2$ and unipotent radical $N \cong \text{Sym}_2 \cong \mathbb{G}_a ^3$. For $g \in GL_2$, define $$m(g)= \begin{bmatrix} g &\\ & ^t g^{-1} \end{bmatrix} \in M.$$ For $X \in \text{Sym}_2$, define $$\begin{aligned} n(X)=\begin{bmatrix} I_2 & X \\ & I_2 \end{bmatrix} \in N.\end{aligned}$$ Let $\delta_P$ be the modular character of $P$. For $m=\begin{bmatrix} g & \\ & ^t g^{-1} \lambda \end{bmatrix} \in M$ and $n \in N$, $\delta_P(mn)= \| \det(g)^3 \cdot \lambda^{-3}\|_{\mathbb{A}} \label{adeq}$. It is possible to extend $\delta_P$ to all of $G$. For $g=nmk$ where $n \in N$, $m \in M$, and $k \in K$, define $\delta_P(g)=\delta_P(m)$. This is well defined because $\delta_P(m)=1$ for $m \in M \cap K$. Bessel Models and Coefficients {#bessel1} ============================== The Bessel Subgroup ------------------- Let $\psi$ be an additive character of $F \backslash \mathbb{A}$. There is a bijection between $\text{Sym}_2(F)$ and the characters of $N(F) \backslash N(\mathbb{A})$. For $T \in \text{Sym}_2(F)$ define $$\begin{aligned} \psi_T : N(F) \backslash N(\mathbb{A}) \rightarrow \mathbb{C} \nonumber \\ \psi_T(n(X))=\psi(tr(TX)). \label{char}\end{aligned}$$ Since $M(F)$ acts on $N(F) \backslash N(\mathbb{A})$, it also acts on its characters. Define $H_T$ to be the connected component of the stabilizer of $\psi_T$ in $M$. For $g \in GL_2$, define $$\begin{aligned} b(g)=\begin{bmatrix} g & \\ &det(g)\cdot \ {^t}g^{-1}\end{bmatrix}.\end{aligned}$$ Then $$\begin{aligned} H_T &=\left\{ b(g) \Big\| \ {^t}gTg=det(g) \cdot T \right\}.\end{aligned}$$ Then $H_T$ is an algebraic group defined over $F$ isomorphic to $GSO(V_T)$ where $V_T$ is defined as above. The adjoint action of $M(F)$ on the characters of $N(F) \backslash N(\mathbb{A})$ has two types of orbits. They are represented by matrices $$\begin{aligned} T_\rho= \begin{bmatrix} 1 & \\ & -\rho \end{bmatrix} \quad \text{with} \, \rho \notin F^{\times, 2}, \, \text{and} \quad T_{\text{split}}=\begin{bmatrix} & 1 \\1 & \end{bmatrix}.\end{aligned}$$ The quadratic spaces corresponding to these matrices have similitude orthogonal groups $$\begin{aligned} GSO(V_{T_\rho}) = \left\{ \begin{bmatrix} x & \rho y \\ y & x \end{bmatrix} \Bigg\| x^2 - \rho y^2 \neq 0 \right\}.\end{aligned}$$ and $$\begin{aligned} GSO(V_{T_{\text{split}}}) = \left\{ \begin{bmatrix} x & \\ & y \end{bmatrix} \Bigg\| xy \neq 0 \right\}. \label{gsosplit}\end{aligned}$$ For the rest of this article assume that $\rho \notin F^{\times, 2}$, and only consider $T=T_\rho$. Define the Bessel subgroup $R=R_T=H_T N$. Consider a character $$\nu : H_T(F) \backslash H_T(\mathbb{A}) \rightarrow \mathbb{C}.$$ Then define $$\begin{aligned} \nu \otimes& \psi_T : R(F) \backslash R(\mathbb{A}) \rightarrow \mathbb{C}&\\ \nu \otimes& \psi_T (tn)= \nu (t) \psi_T(n) & t \in H_T(\mathbb{A}), \ n\in N(\mathbb{A}).\end{aligned}$$ This is well defined since $H_T$ normalizes $\psi_T$. Similarly, for a place $v$ of $F$ there are local characters $$\nu_v \otimes \psi_{T,v} : R(F_v) \rightarrow \mathbb{C}.$$ Non-Archimedean Local Bessel Models ----------------------------------- Let $v$ be a finite place of $F$. Let $\mathcal{B}$ be the space of locally constant functions $\phi : G(F_v) \rightarrow \mathbb{C}$ satisfying $$\begin{aligned} \phi(rg)=\nu_v \otimes \psi_{T,v}(r) \phi(g)\end{aligned}$$ for all $r \in R(F_v)$ and all $g \in G(F_v)$. Let $\pi_v$ be an irreducible admissable representation of $G(F_v)$. Piatetski-Shapiro and Novodvorsky [@Bessel] showed that there is at most one subspace $\mathcal{B}(\pi_v) \subseteq \mathcal{B}$ such that the right regular representation of $G(F_v)$ on $\mathcal{B}(\pi_v)$ is equivalent to $\pi_v$. If the subspace $\mathcal{B}(\pi_v)$ exists, then it is called the $\nu_v \otimes \psi_{T,v}$ Bessel model of $\pi_v$. Archimedean Local Bessel Models ------------------------------- Now suppose $v$ is an infinite place of $F$. Let $K_v$ be the maximal compact subgroup of $G(F_v)$. Let $\mathcal{B}$ be the vector space of functions $\phi : G(F_v) \rightarrow \mathbb{C}$ with the following properties [@pitaleschmidt2009]: 1. $\phi$ is smooth and $K_v$-finite. 2. $\phi(rg)= \nu_v \otimes \psi_{T,v}(r) \phi(g)$ for all $r \in R(F_v)$ and all $g \in G(F_v)$. 3. $\phi$ is slowly increasing on $Z(F_v) \backslash G(F_v)$. Let $\pi_v$ be a $(\mathfrak{g_v}, K_v)$-module with space $V_{\pi_v}$. Suppose that there is a subspace $\mathcal{B}(\pi_v) \subset \mathcal{B}$, invariant under right translation by $\mathfrak{g}_v$ and $K_v$, and is isomorphic as a $(\mathfrak{g}_v, K_v)$-module to $\pi_v$, then $\mathcal{B}(\pi_v)$ is called the $\nu_v \otimes \psi_{T,v}$ Bessel model of $\pi_v$. In some instances the Bessel model at an archimedean place is known to be unique. For example, when $v$ is a real place and $\pi_v$ is a lowest or highest weight representation of $GSp_4(\mathbb{R})$ the Bessel model of $\pi_v$ is unique [@pitaleschmidt2009]. It is also known to be unique when the central character of $\pi_v$ is trivial [@Bessel]. The results of this article do not depend on the uniqueness of the Bessel model at any archimedean place; however, if the model is not unique, then there is no local integral. Bessel Coefficients ------------------- Let $\mathcal{A}_0(G)$ be the space of cuspidal automorphic forms on $G(\mathbb{A})$. Suppose that $\pi$ is an irreducible cuspidal automorphic representation of $G(\mathbb{A})$ with space $V_\pi \subset \mathcal{A}_0(G)$. Let $\omega_\pi$ denote the central character of $\pi$. Let $\phi \in V_\pi$. Suppose that $\nu$ is as above. Denote by $Z_{\mathbb{A}}$ the center of $G(\mathbb{A})$ so $Z_{\mathbb{A}} \subset H_T(\mathbb{A})$. Suppose that $\nu_{\|Z_{\mathbb{A}}}= \omega_\pi^{-1}$. Define the $\nu \otimes \psi_T $ Bessel coefficient of $\phi$ to be $$\begin{aligned} \phi^{T,\nu}(g) = \int \limits_{Z_{\mathbb{A}} R(F) \backslash R(\mathbb{A})} (\nu\otimes \psi_T)^{-1}(r) \phi(rg) dr. \label{besseldef}\end{aligned}$$ Siegel Eisenstein Series {#siegeleisenstein} ======================== For more details about Siegel Eisenstein series of $\Sp_{2n}$ see Kudla and Rallis [@kudlarallis1994] and Section 1.1 of Kudla, Rallis, and Soudry [@kudlarallissoudry1992]. \[induced\] The induced representation of $\delta_P^{\frac{1}{3}(s-\frac{1}{2})}$ to $G(\mathbb{A})$ is defined to be $$\text{Ind}_{P(\mathbb{A})}^{G(\mathbb{A})}( \delta_P ^{\frac{1}{3}(s-\frac{1}{2})})=\left\{ \begin{array}{rl} f:G(\mathbb{A}) \rightarrow \mathbb{C} \Big\| & f \ \text{is smooth, right $K$-finite, and for}\\ \Big\| & p \in P(\mathbb{A}), \, f(pg)=\delta_P^{\frac{1}{3}(s+1)}(p)f(g) \end{array} \right\}.$$ For convenience write Ind$(s)=\text{Ind}_{P(\mathbb{A})}^{G(\mathbb{A})}( \delta_P ^{\frac{1}{3}(s-\frac{1}{2})})$. Ind$(s)$ is a representation of $(\mathfrak{g}_\infty , K_\infty ) \times G(\mathbb{A}_{\text{fin}})$ under right translation. A standard section $f(s, \cdot)$ is one such that its restriction to $K$ is independent of $s$. Let $f(s, \cdot ) \in \text{Ind}(s)$ be a holomorphic standard section. That is for all $g \in G(\mathbb{A})$ the function $s \mapsto f(s, g)$ is a holomorphic function. For a finite place $v$ define $f_v^\circ(s, \cdot)$ to be the function so that $f_v^\circ(s,k)=1$ for $k \in K_v$. There is an intertwining operator $$M(s): \text{Ind}(s) \rightarrow \text{Ind}(1-s).$$ For $Re(s) >2$, $M(s)$ may be defined by means of the integral [@kudlarallis1988]\[4.1]{} $$M(s)f(s,g):= \int \limits_{N(\mathbb{A})} f(s, wng) \, dn$$ where $$w=\begin{bmatrix} & & 1 & \\ & & & 1\\ -1 & & & \\ & -1 & & \end{bmatrix}.$$ The induced representation factors as a restricted tensor product with respect to $f_v^\circ(s, \cdot)$: $$\text{Ind}(s)={\bigotimes_v}^\prime \text{Ind}_v(s),$$ and so does the intertwining operator $$M(s)=\bigotimes_v M_v(s).$$ There is a normalization of $M_v(s)$ $$M^_v(s)=\frac{\zeta_v(s+1) \, \zeta_v(2s)}{\zeta_v(s-1) \, \zeta_v(2s-1)}M_v(s)$$ where $\zeta_v(\cdot)$ is the local zeta factor for $F$ at $v$, so that $$M^_v(s)f^\circ_v(s,g)= f^\circ_v(1-s,g).$$ Define the Siegel Eisenstein series $$E(s ,f,g)= \sum \limits_{\gamma \in P(F)\backslash G(F)} f(s,\gamma g)$$ which converges uniformly for $Re(s)>2$ and has meromorphic continuation to all $\mathbb{C}$ [@kudlarallis1994]. Furthermore, the Eisenstein series satisfies the functional equation $$E(s,f,g)=E(1-s,M(s)f,g)$$ [@kudlarallis1994]\[1.5]{}. Later, it will be useful to work with the normalized Eisenstein series. Let $S$ be a finite set of places, including the archimedean places, such that for $v \notin S$ $f_v=f_v^{\circ}$. Define $$E^(s,f,g)=\zeta^S(s+1) \, \zeta^S (2s) E(s,f,g). \label{normalizing}$$ Kudla and Rallis completely determined the locations of possible poles of Siegel Eisenstein series [@kudlarallis1994]. The normalized Eisenstein series $E^(s,f,g)$ has at most simple poles at $s_0=1,2$ [@kudlarallis1994]\[Theorem 1.1]{}. The Weil Representation ======================= The Schrödinger Model {#schrodinger} --------------------- Consider the orthogonal space $V_T$ with symmetric form $( \, , )_T$, and the four dimensional symplectic space $W$ with symplectic form $<\, ,>$. Let $\mathbb{W}=V_T \otimes W$ be the symplectic space with form $\ll \, , \gg$ defined on pure tensors by $\ll u\otimes v, u' \otimes v' \gg \, =(u,u')_T \, <v,v'>$ and extended to all of $\mathbb{W}$ by linearity. The Weil representation $\omega=\omega_{\psi_T^{-1}}$ is a representation of $\widetilde{Sp}(\mathbb{W})$. However, restricting this representation to $\widetilde{Sp}(W) \times O(V_T) \hookrightarrow \widetilde{Sp}(\mathbb{W})$. Since the dimension of $V_T$ is even, there is a splitting $Sp(W) \times O(V_T) \hookrightarrow \widetilde{Sp}(W) \times O(V_T)$ [@rallis1982]\[Remark 2.1]{}. Suppose that $X$ is a maximal isotropic subspace of $W$. Then $\mathbb{X} = X \otimes_F V_T$ is a maximal isotropic subspace of $\mathbb{W}$. The space of the Schrödinger model, $\mathcal{S} (\mathbb{X})$, is the space of Schwartz-Bruhat functions on $\mathbb{X}$. Let $v$ be a place of $F$. If $v$ is a finite place, then $\mathcal{S}(\mathbb{X}(F_v))$ is the space of locally constant functions with compact support. If $v$ is an infinite place, then $\mathcal{S}(\mathbb{X}(F_v))$ is the space of $C^\infty$ functions all derivatives of which are rapidly decreasing. Identify $\mathbb{X}$ with $V_T^2=\text{Mat}_{2}$. The local Weil representation at a finite place $v$ restricted to $$Sp(W)(F_v) \times O(V_T)(F_v)$$ acts in the following way on the Schrödinger model $$\begin{aligned} \omega_v(1, h) \varphi(x) &=\varphi(h^{-1}x),\\ \omega_v(m(a), 1) \varphi(x) &= \chi_{T,v}\circ det(a) \ \|\mathrm{det}(a)\|_v \ \varphi(xa),\\ \omega_v( n(X), 1) \varphi(x) &= \psi_{ ^tx T x}^{-1}(X) \varphi(x),\\ \omega_v( w, 1) \varphi (x) & = \gamma \cdot \hat{\varphi}(x).\end{aligned}$$ where $\gamma$ is a certain eighth root of unity, and $\hat{\varphi}$ is the Fourier transform of $\varphi$ defined by $$\begin{aligned} \hat{\varphi}(x)= \int \limits_{V_T(F_v)^2} \varphi(x') \psi( (x, x')_1 ) dx'.\end{aligned}$$ Here $( \ , \ )_1$ is defined as follows: for $ x, y \in \mathbb{X}=\text{Mat}_{2}$ define $$(x , y)_1:=tr (x \cdot y).$$ Note that matrices of the form $m(a)$, $n(X)$, and $w$ generate $Sp_4$. The space $\mathcal{S}(\mathbb{X}(\mathbb{A}))$ is spanned by functions $\varphi= \otimes_v \varphi_v$ where $\varphi_v=\varphi_v^\circ$ is the normalized local spherical function for all but finitely many of the finite places $v$. At an unramified place $\varphi_v^\circ=1_{\mathbb{X}(\mathcal{O}_v)}$. The global Weil representation, $\omega={\otimes_v}^\prime \omega_v$, is the restricted tensor product with respect to the normalized spherical functions $\varphi_v^\circ$. Suppose that $F_v=\mathbb{R}$. Assume that $\psi_T= \exp(2\pi i x)$. Let $K_{1,v}=\text{Sp}_4(\mathbb{R}) \cap O_4(\mathbb{R})$. Let $V_T^+$ and $V_T^-$ be positive definite and negative definite, respectively, subspaces of $V_T(F_v)$ such that $V_T(F_v)=V_T^+ \oplus V_T^-$. For $x \in V_T$ define $$(x,x)_+= \left\{ \begin{array}{rl} (x,x) & \text{if } x \in V_T^+ \\ - (x,x) & \text{if } x \in V_T^- \\ \end{array} \right.$$ For $x \in V_T^2$ let $(x,x)=( (x_i, x_j)_{i,j})\in V_T^2$. Define $$\varphi_v^\circ(x)= \exp(-\pi \, \tr((x,x)_+)).$$ Now, suppose $F_v=\mathbb{C}$. Assume that $\psi_T=\exp(4 \pi i (x+\bar{x})$. In this case $K_{1,v}\cong \Sp(4)$, the compact real form of $\Sp(4, \mathbb{C})$. There is a choice of basis so $$(x,x)_+= {}^t \bar{x}x,$$ and $$\varphi_v^\circ(x)=\exp(-2\pi \, \tr((x,x)_+)).$$ The subspace of $K_{1,v}$ finite vectors in the space of smooth vectors, $\mathcal{S}_0(\mathbb{X}(F_v)) \subset \mathcal{S}(\mathbb{X}(F_v))$, consists of functions of the form $p(x) \varphi_v^\circ $ where $p$ is a polynomial on $V_T(F_v)^2$. Extension to Similitude Groups ------------------------------ Harris and Kudla describe how to extend the Weil representation to similitude groups [@harriskudla1992]\[$\S$3]{}. See also [@harriskudla2004] and [@roberts2001]. The Weil representation can be extended to the group $$\begin{aligned} Y=\{ (g, h) \in GSp_4 \times GSO(V_T) \ \| \ \lambda_G(g)=\lambda_{T}(h) \}.\end{aligned}$$ For $(g,h) \in Y$ the action of $\omega_v$ is defined by $$\begin{aligned} \omega_v(g,h) \varphi(x)= \|\lambda_{T}(h)\|_v^{-1} \ \omega_v( g_1 , 1) \varphi(h^{-1}x)\end{aligned}$$ where $$\begin{aligned} g_1=\begin{bmatrix} I_2 & \\ & \lambda_G(g)^{-1} \cdot I_2 \end{bmatrix}g.\end{aligned}$$ Note that the natural projection to the first coordinate $$\begin{aligned} p_1: &Y \rightarrow GSp(4)\\ &(g,h) \mapsto g\end{aligned}$$ is generally not a surjective map. Indeed, $g \in Im(p_1)$ if and only if there is an $h \in GSO(V_T)$ such that $ \lambda_G(g)=\lambda_T(h)$. Define $$G^+ := p_1 (Y).$$ Theta Lifts {#thetalifts} ----------- Let $H=GSO(V_T)$, and $H_1=SO(V_T)$. The theta lift of $\nu^{-1}$ to $G^+(\mathbb{A})$ is given by the integral \[theta\] $$\theta_\varphi(\nu^{-1})(g)= \int \limits_{H_1(F) \backslash H_1(\mathbb{A})} \sum \limits_{x \in V_T^2(F)} \omega(g, h_g h_1) \varphi (x) \nu^{-1} (h_g h_1 ) dh_1.$$ Here, $h_g \in H(\mathbb{A})$ is any element so that $\lambda_{T}(h_g)=\lambda_G(g)$. Note that Definition \[theta\] is independent of the choice $h_g$. Since $H_1(F) \backslash H_1(\mathbb{A})$ is compact the integral is termwise absolutly convergent [@weil1965]. There is a natural inclusion $$G(F)^+ \backslash G(\mathbb{A})^+ \hookrightarrow G(F) \backslash G(\mathbb{A}).$$ Consider $\theta_\varphi(\nu^{-1})$ as a function of $G(F) \backslash G(\mathbb{A})$ by extending it by $0$ [@ganichino]\[$\S$7.2]{}. If $\varphi$ is chosen to be a $K$-finite Schwartz-Bruhat function, then $\theta_\varphi(\nu^{-1})$ is a $K$-finite automorphic form on $G(F) \backslash G(\mathbb{A})$ [@harriskudla1992]. The Degree Five $L$-function {#lfunctionsec} ============================ The connected component of the dual group of $\GSp_4$ is $^L G^\circ= \text{GSp}_4(\mathbb{C})$ [@borel1979]\[I.2.2 (5)]{}. The degree five $L$-function of $\GSp_4$ corresponds to the map of $L$-groups [@soudry1988]\[page 88]{} $$\varrho: \GSp_4(\mathbb{C}) \rightarrow \PGSp_4(\mathbb{C}) \cong \SO_5(\mathbb{C}).$$ I describe the local $L$-factor explicitly when $v$ is finite and $\pi_v$ is equivalent to an unramified principal series. Consider the maximal torus $A_0$ of $G$ and an element $t \in A_0$: $$t=\text{diag}(a_1, a_2, a_0 a_1^{-1}, a_0 a_2^{-1}):=\begin{bmatrix} a_1 & & & \\ & a_2 & & \\ & & a_0 a_1^{-1} & \\ & & & a_0 a_2^{-2} \end{bmatrix}. \label{toruselement}$$ The character lattice of $G$ is $$X=\mathbb{Z}e_0 \oplus \mathbb{Z}e_1 \oplus \mathbb{Z}e_2$$ where $e_i(t)=a_i$. The cocharacter lattice is $$X^{\vee}= \mathbb{Z}f_0 \oplus \mathbb{Z}f_1 \oplus \mathbb{Z}f_2$$ where $$\begin{aligned} &f_0(u)=\text{diag} (1,1,u,u), & f_1(u)=\text{diag}( u, 1, u^{-1}, 1),\\ & f_2(u)=\text{diag}(1,u,1,u^{-1}).\end{aligned}$$ Suppose $$\pi_v \cong \pi_v(\chi)=Ind_{B(F_v)}^{G(F_v)}(\chi)$$ where $$\chi(t)=\chi_1(a_1) \chi_2(a_2) \chi_0(a_0), \label{chidefine}$$ and $t$ is given by . Then $^L G^\circ=\hat{G}$ has character lattice $X^\prime=X^\vee$ and cocharacter lattice $X^{\prime \vee}=X$. Let $f_i^\prime=e_i \in X^{\prime \vee}$. Define $$\hat{t}=\prod_{i=0}^{3} f_i^\prime(\chi_i(\varpi_v)) \in {^L G ^\circ}. \label{satakeparameter}$$ Then $\hat{t}$ is the Satake parameter for $\pi_v(\chi)$ [@asgarischmidt2001]\[Lemma 2]{}. The Langlands $L$-factor is defined in [@borel1979]\[II.7.2 (1)]{} to be [rCl]{} L(s, \_v, ):&=&( I - () q\_v\^[-s]{}) \^[-1]{}\ &=&(1-q\_v\^[-s]{})\^[-1]{}(1- \_1(\_v) q\_v\^[-s]{})\^[-1]{} (1- \_1(\_v)\^[-1]{}q\_v\^[-s]{})\^[-1]{}\ &&(1- \_2(\_v) q\_v\^[-s]{})\^[-1]{} (1- \_2()\^[-1]{} q\_v\^[-s]{})\^[-1]{}. Let $S$ be a finite set of primes, including the archimedean primes, such that if $v \notin S$, then $\pi_v$ is . Then the partial $L$-function is defined to be $$L^S(s, \pi)=L^S(s, \pi, \varrho)=\prod_{v \notin S} L(s, \pi_v, \varrho).$$ The product converges absolutely for $Re(s) \gg 0$ [@langlands1971]. Global Integral Representation {#global} ============================== The main result of this section is Theorem \[eulerproduct\] which states that the integral unfolds as an Euler product of local integrals. As before $G=$GSp$_4$, $G_1=$Sp$_4$, $P=MN$ is the Siegel parabolic subgoup of $G$, and let $P_1=M_1 N=P \cap G_1$ where $M_1 = M \cap G_1$. The global integral is $$\begin{aligned} I(s; f, \phi, \nu)= I(s):&= \int \limits_{Z_\mathbb{A} G(F) \backslash G(\mathbb{A})} E(s,f, g) \phi(g) \theta_\varphi(\nu^{-1})(g) \, dg\\ &=\int \limits_{Z_\mathbb{A} G(F)^+ \backslash G(\mathbb{A})^+} E(s, f, g) \phi(g) \theta_\varphi(\nu^{-1})(g) \, dg \label{10}\end{aligned}$$ where equality holds because $\theta_\varphi(\nu^{-1})$ is supported on $G(F)^+ \backslash G(\mathbb{A})^+$. The central character of $E(s, f, - )$ is trivial, and the central character of $\theta_\varphi (\nu^{-1})=\omega_\pi ^{-1}$, so the integrand is $Z_\mathbb{A}$ invariant. Since $E(s, f, -)$ and $\theta_\varphi(\nu^{-1})$ are automorphic forms, they are of moderate growth. Since $\phi$ is a cuspidal automorphic form, it is rapidly decreasing on a Siegel domain [@moeglinwaldspurger1995]\[I.2.18]{}. Therefore, the integral converges everywhere that $E(s, f, -)$ does not have a pole. Define $$\begin{aligned} \mathbb{A}^{\times, +}:=\lambda_T(H(\mathbb{A})), & & F^{\times, +} := F^\times \cap \mathbb{A}^{\times, +} \subseteq \mathbb{A}^{\times,+},\\ \mathbb{A}^{\times, 2}:=\{a^2 \| \, a \in \mathbb{A}^\times \}, & & \mathcal{C}:=\mathbb{A}^{\times, 2} F^{\times, +} \backslash \mathbb{A}^{\times, +}.\end{aligned}$$ There is an isomorphism $$\begin{aligned} Z_\mathbb{A} G_1(\mathbb{A}) G(F)^+ \backslash G(\mathbb{A})^+ \cong \mathcal{C}. \label{quotient1}\end{aligned}$$ The isomorphism is realized by considering the map from $G(\mathbb{A})^+ \longrightarrow \mathcal{C}$, $g \mapsto \lambda_G(g)$. It has kernel $Z_\mathbb{A} G_1(\mathbb{A}) G(F)^+$. This fact is stated in [@ganichino]. Identify $Z_\mathbb{A}$ with the subgroup of scalar linear transformations in $H(\mathbb{A})$. \[quotient2\] $$Z_\mathbb{A} H_1(\mathbb{A}) H(F) \backslash H(\mathbb{A}) \cong \mathcal{C}. \label{Hiso}$$ Consider the map $H(\mathbb{A}) \rightarrow \mathcal{C}$, $h \mapsto \lambda_T(h)$. This map is onto by definition of $\mathbb{A}^{\times, +}$. I must show that the kernel is $Z_\mathbb{A} H_1(\mathbb{A}) H(F)$. Suppose $\lambda_T(h)=a^2 \mu$ where $a \in \mathbb{A}^{\times, 2}$ and $\mu \in F^{\times, +}$. By Hasse’s norm theorem [@hasse1967] there is an element $h_\mu \in H(F)$ such that $\lambda_T(h)=\mu$. Let $z(a)$ be the scalar matrix with eigenvalue $a$. Since $\lambda_T( z(a)^{-1} h h_\mu^{-1})=1$, $h_1= z(a)^{-1} h h_\mu^{-1} \in H_1(\mathbb{A})$. Therefore, $h= z(a) h_1 h_\mu$. This shows that $Z_\mathbb{A} H_1(\mathbb{A}) H(F)$ contains the kernel of this map. The opposite inclusion is obvious. This proves the proposition. Fix sections $$\begin{aligned} &\mathcal{C} \rightarrow G(\mathbb{A})^+ & & \mathcal{C} \rightarrow H(\mathbb{A}) \nonumber \\ &c \mapsto g_c & & c \mapsto h_c\end{aligned}$$ There is a measure $dc$ on $\mathcal{C}$ and measures $dh_1$ and $dg_1$ on $H_1(F) \backslash H_1(\mathbb{A})$ and $G_1(F) \backslash G_1(\mathbb{A})$, respectively, such that $$\int \limits_{Z_\mathbb{A} H(F) \backslash H(\mathbb{A}) } \, f(h) \, dh= \int \limits_{\mathcal{C} } \int \limits_{H_1(F) \backslash H_1(\mathbb{A})} \, f(h_1 h_c) \, dh_1 \, dc,$$ and $$\int \limits_{Z_\mathbb{A} G(F)^+ \backslash G(\mathbb{A})^+ } \, f(g) \, dg= \int \limits_{\mathcal{C}} \int \limits_{G_1(F) \backslash G_1(\mathbb{A})} \, f(g_1 g_c) \, dg_1 \, dc.$$ Let $dh$ denote the right invariant measure on $Z_\mathbb{A} H(F) \backslash H(\mathbb{A})$. Then by [@prasadtakloobighash2011]\[Lemma 13.2]{} there are measures $dh_1$ and $dh_c$ so that for all $f \in L^{1}(Z_\mathbb{A} H(F) \backslash H(\mathbb{A}))$ $$\int \limits_{Z_\mathbb{A} H(F) \backslash H(\mathbb{A}) }f(h) \, dh= \int \limits_{Z_\mathbb{A} H_1(\mathbb{A}) H(F) \backslash H(\mathbb{A}) } \int \limits_{H_1(F) \backslash H_1(\mathbb{A})} f(h_1 h_c) \, dh_1 \, dh_c. \label{hintegral}$$ Through the isomorphism define a measure $dc := dh_c$ on $\mathcal{C}$. By define a measure $dg_c:=dc$ on $Z_\mathbb{A} G_1(\mathbb{A}) G(F)^+ \backslash G(\mathbb{A})^+$. Then there is a choice of measures $dg$ and $dg_1$ so that for $f \in L^{1}(Z_\mathbb{A} G(F)^+ \backslash G(\mathbb{A})^+)$ $$\int \limits_{Z_\mathbb{A} G(F)^+ \backslash G(\mathbb{A})^+ }f(g) \, dg= \int \limits_{Z_\mathbb{A} G_1(\mathbb{A}) G(F)^+ \backslash G(\mathbb{A})^+ } \int \limits_{G_1(F) \backslash G_1(\mathbb{A})} f(g_1 g_c) \, dg_1 \, dg_c. \label{gintegral}$$ Then equals $$\int \limits_{\mathcal{C}} \int \limits_{G_1(F) \backslash G_1(\mathbb{A})} E(s, f, g_1 g_c) \phi(g_1 g_c) \theta_\varphi(\nu^{-1})(g_1 g_c) \, dg_1 \, dc. \label{9}$$ Denote the theta kernel by $$\theta_\varphi(g_1g_c,h_1h_c)= \sum \limits_{x \in V_T^2(F)} \omega(g_1 g_c, h_1 h_c ) \varphi(x).$$ Then $$\theta_\varphi(\nu^{-1})(g_1 g_c) = \int \limits_{H_1(F) \backslash H_1(\mathbb{A})}\theta_\varphi(g_1 g_c, h_1 h_c) \nu^{-1}(h_1 h_c) dh_1.$$ As noted in section \[thetalifts\] this integral converges absolutely. The following adjoint identity holds for the global theta integral [rCl]{} \_[G\_1(F) \\G\_1()]{} \_[H\_1(F) \\H\_1()]{} E(s, f, g\_1 g\_c) (g\_1 g\_c) \_(g\_1 g\_c, h\_1 h\_c) \^[-1]{}(h\_1 h\_c) dh\_1 dg\_1\ = \_[H\_1(F) \\H\_1()]{} \_[G\_1(F) \\G\_1()]{} E(s, f, g\_1 g\_c) (g\_1 g\_c) \_(g\_1 g\_c, h\_1 h\_c) \^[-1]{}(h\_1 h\_c) dg\_1 dh\_1. \[adjoint\] Since $ P_1(F) \backslash G_1(F) \cong P(F) \backslash G(F)$, then $$E(s, f,g) = \sum \limits_{\gamma \in P(F) \backslash G(F) } f(s, \gamma g) = \sum \limits_{\gamma \in P_1(F) \backslash G_1(F)} f(s, \gamma g).$$ Then the inner integral of becomes $$\int \limits_{P_1(F) \backslash G_1(\mathbb{A})} f(s, g_1 g_c) \, \phi(g_1 g_c) \, \theta_\varphi(g_1 g_c, h_1 h_c) \, \nu^{-1}(h_1 h_c) \, dg_1.$$ Expanding the theta kernel gives $$\begin{aligned} \int \limits_{G_1(F) \backslash G_1(\mathbb{A})} f(s, g_1 g_c) \, \phi(g_1 g_c) \, \sum \limits_{x \in V_T^2(F)} \omega(g_1 g_c, h_1 h_c) \varphi (x) \nu^{-1} (h_1 h_c) \, dg_1 \label{1}\end{aligned}$$ The Levi factor of $P_1$ is $M_1 \cong \GL_2$. The Weil representation restricted to this subgroup acts on $\mathcal{S}(\mathbb{A})$ by $$\omega(m(y), 1) \, \varphi_1(x)= \|\det(y)\|_\mathbb{A} \varphi_1(xy)$$ for $\varphi_1 \in \mathcal{S}(\mathbb{A})$, $y \in \GL_2(\mathbb{A})$, and $x \in Mat_2(\mathbb{A})$. Consider $x \in Mat_2(F)$. If $\det(x)=0$, then $Stab_{\GL_2(\mathbb{A})}(x)$ contains a normal unipotent subgroup. By the cuspidality of $\phi$ this term vanishes upon integration. Therefore [rCl]{} & & \_[ P\_1(F) \\G\_1()]{} f(s, g\_1 g\_c) (g\_1 g\_c) \_[x Mat\_2(F)]{} (m(x) g\_1 g\_c, h\_1 h\_c) (1\_2) \^[-1]{} (h\_1 h\_c ) dg\_1\ &= & \_[ P\_1(F) \\G\_1()]{} f(s, g\_1 g\_c) (g\_1 g\_c) \_[x \_2(F)]{} (m(x) g\_1 g\_c, h\_1 h\_c) (1\_2) \^[-1]{} (h\_1 h\_c ) dg\_1\ &= & \_[ N(F) \\G\_1()]{} f(s, g\_1 g\_c) (g\_1 g\_c) (g\_1 g\_c, h\_1 h\_c) (1\_2) \^[-1]{} (h\_1 h\_c ) dg\_1. \[a1\] Note that the integral $$\int \limits_{ N(F) \backslash G_1(\mathbb{A})} f(s, g_1 g_c) \, \phi(g_1 g_c) \, \omega(g_1 g_c, h_1 h_c) \varphi (1_2) \, \nu^{-1} (h_1 h_c ) \, dg_1$$ is $H_1(F)$ invariant; however, the integrand is not. Then $$\begin{aligned} & \int \limits_{ N(F) \backslash G_1(\mathbb{A})} f(s, g_1 g_c) \, \phi(g_1 g_c) \omega(g_1 g_c, h_1 h_c) \varphi (1_2) \, dg_1\\ =& \int \limits_{ N(\mathbb{A}) \backslash G_1(\mathbb{A})} \int \limits_{N(F) \backslash N(\mathbb{A})} f(s,g_1 g_c) \phi(ng_1 g_c) \omega(n g_1 g_c, h_1 h_c) \, \varphi (1_2) \, dn \, dg_1.\end{aligned}$$ Define $$\phi^T(g) :=\int \limits_{N(F) \backslash N(\mathbb{A}) } \phi(ng) \, \psi_T^{-1}(n) \, dn.$$ Then $$\int \limits_{N(F) \backslash N(\mathbb{A})} \phi(ng)\omega(n g, h_g h) \, \varphi (1_2) \, dn \\ = \phi^{T}(g) \, \omega(g, h_g h) \varphi (1_2).$$ This follows since for $n \in N(\mathbb{A})$ $$\omega(n g_1 g_c, h_1 h_c) \varphi (1_2)= \psi_T^{-1}(n) \, \omega(g_1 g_c, h_1 h_c) \varphi (1_2),$$ so the integral becomes $$\begin{aligned} I(s) = & \int \limits_\mathcal{C} \int \limits_{H_1(F) \backslash H_1(\mathbb{A})} \int \limits_{ N(\mathbb{A}) \backslash G_1(\mathbb{A})} f(s,g_1 g_c) \phi^{T}(g_1 g_c) \nonumber\\ & \times \omega(g_1 g_c, h_1 h_c) \varphi (1_2) \, \nu^{-1} (h_1 h_c ) \, dh_1 dg_1 dg_c, \label{2}\end{aligned}$$ Computing in the Weil representation $$\begin{aligned} \omega(g_1 g_c , h_1 h_c) \varphi(1_2) =& \|\lambda_G(g_c)\|^{-1}_\mathbb{A} \, \omega \left( \ell \left(\lambda_G(g)^{-1}\right) g_1 g_c, 1 \right) \varphi \left( (h_1 h_c)^{-1}\right)\\ =& \chi_V \circ \det(h_1 h_c) \, \|\lambda_G(g_c)\|^{-1}_\mathbb{A} \, \|\det (h_1 h_c)^{-1}\|^{-1}_\mathbb{A} \nonumber \\ &\times \omega \left( m(h_1 h_c )^{-1} \ell\left(\lambda_G(g_c)\right)g_1 g_c, 1 \right) \varphi(1_2). \label{41}\end{aligned}$$ For $h \in H_T$, $\det(h) \in N_{E/F}(E^\times)$. Therefore, $\chi_V \circ \det(h)=1$. Combining this with the fact that $$\|\lambda_G(g_c)\|_\mathbb{A} =\|\det\left(h_1 h_c\right)\|_\mathbb{A}$$ (see Section \[orthog\]) and applying it to $\eqref{41}$ gives $$\begin{aligned} \omega(g_1 g_c , h_1 h_c) \varphi(1_2)&= \omega \left( m(h_1 h_c)^{-1} \ell \left( \lambda_G(g_c)^{-1} \right) g_1 g_c, 1 \right) \varphi(1_2) \nonumber\\ &= \omega \left( b(h_1 h_c)^{-1} g_1 g_c, 1 \right) \varphi(1_2). \label{3}\end{aligned}$$ Since $\lambda_G( b(h_1 h_c))=\lambda_G(g_1 g_c)$, the map $g_1 \mapsto b(h_1 h_c) g_1 g_c^{-1}$ sends $G_1$ to itself. Let $d\bar{g}$ be the right invariant measure on $N(\mathbb{A}) \backslash G_1(\mathbb{A})$, and $g \in P(\mathbb{A}) \subseteq G(\mathbb{A})$. Then $d (\overline{ghg^{-1}})=\|\delta_P(g)^{-1}\|_\mathbb{A} \cdot d \bar{g}$. Suppose $dn$ is Haar measure on $N(\mathbb{A})$ and $d\bar{g}$ is the right invariant measure on $N(\mathbb{A}) \backslash G_1(\mathbb{A})$ normalized so that for $f \in L^1( G_1(\mathbb{A}))$ $$\int \limits_{G_1(\mathbb{A})} f(g) \, dg = \int \limits_{N(\mathbb{A}) \backslash G_1(\mathbb{A})} \int \limits_{N(\mathbb{A})} f(n \bar{g}) \, dn \, d\bar{g}$$ Let $g \in P(\mathbb{A})$. The transformation $h \mapsto ghg^{-1}$ preserves Haar measure on $G_1(\mathbb{A})$. Let $E$ be a measurable subset of $G_1(\mathbb{A})$ such that with finite volume with respect to $dg_1$, and let $vol(E)$ denote this volume. Then the volume of $N(\mathbb{A}) \backslash N(\mathbb{A}) E$ is given by the formula $$vol(N(\mathbb{A}) \backslash N(\mathbb{A}) E) = \dfrac{vol(E)}{\int \limits_{N(\mathbb{A}) \cap E} dn}.$$ Since $d(gng^{-1})=\|\delta_P(g)\|_\mathbb{A} \cdot dn$, then $d(\overline{ghg^{-1}})=\|\delta_P(g)\|_\mathbb{A}^{-1} \cdot d\bar{g}$. Since $\delta_P(b(h_1 h_c))=1$, the map $g_1 \mapsto b(h_1 h_c) g_1 g_c^{-1}$ preserves the right invariant measure on $N(\mathbb{A}) \backslash G_1(\mathbb{A})$. Substituting $\eqref{3}$ into $\eqref{2}$, and making the above change of variables gives [rCl]{} I(s) &= & \_ \_[H\_1(F) \\H\_1()]{} \_[ N() \\G\_1()]{} f(s,b(h\_1 h\_c) g\_1) \^[T]{}(b(h\_1 h\_c) g\_1)\ & & (g\_1, 1 ) (1\_2) \^[-1]{} (h\_1 h\_c ) dg\_1 dh\_1 dg\_c. The $Z_{\mathbb{A}} H_1(\mathbb{A}) H(F) \backslash H(\mathbb{A})$ integral and the $H_1(F) \backslash H_1(\mathbb{A})$ fold together ($H$ is abelian so $h_c h_1 = h_1 h_c$) to produce $$\begin{aligned} \int \limits_{Z_{\mathbb{A}} H(F) \backslash H(\mathbb{A})} \int \limits_{ N(\mathbb{A}) \backslash G_1(\mathbb{A})} f(s,b(h) g_1) \, \phi^{T}(b(h) g_1) \, \omega(g_1, 1 ) \varphi (1_2) \, \nu^{-1} (h_1 h_c ) \, dg_1 dh. \label{42}\end{aligned}$$ Since $b(h) \in P(\mathbb{A})$, and $\delta_P\left( b(h) \right)=1$, $f(s,b(h) g_1)=f(s, g_1)$. Therefore, changing the order of integration in $\eqref{42}$ and applying Proposition \[quotient2\] produces $$\begin{aligned} &\int \limits_{ N(\mathbb{A}) \backslash G_1(\mathbb{A})} f(s, g_1) \, \omega(g_1, 1 ) \varphi (1_2) \int \limits_{Z_{\mathbb{A}} H_T(F) \backslash H_T(\mathbb{A})} \phi^{T}(h g_1) \, \nu^{-1} (h) \, dh \, dg_1 \label{4} \\ = &\int \limits_{N(\mathbb{A}) \backslash G_1(\mathbb{A})} f(s,g_1) \phi^{T, \nu}(g_1) \, \omega(g_1, 1) \varphi(1_2) \, dg_1. \label{blah3}\end{aligned}$$ The next section shows that converges absolutely for $Re(s) >2$, justifying the change in order of integration. \[eulerproduct\] Let $\phi^{T,\nu}=\otimes_v \phi^{T,\nu}_v$, $f(s, \cdot)=\otimes_v f_v(s, \cdot)$, and $\varphi=\otimes_v \varphi_v$. Then for $Re(s)>2$ $$\begin{aligned} & \int \limits_{Z_\mathbb{A} G(F) \backslash G(\mathbb{A})} E(s, f,g) \phi(g) \theta(\nu^{-1}, \varphi)(g) \, dg\\ =& \int \limits_{N(\mathbb{A}) \backslash G_1(\mathbb{A})} f(s,g) \, \phi^{T, \nu}(g) \, \omega(g, 1) \varphi(1_2) \, dg\\ =& \int \limits_{N(\mathbb{A}_\infty) \backslash G_1(\mathbb{A}_\infty)} f(s,g_\infty) \, \phi^{T, \nu}(g_\infty) \, \omega(g_\infty, 1) \varphi(1_2) \, dg_\infty \cdot \prod \limits_{v < \infty} I_v(s) \label{blah1111}\end{aligned}$$ where $$\begin{aligned} I_v(s)=\int \limits_{N(F_v) \backslash G_1(F_v)} f_v(s, g_v) \, \phi_v ^{T, \nu}(g_v) \, \omega_v(g_v, 1) \varphi_v(1_2) \, dg_v. \label{blah2222}\end{aligned}$$ The uniqueness of the Bessel model is used to obtain the factorization in . When the local archimedean Bessel models are unique, the integral factors at these places as in . Absolute Convergence of the Unfolded Integral {#abs} ============================================= The integral $$\int \limits_{N(\mathbb{A}) \backslash G_1(\mathbb{A})} \int \limits_{Z_{\mathbb{A}} H_T(F) \backslash H_T(\mathbb{A}) } f(s,g) \, \phi^{T}(h g) \, \nu^{-1}(h) \, \omega(g, 1) \varphi(1_2) \, dh \, dg \label{absconv}$$ converges absolutely for $Re(s)>2$. This argument follows [@moriyama2004] to show that $\phi$ is bounded on $G_1(\mathbb{A})$. By [@moeglinwaldspurger1995]\[Corollary I.2.12, I.2.18]{}, $\phi$ is rapidly decreasing. To be precise, suppose $\mathfrak{S}$ is a Siegel domain for $G(\mathbb{A})$. Let $G(\mathbb{A})^1 := \cap_{\chi} \ker \|\chi\|_\mathbb{A}$ where $\chi$ range over rational characters of $G$. Then $$G(\mathbb{A})^1=\{ g \in G(\mathbb{A}) \Big\| \|\lambda_G(g)\|_\mathbb{A}=1 \}.$$ Therefore, $G_1(\mathbb{A}) \subset G(\mathbb{A})^1$. \[rapiddecrease\] A function $\phi : \mathfrak{S} \rightarrow \mathbb{C}$ is rapidly decreasing if there exists an $r>0$ such that for all real positive valued characters $\lambda$ of the standard maximal torus $A_0$, there exists $C_0>0$ such that for all $z \in Z_\mathbb{A}$ and $g \in G(\mathbb{A})^1 \cap \mathfrak{S}$ the follwing inequality holds $$\|\phi(zg)\| \leq C_0 \|\|z\|\|^r \lambda( a(g)) \label{siegel}$$ where $\|\| \cdot \|\|$ is the height function on $G(\mathbb{A})$, and $a(g)$ is defined so that if $g=nak$, then $a(g)=a$ where $n \in N_0$, the unipotent radical of the Borel, $a \in A_0$, and $k \in K$. By choosing $z=1$, and $\lambda$ to be the adelic norm of the similitude character in , then the right hand side of the inequality equals $C_0$ for $g \in \mathfrak{S} \cap G_1(\mathbb{A})$. Therefore, $\phi$ is bounded on $\mathfrak{S} \cap G_1(\mathbb{A})$. However, $\phi$ is $G(F)$ invariant, so $\phi$ is bounded on $$G(F)(\mathfrak{S} \cap G_1(\mathbb{A}))\supseteq G_1(\mathbb{A}).$$ The quotients $Z_{\mathbb{A}} H_T(F) \backslash H_T(\mathbb{A})$ and $N(F) \backslash N(\mathbb{A})$ are compact. Therefore, $$\|\nu(r)\|=\|\psi_T(n)\|=1$$ for all $r \in H_T(\mathbb{A}) \cap G_1(\mathbb{A})$, and all $n \in N(\mathbb{A})$. Assume that all representatives $r \in Z_{\mathbb{A}} H_T(F) \backslash H_T(\mathbb{A})$ are chosen so that $r \in G_1(\mathbb{A})$. Then $r g_1 \in G_1(\mathbb{A})$ and $$\|\phi(r g_1)\| \, \|\nu^{-1}(r)\| < C_0. \label{ineq}$$ Furthermore, since $\nu_{\|Z_\mathbb{A}}$ agrees with the central character of $\phi$, holds for all $r \in H_T(\mathbb{A})$. Then $$\begin{aligned} \int \limits_{Z_{\mathbb{A}} H_T(F) \backslash H_T(\mathbb{A}) } \|\phi^T( h g)\| \, \|\nu^{-1}(h)\| \, dh & =\int \limits_{Z_\mathbb{A} R(F) \backslash R(\mathbb{A})} \| (\nu \otimes \psi_T)^{-1}(r)\| \, \| \phi(r g_1) \| \, dr \nonumber \\ &\leq \text{vol}\left(Z_\mathbb{A} R(F) \backslash R(\mathbb{A})\right) \cdot C_0.\end{aligned}$$ Therefore, $$\begin{aligned} \int \limits_{N(\mathbb{A}) \backslash G_1(\mathbb{A})} \int \limits_{Z_{\mathbb{A}} H_T(F) \backslash H_T(\mathbb{A}) } \|f(s,g)\| \, \|\phi^{T}(h g)\| \, \|\nu^{-1}(h)\| \, \|\omega(g, 1) \varphi(1_2)\| \, dh \, dg \nonumber \\ \leq C \int \limits_{N(\mathbb{A}) \backslash G_1(\mathbb{A})} \|f(s,g)\| \, \|\omega(g, 1) \varphi(1_2)\| \, dg.\end{aligned}$$ The Schwartz-Bruhat function $\varphi$ is $K$-finite, as is $f(s, -)$, so there is some open subgroup $K_0 \leq K$ such that $[K : K_0]=n < \infty$, and $\varphi$ and $f(s, -)$ are $K_0$-invariant. Let $\{ k_i \}_{1 \leq i \leq n}$ be a set of irredundant coset representatives for $K / K_0$. We have $$G_1(\mathbb{A})=P_1(\mathbb{A}) K$$ Suppose that $p=m(a) n \in P_1(\mathbb{A})$, and $k \in k_i K_0$. Define $\varphi_i:=\omega(k_i,1)\varphi$. Then we have $$\begin{aligned} \omega(pk, 1)\varphi(1_2)&=\omega(p, 1) \omega(k, 1) \varphi(1_2)\\ &=\omega(p) \varphi_i(1_2)\\ &=\psi_T(n) \, \chi_V \circ \det(a) \, \|\det(a)\|_{\mathbb{A}} \, \varphi_i(a).\end{aligned}$$ Therefore, $$\begin{aligned} &\int \limits_{N(\mathbb{A}) \backslash G_1(\mathbb{A})} \|f(s,g)\| \, \|\phi^{T, \nu}(g)\| \, \|\omega(g, 1) \varphi(1_2)\| \, dg\\ \leq & \int \limits_{N(\mathbb{A}) \backslash P_1(\mathbb{A})} \int \limits_{K} \|\delta_P(p)^{-1}\| \, \|\delta_P(p)^{s/3+1/3}\|\|f(s,k)\| \, \|\omega(pk, 1)\varphi(1_2)\| \, dp \, dk\\ \leq & \text{vol}(K_0) \times \, \sum \limits_{i=1}^{n} \|f(s,k_i)\| \int \limits_{GL_2(\mathbb{A})} \|\varphi_i(a)\| \, \|\det(a)\|^{s-1}_{\mathbb{A}} \, da.\end{aligned}$$ Absolute convergence of $\eqref{absconv}$ depends only on the convergence of $$\int \limits_{GL_2(\mathbb{A})} \|\varphi_i(a)\| \, \|\det(a)\|^{s-1}_{\mathbb{A}} \, da.$$ The Schwartz-Bruhat function $\varphi_i=\otimes \varphi_{i,v}$ is rapidly decreasing, i.e. $\varphi_{i,v}$ is compactly supported at each finite places $v$, and $\varphi_i$ are rapidly decreasing when $v$ is archimedean. Let $Q$ be the Borel subgroup of $GL_2$, and let $L=\prod \limits_v L_v$, where $L_v$ is the maximal compact subgroup of $GL_2(F_v)$ so that GL$_2=QL$. There is a compact finite index open subgroup $L_i \leq L$ such that $\varphi_i$ is $L_i$ invariant. Let $\varphi_{ij}$, $j=1, \ldots, m$, be the $L$ translates of $\varphi_i$. Then $$\begin{aligned} & \quad \int \limits_{GL_2(\mathbb{A})} \|\varphi_i(a)\| \|\det(a)\|^{s-1}_{\mathbb{A}} \, da\\ &= \text{vol}(L_i) \times \sum \limits_{j=1}^{m} \, \int \limits_{Q(\mathbb{A})} \|\varphi_{ij}(b)\| \, \|\det(b)\|^{s-1}_{\mathbb{A}} \, db.\end{aligned}$$ Then $$\begin{aligned} &\int \limits_{Q(\mathbb{A})} \|\varphi_{ij}(b)\| \|\det(b)\|^{s-1}_{\mathbb{A}} db\\ =&\int \limits_{\mathbb{A}^{\times}} \int \limits_{\mathbb{A}^\times} \int \limits_{\mathbb{A}} \, \left\|\varphi_i \begin{pmatrix} a_1 & x\\ & a_2 \end{pmatrix} \right\| \, \|a_1\|^{s-1}_{\mathbb{A}} \|a_2\|_{\mathbb{A}}^{s-1} \, \left\|\frac{a_1}{a_2}\right\|^{-1}_{\mathbb{A}} \, dx \, \frac{da_1}{\|a_1\|_{\mathbb{A}}} \, \frac{da_2}{\|a_2\|_{\mathbb{A}}} \\ =& \int \limits_{\mathbb{A}^{\times}} \int \limits_{\mathbb{A}^\times} \int \limits_{\mathbb{A}} \, \left\|\varphi_i \begin{pmatrix} a_1 & x\\ & a_2 \end{pmatrix} \right\| \, \|a_1\|^{s-3}_{\mathbb{A}} \|a_2\|^{s-1}_{\mathbb{A}} \, dx \, da_1 \, da_2. \label{estimate}\end{aligned}$$ Since $\phi_i$ decreases rapidly as $\|a_1\|_{\mathbb{A}}$, $\|a_2\|_{\mathbb{A}}$, and $\|x\|_{\mathbb{A}}$ become large, the integral $\eqref{estimate}$ converges for $\text{Re}(s) > 2$. \[boundedlemma\] There exists a real number $C$ so that for every $g_1 \in G_1(\mathbb{A})$, $$\|\phi^{T, \nu}(g_1)\| \leq C.$$ Absolute convergence of the integral right of the line Re$(s) =2$ is the best one could hope for since the Eisenstein series $E(s, f, -)$ has a possible pole at $s=2$ by Section \[siegeleisenstein\], and [@kudlarallis1994]\[Theorem 1.1]{}. Computation of the Unramified Integral {#unramifiedchapter} ====================================== The local integral is $$I_v(s)= \int \limits_{N(F_v) \backslash G_1(F_v)} f_v(s, g) \, \phi_v ^{T, \nu}(g) \, \omega_v(g, 1) \varphi_v(1_2) \, dg. \label{11}$$ Let $v$ be a finite place. As before $T= \begin{bmatrix} 1 & \\ & -\rho \end{bmatrix}$. \[unramifieddef\] The data for the integral $I_v(s)$ are unramified if all of the following hold: 1. $K_v = G(\mathcal{O}_v)$, and by the $\mathfrak{p}$-adic Iwasawa decomposition $G(F_v)=P(F_v) K_v$ 2. $\phi_v^{T, \nu}=\phi_v^{T, \nu^\circ}$ is the normalized local spherical Bessel function, i.e. it is right $K_v$ invariant 3. $\varphi_v =\varphi_v^\circ = \mathbf{1}_{Mat_{2, 2}(\mathcal{O}_v)}$ is the normalized spherical function for the Weil representation 4. $f_v(g, s)=f_v^\circ(g,s)=\delta_{P,v} ^{\frac{s}{3}+\frac{1}{3}}(g)$ where the modulus character is extended to the entire group $G(F_v)$ by $\delta_{P,v}(pk)=\delta_{P,v}(p)$ for $p \in P(F_v)$ and $k \in K_v$ 5. $\nu( H_T(\mathcal{O}_v))=1$ 6. $\rho \in \mathcal{O}_v^\times$ Assume that all the data are unramified for $I_v(s)$. This is the case for almost every $v$. Let $P_1=P \cap G_1$, and $M_1=M \cap G_1 \cong GL_2$, and $K_{1,v}=K_v \cap G_1(F_v)$. With these assumptions the integrand of $\eqref{11}$ is constant on double cosets $N(F_v) \backslash G_1(F_v) / K_{1,v}$. By the $\mathfrak{p}$-adic Iwasawa decomposition, $G_1=P_1(F_v) K_{1,v}$, and since $$M_1(F_v) \cong N(F_v) \backslash P_1(F_v)$$ representatives may be found among representatives for $M_1(F_v) / \left( M_1(F_v) \cap K_{1,v} \right).$ By [@furusawa1993] $$\GL_2(F)= \coprod \limits_{m \geq 0} H(F_v) \begin{bmatrix} \varpi^m & \\ & 1 \end{bmatrix} \GL_2(F_v). \label{decompF}$$ Let $m \geq 0$, and define $$H^m(\mathcal{O}_v) :=H(F_v) \cap \begin{bmatrix} \varpi^{m} & \\ & 1 \end{bmatrix} \GL_2(F_v) \begin{bmatrix} \varpi^{-m} & \\ & 1 \end{bmatrix} .$$ Note that $H^m(\mathcal{O}_v) \subseteq H(\mathcal{O}_v)$. Recall that $E$ is the discriminant field of $V_T$. Define $E_v := E \otimes_F F_v$. Let $\left( \frac{E}{v} \right)$ denote the Legendre symbol which equals $-1$, $0$, or $1$ according to whether $v$ is inert, ramifies, or splits in $E$. By elementary number theory, $\left( \frac{E}{v} \right) =0$ for only finitely many primes $v$. This case is not considered. Call the case when $\left( \frac{E}{v} \right) =-1$ as the inert case, and the case when $\left( \frac{E}{v} \right) =1$ as the split case. In each case $E_v^\times \cong H(F_v)$. If $\left( \frac{E}{v} \right)=-1$, then $E_v / F_v$ is an unramified quadratic extension. If $\left( \frac{E}{v}\right)=+1$, then $E_v \cong F_v \oplus F_v$. In this case there is an isomorphism $$\iota : H(F_v) \rightarrow (F_v \oplus F_v)^\times.$$ Let $\Pi_1 := \iota^{-1} ( (\varpi, 1))$, and $\Pi_2 := \iota^{-1}( (1, \varpi))$. Then $\det \Pi_i \in \mathfrak{p}$ for $i=1,2$, and $\Pi_1 \Pi_2 = diag(\varpi, \varpi)$. The Inert Case -------------- If $\left( \frac{E}{v} \right) = -1$, then a complete set of irredundant coset representatives for $N(F_v)\backslash G_1(F_v) / K_1$ is given by $$m\left( h \begin{bmatrix} \varpi^{m+n} & \\ & \varpi^n \end{bmatrix} \right)$$ where $m\geq 0$, $n \in \mathbb{Z}$, and $h$ runs over a set of representatives for $H(\mathcal{O}_v) / H^m(\mathcal{O}_v)$. This follows from the above decomposition and the fact that $H(F_v)=Z(F_v)H(\mathcal{O}_v)$ where $Z$ is the center of $\GL_2$. Since $$\omega_v(m(g),1)\varphi_v^\circ (1_2)=\chi_{T,v}\circ \det(g) \, \|\det(g)\|_v \, \varphi_v^\circ(g),$$ only $m(g)$ with $g \in \text{GL}_2(F_v) \cap \text{Mat}_{2, 2} (\mathcal{O}_v)$ are in the support of the $$\omega_v(m(g) , 1) \varphi_v^\circ (1_2).$$ Hence, a complete set of irredundant representatives for the cosets in the support of $\omega_v(m(g) , 1) \varphi_v(1_2)$ is given by the representatives listed above with $n \geq 0$. The various components of the integrand are computed as follows $$\begin{aligned} &\delta_P ^{\frac{s}{3}+\frac{1}{3}} \left( m \left( h \begin{bmatrix} \varpi_v^{n+m} & \\ & \varpi^n \end{bmatrix} \right) \right)=q_v^{-(2n+m)(s+1)} \label{integrand1}\\ &\omega_v \left( m \left(h \begin{bmatrix} \varpi_v^{n+m} & \\ & \varpi_v^n\\ \end{bmatrix} \right), 1 \right) \varphi_v^\circ(1_2)=\chi_{T,v}(\varpi_v^{2n+m}) q_v^{-(2n+m)} \label{integrand2}\\ &\text{vol}\left(N(F_v) \backslash N(F_v) \, m\left( h \begin{bmatrix} \varpi^{n+m} & \\ & \varpi^n \end{bmatrix} \right) \, K_{1,v}\right)=q_v^{6n+3m} \label{integrand3}\end{aligned}$$ The proof of is just an application of , and follows from Section \[schrodinger\]. To prove observe that there are measures $dn$ on $N$ and $d\bar{g}$ on $N \backslash G_1$ so that $$\int \limits_{G_1} f(g) dg = \int \limits_{N \backslash G_1} \int \limits_{N} f(n \bar{g}) \, dn \, d\bar{g}$$ and $$\int \limits_{G_1} f(nmk) dg = \int \limits_{M_1} \int \limits_{ K} \int \limits_{N} \delta_{P}^{-1}(m) \, f(nmk) \, dn\, dk \, dm.$$ Therefore, $\delta_{P,v}^{-1} \cdot d\dot{g}$ gives a right invariant measure on $N(F_v) \backslash G_1(F_v)$ normalized so that vol $\left( N(F_v) \backslash N(F_v)K_{1,v} \right) =1$. Again, let $h$ be a representative for an element of $H(\mathcal{O}_v) / H^m(\mathcal{O}_v)$, and let $$A(h,m,n)=N(F_v) \backslash N(F_v) \, m\left( h \begin{bmatrix} \varpi_v^{m+n} & \\ & \varpi_v^{n}\\ \end{bmatrix} \right) \, K_{1,v}.$$ Then $\text{vol}(A(h,m,n))=\delta_{P,v}^{-1}\left( m\left( h \begin{bmatrix} \varpi_v^{m+n} & \\ & \varpi_v^n\\ \end{bmatrix} \right) \right)=q_v^{3m+6n}$ Note that the integrand does not depend on the coset of $H(\mathcal{O}_v) / H^{m}(\mathcal{O}_v)$. The index $ [ H(\mathcal{O}_v) : H^{m}(\mathcal{O}_v) ]=q^m(1-\left( \frac{E}{v}\right)\frac{1}{q})$ for $m \geq 1$. Then the local integral $\eqref{11}$ is [rCl]{} I\_v(s) &=& (1+) \_[m,n 0]{} \^[T, ]{} \_v ( m ( \^[n+m]{} &\ & \^n ) ) \_T(\_v\^[m]{}) q\_v\^[2n(1-s)]{}q\_v\^[m(2-s)]{}\ && - \_[n 0]{} \^[T, ]{} \_v ( m ( \^[n]{} &\ & \^n ) ) q\_v\^[2n(1-s)]{}. \[16\] The Split Case -------------- If $\left( \frac{E}{v} \right) = +1$, then a complete set of irredundant coset representatives for $N(F_v)\backslash G_1(F_v) / K_1$ are given by $$m\left( h \, \Pi_i^k \begin{bmatrix} \varpi^{m+n} & \\ & \varpi^{n} \end{bmatrix} \right)$$ where $i=1,2$, $m,k\geq 0$, $n \in \mathbb{Z}$, and $h$ are representatives for $H(\mathcal{O}_v) / H^m(\mathcal{O}_v)$. For every $(x,y) \in (F_v \oplus F_v)^\times$ there are unique integers $k_1$, and $k_2$ such that $(x,y)=(\varpi^{k_1}, \varpi^{k_2}) \cdot u$ where $u \in \mathcal{O}_v^\times \oplus \mathcal{O}_v^\times$. If $k_1 > k_2$, then let $i=1$. Otherwise $i=2$. Let $n=\min\{k_1, k_2\}$, and $k=k_i-n$. The result then follows from the decomposition . The various components of the integrand are computed as follows $$\begin{aligned} &\delta_P ^{\frac{s}{3}+\frac{1}{3}} \left( m \left( h \, \Pi_i^k \begin{bmatrix} \varpi_v^{n+m} & \\ & \varpi^n \end{bmatrix} \right) \right)=q_v^{-(2n+m+k)(s+1)}\\ &\omega_v \left( m \left(h \, \Pi_i^k \begin{bmatrix} \varpi_v^{n+m} & \\ & \varpi_v^n\\ \end{bmatrix} \right), 1 \right) \varphi_v^\circ(1_2)=\chi_{T,v}(\varpi_v^{2n+m+k}) q_v^{-(2n+m)} \\ &\text{vol}\left(N(F_v) \backslash N(F_v) \, m\left( h \, \Pi^k_i\begin{bmatrix} \varpi^{n+m} & \\ & \varpi^n \end{bmatrix} \right) \, K_{1,v}\right)=q_v^{6n+3m+3k}\end{aligned}$$ The only nontrivial part is volume computation which follows from an argument that is similar to the proof of [@furusawa1993]\[Lemma 3.5.3]{}. Then the local integral $\eqref{11}$ is $$\begin{aligned} I_v(s) = (1-\frac{1}{q}) & \sum \limits_{i=1,2} \sum \limits_{m,n \geq 0} \phi^{T, \nu} _v \left( m \left( \Pi_i^k \begin{bmatrix} \varpi^{n+m} & \\ & \varpi^n \end{bmatrix} \right) \right) q_v^{(2n+k)(1-s)}q_v^{m(2-s)} \nonumber \\ + \frac{1}{q} & \sum \limits_{i=1,2} \sum \limits_{n \geq 0} \phi^{T, \nu} _v \left( m \left(\Pi_i^k \begin{bmatrix} \varpi^{n} & \\ & \varpi^n \end{bmatrix} \right) \right) q_v^{(2n+k)(1-s)}. \label{16a}\end{aligned}$$ The expressions and are evaluated using Sugano’s Formula. Sugano’s Formula ================ The results of this section were obtained by Sugano [@sugano1985], but I follow the treatment found in Furusawa [@furusawa1993]. Define $$\begin{aligned} h_v(\ell,m)=\begin{bmatrix} \varpi_v^{2m+\ell} & & & \\ & \varpi_v^{m+\ell} & & \\ & & 1 & \\ & & & \varpi_v^{m} \end{bmatrix}.\end{aligned}$$ The local spherical Bessel function is supported on double cosets $$\coprod \limits_{\ell,m \geq 0} R(F_v) h_v(\ell,m) GSp_4(\mathcal{O}_v).$$ In [@sugano1985] Sugano explicitly computes the following expression when $\phi^{T,\nu}$ is spherical: $$\begin{aligned} C_v(x,y)=\sum \limits_{\ell,m \geq 0} \phi_v^{T,\nu}(h_v(\ell, m)) x^m y^\ell.\end{aligned}$$ Since $\pi_v$ is assumed to be a spherical representation, it is isomorphic to an unramified principal series representation. I describe this more precisely. Let $P_0$ be the standard Borel subgroup of $G$ with Levi component $M_0$. $$\begin{aligned} M_0=\left\{ \begin{bmatrix} a_1 & & & \\ & a_2 & & \\ & & a_3 & \\ & & & a_4 \end{bmatrix} \Bigg\| a_1 a_3 = a_2 a_4 \right\}.\end{aligned}$$ There exists a character $$\gamma_v : M_0(F_v) \rightarrow \mathbb{C}^\times$$ that is trivial on $M_0(\mathcal{O}_v)$ such that $\pi_v \cong \mathrm{Ind}^{G(F_v)}_{P_0(F_v)}(\gamma_v)$. Then $\gamma_v$ is determined by its values $$\begin{aligned} \gamma_{1,v}=\gamma_v \begin{bmatrix} \varpi_v & & & \\ & \varpi_v & & \\ & & 1 &\\ & & & 1 \end{bmatrix}, & \quad \gamma_{2,v}=\gamma_v \begin{bmatrix} \varpi_v & & & \\ & 1 & & \\ & & 1 &\\ & & & \varpi_v \end{bmatrix},\\ \gamma_{3,v}=\gamma_v \begin{bmatrix} 1 & & & \\ & 1 & & \\ & & \varpi_v &\\ & & & \varpi_v \end{bmatrix}, & \quad \gamma_{4,v}=\gamma_v \begin{bmatrix} 1 & & & \\ & \varpi_v & & \\ & & \varpi_v &\\ & & & 1 \end{bmatrix}.\\\end{aligned}$$ Note that $$\gamma_{1,v} \gamma_{3,v} = \gamma_{2,v} \gamma_{4,v}=\omega_{\pi,v}(\varpi_v).$$ Let $$\epsilon_v= \left\{ \begin{array}{ll} 0 & \text{if } \left( \frac{E}{v} \right)=-1,\\ \nu(\varpi_{E,v}) & \text{if } \left( \frac{E}{v} \right)=0,\\ \nu(\varpi_{E,v})+\nu(\varpi_{E,v}^\sigma) \qquad & \text{if } \left( \frac{E}{v} \right)=1. \end{array} \right.$$ (Sugano) $$C_v(x,y)=\frac{H_v(x,y)}{P_v(x)Q_v(y)},$$ where [rCl]{} P\_v(x)&=& (1-\_[1,v]{} \_[2,v]{} q\_v\^[-2]{}x)(1-\_[1,v]{} \_[4,v]{} q\_v\^[-2]{}x)\ && (1-\_[2,v]{} \_[3,v]{} q\_v\^[-2]{}x)(1-\_[3,v]{} \_[4,v]{} q\_v\^[-2]{}x),\ Q\_v(y) &=& \_[i=1]{}\^4 (1-\_[i,v]{} q\_v\^[-3/2]{}y),\ H\_v(x,y) &=& (1+A\_2 A\_3 x y\^2){M\_1(x)(1+A\_2 x)+A\_2 A\_5 A\_1\^[-1]{} x\^2 }\ &&-A\_2 x y {M\_1(x) -A\_5 M\_2(x) } -A\_5 P\_v(x)y -A\_2 A\_4 P\_v(x) y\^2,\ M\_1(x) &=& 1 -A\_1\^[-1]{}(A\_1+A\_4)\^[-1]{}(A\_1 A\_5 + A\_4 -A\_1 A\_5\^2 -2 A\_1 A\_2 A\_4)x\ &&+A\_1\^[-1]{} A\_2\^2 A\_4 x\^2,\ M\_2(x) &=& 1+A\_1\^[-1]{}(A\_1 A\_2 -)x + A\_1\^[-1]{}A\_2 (A\_1 A\_2 -)x\^2+A\_2\^3x\^3, $$\begin{aligned} \alpha=&q_v^{-3/2}\sum \limits_{i=1}^4 \gamma_{i,v},& \quad \beta=&q_v^{-3}\sum \limits_{1 \leq i < j \leq 4} \gamma_{i,v} \gamma_{j,v},\\ A_1=&q_v^{-1},& \quad A_2=&q_v^{-2} \nu(\varpi_v),\\ \quad A_3=&q_v^{-3} \nu(\varpi_v),& \quad A_4=&-q_v^{-2}\left( \frac{E}{v} \right),\\ A_5=&q_v^{-2} \epsilon_v.\end{aligned}$$ The parameters $\gamma_{i,v}$ differ from the parameters of Section \[lfunctionsec\]. One verifies that $$\begin{aligned} \gamma_{1,v}=\chi_1 \chi_2 \chi_0(\varpi_v), & & \gamma_{2,v}=\chi_1 \chi_0(\varpi_v),\\ \gamma_{3,v}=\chi_0(\varpi_v), & & \gamma_{4,v}=\chi_2 \chi_0(\varpi_v),\end{aligned}$$ and $$\omega_{\pi,v} = \chi_1 \chi_2 \chi_0^2.$$ Therefore, $$\begin{aligned} \gamma_{1,v} \gamma_{2,v} = \chi_1^2 \chi_2 \chi_0^2 (\varpi_v)= \chi_1 \omega_{\pi,v}(\varpi_v), \nonumber\\ \gamma_{1,v} \gamma_{4,v} = \chi_1 \chi_2^2 \chi_0^2(\varpi_v) = \chi_1 \omega_{\pi,v}(\varpi_v), \nonumber\\ \gamma_{2,v} \gamma_{3,v} = \chi_1 \chi_0^2(\varpi_v) = \chi_2^{-1} \omega_{\pi,v}(\varpi_v), \nonumber\\ \gamma_{3,v} \gamma_{4,v} = \chi_2 \chi_0^2(\varpi_v) = \chi_1^{-1} \omega_{\pi,v}(\varpi_v).\label{blah124}\end{aligned}$$ \[absconv1\] Let $P_1$, $P_2 \in X \cdot \mathbb{C}[X]$, i.e. $P_i$ have constant coefficient $0$, then $C_v( P_1(q^{-s}) ,P_2( q^{-s}))$ converges absolutely for Re$(s) \gg 0$. Therefore, the terms of this series may be rearranged without affecting the sum. The proposition is a consequence of the following lemma which is the local version of Corollary \[boundedlemma\]. For each place $v$ of $F$, there is a constant $A_v>0$ and a real number $\alpha$ independent of $v$ so that $$\| \phi_v^{T,\nu}(g_v)\| \leq A_v \| \lambda_G(g_v)\|_v ^\alpha.$$ Again, I follow [@moriyama2004]. Pick a place $w \| \infty$. Then for $g \in G(\mathbb{A})$ one may write $g=z_w g_1$ where $g_1 \in G(\mathbb{A})^1$, and $z_w$ is in the center of $G(F_w)$. Then by Corollary \[boundedlemma\], I estimate $$\begin{aligned} \| \phi^{T,\nu} (g)\|=& \| \omega_{\pi,w}(z_w)\|_w \|\phi^{T,\nu}(g_1)\|_\mathbb{A} \nonumber\\ =& A \cdot \|z_w\|_w ^\beta \nonumber \\ =& A \cdot \|\lambda_G(g)\|_\mathbb{A}^{\beta /2}.\end{aligned}$$ Let $g_0 \in G(\mathbb{A})$ so that $\pi^{T,\nu} \neq 0$. Then for each place $v$ define $$A_v := A \times \prod \limits_{v^\prime \neq v} \frac{\|\lambda_G(g_{0,v^\prime})\|_{v^\prime} ^{\beta/2}}{\|\phi^{T,\nu}(g_{0,v^\prime})\|_\mathbb{A}},$$ and let $\alpha= \beta/2$. If $Re(s)$ is sufficiently large (to account for $A_v$ and the coefficients of $P_i$), then comparing the series $C_v(P_1(q^{-s}), P_2(q^{-s}))$ to a doubly geometric series completes the proof of the proposition. Computing the Local Integral ============================ The Inert Case -------------- It is necessary to express $\eqref{16}$ as a linear combination of terms of the form $C_v(x,y)$ where $x$ and $y$ are monomials in $q_v^{-s}$. Since $$\begin{aligned} m \begin{bmatrix} \varpi_v^{m+n} & \\ & \varpi_v^n \end{bmatrix} &= \begin{bmatrix} \varpi_v^{m+n} & & & \\ & \varpi_v^{n} & & \\ & & \varpi_v^{-n-m} & \\ & & & \varpi_v^{-n} \end{bmatrix} \nonumber \\ &=z(\varpi_v^{-n-m}) h_v(2n, m),\end{aligned}$$ then gives $$\begin{aligned} I_v(s) = (1+\frac{1}{q}) & \sum \limits_{m,n \geq 0} \omega_{\pi,v}(\varpi)^{-m-n} \, \phi^{T, \nu} _v (h_v(m, 2n)) \chi_T(\varpi_v)^m q_v^{2n(1-s)}q_v^{m(2-s)} \nonumber \\ - \frac{1}{q} &\sum \limits_{n \geq 0} \omega_{\pi,v}(\varpi)^{-n} \, \phi^{T, \nu} _v (h_v(0, 2 n)) q_v^{2n(1-s)} \end{aligned}$$ When $\left( \frac{E}{v}\right)=-1$ the unramified local integral is $$\begin{aligned} I_v(s)=& (1+\frac{1}{q_v})\sum \limits_{\ell, m \geq 0} (-\omega_{\pi,v}(\varpi_v)^{-1} q_v^{2-s})^m(\omega_{\pi,v}(\varpi_v)^{-1/2}q_v^{1-s})^{2 \ell} \phi^{T,\nu}_v(h_v(2\ell, m))\nonumber \\ &-\frac{1}{q_v}\sum \limits_{\ell \geq 0} (-\omega_{\pi, v}(\varpi_v)^{-1/2}q_v^{1-s})^{2 \ell} \phi^{T,\nu}_v (h_v(2 \ell, 0))\nonumber\\ =&(1+\frac{1}{q_v}) \Big[ \frac{1}{2} C_v\left( -\omega_{\pi,v}(\varpi_v)^{-1}q_v^{2-s}, \omega_{\pi, v}(\varpi_v)^{-1/2}q_v^{1-s} \right)\nonumber \\& \qquad + \frac{1}{2} C_v\left( -\omega_{\pi,v}(\varpi_v)^{-1}q_v^{2-s}, -\omega_{\pi, v}(\varpi_v)^{-1/2}q_v^{1-s} \right) \Big] \nonumber \\& - \frac{1}{q_v} \Big[ \frac{1}{2} C_v\left( 0, \omega_{\pi, v}(\varpi_v)^{-1/2}q_v^{1-s} \right)+ \frac{1}{2} C_v\left( 0, -\omega_{\pi, v}(\varpi_v)^{-1/2}q_v^{1-s} \right) \Big].\end{aligned}$$ This expression was evaluated using Sugano’s formula and Mathematica. When $\left( \frac{E}{v}\right)=-1$ the unramified local integral is [rCl]{} I\_v(s)&=&(1-q\_v\^[-s]{})(1-q\_v\^[-s-1]{})\ && (1+\_1 \_2 \_[,v]{}(\_v)\^[-1]{} q\_v\^[-s]{})\^[-1]{}(1+\_1 \_4 \_[,v]{}(\_v)\^[-1]{} q\_v\^[-s]{})\^[-1]{}\ && (1+\_2 \_3 \_[,v]{}(\_v)\^[-1]{} q\_v\^[-s]{})\^[-1]{}(1+\_3 \_4 \_[,v]{}(\_v)\^[-1]{} q\_v\^[-s]{})\^[-1]{}. \[blah125\] Correcting by the normalizing factor gives \[inertprop1\] When $\left( \frac{E}{v}\right)=-1$, the normalized unramified local integral is $$\begin{aligned} \zeta_v(s+1) \zeta_v(2s) I_v(s)=L(s, \pi_v \otimes \chi_{T,v})\end{aligned}$$ This follows from comparing and . The Split Case -------------- Since $\chi_{T,v}(\varpi_v)=\left( \frac{E}{v} \right)=1$, $\chi_{T,v}$ does not appear in this part of the calculation. Since $$\begin{aligned} m \left(\Pi_i^k \, \begin{bmatrix}\varpi_v^{m+n} & \\ & \varpi_v^n \end{bmatrix}\right) &=b(\Pi_i^k) \begin{bmatrix} \varpi_v^{m+n} & & & \\ & \varpi_v^{n} & & \\ & & \varpi_v^{-n-m-k} & \\ & & & \varpi_v^{-n-k} \end{bmatrix} \nonumber \\ &=b(\Pi_i^k) z(\varpi^{-m-n-k}) \begin{bmatrix}\varpi_v^{2m+2n+k} & & & \\ & \varpi_v^{m+2n+k} & & \\ & & 1 & \\ & & & \varpi_v^{m} \end{bmatrix} \nonumber\\ &=b(\Pi_i^k) z(\varpi_v^{-m-n-k}) h_v(2n+k, m), \end{aligned}$$ so becomes [rCl]{} I\_v(s) &=& (1-) \_[i=1,2]{} \_[m,n,k 0]{} \_[,v]{}()\^[-m-n-k]{} (\_i)\^k \^[T, ]{} \_v (h\_v(m, 2n+k)) q\_v\^[(2n+k)(1-s)]{}q\_v\^[m(2-s)]{}\ & & + \_[i=1,2]{} \_[n,k 0]{} \_[,v]{}()\^[-n-k]{} (\_i)\^k \^[T, ]{} \_v (h\_v(0, 2 n+k)) q\_v\^[(2n+k)(1-s)]{}. \[A\] First, suppose $\nu(\Pi_1) \neq \nu(\Pi_2)$ which is equivalent to $\nu(\Pi_i)^2 \neq \omega_{\pi,v}(\varpi)$. \[splitprop\] $$\frac{\omega_{\pi,v}(\varpi)^{-1} \nu(\Pi_1)}{\omega_{\pi,v}(\varpi)^{-1} \nu(\Pi_1)^2-1}+\frac{\omega_{\pi,v}(\varpi)^{-1} \nu(\Pi_2)}{\omega_{\pi,v}(\varpi)^{-1} \nu(\Pi_2)^2-1}=0 \label{identity1}$$ and $$\frac{\omega_{\pi,v}(\varpi)^{-1}\nu(\Pi_1)^2}{\omega_{\pi,v}(\varpi)^{-1} \nu(\Pi_1)^2-1} + \frac{\omega_{\pi,v}(\varpi)^{-1}\nu(\Pi_2)^2}{\omega_{\pi,v}(\varpi)^{-1} \nu(\Pi_2)^2-1}=1. \label{identity2}$$ Both identities follow from the fact that $\nu(\Pi_1) \cdot \nu(\Pi_2) = \omega_{\pi,v}(\varpi_v)$. Let $$\begin{aligned} \eta_1:= \frac{\omega_{\pi,v}(\varpi)^{-1}\nu(\Pi_1)^2}{\omega_{\pi,v}(\varpi)^{-1} \nu(\Pi_1)^2-1}& & \eta_2:= \frac{\omega_{\pi,v}(\varpi)^{-1}\nu(\Pi_2)^2}{\omega_{\pi,v}(\varpi)^{-1} \nu(\Pi_2)^2-1}.\\ \theta_1:=\frac{\omega_{\pi,v}(\varpi)^{-1}\nu(\Pi_1)}{\omega_{\pi,v}(\varpi)^{-1} \nu(\Pi_1)^2-1}& & \theta_2:= \frac{\omega_{\pi,v}(\varpi)^{-1}\nu(\Pi_2)}{\omega_{\pi,v}(\varpi)^{-1} \nu(\Pi_2)^2-1}.\end{aligned}$$ Then combining terms with $2n+k=\ell$ gives [rCl]{} I\_v(s)&=&\_[i=1,2]{} (1-) \_i \_[,m 0]{} (\_[,v]{}()\^[-1]{} (\_i) q\^[1-s]{} )\^(\_[,v]{} ()\^[-1]{}q\^[2-s]{})\^m \_v\^[T,]{}(h\_v(2 , m))\ && - (1-) (\_i -1) \_[, m 0]{} (\_[,v]{}()\^[-1/2]{} q\^[1-s]{})\^[2 ]{} ( \_[,v]{}()\^[-1]{} q\^[2-s]{})\^m \_v\^[T,]{}(h\_v(2, m))\ && - (1-) \_i \_[,m 0]{} (\_[,v]{}()\^[-1/2]{} q\^[1-s]{})\^[2 +1]{} ( \_[,v]{}()\^[-1]{} q\^[2-s]{})\^m \_v\^[T,]{}(h\_v(2+1, m))\ && + \_i \_[0]{} (\_[,v]{}()\^[-1]{} (\_i) q\^[1-s]{} )\^\_v\^[T,]{}(h\_v( , 0))\ && - (\_i-1) \_[0]{} (\_[,v]{}()\^[-1/2]{} q\^[1-s]{})\^[2 ]{} \_v\^[T,]{}(h\_v(2, 0))\ && - \_i \_[0]{} (\_[,v]{}()\^[-1/2]{} q\^[1-s]{})\^[2 +1]{} \_v\^[T,]{}(h\_v(2+1, 0)). \[splitsum\] Applying Proposition \[splitprop\] to gives When $\left(\frac{E}{v}\right)=+1$ and $\nu(\Pi_1) \neq \nu(\Pi_2)$ the unramified local integral is given by $$\begin{aligned} I_v(s)=&\left(1-\frac{1}{q_v}\right)\eta_1 \cdot C_v(\omega_{\pi,v}(\varpi_v)^{-1} q_v^{2-s}, \omega_{\pi,v}(\varpi_v)^{-1}\nu(\Pi_1)q_v^{1-s}) \nonumber \\ +& \left(1-\frac{1}{q_v}\right) \eta_2 \cdot C_v(\omega_{\pi,v}(\varpi_v)^{-1} q_v^{2-s}, \omega_{\pi,v}(\varpi_v)^{-1}\nu(\Pi_2)q_v^{1-s}) \nonumber\\ -&\frac{1}{q_v} \eta_1 \cdot C_v(0, \omega_{\pi,v}(\varpi_v)^{-1}\nu(\Pi_1)q_v^{1-s})\nonumber \\ -&\frac{1}{q_v} \eta_2 \cdot C_v(0, \omega_{\pi,v}(\varpi_v)^{-1}\nu(\Pi_2)q_v^{1-s}).\label{splitsum1}\end{aligned}$$ When $\left( \frac{E}{v}\right)=+1$ the unramified local integral is [rCl]{} I\_v(s)&=&(1+q\_v\^[-s]{})(1-q\_v\^[-s-1]{})\ &&(1-\_1 \_4 \_[,v]{}(\_v)\^[-1]{} q\_v\^[-s]{})\^[-1]{}(1-\_1 \_2 \_[,v]{}(\_v)\^[-1]{} q\_v\^[-s]{})\^[-1]{}\ &&(1-\_2 \_3 \_[,v]{}(\_v)\^[-1]{} q\_v\^[-s]{})\^[-1]{} (1-\_3 \_4 \_[,v]{}(\_v)\^[-1]{} q\_v\^[-s]{})\^[-1]{}.\[blah127\] When $\nu(\Pi_1) \neq \nu(\Pi_2)$ this follows from applying Sugano’s formula to , and the computation was verified with Mathematica. To extend the identity to all values of $\nu(\Pi_1)$ and $\nu(\Pi_2)$ observe that the right hand side of is equal to $$\begin{aligned} \sum \limits_{i=1,2} \sum \limits_{m,n,k \geq 0} A(m) \phi_{v}^{T,\nu} ( h_v(m, n+2k)) X^m Y^n Z^k,\end{aligned}$$ with $X=\omega_{\pi,v}^{-1}(\varpi) q_v^{2-s}$, $Y=\omega_{\pi,v}^{-1}(\varpi) q_v^{2-2s}$, $Z=\omega_{\pi,v}^{-1}(\varpi) \nu(\Pi_i) q_v^{1-s} $, $A(0)=1$, and $A(m)=1-\frac{1}{q}$ otherwise. This is an absolutely convergent power series for $Re(s) \gg 0$, and convergence is uniform as $\nu(\Pi_1)$ and $\nu(\Pi_2)$ vary in a compact set. Furthermore, for $\nu(\Pi_1) \neq \nu(\Pi_2)$ the sum is equal to a rational function in $q^{-s}$ that remains continuous for all values of $\nu(\Pi_1)$ and $\nu(\Pi_2)$ such that $\omega_{\pi,v}(\varpi) \nu(\Pi_1) \nu(\Pi_2)=1$. Then by uniform convergence the equality holds for all such values of $\nu(\Pi_1)$ and $\nu(\Pi_2)$. \[splitprop1\] When $\left( \frac{K}{v}\right)=+1$, the normalized unramified local integral is $$\begin{aligned} \zeta_v(s+1) \zeta_v(2s) I_v(s)=L(s, \pi_v \otimes \chi_{T,v}).\end{aligned}$$ This follows from comparing and Ramified Integrals at Finite Places {#ramified} =================================== Consider the local integral at a finite place $v$ where some of the data may be ramified. $$I_v(s)= \int \limits_{N(F_v) \backslash G_1(F_v)} f_v(s, g) \, \phi_v ^{T, \nu}(g) \, \omega_v(g, 1) \varphi_v(1_2) \, dg.$$ Let $K_{P,v}=P(F_v) \cap K_v$. Let $K_{P_1,v}=P_1(F_v) \cap K_v= K_1 \cap K_{P,v}$. \[induced2\] Let $h : K_{P,v} \backslash K_v \rightarrow \mathbb{C}$ be a locally constant (i.e. smooth) function. Then there exists $f_v(s,k) \in \text{Ind}(s)$ such that for all $k \in K_v$, $f(s,k)=h(k)$. This proposition follows from [@casselman1995]\[Proposition 3.1.1]{}. \[finiteram\] There exists a $K_v$-finite section $f_v(s,g) \in \text{Ind}(s)$, and a $K_v$-finite Schwartz-Bruhat function $\varphi_v \in \mathcal{S}(\mathbb{X}(F_v))$ such that $I_v(s) \equiv 1$. This argument comes directly from [@piatetski-shapirorallis1988]. Suppose that $K_0$ is an open compact subgroup of $G_1(F_v)$ so that $\phi^{T,\nu}_v$ is right $K_0$ invariant. Consider the isomorphism $p: M_1 \rightarrow \text{GL}_2$, $m(a) \mapsto a$. Let $K_\phi$ be an open compact subgroup of GL$_2(F_v)$ such that $K_\phi \subseteq p(M_1(F_v) \cap \nolinebreak K_0) \cap \ker \chi_T \circ \det$. Let $\varphi_v = 1_{K_\phi}$. Since $\varphi_v$ is a smooth function, there exists an open compact subgroup $K' \subseteq G_1(F_v)$ such that $\omega_v(k,1) \varphi_v = \varphi_v$. Let $\mathcal{K}=K_{P_1,v} \backslash K_{P_1,v} \cdot (K' \cap K_0)$. By Proposition \[induced2\] there is $f_v(s, -) \in \text{Ind}(s)$ so that $f_v(s, -)\|_{K_1}=1_{K_{{P_1},v} \cdot (K^\prime \cap K_0)}.$ Then [rCl]{} I\_v(s)&=& \_[N(F\_v) \\G\_1(F\_v)]{} f\_v(s, g) \_v \^[T, ]{}(g) \_v(g, 1) \_v(1\_2) dg\ &=& \_[K\_[P\_1,v]{} \\K\_[1,v]{}]{} \_[\_2(F\_v)]{} \_P(m(a))\^[-1]{} f\_v(s, m(a)k) \_v\^[T,]{} (m(a)k)\ && \_v(m(a)k, 1) \_v(1\_2) da dk\ &=& \_[K\_[P\_1,v]{} \\K\_[1,v]{}]{} f\_v(s,k) \_[\_2(F\_v)]{} \|(a)\|\_v\^[s-2]{} \_v\^[T,]{} (m(a)k) \_v(m(a)k, 1) \_v(1\_2) da dk\ &=& \_ \_[\_2(F\_v)]{} \|(a)\|\_v\^[s-2]{} \_v\^[T,]{} (m(a)k) \_v(m(a)k, 1) \_v(1\_2) da dk\ &=& \_ \_[\_2(F\_v)]{} \|(a)\|\_v\^[s-1]{} \_v\^[T,]{}(m(a) k) \_T( (a)) 1\_[K\_]{}(a) da dk \[blah2\] For $a \in K_\phi$, $\| \det( a)\|_v=1$, and $\phi_v^{T, \nu}(m(a))=1$ , then $$\begin{aligned} & &\int \limits_{\mathcal{K}} \int \limits_{\text{GL}_2(F_v)} \|\det(a)\|_v^{s-1} \phi_v^{T,\nu}(m(a) k) \, \chi_T( \det(a)) \, 1_{K_\phi}(a) \, da \, dk \\ &=&\int \limits_{K_\phi (K' \cap K_0)} \phi_v^{T,\nu}(k) \, dk.\end{aligned}$$ After normalizing measures and $\phi^{T, \nu}_v$, $I_v(s) \equiv 1$. Ramified Integrals at Infinite Places {#archimedean} ===================================== Consider the integral at the infinite places from Proposition \[eulerproduct\]. $$I_\infty(s)= \int \limits_{N(\mathbb{A}_\infty) \backslash G_1(\mathbb{A}_\infty)} f(s,g) \phi^{T, \nu}(g) \omega(g, 1) \varphi(1_2) \, dg.$$ \[archprop\] For every complex number $s_0$ there is a choice of of data $f( s, g) \in$ $(s)$, and $\varphi=\varphi_\infty \otimes \varphi_{\text{fin}} \in \mathcal{S}( \mathbb{X}(\mathbb{A}))$ such that $I_\infty$ converges to a holomorphic function at $s_0$, and $I_\infty(s_0) \neq 0$. The proof of this proposition is essentially given in [@piatetski-shapirorallis1988]\[§2]{}, but is reproduced here with necessary changes. By the Iwasawa decomposition, $G_1(\mathbb{A}_\infty) = P_1(\mathbb{A}_\infty) K_\infty$, where $P_1$ has Levi factor $M_1 \cong \text{GL}_2$. The integral $I_\infty(s)$ may be broken up as a $M_1(\mathbb{A}_\infty)$ integral and a $K_\infty$ integral: $$\begin{aligned} I_\infty(s)=& \int \limits_{K_\infty} \int \limits_{\text{GL}_2(\mathbb{A}_\infty)} \delta_P(m(a))^{-1} f(s, m(a)k) \phi^{T, \nu}(m(a)k) \omega(m(a)k, 1) \varphi(1_2) \, da \, dk \\ =& \int \limits_{K_\infty} f(k,s) \int \limits_{\text{GL}_2(\mathbb{A}_\infty)} \|\det(a)\|_\infty^{s-2} \phi^{T, \nu}(m(a)k) \omega(m(a)k, 1) \varphi(1_2) \, da \, dk.\end{aligned}$$ Here $\| \, \|_\infty$ denotes the valuation on $ \mathbb{A}_\infty$ defined by $\| x \|_\infty=\prod \limits_v\|x_v\|_v$, and $\det(a) \in \mathbb{A}_\infty$ with $v$ coordinate equal to $\det(a_v)$. Since $f(k ,s)$ is a standard section, it is independent of $s$ when restricted to $K_\infty$. Write $f(k,s)=f(k)$ for $k \in K_\infty$. The integral $$\begin{aligned} A(k,s):= \int \limits_{\text{GL}_2(\mathbb{A}_\infty)} \|\det(a)\|^{s-2} \phi^{T, \nu}(m(a)k) \, \omega(m(a)k, 1) \varphi_\infty(1_2) \, da \label{blah23}\end{aligned}$$ gives a function on $(M_1(\mathbb{A}_\infty) \cap K_\infty) \backslash K_\infty = (P_1(\mathbb{A}_\infty) \cap K_\infty) \backslash K_\infty $. The function $\varphi$ was chosen to be $K$-finite, in particular it is $K_\infty$-finite. Suppose that $\varphi=\otimes_v \varphi_v$, and $\varphi_\infty=\otimes_{v\|\infty} \varphi_v$. Since the integrand of is a smooth function of $k$, in the region of absolute convergence, $A(-,s)$ is smooth function. There is some choice of data so that $A(1,s_0) \neq 0$. The function $\varphi_\infty$ is $K_\infty$-finite. Let $\varphi_\infty^\circ=\bigotimes_{v\|\infty} \phi_v^\circ(X_v)$. According to it is of the form $$\varphi_\infty(X)= p(X) \varphi_\infty^\circ(X)$$ where $X =\bigotimes_{v\|\infty} X_v \in \mathbb{X}(\mathbb{A}_\infty)$ and $p(X)$ is a polynomial in $\mathbb{X}(\mathbb{A}_\infty)$. In particular, $\| \det(X) \|_\infty$ is a polynomial in $\mathbb{X}(\mathbb{A}_\infty)$. Pick $p(X)$ to be of the form $$p(X)=q(X) \cdot \|\det(X)\|_\infty ^n$$ where $q(X)$ is a polynomial in $\mathbb{X}(\mathbb{A}_\infty)$ and $n \in \mathbb{N}$. By Lemma \[boundedlemma\] $\phi^{T, \nu}$ is bounded, so in particular it is bounded on $M_1(\mathbb{A}_\infty)$. Therefore, $$\begin{aligned} A(1,s_0) = \int \limits_{\text{GL}_2(\mathbb{A}_\infty)} & \|\det(a)\|_\infty^{s_0-1+n} \, q(a) \phi^{T, \nu}(m(a)) \, da. \label{integralA}\end{aligned}$$ For $Re(s_0-1+n)>>0$ the integral converges absolutely. Indeed, $\varphi_\infty^\circ$ decays exponentially as the entries of $a$ become large while the rest of the integrand has polynomial growth at infinity. As the entries of $a$ become small, so does $\|\det(a)\|^{s_0-1+n}$. The other terms in the integrand are bounded. By assumption on $\phi$, there is some $g \in G_{1}(\mathbb{A}_\infty)$ so that $\phi^{T,\nu}(g) \neq 0$. By the Iwasawa decomposition I can write $g_\infty=na^\prime k$, where $n \in N(\mathbb{A}_\infty)$, $a^\prime \in M_1(\mathbb{A}_\infty)$, and $k \in K_{1, \infty}$. Since $K_{1, \infty}$ acts on the space of $\pi$, replace $\phi$ with $\pi(k) \phi$ because the action of $\pi$ is compatible with taking Bessel coefficients. Assume $k=1$. Since $\phi^{T,\nu}(n a^\prime)=\psi_T(n) \cdot \phi^{T,\nu}(a^\prime)$, and $\psi_T(n) \neq 0$, then $\phi(a^\prime)\neq 0$. Since polynomials are dense in $L^2(\mathbb{X}(\mathbb{A}_\infty))$, then there is some choice of polynomial $q$ so that $A(1,s_0) \neq 0$. Therefore, $A(1,s)$ is a nonzero holomorphic function in a neighborhood of $s_0$, and $A(k,s)$ is a $K$-finite function of $k$ on $(M_1(\mathbb{A}_\infty) \cap K_\infty) \backslash K_\infty$. There is a bijection between $\bigotimes \limits_{v\|\infty} Ind_{P(F_v)}^{G(F_v)}$ and $K_v$ finite functions in $L^2((M_1(\mathbb{A}_\infty \cap K_\infty) \backslash K_\infty)$ given by restricting $f$ to $K_v$. Therefore, there is a choice of $K$-finite standard section $f(k)$ so that $$\int \limits_{K_\infty} f(k,s_0) \, A(k,s_0) \, dk \neq 0. \qedhere$$ Proof of Theorem 1 ================== This section summarizes the results of previous sections to prove Theorem \[maintheorem\] which is restated here. Let $\pi$ be a cuspidal automorphic representation of GSp$_4(\mathbb{A})$, and $\phi=\otimes_v \phi_v \in V_\pi$ be a decomposable automorphic form. Let $T$ and $\nu$ be such that $\phi^{T,\nu}\neq0$. There exists a choice of section $f(s, -) \in \text{Ind}_{P(\mathbb{A})}^{G(\mathbb{A})}( \delta_P ^{1/3(s-1/2)})$, and some $\varphi=\otimes_v \varphi_v \in \mathcal{S}( \mathbb{X}(\mathbb{A}))$ such that the normalized integral $$I^(s;f, \phi, T, \nu, \varphi)= d(s) \cdot L^{S}(s, \pi \otimes \chi_T)$$ where $S$ is a finite set of bad places including all the archimedean places. Furthermore, for any complex number $s_0$, the data may be chosen so that $d(s)$ is holomorphic at $s_0$, and $d(s_0) \neq 0$. There is a finite set of places $S$ including all the archimedean places, such that for $v \notin S$, the conditions in Definition \[unramifieddef\] are satisfied. Consider the normalized Eisenstein series $E^(s,f,g)=\zeta^S(s+1) \, \zeta^S (2s) E(s,f,g)$ that was described in . Define $I^(s;f, \phi, T, \nu, \varphi)$ to be the global integral defined in except that $E(s,f,g)$ is replaced by $E^(s,f,g)$. By Proposition \[eulerproduct\] the integral factors as $$I^(s;f, \phi, T, \nu, \varphi)=I_\infty(s) \times \prod \limits_{\substack{v < \infty\\ v \in S}} I_v(s) \times \prod \limits_{ v \notin S} I^_v(s).$$ Here, $I^_v(s)= \zeta_v (s+1) \zeta_v (2s) I_v(s)$. According to Proposition \[inertprop1\] and Proposition \[splitprop1\] for $v \notin S$, $$I^*_v(s)=L(s, \pi_v \otimes \chi_{T,v}).$$ By Proposition \[finiteram\] for every finite place $v \in S$, there is a choice of local section $f_v(s, -)$ and local Schwartz-Bruhat function $\varphi_v$ so that $$I_v(s) \equiv 1.$$ By Proposition \[archprop\] there is a choice of data at the infinite places, $f_\infty(s, -)$, and $\varphi_\infty$, so that $I_\infty(s)$ is holomorphic at $s_0$, and $I_\infty(s_0) \neq 0$. Choose $f(s, -)=\otimes_v f_v(s, -)$ so that $f_v(s,-)$ is the choice specified above for $v \in S$, and the local spherical section otherwise. Similarly, choose $\varphi = \otimes_v \varphi_v$ so that $\varphi_v$ is the choice specified for $v \in S$, and the local spherical Schwartz-Bruhat function otherwise. This completes the proof of the theorem. [9999]{} [^1]: This work was done while I was a gruaduate student at Ohio State University as part of my Ph.D. dissertation. I wish to thank my advisor Jim Cogdell for being a patient teacher and for his helpful discussions about this work. I also thank Ramin Takloo-Bighash for many useful conversations.
--- abstract: 'We extend the previously found accelerated Kerr-Schild metrics for Einstein-Maxwell-null dust and Einstein-Born-Infeld-null dust equations to the cases including the cosmological constant. This way we obtain the generalization of the charged de Sitter metrics in static space-times. We also give a generalization of the zero acceleration limit of our previous Einstein-Maxwell and Einstein-Born-Infeld solutions.' author: - 'Metin G[" u]{}rses' - '[" O]{}zg[" u]{}r Sar[i]{}oğlu' title: 'Some Further Properties of the Accelerated Kerr-Schild Metrics' --- \[intro\] Introduction ====================== Using a curve $C$ in $D$-dimensional Minkowski space-time $M_{D}$, we have recently studied the Einstein-Maxwell-null dust [@gur1] and Einstein-Born-Infeld-null dust field equations [@gur2], Yang-Mills equations [@ozg], and Li[' e]{}nard-Wiechert potentials in even dimensions [@gur3]. In the first three works we found some new solutions generalizing the Tangherlini [@tan], Pleba[' n]{}ski [@pleb], and Trautman [@trt] solutions, respectively. The last one proves that the accelerated scalar or vector charged particles in even dimensions lose energy. All of the solutions contain a function $c$ which is assumed to depend on the retarded time $\tau_{0}$ and all accelerations $a_{k}, (k=0,1,2, \cdots)$, see [@gur1]-[@gur3]. We also assumed that when the motion is uniform or the curve $C$ is a straight line in $M_{D}$, this function reduces to a function depending only on the retarded time $\tau_{0}$. In this work we first relax this assumption and give the most general form of the function $c$ when the curve $C$ is a straight line. In addition, we also generalize our previous accelerated Kerr-Schild metrics by including the cosmological constant in arbitrary $D$-dimensions for the Einstein-Maxwell and in four dimensions for the Einstein-Born-Infeld theories. The solutions presented here can be interpreted as the solutions of the Einstein-Maxwell-null perfect fluid field equations with a constant pressure $\Lambda$ or Einstein-Maxwell-null dust field equations with a cosmological constant $\Lambda$. In our treatment we adopt the second interpretation. Our conventions are similar to the conventions of our earlier works [@gur1], [@gur2], [@gur3]. In a $D$-dimensional Minkowski space-time $M_{D}$, we use a parameterized curve $C= \{x^{\mu} \in M_{D}: x^{\mu}=z^{\mu}(\tau)\, , \mu=0,1,2, \cdots , D-1 \, , \tau \in I \}$ such that $\tau$ is a parameter of the curve and $I$ is an interval on the real line ${\mathbb R}$. We define the world function $\Omega$ as $$\Omega=\eta_{\mu \nu}\,(x^{\mu}-z^{\mu}(\tau))\,(x^{\nu}-z^{\nu}(\tau)), \label{dist}$$ where $x^{\mu}$ is a point not on the curve $C$. There exists a point $z^{\mu}(\tau_{0})$ on the non-spacelike curve $C$ which is also on the light cone with the vertex located at the point $x^{\mu}$, so that $\Omega(\tau_{0})=0$. Here $\tau_{0}$ is the retarded time. By using this property we find that $$\lambda_{\mu} \equiv \partial_{\mu}\, \tau_{0} = \frac{x^{\mu}-z^{\mu}(\tau_{0})}{R}$$ where $R \equiv \dot{z}^{\mu}(\tau_{0})\,(x_{\mu}-z_{\mu}(\tau_{0}))$ is the retarded distance. Here a dot over a letter denotes differentiation with respect to $\tau_{0}$. It is easy to show that $\lambda_{\mu}$ is null and satisfies $$\lambda_{\mu, \nu}=\frac{1}{R}\, [\eta_{\mu \nu}-\dot{z}_{\mu}\, \lambda_{\nu}-\dot{z}_{\nu}\, \lambda_{\mu}-(A-\epsilon)\, \lambda_{\mu}\, \lambda_{\nu}]$$ where $A \equiv \ddot{z}^{\mu}\, (x_{\mu}-z_{\mu}(\tau_{0}))$ and $\dot{z}^{\mu}\, \dot{z}_{\mu}=\epsilon= -1,0$. Here $\epsilon=-1$ and $\epsilon=0$ correspond to the time-like and null velocity vectors, respectively. One can also show explicitly that $\lambda^{\mu}\, \dot{z}_{\mu}=1$ and $\lambda^{\mu}\, R_{,\, \mu}=1$. Define $a \equiv \frac{A}{R}=\lambda^{\mu} \ddot{z}_{\mu}$, then $$\lambda^{\mu}\, a_{, \, \mu}=0.$$ Furthermore defining (letting $a_{0}=a$) $$a_{k} \equiv \lambda_{\mu}\, \frac{d^{k+2} \, z^{\mu}(\tau_{0})}{d \tau_{0} ^{k+2}}, \;\; k=0,1,2, \cdots \label{aks}$$ one can show that $$\lambda^{\mu}\, a_{k,\, \mu}=0,\;\;\; \forall k=0,1,2, \cdots \, .$$ Hence any function $c$ satisfying $$\lambda^{\mu}\, c_{,\, \mu}=0, \label{con1}$$ has arbitrary dependence on all $a_{k}$’s and $\tau_{0}$. Using the above curve kinematics, we showed that Einstein-Maxwell-perfect fluid equations with the Kerr-Schild metric give us the following result [Proposition 1]{}. Let the space-time metric and the electromagnetic vector potential be respectively given by $g_{\mu \nu}=\eta_{\mu\nu}-2V \lambda_{\mu} \lambda_{\nu},\,\, A_{\mu}=H\, \lambda_{\mu}$, where $V$ and $H$ are some differentiable functions in $M_{D}$. Let $V$ and $H$ depend on $R$, $\tau_{0}$ and functions $c_{i} \, (i=1,2, \cdots)$ that satisfy (\[con1\]), then the Einstein equations reduce to the following set of equations \[see [@gur1] for details\] $$\begin{aligned} \kappa p+\Lambda & = & \frac{1}{2} V^{\prime \prime}+ \frac{3D-8}{2R} V^{\prime} +\frac{(D-3)^2}{R^2} V, \label{pres1}\\ \kappa (H^{\prime})^2 & = & V^{\prime \prime}+\frac{D-4}{R} V^{\prime}- \frac{2V}{R^2}(D-3), \label{denk1}\\ \kappa (p+\rho) & = & q-\kappa \eta^{\alpha \beta}\, H_{,\alpha} H_{,\beta} \nonumber\\ & & + \, 2\, \left[ \frac{2(A-\epsilon)(D-3)V}{R^2}- \sum_{i=1}(w_{i} \, c_{i, \alpha}\, \dot{z}^{\alpha}) \right] , \label{rho1}\\ \sum_{i=1}\, w_{i} \, c_{i, \alpha} & = & \left[\sum_{i=1}\, (w_{i}\, c_{i, \beta} \,\dot{z}^{\beta}) \right] \, \lambda_{\alpha}, \label{cler1}\end{aligned}$$ where $$w_{i}=V^{\prime}_{,c_{i}}+\frac{D-4}{R}\, V_{,c_{i}}- \kappa H^{\prime}\, H_{,c_{i}} ,$$ and prime over a letter denotes partial differentiation with respect to $R$. Here $\kappa$ is the gravitational constant, $p$ and $\rho$ are, respectively, the pressure and energy density of the fluid, $\Lambda$ is the cosmological constant and the function $q$ is defined by $$q=\eta^{\alpha \beta}\, V_{, \alpha \beta}-\frac{4}{R}\dot{z}^{\alpha}\, V_{,\alpha} +2(\epsilon-A)\, \frac{\lambda^{\alpha} V_{, \alpha}}{R}+ [2\epsilon (-D+3)+2A(D-2)]\, \frac{V}{R^2}.$$ Please refer to [@gur1] for this Proposition. For the case of the Einstein-Born-Infeld field equations with similar assumptions, we have the following proposition (please see [@gur2] for the details of the Proposition) [Proposition 2]{}. Let $V$ and $H$ depend on $R$, $\tau_{0}$ and functions $c_{i} \, (i=1,2, \cdots)$ that satisfy (\[con1\]), then the Einstein equations reduce to the following set of equations $$\begin{aligned} \kappa p+\Lambda & = & V^{\prime \prime}+\frac{2}{R} V^{\prime}- \kappa b^2\, [1-\Gamma_{0}], \label{pres2}\\ \kappa \frac{(H^{\prime})^2}{\Gamma_{0}} & = & V^{\prime \prime}-\frac{2V}{R^2} , \label{denk2}\\ \kappa (p+\rho) & = & \sum_{i=1}\, \left[ V_{,c_{i}}\, (c_{i, \alpha}\,^{,\alpha})-\frac{4}{R} V_{,c_{i}}\,(c_{i,\alpha}\dot{z}^{\alpha}) \right. \nonumber \\ & & \left. -\frac{\kappa}{\Gamma_{0}}\, (H_{,c_{i}})^2 \,(c_{i,\alpha}\, c_{i}\,^{,\alpha}) \right ]-\frac{2A}{R} \left( V^{\prime}-\frac{2V}{R} \right) , \label{pres3}\\ \sum_{i=1}\, w_{i}\,c_{i, \alpha} & = & \left[\sum_{i=1}\, (w_{i}\, c_{i, \beta} \,\dot{z}^{\beta}) \right]\, \lambda_{\alpha}, \label{cler2}\end{aligned}$$ where $$w_{i}=V^{\prime}_{,c_{i}}- \frac{\kappa H^{\prime}}{\Gamma_{0}}\,H_{,c_{i}} , \;\;\; \Gamma_{0}=\sqrt{1-\frac{(H^{\prime})^2}{b^2}},$$ and prime over a letter denotes partial differentiation with respect to $R$. Here $\kappa (=8\pi)$ is the gravitational constant, $p$ and $\rho$ are, respectively the pressure and the energy density of the fluid, $b$ is the Born-Infeld parameter (as $b \rightarrow \infty$ one arrives at the Maxwell limit), and $\Lambda$ is the cosmological constant. \[sect2\] Null-Dust Einstein-Maxwell Solutions in $D$-Dimensions with a Cosmological Constant ============================================================================================= In this section we assume zero pressure. Due to the existence of the cosmological constant $\Lambda$, one may consider this as if there is a constant pressure as the source of the field equations. We shall not adopt this interpretation. Instead, we think of this as if there is a null dust, a Maxwell field and a cosmological constant as the source of the Einstein field equations. Hence assuming that the null fluid has no pressure in Proposition 1, we have the following result: [Proposition 3]{}. Let $p=0$. Then $$\begin{aligned} V&=& \left \{\begin{array}{ll} \frac{\kappa e^2\,(D-3)}{2(D-2)} R^{-2D+6}+m R^{-D+3}+\frac{\Lambda}{(D-2)(D-1)}\, R^2, & (\mbox{$ D \ge 4$})\\ -\frac{\kappa}{2} e^2 \, \ln{R+m} +\frac{\Lambda}{2}\,R^2, &(\mbox{$D=3$}) \end{array} \right. \label{e50}\\ H&=& \left \{\begin{array}{ll} c+\epsilon \,e\, R^{-D+3}, & (\mbox{$D \ge 4$})\\ c+\epsilon e\, \ln{R} . & (\mbox{$D=3$}) \end{array} \right. \label{e51}\end{aligned}$$ The explicit expressions of the energy density $\rho$ and the current vector $J_{\mu}$ do not contain the cosmological constant $\Lambda$ and are identical with the ones given in , so we don’t rewrite those long formulas here. Here $M=m+\epsilon\,\kappa (3-D) e c$ for $D \ge 4$ and $M=m+\frac{\kappa}{2}\,e^2 +\epsilon\, \kappa e c$ for $D=3$. In all cases $e$ is assumed to be a function of $\tau_{0}$ only but the functions $m$ and $c$ which are related through the arbitrary function $M(\tau_{0})$ (depends on $\tau_{0}$ only) do depend on the scalars $a_{k} \, (k \ge 0)$. Of course $\Lambda$ is any real number. In Proposition 3 we have chosen the integration constants ($R$ independent functions) as the functions $c_{i}$ ($i=1,2,3$) so that $c_{1}=m$, $c_{2}=e$, $c_{3}=c$ and $$c=c(\tau_{0}, a_{k}), \;\; e=e(\tau_{0}),$$ $$\begin{aligned} m=\left \{ \begin{array}{ll} M(\tau_{0})+\epsilon\,\kappa (D-3) e c , & (\mbox{$D \ge 4$}) \label{md41}\\ M(\tau_{0})-\frac{\kappa}{2}\,e^2-\epsilon\, \kappa e c , & (\mbox{$D=3$}) \label{md31} \end{array} \right.\end{aligned}$$ where $a_{k}$’s are defined in (\[aks\]). [Remark 1]{}. We can have pure null dust solutions when $e=0$. The function $c$ in this case can be gauged away, that is we can take $c=0$. The Ricci tensor takes its simplest form $R_{\mu \nu}=\rho \lambda_{\mu}\, \lambda_{\nu}+ \Lambda\, g_{\mu \nu}$ then. In this case we have $$\begin{aligned} V & = & m R^{3-D}+\frac{\Lambda}{(D-2)(D-1)}\,R^2 , \nonumber \\ \rho & = & \frac{2-D}{\kappa}\, [a (1-D)m+\dot{m}]\, R^{2-D}\end{aligned}$$ for $D \ge 4$ and $$\begin{aligned} V & = & m+ \frac{\Lambda}{2} \, R^2, \nonumber \\ \rho & = & \frac{2ma -\dot{m}}{\kappa R}\end{aligned}$$ for $D=3$. Such solutions are usually called as the [Photon Rocket]{} solutions [@kin], [@bon]. We give here the $D$ dimensional generalizations of this type of metrics with a cosmological constant. [Remark 2]{}. If $e=m=0$ we obtain a metric $$g_{\mu \nu} = \eta_{\mu \nu}- \frac{2\Lambda \, R^2}{(D-1)(D-2)}\, \, \lambda_{\mu}\, \lambda_{\nu}, \;\;\, (D \ge 3)$$ which clearly corresponds to the $D$-dimensional de Sitter space. [Remark 3]{}. The static limit $a_{0} \equiv a=0$ of our solutions with a constant $c$ are the charged Tangherlini solutions with a cosmological constant. If the function $c$ is not chosen to be a constant, we obtain their generalizations (see Section \[sect4\]). \[sect3\] Null-Dust Einstein-Born-Infeld Solutions in 4-Dimensions with a Cosmological Constant =============================================================================================== Using Proposition 2, and assuming zero pressure, we find the complete solution of the field equations. [Proposition 4]{}. Let $p=0$. Then $$\begin{aligned} V & = & \frac{m}{R}-4 \pi e^2 \frac{F(R)}{R}+\frac{\Lambda}{6}\, R^2 , \\ H & = & c-\epsilon\,e \int^{R}\, \frac{dR}{\sqrt{R^4+e^2/b^2}} ,\end{aligned}$$ where $$\begin{aligned} m & = & M(\tau_{0})+ 8\pi \epsilon e c, \label{mbi41}\\ F(R) & = & \int^{R}\, \frac{dR}{R^2+\sqrt{R^4+e^2/b^2}} .\end{aligned}$$ Here $e$ is assumed to be a function of $\tau_{0}$ only but the functions $m$ and $c$ which are related through the arbitrary function $M(\tau_{0})$ do depend on the scalars $a_{k}$, ($k \ge 0$). We have chosen the integration constants ($R$ independent functions) as the functions $c_{i}$ ($i=1,2,3$) so that $c_{1}=m$, $c_{2}=e$, $c_{3}=c$ and $$c=c(\tau_{0},a_{k}), \;\; e=e(\tau_{0}), \;\; m=M(\tau_{0})+8 \pi \epsilon e c .$$ [Remark 4]{}. In the static limit with a constant $c$, we obtain the Pleba[' n]{}ski solution with a cosmological constant [@fern]. If the function $c$ is not a constant, we can also give a class of solutions of the Einstein-Born-Infeld-null dust equations with a cosmological constant (see Section \[sect4\]). \[sect4\] Straight Line Limits ============================== When the accelerations $a_{k} \, (k \ge 0)$ vanish, the curve $C$ is a straight line in $M_{D}$. In this limit we have the following: $\tau_{0}=t-r$, $z^{\mu}=n^{\mu}\, \tau_{0}$, $n^{\mu} \equiv (1,0,0,0)$, $\dot{z}^{\mu}=n^{\mu}$ and $R=-r$. Moreover, $$x^{\mu}=(t, \vec{x}), \;\; \lambda_{\mu}=(1, -\frac{\vec{x}}{r}), \;\; r^2=\vec{x} \cdot \vec{x}.$$ In this case the function $c$ arising in the metrics introduced in the previous sections can be assumed to depend on some other functions $\xi_{(I)}$ so that $\lambda^{\mu}\, \xi_{(I), \mu}=0$ ($I=1,2, \cdots , D-2$) [@mar]. As an example let $\xi_{(I)}=\vec{l}_{(I)} \cdot \frac{\vec{x}}{r}$, where $\vec{l}_{(I)}$ are constant vectors. It is easy to show that $\lambda^{\mu}\, \xi_{(I),\, \mu}=0$. Hence in this (straight line) limit we assume that $c=c(\tau_{0}, \xi_{(I)})$. From this simple example we may define more general functions satisfying our constraint equation $c_{,\mu}\, \lambda^{\mu}=0$. Let $X_{\mu}$ be a vector satisfying $$X_{\mu, \,\nu}=b_{0}\, \eta_{\mu \nu}+b_{1}(k_{\mu}\, \lambda_{\nu}+ k_{\nu}\, \lambda_{\mu})+ b_{2}\, \lambda_{\mu}\, \lambda_{\nu} + Q_{\mu \nu}, \label{xi}$$ where $b_{0}, b_{1}, b_{2}$ are some arbitrary functions, $k_{\mu}$ is any vector and $Q_{\mu \nu}$ is any antisymmetric tensor in $M_{D}$. Then any vector $X_{\mu}$ satisfying (\[xi\]) defines a scalar $\xi=\lambda^{\mu}\, X_{\mu}$ so that $\lambda^{\mu}\, \xi_{,\, \mu}=0$. The simple example given at the beginning of this section corresponds to a constant vector, $X_{\mu}=l_{\mu}=(l_{0}, \vec{l})$. Hence, $\xi$ becomes a function of the spherical angles. For instance, in four dimensions, $\xi= l_{0}+l_{1}\, \cos{\phi} \, \sin{\theta}+l_{2}\, \sin{\phi} \, \sin{\theta}+ l_{3}\, \cos{\theta}$ where $l_{0}, l_{1}, l_{2}$, and $l_{3}$ are the constant components of the vector $l_{\mu}$. In the straight line limit, the metric can be transformed easily to the form $$ds^2 = -(1+2V)\, dT^2+\frac{1}{1+2V}\, dr^2+r^2\, d \Omega_{D-2}^2,$$ where $$dT=dt- \frac{2V\, dr}{1+2V},$$ and $d\Omega_{D-2}^2$ is the metric on the $D-2$-dimensional unit sphere. The above form of the metric is valid both for the Einstein-Maxwell and for the Einstein-Born-Infeld theories. For the case of the Einstein-Maxwell-null dust with a cosmological constant we have $$V= \left\{ \begin{array}{ll}\frac{\kappa e^2\,(D-3)}{2(D-2)}\, r^{-2D+6}+m (-1)^{D+1}\,r^{-D+3}+ \frac{\Lambda}{(D-2)(D-1)}\, r^2, & \mbox{($ D \ge 4$)}\\ m-\frac{\kappa}{2} e^2 \, \ln{r}+\frac{\Lambda}{2}\,r^2,& \mbox{($D=3$)} \end{array} \right.$$ and the function $H$ defining the electromagnetic vector potential is given by $$H=\left\{ \begin{array}{ll} c+\epsilon \,e\,(-1)^{D+1}\, r^{-D+3},& ( \mbox{$D \ge 4$})\\ c+\epsilon e \, \ln{r} . &(\mbox{$D=3$}) \end{array} \right.$$ This solution is a generalization of the Tangherlini solution [@tan]. The relationship between $c$ and $m$ are given in (\[md41\]), but in this case the function $c$ is a function of the scalars $\xi_{(I)}$ as discussed in the first part of this section. For the case of the Einstein-Born-Infeld-null dust with a cosmological constant, we have $$\begin{aligned} V & = & -\frac{m}{r}+4\pi e^2 \frac{F(r)}{r}+\frac{\Lambda}{6}\, r^2 , \\ H & = & c+\epsilon\,e \int^{r}\, \frac{dr}{\sqrt{r^4+e^2/b^2}}\end{aligned}$$ where $$\begin{aligned} m & = & M(\tau_{0})+8\pi \epsilon e c, \label{mbi42}\\ F(r) & = & -\int^{r}\, \frac{dr}{r^2+\sqrt{r^4+e^2/b^2}}.\end{aligned}$$ This solution is a generalization of the Pleba[' n]{}ski et al [@pleb] static solution of the Einstein-Born-Infeld theory. Our generalization is with the function $c$ depending arbitrarily on the scalars $\xi_{(I)}$. [Remark 5]{}. When the function $c$ is not a constant, the mass $m$ defined through the relations (\[md41\]) or (\[mbi42\]) is not a constant anymore, it depends on the angular coordinates. \[concl\] Conclusion ==================== We have reexamined the accelerated Kerr-Schild geometries for two purposes. One of them is to generalize our earlier solutions of Einstein-Maxwell-null dust [@gur1] and Einstein-Born-Infeld-null dust field equations [@gur2] by including a cosmological constant. The other one is to generalize the static limit (straight line limit) of the above mentioned solutions. Previously in the static limit, we were assuming the function $c$ to be a constant. As long as this function satisfies the condition $\lambda^{\mu}\, c_{,\mu}=0$, as we have seen in Section \[sect4\] (although the acceleration scalars $a_{k} \,(k \ge 0)$ are all zero) we can still obtain the generalization of the charged Tangherlini [@tan] and Pleba[' n]{}ski [@pleb] solutions. We thank Marc Mars for useful suggestions. This work is partially supported by the Scientific and Technical Research Council of Turkey and by the Turkish Academy of Sciences. [99]{} M. G[" u]{}rses and [" O]{}. Sar[i]{}oğlu, [Class. Quantum Grav.]{}, [19]{}, 4249 (2002); Erratum, [ibid]{}, [20]{}, 1413 (2003). M. G[" u]{}rses and [" O]{}. Sar[i]{}oğlu, [Class. Quantum Grav.]{}, [20]{}, 351 (2003). . Sar[i]{}oğlu, [Phys. Rev.]{}, [D 66]{}, 085005 (2002). M. G[" u]{}rses and [" O]{}. Sar[i]{}oğlu, [Li[' e]{}nard-Wiechert potentials in even dimensions]{}, ([hep-th/0303078]{}), accepted for publication in [J. Math. Phys.]{} F. R. Tangherlini, [Nuovo Cimento]{}, [77]{}, 636 (1963). A. Garc[' i]{}a, I. H. Salazar and J. F. Pleba[' n]{}ski, [ Nuovo Cimento]{}, [B 84]{}, 65 (1984). A. Trautman, [Phys. Rev. Lett.]{}, [46]{}, 875 (1981). W. Kinnersley, [Phys. Rev.]{}, [186]{}, 1353 (1969). W. B. Bonnor, [Class. Quantum Grav.]{}, [11]{}, 2007 (1994). S. Fernando, D. Krug, [Gen. Rel. Grav.]{}, [35]{}, 129 (2003). We would like to thank Marc Mars for drawing our attention to such a generalization.
--- abstract: 'We introduce the geodesic flow on the leaves of a holomorphic foliation with leaves of dimension 1 and hyperbolic, corresponding to the unique complete metric of curvature -1 compatible with its conformal structure. We do these for the foliations associated to Riccati equations, which are the projectivisation of the solutions of a linear ordinary differential equations over a finite Riemann surface of hyperbolic type $S$, and may be described by a representation $\rho:\pi_1(S) \rightarrow GL(n,\CC)$. We give conditions under which the foliated geodesic flow has a generic repellor-attractor statistical dynamics. That is, there are measures $\mu^-$ and $\mu^+$ such that for almost any initial condition with respect to the Lebesgue measure class the statistical average of the foliated geodesic flow converges for negative time to $\mu^-$ and for positive time to $\mu^+$ (i.e. $\mu^+$ is the unique SRB-measure and its basin has total Lebesgue measure). These measures are ergodic with respect to the foliated geodesic flow. These measures are also invariant under a foliated horocycle flow and they project to a harmonic measure for the Riccati foliation, which plays the role of an attractor for the statistical behaviour of the leaves of the foliation.' author: - 'Ch. Bonatti, X. Gómez-Mont and R. Vila-Freyer' title: 'Statistical Behaviour of the Leaves of Riccati Foliations [^1]' --- Introduction {#introduction .unnumbered} ============ The objective of this work is to propose a method for understanding the statistical properties of the leaves of a holomorphic foliation, and which we carry out for a simple class of holomorphic foliations: those obtained from the solutions of Riccati Equations. The method consists in using the canonical metric of curvature -1 that the leaves have as Riemann surfaces, the Poincaré metric, and then to flow along foliated geodesics. One is interested in understanding the statistics of this foliated geodesic flow. In particular, in determining if the foliated geodesic flow has an SRB-measure (for Sinaï , Ruelle and Bowen \[21\], \[20\], \[5\]), or physical measure, which means that a set of geodesics of positive Lebesgue measure have a convergent time statistics, which is shared by all the geodesics in this set, called the basin of attraction of the SRB-measure. The SRB-measure is the spatial measure describing this common time statistics of a significant set of geodesics. One then shows that the SRB-measure is invariant also under a foliated horocycle flow (\[2\]) and the projection of the SRB-measure to the $\CC P^{n-1}$-bundle over $S$ is a harmonic measure for the Riccati foliation; in fact, the harmonic measures are in 1-1 correspondance with the measures simultaneously invariant by the foliated geodesic and a horocycle flow (\[1\], \[17\], \[18\]\]). The approach of using harmonic measures to understand the statistical behaviour of the leaves of a foliation started with the work of Garnett(\[11\]) who proved existence of harmonic measures for regular foliations in compact manifolds, containing statistical properties of the behaviour of the leaves of the foliation. In this work we are dealing with singular foliations in compact manifolds (obtained by compactifying the Riccati foliation with a linear model with singularities over each puncture of $S$), which introduces the difficulty that the support of the measures could be contained in the singular set. Our conclusions are related to Fornaess and Sibony’s harmonic currents in $\CC P^2$ (\[8\], \[9\] and \[10\]), where they show existence and uniqueness of harmonic currents using $\bar \partial$-methods for the generic foliations in $\CC P^2$. Their result does not include Riccati foliations in $\CC P^2$, since these have some tangent lines (corresponding to the punctures of $S$) and a non-hyperbolic singular point (arising from the blow down to $\CC P^2$). Our work is also related to Deroin and Kleptsyn \[7\], where they use foliated Brownian motion and heat flow instead of the foliated geodesic and horocycle flows for non-singular transversely holomorphic foliations in compact manifolds to obtain a finite number of attracting harmonic measures and the negativity of the Lyapunov exponent. The Riccati equations are projectivisations of linear ordinary differential equations over a finite hyperbolic Riemann surface $S$ (i.e. compact minus a finite number of points and with universal cover the upper half plane). Locally they have the form $$\frac{dw}{ dz} = A(z)w\hskip 5mm,\hskip5mm w\in \CC^n \ , \ z \in \CC \hskip 5mm, \hskip 5mm A:\CC \to Mat_{n,n}(\CC)$$ with $A$ holomorphic. These equations may be equivalently defined by giving the monodromy representations $$\tilde \rho : \pi_1(S,z_0) \rightarrow GL(n,\CC) \hskip 1cm , \hskip 1cm \rho : \pi_1(S,z_0) \rightarrow PGL(n,\CC) \eqno(1)$$ and suspending them, to obtain flat $\CC^n$ and $\CC P^{n-1}$ bundles over $S$ $$E_{\tilde \rho} \rightarrow S, \hskip 1cm,\hskip1cm M_{ \rho} \rightarrow S. \eqno (2)$$ The graphs of the local flat sections of these bundles are the ‘solutions’ to the linear differential equation defined by the monodromy $(1)$ and define holomorphic foliations $ \cF_{\tilde \rho}$ and $\cF_{ \rho}$ of $ E_{\tilde \rho} $ and $ M_{ \rho}$ whose leaves $\cL$ project as a covering to the base surface $S$. Introduce to the finite hyperbolic Riemann surface $S$ the Poincaré metric, to the unit tangent bundle $q:T^1S\rightarrow S$ the geodesic flow $\varphi:T^1S\times \RR \rightarrow T^1S$ and the Liouville measure $dLiouv$ (hyperbolic area element in $S$ and Haar measure on $T^1_pS$, normalised to volume 1). We may introduce on the leaves $\cL$ of the foliations $\cF_{\tilde \rho}$ and $\cF_\rho$ the Poincaré metric, which is the pull back of the Poincaré metric of $S$ by the covering map $q:\cL \rightarrow S$. The unit tangent bundle $T^1_{{\cF}_{\tilde\rho}}$ to the foliation $\cF_{\tilde\rho}$ in $E_{\tilde \rho}$ is canonically isomorphic to the vector bundle $q^* E_{\tilde\rho}$ over $ T^1S$, that we denote by $ E $. In the same way the unit tangent bundle $T^1_{\cF_\rho}$ of the foliation $\cF_\rho$ is canonically identified to the projectivisation $Proj(E)$ of the vector bundle $E$ over $T^1S$. Introduce on $E$ and on $Proj(E)$ the foliated geodesic flows $\tilde \Phi$ and $\Phi$ (see $(2.2)$), obtained by flowing along the foliated geodesics. Introduce also on $E$ a continuous Hermitian inner product $\|\;.\;\|_v$. Given a vector $v \in T^1S$ we have the geodesic $$\RR \rightarrow T^1S \hskip 1cm,\hskip 1cm t \rightarrow \varphi(v,t)$$ determined by the initial condition $v$ and given $w_0 \in E_v$ we also have the foliated geodesic $$\RR \rightarrow E \hskip 1cm,\hskip 1cm t \rightarrow \tilde \Phi(w_0,t)$$ which is the solution to the linear differential equation defined by $(1)$ along the foliated geodesic determined by $v$ and $w_0$. The function $$t \rightarrow \| \tilde \Phi (w_0,t) \|_{\varphi(v,t)}$$ describes the type of growth of the solution of $(1)$ along the geodesic $\gamma_v$ with initial condition $w_0 \in E_v$ and the function $$t \rightarrow \frac{\| \tilde \Phi (w_1,t) \|_{\varphi(v,t)}}{ \| \tilde \Phi (w_2,t) \|_{\varphi(v,t)}}$$ describes the relative growth of the solution of $(1)$ along the geodesic $\gamma_v$ with initial condition $w_1 \in E_v$ with respect to the growth of the solution of $(1)$ along $\gamma_v$ with the initial condition $w_2 \in E_v$. We say that the Riccati equation has a [section of largest expansion]{} $\sigma^+$ if for Liouville almost any point $v$ on $T^1S$ we may measurably define a splitting $E_v = F_v \oplus G_v$ by linear spaces, which is invariant by the foliated geodesic flow $\tilde \Phi$ with $F_v$ of dimension 1 and with the property that the map $ t \rightarrow \tilde \Phi(w_1,t)$ with initial condition $w_1 \in F_v$ grows more rapidly than the maps $ t \rightarrow \tilde \Phi(w_2,t)$ for any $w_2 \in G_v$. That is, for almost any $v\in T^1S$, for any compact set $K\subset T^1S$ and for any sequence $(t_n)_{n\in\NN}$ of times such that $\varphi(v,t_n)\in K$ and $\lim_{n\to\infty}t_n=+\infty$, one has: $$\lim_{n\to \infty} \frac{ \| \tilde \Phi(w_1,t_n)\|_{\varphi(v,t_n)}}{ \| \tilde \Phi(w_2,t_n)\|_{\varphi(v,t_n)}} = \infty, \quad\mbox{for all non-zero}\quad w_1\in F_v, \quad\mbox{and}\quad w_2\in G_v.$$ So the section of largest expansion is defined as $\sigma^+:= Proj(F):T^1S \rightarrow Proj(E)$. Similarly, we may define a [section $\sigma^-$ of largest contraction]{} (see $(3.1)$). An elementary argument of Linear Algebra suggests that a section $\sigma^+=Proj(F)$ of largest expansion is attracting all the points in $Proj(E) -Proj(G)$ as they flow according to the action of the foliated geodesic flow $ \Phi$. In fact, we prove: Let $S$ be a finite hyperbolic Riemann surface and $\tilde \rho:\pi_1(S,z_0) \rightarrow GL(n,\CC)$ a representation having a section $\sigma^+$ of largest expansion, then $\mu^+=\sigma^+_(dLiouv)$ is a $\Phi-$invariant ergodic measure on $T^1\cF_\rho$ which is an SRB-measure for the foliated geodesic flow $\Phi$ of the Riccati equation, whose basin has total Lebesgue measure in $T^1\cF_\rho$. Similarly, if $\sigma^-$ is the section of largest contraction, then $\mu^-=\sigma^-_(dLiouv)$ is a $\Phi-$invariant ergodic measure which is an SRB-measure whose basin has total Lebesgue measure in $T^1\cF_\rho$, for negative times. \[theo1\] In the case that both $\sigma^\pm$ exist, the foliated geodesic flow has a very simple ‘north to south pole dynamics’: almost everybody is born in $\mu^-$ and is dying on $\mu^+$. If the sections $\sigma^\pm$ are continuous disjoint sections defined on all $T^1S$ then it is easy to imagine this north to south pole dynamics (see section 7 for an example). If $\sigma^\pm$ are only measurable, then they describe more subtle phenomena. The Lyapunov exponents measure the exponential rate of growth (for the metric $\|\;.\;\|_v$ in the vectorial fibers) of the solutions of the linear equation along the geodesics (definition 4.2): $$\lim_{t \rightarrow \pm \infty}\frac{ 1 }{ t} \log \|\tilde \Phi(w_0,t)\|_{\varphi(v,t)}.$$ Let $S$ be a finite hyperbolic Riemann surface, $\tilde\rho\colon\pi_1(S,z_0)\to GL(n,\CC)$ a representation and $E$ the previously constructed bundle. The association of initial conditions to final conditions for the linear equation in $E$ over the geodesic flow of $S$, after a measurable trivialisation of the bundle, gives rise to a measurable multiplicative cocycle over the geodesic flow on $T^1S$ $$\tilde A:T^1S \times \RR \longrightarrow GL(n,\CC)$$ (see $(2.4))$. The integrability condition $$\int_{T^1{S}}log^+\\| \tilde A_{\pm 1} \\| dLiouv < +\infty, \eqno(3)$$ where $\\|\ \\|$ is the operator norm and $\tilde A_t :=\tilde A(\cdot,t)$, asserts that the amount of expansion of $\tilde A_\pm 1$ is Liouville integrable. As a consequence of the multiplicative Ergodic Theorem of Oseledec applied to the foliated geodesic flow we obtain: Let $S$ be a finite hyperbolic Riemann surface, $\tilde\rho\colon\pi_1(S,z_0)\to GL(n,\CC)$ a representation and let $\tilde A$ be the measurable multiplicative cocycle over the geodesic flow on $T^1S$ satisfying the integrability condition $(3)$, then: - The Lyapunov exponents $\lambda_1<\cdots<\lambda_k$ of $\tilde \Phi$ are well defined and are constant on a subset of $T^1S$ of total Liouville measure. Denote by $F_i(v)$ the corresponding Lyapunov spaces. - For every $i\in\{1,\dots,k\}$, $\lambda_{k+1-i}=-\lambda_i$ and $dim(F_{k+1-i})=dim(F_i)$. - If $dim F_k=1$, denote by $\sigma^+$ the section corresponding to $F_k$ and $\sigma^-$ the section correponding to $F_1$, then $\sigma^\pm$ are sections of largest expansion and contraction, respectively. From now on by the [Lyapunov exponents of the linear equation obtained from the representation $\tilde \rho$]{} we will understand the Lyapunov exponents of the above multiplicative cocycle $\tilde A$ over the geodesic flow on $T^1S$ obtained from the foliated geodesic flow on $E$ and satisfying the integrability condition $(3)$. The relationship between the section of largest expansion and the Lyapunov exponents is: Let $S$ be a finite hyperbolic Riemann surface, $\tilde\rho\colon\pi_1(S,z_0)\to GL(n,\CC)$ a representation satisfying the integrability condition $(3)$, then there exists a section of largest expansion if and only if the largest Lyapunov exponent is positive and simple, if and only if the smallest Lyapunov exponent is negative and simple, and if and only if there is a section of largest contraction. \[theo2\] So a section of largest expansion is an extension for non-integrable cocycles $\tilde A$ of the notion of having a simple largest Lyapunov exponent. We give an example of this in section 6. In order to apply Oseledec’s Theorem, the prevailing hypothesis is the integrability condition $(3)$. This condition is always satisfied if the base Riemann surface is compact, and more generally: If $S$ is a finite hyperbolic Riemann surface, $\tilde \rho$ a representation $(1)$ then the multiplicative cocycle $\tilde A$ satisfies the integrability condition $(3)$ if and only if the monodromy $\tilde \rho$ around each of the punctures of $S$ corresponds to a matrix with all its eigenvalues of norm 1. \[t.6\] We then develop two kinds of examples: The ping-pong or Schottky monodromy representations in $SL(2,\CC)$ and the canonical representation obtained from the representation $$\rho_{can}:\pi_1(S,z_0) \rightarrow SL(2,\RR) \subset SL(2,\CC) \eqno (4)$$ on the universal covering of the surface. We obtain: Let $S$ be a finite hyperbolic non-compact Riemann surface and $\rho\colon\pi_1(S ,z_0)\to GL(2,\CC)$ an injective representation onto a Schottky group, then there are sections $s^+$ and $s^-$ of largest expansion and contraction defined and continuous on a subset of $T^1S$ of full Liouville measure. \[t.Schottky\] It follows from Theorem 4 that the Schottky representations in Theorem 5 do not satisfy the integrability condition $(3)$, but we obtain that there are still sections of largest expansion and contraction. We think that the Lyapunov exponents are in this case $\pm \infty$. In fact, we can prove this assertion for specific Schottky representations. For any finite hyperbolic Riemann surface $S$ the foliated geodesic flow associated to the canonical representation $(4)$ admits sections of largest expansion and contraction defined and smooth on all $T^1S$. Moreover, for Lebesgue almost any point of $Proj(E)$ the foliated geodesic starting at this point has $\mu^+$ as its positive statistics and $\mu^-$ as its negative statistics (that is, $\mu^+$ is the unique SRB-measure and its basin has total Lebesgue measure, and similarly $\mu^-$ for negative time). If $S$ is compact then $\sigma^+(T^1S)$ is a hyperbolic attractor and $\sigma^-(T^1S)$ is a hyperbolic repellor with basins of attraction $T^1\cF - \sigma^\mp(T^1S)$. \[t.tautologic\] The statements and arguments presented here extend to the case when the representation $\rho:\pi_1(S) \rightarrow PGL(n,\CC)$ does not admit a lifting to a representation in $GL(n,\CC)$. Restricting now to $n=2$ or $3$, assuming the integrability condition $(3)$ and that the representation $\rho$ does not leave invariant any probability measure (which is a generic condition on $\rho$), it follows from Theorem 3 in \[2\] that the SRB-measure of the geodesic flow $\mu^+$ is the unique measure invariant under the foliated stable horocycle flow $H^{uu}_\rho$ that projects to the Liouville measure on $S$. Furthermore, it follows from the arguments in \[1\] and \[17\] that the projection to $M_\rho$ of $\mu^+$ is the unique harmonic measure $\nu$ of the Riccati foliation ${\cF}_{\rho}$ that projects to the Liouville measure on $S$. It is shown in \[2\] that $\nu$ describes effectively the statistical behaviour of the leaves of the foliation $\cF_\rho$: For any compact set $K \subset M_\rho$, for any sequence $(x_n \in K)_{n\in \NN}$ and any sequence of real numbers $(r_n)_{n\in \NN}$ tending to $+\infty$ the family of probability measures $\nu_{r_n}(x_n)$ obtained by normalizing the area element on the disk $D_{r_n}(x_n)$ in the leafwise Poincaré metric converges towards $\nu$ for the weak topology when $n$ tends to $+\infty$. If $S$ is compact, then the integrability condition $(3)$ is always satisfied and the condition of projecting to the Liouville measure on $S$ is satisfied automatically by Hedlund’s Theorem \[13\]. If $S$ is a compact hyperbolic Riemann surface, then the foliated geodesic flow is a linear or projective multiplicative cocycle over a hyperbolic dynamical system. This led us to think that it could be possible to adapt Fustenberg’s theory of the existence of a positive Lyapunov exponent for random products of matrices. This has been carried out in \[3\]. It seems possible that using a generalization of \[3\] found in \[4\] (simplicity of the Lyapunov spectrum) and \[22\] (generalization for linear cocycles over non-uniform hyperbolic measures), one may extend the above mentioned results for $n\geq 4$ and $S$ a finite hyperbolic Riemann surface. This paper is organised as follows. In section 1 we recall the Riccati equations and in section 2 we set up the foliated geodesic flow on Riccati equations. In section 3 we introduce SRB-measures and prove Theorem 1. In section 4 we prove Corollary 2 and Theorem 3. In section 5 we prove Theorem 4. In sections 6 and 7 we describe the examples, proving Theorems 5 and 6. The Riccati Equation\[s.Riccati\] ================================= Linear Ordinary Differential Equations\[ss.edol\] ------------------------------------------------- [ The classical linear ordinary differential equation]{} is $$\frac{dw}{ dz} = A(z)w \hskip 1cm, \hskip 1cm z\in \CC, \ w\in \CC^n \eqno (1.1)$$ where $A(z)$ is a matrix of rational functions (see \[6\]). The fundamental property of this equation is that locally in $z$ we can find a basis of independent solutions of (1.1) which accept analytic continuation to the universal covering space of $S := \bar\CC - \hbox{poles}(A)$ as holomorphic vector valued functions $w$ satisfying the monodromy relation: $$w(T_\gamma(z)) = \tilde \rho (\gamma)( w(z)) \hskip 1cm ,\hskip 1cm \gamma \in \pi_1(S,z_0)$$ where $T_\gamma$ is the covering transformation corresponding to the close loop $\gamma$ and $$\tilde \rho :\pi_1(S,z_0) \rightarrow GL(n,\CC) \eqno (1.2)$$ is the monodromy representation of the equation. The linear automorphism $\tilde \rho (\gamma):\CC^n \rightarrow \CC^n$ contains the information of how the initial conditions are transformed to final conditions by solving $(1.1)$ along the closed loop $\gamma$ based at $z_0$. Another classical construction of linear ordinary differential equations is the suspension (\[16\]). Assume given a hyperbolic Riemann surface $S$ and a representation $(1.2)$. We construct from these data a vector bundle $E_{\tilde\rho}$ over $S$ and an equation of type $(1.1)$. Let $\HH^+$ be the upper half plane, considered as the universal covering space of $S$, with covering transformations $(4)$ giving rise to the canonical representation $\tilde \rho _{can}$ of the fundamental group of $S$. Consider the trivial bundle $\tilde E :={\HH^+\times \CC^n}$ on the upper half plane $\HH^+$ and the $\pi_1(S,{z_0})$-action on $\tilde E$ $$(z,w) \rightarrow (\tilde \rho _{can}(\gamma) z,\tilde \rho (\gamma) w) \hskip 1cm,\hskip 1cm \gamma \in \pi_1(S,{z_0}). \eqno (1.3)$$ The quotient of $\tilde E$ by this action gives rise to a vector bundle $E_{\tilde\rho}$ over $S$. On $\tilde E$ we can consider the equation given by $\tilde A=0$ (i.e. $\frac {dw}{ dz} =0$). Its solutions are the constant functions. Since this equation $\tilde A$ is invariant under the action in (1.3), it descends to a linear ordinary differential equation on $E_{\tilde\rho}$ which is holomorphic over $S$. The construction gives directly that the monodromy transformation of this equation is the given representation $\tilde \rho $. The graphs of the local solutions to $(1.1)$ form a holomorphic foliation $\cF_{\tilde \rho}$ in $E_{\tilde \rho}$. The Riccati Equation\[ss.Riccati\] ----------------------------------- Riccati equations may be obtained from a linear ordinary differential equation as $(1.1)$ or $(1.2)$ by projectivising the linear variables of the vector bundle $E_{\tilde \rho}$ over the Riemann surface $S$. Denoting $\zeta_j:=\frac{w_j}{ w_1}$ with $j=2,\ldots,n$, the Riccati equation associated to (1.1) in affine coordinates takes the form of a quadratic polynomial in $\zeta_2,\ldots,\zeta_n$ with rational coefficients in $z$: [$$\left(\begin{array}{c} \frac{d\zeta_2}{ dz} \\ \cdots \\ \frac{d\zeta_n}{ dz}\\ \end{array}\right) = \left(\begin{array}{c} a_{21}\\ \cdots\\ a_{n1}\\ \end{array}\right) + \left(\begin{array}{ccc} a_{22} - a_{11} & a_{23} & \cdots\\ a_{32} & a_{33}-a_{11} & \cdots \\ \cdots & \cdots & a_{nn}-a_{11}\\ \end{array}\right) \left(\begin{array}{c} \zeta_2 \\ \cdots \\ \zeta_n \\ \end{array}\right) - (a_{12}\zeta_2 +\cdots + a_{1n}\zeta_n) \left(\begin{array}{c} \zeta_2 \\ \ldots \\ \zeta_n\\ \end{array}\right) \eqno (1.4)$$]{}where $A=(a_{ij}(z))$ is the matrix of rational functions in (1.1). Similarly, given a representation $\tilde\rho$ as in $(1.2)$ we may also construct from the projectivised representation $\rho$ in $(1)$ its suspension $M_\rho=Proj(E_{\tilde\rho})$ which gives a manifold which is a $\CC P^{n-1}$ bundle over $S$ with a flat connection. The set of flat sections form a foliation $\cF_\rho$ of $M_\rho$ which is the projectivisation of the foliation $\cF_{\tilde\rho}$ in $E_{\tilde \rho}$. The foliations so constructed, will be called [Riccati foliations]{}. The Foliated Geodesic Flow on Linear and Riccati Equations\[s.geodesic\] ======================================================================== The Geodesic Flow on Finite Hyperbolic Riemann Surfaces\[ss.geosurface\] ------------------------------------------------------------------------- We say that $S$ is a finite hyperbolic Riemann surface if $S$ is conformally equivalent to $\bar S - \{p_1,\ldots,p_r\}$, where $\bar S$ is a compact Riemann surface of genus $g$ and $g>1$ or $g=1$ with $r\geq1$ or $g=0$ with $r\geq3$. In such a case $S$ has as a universal covering space the Poincaré upper half plane $\HH^+$, with its complete metric of curvature -1 given by $ds =\frac{ \vert dz\vert}{ y}$. We introduce on $S$ the hyperbolic metric induced by the Poincaré metric via the universal covering map. For the measure associated to the hyperbolic metric, the surface $S$ has finite area. Let $T^1S$ be the unit tangent bundle of $S$. The Liouville measure $dLiouv$ on $T^1S$ is the measure obtained from the hyperbolic area element in $S$ and Haar measure $d\theta$ on unit vectors, normalised so as to have volume 1. The geodesic flow $$\varphi:T^1S \times \RR \rightarrow T^1S \hskip 1cm \varphi_t := \varphi(\ ,t) \eqno (2.1)$$ is obtained by flowing along the geodesics (see \[14\] p. 209). The geodesic flow leaves invariant the Liouville measure. [Theorem 2.1 (Hopf-Birkhoff).]{} (\[14\] p. 217, 136) [Let $S$ be a finite hyperbolic Riemann surface, then the Liouville measure is ergodic with respect to the geodesic flow and the generic geodesic of $S$ is statistically distributed in $T^1S$ according to the Liouville measure: For all Liouville integrable functions $h$ on $T^1S$ and for almost any $v_z \in T^1S$ with respect to the Liouville measure]{} $$\lim_{t \rightarrow \infty}\frac {1}{ t} \int_0^th(\varphi(v_z,t))dt = \int_{T^1S}hdLiouv$$ The Foliated Geodesic Flows\[geoRiccati\] ----------------------------------------- Let $S$ be a finite hyperbolic Riemann surface, and $\tilde \rho$ and $\rho$ representation as in $(1)$ and let $\cF_{\tilde\rho}$ and $\cF_\rho$ be the foliations constructed in section 1. If $\cL$ is a leaf of the foliation $\cF_{\tilde\rho}$ or $\cF_\rho$, then the projection map $p:{\cL} \rightarrow S$ is a covering map, and hence the pull back of the Poincaré metric of $S$ induces a metric to the leaves of ${\cF}$, which coincides with the Poincaré metric of each leaf $\cL$ of ${\cF}$. This is the Poincaré metric of the foliations ${\cF}_{\tilde \rho}$ or $\cF_\rho$. Let $T^1{\cF}_{\tilde \rho}$ be the manifolds formed by those tangent vectors to $E_{\tilde\rho}$ and $M_\rho$ which are tangent to ${\cF}_{\tilde \rho} $ and $\cF_\rho$ and are of unit length with respect to the Poincaré metrics of the foliations. The derivative of the projection map $E_{\tilde \rho}, M_\rho \rightarrow S$ induces the commutative diagram $$\begin{array}{ccc} T^1 {\cF}_{\tilde\rho} & {\overset q \to } & E_{\tilde\rho} \\ \downarrow&& \downarrow \\ T^1 {S} & {\overset q \to } & S \\ \end{array} \hskip 2cm \begin{array}{ccc} T^1 {\cF}_\rho & \rightarrow & M_\rho \\ \downarrow&& \downarrow \\ T^1 {S} & \rightarrow & S \\ \end{array}$$ The foliated geodesic flows $\tilde \Phi$ and $\Phi$ are defined by following geodesics along the leaves and is compatible with the geodesic flow $\varphi$ on $S$, giving rise to the commutative diagram $$\begin{matrix} \tilde\Phi:& T^1{{\cF}_{\tilde\rho}}& \times& \RR& \rightarrow& T^1{{\cF}_{\tilde\rho}}\cr & \downarrow& & \downarrow& & \downarrow \cr \varphi:& T^1S& \times & \RR& \rightarrow& T^1S\cr \end{matrix} \hskip 10mm, \hskip 10mm \begin{matrix} \Phi:& T^1{{\cF}_{\rho}}& \times & \RR& \rightarrow& T^1{{\cF}_{\rho}}\cr & \downarrow& & \downarrow& & \downarrow \cr \varphi:& T^1S& \times& \RR& \rightarrow& T^1S\cr \end{matrix}. \eqno (2.2)$$ For any $v \in T^1S$ and $t \in \RR$, the flow $ \tilde\Phi_t:=\tilde\Phi(\ ,t)$ induces a linear isomorphism $$\tilde A(v,t):=\tilde \Phi(v,\ ,t)\|_{E_{\tilde \rho,v}}: E_{\tilde \rho,v} \rightarrow E_{\tilde\rho,\varphi(v,t)} \eqno (2.3)$$ between the $\CC^{n}$-fibres. After a measurable trivialisation of the bundles by choosing measurably an othonormal basis of the fibers, the foliated geodesic flows may be seen as measurable multiplicative cocycles over the geodesic flow on $T^1S$: $$\tilde A: T^1S \times \RR \rightarrow GL(n,\CC) \hskip 5mm,\hskip 5mm \tilde A(v,t_1+t_2) = \tilde A(\varphi(v,t_1),t_2) \tilde A(v,t_1) \hskip 5mm,\hskip 5mm t_1,t_2\in\RR. \eqno(2.4)$$ Moreover the usual operator norm in $GL(n,\CC)$ coincides with the operator norm of $(2.3)$ as Hermitian spaces with the metrics induced from the fibre bundle metric. SRB-measures for Riccati Equations ================================== SRB-measures\[ss.srbdefi\] -------------------------- Let $M$ be a differentiable manifold. The Lebesgue measure class is the set of measures whose restriction on any chart $U$ has a smooth strictly-positive Radom-Nikodyn derivative with respect to $dx_1\wedge dx_2\cdots\wedge dx_n$ where the $x_i$ are coordinates on $U$. A set $E\subset M$ has [zero Lebesgue measure]{} if there is a measure $\mu$ in the Lebesgue class such that $\mu(E)=0$. Let $X$ be a complete vector field on the manifold $M$, and denote by $\varphi_t$ its flow. A probability measure $\mu$ on $M$ is [invariant by $X$]{} if for any $t\in\RR$ one has $\varphi_{t}(\mu) = \mu.$ The [basin]{} $B(\mu)$ of an $X-$invariant probability $\mu$ is the set of points $x \in M$ such that the positive time average along its orbit of any continuous function $h \colon M\to\RR$ with compact support coincides with the integral of the function by $\mu$. In formula: $$\lim_{T\to+\infty}\frac1T\int_0^T h (\varphi_t(x))dt =\int_M h d\mu$$. An $X-$invariant probability measure in $M$ is an [SRB-measure]{} if its basin has non-zero Lebesgue measure in $M$. Key Idea to Build SRB-measures for Riccati Equations ---------------------------------------------------- Let $S$ be a finite hyperbolic Riemann surface and $\tilde \rho$ and $\rho$ representations as in $(1)$ and $\cF_{\tilde \rho}$ and $\cF_\rho$ the foliations in $E_{\tilde\rho}$ and $M_\rho$ constructed in section 2. Consider a continuous Hermitian metric $\|\cdot\|_x$ on the fiber $E_{\tilde\rho,x}$ of $E_{\tilde\rho}$ and for each point $x\in S$ we endow the corresponding Fubini-Study (Hermitian) metric $ \| \cdot \| _x$ on $M_{\rho,x}= Proj(E_{\tilde\rho,x})$. The bundles $ q^E_{\tilde \rho} \simeq T^1 \cF_{\tilde \rho} $ and $ q^M_\rho \simeq T^1 \cF_\rho $ over $T^1S$ are endowed in a natural way with the induced Hermitian or Fubini-Study metric, respectively. Under the above setting, assume that the vector bundle $E \colon= T^1\cF_{\tilde \rho} \to T^1S$ admits a measurable splitting $E_v = F_v\oplus G_v$ , defined for $v$ in a subset $\cA$ of $T^1S$, and verifying the following hypothesis: 1. $\cA$ has total Lebesgue measure in $T^1S$; 2. $\cA$ is invariant by the geodesic flow $\varphi$; 3. the splitting is invariant by the foliated geodesic flow $\tilde \Phi$: for every $t\in\RR$ and every $v\in \cA$, $$F_{\varphi(v,t)}= \tilde \Phi(F_v,t) \quad\mbox{and}\quad G_{\varphi(v,t)}=\tilde\Phi(G_v,t);$$ 4. $dim(F_v)=1;$ 5. for any $v\in\cA$, for any compact set $K\subset T^1S$ and for any sequence $(t_n)_{n\in\NN}$ of times such that $\varphi(v,t_n)\in K$ and $\lim_{n\to\infty}t_n=+\infty$, one has: $$\lim_{n\to \infty} \frac{ \|\tilde \Phi(w_1,t_n) \| _{\varphi(v,t_n)}}{ \| \tilde \Phi(w_2,t_n) \| _{\varphi(v,t_n)}} = \infty, \quad\mbox{for all non-zero}\quad w_1\in F_v, \quad\mbox{and}\quad w_2\in G_v.$$ Under the above hypothesis denote by $\sigma^+\colon\cA\subset T^1S\to T^1\cF_\rho$ the mesurable section defined by letting $\sigma^+(v)$ be the point of $Proj(E_v)$ corresponding to the line $F_v$. A section $\sigma^+$ verifying the above hypothesis is called a [section of largest expansion]{}. Similarly, one defines the [section $\sigma^-$ of largest contraction]{} by requiring $$\lim_{n\to \infty} \frac{ \|\tilde \Phi(w_1,t_n) \| _{\varphi(v,t_n)}}{ \| \tilde \Phi(w_2,t_n) \| _{\varphi(v,t_n)}} = \infty, \quad\mbox{for all non-zero}\quad w_1\in F_v, \quad\mbox{and}\quad w_2\in G_v. \eqno (3.1)$$ with $\lim_{n\to\infty}t_n=-\infty$ where we are imposing the condition that the measurable sub-bundle $F$ is 1 dimensional (i.e. greatest expansion for negative times). [Proof of Theorem 1:]{} $\sigma^+$ induces an isomorphism of the measure $dLiouv$ and $\mu^+ = \sigma^+_dLiouv$, so that the invariance and the ergodicity of $\mu^+$ follow from those of $dLiouv$ and of $\sigma^+$. Let $h\colon T^1\cF_\rho\to \RR$ be a continuous function with compact support, and denote by $K$ the projection of this compact set on $T^1S$. The function $h\circ \sigma^+\colon T^1S\to \RR$ is measurable and bounded, so it belongs in $\cL^1(dLiouv)$. As the Liouville measure is a $\varphi$ ergodic probability on $T^1S$, there is an invariant set $Y_h\subset T^1S$ of total Lebesgue measure such that, for $v\in Y_h$, the average $$\frac1T\int_0^{T} h\circ\sigma^+ (\varphi(v,t))dt \ \rightarrow\ \int_{T^1S} h\circ\sigma^+ dLiouv=\int_{T^1\cF_\rho} h d\mu^+. \eqno (3.2)$$ For each $v\in Y_h$ we denote by $\cY_h(v)$ the set of points in the fiber $y\in Proj(E_v)$ corresponding to a line of $E_v\setminus G_v$. We denote by $\cY_h$ the union $\cY_h=\bigcup_{v\in Y_h} \cY_h(v) \subset M_\rho$.The set $\cY_h$ is invariant by $\Phi$ because $Y_h$ is invariant by $\varphi$ and the bundle $G$ is $\tilde\Phi-$invariant. By Fubini’s theorem, the set $\cY_h$ has total Lebesgue measure in $M_\rho$. For every $w\in \cY_h$, the average $\frac1T\int_0^{T} h(\tilde\Phi(w,t))dt$ converges to $\int_{T^1\cF_\rho} h d\mu^+$ Before proving the claim let us show that this concludes the proof of Theorem 1: There is a countable family $h_i, \quad i\in\NN$ of continuous functions with compact support which is dense (for the uniform topology) in the set of all continuous functions of $T^1\cF$ with compact support. Look now at the set $\cY=\bigcap_0^\infty \cY_{h_j}$ : It is invariant by $\Phi$, has total Lebesgue measure, and is contained in the basin of $\mu^+$ by the claim. This proves Theorem 1. Now we prove the claim: Let $w\in \cY_h(v)$, for some $v\in Y_h$, and denote $w_0=\sigma^+(v)$. As the section $\sigma^+$ is invariant by the foliated geodesic flow, for any $t$, $\Phi(w_0,t)= \sigma^+(\varphi(v,t))$; so for any $T\in \RR$ the averages $\frac1T\int_0^{T} h \circ \Phi(w_0,t)dt$ and $\frac1T\int_0^{T} h\circ\sigma^+(\varphi(v,t))dt$ are equal and we get by $(3.2)$ $$\lim_{T\to\infty}\frac1T\int_0^{T} h(\Phi(w_0,t))dt= \int_{T^1\cF_\rho} hd\mu^+.$$ Consider a non-zero vector $\tilde w$ in the linear space $E_v$ in the line corresponding to $w$. We can write in a unique way $\tilde w= \tilde w_0+\tilde w_1$ where $\tilde w_0\in F_v$ and $\tilde w_1\in G_v$. Notice that $\tilde w_0\neq 0$ projects on $w_0 \in Proj(E_v)$. By hypothesis 5 in Definition 3.2, when $t\in \RR$ is very large, either $\varphi_t(v)\notin K$ or $\frac{ \|\tilde \Phi( w_0,t_n) \| _{\varphi(v,t_n)}}{\| \tilde \Phi ( w_1,t_n) \| _{\varphi(v,t_n)}}$ is very large and so the distance (for the Fubini-Study metrics) between $\Phi(w,t)$ and $\Phi(w_0,t)$ is very small, and goes to zero. Now we decompose the averages $\frac1T\int_0^{T} h(\varphi_t(w))dt$ in two parts, one corresponding to the times $t$ such that $\varphi(v,t)\notin K$, and the other to the times such that $\varphi(v,t)\in K$. The first part is uniformly zero (for both $w$ and $w_0$). Moreover for large $t$ such that $\varphi(v,t)\in K$, the difference $ h(\Phi(w_0,t))-h(\Phi(w,t))$ goes to zero. So the averages of $h$ along the orbits of $w$ and $w_0$ converge to the same limit, which is $\int_{T^1\cF_\rho} hd\mu^+$. \[r.bigger\] The existence of a section of largest expansion does not depend of the choice of the continuous Hermitian metrics on the fibers. Theorem does not use our specific hypotheses (2-dimensional basis, geodesic flow, holomorphic foliation). One has: [Theorem $1^\prime$:*]{} [Let $B$ be a manifold and $\varphi$ a flow on $B$ admiting an ergodic invariant probability $\lambda$ which is absolutely continuous (with strictly positive density) with respect to Lebesgue measure. Let $\tilde\rho\colon\pi_1(B) \to GL(n, \CC)$ be a representation, $(E_{\tilde\rho},\tilde\cF_{\tilde\rho})$ be the vector bundle endowed with the suspension foliation, and $M_\rho=(Proj(E_{\tilde\rho}), \cF_\rho)$ the suspension of the corresponding representation $\rho\colon\pi_1(B)\to PGL(n, \CC)$. Let $\Phi$ be the lift of the flow $\varphi$ to the leaves of $\cF_\rho$. If the bundle $E_{\tilde\rho}$ admits a section $\sigma^+$ of largest expansion then $\sigma^+_(\lambda)$ is an SRB-measure of the flow $\Phi$, whose basin has total Lebesgue measure in $M_\rho$.]{} \[r.sym\] The geodesic flow (and the foliated geodesic flow) have a symmetry: denote by $I$ the involution map on the unit tangent bundle sending each vector $v$ to $-v$ and $\tilde I$ the involution $\tilde I(w_v) = -w_v$ on $T^1\cF_{\tilde\rho}$. Then $I$ is a conjugation between the geodesic flow and its inverse $I\circ\varphi_t\circ I=\varphi_{-t}$. This shows that $\sigma^-=\tilde I\circ \sigma^+\circ I$ is a section of largest expansion for the negative geodesic flow, and $\mu^-=\sigma^-_(dLiouv)$ will be an SRB-measure for the negative orbits of the geodesic flow. Then Lebesgue almost every orbit in $T^1\cF$ has negative average converging to $\mu^-$ and positive average converging to $\mu^+$. Let $E_{\tilde\rho}=F \oplus G$ be a $\tilde \Phi$-invariant measurable splitting giving rise to a section of largest expansion $\sigma^+:=proj(F)$, then the decomposition is measurably unique (i.e. over a set of full Liouville measure in $T^1S$). Let $E_{\tilde\rho}=F_1 \oplus G_1$ be a $\tilde \Phi$-invariant measurable splitting giving rise to a section of largest expansion, $\sigma_1^+:=proj(F_1)$. The line bundle $F_1$ is not contained in $G$, for if it were contained, then the order of growth of $\sigma^+$ would be larger than the order of growth of $\sigma_1^+$. But then $G_1$ would not be a subset of $G$ and any initial condition in $G_1-G$ has the same order of growth than $\sigma^+$, which is larger than the order of growth of sections in $G$, like $\sigma_1^+$, contradicting that the order of growth of $\sigma_1^+$ is larger than the order of growth of any section in $G_1$. Assume that $F \neq F_1$. For $\varepsilon>0$ define the subset $$H_\varepsilon:=\{v \in T^1S\ / \ dist(\sigma^+(v),G_v)>\varepsilon\ , \ dist(\sigma_1^+(v),G_v)>\varepsilon\ , \ dist(\sigma^+(v),\sigma_1^+(v))>\varepsilon\ \}$$ where the distances are measured in the Fubini-Study metrics of $Proj(E_v)$. For small $\varepsilon$ the set $H_\varepsilon$ will have positive Liouville measure. But since the Liouville measure is ergodic, almost all points in $H_\varepsilon$ are recurrent. But this cannot be, since both $\sigma^+$ and $\sigma_1^+$ are invariant and as time increases the component in $F_v$ grows much more than the component on $G_v$ so that in $Proj(E_v)$ the sections $\sigma^+$ and $\sigma_1^+$ are getting closer which contradicts the condition $dist(\sigma^+(v),\sigma_1^+(v))>\varepsilon$. Hence we must have $F=F_1$ (Liouville almost everywhere), as well as $\sigma^+=\sigma_1^+$. Now $G$ is uniquely determined by $\sigma^+$, since any section outside $G$ has the same order of growth as $\sigma^+$, and those on $G$ have smaller order of growth. Using Oseledec’s Theorem\[ss.oseledec\] ======================================= A Corollary of Oseledec’s Theorem ---------------------------------- Let $$f:B\to B \hskip 2cm,\hskip 2cm A\colon B\to GL(n, \CC)$$ be measurable maps. For any $n\in\NN$ and any $x\in B$ we denote $$A^n(x)= A(f^{n-1}(x)) \cdots A(f(x)) A(x) \hbox{ and } A^{-n}(x)= [A^n(f^{-n}(x))]^{-1}.$$ One says that the family $\{A^n\}$ form a multiplicative cocycle over $f$. A point $x\in B$ has Lyapunov exponents for the multiplicative cocycle $\{A^n\}$ over $f$ if there exists $0<k\leq n$ and for all $i\in \{1,\dots,k\}$ there is $\lambda_i\in \RR$ and a subspace $F_i$ of $\RR^n$ such that: 1. $\RR^n = \bigoplus_i F_i$ 2. For any $i$ and any non zero vector $v\in F_i$ one has $$\lim_{n\to\pm\infty}\frac1n \log ( \| A^n(v) \| ) = \pm \lambda_i$$ [Oseledec’s Multiplicative Ergodic Theorem (\[14\],p.666-667):]{} Let $f: B\to B$ be an invertible measurable transformation, $\mu$ an $f-$invariant probability measure and $A$ a mesurable multiplicative cocycle over $f$. Assume that the functions $\log^+\\|A\\|$ and $\log^+\\|A^{-1}\\|$ belong to $\cL^1(\mu)$. Then the set of points for which the Lyapunov exponents of $A$ are well defined has $\mu$-measure $1$. If $\mu$ is ergodic the Lyapunov exponents are independent of the point in a set of total $\mu-$measure. The Lyapunov exponents and the Lyapunov spaces above depend measurably of $x\in B$ on a set of $\mu-$total measure (see \[14\] p.666-667). When the measure $\mu$ in Oseledec’s Theorem is ergodic, we can then speak of the [Lyapunov exponents of the measure $\mu$]{}. We want to use Oseledec’s Theorem for flows when the base manifold is non-compact. Let $\varphi$ be a complete flow on the manifold $B$, $\pi\colon E\to B$ a vector bundle over $B$ and $\tilde \Phi$ be a flow on $E$ inducing a multiplicative cocycle as in $(2.4)$ over $\varphi$. We say that the Lyapunov exponents of $\tilde\Phi$ are well defined at a point $v\in B$ if there is a continuous Euclidean or Hermitian metric on the bundle $E$, a finite sequence $\lambda_1<\cdots<\lambda_k$ and a $\tilde\Phi$-invariant splitting $E(v)=F_1(v)\oplus\cdots \oplus F_k(v)$ such that, for any non zero vector $w\in F_i(v)$, any compact $K\subset B$ and any sequence $\{t_n\}_{n \in \ZZ }$ with $lim_{n \rightarrow \pm \infty} t_n = \pm \infty$ and $\varphi(v,t_n)\in K$ one has: $$\lim_{ n \rightarrow \pm \infty} \frac1{t_{ n}} \log( \| \tilde\Phi(w, t_{ n}) \| )=\pm\lambda_i.$$ \[d.noncompact\] The existence and the value of the Lyapunov exponents does not depend of the continuous metric on the vector bundle $E$; moreover we can allow the metric to be discontinuous if the change of metric to a continuous reference metric is bounded on compact sets of the basis $B$. With the notation above the Lyapunov exponents of $v\in B$ for the flow $\tilde\Phi$ are well defined if and only if they are well defined for the multiplicative cocycle $\{\tilde A_1^n\}$ over $\varphi_1$ defined by the diffeomorphism $\tilde\Phi_1$. Moreover the Lyapunov exponents and spaces are equal for the flow and the diffeomorphism. \[l.noncompact\] One direction is clear, so we will assume that the diffeomorphism $\Phi_1$ has Lyapunov exponents on $v$. As the flow $\varphi$ is complete, for any compact set $K\subset B$ the union $K_1=\bigcup_{t\in [-1,1]} \varphi(K,t)$ is compact. Moreover for each $t_n$ such that $\varphi(v,t_n)\in K$, let $T_n$ be the integer part of $t_n$, then $\varphi_1^{t_n-T_n}(v)\in K_1$. We conclude the proof noticing that $$\tilde A(v,t_n)= \tilde A(\varphi_1^{t_N-T_n}(v),T_n) \tilde A(v,t_n-T_n)$$ and that the norm of $\tilde A(,T_n)$ is uniformly bounded over $K_1$ independently of $t_n-T_n\in[0,1]$. Let $\mu$ be a $\varphi-$invariant probability on $B$. We say that the flow $\tilde\Phi$ defining a measurable multiplicative cocycle $(2.4)$ is $\mu-$integrable if there is a continuous norm $ \| \cdot \| $ on the vector bundle $E$ such that the functions $\log^+\\|\tilde A_1\\|$ and $\log^+\\|\tilde A_{-1}\\|$ belong to $\cL^1(\mu)$, where $\\|\ \\|$ is the operator norm on the normed vector spaces. The condition of integrability of the norm of the multiplicative cocycle is always verified if the manifold $B$ is compact. [Proof of Corollary 2:]{} Consider $f=\varphi_1$, the time 1 of the geodesic flow on $T^1S$, and let $\tilde A(v)\colon E_v\to E_{f(v)}$ the linear multiplicative cocycle induced on the vector bundle $T^1{\cF}_{\tilde\rho}$ by $\tilde\rho$ in Oseledec’s Theorem. By hypothesis, this multiplicative cocycle is integrable so that the Lyapunov exponent of the multiplicative cocycle $\tilde A$ are well defined for a Liouville total measure set by Lemma 4.3. The Lyapunov exponents and spaces depend measurably of $v\in T^1S$ which are invariant respectively by $\varphi$ and $\tilde\Phi$. As the Liouville measure is ergodic, the Lyapunov exponents are constant on a set of total Liouville measure. This ends the proof of item 1. The proof of item 2 is a direct consequence of the symmetry of the flow $\Phi$: $\tilde I\circ \Phi_t\circ \tilde I= \Phi_{-t}$ (see item 3 in remark 3.3). With the hypothesis of item 3 the section $\sigma^+$ is clearly a section of largest expansion so that item 3 is a direct consequence of Theorem 1. A direct corollary of Theorem $1^\prime$ and Oseledec’s Theorem is the following [Corollary $2^\prime$:]{} Let $f$ be a diffeomorphism of a manifold $B$, admitting an invariant ergodic probability $\lambda$ in the class of Lebesgue and let $E$ be an $n-$dimensional vector bundle over the basis $B$ and $M$ the corresponding projective bundle. Assume that $\tilde \Psi$ is a diffeomorphism of $E$ leaving invariant the linear fibration, inducing linear maps on the fibers and whose projection on $B$ is the diffeomorphism $f$. We denote by $\Psi$ the induced diffeomorphism on $M$. Let $U_i$ be a covering of $B$ by trivializing charts of the bundle $E$: then writing $\Psi$ in these charts we get a multiplicative cocycle $\tilde A\colon B\to GL(n,\CC)$. Assume that $log^+\\|\tilde A\\|$ and $log^+\\|\tilde A^{-1}\\|$ belong to $\cL^1(\lambda)$ and that the largest Lyapunov exponent of the measure $\lambda$ for the multiplicative cocycle $\tilde A$ corresponds to a 1 dimensional space. Denote by $\sigma^+$ the corresponding measurable section defined on a Lebegue total measure set of $B$ to $M$. Then $\sigma_^+(\lambda)$ is an SRB-measure for $\Psi$ and its basin has total Lebesgue measure in $M$. Proof of Theorem 3 ------------------- Due to Corollary 2 and the Remark 3.2, the only thing that remains to be proved is that, under the integrablity condition $(3)$, if there is a section of largest expansion then the largest Lyapunov exponent is positive and simple. We begin first with the case that $S$ is compact. So assume that there is a section $\sigma^+$ of largest expansion providing a measurable decomposition $E_{\tilde \rho}=F\oplus G$, $\sigma^+:=Proj(F)$ and let $\lambda_i$ and $F_i$ be the Lyapunov exponents and spaces as in Corollary 2. We have $F \subset F_k$, corresponding to the greatest eigenvalue $\lambda_k$, and denote by $H$ the measurable bundle $F_k \cap G$ of dimension $n_k-1$. Assume that the dimension $n_k$ of $F_k$ is at least 2, and we will argue to obtain a contradiction to this assumption. Since the foliated geodesic flow leaves invariant the measurable bundle $F_k$, after a measurable trivialisation we will obtain a measurable cocycle $$B:T^1S \times \RR \rightarrow GL(n_k,\CC)$$ which carries the information of how initial conditions are transformed into final conditions, when starting from the point $v \in T^1S\ , \ w \in F_{k,v}$, and flowing a time $t$ along the geodesic. Recall that we have introduced a Hermitian metric on the bundle $E_{\tilde{\rho}}$, by pull back in the bundle $q^E_{\tilde \rho} = T^1\cF_{\tilde \rho}$ and by restriction into the bundle $F_k$. Recall also that if we have a $\CC$-linear map $L$ between Hermitian spaces, the determinant $det(L,W)$ of $L$ on a subspace $W$ is by definition the quotient of the volumes of the paralelograms determined by $Lw_1,\ldots,Lw_m,iLw_1,\ldots,iLw_m$ and $w_1,\ldots w_m,iw_1,\ldots,iw_m$ corresponding to any $\CC$-basis $w_1,\ldots,w_m$ of $W$. Define $$\Delta^m:T^1S \rightarrow \RR \hskip 1cm,\hskip 1cm \Delta^m(v):= \frac {det(B(v,m),F_v)^{n_{k}-1}}{det(B(v,m),H_{v})}$$ and note that the cocycle condition $(2.4)$ for $B$ and the $\tilde \Phi$-invariance of $H$ and $F$ gives the multiplicative condition $$\Delta^m(v) = \Delta(\varphi(v,m-1)) \Delta(\varphi(v,m-2)) \cdots \Delta(v) \hskip 5mm , \hskip 5mm \Delta:=\Delta^1. \eqno(4.1)$$ The volume in $H$ has exponential rate of growth $(n_k-1)\lambda_k$, since it is the Lyapunov exponent of $\Lambda^{n_k-1}H$. The exponential rate of growth of $F$ is $\lambda_k$, hence $$\int_{T^1S}\log(\Delta)dLiouv = (n_k-1)\lambda_k - \lambda_k - \ldots - \lambda_k =0. \eqno (4.2)$$ Now we need the following corollary of a general statement from Ergodic Theory, (see \[15\], Corollary 1.6.10): Let $\varphi:B \rightarrow B$ be a measurable transformation preserving a probability measure $\nu$ in $B$, and $g:B \rightarrow \RR$ a $\nu$-integrable function such that $\lim_{n \rightarrow \infty} \sum_{j=0}^n (g \circ \varphi^j) = \infty$ at $\nu$-almost every point, then $\int_B g d\nu >0$. Consider the set $$A:=\{v \in T^1S \ / \ \sum_{j=0}^\ell (g \circ \varphi^j)(v)>0, \ \forall \ell\geq 0 \},$$ and for $v \in A$ let $$S_g(v):= \inf_\ell \{ \sum_{j=0}^\ell (g \circ \varphi^j)(v) \}.$$ $A$ has a strictly positive $\nu$ measure since almost any orbit will have a point in $A$, and $S_g$ is a measurable function on $A$ which is strictly positive. By Corollary 1.6.10 in \[15\] we have $$\int_B g d\nu = \int_A S_g d\nu,$$ but this last number is strictly positive, since we are integrating a strictly positive function over a set of positive measure. We want to apply the above Lemma to $(X,\nu) = (T^1S,dLiouv)$ and $g=\log\Delta$. Note that the multiplicative relation $(4.1)$ implies $$\sum_{j=0}^{m-1}\log\Delta(\varphi_j(v)) = \log \Delta^m(v) \eqno (4.3)$$ The hypothesis on the growth of the section $\sigma^+$ implies that $\lim_{n\rightarrow \infty} \log\Delta^m(v) \rightarrow \infty$. But using $(4.3)$ this is the hypothesis in the Lemma, so as a conclusion of it we obtain that $$\int_{T^1S}\log(\Delta)dLiouv >0,$$ which contradicts $(4.2)$. Hence $F_k$ has dimension 1, so that the largest Lyapunov exponent is simple. Assume now that $S$ is not compact. According to Lemma 4.3, it is sufficient to consider the integrability condition for the time 1 flow $\varphi_1$. Let $K$ be a compact set of positive Liouville measure in $T^1S$ and partition $$K_m:=\{v \in K \ / \ \varphi^j(v) \notin K, j=1,\cdots,m-1,\ \varphi^m(j) \in K \}$$ according to the time of the first return to $K$. Define the multiplicative cocycle generated by $$C:K \rightarrow GL(n,\CC) \hskip 1cm,\hskip 1cm C(v) := \tilde A_1^m(v) \hskip 1cm,\hskip 1cm v \in K_m$$ corresponding to the first return map to $K$. Since $$C(v) = \tilde A_1(\varphi^{m-1}(v)) \ldots \tilde A_1(\varphi(v)) \tilde A_1(v),$$ we have $$\log^+( \\|C(v)\\|) \leq \log^+(\\|\tilde A_1(\varphi^{m-1}(v))\\|)+ \ldots +\log^+(\\| \tilde A_1(\varphi(v))\\|) + \log^+(\\|\tilde A_1(v)\\|)),$$ and hence on $K$ we obtain $$\sum_{m=1}^\infty \int_{K_m}\log^+( \\|C(v)\\|) \leq \sum_{m=1}^\infty[\log^+(\\|\tilde A_1(\varphi^{m-1}(v))\\|)+ \ldots +\log^+(\\| \tilde A_1(\varphi(v))\\|) + \log^+(\tilde A_1(v))]\leq$$ $$\leq \int_{T^1S}\log^+( \\|\tilde A_1(v)\\|)$$ since the sets $$\varphi_j(K_m) \hskip 1cm , \hskip 1cm j=0,\ldots, m-1, \hskip 1cm, \hskip 1cm m=1,\ldots$$ are disjoint. Hence the cocycle generated by $C$ is integrable, and we may repeat the argument presented for the case that $T^1S$ is compact. Using Oseledec’s Theorem in the Non-compact case ================================================ The objective of this paragraph is to prove Theorem 4. The proof of the parts “if” and “only if” are given by some estimates over the punctured disc $\DD^$. As both proofs are long, we will treat them separately. The common argument is the following estimate about the geodesic flow of $\DD^$. Estimates on the Geodesic Flow on a Punctured Disc -------------------------------------------------- Denote by $\DD^$ the punctured disc endowed with the usual complete metric of curvature $-1$, that is, its universal cover is the Poincaré half plane $\HH^+$ with covering group generated by the translation $T(z)= z+1$ and define $$D^:= \frac{\{z\in \HH^+ \ / \ Im(z)>1\} }{ (T^n)} \subset \DD^ \hskip 1cm, \hskip 1cm S^1:=\partial D^= \frac{\{z\in \HH^+ \ / \ Im(z)=1\} }{ (T^n)} \subset \DD^.$$ $$\bar {D^}:= \frac{\{z\in \HH^+ \ / \ Im(z)\geq 1\} }{ (T^n)} \subset \DD^$$ A unit vector $u\in T^1D^$ at a point $z\in D^$ is called a [radial vector]{} if $u \in \RR w \frac{\partial}{\partial w}$. Note that for any non-radial vector $u\in T^1D^$ the geodesic $\gamma_u$ through $u$ in $\bar{ D^}$ is a compact segment $\gamma_u$ whose extremities are on the circle $S^1$. We will denote the tangent vector of the geodesic $\gamma_u$ on $S^1$ by $\alpha(u)$ (the incoming) and $\omega(u)$ (the outgoing), and let $t(u)$ be the lenght of $\gamma_u$. The set of radial vectors has zero Lebesgue measure. We will denote by $M$ the set of nonradial unit vectors on $T^1\DD^\|_{\bar{ D^}}$ and by $N$ the subset of $M$ over the circle $S^1$. We denote $N^+$ the set of vectors in $N$ pointing inside $D^$ and by $$\cA=\{(u,t), u\in N^+, t\in[0,t(u)]\}\subset N^+\times [0,+\infty[.$$ The geodesic flow $\varphi$ on $T^1\DD^$ induces a natural map $F\colon \cA\to M$ defined by $F(u,t)=\varphi(u,t)$. The unit tangent bundle over $S^1$ admits natural coordinates : If $u$ is a unit vector at $w$ we will denote $\theta(u)$ the argument of $w$, and $\eta(u)$ the angle between $u$ and the radial vector $-z\partial/\partial z$. We denote by $\mu$ the measure on $\cA$ defined by $d\mu= \cos(\eta)\cdot d\theta\wedge d\eta\wedge dt$ The Liouville measure on $T^1D^$ is $F_(d\mu)$ (up to a multiplicative constant). The measure $F_^{-1}(dLiouv):=hd\theta\wedge d\eta\wedge dt$ for a certain function $h$. Since the Liouville measure is invariant under the geodesic flow, and in $M$ the geodesic flow has the expression $\frac{\partial }{ \partial t}$, then $h$ is independent of $t$. Since the Liouville measure is invariant under rotations in $\theta$ then $h$ is also independent of $\theta$. Hence $h$ is only a function of $\eta$. To compute the value of $h$ it is enough to compute for an arbitrary $\eta$ at a point in $N^+$. We have $F_(d\theta \wedge dt) = h(\eta) dArea$. The variable $\theta$ is parametrized according to geodesic length and since the angle between the vertical and the geodesic at $Im(z)=1$ is $\eta$, we project the tangent vector to the geodesic to the vertical direction to obtain the weight $cos(\eta)$. We will denote by $\mu_0$ the measure on $N^+$ defined by $d\mu_0= d\theta\wedge d\eta$. Let $\tilde A_t:T^1D^* \times \RR \rightarrow GL(n,\CC)$ be a linear multiplicative cocycle over the geodesic flow of $D^$. For every unit vector $u\in N^+$, we denote $$B:N^+\rightarrow GL(n,\CC) \hskip 1cm,\hskip 1cm B(u) = \tilde A_{t(u)}(u)$$ the matrix corresponding to the geodesic $\gamma_u$ of length $t(u)$ going from $\alpha(u)$ to $\beta(u)$. Then the two following sentences are equivalent: 1. There is a Hermitian metric $\|\cdot\|$ on the vector bundle over $T^1D^$ such that the multiplicative cocycle $\tilde A_1$ is integrable for Liouville, that is $$\int_{T^1D^} \log^+\\| \tilde A_{\pm 1}\\| dLiouv<+\infty. \eqno (5.1)$$ 2. The function $\log^+( \\|B\\|)$ belongs to $\cL^1(\mu_0)$, that is $$\int_{N^+} \log^+(\\| B(u)\\| ) d\mu_0 <+\infty. \eqno (5.2)$$ \[p.mu0\] [ $(5.2)$ does not depend of the choice of the continuous Euclidean metric : Two continuous Hermitian metrics $\|\cdot\|_1$ and $\|\cdot\|_2$ on the bundle over $T^1\DD^\|_{\partial D^}$ are equivalent because $\partial D^$ is compact, so that the difference $\|\log(\\| B(u)\\| _1)\| - \|\log(\\| B(u)\\| _2)\|$ is uniformly bounded on $N^+$.]{} For every $u\in N^+$ set $t_u:=t(u)$, and divide the interval $[0, t_u]$ in $$[0,1]\cup [1,2]\cup \cdots\cup [ E(t_u)-1,E(t_u)]\cup[E(t_u),t_u],$$ so that if $u$ is a vector at a point $x \in\partial D^$ one gets on setting $\varphi:=\varphi_1$ the geodesic flow at time 1: $$B(u)= \tilde A_{t_u-E(t_u)}(\varphi^{E(t_u)}(u)\circ \prod_0^{E(t_u)-1} \tilde A_1(\varphi^i(u))$$ So for any Hermitian norm $\|\cdot \|$ we get $$\\| B(u)\\| \leq \\| \tilde A_{t_u-E(t_u)} (\varphi^{E(t_u)}(u))\\| \prod_0^{E(t_u)-1} \\| \tilde A_1(\varphi^i(u))\\|$$ So $$\log^+(\\| B(u)\\| ) \leq \log^+ \\| \tilde A_{t_u-E(t_u)}(\varphi^{E(t_u)}(u))\\| + \sum_0^{E(t_u)-1} \log^+\\| \tilde A_1(\varphi^i(u))\\|$$ Remark that $\log^+ \\| \tilde A_{t_u-E(t_u)}(\varphi^{E(t_u)}(u))\\| $ is uniformly bounded by a constant $K$ depending on $\tilde A$ and $\|\cdot\|$, because $t_u-E(t_u)\in[0,1[$ and $\varphi^{E(t_u)}(u) = \varphi_{E(t_u)-t_u}(\varphi_{t_u}(u))$ remains in a compact set (recall that $\varphi_{t_u}(u)\in\partial D^$). So we get that there is a constant $K_1$ such that for every $u\in N^+$ one has $$\log^+(\\| B(u)\\| ) \leq K_1 + \int_0^{t_u} \log^+\\| \tilde A_1(\varphi_t(x))\\| dt$$ Notice now that, for any $\varepsilon \in [0,1[$ there is $\delta>0$ such that if $\cos(\eta)\leq \varepsilon$ then $t_u\leq\delta$. So it is equivalent that the function $\log^+(\\| B\\| )$ is integrable for the measure $d\mu_0$ or for $\cos(\eta) d\theta\wedge d\eta$. Hence we obtain that if $\int_{N^+} \log^+(\\| B\\| ) d\mu_0 =+\infty$ then for any Riemannian metric $\|\cdot\|_2$ the function $\log^+(\\| \tilde A_1\\| _2)$ is not Liouville integrable. We have proven that $ \mbox{item 1}\quad \Longrightarrow \quad \mbox{item 2}.$ For the other implication, choose a continuous Riemannian metric on the bundle over $N$, assume the integrability condition $(5.2)$ and let $v \in T^1\DD\|_{\bar D^}$. If $v$ is a radial vector, then push forward the metric over $\alpha(v)$ along the geodesic using the flat structure of the bundle. If $v$ is not a radial vector then push forward the metric on $\alpha(v)$ on the first third of $\gamma_v$, on the last third of the geodesic push forward the metric on $\omega(u)$ and on the middle third of $\gamma_u$ put the corresponding convex combination of the metrics on $\alpha(u)$ and $\omega(u)$. This produces a continuous metric on the bundle over $T^1\DD\|_{\bar D^}$ such that $\\|\tilde A_{\pm 1} \\|$ does not expand except in the middle part, and there it expands in a constant way. Hence for this metric the integral $(5.2)$ coincides with $(5.1)$. To use Proposition 5.2 we will need to estimate $\\| B(u)\\| , \quad u\in N^+$. For that we will use the following estimate of $t_u$ and the estimate of the variation of the argument along the geodesic $\gamma_u$: 1. There is a constant $T$ such that $t_u\in [-2\log \|\eta\|-T,-2\log\|\eta\| +T]$. 2. Denote by $a_u$ the variation of the argument along $\gamma_u$. Then $a_u= 2\frac{\cos \eta}{\sin\eta}$ \[p.tu\] The easiest way is to look at the universal cover $\HH$. Recall that in this model the geodesic for the hyperbolic metric are circles or straight lines (for the Euclidean metric) orthogonal to the real line. Let $u\in E^+_1$ at a point $ x\in \partial D^$. Denote by $u$ the corresponding vector at a point $\tilde x\in \HH$, $Im(x)=1$, where $\tilde x$ is a lift of $x$. The angle $\eta(u)$ is the angle between the vector and the vertical line. Consider the geodesic $\tilde \gamma_u$ throught $u$. The Euclidean radius $R_u$ of this circle verifies $1=\|\sin(\eta)\|\cdot R_u$. Now denote by $\tilde y\neq \tilde x$ the intersection point of $\tilde \gamma_u$ with the boundary $Im(z)=1$ of ${ {D}}^$. Then $a_u = \tilde y -\tilde x = 2\frac{\cos (\eta)}{\sin(\eta)}$. So the second item of Proposition 5.4 is proved. To give an estimate of $t_u$ let us consider the following curve $\sigma_u$ joining the points $\tilde x$ and $\tilde y$: $\tilde \sigma_u$ is the union of the vertical segment $\sigma^1_u$ joining $\tilde x=(\cR e(\tilde x),1)$ to $(\cR e(\tilde x), R_u)$ the horizontal segment $\sigma_2^u$ joining $(\cR e (\tilde x),R_u)$ to $(\cR e(\tilde y), R_u)$ and the vertical segment $\sigma^3_u$ joining $(\cR e(\tilde y),R_u)$ to $(\cR e(\tilde y), 1)=\tilde y$. The hyperbolic length of the vertical segments is $\log (R_u)$. The hyperbolic length of the horizontal segment is $\frac{\|a_u\|}{R_u}= 2cos(\eta)$. So we get: $$\ell(\tilde \gamma_u) < \ell(\sigma_u)= -2\log(\|\sin(\eta)\|)+2\cos(\eta)$$ On the other hand, consider the point $z_u\in\gamma_u$ whose imaginary part is $R_u$. This point is the middle of the horizontal segment of $\sigma_u$. Denote by $\gamma_u^0$ the segment of $\gamma_u$ joining $\tilde x$ to $z_u$ and $\sigma_u^0$ the segment of $\sigma^2_u$ joining $zu$ to the point $(\cR e(\tilde y), R_u)$. The union of these 2 segments is a segment joining the two extremities of $\sigma_u^1$ which is a geodesic. So we get $$-\log(\|\sin(\eta)\|)<\ell(\gamma_u^0)+\ell(\sigma_u^0) =\frac12\ell(\tilde\gamma_u) +\cos(\eta).$$ So we get $$t_u= \ell(\tilde \gamma_u)\in [-2\log(\|\sin(\eta)\|)-2\cos(\eta),-2\log(\|\sin(\eta)\|)+2\cos(\eta)]$$ So $$t_u \in [-2\log(\|\sin(\eta)\|)-2,-2\log(\|\sin(\eta)\|)+2]$$ To conclude the first item it is enought to note that $\|\log (\|\eta\|)-\log(\|\sin(\eta)\|)\|$ is bounded for $\eta\in [-\pi/2,\pi/2]$. The Parabolic Case ------------------ If for each $i$ all the eigenvalues of $\rho(\gamma_i)$ have modulus $1$, then the multiplicative cocycle flow is integrable. \[p.if\] As the function $\log^+ \|\tilde A_1\|$ is continuous, it is integrable for the Liouville measure over every compact set of $T^1S$. So the problem is purely local, in the neighbourhood of the punctures of $S$. So it is enough to look at a multiplicative cocycle $\tilde A_t$ over the geodesic flow of the punctured disc $D^$. The proposition is a direct corollary of the following proposition: Let $B\in GL(n,\CC)$ be a matrix and $\cF_B$ be the corresponding suspension foliation over $\DD^$ (as $B$ is isotopic to identity the foliation $\cF_B$ is on $\DD^\times \CC^n$), and denote by $\tilde A_t$ the linear multiplicative cocycle over the geodesic flow $\varphi$ of $\DD^$ induced by $\cF_B$. Assume that all the eigenvalues of $B$ have modulus equal to $1$. Then the functions $\log^+(\\|\tilde A_{\pm 1}\\|)$ are in $\cL^1(dLiouv\|_{D^})$. \[p.parabolic\] We begin the proof of Proposition 5.6 by the following remarks allowing us to reduce the proof to an easier case: * 1. If two matrices $B_1$ and $B_2$ are conjugate then the corresponding cocycles are both integrable or both non-integrable. 2. If $B$ is a matrix on $\CC^k\times \CC^m$ leaving invariant $\CC^k\times\{0\}$ and $\{0\}\times\CC^m$, then the multiplicative cocycle induce by $B$ is integrable if and only if the cocycles induced by the restrictions of $B$ to $\CC^k\times \{0\}$ and $\{0\}\times\CC^m$ are both integrable. 3. As a consequence of item 2, we can assume that $B$ is a matrix which doesn’t leave invariant any splitting of $\CC^n$ in a direct sum of non-trivial subspaces. In particular $B$ has a unique eigenvalue $\lambda_B$ and by hypothesis $\|\lambda_B\|=1$. Moreover two such matrices are conjugate: their Jordan form is $$\left(\begin{array}{cccccc} \lambda_B&1&&\cdots&0&0\\ 0&\lambda_B&1&\cdots&&0\\ \cdots&&&&&\cdots\\ 0&0&\cdots&0&\lambda_B&1\\ 0&0&\cdots&0&0&\lambda_B\\ \end{array}\right)$$ Using the remarks above, it is enough to prove Proposition 5.6 for the matrices $B_\theta$ define as follows. Let $$A_\theta = \left(\begin{array}{cccccc} i\theta&1&0&\cdots&0&0 \\ 0&i\theta&1&0&\cdots&0 \\ \cdots&&&&&\cdots\\ 0&0&\cdots&0&i\theta&1\\ 0&0&\cdots&0&0&i\theta\\ \end{array}\right).$$ We define $B_\theta=exp(A_\theta)$. Notice that $$exp(t\cdot A_\theta)=e^{i t\theta}\left(\begin{array}{cccccc} 1&t&&\cdots&\frac{t^{n-2}}{(n-2)!}&\frac{t^{n-1}}{(n-1)!} \\ 0&1&t&\cdots&& \frac{t^{n-2}}{(n-2)!}\\ \cdots&&&&&\cdots\\ 0&0&\cdots&0&1&t\\ 0&0&\cdots&0&0&1\\ \end{array}\right)$$ Consider the holomorphic foliation defined by the linear equation $$\begin{pmatrix} \dot z \cr \dot w \end{pmatrix} = \begin{pmatrix} i & 0 \cr 0 & A_\theta \end{pmatrix} \begin{pmatrix} z \cr w \end{pmatrix}$$ on $\DD^\times \CC^n$ such that the holonomy map from $\{e^{-2\pi}\}\times \CC^n\to\{z\}\times\CC^n$ with $z \in S^1$ is $exp(arg(z)A_\theta)$. The monodromy of this foliation is $B_\theta = e^{2i\pi \theta} exp(2\pi A_0)$. The multiplicative cocycle $\tilde A_t$ obtained by lifting the geodesic flow of $\DD^$ on the leaves of $\tilde\cF_\theta$ is integrable over $T^1\DD\|_{D^}$. For any $u\in N^+$ one has $B(u)=A_{t_u}(u)= exp(\frac{a_u}{2\pi} \cdot A_{\theta})$, so that there is a constant $K$ such that $\\| B(u)\\|< K(1+ a_u^{n-1})$, so that $\log^+\\|B(u)\\|$ is integrable if and only if $\log^+(\|a_u\|)$ is integrable for $\mu_0$. By Proposition \[p.tu\] one has $a_u= 2\cos(\eta)/\sin(\eta) $ so that $a_u<2/\eta$. As $\int_{-1}^1 \|\log(\| 1/x\|)\|dx<+\infty$, we get easily that $\int_{N^+_1} \log^+(\|a_u\|) d\mu_0 <+\infty$, concluding the proof. The Hyperbolic Case ------------------- If there is i such that the matrix $B=\rho(\gamma_i)$ has an eigenvalue with modulus different from $1$, then the multiplicative cocycle is not integrable. \[p.hyperbolic\] If $B\in GL(n,\CC)$ has an eigenvalue with modulus different from $1$, we may suppose that its modulus is greater than $1$, since the suspension of $B$ and $B^{-1}$ are isomorphic. As in the parabolic case the proof of Proposition \[p.hyperbolic\] follows directly from a local argument in a neighbourhood of the puncture corresponding to $\gamma_i$. Let $B\in GL(n,\CC)$ having an eigenvalue $\lambda>1$ and $\cF_B$ the suspension folition on $D^$. Then the multiplicative cocycle $\tilde A_t$ induced by $\cF_B$ over the geodesic flow $\varphi$ of $D^$ is not integrable. \[p.localhyp\] We begin by an estimate of the norm of the multiplicative cocycle corresponding to the “in-out” map : There is a constant $K>0$ such that for any $u\in N^+$ one has: $$\| \tilde A_{t_u}(u)\| \geq K\cdot \lambda^{a_u/2}.$$ So $\log^+\|\tilde A_{t_u}(u)\|\geq \log K + \frac{\|a_u\|}2 \log\lambda$. One deduces that $\log^+\|\tilde A_{t_u}(u)\|$ cannot be $\mu_0$-integrable if $\|a_u\|$ is not integrable. By Proposition \[p.tu\] one knows that $a_u = 2\frac{\cos(\eta)}{\sin(\eta)}$ and this function is not integrable for $d\mu_0= d\eta\wedge d\theta$. From Proposition \[p.mu0\] we get that the multiplicative cocycle $\tilde A_1$ is not integrable for Liouville, finishing the proof the Proposition \[p.localhyp\]. [Remark:]{} If $\rho:\pi_1(S) \rightarrow PGL(n,\CC)$ is a representation that does not admit a lifting to a representation in $GL(n,\CC)$ we may still define a flat bundle over $S$ but with fibres $\CC^n/\ZZ_n$ and transition coordinates in $SL(n,\CC)/\ZZ_n\cdot Id$, and hence a foliation $\cF_{\tilde \rho}$ on this singular bundle, where $\ZZ_n$ is the group of $n$ roots of unity. We may introduce a continuous Hermitian norm on this bundle (locally induced from a Hermitian norm in $\CC^n$ as well as choosing a trivialisation of the generator of the discrete dynamics $\tilde A_1$, and the statements and arguments given in the text extend to this situation. Ping-pong and Schottky Monodromy Representations {#s.pingpong} ================================================ The ping-pong is a classical technique used to verify that a finitely generated group of transformation of some space is a free group. When the space is a metric space additional geometric information on the ping-pong allows one to describe almost completely the topological dynamics of this group of transformations. We will use this technique to describe the foliated geodesic flow associated to an injective representation $\rho$ from $\pi_1(S)$ to a Schottky group $\Ga\subset PSL(2,\CC)$. The Ping-pong ------------- Let us first recall some basic properties and definitions on the ping-pong. Let $\cE$ be a set, $k>1$ and for every $i\in\{1,\dots,k\}$ let $f_i\colon\cE\to\cE$ be a bijection. We say that [the group $\Ga\subset \mbox{Bij}(\cE)$ generated by $f_1,\dots,f_k$ is a ping-pong (for this system of generators)]{} if for every $i\in\{1,\dots,k\}$ there exist subsets $A_i$, $B_i$ of $\cE$ such that the following properties are verified: - The family $\{A_i, B_i, i\in\{1,\dots,k\}\}$ is a family of mutualy disjoint subsets of $\cE$, - for every $i\in\{1,\dots,k\}$ one has $f_i(\cE\setminus A_i)\subset B_i$. \[d.pingpong\] Denote by $\FF_k$ the free group with $k$ generators $\{e_1,\dots,e_k\}$. The first result on the ping-pong is: If a group $\Ga\subset\mbox{{\em Bij}}(\cE)$ is a ping-pong group for the generators $f_1,\dots,f_k$ then the morphism $\varphi\colon\FF_k\to \Ga$ defined by $\varphi(e_i)=f_i,\quad i\in\{1,\dots,k\}$ is an isomorphism. Let $i_1,\ldots,i_m \in \{1,\ldots,k\}, $ and $\varepsilon_j\in\{-1,1\}$ be such that the word $e_{i_1}^{\varepsilon_1}\cdots e_{i_m}^{\varepsilon_m}$ is a reduced word in $\FF_k$. We have to prove that the bijection $f= f_{i_m}^{\varepsilon_m}\circ\cdots\circ f_{i_i}^{\varepsilon_1}= \varphi(e_{i_1}^{\varepsilon_1}\cdots e_{i_m}^{\varepsilon_m})$ is different from identity. For instance assume that $\varepsilon_1=1$. Then, using that the word is a reduced word, one easily shows (by induction on $m$) that $f(\cE\setminus A_{i_1})$ is included in one of the sets $A_{i_m}$ or $B_{i_m}$. As $k>1$, $f(\cE\setminus A_{i_1})$ is not included in one element of $\{A_i, B_i, i\in\{1,\dots,k\}\}$, so $f$ is not the identity. Assume now that $(\cE,d)$ is a compact metric space, the $f_i$ are homeomorphisms of $\cE$, every $A_i$, $B_i$ is compact, and for each $i\in\{1,\dots,n\}$ the restrictions of $f_i$ and $f^{-1}_i$ to $\cE\setminus A_i$ and $\cE\setminus B_i$, respectively, are contractions for the distance $d$: we will say that $(\cE,d,\{f_i\})$ is a [compact contracting ping-pong.* ]{} For any $g\in\{f_i, f_i^{-1}, i\in\{1,\dots,n\}$ we denote by $C(g)= B_i, \hbox { and } C'(g)=A_i$ if $g=f_i$ and $C(g)= A_i\hbox{ and } C'(g)=B_i$ if $g=f_i^{-1}$, so that for every $g$ one has $g(\cE\setminus C'(g))\subset C(g)$. Note that if $g_1\neq g_2^{-1}$ then $g_2(C(g_1))\subset C(g_2)$ so that $g_2\circ g_1(\cE\setminus C'(g_1))\subset C(g_2)$. Let $(\cE,d,\{f_i\})$ be a compact contracting ping-pong. For every $\varepsilon>0$ there is $\ell\in \NN$ such that for every reduced word $g_\ell\circ\cdots \circ g_1$, $g_i\in\{f_i, f_i^{-1}, i\in\{1,\dots,n\}\}$ one has $$diam( g_\ell\circ \cdots\circ g_1(\cE\setminus C'(g_1))) <\varepsilon$$ \[l.contract\] Using the compacity of the set of points $x,y$ such that $d(x,y)\geq\varepsilon$ we get that there is $0<\delta<1$ such that if $x,y\in \cE\setminus C'(g)$, and $d(x,y)\geq \varepsilon$ then $d(g(x),g(y))\leq \delta\cdot d(x,y)$. Let $\Si_0=\{f_i, f_i^{-1}, i\in\{1,\dots,n\}\}^{Z}$ be the set of infinite words with letters equal to $f_i^{\pm 1}$, endowed with the product topology. An infinite word $(g_i)_{i\in\ZZ}$ is called [reduced]{} if for any $n$ the finite word $(g_i)_{-n<i<n}$ is reduced. We denote by $\Si=\{(g_i)\in\Si_0, (g_i) \mbox{ is reduced} \}$ the subspace of reduced words, $\De= \Si\times \cE$ and $\Pi\colon \De\to \Si$ the natural projection. Denote by $\sigma$ the shift on $\Si$, that is $\sigma(g_i)= (h_i)$ where $h_i= g_{i+1}$, and by $\tilde \sigma$ the map on $\De$ defined by $\tilde\sigma ((g_i),x))= (\sigma(g_i),g_0(x))$. One verifies easely that $\sigma$ and $\tilde\sigma$ are homeomorphisms. Notice that $\tilde\sigma$ is a multiplicative cocycle over $\sigma$. The topological picture of the ping-pong may be completely understood: With the notation above, there are exactly two continuous sections $s^+\colon \Si\to \De$ and $ s^-\colon\Si\to \De$ which are $\tilde\sigma-$invariant. Moreover, $s^+(\Si)$ is a topological attractor for $\tilde \sigma$ whose basin is $\De- s^-(\Si)$ and $s^-(\Si)$ is a topological repellor for $\tilde \sigma$ with basin $\De- s^+(\Si)$ and these two sections are disjoint. \[4.sections\] Let $(g_i)\in\Si$ be a reduced word. For every $n\in\NN$, consider the compact sets $$K_n^+ = g_{-1}\circ g_{-2}\circ\cdots\circ g_{-n}(\cE\setminus C'(g_{-n})\subset C(g_{-1})$$ and $$K_n^- = g_{0}^{-1}\circ g_1^{-1}\circ\cdots\circ g_{n-1}^{-1}(\cE\setminus C(g_{n-1})\subset C'(g_0)$$ Using the fact that the word $(g_i)$ is reduced, one shows easily that these sequences of compact sets are decreasing with $n$: $K_{n+1}^+\subset K_n^+$ and $K_{n+1}^-\subset K_n^-$. Moreover as $g_0\neq g_{-1}^{-1}$ one has $C(g_{-1})\cap C'(g_0)=\emptyset$, so that $K_n^+\cap K_n^-=\emptyset$. Finally, Lemma \[l.contract\] ensures that the diameter of $K_n^+$ and $K_n^-$ goes uniformly to $0$. We define then $$s^-((g_i))=\bigcap_{n\in\NN} K_n^- \quad\mbox{and}\quad s^+((g_i))=\bigcap_{n\in\NN} K_n^+$$ Schottky Groups --------------- A [Schottky group]{} of rank $n$ is a finitely generated group $\Ga\subset PSL(2,\CC)$ having $2n$ disjoint circles $C_1, C'_1,\dots, C_n, C'_n$ bounding a domain $D\subset \CC P^1=\CC\cup\{\infty\}$, and a system $g_1,\dots,g_n$ of generators such that $g_i(C'_i)=C_i$ and $g_i(D)\cap D=\emptyset$ (see [@Ma]). Using the discs $A_i,B_i$ bounded by the circles $C_i,C'_i$ respectively and disjoint from $D$, one see that $\Ga$ is a ping-pong group of $Aut(\CC P^1)$, moreover it is a compact contracting ping-pong group. Geodesics and Reduced Words --------------------------- Let $S$ be a finite non-compact hyperbolic Riemann surface, endowed with its natural hyperbolic metric. There are $\gamma_1,\dots,\gamma_k$ complete mutually disjoint geodesics whose ends arrive to punctures of $S$, such that the complement $S\setminus\bigcup_1^k \gamma_i$ is connected and simply connected, the $\gamma_i$ bound a fundamental domain of $S^\prime$ in its universal cover $\DD$ and the fundamental domain is a $2k$ sided polygon whose vertices are on the circle at infinity of $\DD$. Let $\beta_1,\dots,\beta_k$ be a maximal set of non-homotopic mutually disjoint curves whose ends arrive to punctures of $S$. Clearly, by removing them from $S$ we obtain a connected simply connected domain (for otherwise we could pick and additional $\beta_{k+1})$. Lift them to the universal cover of $S$ and replace the lifts of $\beta_j$ by the geodesics that have the same endpoints. Pushing down these geodesics to $S$, gives the desired curves $\gamma_i$. Now fix an origin $x_0\in S\setminus \bigcup_1^k\gamma_i$. For each $i$ there is a unique geodesic segment $\alpha_i$ joining $x_0$ to $x_0$ and cutting $\gamma_i$ at exactly one point, with the positive orientation, and not cutting $\gamma_j,\quad j\neq i$. The closed paths $\alpha_i$ build a system of generators of the fundamental group $\pi_1(S,x_0)$. More precisely the fundamental group is the free group generated by the $\alpha_i$. The union of the $\alpha_i$ is a bouquet of circles and we verify easily that $S$ admits a retraction by deformation on this bouquet of circles. Now fix an orientation on each geodesic $\gamma_i$ and call $\gamma_i$ the oriented geodesic. Given any vector $u\in T^1_xS$ at a point $x\in S\setminus\bigcup_1^k\gamma_i$, the geodesic $\gamma_u$ has two possibility: 1. either one of its ends goes to one puncture of $S$, 2. or $\gamma_u$ cuts transversely infinitely many times (in the future and in the past) the geodesics $\gamma_i$. [The itinerary of the geodesic $\gamma_u$]{} is the sequence $b(u)=(b_i)_{i\in\ZZ}$ defined as follows: $b_i$ is $\alpha_i^{\pm 1},\quad i\in\{1,\dots,k\}$ if the $(i-1)^{th}$ intersection of $\gamma_u$ with $\bigcup \gamma_l$ belongs to $\gamma_i$ and the coefficient is $+1$ or $-1$ according if the orientation of $\gamma_u$ followed by the orientation of $\gamma_i$ is a direct or inverse basis of the tangent space. For any $u \in T^1_xS^\prime$ the itinerary $b(u)$ is a (finite or infinite) reduced word in the letters $\alpha_i^{\pm 1}$, where $b_0$ corresponds to the first intersection point. If a segment in the fundamental domain cuts 2 times the same geodesic $\gamma_i$ with opposite direction, then its lift on $\DD$ will cut 2 times the same lift of $\gamma_i$. So this segment cannot be geodesic. Given the geodesic $\gamma_u$, and a time $t_0\in\RR$ such that $\gamma_u([0,t_0])\notin\bigcup_1^k \gamma_i$, we get a closed path $\tilde\gamma_u(t)$ joining respectively $\gamma_u(0)$ and $\gamma_u(t)$ by a geodesic segment in the fundamental domain. Moreover if $t>0$ and if the segment $\gamma_u([0,t])$ cuts $\ell+1$ times the geodesic $\gamma_i$, then the closed path $\tilde \gamma_u(t)$ is homotopic to $\beta_0\cdot \beta_1\cdots\beta_\ell$ where $\beta_j$ is a closed path $\alpha_i^{\pm 1}$ according to the letter $b_j= \alpha_i^{\pm 1}$. The geodesic $\gamma_u$ defines a (finite or infinite) reduced word in $\pi_1(S,x_0)$ for the basis $\alpha_i,\quad i\in\{1,\dots,k\}$. Proof of Theorem \[t.Schottky\] ------------------------------- Let $G_e$ and $G_f$ be free groups generated by $e=\{e_1,\dots,e_k\}$ and $f=\{f_1,\dots,f_\ell\}$, respectively. Denote by $\Ga_e$ and $\Ga_f$ their Cayley graphs for the given basis. Both Cayley graphs are trees. Let $\rho:G_e \rightarrow G_f$ be a group isomorphism. Any infinite word $b=(b_j)_{j\in \ZZ},\quad b_j\in\{e_i^{\pm1}, 1\leq i\leq k\}$ defines an infinite path $\sigma(b)$ in the Cayley graph $\Ga_e$. This path $\sigma(b)$ is a geodesic if and only if the word $b$ is reduced (see \[12\] for background material on hyperbolic groups). We say that an infinite path $\sigma\subset \Ga_e$ is stretchable if it is properly embedded (namely, only a bounded part of the path remains in a given compact set of the Cayley graph). It is strictly stretchable if its 2 ends correspond to two distinct ends $\sigma_-$ and $\sigma_+$ of the Cayley graph. The unique geodesic joining $\sigma_-$ to $\sigma_+$ is the reduction $\sigma^r$ of $\sigma$. Let $b$ an infinite word in the letters $(e_i)$. Let $c:=\rho(b)$ be the corresponding word in the letters $f_i$. Then $b$ is stretchable if and only if $c$ is stretchable. $\rho$ induces a homeomorphism from the boundary of $\Gamma_e$ to the boundary of $\Gamma_f$ by associating to the boundary point $b$ the boundary point $\rho(b)$. Given any word $b$ in the letters $e_i$, $\rho$ produces a reduced word $c:=\rho(b)$ in the letter $f_i$ obtained as follows: Change each letters $b_j=e_i^{\pm1}$ by the reduced word $\rho(b_j)$ written in terms of $f$. Do the appropiate cancellations to obtain the reduced word $c$. By \[12\] p.7, the isomorphism $\rho$ induces a quasi-isometry of the Cayley graphs, hence $b$ is stretchable if and only if $c$ is. A stretchable word $a$ in a free group defines two points $a_-$ and $a_+$ in the boundary of the group. So there is a unique geodesic $c^r$ in the Cayley graph of the group, which corresponds to a reduced word on the group, joining $a_+$ to $a_-$. Using the same notation as in Proposition 6.4, define $s^-(a) =s^-(c^r)$ and $s^+(a)=s^+(c^r)$. Denote by $\hat\Si$ the set of stretchable infinite words whose letters are the generators of the Schottky group $G$. The reduced word corresponding to $c^r$ above belongs to $\hat\Si$. Recalling that in this case, the group acts on $\CC P^1$; $\sigma$ is the shift on $\hat \Si$, being a homeomorphism, because $\hat \Si$ is $\sigma$ invariant. Recall that $\tilde \sigma$ is the map on $\hat\Sigma\times \CC P^1$ defined by $\tilde\sigma(a,x)=(\sigma(a),a_0(x))$. Then, since the Schottky group defines a compact contracting ping pong, Proposition 6.4 implies immediately the following: Let $a$ be a stretchable word in a Schottky group $G\subset SL(2,\CC)$ and $b$ the image of $a$ by the shift. Then $s^\pm(b)=a_0(s^\pm(a))$. The map $s^\pm\colon a\mapsto (a,s^\pm(a))$ defines an $\tilde\sigma-$ measurable section of the trivial fibration $\hat\Si\times\CC P^1\to \hat\Si$. [Proof of Theorem \[t.Schottky\]:]{} Let $\rho\colon \pi_1(S,x_0)\to SL(2,\CC)$ be an injective representation with $G=\rho( \pi_1(S,x_0))$ a Schottky group. Notice that the set of vector $u\in T^1S$ such that the corresponding geodesic $\gamma_u$ goes to a puncture of $S$ has zero Lebesgue measure. For any unit vector $u$ at a point of the fundamental domain such that the geodesic $\gamma_u$ has no end at a puncture of $S$, the word $\rho(b(u))$ is a stretchable word of the Schottky group. For any point $x$ of the fundamental domain of $S$ we denote by $H_x$ the holonomy of the foliation $\cF_\rho$ from the fiber over $x$ to the fiber over $x_0$ by a path contained inside the fundamental domain. This holonomy is well defined because the fundamental domain is simply connected. So we define $s^\pm\colon T^1S\to T^1\cF_\rho$ as $s^\pm(u)=H_x^{-1}(s^\pm(\rho b(\gamma_u)))$. By construction the sections $s^\pm$ are defined Liouville almost everywhere, are measurable, and are the sections of largest expansion and contraction. The continuity of $s^\pm$ follows from the topological way of constructing the sections in Proposition 6.4 and the fact that the map which associates the point at infinity of the Cayley graph of the presentation of $\pi_1(S)$ to the point at infinity of the Cayley graph of the Schottky group is continuous, by Lemma 6.11. This proves Theorem \[t.Schottky\]. [Remark:]{} Observe that Schottky representations over punctured Riemann surfaces never satisfy the integrability condition $(3)$ due to Theorem 3, since all its elements are hyperbolic and so, in particular, the maps corresponding to loops around a puncture. By the way we chose the presentation of the fundamental group (Lemma 6.5) the geodesics give rise to reduced words. Assume now that the image under $\rho$ of these generators of $\pi_1(S)$ are generators of the Schottky group, then we will have that there are no cancellations in the words corresponding to $\rho$(geodesic). For the general geodesic in $S$, the ratio between the number of letters to the length of the geodesic goes to infinity as the length of the geodesic goes to infinity, since by ergodicity of the geodesic flow the average time that the general geodesic spends in a small disk around the puncture is proportional to the area of the disk and the number of turns that the geodesic does around the pucture is $cot(\eta)$ by Proposition 5.4. This shows that the ‘Lyapunov exponents’ of these Schottky representations are $\pm\infty$. [Remark:]{} If $S$ is compact and the group $\tilde\rho(\pi_1(S))$ is non-cyclic but contained in a Schottky group, it follows from the results in \[3\] that there are positive and negative Lyapunov exponents, and hence sections of largest expansion and contraction, but they will only be measurable sections now due to cancellations in the reduced words. Foliation Associated to the Canonical Representation ==================================================== The Geometry of the Bundles --------------------------- Let $S$ be a hyperbolic Riemann surface, and denote by $\pi\colon \HH^+\to S$ its universal cover by the upper half plane $\HH^+$. Fix a point $x_0\in S$, and $\bar x_0\in\pi^{-1}(x_0)$. Denote by $$\rho_{can}\colon\pi_1(S,x_0)\to PSL(2,\RR)\subset PSL(2,\CC)$$ the covariant representation obtained by the covering transformations. We consider now the suspension foliation $\cF_{can}$ associated to the representation $\rho_{can}$ (that is a foliation in $M_{can}$ whose holonomy is given by $Hol(\gamma)=\rho_{can}(\gamma)^{-1}$). The representation $\rho$, the $\CC P^1$ bundle $M_{can}$ and the foliation $\cF_{can}$ are called the [canonical representation, bundle and foliation of the hyperbolic Riemann surface $S$]{}. \[d.tautologic\] Denote by $\iota\colon\HH^+\to\CC P^1$ the usual inclusion of the upper half plane in the projective line. We have the canonical action $$\pi_1(S,x_0) \times [\HH^+ \times \CC P^1] \longrightarrow [\HH^+ \times \CC P^1] \hskip 1cm, \hskip 1cm (\gamma,x,z) \rightarrow (\rho_{can}(\gamma)(x), \rho_{can}(\gamma)(z))$$ corresponding to the representation $$\rho_{can}\times\rho_{can}: \pi_1(S,x_0) \rightarrow PSL(2,\RR)\times PSL(2,\CC)$$ The quotient $\Pi:M_{can} \rightarrow \HH^+/\rho_{can} = S$ is a 2-dimensional complex manifold and the projection to the first factor gives it the structure of a $\CC P^1$ bundle over $S$. For any $\alpha\in PSL(2,\RR)$ one has $\iota\circ \alpha_{\HH^+}=\alpha_{\CC P^1}\circ\iota$. Denote by $\tilde \De$ the diagonal $\tilde\De=\{(z,\iota(z))\| z\in\HH^+\}$. Then for each $\gamma\in\pi_1(S,x_0)$ and each $z\in\HH^+$ one gets: $$(\rho_{can}(\gamma)z,\rho_{can}(\gamma)\iota(z))= (\rho_{can}(\gamma)z, \iota(\rho_{can}(\gamma)(z))\in\tilde\De,$$ so the diagonal $\tilde\De$ is invariant by the action of $\rho_{can}\times\rho_{can}$ and induces in the complex surface $M_{can}$ a Riemann surface $\De$ and the projection $\Pi$ induces a biholomorphism $\De\to S$. The diagonal $\De$ is the image of a holomorphic section of the bundle $M_{can} \rightarrow S$. As the representation $\rho_{can}$ has its values in $PSL(2,\RR)$, the circle bundle $\HH^+\times \RR P^1$ is invariant by the action of $\rho(\gamma),\quad \gamma\in\pi_1(S,x_0)$, so that it defines $M_{can}^\RR \subset M_{can}$ an $\RR P^1-$subbundle. For every point $p$ of $S$ we will denote by $\RR P^1_p\subset \CC P^1_p$ the fiber of these bundles over $p$. $M_{can}^\RR$ is disjoint from the diagonal $\De$. Consider now the unit tangent spaces $\Pi_\colon T^1\cF_\rho\to T^1S$. Notice that every unit vector $u$ at a point $p\in S$ lifts canonically to a unit vector tangent to $\cF$ at any point $\tilde p$ in the fiber $\CC P^1_p$. So the diagonal $\De$ induces canonically a section $\De_ \colon T^1S\to T^1\cF$: $$\begin{matrix} M_{can} & \leftarrow & T^1\cF \cr \Delta\uparrow\downarrow \Pi && \Pi_\downarrow\uparrow \Delta_\cr S & \leftarrow & T^1S \cr \end{matrix}$$ For every unit vector $u \in T^1_p\HH^+$, the geodesic $\gamma_u$ through $p$ tangent to $u$ has its extremities $\tilde \sigma^+(u)$ and $\tilde\sigma^-(u)$ in $\RR P^1$. This defines 2 smooth sections $\tilde\sigma^\pm:T^1\HH^+ \rightarrow T^1\HH^+ \times \CC P^1$. Let $Y_u$ be the holomorphic vector field on $\CC P^1$ vanishing at $\tilde\sigma^\pm(u)$ and having $Y_u(p) = u$. Let $\tilde Y$ be the smooth vector field defined on $T^1\HH^+ \times \CC P^1$ by $\tilde Y(v,.) := Y_v(.)$. $\tilde Y$ is tangent to the fibers $\{u\}\times\CC P^1, \quad u\in T^1\HH^+$. \[r.y\] Note that if $\tilde\sigma^-(u)= 0 \in \CC P^1$, $\tilde\sigma^+(u)=\infty$ and $u$ is the vector $i \in T_i\HH^+$ then $Y_u$ is the vector field $z\frac{\partial}{\partial z}$. So for every $u$, $Y_u$ is conjugate to $z\frac{\partial}{\partial z}$. The hyperbolic norm of $Y_u$ along the geodesic $\gamma_u$ is uniformly $1$. So the flow of $Y_u$ induces the translations along this geodesic. The derivative of $Y_u$ at the point $\tilde\sigma_0^-(u)$ is equal to $1$, and this does not depend on the metrics on $\CC P^1$. The flow lines of the vector field $z\frac{\partial}{\partial z}$ consist of semirays through $0$ having a north to south pole dynamics, with $0$ as a hyperbolic repellor and $\infty$ as a hyperbolic attractor. The vertical ray is a geodesic in $\HH^+$. The sections $\tilde\sigma^+$ and $\tilde \sigma^-$ and the vector field $\tilde Y$ are invariant by every $ T \in PSL(2,\RR)$, i.e.: $$\sigma^\pm(T_(v)) = T(\sigma^\pm(v)) \hskip 1cm, \hskip 1cm (T_\times T)_\tilde Y = \tilde Y$$ \[l.sections\] The endpoints of the geodesic determined by $T_v$ are $T(\sigma^\pm(v))$, so they are invariant, as well as $Y_{T_(v)} = T_Y_v$, by its definition. The sections $\tilde\sigma^\pm$ induce in the quotient bundle sections $\sigma^\pm$ from $T^1S$ to the $\RR P^1-$subbundle of $T^1\cF$, and $\tilde Y$ induces a vector field $Y$ on $T^1\cF$. The sets $\sigma^\pm(T^1S)$ are the zero sets of $Y$. The diagonal $\De$, $\sigma^+$ and $\sigma^-$ are $3$ smooth sections of $T^1\cF \rightarrow T^1S$, pairwise disjoint, and hence define a smooth trivialisation of the $\CC P^1-$fiberbundle $$[T^1\cF\to T^1S ]\sim [T^1S\times \CC P^1\to T^1S]$$ sending $\sigma^+$ to $\infty$, $\sigma^-$ to $0$ and $\De$ to $1$. \[c.trivial\] The unique thing we need to prove is that the sections are two by two disjoint. $\sigma^+$ and $\sigma^-$ are included in the $\RR P^1$ bundle which is disjoint from $\De$, since the image of $\Delta$ is in the upper half plane. The 2 points $\sigma^\pm(u)$ are the extremities in $\RR P^1$ of a geodesic in $\HH^+\subset \CC P^1$, so they are different. We will denote by $\|\cdot\|$ the Fubini Study metric on the fibers of $T^1\cF_{can}$ induced by the trivialisation $T^1\cF=T^1S\times \CC P^1$ given by Corollary \[c.trivial\]. In the trivialisation $T^1\cF_{can} \sim T^1S\times \CC P^1 $ given by Corolary \[c.trivial\] the flow $Y$ admits the sections $T^1S\times\{0\}$ and $T^1S\times\{\infty\}$ as zeros and the vertical derivative on every point $(u,0)$ is $1$. So in this coordinates the vector field $Y$ is $(0,z\frac{\partial}{\partial z})$. The Foliated Geodesic Flow -------------------------- Denote by $X$ and $X_{can}$ the infinitesimal generators of the geodesic and the foliated geodesic flows on $T^1S$ and $T^1\cF_{{can}}$, respectively, and $\varphi$ and $\Phi$ the corresponding flows, as in $(2.2)$. The vector fields $X_{can}$ and $Y$ on $T^1\cF_{{can}}$ commute. In particular, the set $Zero(Y)$ is invariant by $X$, so that $\sigma^+$ and $\sigma^-$ are invariant by $X$. \[p.commute\] It suffices to show that $\Phi_{t}Y=Y$, since $$[X,Y] = \lim_{t \rightarrow \infty} \frac{1 }{ t} [\Phi_{t}Y - Y] = 0.$$ The proof of this is easier on the universal cover $T^1\HH^+\times \CC P^1$. Let $\tilde X$ be the lift of $X$ to the universal covering space $T^1\HH^+ \times \CC P^1$. In this trivialisation, the foliated geodesic flow is generated by $(X,0)$, since the foliation is horizontal. So it is enough to prove the following statment: Let $u$ and $v$ be unit vectors tangent to the same geodesic $\gamma$ of $\HH^+$ at $x$ and $y$, and inducing the same orientation of $\gamma$. Then the vector fields $Y_u$ and $Y_v$ on $\CC P^1$ coincide. To prove the claim it is enough to notice that $\iota_(u)$ and $\iota_(v)$ are unit vectors for the hyperbolic metric of $\HH^+\subset \CC P^1$ tangent at the points $\iota(x)$ and $\iota(y)$ to the geodesic (for the hyperbolic metric) $\iota(\gamma)$. The vector field $Y_u$ is tangent to every point of $\gamma_u$ and its hyperbolic norm is $1$, moreover the orientation induced by $Y_u$ on $\gamma$ cannot change. So $Y_u(y)=\iota_(v)$ and so $Y_u=Y_v$. Hence $\Phi_{t}Y=Y$ as required. The claim shows that for every $u$ and every $v=\phi_t(u)$ the vertical vector field $Y$ on $\{v\}\times\CC P^1$ is $\Phi_{t}(Y\|_{\{u\}\times \CC P^1})$. Hence $\Phi_{t}Y=Y$ as required. The vector field $Z=X+Y$ is tangent to the diagonal $\tilde\De$. \[p.tangent\] The proof is easier on the cover $\HH^+\times \CC P^1$. Consider the following diagram: $$\begin{array}{ccc} &\tilde p&\\ \HH^+\times \CC P^1 & \leftarrow & T^1 {\tilde\cF}=T^1\HH^+\times\CC P^1 \\ \Pi \downarrow\quad\uparrow \De & & \Pi_\downarrow\uparrow \De_ \\ {\HH^+} & \leftarrow & T^1\HH^+ \\ &p&\\ \end{array}$$ To show that $X+Y$ is tangent to the diagonal $\tilde \De$ it is enough to show that, for every $u_x\in T^1\HH^+, x\in\HH^+$ the vector $\tilde p_((X+Y)(u_x,\iota(x))$ is tangent to $\De$ at the point $(x,\iota(x))$. On one hand, $\tilde p_(X(u_x,y))$ is the horizontal vector $(u_x,0)$ at the point $(x,y)$. On the other , $\tilde p_(Y(u_x,\iota(x)))$ is the vertical vector $(0,\iota_(u_x))$ at the point $(x,\iota(x))$. So the vector $\tilde p_((X+Y)(u_x,\iota(x)))$ is the vector $(u_x,\iota_(u_x))$ at the point $(x,\iota(x))$ and is tangent to $\De$. The flow $Z_t$ of $Z$ is horizontal in the trivialisation $T^1\cF$. In particular it induces isometries on the fibers $\CC P^1$ endowed with the metric $\|\cdot\|$. As $X$ and $Y$ commute and all preserve the fibration so does $Z$. Moreover, as $X$ and $Y$ induce on the fiber maps belonging to $SL(2,\RR)$ so does $Z$. To prove the corollary its suffices to show that $Z$ preserves the $3$ sections $\tilde\De$, $\tilde\sigma^+$ and $\tilde\sigma^-$. $Z$ is tangent to $\tilde \De$ by Proposition 7.7. $Y$ vanishes on $\sigma^\pm(T^1S)$ and $X$ is tangent to them by Proposition 7.6. [Proof of Theorem \[t.tautologic\]]{} The foliated geodesic flow is $X=Z-Y$. As these flows commute $X_t= Y_{-t}\circ Z_t$, where the notation corresponds to the flows of the corresponding vector fields. In the trivialisation given by Corollary \[c.trivial\] the flow $Z_t$ induces the identity on the fibers and $Y_{-t}$ is the homothety $z\to e^{-t}z$. Hence we obtain a contraction in the projective space, which may be translated to the affine space. This means that there is a section of largest expansion and contraction. The sections are smooth sections. The geodesic flow is recurrent hence the $\omega$ limit set of any point not in $\sigma^-(T^1S)$ is contained in $\sigma^+(T^1S)$. The $\alpha$ limit set of any point not in $\sigma^+(T^1S)$ is contained in $\sigma^-(T^1S)$. Along $\sigma^\pm(T^1S)$ the foliated geodesic flow $X_\De$ is hyperbolic. This proves the Theorem \[t.tautologic\]. Representation Topologically Equivalent to the Canonical Representation ------------------------------------------------------------------------ Let $$V := \{\rho := (A_1,\ldots,A_g) \in PSL(2,C) \ / \ \Pi_{1}^g[A_{2i-1},A_{2i}] = Id \}$$ be the complex algebraic variety parametrizing representations of the fundamental group $\pi_1(S)$ of the compact Riemann surface of genus $g \geq 2$, where $[A,B]:=ABA^{-1}B^{-1}$. We also have an action $$PSL(2,C) \times V \rightarrow V \eqno (7.1)$$ given by conjugation. Let $\rho_0$ be the representation corresponding to the canonical representation. Bers’s simultaneous uniformisation (\[19\]) implies that there is an open connected set $U \subset V$ containing $\rho_0$ such that all representations in $U$ are quasiconformally conjugate, and there is a surjective map $$U \rightarrow Teich^g \times Teich^g$$ which associates to each representation $\rho \in U$ the Riemann surfaces obtained by quotienting the region of discontinuity of $\rho$ by $\rho$, and its fibers are the $PSL(2,\CC)$ orbits $(7.1)$. For any representation $\rho$ in the above open set $U$, the Riccati equation with monodromy $\rho$ has a unique SRB-measure with basin of attraction of total Lebesque measure for positive and for negative times. By Theorem 6, the assertion is true for the canonical representation $\rho_{can}$. By Bers’s simultaneous uniformization, there is a quasiconformal map $h:\CC P^1 \rightarrow \CC P^1$ conjugating the action of $\rho_{can}$ to the action of $\rho \in U$. We may use this map to obtain a homeomorphism over $T^1S$ of the $\CC P^1$-bundles $H:Proj(E_{{can}}) \rightarrow Proj(E_\rho)$ conjugating the geodesic flows. This homeomorphism is absolutely continuos, since horizontally it is the identity and vertically it is the quasiconformal map $h$, which is absolutely continuos. Hence $ Proj(E_\rho)$ has a unique SRB-measure for positive and negative times, and it is $H_(\mu^\pm)$. [ZZZZZ]{} Y. Bakhtin and M. Martinez, [A characterization of harmonic measures on laminations by hyperbolic riemann surfaces,]{} Preprint Arxiv:0611235, to appear in Annales de l’Institut Hénri Poincaré, (2008). Ch. Bonatti and X. Gómez-Mont [Sur le comportement statistique des feuilles de certains feuilletages holomorphes,]{} Monogr. Enseign. Math, [38]{}, 15-41, (2001). Ch. Bonatti, X. Gómez-Mont and M. Viana [Généricité d’exposants de Lyapunov non-nuls pour des produits déterministes de matrices,]{} Annales Inst. Henri Poincaré, [20]{}, 579-624, (2003). Ch. 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Introduction. ============= The influence of quenched random pinning on the crystalline order in the vortex lattice is a subject of longstanding interest. The early pioneering works by Larkin [@Larkin] and Larkin and Ovchinnikov [@LarkinOvch] introduced the picture of collective pinning in which a vortex lattice breaks up into coherently pinned domains, and these domains adjust themselves to the pinning field independently of each other. The relative displacements of vortices grow fast with the distance within these domains until they reach the characteristic scale of the pinning potential associated with the vortex core size. The fact that different Larkin domains are pinned independently suggested that any weak disorder destroys long range crystalline order. The nature of the weakly pinned vortex state was further investigated in [@mfisher], where a concept of the low temperature vortex glass state with vortices locked near their equilibrium positions by infinite pinning barriers had been put forward. The topological nature and the consistency of the elastic description of the vortex glass has become a subject of extensive debate. It was suggested [@fisher91] that on scales where roughness becomes of the order of the lattice constant, topological defects in the form of dislocations must appear, converting the elastic vortex glass into a liquid-like substance. This claim was however disputed in [@Blatter] where a different point of view was advocated, namely, that vortex glass induced by weak pinning preserves its elastic integrity on all scales. The conclusion of [@fisher91] did not account for the fact that on scales where vortex lattice roughness becomes of the order of the vortex spacing and where according to [@fisher91] the generation of dislocations was supposed to start, the further growth of the roughness slows down and changes from the power-like short-scale behavior [@Feigelman89] to the logarithmic one [@Nattermann]. This ultra-slow logarithmic behavior reflects the fact that as the roughness exceeds the vortex spacing, the periodic structure of the lattice becomes essential and that on the largest spatial scales vortex glass maintains quasi-long-range order. This result of the seminal work by Nattermann [@Nattermann] was later confirmed and developed by Giamarchi and Le Doussal [@BraggGlass] and Korshunov [@korshunov] by using the variational replica method of Bouchaud [et al]{}[@bouchaud] allowing for derivation of relations between the different energy and spatial scales within the unique approach. Structure factor $S({\bf q})$ of quasilattices with quasi-long-range order (logarithmic divergence of roughness) shows power-law Bragg singularities. Owing to this property Giamarchi and Le Doussal proposed to call the glassy state resulting from weak disorder the “Bragg glass”, to distinguish this state from the glassy state without any crystalline order. The possibility of dislocation-free glassy phase has also been suggested by numerical simulations of the disordered XY model by Gingras and Huse [@GH]. A vortex lattice close to ideal has indeed been observed in clean $Bi_{2}SrCa_{2}Cu_{2}O_{x}$ (BiSCCO) crystals at small fields ($\lesssim $ 100 Oe) by decorations [@lat-dec], neutron diffraction [@lat-neut] and Lorentz microscopy [@lat-Lor] suggesting the existence of the topologically ordered quasilattice at low fields. At higher fields, however, the perfect lattice was never observed. This failure has been taken as the indication that elevated magnetic fields promote the action of disorder which destroys the vortex lattice. This view was also supported by numerous transport measurements [@transport] which show that at high fields the resistivity components have long Arrhenius tails indicating pinning-dominated behavior. The conclusion one can draw from the above observations is that there should be a disorder-driven phase transition between the low-field quasilattice or “Bragg glass” phase and the high field topologically disordered phase. Such transition indeed has been observed in numerical simulations of Ryu [et al.]{} [@Ryu] of the model of discrete lines interacting with the randomly distributed identical pinning centers. This understanding also correlates well with the recent theoretical findings by Kierfeld [et al.]{} [@kierf] that the periodic elastic medium should remain stable with respect to weak disorder but sufficiently strong disorder can trigger the formation of topological defects in the system involved. This putative transition can be identified with the so-called second peak field. A second peak in the magnetic hysteresis loops was reported by Kopylov [et al.]{} [@Kopylov] in Tl-based HTSC compounds and by Daeumling [et al ]{}[@Daeumling] for oxygen deficient YBa$_{2}$Cu$_3$O$_{7-x}$ (YBCO). Further numerous investigations [@Chikumoto91; @Chikumoto92; @Angadi91; @Kadowaki92; @Tamegai93; @Xu93; @Hardy94] demonstrated that it is actually a generic feature of anisotropic and relatively clean HTSC crystals. Local magnetization measurements of Zeldov [et al.]{} [@Zeldov-SecPeak] on high quality BiSCCO single crystals reveal that the onset of the peak is very sharp, suggesting that the peak indeed may mark some phase transition. Smoothness of the peak in earlier global measurement is a consequence of spatial average due to inhomogeneous induction inside the samples. Field distribution measurements by $\mu ^{+}$SR method [@muSR] show that vortex line wandering is strongly enhanced above the onset of the peak. Investigations on BiSCCO crystals with different oxygen content [@Khaykovich; @Kishio; @Tamegai] and different Tl-based compounds [@Hardy94] suggest that the peak field scales down with the decrease of the interlayer coupling. However further $\mu ^{+}$SR studies by Aegerter [et al.]{} [@Aegerter] show that in BiSCCO compounds with different doping the peak field correlates with the London penetration depth $\lambda _{ab}$ rather then with the interlayer coupling and scales as $\lambda _{ab}^{-2}$. Controlled irradiation experiments [@Chikumoto92; @Khaykovich1] have shown that the peak field decreases with the increase of pinning strength. These experimental facts support the interpretation of the second peak field as a field at which the vortex lattice is destroyed by pinning. A natural question arises: why should the lattice be more ordered at smaller fields? In fact, the collective pinning theory, which for many years was the only analytical tool to study pinning phenomena, suggests exactly the opposite. By balancing the elastic and random forces, one can immediately see that random forces dominate [ below]{} some typical field $B_{cp}$ and other vortices cannot prevent a given vortex to fall into the local minimum of the random potential at $B<B_{cp}$. Naively, one would expect that the vortex lattice will always be disrupted in this individual pinning regime. This fast conclusion occurs not always to be correct. In the situation when the typical distance between local minima is much smaller than the typical intervortex spacing, vortices have a wide choice of minima which gives them an extra possibility to minimize their interaction energy and restore the lattice. This situation is very typical for HTSC due to the large difference between the core size of the vortices and the intervortex spacing in the experimentally interesting field range. In order to examine vortex lattice stability with respect to weak disorder one has to compare elastic and pinning energies stored in an elementary ”prism” of the vortex lattice with the base made by the elementary cell and the height equal to the characteristic elastic screening length $L_{0}$ along the field direction [@LV], where $L_{0}=a_{0}/\gamma $, $a_{0}$ is the lattice spacing, and $\gamma $ is the anisotropy parameter. The analysis of the stability of the vortex lattice based on the Lindemann criterion has been done recently in [@ErtasNelson; @Vin; @Giamarchi; @Kierfeld] for an anisotropic 3D superconductor. This analysis shows that the lattice is expected to be stable below some typical field and relates this field with the parameters of the superconductor and the pinning strength. Unfortunately, most of these estimates cannot be applied directly to highly anisotropic layered superconductors like BiSCCO, because in [@ErtasNelson; @Giamarchi; @Kierfeld] the discrete nature of the vortex lines imposed by the layered structure is not taken into account and therefore applies only to continuous elastic strings and in the electromagnetic interaction between vortices was neglected. In this paper we extend the analysis of Refs. to the case of very weakly coupled layered superconductors. Two important features distinguish layered superconductors from three dimensional superconductors. Vortex lines in layered superconductors consist of discrete segments (“pancakes”). All properties of superconductors are strongly influenced by this discrete nature of the lines. For continuous elastic strings, which are usually used to model the vortex lines, only segments of the order of the Larkin length $l_{c}$ have the possibility to explore different minima of the random potential. In layered superconductors the description in terms of elastic strings is valid only if $l_{c}$ is much larger than the separation between layers $s$. This condition is violated in the weakly coupled layered superconductors like BiSCCO. When $l_{c}$ drops below $s$, pancakes in neighboring layers acquire the possibility to explore different minima of the random potential. Vortex line wandering in such situation has been discussed in Ref. in conjunction with the problem of disorder induced decoupling. Another important feature of layered compounds is that in the region where magnetic interactions between pancake vortices in different layers dominate coupling, the tilt energy of the pancake stacks becomes strongly non-local, thus modifying significantly the overall physical picture of the pinning-induced lattice destruction. Pinning-induced vortex lines wandering in layered superconductors ================================================================= Consider an isolated vortex line in a disordered layered superconductor oriented orthogonally to the layers (z-direction). Adjustment of pancake vortices to the pinning potential leads to wandering of the vortex line in the z-direction. The key parameter, which determines the amplitude of this wandering, is an elemental wandering distance $r_{{\rm w}}$ [@DID] or typical distance between two pancakes in adjacent layers belonging to the same vortex line. Two distinct types of behavior emerge depending on the relation between the strength of the interlayer coupling and the pinning strength. If the distance $r_{{\rm w}}$ is smaller than the core size $\xi $ (weak pinning) the discreteness of vortices is not relevant, and their lateral wandering is described by the usual elastic string model. In the opposite limit, $r_{{\rm w}}>\xi$, the discreteness plays an essential role. Now pancakes in the neighboring layers have the possibility to stretch their relative displacements to distances much larger than the typical spacing between the minima of the random potential, to explore a whole lot of local minima within the area of $r_{{\rm w}}^{2}$ and thus choose the best of all of them. The distance $r_{{\rm w}}$ is determined by the balance between the elastic tilt energy and the random energy [@DID]. The tilt energy of the vortex line $${\cal E}_{{\rm tilt}}[{\bf u}_{n}]= {\cal E}_{{\rm J}}[{\bf u}_{n}]+{\cal E}_{{\rm M}}[{\bf u}_{n}] \label{En-tilt}$$ consists of the Josephson, $${\cal E}_{{\rm J}}[{\bf u}_{n}]= \sum_{n}{\frac{\pi E_{{\rm J}}}{{2}}}({\bf u}_{n}-{\bf u}_{n-1})^{2} \ln {\frac{r_{{\rm J}}}{\left\| {\bf u}_{n}-{\bf u}_{n-1}\right\| },} \label{Eel-J}$$ and magnetic, $${\cal E}_{{\rm M}}[{\bf u}_{n}]= \frac{E_{{\rm M}}}{2s}\int \left( dk_{z}\right) \ln \left( 1+\frac{r_{{\rm cut}}^{2}Q_{z}^{2}}{1+r_{{\rm w}}^{2}Q_{z}^{2}}\right) \left\| {\bf u}(k_{z})\right\| ^{2}, \label{el-M}$$ contributions. Here ${\bf u}_{n}$ is the pancake displacement in the $n$-th layer and ${\bf u}(k_{z})$ is the corresponding Fourier transform, ${\bf u}(k_{z})=s\sum_{n}\exp (-ik_{z}z_{n}){\bf u}_{n}$. $E_{{\rm J}}=\varepsilon _{0}/\left( \pi \gamma ^{2}s\right) $ and $E_{{\rm M}}=s\varepsilon _{0}/2\lambda_{ab}^{2}$ are the Josephson and magnetic energies per unit area, $\varepsilon _{0}=\Phi _{0}^{2}/\left( 4\pi \lambda _{ab}\right) ^{2}$, $r_{{\rm J}}=\gamma s$ is the Josephson length, $\gamma =\lambda _{c}/\lambda _{ab}$ is the anisotropy ratio of the London penetration depths $\lambda _{ab}$ and $\lambda _{c}$, $s$ is the interlayer spacing, and $Q_{z}^{2}=\frac{2}{s^{2}}\left( 1-\cos k_{z}s\right) $. The expression under the logarithm in the magnetic coupling tilt energy describes the crossovers between different regimes depending on the relations between different length scales of the problem ($\lambda_{ab}$, s, lattice spacing $a$, and $r_{w}$) [@KoshKes93; @Blatter-Magn]. The cut off length $r_{{\rm cut}}$ can be estimated from $r_{{\rm cut}}^{-2}\approx \lambda _{ab}^{-2}+21.3B/\Phi _{0}$. Consider a randomly misaligned vortex line with a typical relative displacement of adjacent pancakes from one stack $r\sim \left\| {\bf u}_{n}-{\bf u}_{n-1}\right\|$. The local energy change $\epsilon (r)$ per pancake caused by this misalignment includes the loss in the coupling energy $\epsilon _{{\rm coup}}(r)$ and the gain in the random energy $\epsilon _{{\rm ran}}(r)$, $\epsilon (r)=\epsilon _{{\rm coup}}(r)+\epsilon _{{\rm ran}}(r)$. The coupling term is determined by the tilt energy (\[En-tilt\]) at large wave vectors $k_{z}\sim \pi /s$ and, as we mentioned above, can be split into the Josephson and magnetic contributions: $$\epsilon _{{\rm coup}}(r)= \frac{\pi }{2}E_{{\rm J}}r^{2}\ln \frac{r_{{\rm J}}}{r} +E_{{\rm M}}r^{2}\ln \frac{r_{{\rm cut}}}{r}. \label{coupling}$$ Note the different physical origin of these terms: while the loss in Josephson energy is related to the interaction of a given pancake with pancakes in neighboring layers, the magnetic energy is determined by the averaged interaction with pancakes from a large number ($r_{{\rm cut}}/s$) of remote layers. The term $\epsilon _{{\rm ran}}(r)$ describes the gain in the pinning energy upon adjusting the position of the given pancake to a best minimum of the random potential within the area $\pi r^{2}$. We consider $r\gg \xi $ (weak coupling) so that pancakes have the possibility to choose among a large number of minima of the random potential. In such a situation, the function $\epsilon _{{\rm ran}}(r)$ is determined by the distribution of the pinning energies, which for pinning due to point defects is natural to expect to have a Gaussian form. Accordingly, the concentration of minima with energies between $\epsilon $ and $\epsilon +d\epsilon $ is $P(\epsilon )d\epsilon $ with $$P(\epsilon )=\frac{\exp \left( -\frac{\epsilon ^{2}}{U_{p}^{2}}\right)} {r_{{\rm p}}^{2}U_{{\rm p}}}, \label{Concentr}$$ where $U_{{\rm p}}$ is the pinning energy and $r_{{\rm p}}$ is the typical pinning size. Numerical investigation of random potential for $\delta T_{c}$-pinning give $r_{{\rm p}}\approx 6.3\xi $, where $\xi $ is the coherence length. The function $\epsilon _{{\rm ran}}(r)$ is determined by the condition that the number of centers with pinning energy less than $\epsilon _{{\rm ran}}$ in an area $\sim r^{2}$ is of the order of one, i.e., $\pi r^{2}\int^{\epsilon }P(\varepsilon )d\varepsilon \sim 1$. For large displacements, $r\gg r_{{\rm p}}$, this condition gives $$\epsilon _{{\rm ran}}=-U_{p}\ln ^{1/2}\left( \frac{r^{2}}{2\sqrt{\pi }r_{p}^{2}}\right). \label{erandom}$$ The optimization of $\epsilon (r)$ with respect to $r$ gives the following equation for $r_{{\rm w}}$: $$r_{{\rm w}}^{2}=\frac{U_{p}}{\left( \pi E_{{\rm J}}\ln \frac{r_{{\rm J}}}{r_{{\rm w}}} +2E_{{\rm M}}\ln \frac{r_{{\rm cut}}}{r_{{\rm w}}}\right) \ln ^{1/2}\left( \frac{r_{{\rm w}}^{2}}{2\sqrt{\pi }r_{p}^{2}}\right) }, \label{rwander}$$ which is valid if $r_{{\rm w}}\gg \xi $. In the limit where the interlayer coupling is not too weak, $\gamma <\lambda _{ab}/s$, the distance $r_{{\rm w}}$ is mainly determined by the Josephson coupling, and an approximate solution of Eq. (\[rwander\]) is given by $$r_{{\rm wJ}}^{2}\approx \frac{U_{p}}{\pi E_{{\rm J}}\ln \frac{s\varepsilon _{0}}{U_{p}}\ln ^{1/2}\left( \frac{U_{p}}{E_{{\rm J}}r_{p}^{2}}\right) }; \label{rwandJos}$$ In the opposite limit of very weak coupling, $\gamma >\lambda _{ab}/s$, the wandering length is determined by the magnetic coupling, $$r_{{\rm wM}}^{2}\approx \frac{U_{p}}{E_{{\rm M}} \ln \frac{E_{{\rm M}}r_{{\rm cut}}^{2}}{U_{p}}\ln ^{1/2} \left( \frac{U_{p}}{E_{{\rm M}}r_{p}^{2}}\right) }. \label{rwanMagn}$$ In the latter case $r_{{\rm w}}$ is mainly controlled by the London penetration depth $\lambda _{ab}$ and does not depend upon the anisotropy ratio at all. Point pinning induces lateral random wandering of the vortex line as it traverses a sample in the z-direction so that the line displacements grow as ${\bf u}_{n}={\bf u}(z_{n})$, which is commonly characterized by the mean squared displacement or [roughness]{}, $w(z_{n})$, $w(z_{n})=\sqrt{\left\langle \left( {\bf u}_{n}-{\bf u}_{0}\right) ^{2}\right\rangle }$. The shape of the function $w(z_{n})$ is determined by the nature of interlayer coupling. In the case of dominating magnetic coupling the energy of the mismatched pancake stack (\[el-M\]) is strongly nonlocal, i.e., the energy cost of pancake displacement in a given layer is not simply determined by the pancake positions in two adjacent layers but it is determined by the whole line configurations at distances $r_{\rm cut}=\min(\lambda _{ab}, a)$ from the given layer. A rough estimate for the displacement field at small distances $z\ll r_{{\rm cut}}$ can be obtained if we neglect weak logarithmic $k_{z}$-dependence of the tilt energy (\[el-M\]) which allows us to describe the collective interaction with a large number of pancakes in terms of an effective “cage potential” $${\cal E}_{{\rm M}}^{{\rm (cage)}}\approx E_{{\rm M}} \ln \left( \frac{r_{{\rm cut}}}{r_{{\rm wM}}}\right) \sum_{n}{\bf u}_{n}^{2}. \label{em-cage}$$ In the “cage” approximation the correlations between the layers are absent and the displacement field $w(z)$ does not increase at all $$w(z)\approx r_{{\rm wM}},\,\text{at }z< r_{\rm cut}. \label{m_displ}$$ At small distances a weak dispersion of the tilt energy is to be taken into account perturbatively and gives rise to a weak increase of $w(z)$ $$w(z)\approx r_{{\rm wM}} \left[ 1-\frac{s}{4z\ln \frac{r_{{\rm cut}}}{r_{{\rm wM}}}}\right]. \label{m_displ1}$$ Only at very large distances $z\gg r_{\rm cut}$ the nonlocality becomes irrelevant and displacements start to grow again. The Josephson interlayer coupling provides an additional contribution to the tilt energy of a misaligned pancake stack (\[Eel-J\]) which is determined by the interactions between neighboring pancakes in adjacent layers. In the case $\gamma <\lambda _{ab}/s$ the Josephson energy dominates at small distances. Due to the locality of interactions the pinning-induced relative displacement $w(z_{n})$ grows with $z_{n}$ similar to the elastic string: $$w(z)=r_{{\rm wJ}}(z/s)^{\zeta }. \label{displ}$$ The rate of growth is determined by the universal wandering exponent $\zeta $. Elaborate numerical analysis [@WandExp] gives $\zeta \approx 5/8$, which is larger than the naively expected ”random walk” exponent $1/2$. It is also slightly larger that the value $3/5$ suggested by the renormalization group analyses [@Halpin-Healy] and by a simple scaling reasoning [@Feigelman89] (see also review [@Halpin-Healy95]). Relative displacements for the layered superconductor given by Eq. (\[displ\]) differ from displacement for weakly pinned vortex line in 3D superconductors $w(z)=\xi (z/l_{c})^{\zeta }$ by the elemental lengths: $r_{{\rm wM}}$ replaces the core size $\xi$ and the interlayer spacing $s$ comes instead of the Larkin length $l_{c}$. The 3D expression for $w(z)$, which has been used in Refs. , is valid only if $l_{c}>s$. The last condition is strongly violated in highly anisotropic layered high-T$_{c}$ materials like BiSSCO and in these materials one has to use Eqs. (\[m\_displ\],\[m\_displ1\]) or Eq. (\[displ\]) depending on relative strength of the Josephson and magnetic couplings. Note that even in the case of relatively strong Josephson coupling ($\gamma <\lambda _{ab}/s$) the growth of $w(z)$ at large distances is restricted by magnetic interactions. Comparison of the Josephson (\[Eel-J\]) and magnetic (\[el-M\]) tilt energies shows that above the typical length $L_{\rm J-M}\approx \pi \sqrt{\frac{2}{\ln \left( \gamma \right) }}\frac{\lambda _{ab}}{\gamma }$ the magnetic coupling takes over again and the displacement field crosses over from dependence (\[displ\]) to the almost $z$ independent behavior $$w(L_{{\rm J-M}})\approx r_{{\rm wJ}}\left( \frac{\lambda _{ab}}{\gamma s}\right) ^{\zeta }. \label{wLJ_M}$$ Different regimes of line wandering are illustrated in Fig. \[Fig-LineWander\]. Destruction of the vortex crystal by disorder ============================================= Interaction length scale $l_{3D}$ --------------------------------- Lateral displacements of an individual vortex line would grow infinitely with $z$, however the presence of other vortices restricts its wandering. A given line starts to feel its neighbors at scales exceeding some typical length $l_{3D}$, which marks the crossover between the single vortex and three dimensional (bundle) regimes. To estimate this length one has to balance the typical tilt energy at the wave vector $\pi /l_{3D}$, $C_{44}(\pi /l_{3D})^{2}$ and the typical shear energy $4\pi C_{66}B/\Phi _{0}$. This gives the following estimate $$l_{3D}\approx \sqrt{\frac{C_{44}\Phi _{0}}{C_{66}B}}. \label{lint}$$ The individual line description applies as long as $l_{3D}$ is larger than the separation between the layers. The shear and tilt moduli for layered superconductor in the field range $\Phi /\lambda _{ab}^{2}<B\ll H_{c2}$ are given by [@GK; @BrandtSudbo; @KoshKes93] $$\begin{aligned} C_{66} &=&\frac{B\Phi _{0}}{(8\pi \lambda _{ab})^{2}}; \label{moduli} \\ C_{44}({\bf k}) &=&\frac{B^{2}/4\pi }{1+\lambda _{c}^{2}k_{\parallel }^{2}+\lambda _{ab}^{2}Q_{z}^{2}}+\frac{B\Phi _{0}}{2(4\pi \lambda _{c})^{2}} \ln \frac{k_{\max }^{2}}{K_{0}^{2}+(Q_{z}/\gamma )^{2}} \nonumber\\ &+&\frac{B\Phi _{0}}{2(4\pi )^{2}\lambda _{ab}^{4}Q_{z}{}^{2}} \ln \left(1+\frac{r_{{\rm cut}}^{2}Q_{z}^{2}}{1+r_{{\rm w}}^{2}Q_{z}^{2}}\right) \nonumber\end{aligned}$$ with $K_{0}=\sqrt{4\pi B/\Phi _{0}}$ being the average radius of the Brillouin zone. Three terms in the tilt modulus correspond to the nonlocal collective, Josephson single vortex and magnetic contributions respectively. In our case the cutoff wave vector $k_{\max }$ in the second term of $C_{44}$ can be estimated as $k_{\max }\approx \pi /r_{{\rm w}}$. The nonlocal tilt modulus in Eq. ( \[lint\]) should be taken at $k_{z}=\pi /l_{3D}$ and $k_{\parallel }=K_{0}$. Two field regimes exist depending upon the strength of the interlayer coupling. For small Josephson coupling $\gamma >\lambda _{ab}/s$ the length $l_{3D}$ is determined by the magnetic coupling in the whole field range and decays exponentially at $B>\Phi /\lambda _{ab}^{2}$ $$l_{3D}\approx a\exp \left( -\frac{B(\pi \lambda _{ab})^{2}}{4\Phi _{0}}\right) . \label{l3DMagn}$$ At field $$B_{{\rm M}cr}\approx \frac{4\Phi _{0}}{(\lambda _{ab}\pi )^{2}}\ln \frac{ \lambda _{ab}}{r_{\rm w}} \label{CrossMagn}$$ the length $l_{3D}$ matches $r_{\rm w}$, indicating the crossover to the quasi-2D regime. In the case of not too weak Josephson coupling $\gamma <\lambda _{ab}/s$, two relevant length scales govern the behavior of a single vortex line: interlayer spacing $s$ and the crossover length $L_{{\rm J-M}}$ introduced in the previous Section. One can distinguish three field regimes corresponding to different relations between $l_{3D}$ and these lengths. At fields smaller than the typical field $B_{{\rm J-M}}=\frac{2\Phi _{0}}{(\lambda _{ab}\pi )^{2}}\ln \left( \frac{0.1\gamma ^{2}}{\ln (\lambda_{ab}/r_{\rm w})}\right)$ the interaction crossover takes place at length scales larger than $L_{{\rm J-M}}$ where magnetic coupling still dominates and the length $l_{3D}$ is again determined by Eq. (\[l3DMagn\]). At higher fields the Josephson coupling dominates and the length $l_{3D}$ is given by $$l_{3D}\approx 2\frac{a}{\gamma }\sqrt{\ln \frac{a}{r_{{\rm w}}}}. \label{l3DJos}$$ The crossover to the quasi-2D regime takes place at the field $B=B_{{\rm J}cr}$ [@fisher91; @GK], $$B_{{\rm J}cr}<\frac{4\Phi _{0}}{\left( \gamma s\right) ^{2}}\ln \frac{\gamma s}{r_{{\rm w}}} \label{CrossJos}$$ The schematic field-anisotropy phase diagram shown in Fig. \[Fig-gam-B\] summarizes the above description of the different regimes. The field at which random pinning destroys the quasilattice or the “Bragg glass” phase can be estimated from the Lindemann-like criterion. [@ErtasNelson; @Vin; @Giamarchi; @Kierfeld] The destruction of the lattice is expected when the displacement$w(z)$ at $z\approx l_{3D}$ reaches some fraction of the lattice spacing $$w(l_{3D})=c_{L}a \label{Criterion}$$ Several regimes exist depending upon the strength of coupling and the strength of pinning. Dominating magnetic coupling ---------------------------- In the regime where magnetic coupling dominates ($\gamma >\lambda _{ab}/s$) the line roughness $w(z)$ almost does not depend on $z$ and the criterion (\[Criterion\]) can be written simply as $$r_{{\rm wM}}=c_{L}a, \label{CriterionM}$$ which leads to the estimate of the field at which the “Bragg glass” is destroyed: $$B_{x}=C_{M}\frac{\Phi _{0}}{\lambda _{ab}^{2}}\frac{T_{m}^{2D}}{U_{p}} \label{BxM}$$ Here $T_{m}^{2D}\approx s\varepsilon _{0}/70$ is the melting temperature for a single pin-free 2D layer and $C_{M}=35c_{L}^{2}\sqrt{\ln \left( \frac{2\lambda _{ab}^{2}U_{p}}{s\varepsilon _{0}r_{p}^{2}}\right) }\approx 1\div 2$ is a numerical constant weakly depending on the parameters. The melting temperature gives a natural scale for the pinning strength. It is important to note that the relative displacement of pancakes in neighboring layers becomes comparable with the lattice spacing about the same field $B_{x}$. This means that misalignment transition and disorder-induced destruction of the vortex lattice merge and that above $B_{x}$ one is going to find the state with both completely misaligned and disordered configuration of pancakes. This is very similar to merging of the lattice disordering and the misalignment transitions induced by the magnetic coupling first pointed out by Blatter [et al.]{} [@Blatter-Magn] in the context of the melting transition. A very unusual feature of the transition imposed by dominating magnetic coupling is that the transition field (\[BxM\]) is almost insensitive to the shear stiffness of the lattice. The reason is that due to the strong nonlocality of magnetic interactions the wanderings of the given line are self-confined, i.e. the mean squared displacements at relevant length scales are mainly determined by the interactions of the pancakes belonging to the same string, and interaction with other strings very weakly influences these displacements. The disorder-induced destruction of the vortex lattice can occur only if $B_{x}$ lies below the dimensional crossover field $B_{{\rm M}cr}$ (\[CrossMagn\]). This gives the following condition for the pinning strength $$U_{p}\gtrsim 0.5T_{m}^{2D}, \label{CondPin}$$ which is simply a condition that pinning is strong enough to destroy the 2D lattice in a single layer. For weaker pinning the “Bragg glass” remains stable in the quasi-2D regime up to fields close to $H_{c2}$. On the other hand, if $U_{p}$ is substantially larger than $T_{m}^{2D}$ then the field $B_{x}$ falls into the region of exponentially weak interacting vortices. In this case the intermediate quasilattice state collapses and the vortex lattice is in a disordered state throughout the whole field range. One can conclude therefore that the intermediate quasilattice state at fields $\sim \Phi _{0}/\lambda _{ab}^{2}$ in magnetically coupled superconductors exists only within the limited range of the pinning strength where $U_{p}\sim T_{m}^{2D}$. Strong Josephson coupling ------------------------- Now we turn to the case $\gamma <\lambda _{ab}/s$. Due to the existence of the two relevant length scales controlling wandering of a single line, $s$ and $L_{\rm J-M}$, the behavior is very rich. Depending on the pinning strength the destruction of the “Bragg glass” phase may take place either in the “3D Josephson-” or in the “3D Magnetic” regimes (see Fig. \[Fig-gam-B\]) or it may not happen at all. When the transition falls into the “3D Josephson” region we obtain from Eqs.(\[Criterion\],\[displ\],\[l3DJos\]) the following estimate for the transition field $$B_{x}=\frac{\Phi _{0}}{\left( \gamma s\right) ^{2}}\left( \frac{c_{w}\gamma s }{r_{{\rm w}}}\right) ^{2\beta} \label{Bx3DJos}$$ with $\beta={\frac{1}{1-\zeta }}\approx 8/3$ and $c_{w}=c_{L}\left( 4\ln \frac{a}{r_{{\rm w}}}\right) ^{-\zeta /2}=0.08 \div 0.1$. Using Eq.(\[rwandJos\]) $B_{x}$ can also be connected with the pinning potential $U_{p}$ $$B_{x}=\frac{\Phi _{0}}{\left( \gamma s\right) ^{2}}\left( \frac{ C_{J}T_{m}^{2D}}{U_{p}}\right) ^\beta \label{BxJos}$$ with $C_{J}=15\ln \frac{s\varepsilon _{0}}{U_{p}} \left( \ln \frac{\Phi_{0}E_{{\rm J}}}{BU_{p}}\right) ^{-\zeta } \sqrt{\ln \left(\frac{U_{p}}{E_{{\rm J}}r_{p}^{2}}\right) }c_{L}^{2} \approx 0.2\div 0.35$. It is interesting to note that in this regime $B_{x}\propto 1/\gamma ^{2}$ in agreement with experimental trends [@Khaykovich; @Kishio; @Tamegai]. Eq.(\[BxJos\]) is valid provided $B_{x}$ falls into the interval $B_{\rm J-M}<B_{x}<B_{\rm Jcr}$. This gives the following conditions for the elemental wandering distance $$0.05\gamma s\lesssim r_{{\rm w}}\lesssim 0.05\gamma s\left( \frac{\pi \lambda _{ab}}{\gamma s}\right) ^{1-\zeta }. \label{rw_cond}$$ and for the pinning strength $$0.5T_{m}^{2D}\lesssim U_{p}\lesssim 0.5T_{m}^{2D}\left( \frac{\pi \lambda _{ab}}{\gamma s}\right) ^{2(1-\zeta )}. \label{Up_cond}$$ For weaker pinning (smaller $r_{{\rm w}}$) the “Bragg Glass” is not destroyed by pinning. For stronger pinning (larger $r_{{\rm w}}$) the transition falls into the “3D Magnetic” regime. In this regime the criterion for lattice destruction can be written as. $$w(L_{{\rm J-M}})=c_{L}a.$$ and leads to the following estimate for the transition field $$B_{x}=C_{{\rm JM}}\frac{\Phi _{0}}{\lambda _{ab}^{2}}\left( \frac{\lambda _{ab}}{\gamma s}\right) ^{2(1-\zeta )}\frac{T_{m}^{2D}}{U_{p}} \label{BxInt}$$ with $C_{{\rm JM}}=70c_{L}^{2}\ln \frac{s\varepsilon _{0}}{U_{p}}\sqrt{\ln \left( \frac{U_{p}}{E_{{\rm J}}r_{p}^{2}}\right) }\approx 10-15$. The “elastic string” regime of the transition considered in Refs. exists at least for some pinning strength only if the crossover from the pancake pinning regime to the string pinning regime of an individual vortex line occurs at $U_{p}\gtrsim T_{m}^{2D}$. This gives the relation $E_{{\rm J}}\xi ^{2}>T_{m}^{2D}$ which can be rewritten as the condition for the anisotropy $\gamma s\lesssim 20\xi $. Different regimes of quasilattice destruction represented by Eqs.(\[BxM\],\[BxJos\],\[BxInt\]) are summarized in the anisotropy-pinning phase diagram (Fig. \[Fig-U-gamma\]). Discussion ========== Taking typical parameters for optimally doped BiSSCO, $\lambda_{ab}\approx 2400\AA$ [@Aegerter], $\gamma=200-300$, $s=15\AA$, and $U_{p}=10-15$K we can conclude that BiSSCO is located in region (3) of the phase diagram of Fig. \[Fig-gam-B\] (region of dominating magnetic coupling) but not too far from the boundary with region (2) so that the Josephson coupling is probably not completely negligible. Estimate for the transition field in region (3) is given by Eq. (\[BxM\]). Substituting the parameters, we obtain $B_{x}=200-300$ G, in reasonable agreement with experiments [@Aegerter; @Khaykovich]. The above approach can specify a position for the transition line from the quasilattice to disordered solid, but can neither prove the existence of such a transition, nor describe the nature of the resulting high field entangled solid phase. The first question requiring understanding the entangled solid state is the very result that enabled us to observe this transition: what is the mechanism of the sharp increase of the persistent current measured in the magnetization experiment? Indeed, one can think that since at the moment of transition pinning in the quasilattice is dominated by the single vortex pinning regime, the critical current can only drop with the increase of the magnetic field. However as we have already mentioned in the Introduction this “classical” collective pinning approach does not account for the possibility that pancakes can explore the deep bound states. A maximum possible jump of the critical current at the transition is expected for the case of dominated magnetic coupling. In this case vortex lines are destroyed at the transition point and it is reasonable to assume that above the transition each pancake has the possibility to choose the optimum location within the area $(\pi/4) a_0^2$. If we assume that the distribution of the energy minima $E_m$ for pancakes obeys Gaussian statistics $P(E_m)\propto \exp(-E_m^2/U_p^2)$, the typical pinning energy that a pancake can find on the available area just [above]{} the transition is estimated as $$E_{pin}^{>}\simeq U_p\sqrt{\ln\frac{a_0^2}{\xi^2}}. \label{crittoka}$$ On the other hand, just [below]{} the transition point pancakes has the possibility to look for the best pinning center only within the area $u^2$, where $u$ is the mean squared deviation of the vortex lines from its average position, which at the transition point is given by the Lindemann relation $u=c_La$. Therefore a typical pinning energy [ below]{} the transition can be estimated as $$E_{pin}^{<}\simeq U_p\sqrt{\ln\frac{c_L^2a_0^2}{\xi^2}}. \label{crittokb}$$ Since the critical current in a single pancake pinning regime is about $J_c\simeq E_{pin}/\xi$, we see that one can expect a substantial growth of the critical current at the transition. For typical BiSCCO parameters the critical current can jump by a factor of 2. However the experimental jump of the persistent magnetization current is only indirectly related to the enlargement of the critical current, because in real experiment the persistent current is strongly reduced by thermal creep. The jump most probably reflects the enhancement of the relaxation barrier above the transition as was indeed observed in recent elaborated creep measurements of Konczykowski [et al.]{} [@koncz]. Quantitative description of this enhancement is a challenging problem due to the collective nature of relaxation both below and above the transition. Another important question is whether the entangled solid is a glass or the glassy state is destroyed by a massive proliferation of dislocations. A very naive expectation is that dislocations convert the vortex solid into a liquid, infinite barriers for vortex motion disappear and collective creep over divergent barriers transforms into a plastic creep governed by the motion of dislocation over the finite Peierls barriers. More attentive (although still naive) analysis shows however, that being linear elastic objects, dislocations may be pinned in turn recovering the glassy response of the entangled solid. Indeed, recent relaxation measurements on YBCO crystals [@Abulafia95] indicate that creep activation barriers show the linear current dependence in a wide range of currents and only at very small currents some upturn in current dependence of energy is observed, suggesting the appearance of diverging barriers. On the other hand, it is not quite clear whether this naive dislocation-based approach applies to highly anisotropic superconductors. One can notice that neglecting the Josephson coupling one arrives at creep governed by two-dimensional pancake diffusion in the entangled phase. Since the diverging barriers do not appear for 2D particle diffusion, the presence of electromagnetic coupling only cannot provide glassiness. One can notice however that switching on even weak Josephson coupling recovers glassy response, giving rise to a [phase glass]{} (or dislocation glass, or plastic glass) phase above the entanglement transition. The glassy properties of the entangled solid will be discussed in detail elsewhere. 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--- abstract: 'Joint user selection and precoding in multiuser MIMO settings can be interpreted as group sparse recovery in linear models. In this problem, a signal with group sparsity is to be reconstructed from an underdetermined system of equations. This paper utilizes this equivalent interpretation and develops a computationally tractable algorithm based on the method of group LASSO. Compared to the state of the art, the proposed scheme shows performance enhancements in two different respects: higher achievable sum-rate and lower interference at the non-selected user terminals.' author: - bibliography: - 'ref.bib' title: Joint User Selection and Precoding in Multiuser MIMO Systems via Group LASSO --- User selection, precoding, group LASSO, massive MIMO. Introduction ============ Performance gains are often achieved in multiuser massive systems with a large number of transmit antennas per user [@hoydis2013massive]. As a result, in dense settings in which the number of available users is comparable to the number of transmit antennas, user selection is required along with downlink beamforming [@dimic2005downlink; @shen2005low; @wang2008user; @huang2013user]. The conventional approach for user selection and precoding is to divide them into two separate problems: First, a subset of users is selected; then, the information signals of the selected users are precoded via a classic precoding scheme [@huang2013user]. Generally, the optimal approach for user selection deals with integer programming. Hence, this problem is often addressed via sub-optimal greedy algorithms [@dimic2005downlink; @shen2005low]. In this work, we deviate from the conventional approach and propose a scheme for joint user selection and downlink beamforming. User Selection and Precoding as Group Sparsity ---------------------------------------------- Joint user selection and beamforming is interpreted as the problem of constructing a signal with group sparsity. To clarify this point, assume a multiuser downlink scenario with $M$ transmit antennas and $K$ users in which we wish to select a subset of $L$ users. A linear precoder in this problem can be seen as a signal with $MK$ entries, such that each block of size $M$ represents an individual beamforming vector. By such a formulation, joint user selection and downlink beamforming with respect to some performance metric, e.g., the achievable sum-rate or , reduces to the problem of finding a signal with group sparsity: [A signal of size $MK$ in which only $L$ blocks of size $M$ have non-zero entries]{}. Following this equivalent interpretation, we employ the framework for precoding, recently developed in [@bereyhi2017nonlinear; @bereyhi2017asymptotics; @bereyhi2018glse], to formulate joint user selection and precoding as the problem of group sparse recovery in a linear model. A computationally tractable algorithm is then developed based on group to address this problem. Our investigations show significant performance enhancements compared to the state of the art. Notations --------- Throughout the paper, scalars, vectors, and matrices are represented by non-bold, bold lower case, and bold upper case letters, respectively. The real axis is denoted by $\setR$ and the complex plane is shown by $\setC$. $\mH^{\her}$, $\mH^{}$, and $\mH^{\trp}$ indicate the Hermitian, complex conjugate, and transpose of $\mH$, respectively. $\log\left(\cdot\right)$ is the binary logarithm. We denote the statistical expectation by $\Ex{\cdot}{}$. $\diag{\bt}$ represents the diagonal matrix constructed from the elements of vector $\bt$. Problem Formulation {#sec:sys} =================== Consider a multiuser system with multiple which are equipped with transmit antenna arrays of size $M$. The system is intended to serve $K$ single-antenna . For mathematical tractability, we focus on a single which aims to transmit information to a group of $L \leq K$ . System Model ------------ The system operates in the mode. Hence, the uplink and downlink channels are reciprocal. In each coherence time interval, the transmit known training sequences. The then utilizes these sequences to estimate the . Let $\bh_k\in\setC^{M}$ denote the vector of uplink channel coefficients between $k$ and the . The signal received by $k$ is hence given by $$\begin{aligned} y_k = \bh_k^\trp \bx + z_k\end{aligned}$$ where $z_k$ is additive complex Gaussian noise with zero mean and variance $\sigma_k^2$, i.e., $z_k \sim \mathcal{CN} \brc{0,\sigma_k^2}$, and $\bx$ is the downlink transmit signal constructed from the information symbols of the selected and the via linear precoding. As a result, the transmit signal is written as $$\begin{aligned} \bx = \sum_{\ell \in \setS} \sqrt{p_\ell} s_\ell \bww_\ell.\end{aligned}$$ where $\setS$, $s_\ell$, $p_\ell$ and $\bww_\ell$ are defined as follows: 1. $\setS\subseteq \set{1, \ldots,K}$ represents the subset of $L$ selected by the for downlink transmission. 2. $s_\ell$ is the information symbol of user $\ell$ which is assumed to be zero-mean and unit-variance. 3. $p_\ell$ denotes the power allocated to $\ell\in\setS$. 4. $\bww_\ell$ is the beamforming vector of $\ell$. The transmit power at the is restricted. It is hence assumed that $\bx$ satisfies the power constraint $\Ex{\bx^\her \bx}{} \leq P$ for some non-negative real $P$. Performance Measure ------------------- There are various metrics characterizing the performance of the downlink transmission in this system. One well-known metric is the weighted average throughput* which is defined as $$\begin{aligned} R_\avg = \frac{1}{L} \sum_{\ell \in \setS} w_\ell R_\ell \label{eq:Throughput}\end{aligned}$$ for some non-negative weights $\set{w_\ell}$ and transmission rates $$\begin{aligned} R_\ell = \log\brc{1+\sinr_\ell}. \label{eq:R_k}\end{aligned}$$ In , $\sinr_\ell$ is defined as $$\begin{aligned} \sinr_\ell = \dfrac{\displaystyle p_\ell\abs{\bh_\ell^\trp \bww_\ell}^2}{\displaystyle \sigma_\ell^2 + \sum_{j=1,j\neq \ell}^K p_j\abs{\bh_\ell^\trp \bww_j}^2}.\end{aligned}$$ From signal processing points of view, precoding can be interpreted as channel inversion. In this problem, the ultimate aim is to construct the transmit signal such that at a selected $\ell$, $\bh_\ell^\trp\bx = \beta s_\ell$, for some scaling factor $\beta$, and at $k$ which has not been selected, we have $\bh_k^\trp\bx = 0$. The former guarantees channel inversion at the selected which results in minimal post-processing load, and the latter restricts the precoder to have zero leakage at the non-selected . By this alternative viewpoint, a suitable performance measure is the ** at the defined as $$\begin{aligned} \rss = \frac{1}{K} \sum_{k=1}^K \Ex{\abs{\bh_k^\trp\bx - \beta a_k s_k}^2}{},\end{aligned}$$ where $a_k=1$ if $k$ is selected and is zero otherwise. Optimal User Selection and Precoding ==================================== Let $\bs = \dbc{s_1,\ldots,s_K}^\trp$ collect the information symbols of all . By defining $p_k=0$ for those which are not selected, the transmit signal is compactly represented as $$\begin{aligned} \bx = \mW \sqrt{\mP} \bs.\end{aligned}$$ where $\mW$ and $\mP$ are defined as follows: 1. $\mW = \dbc{\bww_1,\ldots,\bww_K}$ is the beamforming matrix. 2. $\mP = \diag{\bp}$ with $\bp= \dbc{p_1,\ldots,p_K}^\trp$. The notation $\sqrt{\mP}$ moreover denotes a matrix whose entries are the square root of the entries of $\mP$. Similarly, the vector of receive signals $\by = \dbc{y_1,\ldots,y_K}^\trp$ reads$$\begin{aligned} \by = \mH^\trp \bx + \bz\end{aligned}$$ where $\mH = \dbc{\bh_1,\ldots,\bh_K}$ and $\bz= \dbc{z_1,\ldots,z_K}^\trp$. User Selection and Precoding with Minimum RSS --------------------------------------------- We design the transmit signal by considering the as the performance measure. In this respect, the optimal approach for joint user selection and precoding is to find $\mW$ and $\bp$ such that the is minimized and the signal constraints are satisfied. In the sequel, we formulate this approach in a standard form. ### Objective Function {#objective-function .unnumbered} Following the given representation, the is written as $$\begin{aligned} \rss = \frac{1}{K} \Ex{\norm{\mH^\trp \mW\sqrt{\mP} \bs - \beta \mA \bs}^2}{},\end{aligned}$$ where $\mA=\diag{a_1, \ldots,a_K}$. In this formulation, $\mA$ is ineffective and can be dropped. To show this, note that for any non-selected $k$, $\bx$ is independent of $s_k$ and hence $$\begin{aligned} \hspace{-2mm}\Ex{\abs{\bh_k^\trp\bx - \beta s_k}^2}{} \hspace{-.7mm} &= \hspace{-.7mm}\Ex{\abs{\bh_k^\trp\bx}^2}{} \hspace{-.7mm} + \hspace{-.7mm} \beta^2 \Ex{\abs{ s_k}^2}{} \\ &= \hspace{-.7mm}\Ex{\abs{\bh_k^\trp\bx}^2}{} \hspace{-.7mm}+\hspace{-.7mm} \beta^2.\end{aligned}$$ Therefore, we can write $$\begin{aligned} \rss = \frac{1}{K} D\brc{\mW,\bp} - \brc{1-\frac{L}{K} } \beta^2,\end{aligned}$$ where $D\brc{\mW,\bp}$ is defined as $$\begin{aligned} D\brc{\mW,\bp} &\coloneqq \Ex{\norm{\mH^\trp \mW\sqrt{\mP} \bs - \beta \bs}^2}{}\\ &= \tr{ \mQ^\her \mQ }{}\end{aligned}$$ with $\mQ = {\mH^\trp \mW\sqrt{\mP} - \beta \mI_K }$. We hence set the objective function to $D\brc{\mW,\bp}$. ### Constraints {#constraints .unnumbered} There are two main constraints: 1. The number of selected should be less than $L$. 2. The average transmit power is constrained. Noting that the number of selected in the system is given by the sparsity of $\bp$, i.e., $\norm{\bp}_0$, the first constraint is written as $$\begin{aligned} \norm{\bp}_0 \leq L.\end{aligned}$$ For the second constraint, we note that $$\begin{aligned} \Ex{\bx^\her \bx}{} &= \Ex{\bs^\her \sqrt{\mP} \mW^\her \mW \sqrt{\mP} \bs}{}\\ &\stackrel{\dagger}{=} \Ex{\tr{\sqrt{\mP} \mW^\her \mW \sqrt{\mP} \bs \bs^\her}}{}\\ &= \tr{\mW \mP \mW^\her}\end{aligned}$$ where $\dagger$ follows the fact that $\Ex{{\bs\bs^\her}}{} = \mI_K$. As a result, the transmit power constraint reads $$\begin{aligned} \tr{\mW \mP \mW^\her} \leq P.\end{aligned}$$ ### Optimization Problem {#optimization-problem .unnumbered} Considering the objective function and constraints, the jointly optimal approach for user selection and precoding is formulated as $$\begin{aligned} {6} &\min_{ \mW \in \setC^{M\times K} , \bp \in \setR_+^{K} } &\qquad & D\brc{\mW,\bp} \label{eq:optProb_F}\\ &\mathrm{subject \ to} & &\mathrm{C_1:} \ \norm{\bp}_0 \leq L,\nonumber\\ & & &\mathrm{C_2:} \ \tr{\mW \diag{\bp} \mW^\her} \leq P.\nonumber\end{aligned}$$ The optimization problem in its initial form is not tractable, since both the objective function and constraints are not convex. We address this issue by converting into a group selection problem. We then develop an algorithm based on group to estimate the solution. Precoding via Group LASSO ========================= The optimization problem in can be converted into a group selection problem. To show this, let $\mV \coloneqq \mW\sqrt{\mP}$ be the overall precoding matrix. The objective function is rewritten in terms of $\mV$ as $$\begin{aligned} D\brc{\mW,\bp} &= \tr{ \brc{\mH^\trp \mV- \beta \mI_K}^\her \brc{\mH^\trp \mV- \beta \mI_K} } \nonumber \\ &= \norm{ \mH^\trp \mV- \left. \beta \right. \mI_K }_{F}^2.\end{aligned}$$ The power constraint is further given in terms of $\mV$ as $$\begin{aligned} \tr{\mV^\her \mV} = \norm{\mV}_F^2 \leq P.\end{aligned}$$ To represent constraint $\rm C_1$ in terms of $\mV$, we note that only the column vectors in $\mV$ whose corresponding is selected have non-zero entries. This equivalently means that $$\begin{aligned} \begin{cases} \norm{\bvv_k} \neq 0 &\text{if \ac{ut} $k$ is selected}\\ \norm{\bvv_k} = 0 &\text{otherwise} \end{cases},\end{aligned}$$ where $\bvv_k = \sqrt{p_k} \bww_k$ denotes the $k$-th column vector of $\mV$. As the result, one can write $$\begin{aligned} \norm{\mV}_{2,0} = \norm{\bp}_0,\end{aligned}$$ where $\norm{\mV}_{p,q}$ denotes the $\ell_{p,q}$ norm of $\mV$ defined as $$\begin{aligned} \norm{\mV}_{p,q} \coloneqq \dbc{ \sum_{k=1}^K \brc{\norm{\bvv_k}_p}^q }^{1/q}.\end{aligned}$$ From the above derivations, we conclude that the optimal approach for joint user selection and precoding reduces to the following programming: $$\begin{aligned} {6} &\min_{ \mV \in \setC^{M\times K} } &\qquad &\norm{ \mH^\trp \mV- \left. \beta \right. \mI_K }_{F}^2 \label{eq:optProb_F2}\\ &\mathrm{subject \ to} & &\mathrm{C_1:} \ \norm{\mV}_{2,0} \leq L,\nonumber\\ & & &\mathrm{C_2:} \ \norm{\mV}_F^2 \leq P.\nonumber\end{aligned}$$ The optimization in describes a group selection problem in which a matrix with group sparsity is to be recovered, i.e., a matrix with a certain fraction of column or row vectors being zero. Such a problem raises in several applications, e.g., distributed compressive sensing and machine leaning [@sarvotham2005distributed; @bereyhi2018theoretical; @bach2008consistency]. Group selection in its primitive form is a -hard problem, since it reduces to an integer programming. To address this problem tractably, several suboptimal approaches have been developed in the literature which approximate the solution. Group is one of the most efficient approaches which relaxes the problem of group selection into a convex programming [@yuan2006model; @deng2013group]. In the sequel, we use group to develop a computationally tractable algorithm for joint user selection and precoding. A Tractable Algorithm via Group LASSO ------------------------------------- Group selection is an extension of the basic sparse recovery problem in which a sparse vector is to be recovered from an underdetermined system of equations [@donoho2006compressed; @candes2008restricted]. Group extends Tibshirani’s regularization approach [@tibshirani1996regression] and convexifies the non-convex $\ell_0$-norm with the $\ell_1$-norm. This means that constraint $\rm C_1$ is relaxed as $$\begin{aligned} \mathrm{C_1:} \ \norm{\mV}_{2,1} \leq \eta L\end{aligned}$$ for some $\eta$ which regularizes the relaxation. By doing so, the joint user selection and precoding reduces to $$\begin{aligned} {6} &\min_{ \mV \in \setC^{M\times K} } &\qquad & \norm{ \mH^\trp \mV- \left. \beta \right. \mI_K }_{F}^2 \label{eq:optProb_F3}\\ &\mathrm{subject \ to} & &\mathrm{C_1:} \ \norm{\mV}_{2,1} \leq \eta L,\nonumber\\ & & &\mathrm{C_2:} \ \norm{\mV}_F^2 \leq P.\nonumber\end{aligned}$$ This relaxed program represents a group algorithm which is convex and is posed as a generic linear programming. An Alternative Formulation via RLS ---------------------------------- The joint user selection and precoding scheme in describes least squares with side constraints, where the $\norm{ \mH^\trp \mV- \left. \beta \right. \mI_K }_{F}^2$ is minimized subject to some constraints. Following the method of *, this problem is converted into the following unconstrained optimization[^1] $$\begin{aligned} \hspace{-2mm}\min_{ \mV \in \setC^{M\times K} } \norm{ \mH^\trp \mV- \beta \mI_K }_{F}^2 + \lambda \norm{\mV}_F^2 + \mu \norm{\mV}_{2,1} \label{eq:RLS_final}\end{aligned}$$ for some regularizers $\lambda$ a $\mu$. The key features of this algorithm are as follows: - For given upper bounds on the group sparsity and transmit power of $\mV$, there exists a pair of regularizers $\lambda$ and $\mu$, such that the solution to satisfies the constraints. Hence, by tuning $\lambda$ and $\mu$ different constraints are fulfilled. - Due to its convexity, the problem is tractably solved via generic linear programming. Alternatively, an iterative algorithm based on can be developed to find the solution with minimal computational complexity; see [@rangan2011generalized] for more details on and [@bereyhi2018precoding] for its applicatindons to precoding. Channel matrix $\mH$, average transmit power $P$ and the number of selected users $L$. $\mV = \dbc{\bvv_1,\ldots,\bvv_K}$ $$\begin{aligned} \mV = \mathrm{GroupLASSO}\brc{\mH,P,L,\beta}\end{aligned}$$ subset $\setS \subseteq \set{1,\ldots,K}$ contain indices of the column vectors in $\mV$ which have the $L$ largest $\ell_2$-norms, i.e., $\abs{\setS}=L$ and $$\begin{aligned} \norm{\bvv_\ell}^2 \geq \norm{\bvv_j}^2\end{aligned}$$ for any $\ell\in\setS$ and $j\in \set{1,\ldots,K}-{\setS}$. $\bvv_j = 0$ for $j\in \set{1,\ldots,K}-{\setS}$, and update $\mV$ as $$\begin{aligned} \mV \leftarrow \frac{\sqrt{P}}{\norm{\mV}_F} \left. \mV \right.\end{aligned}$$ $p_k = \norm{\bvv_k}^2$ and $\bww_k = \dfrac{\bvv_k}{ \norm{\bvv_k}}$ for $k\in\set{1,\ldots,K}$. Beamforming matrix $\mW = \dbc{ \bww_1,\ldots,\bww_K }$ and power allocation matrix $\mP = \diag{p_1,\ldots,p_K}$. Using either the algorithm in or the one in , a matrix $\mV$ is tractably found which approximates the optimal solution to . The beamforming and power allocation matrices are then given by decomposing this matrix as $\mV = \mW \sqrt{\mP}$ for a diagonal $\mP$. In the sequel, we investigate the performance of the proposed approach through some numerical simulations. Performance Investigation ========================= We study the performance of the proposed approach by simulating some sample scenarios. To jointly precode and select user via group , Algorithm \[GroupLasso\] is used. In this algorithm, $$\begin{aligned} \mV = \mathrm{GroupLASSO}\brc{\mH,P,L,\beta}\end{aligned}$$ denotes the solution to the minimization in with $\eta=1$. The algorithm finds first the solution $\mV$ to , and selects $L$ with strongest precoding vectors while setting the other column vectors zero. It then scales the precoding vectors of the selected users, such that the downlink transmit signal remains $P$. As a benchmark, we evaluate the performance of beamforming with random user selection, and compare it with the performance of Algorthm \[GroupLasso\]. In this approach, $L$ are selected at random. The precoding vector of selected user $k$ is then set to $$\begin{aligned} \bvv_k = \sqrt{\frac{P}{L}} \left. \frac{ \bh_k^* }{ \norm{\bh_k} } \right. .\end{aligned}$$ Throughout the simulations the standard Rayleigh model is considered for the fading channel. This means that the entries of $\mH$ are generated independently and identically with complex zero-mean and unit-variance Gaussian distribution, i.e., $$\begin{aligned} h_{mk} \sim \mathcal{CN}\brc{0,1}\end{aligned}$$ for $m\in\set{1,\ldots,M}$ and $k\in\set{1,\ldots,K}$. Performance Metrics ------------------- To quantify the performance, the following metrics are considered: 1. The weighted average throughput $R_\avg$ defined in for uniform wights, i.e., $w_1, \ldots,w_K = 1$. This metric determines the average achievable rate per selected which is widely used in this literature. 2. The power leakage to the non-selected which is given by $$\begin{aligned} Q_{\rm Leak} &\coloneqq \Ex{ \sum_{k=1, k\notin \setS}^K \abs{\bh_k^\trp\bx}^2 }{}\\ &= \sum_{k=1, k\notin \setS}^K \left. \sum_{\ell\in\setS} \right. \abs{\bh_k^\trp\bvv_\ell}^2.\end{aligned}$$ This metric calculates the total amount of interference at the non-selected from the downlink transmission to the selected . Scenario A: Fixed Loads ----------------------- We first consider a scenario in which the total number of , as well as the number of selected ones, is a fixed fraction of the transmit array size $M$. More precisely, a downlink transmission scenario is considered in which $K=\lceil \alpha_{ K} M \rceil$ number of users are available and we intend to select $L = \lceil \alpha_{ L} M \rceil$ . Here, $\alpha_{ K}$ and $\alpha_{ L}$ are fixed numbers. For this scenario, both the performance metrics are sketched for fixed transmit power $P$ and noise variance in Fig. \[fig:1\] and Fig. \[fig:2\] in terms of the downlink transmit array size $M$. Fig. \[fig:1\] shows the weighted average throughput[^2] against $M$. Here, $P=1$ and the noise variances are set to $\sigma_k = 0.1$ for $k\in\set{1,\ldots,K}$. Moreover, the scaling factor reads $\beta=1$. The results are sketched for $\alpha_K = 1$ and two different values of $\alpha_L$; namely, $\alpha_L\in\set{0.3,0.5}$. As the figure depicts, the proposed approach considerably outperforms the conventional technique. Such an enhancement comes from the joint selection and precoding approach. The convergence of $R_\avg$ to a constant in both the techniques follows hardening of the channel in large dimensions for fixed loads [@asaad2018massive; @hoydis2013massive]. The power leakage for this scenario is plotted in Fig. \[fig:2\] versus $M$. Here, the parameters are set exactly to the ones considered in Fig. \[fig:1\]. The figure demonstrates the following two observations: 1. The proposed algorithm imposes significantly less interference to the non-selected . This observation comes from the fact that the objective function in contains the power leakage as a penalty term. 2. The power leakage in both techniques converges to a constant value. Such a behavior is naturally following the fact that the loads $\alpha_K$ and $\alpha_L$ are kept fixed. Scenario B: Fixed Number of UTs ------------------------------- As another scenario, we consider a case in which the total number of , as well as the number of selected ones, does not grow with $M$. For this case, we study a settings in which a downlink array of size $M$ is employed to service $L$ users out of $K=16$ available . Similar to Scenario A, we set $P$ and noise variances to fixed numbers and sketch the average throughout, as well as the power leakage, against the transmit array size $M$ in Fig. \[fig:3\] and Fig. \[fig:4\]. In Fig. \[fig:3\], the average throughput $R_\avg$ is sketched against $M$ assuming $\beta=1$, $P=1$ and $\sigma_k=0.1$ for $k\in\set{1,\ldots,K}$. The results are given for $L\in\set{4,8}$. Similar to Scenario A, the figure depicts performance enhancement achieved by using the proposed algorithm based on the group . In contrast to Scenario A, the throughput in this case grows logarithmically with $M$. Such a behavior follows the fact that in this case, the number of is constant and does not grow with $M$. Fig. \[fig:4\] shows the variation of the power leakage against $M$. As the figure demonstrate, in the proposed algorithm, $Q_{\rm Leak}$ vanishes significantly fast as $M$ grows, such that at $M=64$ it imposes almost no interference to the non-selected . Such a behavior follows the fact that in the joint approach based on the group , the beamforming vectors are constructed, such that the power leakage is suppressed at non-selected . For a fixed number of , the suppression is performed more accurately by narrow beamforming towards the selected users, as the array size grows large [@bereyhi2018robustness]. Conclusions =========== A joint user selection and precoding scheme has been proposed for multiuser systems based on group . The scheme depicts performance enhancement in two different aspects: The throughput of the system, defined as the sum-rate divided by the number of active users, shows some gains. The interference imposed by downlink transmission at the non-selected is significantly reduced. For instance, when $L=8$ are selected out of $K=16$ users, there is almost zero interference, when the is equipped with $M=64$ antennas. These observations indicate that the proposed scheme is a good candidate for massive settings. The current work can be pursued in various directions. For example, considering the -based derivation in , an iterative algorithm can be developed via implementing the proposed scheme with low computational complexity. Another direction is to extend the current framework to wiretap settings following the approach in [@asaad2019asilomar]. The work in these directions is currently ongoing. \[MIMO\][multiple-input multiple-output]{} \[MIMOME\][multiple-input multiple-output multiple-eavesdropper]{} \[CSI\][channel state information]{} \[AWGN\][additive white Gaussian noise]{} \[i.i.d.\][independent and identically distributed]{} \[UT\][user terminal]{} \[BS\][base station]{} \[MT\][mobile terminal]{} \[Eve\][eavesdropper]{} \[LSE\][least squared error]{} \[MSE\][mean squared error]{} \[GLSE\][generalized least squared error]{} \[RLS\][regularized least-squares]{} \[r.h.s.\][right hand side]{} \[l.h.s.\][left hand side]{} \[w.r.t.\][with respect to]{} \[TDD\][time-division duplexing]{} \[PAPR\][peak-to-average power ratio]{} \[MRT\][maximum ratio transmission]{} \[ZF\][zero forcing]{} \[RZF\][regularized zero forcing]{} \[SRZF\][secure ]{} \[SNR\][signal to noise ratio]{} \[SINR\][signal to interference plus noise ratio]{} \[RF\][radio frequency]{} \[MF\][match filtering]{} \[MMSE\][minimum mean squared error]{} \[RSS\][residual sum of squares]{} \[AMP\][approximate message passing]{} \[NP\][non-deterministic polynomial time]{} \[DCA\][DC programming algorithm]{} \[LASSO\][least absolute shrinkage and selection operator]{} [^1]: Alternatively, one could use the method of Lagrange multipliers to conclude the similar unconstrained form. [^2]: Remember that the average throughput in this case is defined as the sum-rate divided by the number of selected users.
--- abstract: 'Nonsingular Bianchi type I solutions are found from the effective action with a superstring-motivated Gauss-Bonnet term. These anisotropic nonsingular solutions evolve from the asymptotic Minkowski region, subsequently super-inflate, and then smoothly continue either to Kasner-type (expanding in two directions and shrinking in one direction) or to Friedmann-type (expanding in all directions) solutions. We also find a new kind of singularity which arises from the fact that the anisotropic expansion rates are a multiple-valued function of time. The initial singularity in the isotropic limit of this model belongs to this new kind of singularity. In our analysis the anisotropic solutions are likely to be singular when the superinflation is steep. As for the cosmic no-hair conjecture, our results suggest that the kinetic-driven superinflation of our model does not isotropize the space-time.' address: - 'Graduate School of Human and Environmental Studies, Kyoto University, Kyoto 606-8501, Japan' - 'Department of Fundamental Sciences, FIHS, Kyoto University, Kyoto 606-8501, Japan' author: - 'Shinsuke [Kawai]{}[^1]' - 'Jiro [Soda]{}[^2]' date: Revised Jan 1999 title: \| Nonsingular Bianchi type I cosmological solutions\ from the 1-loop superstring effective action --- INTRODUCTION ============ Stimulated by the developments of superstring theory, various cosmological solutions based on string theory have been proposed. Although the theory has not developed enough to present a unique history of our universe in its earliest stage, one may now imagine that the big bang is no longer just a point of enigma named the initial singularity but has a rich and complex structure. Among these string-based universe models, the most widely studied would be the so-called pre-big-bang model[@pbb] (in the literature it is often called string cosmology). The remarkable aspect of this model is that it tries to explain the inflationary behavior of the early universe by introducing a polelike acceleration phase (superinflation) which is driven by the kinetic term of the dilaton. This superinflationary branch of the solution has a dual relation named the scale factor duality with the Friedmann branch, which is the usual decelerating expansion of the universe. The biggest problem arising in the pre-big-bang model is the difficulty of connecting the superinflationary branch and the Friedmann branch (termed the graceful exit problem), and there are no-go theorems proved under some assumptions[@gexit]. Antoniadis, Rizos, and Tamvakis[@art94] proposed a nonsingular (i.e. free of the graceful exit problem) cosmological model by including the 1-loop (genus) correction term in the low-energy string effective action. This 1-loop model is excellent in that it gives a simple example of the smooth transition between the superinflationary branch and the Friedmann branch. This original 1-loop model involves dilaton and modulus fields, and there is a simplified version[@rt94] by Rizos and Tamvakis with a modulus field only. This metric-modulus system gives essentially the same nonsingular solution as the full metric-dilaton-modulus system, since the behavior of the solution mainly depends only on the modulus field. The existence of the nonsingular solution is analytically shown by Rizos and Tamvakis[@rt94] for the Friedmann-Robertson-Walker metric, i.e., assuming the homogeneity and isotropy of the universe. We studied whether the nature of this nonsingular solution is affected if anisotropy is included. The purpose of this paper is to extend the solution of this metric-modulus system to include anisotropy and to observe the behavior of its solutions, particularly of nonsingular ones. The following sections of this paper are organized as follows. In the Sec. II we briefly review the isotropic case studied by Rizos and Tamvakis[@rt94], and then derive the basic equations of motion from the effective action for the Bianchi type I metric. We also solve these equations analytically in the asymptotic region. In Sec. III we solve these equations numerically. We study the solutions through several cross sections of the parameter space. The existence of nonsingular anisotropic solutions is shown, and the nature of the singularity is also examined. Implications of our results are discussed in the last section. MODEL AND EQUATIONS OF MOTION ============================= We start with the action[^3] given by[@rt94] $${\cal S}=\int d^4x\sqrt{-g}\left\{\frac 12 R-\frac 12 (D\sigma)^2 -\frac{\lambda}{16}\xi(\sigma) R_{GB}^2\right\}, \label{eqn:rt94act}$$ which is essentially the same as the 1-loop-corrected 4-dimensional effective action of orbifold-compactified heterotic string[@art94], $$\begin{aligned} {\cal S}=\int d^4x\sqrt{-g}\left\{\frac 12 R-\frac 14(D\Phi)^2-\frac 34 (D\sigma)^2\right. \nonumber\\ +\left.\frac 1{16} [\lambda_1 e^\Phi-\lambda_2\xi(\sigma)]R^2_{GB}\right\}, \label{eqn:ss1leAct}\end{aligned}$$ except that the dilaton field $\Phi$ is neglected. $R$, $\Phi$, and $\sigma$ are the Ricci scalar curvature, the dilaton, and the modulus field, respectively. Our convention is $g_{\mu\nu}=(-,+,+,+)$, $R^\mu{}_{\alpha\nu\beta}=\Gamma^\mu{}_{\alpha\beta,\nu}+\cdots$, $R_{\alpha\beta}=R^\mu{}_{\alpha\mu\beta}$, and $8\pi G=1$. The Gauss-Bonnet curvature is defined as $ R^2_{GB}=R^{\mu\nu\kappa\lambda}R_{\mu\nu\kappa\lambda} -4 R^{\mu\nu}R_{\mu\nu}+R^2, $ and $\xi(\sigma)$ is a function determining the coupling of $\sigma$ and the geometry, written in terms of the Dedekind $\eta$ function as $$\begin{aligned} \xi(\sigma)&=&-\ln[2e^\sigma\eta^4 (i e^\sigma)]\nonumber\\ &=&-\ln 2-\sigma+\frac{\pi e^\sigma}{3}-4\sum_{n=1}^{\infty}\ln (1-e^{-2n\pi e^\sigma}). \label{eqn:ssxi}\end{aligned}$$ This $\xi(\sigma)$, an even function of $\sigma$, has a global minimum at $\sigma=0$ and increases exponentially as $\sigma\rightarrow\pm\infty$. $\lambda_1$ is the 4-dimensional string coupling and takes a positive value. $\lambda_2$ is proportional to the 4-dimensional trace anomaly of the $N=2$ sector and determined by the number of chiral, vector, and spin-$\frac 32$ supermultiplets. It is important that $\lambda_2$ can take positive values, since nonsingular solutions arise only when $\lambda_2>0$. In our simplified model (\[eqn:rt94act\]), therefore, we assume $\lambda$ to be positive (in actual numerical calculations we set $\lambda=1$) and adopt the form of the $\xi$ function (\[eqn:ssxi\]). Isotropic solutions ------------------- First, we review the homogeneous and isotropic case which is discussed in [@rt94]. We neglect the spatial curvature and write the metric in the flat FRW form $$ds^2=-N(t)^2dt^2+a(t)^2(dx^2+dy^2+dz^2). \label{eqn:frwmetric}$$ Variation of the action (\[eqn:rt94act\]) with respect to the lapse $N$, the scale factor $a$, and the modulus field $\sigma$ gives three equations of motion as $$\begin{aligned} &&\dot\sigma^2=6H^2 \left(1-\frac \lambda 2 H\dot\xi\right), \label{eqn:bg1}\\ &&(2\dot H + 5H^2)\left(1-\frac \lambda 2 H \dot \xi\right) +H^2\left(1-\frac \lambda 2 \ddot\xi\right)=0, \label{eqn:bg2}\\ &&\ddot\sigma+3H\dot\sigma+\frac {3\lambda}{2} (\dot H+H^2)H^2\frac{\partial\xi}{\partial\sigma}=0, \label{eqn:bg3}\end{aligned}$$ where $H$ is the Hubble parameter $\dot a/a$, the overdot means derivative with respect to physical time $t$, and we have set $N(t)=1$. As a result of the absence of scales \[we are only considering spatially flat metric (\[eqn:frwmetric\])\] and the existence of the constraint (\[eqn:bg1\]), the solutions are completely determined by a couple of first order differential equations for two variables $H$ and $\sigma$; that is, if we give values of $H$ and $\sigma$ at some time $t$, the preceding and following evolutions of the solution are automatically determined by these equations. Figure 1 shows the $H$-$\sigma$ phase diagram of the isotropic system solved with initial conditions $H>0$ and $\dot\sigma>0$. These solution flows are distinguished by only one degree of freedom (for example, the value of $H$ at some fixed $\sigma>0$). There are singular solutions and nonsingular solutions, and it is shown in [@rt94] that all flows in the $H>0$, $\sigma<0$ quarter-plane continue smoothly to $H>0$, $\sigma>0$ quarter-plane, but some flows in $H>0$, $\sigma>0$ quarter-plane go into singularity and do not continue to the $\sigma<0$ region. It is also shown in [@rt94] that the signs of $H$ and $\dot\sigma$ are conserved throughout the evolution of the system. Since $\dot\sigma$ is always positive in Fig. 1, time flows from left to right. We consider that our Friedmann universe corresponds to the “future” region ($\sigma>0$) in Fig. 1, and we regard the “past” region ($\sigma<0$) with increasing Hubble parameter as a superinflation, which is expected to solve the shortcomings of the big-bang model. Equations of motion ------------------- We now extend the above model to anisotropic Bianchi type I space-time. We write the metric as $$ds^2=-N(t)^2dt^2+e^{2\alpha(t)}dx^2+e^{2\beta(t)}dy^2+e^{2\gamma(t)}dz^2,$$ and define the anisotropic expansion rates as $$p=\dot\alpha, \hspace{1cm}q=\dot\beta, \hspace{1cm}r=\dot\gamma.$$ The average expansion rate, which coincides with the Hubble parameter $H$ in the isotropic limit, is $$H_{\mbox{\scriptsize avr}}=\frac 13(p+q+r).$$ The equations of motion are obtained by variation of the action (\[eqn:rt94act\]) with respect to $N$, $\alpha$, $\beta$, $\gamma$, and $\sigma$, viz., $$\begin{aligned} &&pq+qr+rp-\frac 12{\dot\sigma}^2 -\frac 32\lambda\frac{\partial\xi}{\partial\sigma}\dot\sigma pqr=0 \label{eqn:b1eom1},\\ &&\dot p=\frac{(CA-EB)G+(EF-A^2)H+(AB-FC)Q}\Delta \label{eqn:b1eom2},\\ &&\dot q=\frac{(BC-DA)G+(BA-FC)H+(FD-B^2)Q}\Delta \label{eqn:b1eom3},\\ &&\dot r=\frac{(DE-C^2)G+(AC-BE)H+(BC-AD)Q}\Delta \label{eqn:b1eom4},\\ &&\ddot\sigma+(p+q+r)\dot\sigma\nonumber\\ &&\mbox{\hspace{1mm}}+\frac 12 \lambda \frac{\partial\xi}{\partial\sigma}\left\{\dot p qr+p\dot q r+pq\dot r +pqr(p+q+r)\right\}=0 \label{eqn:b1eom5},\end{aligned}$$ where $$\begin{aligned} A&=&1-\frac\lambda 2\frac{\partial\xi}{\partial\sigma}\dot\sigma p +\frac {\lambda^2}4 \left(\frac{\partial\xi}{\partial\sigma}\right)^2p^2qr,\\ B&=&1-\frac\lambda 2\frac{\partial\xi}{\partial\sigma}\dot\sigma q +\frac {\lambda^2}4 \left(\frac{\partial\xi}{\partial\sigma}\right)^2pq^2r,\\ C&=&1-\frac\lambda 2\frac{\partial\xi}{\partial\sigma}\dot\sigma r +\frac {\lambda^2}4 \left(\frac{\partial\xi}{\partial\sigma}\right)^2pqr^2,\\ D&=&\frac {\lambda^2}4 \left(\frac{\partial\xi} {\partial\sigma}\right)^2q^2r^2,\\ E&=&\frac {\lambda^2}4 \left(\frac{\partial\xi} {\partial\sigma}\right)^2r^2p^2,\\ F&=&\frac {\lambda^2}4 \left(\frac{\partial\xi} {\partial\sigma}\right)^2p^2q^2,\\ G&=&-p^2-q^2-pq-\frac 12{\dot\sigma}^2 +\frac\lambda 2\frac{\partial^2\xi}{\partial\sigma^2}{\dot\sigma}^2 pq \nonumber\\ &&-\frac\lambda 2\frac{\partial\xi}{\partial\sigma}\dot\sigma pqr -\frac\lambda 4\left(\frac{\partial\xi} {\partial\sigma}\right)^2p^2q^2r(p+q+r),\\ H&=&-q^2-r^2-qr-\frac 12{\dot\sigma}^2 +\frac\lambda 2\frac{\partial^2\xi}{\partial\sigma^2}{\dot\sigma}^2 qr \nonumber\\ &&-\frac\lambda 2\frac{\partial\xi}{\partial\sigma}\dot\sigma pqr -\frac\lambda 4\left(\frac{\partial\xi} {\partial\sigma}\right)^2pq^2r^2(p+q+r),\\ Q&=&-r^2-p^2-rp-\frac 12{\dot\sigma}^2 +\frac\lambda 2\frac{\partial^2\xi}{\partial\sigma^2}{\dot\sigma}^2 rp \nonumber\\ &&-\frac\lambda 2\frac{\partial\xi}{\partial\sigma}\dot\sigma pqr -\frac\lambda 4\left(\frac{\partial\xi}{\partial\sigma}\right)^2p^2qr^2(p+q+r),\end{aligned}$$ and $$\Delta=2 ABC+DEF-FC^2-DA^2-EB^2 \label{eqn:delta}.$$ In the isotropic limit, Eqs. (\[eqn:b1eom1\]) and (\[eqn:b1eom5\]) become Eqs. (\[eqn:bg1\]) and (\[eqn:bg3\]) respectively, and Eqs. (\[eqn:b1eom2\]), (\[eqn:b1eom3\]), and (\[eqn:b1eom4\]) reduce to one equation (\[eqn:bg2\]). Similar to the isotropic case, once the values of $\sigma$, $p$, $q$, and $r$ are given at some time $t$, and $\dot\sigma$ is given according to the constraint (\[eqn:b1eom1\]), the system evolves along the flow of the solution following Eqs. (\[eqn:b1eom2\])$-$(\[eqn:b1eom5\]). In our numerical analysis we integrated Eqs. (\[eqn:b1eom2\])$-$(\[eqn:b1eom5\]) with the initial condition satisfying the constraint (\[eqn:b1eom1\]). This constraint equation (\[eqn:b1eom1\]) is also used to estimate the numerical error. There are five parameters ($\sigma$, $\dot\sigma$, $p$, $q$, and $r$) and one constraint (\[eqn:b1eom1\]); so the solution flows are drawn in a 4-dimensional space, for example, $\sigma$-$p$-$q$-$r$ space. Since the aim of this paper is to examine the effect of anisotropy on the nonsingular solution, we observe the change of the solution as we increase the anisotropy of the metric. To facilitate this we introduce two parameters indicating the anisotropy of the metric, viz., $$X=\frac{p-r}{p+q+r},\hspace{5mm} Y=\frac{q-r}{p+q+r}. \label{eqn:defxy}$$ In this notation, $(X,Y)=(0,0)$ corresponds to the isotropic (FRW) metric, and postulating $H_{\mbox{\scriptsize avr}}$ to be positive, the region surrounded by the lines $Y=2X+1$, $Y=\frac 12 X-\frac 12$, $Y=-X+1$ indicates the universe expanding in all directions (these lines are drawn in Figs. 3 and 6). The regions $Y<2X+1$, $Y>\frac 12 X-\frac 12$, $Y>-X+1$, etc., are the “Kasner-like” universe expanding in two directions and contracting in one direction. The regions $Y>2X+1$, $Y>\frac 12 X-\frac 12$, $Y>-X+1$, etc., describe the universe expanding in one direction, contracting in two directions. The universe shrinking in all directions cannot be included if we use Eqs. (\[eqn:defxy\]) and assume $H_{\mbox{\scriptsize avr}}>0$. Instead of $\sigma$-$p$-$q$-$r$ we use $\sigma$-$H_{\mbox{\scriptsize avr}}$-$X$-$Y$ as four variables describing our system. Asymptotic solutions -------------------- Before going to the numerical analysis, we study the asymptotic form of the solutions at $t\rightarrow\pm\infty$, $\vert\sigma\vert\rightarrow\infty$. We assume $\|\sigma\|$ to become large when $t\rightarrow\pm\infty$. The asymptotic form of the derivatives of the function $\xi$ is $$\begin{aligned} \frac{\partial \xi}{\partial \sigma}&\sim&\mbox{sgn}(\sigma)\frac\pi 3 e^{\vert\sigma\vert} \label{eqn:axi1},\\ \frac{\partial^2 \xi}{\partial \sigma^2}&\sim&\frac\pi 3e^{\vert\sigma\vert} \label{eqn:axi2}.\end{aligned}$$ If we assume the power-law ansatz for the expansion rates, the asymptotic form of the modulus has to be logarithmic in order to cancel the exponential dependence of Eqs. (\[eqn:axi1\]) and (\[eqn:axi2\]). Thus we choose the following forms for the asymptotic solutions: $$\begin{aligned} p&\sim&\omega_1\vert t \vert^\rho,\\ q&\sim&\omega_2\vert t \vert^\rho,\\ r&\sim&\omega_3\vert t \vert^\rho,\\ \sigma&\sim&\sigma_0 +\omega_4\ln\vert t\vert.\end{aligned}$$ Putting all these into Eqs. (\[eqn:b1eom1\])$-$(\[eqn:b1eom5\]), we obtain two possible asymptotic solutions: $$\begin{aligned} {\cal A}&:&\rho=-1,\omega_1+\omega_2+\omega_3=\mbox{sgn}(t),\nonumber\\ &&\omega_1{}^2+\omega_2{}^2+\omega_3{}^2+\omega_4{}^2=1, \label{eqn:assola}.\\ {\cal B}&:&\rho=-2,\vert\omega_4\vert=5,\nonumber\\ &&\omega_1\omega_2\omega_3=-\mbox{sgn}(t)\frac{5\exp[-\sigma_0\mbox{sgn} (\omega_4)]}{\lambda\pi} \label{eqn:assolb}.\end{aligned}$$ The solution ${\cal A}$ is obtained by balancing terms that do not include the 1-loop effect, i.e., the Gauss-Bonnet term, and thus describes the asymptotic behavior where the Gauss-Bonnet effect is negligible. In the absence of the modulus field, ${\cal A}$ is nothing but the Bianchi type I vacuum (Kasner) solution. The solution ${\cal B}$ is obtained by balancing the kinetic term of the modulus and the Gauss-Bonnet term, and this solution corresponds to the phase where the Gauss-Bonnet term is important. Other possibilities of solutions are excluded as long as we impose $\lambda>0$, which is, in the isotropic limit, a necessary condition for the existence of the nonsingular solutions. Following the isotropic case[@art94; @rt94; @maeda96], we choose $\dot\sigma>0$ and assume the solution ${\cal A}$ in the future asymptotic region and ${\cal B}$ in the past asymptotic region. Then, in the region $t\rightarrow\infty$, the conditions (\[eqn:assola\]) can be seen in the $\omega_1$-$\omega_2$-$\omega_3$ space as the cross section of the sphere of radius $\sqrt{1-\omega_4^2}$ centered at the origin with the $\omega_1+\omega_2+\omega_3=1$ plane. Depending on the asymptotic value of $\omega_4$, the asymptotic solutions of $t\rightarrow\infty$ are categorized into two cases. ${\cal A}$1: $0\leq\omega_4<\sqrt\frac 12$. The asymptotic solution is either expanding in all directions (Friedmann type) or expanding in two directions, shrinking in one direction (Kasner type). ${\cal A}$2: $\sqrt\frac 12\leq\omega_4\leq\sqrt\frac 23$. The asymptotic solution is Friedmann type only. In terms of $X$ and $Y$ introduced in Eq. (\[eqn:defxy\]), the asymptotic solution is represented by a point on an arc of the ellipse $X^2+Y^2-XY=1-\frac 32\omega_4^2$. Thus ${\cal A}$2 falls into the region inside the oval $X^2+Y^2-XY=\frac 14$, and ${\cal A}$1 is in the region between two ellipses $X^2+Y^2-XY=\frac 14$ and $X^2+Y^2-XY=1$. No asymptotic solutions exist in the region outside the ellipse $X^2+Y^2-XY=1$, where the constraint equation (\[eqn:b1eom1\]) is not satisfied. Therefore, at least in the far enough future region, it is sufficient to examine solutions near the isotropic one. The difference in our model from the Kasner (Bianchi type I vacuum) solution is the existence of the field $\sigma$, which allows the existence of a Friedmann-type solution in the future asymptotic region. In the past asymptotic region, the condition (\[eqn:assolb\]) indicates $\omega_1\omega_2\omega_3>0$, i.e., one of the following. ${\cal B}$1: $\omega_1$, $\omega_2$, $\omega_3>0$. ${\cal B}$2: One of $\omega_i$ ($i=1,2,3$) is positive; two are negative. This means that in the past asymptotic region the universe is either expanding in all directions or expanding in one direction, contracting in two directions. NUMERICAL RESULTS ================= Once the anisotropy is included, the behavior of the solution deviates substantially from the isotropic case. Figure 2a shows the average expansion rate $H_{\mbox{\scriptsize avr}}=(p+q+r)/3$ versus the modulus $\sigma$ in the anisotropic (Bianchi type I) case, solved with initial anisotropy $X=0.1$, $Y=0.2$ at $\sigma=-10$. Unlike the isotropic case (Fig. 1), some solution flows in the $\sigma<0$ region do not continue smoothly to the $\sigma>0$ region, but terminate suddenly with finite values of $\sigma$ and $H_{\mbox{\scriptsize avr}}$. At these unusual singularities the time derivatives of $p$, $q$, and $r$ become infinite, although $p$, $q$, and $r$ themselves stay finite (see Fig. 2b). This is because the value of $\Delta$, Eq. (\[eqn:delta\]), approaches zero, while the numerators of Eqs. (\[eqn:b1eom2\]), (\[eqn:b1eom3\]), and (\[eqn:b1eom4\]) stay finite. Thus, the function $\Delta$ plays an important role in the anisotropic case, and the regularity of the solutions depends largely on its behavior. In the equations in our model there are four independent variables $\sigma$-$H_{\mbox{\scriptsize avr}}$-$X$-$Y$. Since our interest is mainly in the vicinity of the isotropic solution, we examine the solutions which pass near the origin of the $X$-$Y$ plane, first at the $\sigma=-10$ section with several different values of $H_{\mbox{\scriptsize avr}}$ and next at the $\sigma=2.5$ section. Solutions through the $\sigma=-10$ cross section ------------------------------------------------ It is helpful to consider the general behavior of $\Delta$ and the region prohibited by the constraint equation before solving the equations numerically. Rewriting $\Delta$, Eq. (\[eqn:delta\]), and the constraint equation (\[eqn:b1eom1\]) in terms of $\sigma$, $H_{\mbox{\scriptsize avr}}$, $X$, and $Y$, we can specify $\Delta>0$, $\Delta<0$, and prohibited regions on the $X$-$Y$ plane for fixed $\sigma$ and $H_{\mbox{\scriptsize avr}}$, which is shown in Fig. 3. The dark-shaded region is the prohibited region, the light-shaded region is where $\Delta<0$, and the white region is where $\Delta>0$. Since $\Delta=0$ is not physically allowed \[$\dot p$, $\dot q$, and $\dot r$ diverge from Eqs. (\[eqn:b1eom2\]), (\[eqn:b1eom3\]), and (\[eqn:b1eom4\])\], the solutions in the white region cannot go smoothly to the light-shaded region. Also indicated in Fig. 3 are the lines $X+Y=1$, etc., discussed in Sec. II B. We can see that the universe expanding in all directions always lies in the $\Delta>0$ region. For larger $H_{\mbox{\scriptsize avr}}$ the prohibited region becomes thinner, and the white and light gray regions will be separated by the lines $X+Y=1$, etc. Starting from the initial conditions $\sigma=-10$ and $H_{\mbox{\scriptsize avr}}=0.001$, $0.005$, $0.01$, we solved the equations futureward and indicated the behavior of the solutions in the $X$-$Y$ plane (Fig. 4). Because of the symmetry of the axes, we restrict the region to $X>0$, $Y>0$, and since we are not interested in the prohibited region, we only examined the vicinity of the origin. The black region in Fig.4 is prohibited by the Hamiltonian constraint, and the regions marked NS means nonsingular solutions. The difference between NSa and NSb is in their form in the future asymptotic region, where NSa has Friedmann-type (expanding in all directions) and NSb has Kasner-type (expanding in two directions and contracting in one direction) asymptotic solutions. Examples of NSa and NSb solutions are shown in the first and the second panels of Fig. 5. The solutions in the region marked S1 in Fig.4 lead to singularities where $\Delta\rightarrow0$ (we term this singularity type I) and these singular solutions are the same as those appearing in Fig.2. We divided the S1 solutions into two classes, S1a and S1b. S1a approaches the singularity as $p\dot p>0$, $q\dot q>0$, $r\dot r>0$, while S1b as $p\dot p<0$, $q\dot q<0$, $r\dot r<0$. This singularity, since it arises because $\Delta$ crosses zero, can be overpassed if we introduce a new “time” variable $$d\tau=dt/\Delta \label{eqn:tau}.$$ Using this $\tau$, two solutions S1a and S1b can be joined via the singularity, which is shown in the third panel of Fig. 5 (S1a,b). The solution S1b, solved backwards in time, goes into another singularity which has different property from the one between S1a and S1b. At this singularity, which we call type II, $\Delta$ goes to $-\infty$, and at least one of the expansion rates ($q$ in the case of Fig. 5, S1a,b) diverges. We can say that this solution (S1a and S1b joined together) comes regularly from $t=-\infty$, turns back at type I singularity, and then goes backwards in time into a type II singularity. Or we can also see this as two solutions, one coming from $t= -\infty$ and the other from the type II singularity, “pair-annihilate” at one type I singularity. S2 is yet another solution, which comes from one type II singularity and disappears into another type II singularity. As $H_{\mbox{\scriptsize avr}}$ becomes larger, the boundary between S1a and S1b ($\Delta =0$ line in Fig. 3) gets pushed to approach the line $X+Y=1$, and accordingly the nonsingular regions NSa and NSb become smaller. This is consistent with Fig. 2, which shows the existence of the upper limit of $H_{\mbox{\scriptsize avr}}$ for the regular solution through the $\sigma<0$ region. Solutions through the $\sigma=2.5$ cross section ------------------------------------------------ As is expected from the isotropic case discussed in the previous section, the solutions through the $\sigma>0$ cross section are quite different from those through the $\sigma<0$ section. In Fig. 6 we show the prohibited region (dark shaded), $\Delta>0$ region (white), and $\Delta<0$ region (light shaded) in the $X$-$Y$ plane, with $\sigma$ fixed to $2.5$. The elliptic allowed region of small $H_{\mbox{\scriptsize avr}}$ is the one discussed in relation to the future asymptotic form of the solution, which is expressed as $X^2+Y^2-XY=1$. As $H_{\mbox{\scriptsize avr}}$ takes large values the $\Delta=0$ contour takes complicated forms, and the region connected to the isotropic solution becomes small. Figure 7 shows the solutions passing through the $X$-$Y$ plane of the $\sigma=2.5$ cross section, and the time evolution of each type is shown in Fig. 8. $H_{\mbox{\scriptsize avr}}$ is chosen to be 0.01, 0.05, and 0.1. As $H_{\mbox{\scriptsize avr}}$ increases the nonsingular region becomes smaller, and for $H_{\mbox{\scriptsize avr}}$ larger than 0.1 the nonsingular solution completely disappears from the $X$-$Y$ plane. All the singularities appearing in Fig.7 are type I, and these singular solutions can be extended further by using $\tau$ defined by Eq. (\[eqn:tau\]). Just like the S1a and S1b solutions in the previous subsection, S1c and S1d, extended beyond the type I singularity, turn back futureward and then go into the type II singularity. The only difference between these and S1a,b is the direction of time, and the former can be regarded as the “pair creation” of cosmological solutions, while the latter is the “pair annihilation.” One of the nontrivial results of our analysis, and what makes this model very different from ordinary universe models, is that the “initial singularity” in the isotropic limit is categorized into the type I singularity (see the third panel of Fig. 7 and compare S1d and isotropic solutions in Fig. 8). This means that the singular solutions in the model proposed by Antoniadis, Rizos, and Tamvakis[@art94] or Rizos and Tamvakis [@rt94] will, if small anisotropy is included, terminate suddenly at finite past with finite Hubble parameter or, if extended using $\tau$, turn back towards the future. All nonsingular solutions in Figs. 7 and 8 continue to the asymptotic solutions expanding in all directions in the past asymptotic region. That is, the asymptotic form ${\cal B}$1 (discussed in Sec. II C) can be reached from $\sigma=2.5$ but ${\cal B}$2 cannot. There exist solutions having the past asymptotic form ${\cal B}$2. For example, solutions through the outer white ($\Delta>0$) region in the third panel of Fig. 3 behave as ${\cal B}$2 in the $t\rightarrow -\infty$ region. These solutions, however, go into singularities between $\sigma=-10$ and $\sigma=2.5$, and do not appear in Fig. 7 or 8. Nature of the singularity ------------------------- In our numerical analysis there appear two types of singularities, which we called type I and type II. We discuss the nature of these singularities briefly. At type I singularities the expansion rates ($p,q,r$) stay finite whereas the time derivatives of them diverge. These situations happen when $p,q,r\sim \vert t-t_{sing}\vert ^\rho$ with $0<\rho<1$. This is actually the case, and can be verified by analyzing solutions near the singularity. Although the type I singularity ($\Delta=0$) is a physical one, the equations of motions can be integrated regularly by using the new “time” parameter $\tau$, Eq. (\[eqn:tau\]), through the type I singularity. At the singularity $p,q,r$ “turn around” (see the solutions S1 in Figs. 5 and 8), meaning that $p$, $q$, and $r$ are multiple-valued function of $t$. Since $dt=\Delta d\tau=0$ at the singularity, the solutions are tangential to $t=t_{sing}$. Also, in order that the solutions change the chronological direction, the leading power of $p,q,r$ in the expansion of $t$ near the singularity must be even. Thus we can express $t$ using $p,q,r$ as $$\begin{aligned} t&-&t_{sing}\nonumber\\ &=&t_{p,2l}(p-p_{sing})^{2l}+t_{p,2l+1}(p-p_{sing})^{2l+1}+\cdots\nonumber\\ &=&t_{q,2m}(q-q_{sing})^{2m}+t_{q,2m+1}(q-q_{sing})^{2m+1}+\cdots\nonumber\\ &=&t_{r,2n}(r-r_{sing})^{2n}+t_{r,2n+1}(r-r_{sing})^{2n+1}+\cdots,\end{aligned}$$ with $l,m,n$ being positive integers. By solving these with respect to $p$, $q$, and $r$, we have $$\begin{aligned} p&=&p_{sing}+p_1(t-t_{sing})^{1/{2l}}+\cdots\\ q&=&q_{sing}+q_1(t-t_{sing})^{1/{2m}}+\cdots\\ r&=&r_{sing}+r_1(t-t_{sing})^{1/{2n}}+\cdots.\end{aligned}$$ Thus, $0<1/2l, 1/2m, 1/2n<1$, and $\dot p=(p_1/2l)(t-t_{sing})^{1/2l-1}+\cdots $, etc., will diverge. The behavior of $\dot \sigma$ is similar to that of $p$, $q$, and $r$. In our numerical calculations (S1 of Figs. 5 and 8), $l,m,n$ take the values $l=m=n=1$, which is the most generic case. In the vicinity of other types of singularities, analytic expressions of the solutions are obtained by assuming power-law behavior of the scale factors and the regularity of the modulus field, just as in the isotropic case [@art94]. Because of the anisotropy, there are 3 cases of singular solutions other than the type I. $$\begin{aligned} {\cal C}1&:&p\sim p_1/t,q\sim q_0, r\sim r_0, \sigma\sim\sigma_0, \dot\sigma\sim\sigma_1,\ddot\sigma\sim\sigma_2\nonumber,\\ &&p_1=1, q_0+r_0-\frac 32 \lambda\left.\frac{\partial\xi}{\partial\sigma} \right\|_{\sigma_0}\sigma_1q_0r_0=0 \label{eqn:assolc1},\\ {\cal C}2&:&p\sim p_1/t,q\sim q_1/t, r\sim r_0, \sigma\sim\sigma_0, \dot\sigma\sim\sigma_1,\ddot\sigma\sim\sigma_2\nonumber,\\ &&p_1=q_1=1, 1-\frac 32 \lambda\left.\frac{\partial\xi}{\partial\sigma} \right\|_{\sigma_0}\sigma_1r_0=0 \label{eqn:assolc2},\\ {\cal C}3&:&p\sim p_1/t,q\sim q_1/t, r\sim r_1/t, \sigma\sim\sigma_0, \dot\sigma\sim\sigma_1 t,\ddot\sigma\sim\sigma_1\nonumber,\\ &&p_1=q_1=r_1=1, 1-\frac 12 \lambda\left.\frac{\partial\xi}{\partial\sigma} \right\|_{\sigma_0}\sigma_1=0 \label{eqn:assolc3},\end{aligned}$$ where we have chosen the origin of $t$ at the singularity. ${\cal C}1$ is the case where only one of the three expansion rates ($p,q,r$) is singular, ${\cal C}2$ two, and ${\cal C}3$ all three. The solution ${\cal C}1$ agrees well with our numerical results (S2 of Fig. 5). Since $p_1,q_1,r_1=1$ (if not zero), the divergent behavior is determined by the sign of $t$; i.e., if a solution goes into a singularity futureward ($t\rightarrow -0$), then $p\rightarrow-\infty$, etc., and if pastward ($t\rightarrow +0$), $p\rightarrow+\infty$, etc. Putting Eqs. (\[eqn:assolc1\])$-$(\[eqn:assolc3\]) into Eq. (\[eqn:delta\]), we have $\Delta\rightarrow-\infty$ for ${\cal C}1$ and ${\cal C}2$ but $\Delta\rightarrow +0$ for ${\cal C}3$. Therefore, according to our definition of the singularities ($\Delta\rightarrow 0$ for type I and $\Delta\rightarrow -\infty$ for type II), ${\cal C}1$ and ${\cal C}2$ will be categorized into type II and ${\cal C}3$ will be categorized into type I. In the isotropic case ($p=q=r$), we can show that $\Delta>0$ is always satisfied, and ${\cal C}3$ can be seen as a type I singularity “pushed towards the infinity.” All of these singularities, both types I and II, are physical singularities. This can be shown by putting $p,q,r$ and $\dot p,\dot q,\dot r$ into the curvature scalar $R=2(\dot p+p^2+\dot q+q^2+\dot r+r^2+pq+qr+rp)$. Summary of numerical results ---------------------------- We extended the nonsingular universe model proposed by Rizos and Tamvakis [@rt94] to include anisotropy, and examined solutions in the vicinity the of isotropic solution in both $\sigma<0$ and $\sigma>0$ regions. We found both nonsingular solutions and singular solutions. Nonsingular solutions inhabit the region where the anisotropy is small, and they evolve from the past asymptotic region, superinflate, and then lead either to Friedmann-type or to Kasner-type solutions in the future. Singularities appearing in our analysis are classified into two types, namely, type I and type II. The type I singularity corresponds to the crossing of $\Delta=0$, and $\dot p$, $\dot q$, and $\dot r$ diverge, while $p$, $q$, $r$, and $\sigma$, $\dot\sigma$ stay finite. At the type II singularity, on the other hand, $\Delta$ tends to $-\infty$, and $p$, $q$, and $r$ will diverge. The evolution of the singular solutions is characterized by the behavior of $\Delta$. At the origin of the $X$-$Y$ plane (isotropic solution) $\Delta$ is always positive regardless of the values of $\sigma$ or $H_{\mbox{\scriptsize avr}}$, and as anisotropy increases there appear $\Delta<0$ regions or regions prohibited by the constraint equation (\[eqn:b1eom1\]), which is shown in Figs. 3 and 6. There are three types of singular solutions appearing in our analysis (if two branches connected by a type I singularity are counted as one solution). The first type of singular solution is marked by the crossing of the $\Delta=0$ line, which we termed the type I singularity, in the future. This includes S1a and S1b in Figs. 4 and 5, and if we continue the solution beyond $\Delta=0$ by changing the variable, this singular solution can be seen as a pair annihilation of the $\Delta>0$ and $\Delta<0$ branches of solutions. The $\Delta>0$ branch continues from the infinite past, while the $\Delta<0$ branch leads to a type II singularity at the finite past. The second singular solution is very similar to the first one, except it crosses the $\Delta=0$ line in the past. This solution, examples of which are S1c and S1d of Figs.7 and 8, can be regarded as a pair creation of two branches. The singular solution in the isotropic model[@art94; @rt94] is a special case of this second singular solution. The third singular solution lies always in the $\Delta<0$ region and never crosses the $\Delta=0$ line, i.e., includes no type I singularity. This solution is born in the type II singularity, and disappears into the type II singularity (see S2 of Figs. 4 and 5). CONCLUSION ========== In this paper we presented anisotropic nonsingular cosmological solutions derived from the 1-loop effective action of the heterotic string. We found nonsingular solutions which evolve from the infinite past asymptotic region, superinflate, and then continue either to Friedmann-type or Kasner-type solutions. The singular solutions of moderate anisotropy are classified into three types, and involved in these solutions are two types of singularities: one corresponds to $\Delta=0$ and the other to $\Delta=-\infty$. The initial singularity in the isotropic limit is a special case of the $\Delta=0$ singularity. Violation of the energy conditions, which is necessary to avoid the singularity, is achieved by the existence of a Gauss-Bonnet term coupled to a modulus field. This can be confirmed by using the asymptotic forms (\[eqn:assola\]) and (\[eqn:assolb\]) for nonsingular solutions. We define the effective energy density and effective pressure as $$\begin{aligned} \epsilon&:=&-G^0{}_0=pq+qr+rp,\\ p_1&:=&G^1{}_1=-(\dot q+\dot r+q^2+r^2+qr),\\ p_2&:=&G^2{}_2=-(\dot r+\dot p+r^2+p^2+rp),\\ p_3&:=&G^3{}_3=-(\dot p+\dot q+p^2+q^2+pq).\end{aligned}$$ Assuming the asymptotic solution ${\cal A}$ in the region $t\rightarrow\infty$ and that $\dot\sigma$ is always positive (in our numerical analysis $\dot\sigma$ keeps its sign except the singular solution S2), the effective energy density and pressure behave in the future asymptotic region as $\epsilon$, $p_i\sim \frac 12 \omega_4^2\vert t\vert ^{-2}$ ($i=1,2,3$). Thus, the weak energy condition $\epsilon+p_i\sim\omega_4^2\vert t\vert^{-2}>0$ and the strong energy condition $\epsilon+\sum p_i\sim 2\omega_4^2\vert t \vert^{-2}>0$ are satisfied. In the past asymptotic region, on the other hand, the asymptotic solution becomes ${\cal B}$ in which the Gauss-Bonnet term is dominant. Then $\epsilon\sim (\omega_1\omega_2+\omega_2\omega_3 +\omega_3\omega_1)\vert t\vert ^{-4}$ and $p_1\sim-2(\omega_2+\omega_3)\vert t\vert^{-3} -(\omega_2^2+\omega_3^2+\omega_2\omega_3)\vert t\vert^{-4}$, etc. If the weak energy condition is satisfied, $\epsilon+p_i>0$ for $i=1,2,3$, and so $3\epsilon+\sum p_i\sim-4(\omega_1+\omega_2+\omega_3)\vert t\vert^{-3}$ must be positive. If the strong energy condition is satisfied, $\epsilon+\sum p_i\sim-4(\omega_1+\omega_2+\omega_3)\vert t\vert^{-3}$ must be positive. Neither of them is possible as long as $H_{\mbox{\scriptsize avr}} \sim(\omega_1+\omega_2+\omega_3)\vert t\vert^{-2}/3>0$. Therefore, weak and strong energy conditions are violated in the past asymptotic region. One of the biggest advantages of our model is that it includes a rather long period of (super)inflationary stage in a natural form. However, our result, which admits the evolution of an almost-isotropic superinflating solution into a Kasner-type anisotropic solution (see the panel “NSb” of Fig. 5 for example), suggests that the superinflation in our model does not isotropize the space-time. Our recent study[@kss98] of the cosmological perturbation for the homogeneous and isotropic background shows the existence of exponential growth in graviton-mode perturbation during the superinflationary stage. Together with this we can conclude that the cosmic no-hair hypothesis [@cnohair] does not hold in this superinflationary model. We are interested in whether this result is common to all kinetic-driven superinflation. This work of J.S. is supported by the Grant-in-Aid for Scientific Research No. 10740118. M. Gasperini and G. Veneziano, Astropart. Phys. [1]{}(1993) 317, Mod. Phys. Lett. [A8]{}(1993) 3701, Phys. Rev. [D50]{}(1994) 2519. R. Brustein and G. Veneziano, Phys. Lett. [B329]{}(1994) 429; N. Kaloper, R. Madden and K. A. Olive, Nucl. Phys. [B452]{}(1995)677; M. Gasperini, M. Maggiore and G. Veneziano, Nucl. Phys. [B494]{}(1997) 315; R. Brustein and R. Madden, Phys.Lett. [B410]{}(1997) 110. I. Antoniadis, J. Rizos and K. Tamvakis, Nucl. Phys.[B415]{}(1994) 497. J. Rizos and K. Tamvakis, Phys. Lett. [B326]{}(1994) 57. R. Easther and K. Maeda, Phys. Rev. [D54]{}(1996) 7252. J. A. Harvey and G. Moore, Phys. Rev. [D57]{}(1998) 2323. S. Kawai, M. Sakagami and J. Soda, Phys. Lett. [B437]{} (1998) 284. R. Wald, Phys. Rev. [D28]{}(1983) 2118, M. S. Turner and L. M. Widrow, Phys. Rev. Lett. [57]{}(1986) 2237, L. G. Jensen and J. A. Stein-Schabes, Phys. Rev. [D34]{}(1986) 931. epsf.sty $H_{\mbox{\scriptsize avr}}=0.001, \hspace{2mm}\sigma=-10$ $H_{\mbox{\scriptsize avr}}=0.005, \hspace{2mm}\sigma=-10$ $H_{\mbox{\scriptsize avr}}=0.01, \hspace{2mm}\sigma=-10$ $H_{\mbox{\scriptsize avr}}=0.001$, $\sigma=-10$ $H_{\mbox{\scriptsize avr}}=0.005$, $\sigma=-10$ $H_{\mbox{\scriptsize avr}}=0.01$, $\sigma=-10$ NSaNSb S1a,bS2 $H_{\mbox{\scriptsize avr}}=0.1$, $\sigma=2.5$ $H_{\mbox{\scriptsize avr}}=0.2$, $\sigma=2.5$ $H_{\mbox{\scriptsize avr}}=0.3$, $\sigma=2.5$ $H_{\mbox{\scriptsize avr}}=0.01$, $\sigma=2.5$ $H_{\mbox{\scriptsize avr}}=0.05$, $\sigma=2.5$ $H_{\mbox{\scriptsize avr}}=0.1$, $\sigma=2.5$ NSS1c S1dIsotropic [^1]: E-mail:kawai@phys.h.kyoto-u.ac.jp [^2]: E-mail:jiro@phys.h.kyoto-u.ac.jp [^3]: This action is also justifiable in more solid grounds. In the $N=4$ superstring compactifications (heterotic on $T^6$ or type IIA on $K3\times T^2$) the function $\xi(\sigma)$ given in Eq.(\[eqn:ssxi\]) describes the exact $R^2$ couplings including nonperturbative effects[@hm96]. These vacua exhibit an exact $S$ duality, which is identical with the SL$(2,Z)$ modular invariance on $\sigma$.
--- author: - Jorge Mastache - Axel de la Macorra bibliography: - 'cmbbdm.bib' title: 'Constraining Dark Matter transition velocity with CMB Plank Data and Large Scale Structure.' --- Introduction {#sec:introduction} ============ The measurement of the Cosmic Microwave Background (CMB) anisotropies [@Aghanim:2018eyx] and the mapping of the large-scale structures (LSS), through galaxy redshift surveys [@Abbott:2016ktf; @Percival:2006gt] and type Ia Super Novae [@Betoule:2014frx] have had a great impact on our knowledge of the Universe. In particular, they have established the standard model for cosmology, the $\Lambda$CDM model, in which the content of the Universe consist on 65% dark energy driving the expansion of the Universe, 31% dark matter (DM) whose clustering properties influence the large scale structure formation. This two component which account for up to $96\%$ of the energy content of the Universe, and unfortunately its nature is unknown. In this work we explore the parameters that could constrain the essence of dark matter. A large number of candidates have been proposed for DM [@Bertone:2004pz] of which cold dark matter (CDM) has been the most popular. The CDM model has been successful on explaining large scale structure formation in the early Universe as well as abundances of galaxy clusters [@Springel:2005nw; @Cooray:2002dia]. However, the clustering property of structure at scales of the order of galaxy scales is not well understood. For instance, the number of satellite galaxies in our Galaxy is smaller than the expected from $\Lambda$CDM model [@Kazantzidis:2003hb], the so-called missing satellite problem. In a CDM scenario the DM particles become non-relativistic at very early stages of the evolution of the Universe (e.g. for a masses larger than $\mathcal{O}$(MeV)) and form structure at all relevant scales, small halos merge to form larger ones, a process that spans a wide of range of scales, from galaxy clusters down to microhaloes with masses down to the Earth’s mass. However, his standard scenario seems to disagree with a number of observations. First, the number of sub-haloes around a typical Milky Way galaxy, as identified by satellite galaxies, is an order of magnitude smaller than predicted by $\Lambda$CDM model [@Kazantzidis:2003hb; @BoylanKolchin:2011de]. At the same time, $\Lambda$CDM also predicts steeply cusped density profiles, causing a large fraction of haloes to survive as substructure inside larger haloes [@Navarro:1996gj]. Observed rotation curves for dwarf spherodial dSph and low surface brightness (LSB) galaxies seem to indicate that their DM haloes prefer constant density cores [@Moore:1994yx] instead of steep cusps as predicted by the Navarro-Frank-White profile [@Navarro:1996gj]. LSB galaxies are diffuse, low luminosity systems, which kinematics is believed to be dominated by their host DM halos [@deBlok:2001hbg]. Assuming that LSB galaxies are in virial equilibrium, the stars act as tracers of the gravitational potential, therefore, can be used as probe of the DM density profile. Much better fits to dSph and LSB observations are found when using a cored halo model [@deBlok:2001hbg]. Cored halos have a mass-density that remains at an approximately constant value towards the center. Many solutions to both of this problems have been proposed, and there are two main branches. First, the solution through baryon physics - star formation and halo evolution in the galaxy may be suppressed due to some baryonic process [@Penarrubia:2012bb]. Second, the DM solution - the number of satellite galaxies are suppressed due to the kinematic properties of the DM particles [@Bode:2000gq]. The discussion is still in progress. Here we take the second branch by constraining the parameters of DM particles that influence structure formation. Regarding the cosmological impact of DM particles, perhaps the two most important quantities to determine the cosmological effects, besides the amount of energy density, $\Omega_{\rm dm}$, is the time when the DM particles became non relativistic, defined by the scale factor $a_{nr}$, and the velocity dispersion of these particles at that moment, $v_{nr}$. For decoupled warm dark matter (WDM) particles we compute a one to one relationship between the WDM mass $m_{wdm}$ and $a_{nr}$ and a standard lower limit is given by $m_{wdm} \gtrsim $ 3keV, using Ly-$\alpha$ forest observations [@Viel:2013apy]. It is common to define the time when a particle became non-relativistic when the momentum equals its mass, i.e. $p^2=m^2$ and $E^2=p^2+m^2=2m^2$, which correspond to a velocity $v_{nr}= 1/\sqrt{2}$ [@Bode:2000gq]. A WDM with $m_{\rm wdm}=3$ keV mass we obtain a $a_{nr}=3.14\times 10^{-7}$. However the connection between the DM mass and $a_{nr}$ can vary depending on the properties of the DM particle. For example axion particles may have very low masses (as low as $m<\mathcal{O}$(eV)) and are still considered CDM because these axions have a small velocity dispersion at high energies and therefore at all relevant scales for structure formation. Furthermore, it is usually assumed a smooth evolution for the velocity dispersion $v(a)$ as for thermal WDM candidates. However this assumption is based that the mass of the particle is constant, which is not necessarily true in all cases. For example, we could have DM with an abrupt evolution of $v(a)$ if the mass of the DM particle is due to the binding energy of elementary particles, as for example the mass of neutrons or protons in the Standard Model of particles, or in a Bound Dark Matter (BDM) model [@delaMacorra:2009yb] analyzed here. The BDM model, motivated by particle physics, assumes that elementary particles contained in a gauge group are nearly massless at high energies but once the energy decreases the gauge force becomes strong (at a scale factor defined by $a_c$ and energy $\Lambda_c$) and forms neutral bound states (neutron like particles) which acquire a non-perturbative mass proportional to $\Lambda_c$. The BDM particles are by hypothesis not contained in the standard model of particles physics. Indeed, at high energies the elementary particles (quarks) of the Standard Model (SM) of particle physics are weakly coupled however the strength of the gauge coupling constant increases for lower energies and eventually becomes strong at the condensation energy scale $\Lambda_{\rm QCD}$ and scale factor denoted by $a_{\rm QCD}$. At this scale, gauge invariant states are created forming gauge neutral composite particles, mesons (e.g. pions $\pi$) and baryons (e.g. protons and neutrons), at $\Lambda_{\rm QCD}= (210 \pm 14)$MeV [@Tanabashi:2018oca], with non perturbative masses generated and being proportional to $\Lambda_{\rm QCD}$ (e.g. $m_\pi \simeq 140$MeV, $m_{n}\simeq 940$MeV) much larger the quark masses ($m_u\simeq 2.3$MeV, $m_d\simeq 4.8$MeV). The mass of the bound states (mesons and baryons) are due to the underlying gauge force and are independent of the bare mass of the original quarks. Since the mass of the bounds states is much larger then the mass of the elementary particles the resulting velocity dispersion, $v_c$, of these bounds states is significantly reduced to the velocity of the original elementary particles. Therefore we expect to have an abrupt transition for the velocity dispersion $v_c$ at $a_c$ with $v_c\ll v_{nr}\equiv1/\sqrt{2}$. As in QCD, where the transition from fundamental elementary particles (quarks) to bound states (mesons and baryons) takes place from high to low the energies, our BDM model also form bounds states at lower energies. The transition from high to low energy densities can be encountered in different cosmological scenarios. One case is as a consequence of the expansion of the Universe, as it grows the energy density dilutes, and secondly inside massive structures (e.g. galaxies) where the energy density increases with decreasing radius. Therefore our BDM could help to ameliorate two of the main CDM problems, namely the missing satellite problem and the the cuspy energy density profiles in low density galaxies [@delaMacorra:2011df]. Indeed, the free streaming of the BDM particles prevents small halos and will also have an impact in the center region of galaxies rendering a core galactic profile. Here we study the cosmological properties of BDM particles by constraining these two parameters, the scale factor $a_c$ and the velocity dispersion $v_c$ when the bound states acquire their non-perturbative mass, and by taking $a_c=a_{nr}$ and $v_c=v_{nr}=1/\sqrt{2}$ we recover WDM as a limit of BDM. We find that a BDM model that becomes non-relativistic at a scale factor 10 times larger than in a WDM model can have an equivalent free streaming scale $\lambda_{fs}$ and $k_{1/2}$, the mode where the power spectrum is suppressed by $50\%$ with respect to $\Lambda$CDM model, therefore rendering the same cosmological properties as WDM. We organize the work as follows: in Section \[sec:framework\] we present the theoretical dark matter framework, introduce the BDM model in sub-section \[subsec:BDM\] and compute the free streaming scale for different dark matter models in sub-section \[ssec:freestreaming\]. In Section \[sec:lls\] we compute the CMB power spectrum using the perturbations of the BDM model, \[ssec:cmb\_ps\], and show the repercussions of the BDM model on structure formation by showing the matter power spectrum in sub-section \[ssec:mps\], the mass function in sub-section \[ssec:preschechter\] . We present an analysis of our results and discussion on the conection found with WDM in Section \[sec:wdm\]. We conclude in Section \[sec:conclusion\]. Finally, we present the standard perturbations equations for BDM in appendix \[appendix:perturbation\] General Dark Matter Framework {#sec:framework} ============================= DM particles are usually classified by their velocity dispersion given in terms of three broad categories: hot (HDM), warm and cold dark matter. The main difference between these three cases is the scale factor, $a_{nr}$, when the DM particles become nonrelativistic. In principle HDM are relativistic at all cosmological relevant scales, e.g. neutrions, CDM have a small $a_{nr}$ (with $a_{nr}\ll a_{eq}$) while WDM are particles in between. For thermal DM particles one can relate their mass to the scale factor $a_{nr}$ and in such a case recent studies give a lower limit $m_{wdm} \approx 3$ keV for WDM particles . Relativistic particles with mass $m$ and a peculiar velocity, $v$, have a momentum ${p}=\gamma mv$, and energy $E^2=p^2+m^2$, where $\gamma\equiv 1/\sqrt{1-v^2}$. Solving for $v$ we get, $$\label{eq:vel_gral_eq} v = \frac{p^2}{E^2} =\frac{p^2}{\sqrt{m^2 + p^2}}.$$ For $p^2 \gg m^2$ the velocity is $v \sim 1$ and the particle is relativistic, while for $p^2\ll m^2$ we have $v \ll 1$ and the particle is then non-relativistic. In an expanding FRW Universe, the momentum of a relativistic particle redshift as $p(a)= p_{\star} (a_\star/a)$, where $p_\star$ and $v_\star$, with the correspondent parameter $\gamma_\star \equiv (1 - v_\star^2)^{-1/2}$, are a pivotal point condition for the momentum and the velocity at $a = a_\star$. Therefore, the velocity at all times in an expanding Universe evolves as $$\label{eq:veldm} v(a) = \frac{ \gamma_\star v_\star(a_\star/a)}{\sqrt{1 + \gamma^2_\star v^2_\star(a_\star/a)^2}}.$$ Eq.(\[eq:veldm\]) describes the velocity evolution of a decoupled relativistic massive particle having a smooth transition from the relativistic limit $v\simeq 1$, for $a \ll a_\star$, to a non-relativistic behavior $v\simeq 0$, in the limit $a \gg a_\star$. This is a general evolution and it is valid for massive particles (WDM, CDM or massive HDM). It is common to set the epoch when the particle becomes non relativistic when $p^2=m^2$, with $E^2=p^2+m^2=2m^2$, from Eq. the velocity is simply $v(a_{nr})\equiv v_{nr}=1/\sqrt{2}$ with $\gamma_{nr}=\sqrt{2} $ and $\gamma_{nr} v_{nr}=1$. We have set the pivotal time at this epoch. i.e. $a_\star=a_{nr}, v_\star=v_{nr}$. For a massive particle that becomes non-relativistic at $a_{nr}$ the velocity at any time evolve then as $$\label{eq:wdm_vel} v(a) = \frac{ (a_{nr}/a)}{\sqrt{1 + (a_{nr}/a)^2}}.$$ Notice here that for thermal particles the value of $a_{nr}$ can be directly related to the mass of the particle, a larger mass gives a smaller $a_{nr}$ and become non-relativistic earlier. In order to determine the evolution of the energy density $\rho$, we take $\rho= \langle E\rangle\, n$ and the pressure $P=\langle \|\bar p\|^2 \rangle n / 3\langle E \rangle$, with $n$ the particle number density, $\langle\|\bar p\|^2\rangle$ is the average quadratic momentum and $\langle E\rangle$ the average energy of the particles. The equation of state (EoS) $w\equiv \rho/P$ is given then by $$\label{eq:eos} \omega = \frac{ \langle \|\bar p\|^2\rangle}{3\langle E\rangle^2}= \frac{v(a)^2}{3}$$ Integrating the continuity equation $\dot\rho=-3H(\rho+P)$ and using Eq. and we obtain the analytic evolution of the background $\rho_{\rm dm}(a)$ $$\la{rm} \rho_{\rm dm}(a) = \rho_{\rm dm\star} \left( \frac{a}{a_\star} \right)^{-3} f(a)$$ with f(a)= () \^[-1]{}() = \_\^[-1]{} . Eqs.(\[rm\]) and (\[f\]) are valid for any value of $v_\star$ including the case of a standard massive particle with $ v_\star=v_{nr}=1/\sqrt{2}$ at $a_\star=a_{nr}$. From Eq. we clearly see that $\rho_{\rm dm}(a_\star)=\rho_{\rm dm\star} $ since $f(a_\star)=\gamma_\star^{-1}\sqrt{1+\gamma_\star^2v_\star^2} =1$. In the limit $a\ll a_\star$ we have $\rho_{\rm dm}(a)=\rho_{\rm dm\star} (a/a_\star)^{-4} $, since $f\simeq (a_\star/a)$ and $v_\star/ v \simeq 1$, showing that $\rho_{\rm dm}(a)$ evolves as radiation, while in the late time limit $a\gg a_\star$ we have $f\simeq 1/\gamma_\star$ and $\rho_{\rm dm}(a)=\rho_{\rm dm\star} (a/a_\star)^{-3} /\gamma_\star $. Finally, in terms of present day values we obtain $$\la{rmo} \rho_{\rm dm}(a) = \rho_{\rm dm o} \left( \frac{a}{a_o} \right)^{-3} \fr{f(a)}{f(a_o)}$$ with $f(a)/f(a_o)\simeq 1$ for $a\gg a_\star$, thus $\rho_{\rm dmo}$ is the amount for dark matter density today. ![[]{data-label="fig:eos"}](eos.eps){width="\textwidth"} BDM Model {#subsec:BDM} --------- An interesting model not contained in the above description is our BDM model, previously introduced in [@delaMacorra:2009yb]. Here we just summarize the most important characteristics that will help us developed the present work. In the model of interest, the particles are relativistic for $a< a_c$ and they go through a phase transition at $a_c$, where the original elementary (massless or nearly massless) particles form bound states which we call Bound Dark Matter BDM, similar as in QCD where quarks form baryons and mesons. Clearly the mass of the mesons and baryons, with masses of the order $\mathcal{O}$(GeV), do not correspond to the sum of the of the constituent quark masses, for which are of the order of $\mathcal{O}$(MeV). We propose that BDM particles are relativistic and massless (or with a very small mass) for $a<a_c$ and acquire a nonperturbative mass $m_{\rm bdm} \propto \Lambda_c$ at $a_c$, due to the non perturbative effects of the underlying force with transition energy $\Lambda_c$. Since this effect is a non-smooth transition we expect the BDM particles to go from being masless, for $a<a_c$, to massive at $a_c$ (with a corresponding time $t_c$). Therefore, the velocity of the particle goes from $v=1$, for $a<a_c$, to $v \rightarrow v(a_c) \equiv v_c$ (with $\gamma(a_c) \equiv \gamma_c$) and the evolution of its velocity and the EoS is given by $$\begin{aligned} \begin{alignedat}{3} \omega_{\rm bdm} & = \frac{1}{3}, & \ v_{\rm bdm} &=1, & \quad {\rm for} \, a < a_c \\ \omega_{\rm bdm} &= \frac { v_{\rm bdm}^2} {3}, & v_{\rm bdm} &= \frac{ \gamma_{c} v_{c}(a_{c}/a)}{\sqrt{1 + \gamma^2_{c} v^2_{c}(a_{c}/a)^2}} & \quad {\rm for} \, a \geq a_c, \end{alignedat}\label{eq:eos_bdm} \end{aligned}$$ where subindices $c$ denotes quantities evaluated at $a_c$. The case where $1/\sqrt{2} > v_c > 0$ describe a particle whose velocity has been suddenly suppressed due to bounding nature of the particle. The case where $1> v_c > 1/\sqrt{2} $, the particle acquired mass and the velocity is suppressed, but still being a relativistic particle, this particle become non-relativistic at $a_{nr} = \gamma_c v_c a_c $ and this is not an interesting since it emulates WDM. We plot $\omega_{\rm bdm}$ in Figure\[fig:eos\]. The evolution of $\rho_{\rm bdm}$ before the phase transition at $a_c$ is that of relativistic energy density while after the transition we have from Eq.(\[rm\]) $$\begin{aligned} \begin{alignedat}{2} \rho_{\rm bdm}(a) & = \rho_{\rm bdm c} \left( \frac{a}{a_c} \right)^{-4} & \quad {\rm for} \,\, a < a_c \\ \rho_{\rm bdm}(a) & = \rho_{\rm bdm o} \left( \frac{a}{a_o} \right)^{-3} \fr{f(a)}{f(a_o)} & \quad {\rm for} \,\, a \geq a_c, \end{alignedat}\label{eq:rho_bdm} \end{aligned}$$ with $ f(a)= \gamma_c^{-1} \sqrt{ 1+\gamma_c^2 v_c^2(a_c/a)^2}$, with $f(a_c)=1, f(v_c=0)=1$ and $f(a\gg a_c)=1/\gamma_c$. As seen in Eq. a massive particle (WDM or CDM) becomes non-relativistic at $a_{nr}$ with $v_{nr}=1/\sqrt{2}$ and has only one free parameter, the scale factor $a_{nr}$, however in BDM the EoS has two free parameters, the moment of the transition $a_c$ and the velocity dispersion $v_c$ at that time. We recover CDM if the transition happens at $a_c = \mathcal{O}$(MeV), in this case the velocity parameter is less important because the particle become non-relativistic in a very early stage of the Universe. HDM can also be describe if the transition is of the order of $a_c = \mathcal{O}$(eV) and the particle is highly relativistic. Free-streaming scale {#ssec:freestreaming} -------------------- In this section we focus on studying the imprints that the BDM model has upon the statistical properties of the LSS. The thermal velocities of the BDM particles have a direct influence on structure formation. While DM particles are still relativistic, primordial density fluctuations are suppressed on scales of order the Hubble horizon at that time. This is call the free-streaming scale and depends on the moment when a massive particle becomes non-relativistic ($a_{nr}$) or in our BDM when the phase transition takes place, $a_{c}$. We study first the free-streaming of the BDM particles and later in Section(\[ssec:preschechter\]) the structure abundances for masses using the Press-Schechter formalism. The comoving free streaming scale $\lambda_{fs}$ is defined by $$\lambda_{fs} = \int_0^{t_{eq}} \frac{v(t) dt}{a(t)} = \fr{2t_{c}}{a_{c}^2} \int_{0}^{a_{eq}} v(a) da, \label{eq:fs_general}$$ where we have assume a radiation dominated Universe with $t \propto a^2$ for $a\leq a_{eq}$. The free-streaming scale is defined by th e mode $k_{fs}$ and the Jeans mass $M_{fs} $ contained in sphere of radius $\lambda_{fs}/2$ given by k\_[fs]{}=, M\_[fs]{} = ()\^3 \_[mo]{}. Haloes with masses below the free-streaming mass scale will be suppressed. ### WDM scenario ![[]{data-label="fig:power_spec"}](CMBPowerSpectrumAll_ac.eps){width="\textwidth"} Let us now determine the comoving free streaming scale $\lambda_{fs}$, first for the fiducial CDM case. It is standard to separate the integral in the relativistic regime with a constant speed $v=1$ for $a<a_{nr}$ and in the non-relativistic regime $a>a_{nr}$ to take $v = a_{nr}/a$. With these choices of $v$ one gets the usual free streaming scale \_[fs]{} (a\_[eq]{})&=& \_0\^[t\_[nr]{}]{} + \_[t\_[nr]{}]{}\^[t\_[eq]{}]{}\ &=& where we used that in radiation domination $t \propto a^2$ and $t/t_{nr}=a^2/a^2_{nr}$. The first term in eq.(\[fs1\]) corresponds to the integration from $a=0$ to $a_{nr}$, while the second from $a_{nr}$ to $a_{eq}$. However, it is more accurate to us Eq. for the velocity, since it is valid for all $a$. In this case we obtain a free streaming scale \_[fs]{}(a) = valid for all $a$. Let us now evaluate Eq.(\[fs2a\]) at $a_{eq}$ and take the limit $a_{nr}/a_{eq} \ll 1$ to get \_[fs]{}(a\_[eq]{}) (\[2\] + ) = with $\operatorname{Log}[2]\simeq 0.69$. Eq.(\[fs2a\]) or its limit Eq.(\[fs4\]) should be used instead of Eq.(\[fs1\]) since they capture the full evolution of the velocity $v(a)$ of a massive particle given in Eq.(\[eq:wdm\_vel\]). ![image](mps_all_vc_fix.eps){width="\textwidth"} ![image](mps_all_ac_fix.eps){width="100.00000%"} ### BDM scenario Let us now determine the free streaming scale for our BDM model. The velocity of the particle is given by Eq.(\[eq:eos\_bdm\]), which takes into account the transition for BDM, this leads to the free streaming scale \_[fs]{}\^[bdm]{} (a\_[eq]{})= \_0\^[t\_c]{} + \_[t\_c]{}\^[t\_[eq]{}]{} giving $$\begin{aligned} \lambda_{fs}^{\rm bdm} (a_{eq})&=& \fr{2t_{c}}{a_{c}^2} \left[ \int_0^{a_c} da + \int_{a_c}^{a_{eq}} v(a) da \right] \label{eq:fs_bdm0} \\ &=& \fr{2t_{c}}{a_{c}} \le( 1 + \gamma_c v_c \textrm{Log} \le[\fr{ 1+ \sqrt{1 + \gamma_c^2 v_c^2(a_c/a_{eq})^2 }}{ {(1 + \gamma_c)(a_c/a_{eq})} } \ri] \ri) \la{fs2} \nonumber \\ &\simeq& \fr{2t_{c}}{a_{c}} \le( 1 + v_c \gamma_c \textrm{Log} \le[ \frac{2}{(1 + \gamma_c)}\frac{a_{eq}}{a_c} \ri] \ri), \la{fs3}\end{aligned}$$ where $t_c$ is the time corresponding to the transition $a_c$. In the last equation we assume that $a_c \ll a_{eq}$. Clearly the value of $v_c$ in eq.(\[fs3\]) has a huge impact on the resulting free streaming scale and the corresponding mass contained within a sphere of radius $R=\lambda_{fsb}/2$. For example, if we take a BDM that becomes no-relativistic at the same scale factor as a 3 keV mass WDM we find for $v_c=0$ that $\lambda_{fs} = 0.09$ Mpc/h, with a Jeans mass of $M_{fs} = 1.31 \times 10^{7} \ M_\odot/h^3$, in contrast with the $\lambda_{fs} = 0.7$ Mpc/h and $M_{fs} = 5.93 \times 10^{9} \ M_\odot/h^3$ for WDM. Clearly the amount of structure can be severely reduced in BDM compared to a WDM model depending on the value of $v_c$. Large Scale Structure in BDM scenario {#sec:lls} ===================================== To have a better understanding in the cut-off scale we need to compute the matter power spectrum, in order to achieve this goal we show in Appendix \[appendix:perturbation\] the synchronous perturbed equations a BDM particle which let us to compute the power spectrum using the code CLASS. Then we compute the photon and matter power spectrum in subsection \[ssec:cmb\_ps\] and \[ssec:mps\], respectively. Throughout this paper, we adopt Planck 2018 cosmological parameters [@Aghanim:2018eyx]. For the several simulations we adopt a flat Universe with $\Omega_c h^2 = 0.12$, and $\Omega_b h^2 = 0.02237$ as the CDM matter and baryonic omega parameter. $h = 0.6736$ is the Hubble constant in units of 100 km/s/Mpc, $n_s = 0.965$ is the tilt of the primordial power spectrum. $z_{\rm reio} = 7.67$ is the redshift of reonization and $\ln(10^{10} A_s) = 3.044$, where $A_s$ is the amplitud of primordial fluctuations. CMB Power Spectrum {#ssec:cmb_ps} ------------------ In Figure\[fig:power\_spec\] we show the CMB power spectrum obtained with CLASS code [@Blas:2011rf] taking into account the BDM perturbations, see Appendix \[appendix:perturbation\]. We show the power spectrum for two different values of $a_c = \{10^{-6}, 10^{-7}\}$ and $v_c = \{0.01, 1/\sqrt{2}\}$. The smaller the value for $a_c$ and the smaller the velocity $v_c$ implies that BDM is more like CDM, therefore the difference between curves is less notorious. We also compare the curves with a WDM particle with mass of 3 keV. We notice that the effect of changing the initial velocity, $v_c$, is negligible for the CMB power spectrum. The percentage difference between $\Lambda$-BDM and $\Lambda$-CDM power spectrum, show in the bottom panel of Figure\[fig:power\_spec\] is less than 0.1% for the BDM cases. The CMB power spectrum barely increased the height of the acoustic peaks, mainly because the increase in the free-streaming smooth out perturbations and increase the acoustic oscillations. Matter Power Spectrum {#ssec:mps} ---------------------- In Fig \[fig:mps\_fix\_vc\] and \[fig:mps\_fix\_ac\] we plot the linear dimensionaless power spectrum. The effect of the free-streaming (computed in subsection \[ssec:freestreaming\]) for BDM particles is to suppress structure formation below a threshold scale, therefore the matter power spectrum show a cut-off at small scales depending the value of $a_{c}$ and $v_{c}$. From Figure\[fig:mps\_fix\_vc\] one can notice that smaller the scale of the transition, $a_c$, the power is damped at smaller scales, for transitions at $a_c \lesssim 10^{-8}$ BDM model is indistinguishable from CDM at observable scales, $k \sim \mathcal{O} (10)$ Mpc/h. The scale of the transition would correspond to a WDM mass of $m_{\rm wdm} \sim 300$ keV. The novelty is this work comes with the relevance that the velocity $v_c$ takes for LSS, in Figure\[fig:mps\_fix\_ac\] we show that smaller values of $v_c$ implies a cooler dark matter, therefore a cut-off at smaller scales, and for a single transition at fixed $a_c$ the cut-off scale in the matter power spectrum can vary an order of magnitud. For instance, we can have the same free-streaming scale for two different massive particles, for instance, a particle having a transition with $(a_c = 10^{-7}, v_c=1/\sqrt{2})$ is similar to a different particle having a transition with $(a_c = 10^{-6}, v_c \sim 0)$. Notice that the WDM matter power spectrum is also replicable from the BDM model, in particular we show the case for a WDM of 3 keV mass, for which we can reproduce the cut-off scale with a BDM particle with a transition of $a_c = 2.73\times 10^{-7}$ and $v_c = 1/\sqrt{2}$, see Figure\[fig:WDM\_as\_BDM\]. ![image](khalf_ac.eps){width="\textwidth"} ![image](PressSchechter.png){width="\textwidth"} The parametrization of the MPS along with the cut-off scale can be found for WDM [@Bode:2000gq]. The same parametrization can be use for BDM particles, this is define $$\label{eq:fitted_viel} T _ { \mathrm { X} } ( k ) = \left[ \frac { P _ { \mathrm { lin } } ^ { \mathrm { X } } } { P _ { \mathrm { lin } } ^ { \mathrm { cdm } } } \right] ^ { 1 / 2 } = \left[ 1 + ( \alpha k ) ^ { 2 \mu } \right] ^ { - 5 / \mu }$$ with $\mu = 1.12$ and $X$ being the dark matter particle, either WDM or BDM. The cut-off of the power spectrum depends on the parameter $\alpha$, for the BDM case its value depends on $a_c$ and $v_c$, and it can be computed by the fitting function $$\alpha = 0.037 \left( \frac{a_c}{10^{-7}} \right)^{0.85} \left( \frac{v_c}{0.7} \right)^{0.97}$$ We found that this parametrization is valid for$k \leq k_{1/2}$, where $k_{1/2}$ is obtained by setting $T_X(k)^2 =1/2$, we therefore have $$\label{eq:k_half} k _ { 1 / 2 } = \frac { 1 } { \alpha } \left[ \left( \frac { 1 } { \sqrt { 2 } } \right) ^ { -\mu / 5 } - 1 \right] ^ { 1 / 2\mu },$$ for smaller scales the difference between numerical MPS and the parametrization of Eq. became bigger, but less than 50%, mainly because the cut-off of the BDM model are stepper than the ones obtained from WDM. Constraints on the BDM mass can be computed using Eq.(\[eq:k\_half\]), In Figure \[fig:k\_half\] we show lines that fit to the numerical values of $k_{1/2}$ obtained from the numerical code CLASS, their value depends on $a_c$ for the BDM case, and the mass, $m_{\rm wdm}$, for the WDM case. Press-Schecter {#ssec:preschechter} -------------- The change in the matter power spectrum is known to strongly affect large scale structure, we include the effects of the abundance of structure in the BDM cosmological model, we adopt the PressSchechter (PS) approach [@Press:1973iz]. First, we compute the linear matter power spectrum for the BDM, as described above, and compute the halo mass function as $$\frac{dn}{d\operatorname{Log}M} = M \frac{dm}{dM} = \frac{1}{2} \frac{\overline{\rho}}{M} \mathcal{F}(\nu) \frac{d \operatorname{Log}\sigma^2}{ d \operatorname{Log}M}$$ where $n$ is the number density of haloes, $M$ the halo mass and the the peak-height of perturbations is $$\begin{aligned} \nu = \frac{\delta_c^2(z) }{\sigma^2(M)}, \end{aligned}$$ where $\delta_c(z) = \frac{1.686}{D(z)}$ is the overdensity required for spherical collapse model at redshift z in a $\Lambda$CDM cosmology and and $D(z)$ is the linear theory growth function. The evolution of $\delta_{c}(z)$ and $D(z)$ evolve accordingly to the perturbation formalism for BDM introduced in Section \[sec:framework\]. The average density $\overline{\rho} = \Omega_m \rho_c$ where $\rho_c$ is the critical density of the Universe. Here $\Omega_m = \Omega_{c} + \Omega_b $. The variance of the linear density field on mass-scale, $\sigma^2(M)$, can be computed from the following integrals $$\label{eq:sigma_8} \sigma^2(M) = \int_0^\infty dk \frac{k^2 P_{\rm lin}(k)}{2\pi^2}\| W(kR) \|^2$$ we use the sharp-k window function $W(x) = \Theta( 1 - kR)$, with $\Theta$ being a Heaviside step function, and $R = (3cM/4\pi \overline{\rho})^{1/3}$, where the value of $c = 2.5$ is proved to be best for cases similar as the WDM [@Benson:2012su]. The sharp–k window function has also been prove to better work on models that show cut-off scale al large scales. Finally for the first crossing distribution $\mathcal{F}(\nu)$ we adopt [@Bond:1990iw], that has the form $$\mathcal{F}(\nu) = A\left( 1 + \frac{1}{\nu^{\prime p}} \right)\sqrt{ \frac{\nu^\prime}{2\pi} } e^{-\nu^\prime/2}$$\[eq:pressschechter\] with $\nu^{\prime} = 0.707\nu$, $p = 0.3$, and $A = 0.322$ determined from the integral constraint $\int f(\nu)d\nu = 1$. ![[]{data-label="fig:fs8"}](BDM_fs8.eps){width="\textwidth"} For mass-scales $M < M_{\rm fs}$, free-streaming erases all peaks in the initial density field, and hence peak theory should tell us that there are no haloes below this mass scale, therefore, significant numbers of haloes below the cut-off mass are suppressed. We show this behaviour more schematically in Figure \[fig:pressschechter\], where we compare CDM and BDM mass functions. For large halo masses $M > 10^{13} M_{\odot } h^{-1}$ the models are indistinguishable for a BDM particle with early transition. However, for smaller halo masses and late transitions, we can see significant suppression in the number of structure. To compute the value of $f\sigma_8$ first we compute $\sigma_8$ with Eq.(\[eq:sigma\_8\]) for $R = 8$ Mpc/h. For the sake of comparison with previous results, we adopt the top-hat window function to compute $\sigma_8$, using the sharp-k window function we obtained the same behavior but with a 58% difference respect the top-hat window function. As mention before the spherical top-hat window filter is not perfect for a truncated power spectrum [@Benson:2012su], but is a conservative choice that would result in weaker bound on the model. The growth rate of structure, $f$, is well defined by $$f \equiv \frac { \mathrm { d } \ln \delta _ { \mathrm { m } } } { \mathrm { d } \ln a }$$ The growth rate of structure can be approximated by the parametrization $f = \Omega _ { \mathrm { m } } ( a ) ^ { \gamma }$, where $\gamma$ is commonly referred to as growth index, which is approximately a constant in the range of observations. The definition of the parameter $\Omega_{ \mathrm { m } } (a) \equiv \rho_{ \mathrm { m } } (a) / 3 M_{ P }^{ 2 }H^{ 2 }( a )$ and $\rho_m$ is the density of matter evolution. For the BDM case $$f = \Omega_m^{0.58}$$ for $z<1$, in contrast with the value from $\Lambda$CDM which has an $\alpha = 6/11 \sim 0.545$. Because the evolution of the perturbations. In Figure\[fig:fs8\] we plot $f\sigma_8$ for CDM and BDM models. We obtain the $1\sigma$ tolerance for $f\sigma_8$ from Montecarlo simulations, we compare this curves with the ones obtained for BDM for different values of $a_c$, we notice that the velocity parameter, $v_c$, has no physical implications on the value of $f\sigma_8$ and it is important to notice that BDM and CDM deviates from one another at large redshift. This could be important for future observations. z $f\sigma_8$(z) 1/k Reference ------- ----------------- --------- ----------------------------------- 0.067 $0.42 \pm 0.05$ 16.0-30 6dFGRS(2012) [@Beutler:2012px] 0.17 $0.51\pm0.06$ 6.7-50 2dFGRS(2004) [@Percival:2004fs] 0.22 $0.42\pm0.07$ 3.3-50 WiggleZ(2011) [@Jennings:2010uv] 0.25 $0.35\pm0.06$ 30-200 SDSS [@Samushia:2011cs] 0.37 $0.46\pm0.04$ 30-200 SDSS [@Samushia:2011cs] 0.41 $0.45\pm0.04$ 3.3-50 2dFGRS(2004) [@Percival:2004fs] 0.57 $0.462\pm0.041$ 25-130 BOSS [@Alam:2015qta] 0.6 $0.43\pm0.04$ 3.3-50 WiggleZ(2011) [@Jennings:2010uv] 0.78 $0.38\pm0.04$ 3.3-50 WiggleZ(2011) [@Jennings:2010uv] 0.8 $0.47\pm0.08$ 6.0-35 VIPERS(2013) [@delaTorre:2013rpa] : $f\sigma_8$ table data. \[table2\] Halo Density Profiles {#ssec:bdmprofile} --------------------- ![image](mps_all_ac_bdm.eps){width="\textwidth"} ![image](2D_ac_vi.eps){width="\textwidth"} There are essentially two types of profiles, the ones stemming from cosmological $N$-body simulations that have a cusp in its inner region, e.g. Navarro-Frenk-White (NFW) profile [@Navarro:1995iw; @Navarro:1996gj]. On the other hand, the phenomenological motived cored profiles, such as the Burkert or Pseudo-Isothermal (ISO) profiles [@Burkert:1995yz; @vanAlbada:1984js]. Cuspy and cored profiles can both be fitted to most galaxy rotation curves, but with a marked preference for a cored inner region with constant density [@deBlok:2009sp]. As mention before, BDM particles at high densities are relativistic, the galaxy central regions could concentrate high amount of dark matter that BDM could behave as HDM, while in the outer galaxy regions BDM will behave as a non-relativistic particle, this is, as a standard CDM. To come forward this idea in galaxies we introduce a core radius ($r_{\rm core}$) stemming from the relativistic nature of the BDM, besides the scale length ($r_s$) and core density ($\rho_{\rm core}$) typical halo parameters. The galactic core density is going to be proportional to energy of the transition $\rho_{\rm core} \propto \Lambda_c^4$ and the profile properties determine the energy scale of the particle physics model. The average energy densities in galactic halos is of the order $\rho_g \sim 10^{5} \rho_{\rm cr}$ ($\rho_{\rm cr}$, being the critical Universe’s background density) and as long as $\rho_g<\rho_{\rm core}$ we expect a standard CDM galaxy profile (given by the NFW profile, $\rho_{{\rm nfw}}$). The NFW profile has a cuspy inner region with $\rho_{{\rm nfw}}$ diverging in the center of the galaxy. Therefore, once one approaches the center of the galaxy the energy density increases in the NFW profile and once it reaches the point $\rho_g=\rho_{\rm core}$ we encounter the BDM phase transition. Therefore, inside $r<r_{\rm core}$ the BDM particles are relativistic and the DM energy density $\rho_{\rm core}$ remains constant avoiding a galactic cusp. Of course we would expect a smooth transition region between these two distinct behaviors but we expect the effect of the thickness of this transition region to be small and we will not consider it here. Since our BDM behaves as CDM for $\rho < \rho_{\rm core}$ we expect to have a NFW type of profile. Therefore, the BDM profile is assume to be [@delaMacorra:2009yb] $$\begin{aligned} \label{eq:rhobdm} \rho_{bdm} = \frac{2 \rho_{\rm core}}{\left( 1+\frac{r}{r_{\rm core}}\right)\left( 1+\frac{r}{r_s}\right)^2}, \label{eq:fix-BDM}\end{aligned}$$ with $r_{\rm core}< r_s$ The BDM profile coincides with $\rho_{{\rm nfw}}$ at large radius but has a core inner region, then we can find a conection with NFW parameters. When the galaxy energy density $\rho_{\rm bdm}$ reaches the value $\rho_{\rm core}=\Lambda_c^4$ at $r\simeq r_{\rm core} $ and for $r_{\rm core} \ll r_s$ we have $$\label{eq:relation ec_free_bdm_params} \rho_{core} = \frac{\rho_0 r_s}{2r_{\rm core}},$$ where $r_s$ and $\rho_0$ are typical NFW halo parameters. The value of $r_{c}$ The parameters $r_{\rm core}$, $\rho_{\rm core}$ and $r_s$ had been estimated fitting galaxy rotation curves [@delaMacorra:2011df; @Mastache:2011cn] finding the lower limit of the BDM transition to be $\Lambda_c \gtrsim 0.1 $ eV, this resul also coincides with the lower energy transition constrains with Big Bang nucleosynthesis and extra degrees of freedom, that put lower bound of the order of $\lambda_c = 2.3$ eV [@Mastache:2013iha]. Results and Discussion {#sec:wdm} ====================== Bearing the high dependance of the free-streaming on $a_c$ and $v_c$, and the computation of the the cut-off scale from the matter power spectrum let us discuss some interesting results regarding BDM and WDM. Several constraints has been placed around the mass of the WDM based on different methods. Based on the abundance of redshift $z = 6$ galaxies in the Hubble Frontier Fields put constrains of $m_{\rm wdm} > 2.4$ keV [@Menci:2016eui]. Based on the galaxy luminosity function at $z \sim 6 - 8$ put constrains on $m_{\rm wdm} > 1.5$ keV [@Corasaniti:2016epp]. Lensing surveys such as CLASH provide $m_{\rm wdm} > 0.9$ keV lower bounds [@Pacucci:2013jfa]. The highest lower limit is given by the high redshift Ly-$\alpha$ forest data which put lower bounds of $m_{\rm wdm} > 3.3 keV$ [@Viel:2013apy]. In this section we compare BDM and WDM particles. The main characteristic for these dark matter models is the moment when they become non-relativistic, for WDM defined as $a_\textrm{nr}$, for BDM as $a_c$, moreover for BDM it is also important the initial velocity dispersion $v_c$ at this time. For a WDM we can find the relation between $a_{nr}$ and its mass, $m_{\rm wdm}$, knowing that WDM became non-relativistic when $p^2 = m^2$, therefore $v_c(a_{nr})=1/\sqrt{2}$. As the Universe expands the temperature redshifts as $T\propto 1/a$ and eventually the WDM particle becomes non-relativistic at $a_\textrm{nr}$ with an EoS $w_\textrm{wdm}(a)=v(a)^2/3$ with velocity $v(a)$ given by Eq.. The evolution of energy density of WDM particle of mass M for all the evolution of the Universe is, the same as BDM, given by Eq. but the velocity given by Eq., for $a_{nr} \ll a_0$, the WDM energy density evolve as matter with $\rho \propto a^{-3}$ valid for $a \gg a_{nr} $. In order to relate the mass of the WDM particle to the scale factor $a_{nr}$ let us take the non-relativist limit of the EoS $w\simeq 3T/M$ at $a_{nr}$, and use Eq. to approximate $w=3T/M=v^2/3$ to obtain the relationship $T=M v^2/9$ at the scale factor $a_{nr}$. The relativistic energy density is given by $\rho =(\pi^2 g_x/30) T^4$ valid $a\leq a_\textrm{nr}$ which becomes $\rho (a_{nr}) \simeq (\pi^2 g_x/30) (M v^2/9)^4$. We equate the energy density of these two region at $a_{nr}$ and obtain $\rho^{\rm wdm}_o(a_o/a_{nr})^3=(\pi^2 g_x/30) (M v(a_{nr})^2/9)^4$. We know that $v(a_{nr})=1/\sqrt{2}$ for WDM, and we assume $g_x=7/4$ for a neutrino type fermion, the scale factor where WDM becomes non-relativistic is then = 3.14 10\^[-7]{}()\^[1/3]{} ()\^[4/3]{} ()\^[1/3]{} . With the numerical code CLASS we obtained the EoS for a 3 keV WDM particle and look at the moment where it becomes non-relativistic, when $\omega = 1/6$, to find that the numerical value $a_{nr}^{\rm wdm} = 2.73 \times 10^{-7}$, is just a $13\%$ different with respect to the value obtained with Eq.. In Table.\[tab:results\] we campare the analytical result with the one obtained from CLASS. We can relate the time when two different WDM become non-relativistic, from Eq.(\[eq:anrao\]) and we find =( )\^[4/3]{}. [l\|ccccc]{} & $a_{nr} \, (a_c)$ & $k_{1/2}$ & $\lambda_{fs}$ & $k_{fs}$ & $M_{fs}$\ WDM CLASS & $2.73\times 10^{-7}$ & 3.46 & 0.70 & 9.00 & $5.93\times 10^9$\ WDM $m_{\rm wdm} = 3$ keV & $3.14\times 10^{-7}$ & 3.46 & 0.79 & 7.96 & $8.56\times 10^9$\ BDM $v_c = 1/\sqrt{2}$ & $2.73\times 10^{-7}$ & 4.36 & 0.71 & 8.86 & $6.21\times 10^9$\ BDM $v_c = 0$ & $2.09\times 10^{-6}$ & 3.47 & 0.70 & 9.00 & $5.93\times 10^9$\ BDM $v_c = 0$ & $2.73\times 10^{-7}$ & 27.52 & 0.09 & 69.10 & $1.31\times 10^7$\ Let us now compare $\lambda_{fs}$ in WDM and BDM models given in Eq. and Eq. both with the same $a_{eq}$ and using $t_{nr}=(a_{nr}^2/a_{eq}^2) t_{eq}, \;t_c=(a_c^2/a_{eq}^2) t_{eq}$, = . Taking the limiting case $v_c=0$ and the condition $ \lambda_{fs}^{wdm}= \lambda_{fs}^{bdm}$ gives a\_c=a\_[nr]{}\[0.69+ ( a\_[eq]{}/a\_[nr]{})\]. For example for a WDM with $M=3$ keV we use Eq. along with the value $a_{nr} = 2.73\times 10^{-7}$ obtained from CLASS to get =2.0910\^[-6]{}, \[eq:bdm\_equivalent\] this is equivalent to have a mass $M_{\rm wdm}$ of M\_[wdm]{}= 723.25 ( ) ()\^[-3/4]{} ()\^[-3/4]{} Notice that $a_c$ is a factor of 10 smaller than $a_{nr}$ in Eq.. The mass of a WDM particle becoming non-relativistic at $a_c$ is reduced by a factor of $4.61$. To conclude, a BDM particle that becomes non-relativistic at $a_c/a_o=2.09\times10^{-6}$, with $v_c=0$ has the same free streaming scale $\lambda_{fs} $ and suppression of halo mass $M_{fs}$ equally as a WDM particle with a mass $m_{wdm} = 3 \, keV$ and becoming non-relativistic much earlier, at $a_{nr}/a_o=3.14 \times 10^{-7} $. While at large scales $a\gg a_c$ ($a\gg a_{nr}$) for BDM (WDM), the structure formation is the same as CDM, at scales below the free-streaming scale is suppressed and modulated by its velocity dispersion $v_c$ ($v_{nr}$) at the moment when the particles become non relativistic, $a_c$ ($a_{nr}$). For example, a WDM with $m_{\rm wdm} = 3$ keV has a free-streaming scale $\lambda_{fs}=0.79$ Mpc/h with its respective Jeans mass $M_{fs}= 8.56\times10^9 M_\odot/h^3$ while a BDM with $a_c=2.09\times 10^{-6}$ and $v_c=0$ has the same free-streaming scale and contained mass, see Table \[tab:results \]. This shows that the scale of transition in BDM is 10 times larger than in WDM, i.e occurs at a later time in BDM than in WDM. Notice that a thermal WDM that becomes non-relativistic at $a_c=2.09\times 10^{-6}$ would have a mass $m=723$eV. In Figure\[fig:likelihood\] we show the likelihood for the Montecarlo run using Montepython. The shadow areas corresponds to $1\sigma$ and $2\sigma$ likelihoods. Smaller values of $a_c < 10^{-7}$ are within $1\sigma$ likelihood, and this is reasonable because smaller values of $a_c$ implies early transitions and bigger masses for the dark matter particle, just as CDM. This plot show the smaller bounds in the $a_c - v_c$ parameter space, nevertheless show an interesting connection with WDM. We can compute $a_{nr}$ for a given mass for the WDM particle using Eq., and Eq. gives a function of $a_c$ and $v_c$ for a given $a_{nr}$, therefore The different lines in Figure\[fig:likelihood\] represent the different values of $a_c$ and $v_c$ that preserve the free-streaming scale of a specific WDM particle. When BDM has $v_c = 1/\sqrt{2}$ it is when is most similar to WDM, and we can assume $a_c \sim a_{nr}$ to find that the boundary of the $1\sigma$ ($2\sigma$) likelihood for the $a_c - v_c$ parameter space corresponds to a WDM mass $m_{\rm wdm} = 2.3$ keV (1.4 keV). Our constrain agrees with previos work that had placed lower limits to $m_{\rm wdm}$ using different methods. Based on the abundance of redshift $z = 6$ galaxies in the Hubble Frontier Fields put constrains of $m_{\rm wdm} > 2.4$ keV [@Menci:2016eui]. Based on the galaxy luminosity function at $z \sim 6 - 8$ put constrains on $m_{\rm wdm} > 1.5$ keV [@Corasaniti:2016epp]. Lensing surveys such as CLASH provide $m_{\rm wdm} > 0.9$ keV lower bounds [@Pacucci:2013jfa]. The highest lower limit is given by the high redshift Ly-$\alpha$ forest data which put lower bounds of $m_{\rm wdm} > 3.3 keV$ [@Viel:2013apy]. Regarding the power spectrum and the evolution of the matter density perturbation a useful and standard quantity to compare different DM models is the value of $k_{1/2}$ corresponding to the mode when WDM power spectrum is suppressed by 50% compared to CDM. We obtain a $k_{1/2}=5.14$Mpc$^{-1}$ for a mass $m=3$keV for WDM and $a_c=2.09\times 10^{-6}$ for BDM with $v_c=0$. Conclusion {#sec:conclusion} ========== We have presented the BDM model which novelty introduce the velocity dispersion parameter, $v_c$ as one of the main characteristic in the particle model, along with the moment to the transition to non-relativistic behavior $a_c$, as an important characteristic to study in the nature of the dark matter. This velocity introduce different effects to the CMB power spectrum, linear matter power spectrum, dark matter halo density profile, halo mass function and growth rate of structure in the case of BDM. The effect of introducing a non-trivial initial velocity dispersion, $v_c$, at the moment of transition, $a_c$, prevent clustering inside the Jeans length. We perform the analysis by constraining our model using the 2018 Planck CMB likelihoods [@Aghanim:2018eyx] and included the BAO measurements, and the JLA SNe Ia catalog in the MonteCarlo run in order to provide a reasonable representation of the degeneracies. We find that the relation between $a_c$ and $v_c$ should preserve the free-streaming equivalent to a $m_{\rm wdm}> 2.3$ KeV WDM at $1\sigma$ likelihood. For instance, dark matter could have a late non-relativistic transition at $a_c = 2.09 \times 10^{-6}$ and preserve large structure formation if the dispersion velocity abruptly decrease from $v = c$ to $v_c = 0$, this result is in agreement with previous results [@Menci:2016eui; @Corasaniti:2016epp]. We also find the relation between WDM mass, $m_{\rm wdm},$ and the moment where the particle become non-relativistic $a_{nr}$ using a fundamental velocity evolution coming from the relativistic behaviour. i.e. a 3 keV WDM become non-relativistic at $3.14 \times 10^{-7}$, this differ only 15% in respect with the value obtained from solving the complete Boltzmann equations. This framework where we include the dispersion velocity of the dark matter particle may be incorporated in a broad number of observational cosmological probes, including forecasts for large scale structure measures, i.e. weak lensing [@Markovic:2010te], future galaxy clustering two-point function measures of the power spectrum [@vandenBosch:2003nk]. Future observation from large to small-scale clustering of dark and baryonic matter may be able to clear the nature of dark matter and its primordial origin. Perturbations {#appendix:perturbation} ============= ![[]{data-label="fig:pert_evol"}](pert_density_evol.png){width="\textwidth"} In this appendix we show the first order equations of the perturbations of the BDM. We follow [@Ma:1995ey] to compute the fluid approximation to the perturbed equations in k-space in the synchronous gauge for the BDM model. Before the transition, $a<a_c$, the perturbed equations are: $$\begin{aligned} \label{eq:BDM_rad} \dot{\delta} &=& -\frac{4}{3}\left( \theta + \frac{\dot {h}}{2} \right) \label{eq:delta_rad} \\ \dot{\theta} &=& \frac{1}{4}k^{2}\delta - k^2\sigma, \\ \dot { \sigma } &=& - 3 \frac{\sigma}{\tau} + \frac{1}{3} \left( 2\theta + \dot{h} \right) \, ,\end{aligned}$$ where $\sigma$ is the anisotropic stress perturbations. The dot represent the derivative respect to the conformal time $\tau \equiv \int d t/a(t)$, $H$ is the Hubble parameter, $H \equiv \dot{a}/a$, and $\theta$ is the velocity of the perturbation. Until this point the behavior of the BDM particles are similar as the ultra-relativistic massless neutrinos [@Ma:1995ey]. After the transition, $a>a_c$, the BDM particles goes through the transition. Using Eqs.(\[eq:eos\_bdm\]) and (\[eq:rho\_bdm\]) we are able to compute the perturbation equations: $$\begin{aligned} \dot{\delta_c} &= -\left(1+\omega\right)\left(\theta + \frac{\dot{h}}{2} \right) - \frac{4\omega(1-\omega)}{1+\omega} H\delta_c \label{delta_mat} \\[1ex] \dot{\theta} &= -H\theta \frac{ (1 - \omega)(1-3\omega) }{ 1+\omega } + k^2\delta \, \frac{\omega(5-3\omega) }{ 3(1+\omega)^2} - k^2\sigma \\[1ex] \dot { \sigma } &= - 3 \left( \frac { 1 } { \tau } +\frac { 2 H} { 3 } \left[ \frac{1-3\omega}{1+3\omega} \right] \right) \sigma + \frac { 4 } { 9 } \left( 2 \theta + \dot { h } \right) \frac{\omega (5 - 3\omega)}{ \left( 1+\omega \right)^2} \label{eq:sigma_matter}\end{aligned}$$ In Eq.(\[eq:sigma\_matter\]) we have taken the anisotropic stress approximation for massive neutrinos [@Hu:1998kj] and ignore the $\dot{\eta}$ term that slightly improve the computation of the matter power spectrum [@Lesgourgues:2011rh]. We have also used the relation ${\dot \omega} = -2 H \omega$. The perturbation evolution for different components of the Universe is shown is Figure \[fig:pert\_evol\] as a function of the scale factor. When $v_c$ takes values $v_c < 1/\sqrt{2}$ the EoS is a non-continuos function, as well as $\delta_{\rm bdm}$, $\theta_{\rm bdm}$ and $\sigma_{\rm bdm}$, therefore has no good numerical solution to the set of equation that describe the perturbation evolution. To overcome this problem we implement a step function in order to smooth the transition and compute a solution for the perturbation. From Figure \[fig:pert\_evol\] we notice that BDM perturbation behaves always as radiation at early times, the evolution is similar to the massless neutrinos, after the transition start behaving as CDM and only after matter-radiation equality BDM, CDM and WDM (with a mass of 3 keV) has the same behavior. This work is partially supported by CONACYT. We acknowledge support from PASAP-DGAPA, UNAM and Project IN103518 PAPIIT-UNAM.
--- abstract: 'We present a simple classification of the different liquid and solid phases of quantum Hall systems in the regime where the Coulomb interaction between electrons is significant, [i.e.]{} away from integral filling factors. This classification, and a criterion for the validity of the mean-field approximation in the charge-density-wave phase, is based on scaling arguments concerning the effective interaction potential of electrons restricted to an arbitrary Landau level. Finite-temperature effects are investigated within the same formalism, and a good agreement with recent experiments is obtained.' author: - 'M. O. Goerbig and C. Morais Smith' title: Scaling approach to the phase diagram of quantum Hall systems --- Two-dimensional electron systems (2DES) in a perpendicular magnetic field exhibit a rich variety of phases, ranging from incompressible quantum liquids, which are responsible for the integral and fractional quantum Hall effects (IQHE and FQHE), to electron-solid phases such as charge density waves (CDWs) and the Wigner crystal (WC). The IQHE is found when the electron density $n_{el}$ is an integral multiple of the density of states per Landau level (LL) $n_B=1/2\pi l_B^2$, where $l_B=\sqrt{\hbar/eB}$ is the magnetic length [@KvK], and can be described within a single-particle picture. On the other hand, when the filling factor $\nu=n_{el}/n_B$ is non-integral, the highest filled LL has only a partial filling $\bar{\nu}=\nu-n$, and it becomes essential to include the Coulomb interaction [@note1]. It lifts the LL degeneracy and leads to a rich phase diagram. At extremely low electron densities, an insulating phase has been observed whose properties are attributed to WC formation [@jiang]. At $\nu=p/q$, with $p,q$ integral and $q$ odd, the FQHE is observed in the two lowest LLs [@TSG]. The corresponding ground state is an incompressible liquid, analogous to the IQHE if described in terms of composite fermions (CFs) [@jain]. At $\bar{\nu}= 1/2$ in higher LLs, huge anisotropies have been detected in the longitudinal magnetoresistance, indicating the formation of a unidirectional CDW [@exp]. From the theoretical point of view, Hartree-Fock calculations predict correctly a CDW formation around $\bar{\nu}=1/2$ in higher LLs [@FKS; @MC], but have failed to describe the FQHE regime. In this letter we propose a new scaling approach, which classifies the liquid and solid phases observed in quantum Hall systems in terms of length scales, and remains valid in all LLs. In addition, we clarify the reasons for breakdown of the Hartree-Fock approximation in the lowest LLs and establish a criterion for the appearance of recently observed FQHE features in the WC regime at finite temperatures [@pan]. Because in the high-magnetic-field limit the Coulomb interaction between the electrons constitutes a smaller energy scale than the gap between LLs, inter-LL excitations are high-energy degrees of freedom. At integral filling, they are the only excitations, but may be neglected when considering a system at $\nu\neq n$ because in this case low-energy excitations become possible within the same LL [@AG]. The kinetic energy of the electrons may therefore be set to zero, and the Coulomb interaction remains as the only energy scale in the problem if the electrostatic potential due to underlying impurities is small. One obtains a system of strongly correlated electrons described by the Hamiltonian $$\label{equ001} \hat{H}_n=\frac{1}{2}\sum_{\bf q}v(q)\left[F_n(q)\right]^2\bar{\rho}(-{\bf q})\bar{\rho}({\bf q}),$$ where $v(q)=2\pi e^2/\epsilon q$ is the two-dimensional Fourier transform of the Coulomb interaction. In this model, one considers only interactions between spinless electrons within the $n$th LL described by the density operators $\langle \rho({\bf q})\rangle_n=F_n(q) \bar{\rho}({\bf q})$, where $\rho({\bf q})$ is the usual electron density in reciprocal space. The factors $F_n(q)=L_n(q^2/2)\exp(-q^2/4)$, with the Laguerre polynomials $L_n(x)$, arise from the wave functions of electrons in the $n$th LL and may be absorbed into an effective interaction potential $v_n(q)=v(q)[F_n(q)]^2$. The quantum nature of the problem is contained in unusual commutation relations for the electron density operators [@GMP], $$\label{equ003} [\bar{\rho}({\bf q}),\bar{\rho}({\bf k})]=2i\sin\left(\frac{({\bf q}\times{\bf k})_zl_B^2}{2}\right)\bar{\rho}({\bf q}+{\bf k}).$$ The Hamiltonian (\[equ001\]) together with the commutation relations (\[equ003\]) defines the full model, which was used recently as a starting point for the description of the FQHE in the lowest LL [@MS] and for the formation of CDWs in higher LLs [@FKS; @MC]. Note the formal equivalence between electrons in the lowest LL at $\nu=\nu_0$ and electrons in a higher LL at $\bar{\nu}=\nu_0$ within this model. Deeper insight into the stucture of the model is obtained by transforming the effective interaction potential back to real space. In appropriate units, one may derive a universal scaling function $\tilde{v}(r)$, $$\label{equ004} v_n(r)=\sum_{\bf q}e^{-i{\bf q}\cdot{\bf r}}v_n(q)\approx\frac{\tilde{v}\left(r/R_C\right)}{\sqrt{2n+1}}\,,$$ where $R_C=l_B\sqrt{2n+1}$ is the cyclotron radius. This scaling form becomes exact in the limit $n\rightarrow\infty$ because then $F_n(q)\rightarrow J_0(q\sqrt{2n+1})$. However, it is valid also at low values of $n$ as can be seen from fig.\[fig02\], where the rescaled results of $v_n(r)$ are shown for the LLs $n=1,...,5$. The universal function exhibits a plateau of width $2R_C$ superimposed on the bare $1/r$ Coulomb potential, which is retrieved at large distances. +5cm Although the bare Coulomb potential possesses no characteristic length scale, a component whose range is characterised by $R_C$ is introduced in the effective interaction potential. This permits a classification of the different phases according to the ratio of the average electronic separation $d\sim l_B/\sqrt{\bar{\nu}}$ in the $n$th LL and the range $2 R_C$ of the effective interaction. [Quasi-classical limit (WC): $d\gg 2R_C$.]{} In the limit of low density, the average distance between electrons interacting via the $1/r$ Coulomb potential is much larger than the spatial extent $R_C$ of their wave functions. Quantum corrections to the classical result are of order $\mathcal{O}(R_C/d)$ and may thus be neglected. Classically, the electrons are arranged in a triangular WC in order to minimise the repulsive Coulomb interaction. The transition line which separates the WC phase from other phases is obtained by comparing $d$ and $R_C$ as functions of $\bar{\nu}$ and $n$ (gray line in fig.\[fig01\]). It is given by $\bar{\nu}_n^{WC}=\nu_0^{WC}/(2n+1)$, where $\nu_0^{WC}$ is the critical filling factor below which the WC is found in the lowest LL. Theoretical calculations predict $\nu_0^{WC}\approx 1/6.5$ for clean samples [@lam], while experimentally the onset of WC behaviour is observed around $\nu=1/5$ [@jiang]. Electrical transport in the WC phase arises either by a collective sliding mode or by the propagation of crystal dislocations. Sliding is suppressed by pinning of the WC due to residual impurities in the sample, and the number of dislocations is reduced by lowering the temperature. The experimental evidence for a WC phase arises from transport measurements [@jiang], which indicate an insulating phase. In principle, this insulating behaviour could be attributed also to the localization of electrons by impurities [@AndersonLoc]. However, because the samples used for the measurements are extremely clean, this is unlikely to be the case. [Mean-field limit (CDWs): $d\ll 2R_C$.]{} Within the effective-potential framework, an arbitrarily chosen particle interacts strongly with a number of neighbours which can be estimated as $N_{n.n.}\approx \pi(2R_C)^2\bar{n}_{el}= 2(2n+1)\bar{\nu}$, where $\bar{n}_{el}$ is the density of electrons in the $n$th LL. A mean-field approximation, such as Hartree-Fock, is valid for large $N_{n.n.}$, where each particle may be considered to interact only with an averaged background, without being influenced by the individual motions of its neighbours. This limit cannot be obtained in the lowest LL, where the cyclotron radius coincides with the magnetic length $l_B$. The average electronic separation would have to be shorter than this length, which constitutes the smallest possible spacing because each electronic state occupies a minimal surface $\sigma=2\pi l_B^2$. We note in addition that in the lowest LL the effective interaction potential does not exhibit a plateau as for $n\geq 1$. The dark gray line in fig.\[fig01\] shows the relation $\bar{\nu}_n^{CDW}=N_c/2(2n+1)$. Comparison with experiment yields $N_{n.n.}=N_c\sim 5$ for the limiting value above which the mean-field approximation is justified [@exp]. The validity of the mean-field approximation in the regime $d\ll 2R_C$ also becomes apparent in Fourier space, where the wave vectors are renormalised in the same manner as the distances in eq.(\[equ004\]). At large $n$, only the small-wave-vector limit remains important, and the commutation relation (\[equ003\]) for the density operators becomes $$\begin{aligned} \nonumber \left[\bar{\rho}\left(\frac{\bf q}{\sqrt{2n+1}}\right)\right.&,&\left.\bar{\rho}\left(\frac{\bf k}{\sqrt{2n+1}}\right)\right]\\ \nonumber \approx &i& \frac{({\bf q}\times{\bf k})_zl_B^2}{2n+1}\ \bar{\rho}\left(\frac{{\bf q}+{\bf k}}{\sqrt{2n+1}}\right)=\mathcal{O}\left(\frac{1}{n}\right),\end{aligned}$$ after rescaling and expansion of the sine function. The complicated algebraic structure of the density operators, and thus the quantum mechanical nature of the problem, become less important in higher LLs (larger $n$). This quasi-classical limit is different from the WC regime, in which the overlap of different electronic wave functions may be neglected. Here their overlap is sufficiently strong that the exchange interaction, which is included at the mean-field level, is essential. The mean-field solution of the Hamiltonian (\[equ001\]) reveals that the ground state in this limit is a CDW with a characteristic period on the order of the cyclotron radius $R_C$ [@FKS; @MC]. This clustering of electrons, in spite of their repulsive interaction, may be understood qualitatively from the form of the effective interaction potential in fig.\[fig02\]: if two electrons approach more than their average separation $d$, only a small additional energy cost is incurred because of the plateau in the region $r<2R_C$. However, a large energy on the order of the height of the plateau may be gained if the clustered electrons thus reduce the number of other electrons with which they interact strongly. The boundary above which the mean-field approximation becomes valid need not necessarily coincide with the CDW-FQHE phase transition. A detailed calculation of the ground-state energy would be needed to determine the exact transition line [@FKS; @MC; @fogler], but the present scaling investigations serve as an upper limit for this boundary. +6.0cm In the region $\bar{\nu}< 1/2$, charged clusters (or “bubbles”) of several electrons form a super-WC to minimise their Coulomb repulsion (triangular CDW). Exactly at $\bar{\nu}=1/2$, however, these bubbles percolate to form lines and thus give rise to a “stripe” pattern (unidirectional CDW). This breaking of rotational symmetry arises because of a competing symmetry: the particle-hole symmetry becomes exact at half filling. A phase of bubbles would violate this symmetry, and a very small residual anisotropy in the underlying crystal suffices to fix the direction of the stripes [@scheidel]. However, there are other inhomogeneous charge configurations, which would also account for these symmetries, and there is theoretical evidence that a stripe pattern becomes unstable to the formation of an anisotropic WC at very low temperatures, breaking the particle-hole symmetry [@MF]. The experimental evidence for stripe phases in quantum Hall systems consists of very large anisotropies in the magnetoresistance with respect to two orthogonal directions [@exp]. This suggests an interpretation in terms of charge stripes, where easy electron transport along the stripe edges leads to a small resistance, but transport across the stripes involves relatively rare tunneling processes, thus explaining the large resistance in the orthogonal direction [@MF; @OHS]. [Quantum limit (FQHE): $d\sim 2R_C$.]{} In the regime of the phase diagram intermediate between the WC and CDW phases, the FQHE is observed in the lowest two LLs [@TSG]. Recent theoretical and experimental studies [@MS; @goldman] support the description of the FQHE in terms of an incompressible liquid of CFs, each of which consists of a bound state of an electron and a vortex-like collective excitation which carries a charge of opposite sign [@jain]. The scaling arguments presented above show that in the quantum limit only one length scale is present in the problem, and thus no perturbative approaches are applicable. [Effect of impurities and finite temperature.]{} The arguments presented above suggest that the observation of the FQHE is possible in higher LLs, e.g. at $\bar{\nu}=1/5$ for $n=2$. However, this phase has not yet been observed, and some experiments even suggest a CDW ground state in this regime [@lewis]. Failure to observe the FQHE in higher LLs is likely to be due to the residual impurities in the samples, which favour crystalline structures such as the WC and CDWs [@jiang]. Deformation of the inhomogeneous charge structure makes these phases better adapted to follow an underlying electrostatic potential than is an incompressible, homogeneous liquid. The two transition lines therefore move towards the quantum-liquid phase, as shown by the arrows and broken lines in fig.\[fig01\]. The region where the FQHE is observed can become extremely narrow in higher LLs, and may even vanish, leading to a direct crossover between the triangular CDW and WC phases with no quantum melting in the intermediate regime $d\sim 2R_C$. However, samples of higher mobility and even lower impurity concentrations are expected to reveal FQHE states in the LL $n=2$ in the future [@morf]. So far we have discussed the different phases determined by the two length scales $d$ and $R_C$ only at $T=0$. Finite temperatures will introduce an additional thermal length scale defined by comparing the thermal energy $k_BT$ and the Coulomb interaction between electrons. One thus obtains $l_T=e^2/\epsilon k_BT$, which corresponds to the de Broglie wavelength of the free electron gas. If $l_T\gg R_C$, [i.e.]{} at low temperatures, thermal fluctuations may be neglected compared to quantum fluctuations of the correlated electron liquid. At $l_T\sim R_C$ thermal fluctuations destroy the quantum correlations and the FQHE disappears. In the mean-field limit, local crystalline structure of the CDW vanishes when thermal fluctuations become important on the length scale of the CDW periodicity $l_T\sim R_C$. This leads to an estimate of the CDW melting temperature $T_{CDW}(n)=Ce^2/\epsilon k_Bl_B\sqrt{2n+1}$, with $C$ a dimensionless constant, in agreement with previous work [@FKS]. Because CDW states in high LLs are observed at relatively low magnetic fields (decreasing LL separation), inter-LL excitations have to be included, giving rise to a screening of the Coulomb interaction. The dielectric constant $\epsilon$ may then be replaced by $\epsilon(n)\sim 2n+1$ [@FKS; @AG]. We stress that these scaling arguments provide an estimate of the melting temperature for [local]{} CDW order. The anisotropy observed at half-filling in higher LLs [@exp] vanishes at lower temperatures, indicating an isotropic distribution of local stripes, as proposed in a liquid-crystal picture [@fradkin1]. Finite-temperature effects are most complex in the WC phase. Minima in the longitudinal magnetoresistance, similar to the ones arising in the FQHE regime, are experimentally observed above a temperature $T_1$ at lowest-LL filling fractions, where the WC phase is expected at $T=0$ [@pan]. These minima vanish above a second temperature $T_2>T_1$. An estimate for $T_1$ can be obtained from the Lindemann criterion [@lindemann]: the WC melts when the average displacement of an electron due to thermal fluctuations is a substantial fraction of the lattice constant $\langle \Delta r^2\rangle=c_L^2 d^2$ with $c_L\sim 0.1$. Equating the potential energy for a small displacement of an electron in a WC, $e^2\langle \Delta r^2\rangle /\epsilon d^3$, to the thermal energy yields a melting temperature $T_1\approx c_L^2e^2/\epsilon k_B d$, which is independent of the magnetic field, in good agreement with recent experiments [@pan]. This temperature corresponds to a thermal length $l_T\sim d/c_L^2\gg l_B$. The liquid phase therefore exhibits quantum coherence on a length scale $l_T$, and may be described locally by the Laughlin wave function for temperatures such that $l_T> l_B$, thus displaying features of the FQHE. The scaling estimate $l_T\sim l_B$ for the definition of a temperature $T_2$ at which this coherence is lost turns out, however, to be rather crude because it neglects a renormalization of the magnetic length due to CF formation in the quantum limit. In the CF picture, the minima disappear when the temperature reaches the activation gap [@MS]. This leads to the relation $T_2\approx T_C/(2ps\pm 1)$, where $T_C$ is a constant, and the integers $p,s$ are related to the filling factor $\bar{\nu}=p/(2ps\pm 1)$ for the FQHE states. Comparison with experiment [@pan] suggests a value $T_C\sim 2$K around $\nu=1/6$. [Conclusions.]{} We have discussed the different solid and liquid phases of quantum Hall systems using straightforward scaling arguments. Although the bare Coulomb potential is scale-free, the effective potential of electrons restricted to the $n$th LL has a range of strong interaction characterised by the cyclotron radius. The ratio between this range and the average separation $d$ of the electrons classifies the different phases at $T=0$. At finite temperature, a further length scale $l_T$ enters. For $d\gg 2R_C$ a WC is formed, which melts at a temperature $T_1$ into a quantum liquid showing features of FQHE states. Our estimates for the melting temperature $T_1$ show that it does not vary with $B$, in agreement with recent experiments [@pan]. The quantum coherence of the electrons is not destroyed until a higher temperature $T_2$. In the opposite limit $d\ll 2R_C$ the mean-field approximation is justified and predicts a CDW ground state. We present a criterion for the validity of this approximation, which excludes CDW formation in the lowest LLs, where the required condition cannot be satisfied because $R_C$ coincides with the smallest length $l_B$ in the system. In an intermediate regime $d\sim 2R_C$ quantum melting of the crystalline structures leads to a liquid phase, which is incompressible at certain filling factors where the FQHE is observed. The presence of impurities reduces the FQHE regime in the phase diagram because solid phases are better adapted to follow an underlying impurity potential. We acknowledge fruitful discussions with D. Baeriswyl, L. Benfatto, K. Borejsza, P. Lederer, R. Morf, B. Normand, V. Pasquier, and J. Wakeling. This work was supported by the Swiss National Foundation for Scientific Research under grant No. 620-62868.00. [99]{} . If one takes into account the Zeeman splitting of each LL, the partial filling factors of the lower and upper spin branches of the $n$th LL are, respectively, $\bar{\nu}=\nu-2n$ and $\bar{\nu}=\nu-(2n+1)$. ; ; . ; . ; ; [Composite Fermions]{}, edited by HEINONEN O. (World Scientific) 1998. ; . ; . . . Inter-LL excitations may be included in the random-phase approximation. This leads to a screened Coulomb interaction in a certain wave vector range but has only a minor influence on the physical properties of the system; see: . . cond-mat/0205326 Preprint, 2002; . . . . . . . . . . ; . .
--- abstract: \| In 2006, Dafermos and Holzegel [@DafHol; @DafermosTalk] formulated the so-called AdS instability conjecture, stating that there exist arbitrarily small perturbations to AdS initial data which, under evolution by the Einstein vacuum equations for $\Lambda<0$ with reflecting boundary conditions on conformal infinity $\mathcal{I}$, lead to the formation of black holes. The numerical study of this conjecture in the simpler setting of the spherically symmetric Einstein–scalar field system was initiated by Bizon and Rostworowski [@bizon2011weakly], followed by a vast number of numerical and heuristic works by several authors. In this paper, we provide the first rigorous proof of the AdS instability conjecture in the simplest possible setting, namely for the spherically symmetric Einstein–massless Vlasov system, in the case when the Vlasov field is moreover supported only on radial geodesics. This system is equivalent to the Einstein–null dust system, allowing for both ingoing and outgoing dust. In order to overcome the break down of this system occuring once the null dust reaches the centre $r=0$, we place an inner mirror at $r=r_{0}>0$ and study the evolution of this system on the exterior domain $\{r\ge r_{0}\}$. The structure of the maximal development and the Cauchy stability properties of general initial data in this setting are studied in our companion paper [@MoschidisMaximalDevelopment]. The statement of the main theorem is as follows: We construct a family of mirror radii $r_{0\text{\textgreek{e}}}>0$ and initial data $\mathcal{S}_{\text{\textgreek{e}}}$, $\text{\textgreek{e}}\in(0,1]$, converging, as $\text{\textgreek{e}}\rightarrow0$, to the AdS initial data $\mathcal{S}_{0}$ in a suitable norm, such that, for any $\text{\textgreek{e}}\in(0,1]$, the maximal development $(\mathcal{M}_{\text{\textgreek{e}}},g_{\text{\textgreek{e}}})$ of $\mathcal{S}_{\text{\textgreek{e}}}$ contains a black hole region. Our proof is based on purely physical space arguments and involves the arrangement of the null dust into a large number of beams which are successively reflected off $\{r=r_{0\text{\textgreek{e}}}\}$ and $\mathcal{I}$, in a configuration that forces the energy of a certain beam to increase after each successive pair of reflections. As $\text{\textgreek{e}}\rightarrow0$, the number of reflections before a black hole is formed necessarily goes to $+\infty$. We expect that this instability mechanism can be applied to the case of more general matter fields. author: - Georgios Moschidis bibliography: - 'DatabaseExample.bib' title: \| A proof of the instability of AdS\ for the Einstein–null dust system with an inner mirror --- Princeton University, Department of Mathematics, Fine Hall, Washington Road, Princeton, NJ 08544, United States, gm6@math.princeton.edu \[sec:Introduction\]Introduction ================================ Anti-de Sitter spacetime $(\mathcal{M}_{AdS}^{n+1},g_{AdS})$, $n\ge3$, is the simplest solution of the Einstein vacuum equations $$Ric_{\text{\textgreek{m}\textgreek{n}}}-\frac{1}{2}Rg_{\text{\textgreek{m}\textgreek{n}}}+\Lambda g_{\text{\textgreek{m}\textgreek{n}}}=0\label{eq:VacuumEinsteinEquations}$$ with a negative cosmological constant $\Lambda$. In the standard polar coordinate chart on $\mathcal{M}_{AdS}$, the AdS metric takes the form $$g_{AdS}=-\big(1-\frac{2}{n(n-1)}\Lambda r^{2}\big)dt^{2}+\big(1-\frac{2}{n(n-1)}\Lambda r^{2}\big)^{-1}dr^{2}+r^{2}g_{\mathbb{S}^{n-1}},\label{eq:AdSMetricPolarCoordinates}$$ where $g_{\mathbb{S}^{n-1}}$ is the round metric on the $n-1$ dimensional sphere (see [@HawkingEllis1973]). Despite being geodesically complete, $(\mathcal{M}_{AdS},g_{AdS})$ fails to be globally hyperbolic. In particular, it can be conformally identified with the interior of $(\mathbb{R}\times\mathbb{S}_{+}^{n},g_{E})$, where $\mathbb{S}_{+}^{n}$ is the closed upper hemisphere of $\mathbb{S}^{n}$ and $g_{E}$ is the metric $$g_{E}=-d\bar{t}^{2}+g_{\mathbb{S}^{n}}.\label{eq:EinsteinMetric}$$ Through this identification, the timelike boundary $$\mathcal{I}^{n}=\mathbb{R}\times\partial\mathbb{S}_{+}^{n}\simeq\mathbb{R}\times\mathbb{S}^{n-1}\label{eq:ConformalBoundaryAtInfinity}$$ of $(\mathbb{R}\times\mathbb{S}_{+}^{n},g_{E})$ is naturally attached to $(\mathcal{M}_{AdS},g_{AdS})$ as a “conformal boundary at infinity” (see [@HawkingEllis1973]). In 1998, Maldacena, Gubser–Klebanov–Polyakov and Witten [@Maldacena; @gubser1998gauge; @witten1998anti] proposed the AdS/CFT conjecture, suggesting a correspondence between certain conformal field theories defined on $\mathcal{I}^{n}$ (in the strongly coupled regime) and supergravity on spacetimes asymptotically of the form $(\mathcal{M}_{AdS}^{n+1}\times S^{k},g_{AdS}+g_{S^{k}})$, where $(S^{k},g_{S^{k}})$ is a suitable compact Riemannian manifold of dimension $k$. Following the introduction of this conjecture, asymptotically AdS spacetimes (i.e. spacetimes $(\mathcal{M},g)$ with an asymptotic region with geometry resembling that of $(\mathcal{M}_{AdS},g_{AdS})$ in the vicinity of $\mathcal{I}$) became a subject of intense study in the high energy physics literature (see e.g. [@AGMOO2000; @Hartnoll2009; @AmmonErdmenger] and references therein). The correct setting for the study of the dynamics of asymptotically AdS solutions $(\mathcal{M},g)$ to (\[eq:VacuumEinsteinEquations\]) is that of an initial value problem with appropriate boundary conditions prescribed asymptotically on $\mathcal{I}$. The issue of the right boundary conditions on $\mathcal{I}$ leading to well posedness for the resulting initial-boundary value problem for (\[eq:VacuumEinsteinEquations\]) was first addressed by Friedrich in [@Friedrich1995]. Well posedness for more general boundary conditions and matter fields in the spherically symmetric case was obtained in [@HolzSmul2012; @HolzWarn2015] (see also [@HolzegelLukSmuleviciWarnick; @Friedrich2014]). In general, most physically interesting boundary conditions on $\mathcal{I}$ leading to a well posed initial-boundary value problem can be classified as either reflecting (for which an appropriate “energy flux” for $g$ through $\mathcal{I}$ vanishes) or dissipative (allowing for a non-vanishing outgoing “energy flux” for $g$ through $\mathcal{I}$), with substantially different global dynamics associated to each case; see the discussion in [@HolzegelLukSmuleviciWarnick]. In 2006, Dafermos and Holzegel [@DafHol; @DafermosTalk] suggested the following conjecture: AdS instability conjecture. There exist arbitrarily small perturbations to the initial data of $(\mathcal{M}_{AdS},g_{AdS})$ for the vacuum Einstein equations (\[eq:VacuumEinsteinEquations\]) with a reflecting boundary condition on $\mathcal{I}$ which lead to the development of trapped surfaces and, thus, black hole regions. In particular, $(\mathcal{M}_{AdS},g_{AdS})$ is non-linearly unstable. This conjecture was motivated in [@DafHol] by the study of asymptotically AdS solutions to (\[eq:VacuumEinsteinEquations\]) with biaxial Bianchi IX symmetry in $4+1$ dimensions, a symmetry class in which the vacuum Einstein equations (\[eq:VacuumEinsteinEquations\]) reduce to a $1+1$ hyperbolic system with non-trivial dynamics. This model was introduced in [@BizonChmajSchmidt2005]. In this setting, it was observed in [@DafHol] that perturbations of the initial data of $(\mathcal{M}_{AdS},g_{AdS})$ (which, if not trivial, necessarily have strictly positive ADM mass $M_{ADM}$, in view of [@GibbonsEtAl1983]) can not settle down to a horizonless static spacetime, since $M_{ADM}$ is conserved along $\mathcal{I}$ under reflecting boundary conditions and no static asymptotically AdS solution of (\[eq:VacuumEinsteinEquations\]) with $M_{ADM}>0$ exists (according to [@BoucherGibbonsHorowitz]). This picture was supported by results of Anderson [@AndersonAdS]. The following remarks should be made regarding the statement of the AdS instability conjecture: - [ The perturbations referred to in the conjecture are assumed to be small with respect to a norm for which (\[eq:VacuumEinsteinEquations\]) is well-posed and $(\mathcal{M}_{AdS},g_{AdS})$ is Cauchy stable as a solution to (\[eq:VacuumEinsteinEquations\]) (otherwise, the conjecture is trivial).[^1] For such perturbations, Cauchy stability implies that the “time” elapsed before the formation of a trapped surface tends to $+\infty$ as the size of the initial perturbation shrinks to $0$.]{} - [ The AdS instability conjecture stands in contrast to the non-linear stability of Minkowski space $(\mathbb{R}^{3+1},\text{\textgreek{h}})$, in the case $\Lambda=0$ (see Christodoulou–Klainerman [@Christodoulou1993]), or de Sitter space $(\mathcal{M}_{dS},g_{dS})$, in the case $\Lambda>0$ (see Friedrich [@Friedrich1986]). The proof of the non-linear stability of $(\mathbb{R}^{3+1},\text{\textgreek{h}})$ and $(\mathcal{M}_{dS},g_{dS})$ is based on a stability mechanism related to the fact that linear fields on those spacetimes satisfy sufficiently strong decay rates. The decay rates are, however, borderline in the case $\Lambda=0$, and thus the stability of $(\mathbb{R}^{3+1},\text{\textgreek{h}})$ is a deep fact depending on the precise non-linear structure of the system (\[eq:VacuumEinsteinEquations\]), whereas, in the case $\Lambda>0$, the decay is exponential and stability can be inferred relatively easily. In contrast, on $(\mathcal{M}_{AdS},g_{AdS})$, it can be shown that linear fields satisfying a reflecting boundary condition on $\mathcal{I}$ remain bounded, but do not decay in time. It is precisely the lack of a sufficently fast decay rate at the linear level which is associated to the possibility of non-linear instability.]{} - [ The prescription of a reflecting boundary condition on $\mathcal{I}$ is essential for the conjecture: For maximally dissipative boundary conditions, it is expected that $(\mathcal{M}_{AdS},g_{AdS})$ is non-linearly stable, in view of the quantitative decay rates obtained for the linearised vacuum Einstein equations (and other linear fields) around $(\mathcal{M}_{AdS},g_{AdS})$ by Holzegel–Luk–Smulevici–Warnick in [@HolzegelLukSmuleviciWarnick]. ]{} - [ In the biaxial Bianchi IX symmetry class, all perturbations of $(\mathcal{M}_{AdS},g_{AdS})$ leading to the formation of a trapped surface can be shown to possess a complete conformal infinity $\mathcal{I}$ and are expected to settle down to a member of the Schwarzschild–AdS family (see [@DafHol; @DafHolStability2006]). However, in the absence of any symmetry, the picture regarding the end state of the evolution of general vacuum perturbations of $(\mathcal{M}_{AdS},g_{AdS})$ is complicated; see the discussion in the next section.]{} Starting from the pioneering work [@bizon2011weakly] of Bizon and Rostworoski in 2011, a plethora of numerical and heuristic results have been obtained in the direction of establishing the AdS instability conjecture, mainly in the context of the spherically symmetric Einstein–scalar field system. See the discussion in Section \[sub:Numerics\]. In this paper, we will prove the AdS instability conjecture in the simplest possible setting, namely for the Einstein–massless Vlasov system in spherical symmetry, further reduced to the case when the Vlasov field $f$ is supported only on radial geodesics. We will call this system the spherically symmetric Einstein–radial massless Vlasov system. In fact, this is a singular reduction; the resulting system is equivalent to the spherically symmetric Einstein–null dust system, allowing for both ingoing and outgoing dust. This system has been studied in the $\Lambda=0$ case by Poisson and Israel [@PoissonIsrael1990]. A serious problem with the spherically symmetric Einstein–null dust system is that it suffers from a severe break down when the null dust reaches the centre $r=0$. In particular, in any reasonable initial data topology, the spherically symmetric Einstein–null dust system is well posed and $(\mathcal{M}_{AdS},g_{AdS})$ is a Cauchy stable solution of it. One way to restore the well posedness of this system (a necessary step for the study of the AdS instability conjecture in this setting) is to place an inner mirror at some radius sphere $\{r=r_{0}\}$ with $r_{0}>0$ and study the evolution of the system in the exterior region $\{r\ge r_{0}\}$. However, fixing the mirror radius $r_{0}$ results in a trivial global stability statement for $(\mathcal{M}_{AdS},g_{AdS})$, as initial data perturbations with total ADM mass $\tilde{m}_{ADM}<\frac{1}{2}r_{0}$ cannot form a black hole. Thus, it is necessary to allow the radius $r_{0}$ to shrink to $0$ as the total ADM mass of the initial data shrinks to $0$, in order to address the AdS instability conjecture in this setting. See the discussion in Section \[sub:NeedOfAMirror\]. A non-technical statement of our result is the following: [1]{}\[rough version\]\[Theorem\_Intro\_Rough\] The AdS spacetime $(\mathcal{M}_{AdS}^{3+1},g_{AdS})$ is non-linearly unstable under evolution by the spherically symmetric Einstein–radial massless Vlasov system with a reflecting boundary condition on $\mathcal{I}$ and an inner mirror, in the following sense: There exists a one parameter family of spherically symmetric initial data $\mathcal{S}_{\text{\textgreek{e}}}$, $\text{\textgreek{e}}\in(0,1]$ and a family of inner mirror radii $r=r_{0\text{\textgreek{e}}}$ (with $r_{0\text{\textgreek{e}}}\xrightarrow{\text{\textgreek{e}}\rightarrow0}0$) satisfying the following properties: 1. As $\text{\textgreek{e}}\rightarrow0$, $\|\|\mathcal{S}_{\text{\textgreek{e}}}\|\|_{\mathcal{CS}}\rightarrow0$, i.e. $\mathcal{S}_{\text{\textgreek{e}}}$ converge to the initial data $\mathcal{S}_{0}$ of $(\mathcal{M}_{AdS},g_{AdS})$. 2. For any $\text{\textgreek{e}}>0$, the maximal future development $(\mathcal{M}_{\text{\textgreek{e}}},g_{\text{\textgreek{e}}})$ of $\mathcal{S}_{\text{\textgreek{e}}}$ contains a trapped surface and, thus, a black hole region. Moreover, $(\mathcal{M}_{\text{\textgreek{e}}},g_{\text{\textgreek{e}}})$ possesses a complete conformal infinity $\mathcal{I}$. The norm $\|\|\cdot\|\|_{\mathcal{CS}}$ in 1 measures the concentation of the energy of $\mathcal{S}_{\text{\textgreek{e}}}$ in annuli of width $\sim r_{0\text{\textgreek{e}}}$ and has the property that the radial Einstein–massless Vlasov system is well-posed and $(\mathcal{M}_{AdS},g_{AdS})$ is Cauchy stable with respect to $\|\|\cdot\|\|_{\mathcal{CS}}$ . For progressively more detailed statements of Theorem \[Theorem\_Intro\_Rough\], see Sections \[sub:The-main-result:Intro\] and \[sec:The-main-result:Details\]. For further discussion on the need of an inner mirror at $r\sim r_{0\text{\textgreek{e}}}$ and its relation to natural dispersive mechanisms appearing in other matter models, see Section \[sub:NeedOfAMirror\]. We should also remark the following: - Except for the condition $r_{0\text{\textgreek{e}}}<2(M_{ADM})_{\text{\textgreek{e}}}$ referred to earlier, where $(M_{ADM})_{\text{\textgreek{e}}}$ is the ADM mass of $\mathcal{S}_{\text{\textgreek{e}}}$, there is considerable flexibility in the choice of the mirror radii $r_{0\text{\textgreek{e}}}$ in the statement of Theorem \[Theorem\_Intro\_Rough\] and this can be exploited to one’s advantage. For simplicity, we choose $r_{0\text{\textgreek{e}}}$ to satisfy $r_{0\text{\textgreek{e}}}\sim(M_{ADM})_{\text{\textgreek{e}}}$ (see also the discussion in Section \[sub:The-main-result:Intro\]). - While we do not address the issue of the end state of the evolution of $\mathcal{S}_{\text{\textgreek{e}}}$, it can be easily inferred from our proof of Theorem \[Theorem\_Intro\_Rough\] that the spacetimes $(\mathcal{M}_{\text{\textgreek{e}}},g_{\text{\textgreek{e}}})$ settle down to a member of the Schwarzschild–AdS family (see also [@MoschidisMaximalDevelopment]). The trivial instability at $r=0$ occuring for the spherically symmetric Einstein–null dust system is absent in the case of smooth solutions to the general spherically symmetric Einstein–massless Vlasov system (not reduced to the radial case). In particular, the smooth initial value problem for the spherically symmetric Einstein–massless Vlasov system is well-posed, and placing an inner mirror at $r=r_{0}>0$ is not necessary.[^2] For a proof of the AdS instability in this setting, see our forthcoming [@MoschidisVlasov]. \[sub:Numerics\]Earlier numerical and heuristic works ----------------------------------------------------- Restricted under spherical symmetry, all solutions to the Einstein vacuum equations (\[eq:VacuumEinsteinEquations\]) are locally isometric to a member of the Schwarzschild–AdS family (see [@Eiesland1925]). Thus, any attempt to search for unstable vacuum perturbations of $(\mathcal{M}_{AdS},g_{AdS})$ for (\[eq:VacuumEinsteinEquations\]) in $3+1$ dimensions can not be reduced to a problem for a $1+1$ hyperbolic system (where the wide variety of available tools would make the problem more tractable).[^3] For this reason, instead of (\[eq:VacuumEinsteinEquations\]), numerical and heuristic works on the AdS instability have so far mainly focused on the Einstein–scalar field system $$\begin{cases} Ric_{\text{\textgreek{m}\textgreek{n}}}-\frac{1}{2}Rg_{\text{\textgreek{m}\textgreek{n}}}+\Lambda g_{\text{\textgreek{m}\textgreek{n}}}=8\pi T_{\text{\textgreek{m}\textgreek{n}}}[\text{\textgreek{f}}],\\ \square_{g}\text{\textgreek{f}}=0,\\ T_{\text{\textgreek{m}\textgreek{n}}}[\text{\textgreek{f}}]\doteq\partial_{\text{\textgreek{m}}}\text{\textgreek{f}}\partial_{\text{\textgreek{n}}}\text{\textgreek{f}}-\frac{1}{2}g_{\text{\textgreek{m}\textgreek{n}}}\partial^{\text{\textgreek{a}}}\text{\textgreek{f}}\partial_{\text{\textgreek{a}}}\text{\textgreek{f}}. \end{cases}\label{eq:EinsteinScalarField}$$ The system (\[eq:EinsteinScalarField\]), whose mathematical study in the case $\Lambda=0$ was pioneered by Christodoulou [@Christodoulou1999], admits non-trivial dynamics in spherical symmetry and spherically symmetric solutions to (\[eq:EinsteinScalarField\]) share many qualitative properties with general solutions of (\[eq:VacuumEinsteinEquations\]). Reduced under spherical symmetry in a double null gauge $(u,v)$ in $3+1$ dimensions, i.e. a gauge where $$g=-\text{\textgreek{W}}^{2}dudv+r^{2}g_{\mathbb{S}^{2}},$$ the system (\[eq:EinsteinScalarField\]) takes the form $$\begin{cases} \partial_{u}\partial_{v}(r^{2}) & =-\frac{1}{2}(1-\Lambda r^{2})\text{\textgreek{W}}^{2},\\ \partial_{u}\partial_{v}\log(\text{\textgreek{W}}^{2}) & =\frac{\text{\textgreek{W}}^{2}}{2r^{2}}\big(1+4\text{\textgreek{W}}^{-2}\partial_{u}r\partial_{v}r\big)-8\pi\partial_{u}\text{\textgreek{f}}\partial_{v}\text{\textgreek{f}},\\ \partial_{v}(\text{\textgreek{W}}^{-2}\partial_{v}r) & =-4\pi r\text{\textgreek{W}}^{-2}(\partial_{v}\text{\textgreek{f}})^{2},\\ \partial_{u}(\text{\textgreek{W}}^{-2}\partial_{u}r) & =-4\pi r\text{\textgreek{W}}^{-2}(\partial_{u}\text{\textgreek{f}})^{2},\\ \partial_{u}\partial_{v}(r\text{\textgreek{f}}) & =-\frac{\text{\textgreek{W}}^{2}-4\partial_{u}r\partial_{v}r}{4r^{2}}\cdot r\text{\textgreek{f}}. \end{cases}\label{eq:EinsteinScalarFieldInDoubleNull}$$ The well-posedness of the asymptotically AdS initial-boundary value problem for the system (\[eq:EinsteinScalarFieldInDoubleNull\]) with reflecting boundary conditions on $\mathcal{I}$ was established by Holzegel and Smulevici in [@HolzSmul2012]. Numerical results in the direction of establishing the AdS instability conjecture were first obtained in 2011 by Bizon and Rostworowski in [@bizon2011weakly], who studied the evolution of spherically symmetric perturbations of $(\mathcal{M}_{AdS},g_{AdS})$ for (\[eq:EinsteinScalarField\]) in Schwarzschild-type coordinates. More precisely, [@bizon2011weakly] numerically simulated the evolution of initial data for (\[eq:EinsteinScalarField\]) with $\text{\textgreek{f}}$ initially arranged into small amplitude wave packets. It was found that, for certain families of initial arrangements of this form (of “size” $\text{\textgreek{e}}$), after a finite number of reflections on $\mathcal{I}$ (proportional to $\text{\textgreek{e}}^{-2}$), the energy of the wave packets becomes substantially concentrated, leading to a break down of the coordinate system associated with the threshold of trapped surface formation. Following [@bizon2011weakly], a vast amount of numerical and heuristic works have been dedicated to the understanding of the global dynamics of perturbations of $(\mathcal{M}_{AdS},g_{AdS})$ for the system (\[eq:EinsteinScalarField\]) (see, e.g., [@DiasHorSantos; @BuchelEtAl2012; @DiasEtAl; @MaliborskiEtAl; @BalasubramanianEtAl; @CrapsEtAl2014; @CrapsEtAl2015; @BizonMalib; @DimitrakopoulosEtAl; @GreenMailardLehnerLieb; @HorowitzSantos; @DimitrakopoulosEtAl2015; @DimitrakopoulosEtAl2016]). In these works, the picture that arises regarding the long time dynamics of generic spherically symmetric perturbations is rather complicated: Apart from perturbations that lead to instability and trapped surface formation ([@DiasHorSantos; @BuchelEtAl2012]), it appears that there exist certain types of perturbations (dubbed “islands of stability”) which remain close to $(\mathcal{M}_{AdS},g_{AdS})$ for long times; see [@DiasEtAl; @MaliborskiEtAl; @BalasubramanianEtAl; @DimitrakopoulosEtAl2015]. Perturbations of the latter type might in fact occupy an open set in the moduli space of spherically symmetric initial data for (\[eq:EinsteinScalarField\]) (see [@BalasubramanianEtAl; @DimitrakopoulosEtAl2015]). The question of existence of open “corners” of initial data around $(\mathcal{M}_{AdS},g_{AdS})$ leading to trapped surface formation has also been studied (see, e.g. [@DimitrakopoulosEtAl]). Another interesting problem in this context is the characterization of the possible end states of the evolution of unstable perturbations of $(\mathcal{M}_{AdS},g_{AdS})$. In [@HolSmul2013], Holzegel–Smulevici established that the Schwarzschild–AdS spacetime $(\mathcal{M}_{Sch},g_{Sch})$ is an asymptotically stable solution of the system (\[eq:EinsteinScalarField\]) in spherical symmetry, with perturbations decaying at an exponential rate.[^4] This result supports the expectation that all spherically symmetric perturbations of $(\mathcal{M}_{AdS},g_{AdS})$ for the system (\[eq:EinsteinScalarField\]) leading to the formation of a trapped surface eventually settle down to a member of the Schwarzschild–AdS family (see [@DafHol; @DafHolStability2006]). However, beyond spherical symmetry, Holzegel–Smulevici [@Holzegel2013; @Holzegel2013a] showed that solutions to the linear scalar wave equation $$\square_{g_{Sch}}\text{\textgreek{f}}=0\label{eq:ScalarWaveEquation}$$ on $(\mathcal{M}_{Sch},g_{Sch})$ (and, more generally, on Kerr–AdS) decay at a slow (logarithmic) rate, which is insufficient in itself to yield the non-linear stability of $(\mathcal{M}_{Sch},g_{Sch})$ (cf. our remark below the statement of the AdS instability conjecture). Thus, [@Holzegel2013a] conjectured that $(\mathcal{M}_{Sch},g_{Sch})$ is non-linearly unstable. On the other hand, based on a detailed analysis of quasinormal modes on $(\mathcal{M}_{Sch},g_{Sch})$, Dias–Horowitz–Marolf–Santos [@DiasEtAl] suggested that sufficiently regular, non-linear perturbations of $(\mathcal{M}_{Sch},g_{Sch})$ still remain small, at least for long times. As a result, the picture regarding the end state of the evolution of generic perturbations of $(\mathcal{M}_{AdS},g_{AdS})$ outside spherical symmetry remains unclear (see also [@HorowitzSantos; @DiasSantos2016; @Rostworowski2017]). Following [@bizon2011weakly], the bulk of heuristic works have implemented a frequency space analysis in the study of the AdS instability conjecture. A notable exception is the work [@DimitrakopoulosEtAl] of Dimitrakopoulos–Freivogel–Lippert–Yang, where a physical space mechanism possibly leading to instability for the system (\[eq:EinsteinScalarFieldInDoubleNull\]) is suggested. We will revisit the mechanism of [@DimitrakopoulosEtAl] and compare it with the results of this paper at the and of Section \[sub:Sketch-of-the-proof\]. \[sub:NeedOfAMirror\]The Einstein–null dust system in spherical symmetry ------------------------------------------------------------------------ A spherically symmetric model for (\[eq:VacuumEinsteinEquations\]) which is even simpler than (\[eq:EinsteinScalarField\]) is the Einstein–massless Vlasov system (see [@Andreasson2011; @Rein1995]). The case where the Vlasov field is supported only on radial geodesics is a singular reduction of this system which is equivalent to the Einstein–null dust system, allowing for both ingoing and outgoing dust (see [@Rendall1997]). This system was studied by Poisson and Israel in their seminal work on mass inflation [@PoissonIsrael1990]. In $3+1$ dimensions, it takes the form (in double null coordinates $(u,v)$): $$\begin{cases} \partial_{u}\partial_{v}(r^{2}) & =-\frac{1}{2}(1-\Lambda r^{2})\text{\textgreek{W}}^{2},\\ \partial_{u}\partial_{v}\log(\text{\textgreek{W}}^{2}) & =\frac{\text{\textgreek{W}}^{2}}{2r^{2}}\big(1+4\text{\textgreek{W}}^{-2}\partial_{u}r\partial_{v}r\big),\\ \partial_{v}(\text{\textgreek{W}}^{-2}\partial_{v}r) & =-4\pi r^{-1}\text{\textgreek{W}}^{-2}\bar{\text{\textgreek{t}}},\\ \partial_{u}(\text{\textgreek{W}}^{-2}\partial_{u}r) & =-4\pi r^{-1}\text{\textgreek{W}}^{-2}\text{\textgreek{t}},\\ \partial_{u}\bar{\text{\textgreek{t}}} & =0,\\ \partial_{v}\text{\textgreek{t}} & =0. \end{cases}\label{eq:EinsteinNullDust}$$ In certain cases, the Einstein–null dust system (\[eq:EinsteinNullDust\]) can be formally viewed as a high frequency limit of the Einstein–scalar field system (\[eq:EinsteinScalarFieldInDoubleNull\]) (as was already discussed in [@PoissonIsrael1990]): Setting $$\text{\textgreek{t}}\doteq r^{2}(\partial_{u}\text{\textgreek{f}})^{2},\mbox{ }\bar{\text{\textgreek{t}}}\doteq r^{2}(\partial_{v}\text{\textgreek{f}})^{2}$$ in (\[eq:EinsteinScalarField\]) and dropping all lower order terms from the wave equation for $\text{\textgreek{f}}$, one formally obtains (\[eq:EinsteinNullDust\]) in the region where $\partial_{u}\text{\textgreek{f}}\partial_{v}\text{\textgreek{f}}$ is negligible, i.e. outside the intersection of the supports of $\text{\textgreek{t}},\bar{\text{\textgreek{t}}}$. While this formal limiting procedure can be rigorously justified away from $r=0$, the dynamical similarities between (\[eq:EinsteinScalarFieldInDoubleNull\]) and (\[eq:EinsteinNullDust\]) break down close to $r=0$. A fundamental difference between these systems is the fact that, while small data asymptotically AdS solutions to (\[eq:EinsteinScalarFieldInDoubleNull\]) satisfying a reflecting boundary condition at $\mathcal{I}$ remain regular (and “small”) for large times, all non-trivial solutions to the system (\[eq:EinsteinNullDust\]) break down once the support of $\bar{\text{\textgreek{t}}}$ reaches the axis $\text{\textgreek{g}}$ (i.e. the timelike portion of $\{r=0\}$), independently of the boundary conditions imposed at $\mathcal{I}$. This is an ill-posedness statement for (\[eq:EinsteinNullDust\]), which needs to be addressed before any attempt to study the AdS instability conjecture in the setting of (\[eq:EinsteinNullDust\]). We will now proceed to discuss this difference of (\[eq:EinsteinScalarFieldInDoubleNull\]) and (\[eq:EinsteinNullDust\]) in more detail. ### Cauchy stability for the Einstein–scalar field system {#cauchy-stability-for-the-einsteinscalar-field-system .unnumbered} The following Cauchy stability result holds for the system (\[eq:EinsteinScalarFieldInDoubleNull\]): [1]{}\[Cauchy stability for ; see [@HolzSmul2012]\]\[prop:CauchyStabilityIntro\] For a suitable initial data norm $\|\|\cdot\|\|_{initial}$, $(\mathcal{M}_{AdS},g_{AdS})$ is Cauchy stable as a solution of the system (\[eq:EinsteinScalarFieldInDoubleNull\]) with reflecting boundary conditions on $\mathcal{I}$. That is to say, for all fixed times $T_{}>0$, any perturbation of the initial data of $(\mathcal{M}_{AdS},g_{AdS})$ which is small enough (when measured in terms of $\|\|\cdot\|\|_{initial}$) with respect to $T_{}$ gives rise to a solution of (\[eq:EinsteinScalarFieldInDoubleNull\]) which is regular and close to $(\mathcal{M}_{AdS},g_{AdS})$ for times up to $T_{}$. In the statement of Proposition \[prop:CauchyStabilityIntro\], Cauchy stability of $(\mathcal{M}_{AdS},g_{AdS})$ refers to stability over fixed compact subsets of the conformal compactification* of $(\mathcal{M}_{AdS},g_{AdS})$, such as subsets of the form $\{0\le t\le T_{}\}$ in the $(t,r,\text{\textgreek{j}},\text{\textgreek{f}})$ coordinate chart. Any such subset contains, in particular, a compact subset of the timelike boundary $\mathcal{I}$. The initial data norm $\|\|\cdot\|\|_{initial}$, for which the Cauchy stability of $(\mathcal{M}_{AdS},g_{AdS})$ follows from [@HolzSmul2012], is a higher order, suitably weighted $C^{k}$ norm. However, this is not the only norm for which $(\mathcal{M}_{AdS},g_{AdS})$ can be shown to be Cauchy stable: An additional, highly non-trivial example of such a norm is the bounded variation norm of Christodoulou [@ChristodoulouBoundedVariation] (modified with suitable $r$-weights near $r=\infty$). Similar low-regularity norms will also play an important role in this paper (see Section \[sub:NeedOfAMirror\]). Assuming, for simplicity, that initial data are prescribed on the outgoing null hypersurface corresponding to $u=0$, for $0\le v\le v_{}$, a necessary condition for Cauchy stability of $(\mathcal{M}_{AdS},g_{AdS})$ for the system (\[eq:EinsteinScalarFieldInDoubleNull\]) with respect to an initial data norm $\|\|\cdot\|\|_{initial}$ is that, for any given $R_{0}>0$, $\|\|\cdot\|\|_{initial}$ controls the quantity $$\mathscr{M}\doteq\sup_{\substack{0\le v_{1}<v_{2}\le v_{},\\ \frac{r(0,v_{2})}{r(0,v_{1})}<\frac{3}{2},\mbox{ }r(0,v_{2})\le R_{0} } }\frac{\tilde{m}(0,v_{2})-\tilde{m}(0,v_{1})}{(r(0,v_{2})-r(0,v_{1}))\big\|\log\big(\frac{r(0,v_{2})}{r(0,v_{1})}-1\big)\big\|},\label{eq:EnergyConcentration}$$ where $\tilde{m}$ is the renormalised Hawking mass, defined in terms of the Hawking mass $m$, $$m\doteq\frac{r}{2}\Big(1-4\text{\textgreek{W}}^{-2}\partial_{u}r\partial_{v}r\Big),\label{eq:HawkingMassIntro}$$ by the relation $$\tilde{m}\doteq m-\frac{1}{6}\Lambda r^{3}.\label{eq:RenormalisedhawkingMassIntro}$$ This is a consequence of the fact that, when $\mathscr{M}$ exceeds a certain threshold (depending on $R_{0}$), there exists a $u_{\dagger}\in(0,v_{})$ and a point $p=(u_{\dagger},v_{\dagger})$ in the development of the initial data such that $$\frac{2m}{r}(u_{\dagger},v_{\dagger})>1,\label{eq:IndicatorBlackHole}$$ a result proven by Christodoulou in [@Christodoulou1991].[^5] The bound (\[eq:IndicatorBlackHole\]) implies that $$\partial_{u}r(u_{\dagger},v_{\dagger})<0\mbox{, }\partial_{v}r(u_{\dagger},v_{\dagger})<0,\label{eq:TrappedSurface}$$ i.e. that the symmetry sphere associated to $(u_{\dagger},v_{\dagger})$ is a trapped surface. In particular, $(u_{\dagger},v_{\dagger})$ is contained in a black hole.[^6] As a corollary, it follows that the total ADM mass of the initial data, though expressible as a coercive functional on the space of initial data of (\[eq:EinsteinScalarFieldInDoubleNull\]), does not yield a norm for which $(\mathcal{M}_{AdS},g_{AdS})$ is Cauchy stable for (\[eq:EinsteinScalarFieldInDoubleNull\]), since the ADM mass manifestly fails to control (\[eq:EnergyConcentration\]). ### Break down at $r=0$ and “trivial” Cauchy instability for the Einstein–null dust system {#break-down-at-r0-and-trivial-cauchy-instability-for-the-einsteinnull-dust-system .unnumbered} The following instability result holds for the system (\[eq:EinsteinNullDust\]) (see [@MoschidisMaximalDevelopment]): [2]{}\[Cauchy instability for \]\[prop:C0InextendibilityIntroduction\] Any globally hyperbolic spherically symmetric solution $(\mathcal{M},g;\text{\textgreek{t}},\bar{\text{\textgreek{t}}})$ of (\[eq:EinsteinNullDust\]) with non-empty axis $\text{\textgreek{g}}$ “breaks down” at the first point when a radial geodesic in the support of $\bar{\text{\textgreek{t}}}$ reaches $\text{\textgreek{g}}$: Beyond that point, $(\mathcal{M},g;\text{\textgreek{t}},\bar{\text{\textgreek{t}}})$ is $C^{0}$ inextendible as a spherically symmetric solution to (\[eq:EinsteinNullDust\]). As a result, $(\mathcal{M}_{AdS},g_{AdS})$ is not a Cauchy stable solution of (\[eq:EinsteinNullDust\]) for any “reasonable” initial data topology. For the precise definition of the notion of $C^{0}$ inextendibility as a spherically symmetric solution to (\[eq:EinsteinNullDust\]), see [@MoschidisMaximalDevelopment]. Note that this a stronger statement than $(\mathcal{M},g;\text{\textgreek{t}},\bar{\text{\textgreek{t}}})$ breaking down as a smooth solution of (\[eq:EinsteinNullDust\]). We should also remark the following regarding Proposition \[prop:C0InextendibilityIntroduction\]: - Proposition \[prop:C0InextendibilityIntroduction\] holds independently of the value of the cosmological constant $\Lambda$. In particular, Minkowski spacetime $(\mathbb{R}^{3+1},\text{\textgreek{h}})$ is not Cauchy stable for (\[eq:EinsteinNullDust\]) with $\Lambda=0$ for any “reasonable” initial data topology. - Proposition \[prop:C0InextendibilityIntroduction\] yields a uniform upper bound on the time of existence of solutions $(\mathcal{M},g)$ to (\[eq:EinsteinNullDust\]) for any initial data set for which $\bar{\text{\textgreek{t}}}$ is not identically equal to $0$, depending only on the distance of the initial support of $\bar{\text{\textgreek{t}}}$ from the axis and, thus, independently of the proximity of the initial data to the trivial data (in any reasonable initial data norm). We should also highlight that the instability of Proposition \[prop:C0InextendibilityIntroduction\] has nothing to do with trapped surface formation: Up to the first retarded time when a radial geodesic in the support of $\bar{\text{\textgreek{t}}}$ reaches $\text{\textgreek{g}}$, any solution $(\mathcal{M},g)$ to (\[eq:EinsteinNullDust\]) arising from smooth initial data close to $(\mathcal{M}_{AdS},g_{AdS})$ remains smooth and close to $(\mathcal{M}_{AdS},g_{AdS})$, and $(\mathcal{M},g)$ contains no trapped surface. In fact, in this case, despite being $C^{0}$ inextendible as a globally hyperbolic spherically symmetric solution to (\[eq:EinsteinNullDust\]), $(\mathcal{M},g)$ is globally $C^{\infty}$-extendible as a spherically symmetric Lorentzian manifold; see [@MoschidisMaximalDevelopment]. - The Cauchy stability statement for $(\mathcal{M}_{AdS},g_{AdS})$ for the system (\[eq:EinsteinScalarFieldInDoubleNull\]) stated in Proposition \[prop:CauchyStabilityIntro\] can be informally interpreted as the result of a natural dispersive mechanism close to the axis $\text{\textgreek{g}}$ displayed by the system (\[eq:EinsteinScalarFieldInDoubleNull\]), which does not allow the energy of $\text{\textgreek{f}}$ to concentrate on scales smaller than $\tilde{m}$ in $O(1)$ time, provided a suitable initial norm of $\text{\textgreek{f}}$ (controlling at least (\[eq:EnergyConcentration\])) is small enough. No such mechanism is present for the system (\[eq:EinsteinNullDust\]), as is illustrated by Proposition \[prop:C0InextendibilityIntroduction\]. ### Resolution of the “trivial” instability of (\[eq:EinsteinNullDust\]) through an inner mirror {#resolution-of-the-trivial-instability-of-eqeinsteinnulldust-through-an-inner-mirror .unnumbered} In order to turn the spherically symmetric Einstein–null dust system (\[eq:EinsteinNullDust\]) into a well-posed, Cauchy-stable system (a necessary step for converting (\[eq:EinsteinNullDust\]) into an effective model of the vacuum Einstein equations (\[eq:VacuumEinsteinEquations\])), it is necessary to explicitly add to (\[eq:EinsteinNullDust\]) a mechanism that prevents the break down at $r=0$ described by Proposition \[prop:C0InextendibilityIntroduction\], so that, moreover, an analogue of Proposition \[prop:CauchyStabilityIntro\] holds for (\[eq:EinsteinNullDust\]). This can be achieved by by placing an inner mirror at $r=r_{0}>0$, i.e. by restricting (\[eq:EinsteinNullDust\]) on $\{r\ge r_{0}\}$, for some $r_{0}>0$, and imposing a reflecting boundary condition on the portion $\text{\textgreek{g}}_{0}$ of the set $\{r=r_{0}\}$ which is timelike. The reflecting boundary condition on $\text{\textgreek{g}}_{0}$ can be motivated by the fact that, for smooth spherically symmetric solutions $(\mathcal{M},g;\text{\textgreek{f}})$ to (\[eq:EinsteinScalarField\]), the function $\text{\textgreek{f}}$, viewed as a function on the quotient of $(\mathcal{M},g)$ by the spheres of symmetry, satisfies a reflecting boundary condition on the axis. The well-posedness and the properties of the maximal development for the system (\[eq:EinsteinNullDust\]) with reflecting boundary conditions on $\mathcal{I}$ and $\text{\textgreek{g}}_{0}$ are addressed in the companion paper [@MoschidisMaximalDevelopment]. The following result is established in [@MoschidisMaximalDevelopment]: [2]{}\[Well posedness for with an inner mirror\]\[thm:MaximalDevelopmentIntro\] For any $r_{0}>0$ and any smooth asymptotically AdS initial data set $(r,\text{\textgreek{W}}^{2},\text{\textgreek{t}},\bar{\text{\textgreek{t}}})\|_{u=0}$ on $u=0$, there exists a unique smooth maximal future development $(r,\text{\textgreek{W}}^{2},\text{\textgreek{t}},\bar{\text{\textgreek{t}}})$ on $\{r\ge r_{0}\}$, solving (\[eq:EinsteinNullDust\]) with reflecting boundary conditions on $\mathcal{I}$ and $\text{\textgreek{g}}_{0}$, where $r\|_{\text{\textgreek{g}}_{0}}=r_{0}$ and $\text{\textgreek{g}}_{0}$ coincides with the portion of the curve $\{r=r_{0}\}$ which is timelike (fixing the gauge freedom by imposing a reflecting gauge condition on both $\mathcal{I}$ and $\text{\textgreek{g}}_{0}$). For this development, $\mathcal{I}$ is complete and $\{r=r_{0}\}$ is timelike in the past of $\mathcal{I}$ (see Figure 1.2). In the case when the future event horizon $\mathcal{H}^{+}$ is non-empty, it is smooth and future complete. A necessary condition for $\mathcal{H}^{+}$ to be non-empty is the existence of a point $(u_{\dagger},v_{\dagger})$ where (\[eq:IndicatorBlackHole\]) holds. If the total mass $\tilde{m}\|_{\mathcal{I}}$ and the mirror radius $r_{0}$ satisfy $$\frac{2\tilde{m}\|_{\mathcal{I}}}{r_{0}}\le1-\frac{1}{3}\Lambda r_{0}^{2},\label{eq:BoundForTotalMassHorizon}$$ then necessarily $\mathcal{H}^{+}=\emptyset$. For a more detailed statement of Theorem \[thm:MaximalDevelopmentIntro\], see Section \[sec:ResultsFromTheOtherPaper\] and [@MoschidisMaximalDevelopment]. In view of the fact that $\mathcal{H}^{+}=\emptyset$ in the case when the total mass $\tilde{m}\|_{\mathcal{I}}$ and the mirror radius $r_{0}$ satisfy (\[eq:BoundForTotalMassHorizon\]), in order to address the AdS instability conjecture for the system (\[eq:EinsteinNullDust\]) with reflecting boundary conditions on $\mathcal{I}$ and $\text{\textgreek{g}}_{0}$, it is necessary to allow $r_{0}$ to shrink to $0$ with the size of the data. Thus, addressing the AdS instability conjecture in this setting requires establishing a Cauchy stability statement for $(\mathcal{M}_{AdS},g_{AdS})$ which is independent of the precise value of the mirror radius $r_{0}$. This is the statement of the following result, proved in our companion paper [@MoschidisMaximalDevelopment]: [3]{}\[Cauchy stability for uniformly in $r_0$\]\[thm:CauchyStabilityIntro\] Given $\text{\textgreek{e}}>0$, $u_{}>0$, there exists a $\text{\textgreek{d}}>0$ such that the following statement holds: For $r_{0}>0$ and initial data set $(r,\text{\textgreek{W}}^{2},\text{\textgreek{t}},\bar{\text{\textgreek{t}}})\|_{u=0}$ satisfying $$\|\|(r,\text{\textgreek{W}}^{2},\text{\textgreek{t}},\bar{\text{\textgreek{t}}})\|\|_{u=0}\doteq\sup_{\bar{v}}\int_{u=0}\frac{\bar{\text{\textgreek{t}}}(0,v)}{\|\text{\textgreek{r}}(0,v)-\text{\textgreek{r}}(0,\bar{v})\|+\tan^{-1}(\sqrt{-\Lambda}r_{0})}\,\frac{\sqrt{-\Lambda}dv}{\partial_{v}\text{\textgreek{r}}(0,v)}+\sup_{u=0}\Big(\Big\|\big(1-\frac{2\tilde{m}}{r}\big)^{-1}-1\Big\|+\sqrt{-\Lambda}\tilde{m}\Big)\le\text{\textgreek{e}},\label{eq:NormCauchyStabilityIntro}$$ where $$\text{\textgreek{r}}(0,v)\doteq\tan^{-1}(\sqrt{-\Lambda}r)(0,v),$$ the corresponding solution $(r,\text{\textgreek{W}}^{2},\text{\textgreek{t}},\bar{\text{\textgreek{t}}})$ to (\[eq:EinsteinNullDust\]) with reflecting boundary conditions on $\mathcal{I}$ and $\text{\textgreek{g}}_{0}$ will satisfy $$\sup_{0\le\bar{u}\le u_{}}\|\|(r,\text{\textgreek{W}}^{2},\text{\textgreek{t}},\bar{\text{\textgreek{t}}})\|\|_{u=\bar{u}}\le\text{\textgreek{d}}.$$ For a more detailed statement of Theorem \[thm:CauchyStabilityIntro\], see Section \[sec:ResultsFromTheOtherPaper\] and [@MoschidisMaximalDevelopment]. Notice that the norm (\[eq:NormCauchyStabilityIntro\]) vanishes only for the trivial initial data $(r,\text{\textgreek{W}}^{2},0,0)$. Informally, Theorem \[thm:CauchyStabilityIntro\] implies that, if the energy of the initial data concentrated on scales proportional to the mirror radius $r_{0}$ is small enough, then the energy of the solution to (\[eq:EinsteinNullDust\]) (with reflecting boundary conditions on $\mathcal{I}$ and $\text{\textgreek{g}}_{0}$) will remain similarly dispersed for times less than any given constant. In particular, no trapped surface can form in this timescale if $\text{\textgreek{d}}$ is chosen sufficiently small. In Section \[sec:ResultsFromTheOtherPaper\], we will also present a Cauchy stability statement for general solutions of (\[eq:EinsteinNullDust\]) with reflecting boundary conditions on $\mathcal{I}$ and $\text{\textgreek{g}}_{0}$, which will be used in the proof of Theorem \[Theorem\_Intro\_Rough\] (see Theorem \[prop:CauchyStability\]). \[sub:The-main-result:Intro\]Statement of Theorem \[Theorem\_Intro\_Rough\]: the non-linear instability of AdS -------------------------------------------------------------------------------------------------------------- According to Theorem \[thm:CauchyStabilityIntro\], a Cauchy stability statement holds for $(\mathcal{M}_{AdS},g_{AdS})$ for time intervals which are independent of the precise value of the mirror radius $r_{0}$, depending only on the smallness of the initial data norm (\[eq:NormCauchyStabilityIntro\]). As a result, it is possible to study the AdS instability conjecture for the system (\[eq:EinsteinNullDust\]) with reflecting boundary conditions on $\mathcal{I}$ and $\text{\textgreek{g}}_{0}$, for perturbations which are small with respect to (\[eq:NormCauchyStabilityIntro\]), allowing the mirror radius $r_{0}$ to shrink with the size of the data. In this paper, we will prove the following result: [1]{}\[more precise version\]\[thm:TheoremDetailedIntro\] There exists a family of positive numbers $r_{0\text{\textgreek{e}}}$ (satisfying $r_{0\text{\textgreek{e}}}\xrightarrow{\text{\textgreek{e}}\rightarrow0}0$) and smooth initial data $(r,\text{\textgreek{W}}^{2},\text{\textgreek{t}},\bar{\text{\textgreek{t}}})^{(\text{\textgreek{e}})}\|_{u=0}$ for the system (\[eq:EinsteinNullDust\]) satisfying the following properties: 1. In the norm $\|\|\cdot\|\|_{u=0}$ defined by (\[eq:NormCauchyStabilityIntro\]): $$\|\|(r,\text{\textgreek{W}}^{2},\text{\textgreek{t}},\bar{\text{\textgreek{t}}})^{(\text{\textgreek{e}})}\|\|_{u=0}\xrightarrow{\text{\textgreek{e}}\rightarrow0}0.\label{eq:NormToZeroIntro}$$ 2. For any $\text{\textgreek{e}}>0$, the maximal development $(r,\text{\textgreek{W}}^{2},\text{\textgreek{t}},\bar{\text{\textgreek{t}}})^{(\text{\textgreek{e}})}$ of $(r,\text{\textgreek{W}}^{2},\text{\textgreek{t}},\bar{\text{\textgreek{t}}})^{(\text{\textgreek{e}})}\|_{u=0}$ for the system (\[eq:EinsteinNullDust\]) with reflecting boundary conditions on $\mathcal{I}$ and $\text{\textgreek{g}}_{0}$, $r\|_{\text{\textgreek{g}}_{0}}=r_{0\text{\textgreek{e}}}$, contains a trapped sphere, i.e. there exists a point $(u_{\text{\textgreek{e}}},v_{\text{\textgreek{e}}})$ such that: $$\frac{2m^{(\text{\textgreek{e}})}}{r^{(\text{\textgreek{e}})}}(u_{\text{\textgreek{e}}},v_{\text{\textgreek{e}}})>1.\label{eq:TrappedSurfaceIntro}$$ Thus, in view of Theorem \[thm:MaximalDevelopmentIntro\], $(r,\text{\textgreek{W}}^{2},\text{\textgreek{t}},\bar{\text{\textgreek{t}}})^{(\text{\textgreek{e}})}$ contains a non-empty, smooth and future complete event horizon $\mathcal{H}^{+}$ and a complete conformal infinity $\mathcal{I}$. For the definitive statement of Theorem \[thm:TheoremDetailedIntro\], see Section \[sec:The-main-result:Details\]. The following remarks should be made concerning Theorem \[thm:TheoremDetailedIntro\]: - In view of the Cauchy stability of $(\mathcal{M}_{AdS},g_{AdS})$ with respect to (\[eq:NormCauchyStabilityIntro\]) (see Theorem \[thm:CauchyStabilityIntro\]), the time[^7] required to elapse before $\big(1-\frac{2m^{(\text{\textgreek{e}})}}{r^{(\text{\textgreek{e}})}}\big)$ becomes negative necessarily tends to $+\infty$ as $\text{\textgreek{e}}\rightarrow0$. - In view of the fact that $\mathcal{H}^{+}=\emptyset$ when (\[eq:BoundForTotalMassHorizon\]) holds, in order for $(r,\text{\textgreek{W}}^{2},\text{\textgreek{t}},\bar{\text{\textgreek{t}}})^{(\text{\textgreek{e}})}$ to satisfy both (\[eq:NormToZeroIntro\]) and (\[eq:TrappedSurfaceIntro\]), it is necessary that $r_{0\text{\textgreek{e}}}\rightarrow0$ as $\text{\textgreek{e}}\rightarrow0$, at a rate which is at least as fast as that of $2\tilde{m}^{(\text{\textgreek{e}})}\|_{\mathcal{I}}$,[^8] i.e.: $$r_{0\text{\textgreek{e}}}\le2\tilde{m}^{(\text{\textgreek{e}})}\|_{\mathcal{I}}.\label{eq:BoundForMirroRadius}$$ In fact, we choose the family $r_{0\text{\textgreek{e}}}$, $(r,\text{\textgreek{W}}^{2},\text{\textgreek{t}},\bar{\text{\textgreek{t}}})^{(\text{\textgreek{e}})}\|_{u=0}$ of Theorem \[thm:TheoremDetailedIntro\] to saturate the bound (\[eq:BoundForMirroRadius\]) in the limit $\text{\textgreek{e}}\rightarrow0$, i.e. $$\lim_{\text{\textgreek{e}}\rightarrow0}\frac{r_{0\text{\textgreek{e}}}}{2\tilde{m}^{(\text{\textgreek{e}})}\|_{\mathcal{I}}}=1.\label{eq:SaturatedBound}$$ For the proof of Theorem \[thm:TheoremDetailedIntro\], (\[eq:SaturatedBound\]) is not essential and can be relaxed; however, it is fundamental for our proof that $r_{0\text{\textgreek{e}}}$ is bounded from below by some small multiple of $\tilde{m}^{(\text{\textgreek{e}})}\|_{\mathcal{I}}$. - It follows from the proof of Theorem \[thm:MaximalDevelopmentIntro\] that, in the case $\Lambda=0$, Minkowski spacetime $(\mathbb{R}^{3+1},\text{\textgreek{h}})$ is globally stable (for the system (\[eq:EinsteinNullDust\]) with reflecting boundary conditions on the inner mirror $\{r=r_{0}\}$) to initial data perturbations which are small with respect to the norm (\[eq:NormCauchyStabilityIntro\]), independently of the precise choice of $r_{0}$. This fact further justifies the choice of the matter model and the norm (\[eq:NormCauchyStabilityIntro\]) as a setting for establishing the AdS instability conjecture. - The proof of Theorem \[thm:TheoremDetailedIntro\] also applies in the case $\Lambda=0$ when placing an outer mirror at $r=R_{0}\gg r_{0}$ (in addition to the inner mirror at $r=r_{0}$), i.e. restricting the solutions of (\[eq:EinsteinNullDust\]) in the region $\{r_{0}\le r\le R_{0}\}$ and imposing reflecting boundary conditions on both $\{r=r_{0}\}$ and $\{r=R_{0}\}$. This is in accordance with the numerical results of [@BuchelEtAl2012] for the system (\[eq:EinsteinScalarField\]). - It can be readily inferred by Cauchy stability (see Theorem \[prop:CauchyStability\] in Section \[sub:Cauchy-stability-inCauchyStability\]) that, for any $y_{\text{\textgreek{e}}}\doteq(r,\text{\textgreek{W}}^{2},\text{\textgreek{t}},\bar{\text{\textgreek{t}}})^{(\text{\textgreek{e}})}\|_{u=0}$ in the family of initial data of Theorem \[thm:TheoremDetailedIntro\], there exists an open neighborhood $\mathcal{W}_{\text{\textgreek{e}}}$ of initial data around $y_{\text{\textgreek{e}}}$ such that, for all $y\in\mathcal{W}_{\text{\textgreek{e}}}$, the maximal future development of $y$ also contains a trapped surface. In particular, the set of initial data leading to trapped surface formation is open. An even stronger genericity statement would be the existence of an open instability corner in the space of initial data around $(\mathcal{M}_{AdS},g_{AdS})$ (see [@DimitrakopoulosEtAl]), i.e. the existence of a $c_{1}>0$ such that $\big\{ y:\, dist(y,y_{\text{\textgreek{e}}})\le c_{1}\|\|y_{\text{\textgreek{e}}}\|\|_{u=0}\big\}\subset\mathcal{W}_{\text{\textgreek{e}}}$ for all $\text{\textgreek{e}}>0$ (with $dist(\cdot,\cdot)$ being the distance function associated to (\[eq:NormCauchyStabilityIntro\]) for $r_{0}=r_{0\text{\textgreek{e}}}$). While we have not addressed the issue of genericity of the unstable initial data in this paper, we expect that the proof of Theorem \[thm:TheoremDetailedIntro\] can be adapted to yield the existence of an instability corner around $(\mathcal{M}_{AdS},g_{AdS})$. - A plethora of numerical works (see, e.g. [@bizon2011weakly; @BuchelEtAl2012; @BizonMalib; @DimitrakopoulosEtAl]) suggest that, in the case of the Einstein–scalar field system (\[eq:EinsteinScalarField\]), for families of initial data $(\text{\textgreek{f}}_{\text{\textgreek{e}}}^{(0)},\text{\textgreek{f}}_{\text{\textgreek{e}}}^{(1)})$ for the scalar field $\text{\textgreek{f}}$ of the form $$(\text{\textgreek{f}}_{\text{\textgreek{e}}}^{(0)},\text{\textgreek{f}}_{\text{\textgreek{e}}}^{(1)})=(\text{\textgreek{e}}\text{\textgreek{f}}^{(0)},\text{\textgreek{e}}\text{\textgreek{f}}^{(1)})\label{eq:RescaledInitialData}$$ (where $(\text{\textgreek{f}}^{(0)},\text{\textgreek{f}}^{(1)})$ is a fixed initial profile), trapped surface formation occurs at time $\sim\text{\textgreek{e}}^{-2}$. However, any rigorous formulation of this statement for general families of initial data, not necessarily of the form (\[eq:RescaledInitialData\]), requires fixing an initial data norm for which the initial data size is measured, with different choices of (scale invariant) norms possibly leading to different timescales of trapped surface formation for initial data of size $\sim\text{\textgreek{e}}$. For this reason, given that the initial data $(\text{\textgreek{t}},\bar{\text{\textgreek{t}}})^{(\text{\textgreek{e}})}\|_{u=0}$ in Theorem \[thm:TheoremDetailedIntro\] can not be viewed as a rescaling of a fixed profile of the form (\[eq:RescaledInitialData\]), we have not tried to optimize the time required for trapped surface formation in Theorem \[thm:TheoremDetailedIntro\] in terms of the initial norm (\[eq:NormCauchyStabilityIntro\]). \[sub:Sketch-of-the-proof\]Sketch of the proof and remarks on Theorem \[thm:TheoremDetailedIntro\] --------------------------------------------------------------------------------------------------- We will now proceed to sketch the main arguments involved in the proof of Theorem \[thm:TheoremDetailedIntro\]. ### Construction of the initial data {#construction-of-the-initial-data .unnumbered} The family of initial data $(r,\text{\textgreek{W}}^{2},\text{\textgreek{t}},\bar{\text{\textgreek{t}}})^{(\text{\textgreek{e}})}\|_{u=0}$ in Theorem \[thm:TheoremDetailedIntro\] is chosen so that its total ADM mass $\tilde{m}^{(\text{\textgreek{e}})}\|_{\mathcal{I}}$ and the mirror radius $r_{0\text{\textgreek{e}}}$ satisfy (for $\text{\textgreek{e}}\ll1$) $$r_{0\text{\textgreek{e}}},\tilde{m}^{(\text{\textgreek{e}})}\|_{\mathcal{I}}\sim\text{\textgreek{e}}(-\Lambda)^{-\frac{1}{2}}.$$ In particular, fixing a function $h(\text{\textgreek{e}})$ in terms of $\text{\textgreek{e}}$ such that $$\text{\textgreek{e}}\ll h(\text{\textgreek{e}})\ll1,$$ the initial data $(r,\text{\textgreek{W}}^{2},\text{\textgreek{t}},\bar{\text{\textgreek{t}}})^{(\text{\textgreek{e}})}\|_{u=0}$ are constructed so that the null dust initially forms a bundle of narrow ingoing beams emanating from the region $r\sim1$; see Figure 1.3. The number of the beams is chosen to be large, i.e. of order $\sim(h(\text{\textgreek{e}}))^{-1}$, and the beams are initially separated by gaps of $r$-width $\sim(h(\text{\textgreek{e}}))^{-1}\text{\textgreek{e}}(-\Lambda)^{-\frac{1}{2}}$. The large number of beams and their initial separation are chosen so that $$\|\|(r,\text{\textgreek{W}}^{2},\text{\textgreek{t}},\bar{\text{\textgreek{t}}})^{(\text{\textgreek{e}})}\|\|_{u=0}\sim h(\text{\textgreek{e}})\xrightarrow{\text{\textgreek{e}}\rightarrow0}0.\label{eq:InitialDataNorm}$$ ### Remarks on the configuration of the null dust beams {#remarks-on-the-configuration-of-the-null-dust-beams .unnumbered} As the solution $(r,\text{\textgreek{W}}^{2},\text{\textgreek{t}},\bar{\text{\textgreek{t}}})^{(\text{\textgreek{e}})}$ arising fom the initial data set $(r,\text{\textgreek{W}}^{2},\text{\textgreek{t}},\bar{\text{\textgreek{t}}})^{(\text{\textgreek{e}})}\|_{u=0}$ evolves according to equations (\[eq:EinsteinNullDust\]), the null dust beams are reflected successively off $\text{\textgreek{g}}_{0}=\{r=r_{0\text{\textgreek{e}}}\}$ and $\mathcal{I}$, as depicted in Figure 1.3. The beams separate the spacetime into vacuum regions (the larger rectangular regions between the beams in Figure 1.3), where the renormalised Hawking mass $\tilde{m}^{(\text{\textgreek{e}})}$ is constant (recall the definition of the Hawking mass and the renormalised Hawking mass by (\[eq:HawkingMassIntro\]) and (\[eq:RenormalisedhawkingMassIntro\]), respectively). The interaction set of the beams consists of all the points in the spacetime where two different beams intersect (depicted in Figure 1.3 as the union of all the smaller dark rectangles, lying in the intersection of any two beams). As long as the total width of the bundle of beams remains small, the interaction set can be split into two sets, one consisting of the intersections occuring close to the mirror $\text{\textgreek{g}}_{0}$ and one consisting of the intersections near $\mathcal{I}$ (see Figure 1.3). Every beam is separated by the interaction set into several components. To each such component, we can associate the mass difference $\mathfrak{D}\tilde{m}$ between the two vacuum regions which are themselves separated by that beam component. The mass difference $\mathfrak{D}\tilde{m}$ measures the energy content of each beam component and, in view of the non-linearity of the system (\[eq:EinsteinNullDust\]), it is not necessarily conserved along the beam after an intersection with another beam. Precisely determining the resulting change in the mass difference after the interaction of two beams will be the crux of the proof of Theorem \[thm:TheoremDetailedIntro\]. ### Beam interactions and change in mass difference {#beam-interactions-and-change-in-mass-difference .unnumbered} In Figure 1.4, the region around the intersection of an incoming null dust beam $\text{\textgreek{z}}_{in}$ and an outgoing null dust beam $\text{\textgreek{z}}_{out}$ is depicted. This region is separated by the beams into 4 vacuum subregions $\mathcal{R}_{1},\ldots,\mathcal{R}_{4}$ with associated renormalised Hawking masses $\tilde{m}_{1},\ldots,\tilde{m}_{4}$ (see Figure 1.4). Before the intersection of the two beams, the mass difference of the incoming beam $\text{\textgreek{z}}_{in}$ is $$\overline{\mathfrak{D}}_{-}\tilde{m}=\tilde{m}_{3}-\tilde{m}_{4},$$ while the mass difference of the outgoing beam $\text{\textgreek{z}}_{out}$ is $$\mathfrak{D}_{-}\tilde{m}=\tilde{m}_{4}-\tilde{m}_{2}.$$ After the intersection of the beams, the mass differences associated to $\text{\textgreek{z}}_{in}$ and $\text{\textgreek{z}}_{out}$ become $$\overline{\mathfrak{D}}_{+}\tilde{m}=\tilde{m}_{1}-\tilde{m}_{2}$$ and $$\mathfrak{D}_{+}\tilde{m}=\tilde{m}_{3}-\tilde{m}_{1},$$ respectively. Assuming that $$\frac{2m}{r}<1$$ and $$\partial_{u}r<0<\partial_{v}r,$$ we can readily obtain the following differential relations for $r$ and $\tilde{m}$ from (\[eq:EinsteinNullDust\]): $$\begin{aligned} \partial_{u}\log\big(\frac{\partial_{v}r}{1-\frac{2m}{r}}\big)= & -\frac{4\pi}{r}\frac{\text{\textgreek{t}}}{-\partial_{u}r},\label{eq:IntroConstraints}\\ \partial_{v}\log\big(\frac{-\partial_{u}r}{1-\frac{2m}{r}}\big)= & \frac{4\pi}{r}\frac{\bar{\text{\textgreek{t}}}}{\partial_{v}r}\nonumber \end{aligned}$$ and $$\begin{aligned} \partial_{u}\tilde{m}= & -2\pi\Big(\frac{1-\frac{2m}{r}}{-\partial_{u}r}\Big)\text{\textgreek{t}},\label{eq:IntroMassEquation}\\ \partial_{v}\tilde{m}= & 2\pi\Big(\frac{1-\frac{2m}{r}}{\partial_{v}r}\Big)\bar{\text{\textgreek{t} }}.\nonumber \end{aligned}$$ We will also assume that: - [ The null dust beams $\text{\textgreek{z}}_{in}$ and $\text{\textgreek{z}}_{out}$ are sufficiently narrow so that, on their intersection $\text{\textgreek{z}}_{in}\cap\text{\textgreek{z}}_{out}$, $r$ can be considered nearly constant:[^9] $$\sup_{\text{\textgreek{z}}_{in}\cap\text{\textgreek{z}}_{out}}r-\inf_{\text{\textgreek{z}}_{in}\cap\text{\textgreek{z}}_{out}}r\ll\text{\textgreek{e}}(-\Lambda)^{-\frac{1}{2}},$$ ]{} - [ $\overline{\mathfrak{D}}_{+}\tilde{m}-\overline{\mathfrak{D}}_{-}\tilde{m}$ and $\mathfrak{D}_{+}\tilde{m}-\mathfrak{D}_{-}\tilde{m}$ are relatively small.[^10] ]{} Then, equations (\[eq:IntroConstraints\])–(\[eq:IntroMassEquation\]), combined with the conservation laws $$\begin{aligned} \partial_{u}\bar{\text{\textgreek{t}}} & =0,\\ \partial_{v}\text{\textgreek{t}} & =0,\end{aligned}$$ yield the following relations for the change in the mass difference associated to $\text{\textgreek{z}}_{in}$ and $\text{\textgreek{z}}_{out}$ after their intersection: $$\overline{\mathfrak{D}}_{+}\tilde{m}=\overline{\mathfrak{D}}_{-}\tilde{m}\cdot\exp\Big(\frac{2}{r}\frac{\mathfrak{D}_{-}\tilde{m}}{1-\frac{2m_{2}}{r}}+\mathfrak{Err}_{in}\Big)\label{eq:MassDifferenceIncreaseIngoing}$$ and $$\mathfrak{D}_{+}\tilde{m}=\mathfrak{D}_{-}\tilde{m}\cdot\exp\Big(-\frac{2}{r}\frac{\overline{\mathfrak{D}}_{-}\tilde{m}}{1-\frac{2m_{2}}{r}}+\mathfrak{Err}_{out}\Big),\label{eq:MassDecreaseOutgoing}$$ where the error terms $\mathfrak{Err}_{in},\mathfrak{Err}_{out}$ are negligible compared to the other terms in (\[eq:MassDifferenceIncreaseIngoing\]), (\[eq:MassDecreaseOutgoing\]) (see also the relations (\[eq:MassDifferenceIncreaseInUDirection\]) and (\[eq:MassDifferenceDecreaseInVDirection\]) in Section \[sub:Proof-of-Proposition\]). In particular, whenever an ingoing and an outgoing null dust beam intersect, the mass difference of the ingoing beam increases, while that of the outgoing beam decreases. Notice that, according to (\[eq:MassDifferenceIncreaseIngoing\]) and (\[eq:MassDecreaseOutgoing\]), the change in the mass difference of each of the beams $\text{\textgreek{z}}_{in},\text{\textgreek{z}}_{out}$ after their intersection can be estimated in terms of the mass difference of the other beam and the value of $r$ and $\inf(1-\frac{2m}{r})$ in the region of intersection. A relation for the change of the mass difference of two infinitely thin, intersecting null dust beams was also obtained in [@PoissonIsrael1990]. ### The instability mechanism {#the-instability-mechanism .unnumbered} Let us now consider, among the null dust beams arising from the initial data $(r,\text{\textgreek{W}}^{2},\text{\textgreek{t}},\bar{\text{\textgreek{t}}})^{(\text{\textgreek{e}})}\|_{u=0}$, the beam $\text{\textgreek{z}}_{0}$ which initially lies to the future of the rest (this is the beam marked with a red dashed line in Figure 1.3). Denoting $$\mathcal{E}_{\text{\textgreek{z}}_{0}}[t_{}]\doteq\mbox{ mass difference associated to }\text{\textgreek{z}}_{0}\mbox{ at }\text{\textgreek{z}}_{0}\cap\{u+v=t_{}\},\label{eq:MassDifferenceIntro}$$ we will examine how $\mathcal{E}_{\text{\textgreek{z}}_{0}}$ changes along $\text{\textgreek{z}}_{0}$, after each successive intersection of $\text{\textgreek{z}}_{0}$ with the rest of the beams: 1. Starting from $u=0$ up to the first reflection of $\text{\textgreek{z}}_{0}$ off the inner mirror $\text{\textgreek{g}}_{0}$, the beam $\text{\textgreek{z}}_{0}$ is ingoing and intersects all the other beams after they are reflected off $\text{\textgreek{g}}_{0}$. Thus, applying (\[eq:MassDifferenceIncreaseIngoing\]) successively at each intersection of $\text{\textgreek{z}}_{0}$ with an outgoing beam, we infer that $\mathcal{E}_{\text{\textgreek{z}}_{0}}$ at this step by a multiplicative factor $$A_{\text{\textgreek{g}}_{0}}\ge\exp\Bigg(\frac{2\big(\tilde{m}^{(\text{\textgreek{e}})}\|_{\mathcal{I}}-\mathcal{E}_{\text{\textgreek{z}}_{0}}\|_{u=0}\big)}{r_{\text{\textgreek{g}}_{0}}}(1-\text{\textgreek{e}})\Bigg),\label{eq:IncreaseFactor}$$ where $r_{\text{\textgreek{g}}_{0}}$ is the value of $r$ at the region of intersection of $\text{\textgreek{z}}_{0}$ with the first beam which is reflected off $\{r=r_{0\text{\textgreek{e}}}\}$ (note that $r_{\text{\textgreek{g}}_{0}}$ is also the $r$-width of the bundle of beams when $\text{\textgreek{z}}_{0}$ first reaches the mirror $\text{\textgreek{g}}_{0}$). In obtaining (\[eq:IncreaseFactor\]), we have assumed that $r_{0\text{\textgreek{e}}}\ll r_{\text{\textgreek{g}}_{0}}\ll(-\Lambda)^{-\frac{1}{2}}$, $\tilde{m}^{(\text{\textgreek{e}})}\|_{\mathcal{I}}\sim r_{0\text{\textgreek{e}}}$ and $\mathcal{E}_{\text{\textgreek{z}}_{0}}\|_{u=0}\ll\tilde{m}^{(\text{\textgreek{e}})}\|_{\mathcal{I}}$ (which holds in view of the way the initial data where chosen). 2. The mass difference $\mathcal{E}_{\text{\textgreek{z}}_{0}}$ right before and right after the reflection of $\text{\textgreek{z}}_{0}$ off $\text{\textgreek{g}}_{0}$ is the same, in view of the reflecting boundary conditions on $\text{\textgreek{g}}_{0}$. 3. From its first reflection off $\text{\textgreek{g}}_{0}$ up to its first reflection off $\mathcal{I}$, the beam $\text{\textgreek{z}}_{0}$ is outgoing and intersects (again) the rest of the beams in the region close to $\mathcal{I}$ (after these beams are reflected off $\mathcal{I}$). Applying (\[eq:MassDecreaseOutgoing\]) successively at each intersection, we infer that $\mathcal{E}_{\text{\textgreek{z}}_{0}}$ at this step, being multiplied by a factor $$1>A_{out}\ge\exp\Bigg(-\frac{2\big(\tilde{m}^{(\text{\textgreek{e}})}\|_{\mathcal{I}}-\mathcal{E}_{\text{\textgreek{z}}_{0}}\|_{u=0}\big)}{r_{\mathcal{I}}}\Big(\frac{1}{(1-\text{\textgreek{e}}-\frac{1}{3}\Lambda r_{\mathcal{I}}^{2})}+\text{\textgreek{e}}\Big)\Bigg),\label{eq:DecreaseFactor}$$ where $r_{\mathcal{I}}$ is the value of $r$ at the region of intersection of $\text{\textgreek{z}}_{0}$ with the first beam which is reflected off $\mathcal{I}$. In obtaining (\[eq:DecreaseFactor\]), we have asumed that $r_{\mathcal{I}}\gg(-\Lambda)^{-\frac{1}{2}}$ (which holds in view of the way the initial data where chosen). 4. The mass difference $\mathcal{E}_{\text{\textgreek{z}}_{0}}$ right before and right after the reflection of $\text{\textgreek{z}}_{0}$ off $\mathcal{I}$ is the same, in view of the reflecting boundary conditions on $\mathcal{I}$. Therefore, provided $r_{\text{\textgreek{g}}_{0}}\ll(-\Lambda)^{-\frac{1}{2}}\ll r_{\mathcal{I}}$, we infer that, after the first reflection of $\text{\textgreek{z}}_{0}$ off $\text{\textgreek{g}}_{0}$ and $\mathcal{I}$, the mass difference $\mathcal{E}_{\text{\textgreek{z}}_{0}}$ increases by a factor $$\begin{aligned} A_{tot}=A_{in}\cdot A_{out}\ge & \exp\Bigg(\frac{2\big(\tilde{m}^{(\text{\textgreek{e}})}\|_{\mathcal{I}}-\mathcal{E}_{\text{\textgreek{z}}_{0}}\|_{u=0}\big)}{r_{\text{\textgreek{g}}_{0}}}(1-\text{\textgreek{e}})-\frac{2\big(\tilde{m}^{(\text{\textgreek{e}})}\|_{\mathcal{I}}-\mathcal{E}_{\text{\textgreek{z}}_{0}}\|_{u=0}\big)}{r_{\mathcal{I}}}\Big(\frac{1}{(1-\text{\textgreek{e}}-\frac{1}{3}\Lambda r_{\mathcal{I}}^{2})}+\text{\textgreek{e}}\Big)\Bigg)\label{eq:TotalInceaseFactorIntro}\\ & \ge\exp\Big(\frac{\tilde{m}^{(\text{\textgreek{e}})}\|_{\mathcal{I}}-\mathcal{E}_{\text{\textgreek{z}}_{0}}\|_{u=0}}{r_{\text{\textgreek{g}}_{0}}}\Big).\nonumber \end{aligned}$$ The steps 1–4 in the above procedure can then be repeated for each successive reflection of $\text{\textgreek{z}}_{0}$ off $\text{\textgreek{g}}_{0}$ and $\mathcal{I}$, as long as $$r_{0\text{\textgreek{e}}}\ll r_{\text{\textgreek{g}}_{0};n}\ll(-\Lambda)^{-\frac{1}{2}}\ll r_{\mathcal{I};n},\label{eq:BoundForNearAndAwayRegionIntro}$$ where $r_{\text{\textgreek{g}}_{0};n},r_{\mathcal{I};n}$ are the values of $r_{\text{\textgreek{g}}_{0}},r_{\mathcal{I};n}$ after the $n$-th reflection of $\text{\textgreek{z}}_{0}$ on $\text{\textgreek{g}}_{0}$ and $\mathcal{I}$ (note that $r_{\text{\textgreek{g}}_{0};n}$ is also the $r$-width of the bundle of beams at the $n$-th reflection of $\text{\textgreek{z}}_{0}$ off $\text{\textgreek{g}}_{0}$). Thus, as long as (\[eq:BoundForNearAndAwayRegionIntro\]) holds, denoting with $\mathcal{E}_{\text{\textgreek{z}}_{0};n}$ the value of $\mathcal{E}_{\text{\textgreek{z}}_{0}}$ at the $n$-th reflection of $\text{\textgreek{z}}_{0}$ off $\text{\textgreek{g}}_{0}$, the following inductive bound holds: $$\mathcal{E}_{\text{\textgreek{z}}_{0};n}\ge A_{tot;n}\cdot\mathcal{E}_{\text{\textgreek{z}}_{0};n-1},\label{eq:InductiveEnIntro}$$ where the multiplicative factor $$A_{tot;n}\doteq\exp\Big(\frac{\tilde{m}^{(\text{\textgreek{e}})}\|_{\mathcal{I}}-\mathcal{E}_{\text{\textgreek{z}}_{0};n}}{r_{\text{\textgreek{g}}_{0};n}}\Big)\label{eq:MultiplicativeFactor}$$ is always greater than $1$, since $\mathcal{E}_{\text{\textgreek{z}}_{0};n}<\tilde{m}^{(\text{\textgreek{e}})}\|_{\mathcal{I}}$ (see also the relation (\[eq:BoundForMassIncrease\]) in Section \[sub:Inductive-bounds\]). This is the main mechanism driving the instability, and the proof of Theorem \[thm:TheoremDetailedIntro\] is aimed at showing that, for some large enough $n(\text{\textgreek{e}})$ depending on $\text{\textgreek{e}}$, $$\prod_{n=0}^{n(\text{\textgreek{e}})}A_{tot;n}>\frac{r_{0\text{\textgreek{e}}}}{2\mathcal{E}_{\text{\textgreek{z}}_{0}}\|_{u=0}}.\label{eq:BigEnoughFactor}$$ Inequality (\[eq:BigEnoughFactor\]) implies (in view of (\[eq:InductiveEnIntro\])) that $$\frac{2\mathcal{E}_{\text{\textgreek{z}}_{0};n(\text{\textgreek{e}})}}{r_{0\text{\textgreek{e}}}}>1,\label{eq:BoundForTrappedSurfaceFormationIntro}$$ i.e. that, after the $n(\text{\textgreek{e}})$’th successive reflection of $\text{\textgreek{z}}_{0}$ on $\text{\textgreek{g}}_{0}$ and $\mathcal{I}$, the mass difference $\mathcal{E}_{\text{\textgreek{z}}_{0}}$ has become so large that a trapped surface (in particular, a point where $\frac{2m}{r}>1$) necessarily forms before $\text{\textgreek{z}}_{0}$ reaches the mirror $\text{\textgreek{g}}_{0}=\{r=r_{0\text{\textgreek{e}}}\}$ for the $n(\text{\textgreek{e}})+1$-th time (provided $\text{\textgreek{z}}_{0}$ was initially chosen sufficiently “narrow”).[^11] ### Control of $r_{\text{\textgreek{g}}_{0};n}$ and the final step before trapped surface formation {#control-of-r_texttextgreekg_0n-and-the-final-step-before-trapped-surface-formation .unnumbered} The main obstacle to establishing (\[eq:BigEnoughFactor\]) (and, thus, Theorem \[thm:TheoremDetailedIntro\]) is the following: Once $\mathcal{E}_{\text{\textgreek{z}}_{0}}$ exceeds $c\cdot r_{0\text{\textgreek{e}}}$ for some fixed (small) $c>0$, the total $r$-width of the bundle of beams close to $\text{\textgreek{g}}_{0}$, i.e. $r_{\text{\textgreek{g}}_{0};n}$ in (\[eq:MultiplicativeFactor\]), increases after each successive reflection off $\text{\textgreek{g}}_{0}$ and $\mathcal{I}$. Thus, the multiplicative factor (\[eq:MultiplicativeFactor\]) decreases as $n$ grows. The increase in $r_{\text{\textgreek{g}}_{0};n}$ is more dramatic when the spacetime is close to having a trapped surface, i.e. when $\frac{2m}{r}$ is close to $1$.[^12] Controlling the growth of $r_{\text{\textgreek{g}}_{0};n}$ is achieved by establishing an inductive bound of the following form: $$r_{\text{\textgreek{g}}_{0};n}\le r_{\text{\textgreek{g}}_{0};n-1}\cdot\Big(1+C_{0}\frac{r_{0\text{\textgreek{e}}}}{r_{\text{\textgreek{g}}_{0};n-1}}\big(\Big\|\log\big(1-\frac{2\mathcal{E}_{\text{\textgreek{z}}_{0};n-1}}{r_{0\text{\textgreek{e}}}}\big)\Big\|+1\big)\Big)\label{eq:InductiveBoundRinIntro}$$ (see also the relation (\[eq:BoundForMaxBeamSeparation\]) in Section \[sub:Inductive-bounds\]). Obtaining the bound (\[eq:InductiveBoundRinIntro\]) is one of the most demanding parts in the proof of Theorem \[thm:TheoremDetailedIntro\] and requires controlling the $r$-distance $r_{\text{\textgreek{g}}_{0};n}^{(1)}$ of $\text{\textgreek{z}}_{0}$ from the second-to-top beam $\text{\textgreek{z}}_{1}$ at the $n$-th reflection off $\text{\textgreek{g}}_{0}$ for all $n\le n(\text{\textgreek{e}})$, i.e. establish a bound of the form $$\frac{r_{\text{\textgreek{g}}_{0};n}^{(1)}}{r_{0\text{\textgreek{e}}}}\ge1+c_{0}(\mathcal{E}_{\text{\textgreek{z}}_{0};0}/r_{0\text{\textgreek{e}}}).\label{eq:LowerBoundSecondBeamIntro}$$ (see (\[eq:BoundSecondBeamchanged\]) in Section \[sub:Inductive-bounds\]). The bound (\[eq:LowerBoundSecondBeamIntro\]) is in turn obtained by establishing an inductive bound of the form $$\log\Big(\frac{r_{\text{\textgreek{g}}_{0};n-1}^{(1)}}{r_{\text{\textgreek{g}}_{0};n}^{(1)}}\Big)\le C_{0}\log\Big(\frac{\mathcal{E}_{\text{\textgreek{z}}_{0};n}}{\mathcal{E}_{\text{\textgreek{z}}_{0};n-1}}\Big),\label{eq:InductiveBoundSecondBeam}$$ estimating the decrease of $r_{\text{\textgreek{g}}_{0};n}^{(1)}$ by the increase of $\mathcal{E}_{\text{\textgreek{z}}_{0};n}$ at each reflection (see (\[eq:UsefulBoundAlmostThere\]) in Section \[sub:Proof-of-Proposition\]). The bound (\[eq:LowerBoundSecondBeamIntro\]) is inferred from (\[eq:InductiveBoundSecondBeam\]), in view of the fact that $\mathcal{E}_{\text{\textgreek{z}}_{0};n}\ge\mathcal{E}_{\text{\textgreek{z}}_{0};n-1}$ and $$\sum_{n=1}^{n(\text{\textgreek{e}})}\log\Big(\frac{\mathcal{E}_{\text{\textgreek{z}}_{0};n}}{\mathcal{E}_{\text{\textgreek{z}}_{0};n-1}}\Big)=\log\Big(\frac{\mathcal{E}_{\text{\textgreek{z}}_{0};n(\text{\textgreek{e}})}}{\mathcal{E}_{\text{\textgreek{z}}_{0};0}}\Big)\le\log\Big(\frac{r_{0\text{\textgreek{e}}}}{2\mathcal{E}_{\text{\textgreek{z}}_{0};0}}\Big).$$ At the level of the initial data, obtaining (\[eq:InductiveBoundRinIntro\]) and (\[eq:InductiveBoundSecondBeam\]) requires introducing a certain hierarchy for the scales of the $r$-distances and mass differences associated to the beams initially (see (\[eq:h\_1\_h\_0\_definition\]) and (\[eq:h\_2definition\]) in Section \[sub:Parameters-and-auxiliary\]). Combining (\[eq:InductiveEnIntro\]) and (\[eq:InductiveBoundRinIntro\]), we can show that there exists a large $n(\text{\textgreek{e}})$ such that, after $n(\text{\textgreek{e}})$ reflections of $\text{\textgreek{z}}_{0}$ off $\text{\textgreek{g}}_{0}$ (but not earlier!), we have $$\frac{2\mathcal{E}_{\text{\textgreek{z}}_{0};n(\text{\textgreek{e}})}}{r_{0\text{\textgreek{e}}}}>1-c(\text{\textgreek{e}}),\label{eq:AlmostTrappdIntro}$$ where $c(\text{\textgreek{e}})\ll h(\text{\textgreek{e}})$ is a fixed function of $\text{\textgreek{e}}$. Note that, compared to (\[eq:BoundForTrappedSurfaceFormationIntro\]), (\[eq:AlmostTrappdIntro\]) is a slightly weaker bound, which just stops short of implying that a trapped surface is formed. In order to complete the proof of Theorem \[thm:TheoremDetailedIntro\], we therefore have to consider two different scenarios for $\mathcal{E}_{\text{\textgreek{z}}_{0};n(\text{\textgreek{e}})}$: In the case when (\[eq:BoundForTrappedSurfaceFormationIntro\]) holds, the proof of Theorem \[thm:TheoremDetailedIntro\] follows readily, since (\[eq:BoundForTrappedSurfaceFormationIntro\]) implies that, before $\text{\textgreek{z}}_{0}$ reaches $\{r=r_{0\text{\textgreek{e}}}\}$ for the $n(\text{\textgreek{e}})+1$-th time, a point arises where $\frac{2m}{r}>1$. In the case when (\[eq:AlmostTrappdIntro\]) holds but (\[eq:BoundForTrappedSurfaceFormationIntro\]) is violated, we can bound $$1-c(\text{\textgreek{e}})<\frac{2\mathcal{E}_{\text{\textgreek{z}}_{0};n(\text{\textgreek{e}})}}{r_{0\text{\textgreek{e}}}}\le1.\label{eq:SecondScenario}$$ In this case, $\text{\textgreek{z}}_{0}$ reaches $\{r=r_{0\text{\textgreek{e}}}\}$ for the $n(\text{\textgreek{e}})+1$-th time before a trapped surface has formed. One would be tempted to repeat the above procedure for one more reflection, in an attempt to establish that a trapped surface has formed before the $n(\text{\textgreek{e}})+2$-th reflection of $\text{\textgreek{z}}_{0}$ off $\text{\textgreek{g}}_{0}$. However, the bound (\[eq:SecondScenario\]) implies that most of the bootstrap assumptions needed for the proof of Theorem \[thm:TheoremDetailedIntro\] (which we have supressed in this sketch for the sake of simplicity) are violated beyond the $n(\text{\textgreek{e}})+1$-th reflection and, thus, the above procedure can not be repeated. For this reason, we choose a different path: Applying a Cauchy stability statement backwards in time (see Theorem \[prop:CauchyStability\]), we show that there exists a small perturbation $(r',(\text{\textgreek{W}}^{\prime})^{2},\text{\textgreek{t}}',\bar{\text{\textgreek{t}}}')^{(\text{\textgreek{e}})}\|_{u=0}$ of the initial data $(r,\text{\textgreek{W}}^{2},\text{\textgreek{t}},\bar{\text{\textgreek{t}}})^{(\text{\textgreek{e}})}\|_{u=0}$(satisfying (\[eq:InitialDataNorm\])), such that the perturbed solution $(r',(\text{\textgreek{W}}^{\prime})^{2},\text{\textgreek{t}}',\bar{\text{\textgreek{t}}}')^{(\text{\textgreek{e}})}$ to (\[eq:EinsteinNullDust\]) satisfies (\[eq:BoundForTrappedSurfaceFormationIntro\]) and, furthermore, $$\frac{2\mathcal{E}_{\text{\textgreek{z}}_{0};n(\text{\textgreek{e}})}^{\prime}}{r_{0\text{\textgreek{e}}}}>1$$ (where $\mathcal{E}{}_{\text{\textgreek{z}}_{0}}^{\prime}$ is similarly defined by the relation \[eq:MassDifferenceIntro\] for $(r',(\text{\textgreek{W}}^{\prime})^{2},\text{\textgreek{t}}',\bar{\text{\textgreek{t}}}')^{(\text{\textgreek{e}})}$ in place of $(r,\text{\textgreek{W}}^{2},\text{\textgreek{t}},\bar{\text{\textgreek{t}}})^{(\text{\textgreek{e}})}$). Thus, we end up in the scenario of Case 1, and the proof of Theorem \[thm:TheoremDetailedIntro\] follows readily. ### Further remarks on the proof of Theorem \[thm:TheoremDetailedIntro\] {#further-remarks-on-the-proof-of-theorem-thmtheoremdetailedintro .unnumbered} The proof of Theorem \[thm:TheoremDetailedIntro\] involves many technical issues related to the final step of the evolution before a trapped surface is formed. Most of these technical issues simplify considerably in the case when one restricts to showing a weaker instability statement for $(\mathcal{M}_{AdS},g_{AdS})$, e.g. by replacing (\[eq:TrappedSurfaceIntro\]) with $$\big(1-\frac{2m}{r}\big)^{(\text{\textgreek{e}})}\big\|_{(u_{\text{\textgreek{e}}},v_{\text{\textgreek{e}}})}<\frac{1}{2}.$$ See Sections \[sec:The-main-result:Details\] and \[sub:Remark-on-Proposition\] for more details. The mechanism leading to trapped surface formation in the proof of Theorem \[thm:TheoremDetailedIntro\] only made use of the fact that we chose the initial data $(r,\text{\textgreek{W}}^{2},\text{\textgreek{t}},\bar{\text{\textgreek{t}}})^{(\text{\textgreek{e}})}\|_{u=0}$ so that the matter was supported in narrow null beams, successively reflected off $\text{\textgreek{g}}_{0}$ and $\mathcal{I}$, while the matter model satisfied the condition $$T_{uv}=\text{\textgreek{W}}^{2}g^{AB}T_{AB}=0.$$ Thus, we expect that the same mechanism can be adapted to the case of more general matter fields, which allow for matter to be arranged into narrow and suficiently localised null beams, satisfying (in a region around the set of intersection of the beams) $$T_{uv},\|\text{\textgreek{W}}^{2}g^{AB}T_{AB}\|\ll T_{uu}+T_{vv},$$ with such a configuration arising moreover from inital data which are small in a norm for which $(\mathcal{M}_{AdS},g_{AdS})$ is Cauchy stable. For an application of this mechanism in the case of the spherically symmetric Einstein–massless Vlasov system (without reducing to the radial case and without an inner mirror), see our forthcoming [@MoschidisVlasov]. Finally, let us remark that the general mechanism of instability suggested by the proof of Theorem \[thm:TheoremDetailedIntro\] can be summarized as follows: In a configuration consisting of a relatively narrow bundle of nearly-null beams of matter that are successively reflected on $\mathcal{I}$ and $r=0$ (on an approximately $(\mathcal{M}_{AdS},g_{AdS})$ background), the energy content of the “top” beam will increase after each pair of reflections. A similar physical space mechanism was described for the Einstein–scalar field system (\[eq:EinsteinScalarFieldInDoubleNull\]) in [@DimitrakopoulosEtAl], where it was suggested that, on a nearly-null scalar field beam successively reflected off $\mathcal{I}$ and the center $r=0$, the energy density on the top part of the beam tends to increase. Outline of the paper -------------------- This paper is organised as follows: In Section \[sec:The-Einstein–Vlasov-system\], we will introduce the spherically symmetric Einstein–radial massless Vlasov system in double null coordinates. We will also formulate the notion of reflecting boundary conditions for this system on $\mathcal{I}$ and on timelike hypersurfaces of the form $\{r=r_{0}\}$. In Section \[sec:ResultsFromTheOtherPaper\], we will formulate the asymptotically AdS characteristic initial-boundary value problem for the spherically symmetric Einstein–radial massless Vlasov system. We will then recall the main results established in [@MoschidisMaximalDevelopment] regarding the structure of the maximal development and the Cauchy stability properties for this system. In Section \[sec:The-main-result:Details\], we will provide a technical statement of the main result of this paper, namely the instability of AdS for the Einstein–radial massless Vlasov system with reflecting boundary conditions on $\{r=r_{0}\}$ and $\mathcal{I}$. The proof of this result will occupy Sections \[sec:Preliminary-constructions\] and \[sec:Proof\]. Acknowledgements ---------------- I would like to thank my advisor Mihalis Dafermos for suggesting this problem to me, as well for numerous fruitful discussions and crucial suggestions. I would also like to thank Igor Rodnianski for many additional comments and suggestions. This work was completed while the author was a visitor at DPMMS and King’s College, Cambridge. \[sec:The-Einstein–Vlasov-system\]The Einstein–massless Vlasov system in spherical symmetry =========================================================================================== In this Section, we will review the basic properties of the spherically symmetric Einstein–massless Vlasov system in $3+1$ dimensions, expressed in double null coordinates, following the conventions introduced in [@DafermosRendall]. We will also introduce the notion of the reflecting boundary condition on timelike hypersurfaces for the radial massless Vlasov equation. To this end, we will follow the conventions adopted in our companion paper [@MoschidisMaximalDevelopment]. \[sub:Spherically-symmetric-spacetimes\]Spherically symmetric spacetimes in double null coordinates --------------------------------------------------------------------------------------------------- Let $(\mathcal{M}^{3+1},g)$ be a smooth Lorentzian manifold, such that $\mathcal{M}$ is of the form $$\mathcal{M}\simeq\mathcal{U}\times\mathbb{S}^{2}\label{eq:SphericallySymmetricmanifold}$$ where $\mathcal{U}$ is an open domain of $\mathbb{R}^{2}$ with piecewise Lipschitz boundary $\partial\mathcal{U}$ and, in the standard $(u,v)$ coordinates on $\mathcal{U}$, $g$ takes the form $$g=-\text{\textgreek{W}}^{2}(u,v)dudv+r^{2}(u,v)g_{\mathbb{S}^{2}},\label{eq:SphericallySymmetricMetric}$$ where $g_{\mathbb{S}^{2}}$ is the standard round metric on $\mathbb{S}^{2}$ and $\text{\textgreek{W}},r:\mathcal{U}\rightarrow(0,+\infty)$ are smooth functions. In addition, we will assume that $$\inf_{\mathcal{U}}r>0.$$ We will also fix a time orientation on $\mathcal{M}$ by requiring that the timelike vector field $N=\partial_{u}+\partial_{v}$ is future directed. Notice that the action of $SO(3)$ on $(\mathcal{M},g)$ through rotations of the $\mathbb{S}^{2}$ factor of (\[eq:SphericallySymmetricmanifold\]) is an isometric action. We will also define the Hawking mass $m:\mathcal{M}\rightarrow\mathbb{R}$ by the expression $$m=\frac{r}{2}\big(1-g(\nabla r,\nabla r)\big).$$ Viewed as a function on $\mathcal{U}$, $m$ takes the form: $$m=\frac{r}{2}\big(1+4\text{\textgreek{W}}^{-2}\partial_{u}r\partial_{v}r\big).\label{eq:DefinitionHawkingMass}$$ Equivalently, we have $$\text{\textgreek{W}}^{2}=4\frac{(-\partial_{u}r)\partial_{v}r}{1-\frac{2m}{r}}.\label{eq:RelationHawkingMass}$$ In any local coordinate chart $(y^{1},y^{2})$ on $\mathbb{S}^{2}$, the non-zero Christoffel symbols of (\[eq:SphericallySymmetricMetric\]) in the $(u,v,y^{1},y^{2})$ local coordinate chart on $\mathcal{M}$ are computed as follows: $$\begin{gathered} \text{\textgreek{G}}_{uu}^{u}=\partial_{u}\log(\text{\textgreek{W}}^{2}),\hphantom{A}\text{\textgreek{G}}_{vv}^{v}=\partial_{u}\log(\text{\textgreek{W}}^{2}),\label{eq:ChristoffelSymbols}\\ \text{\textgreek{G}}_{AB}^{u}=\text{\textgreek{W}}^{-2}\partial_{v}(r^{2})(g_{\mathbb{S}^{2}})_{AB},\hphantom{\,}\text{\textgreek{G}}_{AB}^{v}=\text{\textgreek{W}}^{-2}\partial_{u}(r^{2})(g_{\mathbb{S}^{2}})_{AB},\nonumber \\ \text{\textgreek{G}}_{uB}^{A}=r^{-1}\partial_{u}r\text{\textgreek{d}}_{B}^{A},\hphantom{\,}\text{\textgreek{G}}_{vB}^{A}=r^{-1}\partial_{v}r\text{\textgreek{d}}_{B}^{A},\nonumber \\ \text{\textgreek{G}}_{BC}^{A}=(\text{\textgreek{G}}_{\mathbb{S}^{2}})_{BC}^{A},\nonumber \end{gathered}$$ where the latin indices $A,B,C$ are associated to the spherical coordinates $y^{1},y^{2}$, $\text{\textgreek{d}}_{B}^{A}$ is Kronecker delta and $\text{\textgreek{G}}_{\mathbb{S}^{2}}$ are the Christoffel symbols of the round sphere in the $(y^{1},y^{2})$ coordinate chart. For any pair of smooth functions $f_{1},f_{2}:\mathbb{R}\rightarrow\mathbb{R}$ with $f_{1}^{\prime},f_{2}^{\prime}\neq0$, the coordinate transformation $$(\bar{u},\bar{v})=(f_{1}(u),f_{2}(v)),\label{eq:GeneralCoordinateTransformation}$$ mapping $\mathcal{U}$ to $\bar{\mathcal{U}}\subset\mathbb{R}^{2}$, can be used to diffeomorphically identify $\mathcal{M}$ with $\bar{\mathcal{U}}\times\mathbb{S}^{2}$. In these new coordinates, the metric $g$ takes the form $$g=-\bar{\text{\textgreek{W}}}^{2}(\bar{u},\bar{v})d\bar{u}d\bar{v}+r^{2}(\bar{u},\bar{v})g_{\mathbb{S}^{2}},\label{eq:SphericallySymmetricMetricNewGauge}$$ where $$\begin{gathered} \bar{\text{\textgreek{W}}}^{2}(\bar{u},\bar{v})=\frac{1}{f_{1}^{\prime}f_{2}^{\prime}}\text{\textgreek{W}}^{2}(f_{1}^{-1}(\bar{u}),f_{2}^{-1}(\bar{v})),\label{eq:NewOmega}\\ r(\bar{u},\bar{v})=r(f_{1}^{-1}(\bar{u}),f_{2}^{-1}(\bar{v})).\label{eq:NewR}\end{gathered}$$ We will frequently make use of such coordinate transformations, without renaming the coordinates each time. Note that $m$ is invariant under coordinate transformations of the form $(u,v)\rightarrow(f_{1}(u),f_{2}(v))$, i.e. $$m(\bar{u},\bar{v})=m(f_{1}^{-1}(\bar{u}),f_{2}^{-1}(\bar{v})).$$ \[sub:VlasovEquations\]The radial massless Vlasov equation ---------------------------------------------------------- Let $(\mathcal{M},g)$ be as in Section \[sub:Spherically-symmetric-spacetimes\]. Let $f\ge0$ be a measure on $T\mathcal{M}$ which is constant along the geodesic flow, that is to say, in any local coordinate chart $(x^{0},x^{1},x^{2},x^{3})$ on $\mathcal{M}$ with associated momentum coordinates $(p^{0},p^{1},p^{2},p^{3})$ on the fibers of $T\mathcal{M}$, $f$ satisfies (as a distribution) the first order equation $$p^{\text{\textgreek{a}}}\partial_{x^{\text{\textgreek{a}}}}f-\text{\textgreek{G}}_{\text{\textgreek{b}\textgreek{g}}}^{\text{\textgreek{a}}}p^{\text{\textgreek{b}}}p^{\text{\textgreek{g}}}\partial_{p^{\text{\textgreek{a}}}}f=0,\label{eq:VlasovEquation}$$ where $\text{\textgreek{G}}_{\text{\textgreek{b}\textgreek{g}}}^{\text{\textgreek{a}}}$ are the Christoffel symbols of $g$ in the chart $(x^{0},x^{1},x^{2},x^{3})$. We will call $f$ a massless Vlasov field if it is supported on the set $P\subset T\mathcal{M}$ of null vectors, i.e. on the set $$g_{\text{\textgreek{a}\textgreek{b}}}(x)p^{\text{\textgreek{a}}}p^{\text{\textgreek{b}}}=0.$$ Associated to $f$ is a symmetric $(0,2)$-form on $\mathcal{M}$ (possibly defined only in the sense of distributions), the energy momentum tensor of $f$, given by the expression $$T_{\text{\textgreek{a}\textgreek{b}}}(x)=\int_{\text{\textgreek{p}}^{-1}(x)}p_{\text{\textgreek{a}}}p_{\text{\textgreek{b}}}f,\label{eq:EnergyMomentumTensor}$$ where $\text{\textgreek{p}}^{-1}(x)$ denotes the fiber of $T\mathcal{M}$ over $x\in\mathcal{M}$ and the indices of the momentum coordinates are lowered with the use of the metric $g$, i.e. $$p_{\text{\textgreek{g}}}=g_{\text{\textgreek{g}}\text{\textgreek{d}}}(x)p^{\text{\textgreek{d}}}.$$ In this paper, we will only consider distributions $f$ for which the expression (\[eq:EnergyMomentumTensor\]) is finite for all $x\in\mathcal{M}$ and depends smoothly on $x\in\mathcal{M}$. We will consider only distributions $f$ which are spherically symmetric, i.e. invariant under the action of $SO(3)$ on $\mathcal{M}$. In that case, in any $(u,v,y^{1},y^{2})$ local coordinate chart as in Section \[sub:Spherically-symmetric-spacetimes\], the energy-momentum tensor $T$ is of the form $$T=T_{uu}(u,v)du^{2}+2T_{uv}(u,v)dudv+T_{vv}(u,v)dv^{2}+T_{AB}(u,v)dy^{A}dy^{B}.\label{eq:SphericallySymmetricTensor}$$ Furthermore, we will restrict to radial Vlasov fields $f$, i.e. fields supported only on radial null vectors which are normal to the orbits of the action of $SO(3)$ on $\mathcal{M}$. In any $(u,v,y^{1},y^{2})$ local coordinate chart as in Section \[sub:Spherically-symmetric-spacetimes\] (with associated momentum coordinates $(p^{u},p^{v},p^{1},p^{2})$), a spherically symmetric, radial massless Vlasov field $f$ has the form $$f(u,v,y^{1},y^{2};p^{u},p^{v},p^{1},p^{2})=\big(\bar{f}_{in}(u,v;p^{u})+\bar{f}_{out}(u,v;p^{v})\big)\text{\textgreek{d}}\big(\sqrt{(g_{\mathbb{S}^{2}})_{AB}p^{A}p^{B}}\big)\text{\textgreek{d}}(\text{\textgreek{W}}^{2}p^{u}p^{v}),\label{eq:RadialVlasovField}$$ where $\bar{f}_{in},\bar{f}_{out}\ge0$ and $\text{\textgreek{d}}$ is the Dirac delta funcion on $\mathbb{R}$. In this case, the only non-zero components of the energy momentum tensor (\[eq:EnergyMomentumTensor\]) are the $T_{uu}$ and $T_{vv}$ components. In particular, in terms of $\bar{f}_{in},\bar{f}_{out}$, we (formally) compute that $$\begin{gathered} T_{uu}(u,v)=\int_{0}^{+\infty}\text{\textgreek{W}}^{4}(p^{v})^{2}\bar{f}_{out}(u,v;p^{v})\, r^{2}\frac{dp^{v}}{p^{v}},\label{eq:T_uuComponent}\\ T_{vv}(u,v)=\int_{0}^{+\infty}\text{\textgreek{W}}^{4}(p^{u})^{2}\bar{f}_{in}(u,v;p^{u})\, r^{2}\frac{dp^{u}}{p^{u}}.\label{eq:T_vvComponent}\end{gathered}$$ In this paper, we will only consider the case when $\bar{f}_{in},\bar{f}_{out}$ are smooth and compactly supported in the $p^{u},p^{v}$ variables, respectively. In the case when $f$ is of the form (\[eq:RadialVlasovField\]), equation (\[eq:VlasovEquation\]) is equivalent to the following system for $\bar{f}_{in}$ and $\bar{f}_{out}$: $$\begin{gathered} \partial_{u}(\text{\textgreek{W}}^{4}r^{4}p^{u}\bar{f}_{in})+p^{u}\partial_{p^{u}}(\text{\textgreek{W}}^{4}r^{4}p^{u}\bar{f}_{in})=0,\label{eq:IngoingEquation}\\ \partial_{v}(\text{\textgreek{W}}^{4}r^{4}p^{v}\bar{f}_{out})+p^{v}\partial_{p^{v}}(\text{\textgreek{W}}^{4}r^{4}p^{v}\bar{f}_{out})=0.\label{eq:OutgoingEquation}\end{gathered}$$ The equations (\[eq:IngoingEquation\])–(\[eq:OutgoingEquation\]) readily yield the following transport equations for $T_{uu}$, $T_{vv}$: $$\begin{gathered} \partial_{v}(r^{2}T_{uu})=0,\label{eq:EquationT_uu}\\ \partial_{u}(r^{2}T_{vv})=0.\label{eq:EquationT_vv}\end{gathered}$$ Under a coordinate transformation of the form (\[eq:GeneralCoordinateTransformation\]), $\bar{f}_{in},\bar{f}_{out}$ transform as $$\bar{f}_{in}^{(new)}(f_{1}(u),f_{2}(v);f_{1}^{\prime}(u)p)=\bar{f}_{in}\big(u,v;p\big)\label{eq:NewIngoingVlasov}$$ and $$\bar{f}_{out}^{(new)}(f_{1}(u),f_{2}(v);f_{2}^{\prime}(v)p)=\bar{f}_{out}\big(u,v;p\big).\label{eq:NewOutgoingVlasov}$$ \[sub:The-Einstein-equations\]The spherically symmetric Einstein–radial massless Vlasov system ---------------------------------------------------------------------------------------------- Let $(\mathcal{M},g)$ be a smooth Lorentzian manifold and let $\text{\textgreek{L}}<0$. Let also $f$ be a non-negative measure on $T\mathcal{M}$. The Einstein–Vlasov system for $(\mathcal{M},g;f)$ with cosmological constant $\Lambda$ is $$\begin{cases} Ric_{\text{\textgreek{m}\textgreek{n}}}(g)-\frac{1}{2}R(g)g_{\text{\textgreek{m}\textgreek{n}}}+\text{\textgreek{L}}g_{\text{\textgreek{m}\textgreek{n}}}=8\text{\textgreek{p}}T_{\text{\textgreek{m}\textgreek{n}}},\\ p^{\text{\textgreek{a}}}\partial_{x^{\text{\textgreek{a}}}}f-\text{\textgreek{G}}_{\text{\textgreek{b}\textgreek{g}}}^{\text{\textgreek{a}}}p^{\text{\textgreek{b}}}p^{\text{\textgreek{g}}}\partial_{p^{\text{\textgreek{a}}}}f=0, \end{cases}\label{eq:EinsteinVlasovEquations}$$ where $T_{\text{\textgreek{m}\textgreek{n}}}$ is expressed in terms of $f$ by (\[eq:EnergyMomentumTensor\]). Restricting to the case where $(\mathcal{M},g)$ is a spherically symmetric spacetime as in Section \[sub:Spherically-symmetric-spacetimes\] and $f$ is a radial massless Vlasov field (i.e. has the form (\[eq:RadialVlasovField\])), the system (\[eq:EinsteinVlasovEquations\]) is equivalent to the following system for $(r,\text{\textgreek{W}}^{2},\bar{f}_{in},\bar{f}_{out})$: $$\begin{aligned} \partial_{u}\partial_{v}(r^{2})= & -\frac{1}{2}(1-\Lambda r^{2})\text{\textgreek{W}}^{2},\label{eq:RequationFinal}\\ \partial_{u}\partial_{v}\log(\text{\textgreek{W}}^{2})= & \frac{\text{\textgreek{W}}^{2}}{2r^{2}}\big(1+4\text{\textgreek{W}}^{-2}\partial_{u}r\partial_{v}r\big),\label{eq:OmegaEquationFinal}\\ \partial_{v}(\text{\textgreek{W}}^{-2}\partial_{v}r)= & -4\pi rT_{vv}\text{\textgreek{W}}^{-2},\label{eq:ConstrainVFinal}\\ \partial_{u}(\text{\textgreek{W}}^{-2}\partial_{u}r)= & -4\pi rT_{uu}\text{\textgreek{W}}^{-2},\label{eq:ConstraintUFinal}\\ \partial_{u}(\text{\textgreek{W}}^{4}r^{4}p^{u}\bar{f}_{in})= & -p^{u}\partial_{p^{u}}(\text{\textgreek{W}}^{4}r^{4}p^{u}\bar{f}_{in}),\label{eq:IngoingVlasovFinal}\\ \partial_{v}(\text{\textgreek{W}}^{4}r^{4}p^{v}\bar{f}_{out})= & -p^{v}\partial_{p^{v}}(\text{\textgreek{W}}^{4}r^{4}p^{v}\bar{f}_{out}),\label{eq:OutgoingVlasovFinal}\end{aligned}$$ where $T_{uu},T_{vv}$ are expressed in terms of $\bar{f}_{out},\bar{f}_{in}$ by (\[eq:T\_uuComponent\]), (\[eq:T\_vvComponent\]), respectively. Notice that the system (\[eq:RequationFinal\])–(\[eq:OutgoingVlasovFinal\]) reduces to the following system for $(r,\text{\textgreek{W}}^{2},T_{uu},T_{vv})$: $$\begin{aligned} \partial_{u}\partial_{v}(r^{2})= & -\frac{1}{2}(1-\Lambda r^{2})\text{\textgreek{W}}^{2},\label{eq:RequationFinal-2}\\ \partial_{u}\partial_{v}\log(\text{\textgreek{W}}^{2})= & \frac{\text{\textgreek{W}}^{2}}{2r^{2}}\big(1+4\text{\textgreek{W}}^{-2}\partial_{u}r\partial_{v}r\big),\label{eq:OmegaEquationFinal-2}\\ \partial_{v}(\text{\textgreek{W}}^{-2}\partial_{v}r)= & -4\pi rT_{vv}\text{\textgreek{W}}^{-2},\label{eq:ConstrainVFinal-1}\\ \partial_{u}(\text{\textgreek{W}}^{-2}\partial_{u}r)= & -4\pi rT_{uu}\text{\textgreek{W}}^{-2},\label{eq:ConstraintUFinal-1}\\ \partial_{u}(r^{2}T_{vv})= & 0,\label{eq:IngoingConservationClosed}\\ \partial_{v}(r^{2}T_{uu})= & 0.\label{eq:OutgoingConservationClosed}\end{aligned}$$ The system (\[eq:RequationFinal-2\])–(\[eq:OutgoingConservationClosed\]) is the Einstein–null dust system with both ingoing and outgoing dust (used as a model for self-gravitating radiation already in [@PoissonIsrael1990]). In the notation of Section \[sub:NeedOfAMirror\] of the introduction, $$r^{2}T_{vv}=\bar{\text{\textgreek{t}}}$$ and $$r^{2}T_{uu}=\text{\textgreek{t}}.$$ Defining the renormalised Hawking mass as $$\tilde{m}\doteq m-\frac{1}{6}\Lambda r^{3},\label{eq:RenormalisedHawkingMass}$$ and using the relation (\[eq:DefinitionHawkingMass\]), equations (\[eq:RequationFinal-2\])–(\[eq:OutgoingConservationClosed\]) formally give rise to the following system for $(r,\tilde{m},T_{uu},T_{vv})$ (valid in the region of $\mathcal{U}$ where $\partial_{v}r>0$, $\partial_{u}r<0$ and $1-\frac{2m}{r}>0$): $$\begin{aligned} \partial_{u}\log\big(\frac{\partial_{v}r}{1-\frac{2m}{r}}\big)= & -4\pi r^{-1}\frac{r^{2}T_{uu}}{-\partial_{u}r},\label{eq:DerivativeInUDirectionKappa}\\ \partial_{v}\log\big(\frac{-\partial_{u}r}{1-\frac{2m}{r}}\big)= & 4\pi r^{-1}\frac{r^{2}T_{vv}}{\partial_{v}r},\label{eq:DerivativeInVDirectionKappaBar}\\ \partial_{u}\partial_{v}r= & -\frac{2\tilde{m}-\frac{2}{3}\Lambda r^{3}}{r^{2}}\frac{(-\partial_{u}r)\partial_{v}r}{1-\frac{2m}{r}},\label{eq:EquationRForProof}\\ \partial_{u}\tilde{m}= & -2\pi\frac{\big(1-\frac{2m}{r}\big)}{-\partial_{u}r}r^{2}T_{uu},\label{eq:DerivativeTildeUMass}\\ \partial_{v}\tilde{m}= & 2\pi\frac{\big(1-\frac{2m}{r}\big)}{\partial_{v}r}r^{2}T_{vv}\label{eq:DerivativeTildeVMass}\\ \partial_{u}(r^{2}T_{vv})= & 0,\label{eq:ConservationT_vv}\\ \partial_{v}(r^{2}T_{uu})= & 0.\label{eq:ConservationT_uu}\end{aligned}$$ \[sub:Reflective-boundary-conditions\]The reflecting boundary condition for the Vlasov equation ----------------------------------------------------------------------------------------------- Let $(\mathcal{M},g)$ be as in Section \[sub:Spherically-symmetric-spacetimes\]. Recall that $\mathcal{M}$ splits topologically as the product $$\mathcal{M}\simeq\mathcal{U}\times\mathbb{S}^{2}.$$ Let $\partial_{tim}\mathcal{U}$ be the subset of the boundary $\partial\mathcal{U}$ of $\mathcal{U}\subset\mathbb{R}^{2}$ consisting of a union of connected, timelike Lipschitz curves with respect to the comparison metric $$g_{comp}=-dudv\label{eq:ComparisonUVMetric}$$ on $\mathbb{R}^{2}$. Recall that a connected Lipschitz curve $\text{\textgreek{g}}$ in $\mathbb{R}^{2}$ is said to be timelike with respect to (\[eq:ComparisonUVMetric\]) if, for every point $p=(u_{},v_{})\in\text{\textgreek{g}}$, we have $$\text{\textgreek{g}}\backslash p\subset I^{+}(p)\cup I^{-}(p)\doteq\big(\{u>u_{}\}\cap\{v>v_{}\}\big)\cup\big(\{u<u_{}\}\cap\{v<v_{}\}\big).$$ Let us fix $w:\mathcal{U}\cup\partial_{tim}\mathcal{U}\rightarrow\mathbb{R}$ to be a smooth boundary defining function of $\partial_{tim}\mathcal{U}$, i.e. $$w\|_{\partial_{tim}\mathcal{U}}=0,$$ $$dw\|_{\partial_{tim}\mathcal{U}}\neq0$$ and $$w\|_{\mathcal{U}}>0.$$ We can split $\partial_{tim}\mathcal{U}$ into its “left” and “right” components as $$\partial_{tim}\mathcal{U}=\partial_{tim}^{\vdash}\mathcal{U}\cup\partial_{tim}^{\dashv}\mathcal{U},$$ where $$\begin{gathered} \partial_{tim}^{\vdash}\mathcal{U}=\big\{(u_{0},v_{0})\in\partial_{tim}\mathcal{U}\,:\,\partial_{v}w(u_{0},v_{0})>0\big\},\\ \partial_{tim}^{\dashv}\mathcal{U}=\big\{(u_{0},v_{0})\in\partial_{tim}\mathcal{U}\,:\,\partial_{v}w(u_{0},v_{0})<0\big\}.\end{gathered}$$ Notice that any future directed radial null geodesic of $\mathcal{M}=\mathcal{U}\times\mathbb{S}^{2}$ with a future limiting point on $\partial_{tim}^{\vdash}\mathcal{U}\times\mathbb{S}^{2}$ (in the ambient $\mathbb{R}^{2}\times\mathbb{S}^{2}$ topology of $\bar{\mathcal{U}}\times\mathbb{S}^{2}$) is necessarily ingoing. Similarly, future directed radial null geodesics “terminating” at $\partial_{tim}^{\dashv}\mathcal{U}\times\mathbb{S}^{2}$ are necessarily outgoing. In the next sections, we will only consider the reflection of radial null geodesics on parts of $\partial_{tim}\mathcal{U}$ for which either $r-r_{0}$ (for some constant $r_{0}>0$) or $1/r$ is a boundary defining function. Following [@MoschidisMaximalDevelopment], we will define the reflecting boundary condition on $\partial_{tim}\mathcal{U}$ for the radial massless Vlasov equation as follows: A radial massless Vlasov field $f$ on $T\mathcal{M}$ will be said to satisfy the reflecting boundary condition on $\partial_{tim}\mathcal{U}\times\mathbb{S}^{2}$ if and only if - [For any $(u_{0},v_{0})\in\partial_{tim}^{\vdash}\mathcal{U}$ and any $p>0$: $$\lim_{h\rightarrow0^{+}}\Bigg(\frac{\bar{f}_{out}\big(u_{0},v_{0}+h;\,\frac{-\partial_{u}w}{\partial_{v}w}(u_{0},v_{0})\cdot\text{\textgreek{W}}^{-2}(u_{0},v_{0}+h)\cdot p\big)}{\bar{f}_{in}\big(u_{0}-h,v_{0};\,\text{\textgreek{W}}^{-2}(u_{0}-h,v_{0})\cdot p\big)}\Bigg)=1.\label{eq:LeftBoundaryCondition}$$ ]{} - [For any $(u_{1},v_{1})\in\partial_{tim}^{\dashv}\mathcal{U}$ and any $p>0$: $$\lim_{h\rightarrow0^{+}}\Bigg(\frac{\bar{f}_{in}\big(u_{1}+h,v_{1};\,\frac{-\partial_{v}w}{\partial_{u}w}(u_{1},v_{1})\cdot\text{\textgreek{W}}^{-2}(u_{1}+h,v_{1})\cdot p\big)}{\bar{f}_{out}\big(u_{1},v_{1}-h;\,\text{\textgreek{W}}^{-2}(u_{1},v_{1}-h)\cdot p\big)}\Bigg)=1.\label{eq:RightBoundaryCondition}$$ ]{} Note that the relations (\[eq:LeftBoundaryCondition\]) and (\[eq:RightBoundaryCondition\]) for $\bar{f}_{in},\bar{f}_{out}$ imply the following boundary relations for the components (\[eq:T\_uuComponent\])–(\[eq:T\_vvComponent\]) of the energy momentum tensor $T$: - For any $(u_{0},v_{0})\in\partial_{tim}^{\vdash}\mathcal{U}$: $$\lim_{h\rightarrow0^{+}}\frac{r^{2}T_{uu}(u_{0},v_{0}+h)}{r^{2}T_{vv}(u_{0}-h,v_{0})}=\Big(\frac{-\partial_{u}w}{\partial_{v}w}(u_{0},v_{0})\Big)^{2}.\label{eq:LeftBoundaryConditionT}$$ - For any $(u_{1},v_{1})\in\partial_{tim}^{\dashv}\mathcal{U}$: $$\lim_{h\rightarrow0^{+}}\frac{r^{2}T_{vv}(u_{1}+h,v_{1})}{r^{2}T_{uu}(u_{1},v_{1}-h)}=\Big(\frac{-\partial_{v}w}{\partial_{u}w}(u_{0},v_{0})\Big)^{2}.\label{eq:RightBoundaryConditionT}$$ \[sec:ResultsFromTheOtherPaper\]The boundary–characteristic initial value problem: well-posedness and Cauchy stability ======================================================================================================================= In this Section, we will formulate the asymptotically AdS initial value problem for the system (\[eq:RequationFinal\])–(\[eq:OutgoingVlasovFinal\]) with reflecting boundary conditions on $\{r=r_{0}\}$ and $\mathcal{I}$, for some $r_{0}>0$. We will then recall the main results established in [@MoschidisMaximalDevelopment], regarding the well-posedness and the structure of the maximal development for this system. Asymptotically AdS characteristic initial data ---------------------------------------------- The following definition was introduced in [@MoschidisMaximalDevelopment]: [3.1]{}\[Definition 3.1 in [@MoschidisMaximalDevelopment]\]\[def:TypeII\] For any $v_{1}<v_{2}$ and any $r_{0}>0$, let $r_{/}:[v_{1},v_{2})\rightarrow[r_{0},+\infty)$, $\text{\textgreek{W}}_{/}:[v_{1},v_{2})\rightarrow(0,+\infty)$ and $\bar{f}_{in/},\bar{f}_{out/}:[v_{1},v_{2})\times(0,+\infty)\rightarrow[0,+\infty)$ be $C^{\infty}$ functions, such that $$r_{/}(v_{1})=r_{0}$$ and $$\lim_{v\rightarrow v_{2}}r_{/}(v)=+\infty.\label{eq:RGoesToInfinity}$$ Let us define $(\partial_{u}r)_{/}:[v_{1},v_{2})\rightarrow(-\infty,0)$ by the relation $$(\partial_{u}r)_{/}(v)=\frac{1}{r_{/}(v)}\Big(-r_{/}\partial_{v}r_{/}(v_{1})-\frac{1}{4}\int_{v1}^{v}(1-\Lambda r_{/}^{2}(\bar{v}))\text{\textgreek{W}}_{/}^{2}(\bar{v})\, d\bar{v}\Big).\label{eq:TransversalDerivativeU-1}$$ We will call $(r_{/},\text{\textgreek{W}}_{/}^{2},\bar{f}_{in/},\bar{f}_{out/})$ an asymptotically AdS boundary-characteristic initial data set on $[v_{1},v_{2})$ for the system (\[eq:RequationFinal\])–(\[eq:OutgoingVlasovFinal\]) satisfying the reflecting gauge condition at $r=r_{0},+\infty$ if: - [ $(r_{/},\text{\textgreek{W}}_{/})$ satisfies the constraint equation $$\partial_{v}(\text{\textgreek{W}}_{/}^{-2}\partial_{v}r_{/})=-4\pi r_{/}(T_{vv})_{/}\text{\textgreek{W}}_{/}^{-2},\label{eq:ConstraintVDef}$$ where $$(T_{vv})_{/}(v)\doteq\int_{0}^{+\infty}\text{\textgreek{W}}_{/}^{4}(v)(p^{u})^{2}\bar{f}_{in/}(v;p^{u})\, r_{/}^{2}(v)\frac{dp^{u}}{p^{u}}.\label{eq:EnergyMomentumIntialRight}$$ ]{} - [ $\bar{f}_{out/}$ solves the massless radial Vlasov equation $$\begin{gathered} \partial_{v}\big(\text{\textgreek{W}}_{/}^{4}(v)r_{/}^{4}(v)p^{v}\bar{f}_{out/}(v,p^{v})\big)+p^{v}\partial_{p^{v}}\big(\text{\textgreek{W}}_{/}^{4}(v)r_{/}^{4}(v)p^{v}\bar{f}_{out/}(v,p^{v})\big)=0.\label{eq:OutgoingEquationCombatibility}\end{gathered}$$ ]{} - [ $(\partial_{u}r)_{/}$ satisfies $$\lim_{v\rightarrow v_{2}^{-}}\frac{(\partial_{u}r)_{/}}{\partial_{v}r_{/}}=1.\label{eq:GaugeInfinityInitialData}$$ ]{} - [ $\bar{f}_{out/},\bar{f}_{in/}$ satisfy the following compatibility conditions at $v=v_{1},v_{2}$ for any $p>0$: $$\frac{\bar{f}_{out/}\big(v_{1};\,\frac{-(\partial_{u}r)_{/}}{\partial_{v}r_{/}}(v_{1})\cdot\text{\textgreek{W}}_{/}^{-2}(v_{1})\cdot p\big)}{\bar{f}_{in/}\big(v_{1};\,\text{\textgreek{W}}_{/}^{-2}(v_{1})\cdot p\big)}=1\label{eq:LeftBoundaryConditionInitialData}$$ and $$\lim_{h\rightarrow0^{+}}\Bigg(\frac{\bar{f}_{in/}\big(v_{2}-h;\,\frac{\partial_{v}r_{/}}{-(\partial_{u}r)_{/}}(v_{2}-h)\cdot\text{\textgreek{W}}_{/}^{-2}(v_{2}-h)\cdot p\big)}{\bar{f}_{out/}\big(v_{2}-h;\,\text{\textgreek{W}}_{/}^{-2}(v_{2}-h)\cdot p\big)}\Bigg)=1.\label{eq:RightBoundaryConditionInitialData}$$ ]{} Notice that the constraint equation (\[eq:ConstraintVDef\]) implies $$\partial_{v}(\text{\textgreek{W}}_{/}^{-2}\partial_{v}r_{/})\le0.$$ Thus, (\[eq:RGoesToInfinity\]) yields $$\partial_{v}r_{/}>0\label{eq:NonTrappedInitialData}$$ everywhere on $[v_{1},v_{2})$. Given any asymptotically AdS boundary-characteristic initial data set $(r_{/},\text{\textgreek{W}}_{/}^{2},\bar{f}_{in/},\bar{f}_{out/})$ on $[v_{1},v_{2})$ with reflecting gauge conditions at $r=r_{0},+\infty$, we will also define the initial Hawking mass $m_{/}$ and initial renormalised Hawking mass $\tilde{m}_{/}$ on $[v_{1}.v_{2})$ by the relations $$m_{/}\doteq\frac{r_{/}}{2}\big(1-4\text{\textgreek{W}}_{/}^{-2}(\partial_{u}r)_{/}\partial_{v}r_{/}\big),\label{eq:DefinitionHawkingMassCharacteristic}$$ and $$\tilde{m}_{/}\doteq m_{/}-\frac{1}{6}\Lambda r_{/}^{3},$$ in accordance with (\[eq:RelationHawkingMass\]), (\[eq:RenormalisedHawkingMass\]). Developments with reflecting boundary conditions on $r=r_{0},+\infty$ --------------------------------------------------------------------- We will only consider solutions $(r,\text{\textgreek{W}}^{2},\bar{f}_{in},\bar{f}_{out})$ to (\[eq:RequationFinal\])–(\[eq:OutgoingVlasovFinal\]) satisfying a reflecting gauge condition on $\partial_{tim}\mathcal{U}$, which fixes $\partial_{tim}\mathcal{U}$ to be a union of vertical straight lines in the $(u,v)$-plane. This motivates defining the following class of domains $\mathcal{U}$ in the plane (see [@MoschidisMaximalDevelopment]): [3.2]{}\[Definition 3.3 in [@MoschidisMaximalDevelopment]\]\[def:DevelopmentSets\] For any $v_{0}>0$, let $\mathscr{U}_{v_{0}}$ be the set of all connected open domains $\mathcal{U}$ of the $(u,v)$-plane with piecewise Lipschitz boundary $\partial\mathcal{U}$, with the property that $\partial\mathcal{U}$ splits as the union $$\partial\mathcal{U}=\text{\textgreek{g}}_{0}\cup\mathcal{I}\cup\mathcal{S}_{v_{0}}\cup clos(\text{\textgreek{g}}),\label{eq:BoundaryOfU}$$ where, for some $0<u_{\text{\textgreek{g}}_{0}},u_{\mathcal{I}}\le+\infty$, $$\text{\textgreek{g}}_{0}=\{u=v\}\cap\{0\le u<u_{\text{\textgreek{g}}_{0}}\},\label{eq:AxisForm}$$ $$\mathcal{I}=\{u=v-v_{0}\}\cap\{0\le u<u_{\mathcal{I}}\},\label{eq:InfinityForm}$$ $$\mathcal{S}_{v_{0}}=\{0\}\times[0,v_{0}],$$ and $\text{\textgreek{g}}:(x_{1},x_{2})\rightarrow\mathbb{R}^{2}$ is a Lipschitz, achronal (with respect to the reference Lorentzian metric (\[eq:ComparisonUVMetric\])) curve, which is allowed to be empty (the closure $clos(\text{\textgreek{g}})$ of $\text{\textgreek{g}}$ in (\[eq:BoundaryOfU\]) is considered with respect to the standard topology of $\mathbb{R}^{2}$). It follows readily from Definition \[def:DevelopmentSets\] that $\mathcal{U}$ is necessarily contained in the future domain of dependence of $\mathcal{S}_{v_{0}}\cup\text{\textgreek{g}}_{0}\cup\mathcal{I}$ (with respect to the comparison metric (\[eq:ComparisonUVMetric\])). In the case when $\text{\textgreek{g}}=\emptyset$ in (\[eq:BoundaryOfU\]), it is necessary that both $\text{\textgreek{g}}_{0}$ and $\mathcal{I}$ extend all the way to $u+v=+\infty$. A development of an asymptotically AdS boundary-characteristic initial data set for the system (\[eq:RequationFinal\])–(\[eq:OutgoingVlasovFinal\]) with reflecting boundary conditions on $r=r_{0},+\infty$ is defined as follows (see [@MoschidisMaximalDevelopment]): [3.3]{}\[Definition 3.4 in [@MoschidisMaximalDevelopment]\]\[def:Development\] For any $v_{0}>0$ and $r_{0}>0$, let $(r_{/},\text{\textgreek{W}}_{/}^{2},\bar{f}_{in/},\bar{f}_{out/})$ be a smooth asymptotically AdS boundary-characteristic initial data set on $[0,v_{0})$ for the system (\[eq:RequationFinal\])–(\[eq:OutgoingVlasovFinal\]) satisfying the reflecting gauge condition at $r=r_{0},+\infty$, according to Definition \[def:TypeII\]. A of $(r_{/},\text{\textgreek{W}}_{/}^{2},\bar{f}_{in/},\bar{f}_{out/})$ will consist of an open set $\mathcal{U}\in\mathscr{U}_{v_{0}}$ (see Definition \[def:DevelopmentSets\]) and smooth functions $r:\mathcal{U}\rightarrow(r_{0},+\infty)$, $\text{\textgreek{W}}^{2}:\mathcal{U}\rightarrow(0,+\infty)$ and $\bar{f}_{in},\bar{f}_{out}:\mathcal{U}\times(0,+\infty)\rightarrow[0,+\infty)$ satisfying the following properties: 1. The functions $r,\text{\textgreek{W}}^{2},\bar{f}_{in},\bar{f}_{out}$ solve the system (\[eq:RequationFinal\])–(\[eq:OutgoingVlasovFinal\]) on $\mathcal{U}$. 2. The functions $r,\text{\textgreek{W}}^{2},\bar{f}_{in},\bar{f}_{out}$ satisfy the given initial conditions on $\mathcal{S}_{v_{0}}=\{0\}\times[0,v_{0})$, i.e.: $$(r,\text{\textgreek{W}}^{2},\bar{f}_{in},\bar{f}_{out})\|_{\mathcal{S}_{v_{0}}}=(r_{/},\text{\textgreek{W}}_{/}^{2},\bar{f}_{in/},\bar{f}_{out/}).\label{eq:InitialDataRightMaximal}$$ 3. The functions $(r,\bar{f}_{in},\bar{f}_{out})$ satisfy on $\text{\textgreek{g}}_{0}$ the boundary conditions $$r\|_{\text{\textgreek{g}}_{0}}=r_{0}\label{eq:MirrorRMaximal}$$ and $$\bar{f}_{out}\big(u_{},v_{};\, p\big)=\bar{f}_{in}\big(u_{},v_{};\, p\big),\label{eq:ReflectionMirrorMaximal}$$ for all $(u_{},v_{})\in\text{\textgreek{g}}_{0}$ and $p>0$, and on $\mathcal{I}$ the boundary conditions $$(1/r)\|_{\mathcal{I}}=0\label{eq:InfinityRMaximal}$$ and $$\lim_{h\rightarrow0^{+}}\Bigg(\frac{\bar{f}_{in}\big(u_{}+h,v_{};\,\text{\textgreek{W}}^{-2}(u_{}+h,v_{})\cdot p\big)}{\bar{f}_{out}\big(u_{},v_{}-h;\,\text{\textgreek{W}}^{-2}(u_{},v_{}-h)\cdot p\big)}\Bigg)=1,\label{eq:ReflectionInfinityMaximal}$$ for all $(u_{},v_{})\in\mathcal{I}$ and $p>0$. 4. The following are satisfied on $\text{\textgreek{g}}_{0}$ and $\mathcal{I}$: $$\partial_{u}r\|_{\text{\textgreek{g}}_{0}}=-\partial_{v}r\|_{\text{\textgreek{g}}_{0}}\label{eq:GaugeMirrorMaximal}$$ and $$\partial_{u}(1/r)\|_{\mathcal{I}}=-\partial_{v}(1/r)\|_{\mathcal{I}}.\label{eq:GaugeInfinityMaximal}$$ Notice that the boundary conditions (\[eq:MirrorRMaximal\]) and (\[eq:InfinityRMaximal\]), combined with the form (\[eq:AxisForm\]) and (\[eq:InfinityForm\]) of $\text{\textgreek{g}}_{0}$ and $\mathcal{I}$, respectively, imply the relations (\[eq:GaugeMirrorMaximal\]) and (\[eq:GaugeInfinityMaximal\]). However, the relations (\[eq:GaugeMirrorMaximal\]) and (\[eq:GaugeInfinityMaximal\]) should be viewed as gauge conditions fixing, in conjuction with (\[eq:MirrorRMaximal\]) and (\[eq:InfinityRMaximal\]), the form (\[eq:AxisForm\]) and (\[eq:InfinityForm\]) of $\text{\textgreek{g}}_{0}$ and $\mathcal{I}$. If $\mathscr{D}=(\mathcal{U};r,\text{\textgreek{W}}^{2},\bar{f}_{in},\bar{f}_{out})$ and $\mathscr{D}^{\prime}=(\mathcal{U}^{\prime};r^{\prime},(\text{\textgreek{W}}^{\prime})^{2},\bar{f}_{in}^{\prime},\bar{f}_{out}^{\prime})$ are two future developments of the same initial data $(r_{/},\text{\textgreek{W}}_{/}^{2},\bar{f}_{in/},\bar{f}_{out/})$, we will say that $\mathscr{D}^{\prime}$ is an extension of $\mathscr{D}$, writing $\mathscr{D}\subseteq\mathscr{D}^{\prime}$, if $\mathcal{U}\subseteq\mathcal{U}^{\prime}$ and the restriction of $(r^{\prime},(\text{\textgreek{W}}^{\prime})^{2},\bar{f}_{in}^{\prime},\bar{f}_{out}^{\prime})$ on $\mathcal{U}$ coincides with $(r,\text{\textgreek{W}}^{2},\bar{f}_{in},\bar{f}_{out})$. If $\mathscr{D}=(\mathcal{U};r,\text{\textgreek{W}}^{2},\bar{f}_{in},\bar{f}_{out})$ and $\mathscr{D}^{\prime}=(\mathcal{U}^{\prime};r^{\prime},(\text{\textgreek{W}}^{\prime})^{2},\bar{f}_{in}^{\prime},\bar{f}_{out}^{\prime})$ are two future developments of the same initial data $(r_{/},\text{\textgreek{W}}_{/}^{2},\bar{f}_{in/},\bar{f}_{out/})$, then $$(r,\text{\textgreek{W}}^{2},\bar{f}_{in},\bar{f}_{out})\|_{\mathcal{U}\cap\mathcal{U}^{\prime}}=(r^{\prime},(\text{\textgreek{W}}^{\prime})^{2},\bar{f}_{in}^{\prime},\bar{f}_{out}^{\prime})\|_{\mathcal{U}\cap\mathcal{U}^{\prime}}$$ (see [@MoschidisMaximalDevelopment]). The maximal development ----------------------- The following result was established in [@MoschidisMaximalDevelopment]: [3.1]{}\[Theorem 1 in [@MoschidisMaximalDevelopment]\]\[thm:maximalExtension\] For any $v_{0}>0$ and $r_{0}>0$, let $(r_{/},\text{\textgreek{W}}_{/}^{2},\bar{f}_{in/},\bar{f}_{out/})$ be a smooth asymptotically AdS boundary-characteristic initial data set on $[0,v_{0})$ for the system (\[eq:RequationFinal\])–(\[eq:OutgoingVlasovFinal\]) satisfying the reflecting gauge condition at $r=r_{0},+\infty$, according to Definition \[def:TypeII\], such that the quantities $\frac{\text{\textgreek{W}}_{/}^{2}}{1-\frac{1}{3}\Lambda r_{/}^{2}},r_{/}^{2}(T_{vv})_{/}$ and $\tan^{-1}r_{/}$ extend smoothly on $v=v_{0}$. Then, there exists a unique, smooth future development $(\mathcal{U};r,\text{\textgreek{W}}^{2},\bar{f}_{in},\bar{f}_{out})$ of $(r_{/},\text{\textgreek{W}}_{/}^{2},\bar{f}_{in/},\bar{f}_{out/})$ which is , i.e. any other future development $(\mathcal{U}^{\prime};r^{\prime},(\text{\textgreek{W}}^{\prime})^{2},\bar{f}_{in}^{\prime},\bar{f}_{out}^{\prime})$ of $(r_{/},\text{\textgreek{W}}_{/}^{2},\bar{f}_{in/},\bar{f}_{out/})$ with $r^{\prime}\ge r_{0}$ everywhere on $\mathcal{U}^{\prime}$satisfies $\mathcal{U}^{\prime}\subseteq\mathcal{U}$ and $r^{\prime},(\text{\textgreek{W}}^{\prime})^{2},\bar{f}_{in}^{\prime},\bar{f}_{out}^{\prime}$ are the restrictions of $r,\text{\textgreek{W}}^{2},\bar{f}_{in},\bar{f}_{out}$ on $\mathcal{U}^{\prime}$. The maximal future development $(\mathcal{U};r,\text{\textgreek{W}}^{2},\bar{f}_{in},\bar{f}_{out})$ satisfies the following properties (for the definition of the curves $\text{\textgreek{g}}_{0},\mathcal{I},\text{\textgreek{g}}$, see Definition \[def:DevelopmentSets\]): 1. The renormalised Hawking mass $\tilde{m}$ is conserved on $\text{\textgreek{g}}_{0}$ and $\mathcal{I}$, i.e.: $$\tilde{m}\|_{\text{\textgreek{g}}_{0}}=\tilde{m}\|_{\text{\textgreek{g}}_{0}\cap\{u=0\}}\label{eq:ConstantMassMirror}$$ and $$\tilde{m}\|_{\mathcal{I}}=\tilde{m}\|_{\mathcal{I}\cap\{u=0\}}.\label{eq:ConstantMassInfinity}$$ 2. The curve $\mathcal{I}$ is conformally complete, i.e. $\text{\textgreek{W}}^{2}/(1-\frac{1}{3}\Lambda r^{2})$ has a finite limit on $\mathcal{I}$ and: $$\int_{\mathcal{I}}\sqrt{\frac{\text{\textgreek{W}}^{2}}{1-\frac{1}{3}\Lambda r^{2}}}\Big\|_{\mathcal{I}}\, du=+\infty.\label{eq:CompleConformalInfinity}$$ 3. We have $$\partial_{u}r<0,\label{eq:NegativeDerivativeRMaximal}$$ $$\big(1-\frac{2m}{r}\big)\big\|_{J^{-}(\mathcal{I})\cup J^{-}(\text{\textgreek{g}}_{0})}>0\label{eq:NonTrappigMaximal}$$ and $$\partial_{v}r\|_{J^{-}(\mathcal{I})\cup J^{-}(\text{\textgreek{g}}_{0})}>0,\label{eq:AlternativeNonTrappingMaximal}$$ where $$J^{-}(\mathcal{I})=\big\{0\le u<\sup_{\mathcal{I}}u\big\}\cap\mathcal{U}\label{eq:PastOfInfinity}$$ is the causal past of $\mathcal{I}$ and $$J^{-}(\text{\textgreek{g}}_{0})=\big\{0\le v<\sup_{\text{\textgreek{g}}_{0}}v\big\}\cap\mathcal{U}$$ is the causal past of $\text{\textgreek{g}}_{0}$ (with respect to the reference Lorenztian metric (\[eq:ComparisonUVMetric\])). 4. In the case $\mathcal{U}\backslash J^{-}(\mathcal{I})\neq\emptyset$, the future event horizon $$\mathcal{H}^{+}\doteq\mathcal{U}\cap\partial J^{-}(\mathcal{I})=\big\{ u=\sup_{\mathcal{I}}u\big\}\cap\mathcal{U}\label{eq:DefinitionHorizon}$$ has the following properties: 1. $\mathcal{H}^{+}$ has infinite affine length, i.e.: $$\int_{\mathcal{H}^{+}}\text{\textgreek{W}}^{2}\, dv=+\infty.\label{eq:InfiniteLengthHorizon}$$ 2. We have $$\sup_{\mathcal{H}^{+}}r=r_{S}\label{eq:UpperBoundRHorizon}$$ and $$\inf_{\mathcal{H}^{+}}\Big(1-\frac{2m}{r}\Big)=0,\label{eq:TrappingAsymptoticallyHorizon}$$ where $r_{S}$ defined by the relation $$1-2\frac{\lim_{v\rightarrow v_{0}^{-}}\tilde{m}_{/}(v)}{r_{S}}-\frac{1}{3}\Lambda r_{S}^{2}=0.\label{eq:DefinitionRs}$$ 5. In the case $\mathcal{H}^{+}\neq\emptyset$, the curve $\text{\textgreek{g}}_{0}$ is bounded and satisfies $$\text{\textgreek{g}}_{0}\nsubseteq J^{-}(\mathcal{I}),\label{eq:MirrorExtendsBeyondHorizon}$$ i.e. $\text{\textgreek{g}}_{0}$ contains points lying to the future of $\mathcal{H}^{+}$. 6. In the case $\mathcal{H}^{+}\neq\emptyset$, the curve $\text{\textgreek{g}}$ is non-empty, piecewise smooth and $r$ extends continuously on $\text{\textgreek{g}}$ with $r\|_{\text{\textgreek{g}}_{0}}=r_{0}$. Furthermore, for any point $(u_{1},v_{1})\in\text{\textgreek{g}}$, the line $\{v=v_{1}\}$ intersects $\mathcal{I}$.[^13] In the case when $\mathcal{U}\backslash J^{-}(\mathcal{I})\neq\emptyset$ (and thus $\mathcal{H}^{+}\neq\emptyset$), in view of (\[eq:UpperBoundRHorizon\]), (\[eq:DefinitionRs\]) and the fact that $r>r_{0}$ on $\mathcal{U}$, it is necessary that $$2\frac{\lim_{v\rightarrow v_{0}^{-}}\tilde{m}_{/}(v)}{r_{0}}>1-\frac{1}{3}\Lambda r_{0}^{2}.\label{eq:LowerBoundMass}$$ In a similar way, we can uniquely define the maximal past development $(\mathcal{U};r,\text{\textgreek{W}}^{2},\bar{f}_{in},\bar{f}_{out})$ of $(r_{/},\text{\textgreek{W}}_{/}^{2},\bar{f}_{in/},\bar{f}_{out/})$, satisfying the properties outlined by Theorem \[thm:maximalExtension\] after performing a “time reversal” transformation $(u,v)\rightarrow(-v,-u)$. Notice that such a coordinate transformation turns an asymptotically AdS boundary-characteristic initial data set on $u=0$ into an asymptotically AdS boundary-characteristic initial data set on $v=0$. However, Theorem \[thm:maximalExtension\] also holds (with exactly the same proof) for such initial data sets. \[sub:Cauchy-stability-inCauchyStability\]Cauchy stability in a rough norm, uniformly in $r_{0}$ ------------------------------------------------------------------------------------------------ In [@MoschidisMaximalDevelopment], the following “norm” was introduced for smooth asymptotically AdS boundary-characteristic initial data sets $(r_{/},\text{\textgreek{W}}_{/}^{2},\bar{f}_{in/},\bar{f}_{out/})$ on $[0,v_{0})$ for the system (\[eq:RequationFinal\])–(\[eq:OutgoingVlasovFinal\]): $$\begin{aligned} \|\|(r_{/},\text{\textgreek{W}}_{/}^{2},\bar{f}_{in/},\bar{f}_{out/})\|\|_{\mathcal{C\mathcal{S}}}\doteq\sqrt{-\Lambda} & \sup_{0\le v<v_{0}}\|\tilde{m}_{/}(v)\|+(-\Lambda)\sup_{0\le v<v_{0}}\int_{0}^{v_{0}}\frac{1}{\text{\textgreek{r}}_{/}(v)-\text{\textgreek{r}}_{/}(\bar{v})+\text{\textgreek{r}}_{/}(0)}\Big(\frac{r_{/}^{2}(T_{vv})_{/}}{\partial_{v}\text{\textgreek{r}}_{/}}\Big)(\bar{v})\, d\bar{v}+\label{eq:GeometricNormForCauchyStability}\\ & +\sup_{0\le v<v_{0}}\max\big\{\frac{2m_{/}}{r_{/}},0\big\},\nonumber \end{aligned}$$ where $$\text{\textgreek{r}}_{/}(v)\doteq\tan^{-1}\big(\sqrt{-\Lambda}r_{/}(v)\big).$$ Note that, in (\[eq:GeometricNormForCauchyStability\]), $$\text{\textgreek{r}}_{/}(0)=\tan^{-1}\big(\sqrt{-\Lambda}r_{0}\big).$$ The expression (\[eq:GeometricNormForCauchyStability\]) is invariant under gauge transformations, as well as scale transformations of the form $(u,v)\rightarrow(\text{\textgreek{l}}u,\text{\textgreek{l}}v)$, $(r,\tilde{m},\Lambda)\rightarrow(\text{\textgreek{l}}r,\text{\textgreek{l}}\tilde{m},\text{\textgreek{l}}^{-2}\Lambda)$, $r_{0}\rightarrow\text{\textgreek{l}}r_{0}$, $(\bar{f}_{in},\bar{f}_{out})\rightarrow(\text{\textgreek{l}}^{-4}\bar{f}_{in},\text{\textgreek{l}}^{-4}\bar{f}_{out})$. Moreover, $\|\|(r_{/},\text{\textgreek{W}}_{/}^{2},\bar{f}_{in/},\bar{f}_{out/})\|\|_{\mathcal{C\mathcal{S}}}=0$ if and only if $(r_{/},\text{\textgreek{W}}_{/}^{2},\bar{f}_{in/},\bar{f}_{out/})$ are the initial data for pure AdS spacetime on $\{r\ge r_{0}\}$, i.e. if $\bar{f}_{in/}=\bar{f}_{out/}=0$ and $\tilde{m}=0$. The following Cauchy stability result for the trivial initial data was established in [@MoschidisMaximalDevelopment]: [3.1]{}\[Corollary 1 in [@MoschidisMaximalDevelopment]\]\[prop:CauchyStabilityOfAdS\] For any (possibly large) $l_{}>0$, there exists a (small) $\text{\textgreek{e}}_{0}>0$ and a constant $C_{l_{}}>0$ depending only on $l_{}$, so that the following statement holds: For any $v_{0}>0$ and $0<r_{0}<(-\Lambda)^{-\frac{1}{2}}$, if $(r_{/},\text{\textgreek{W}}_{/}^{2},\bar{f}_{in/},\bar{f}_{out/})$ is a smooth asymptotically AdS boundary-characteristic initial data set on $[0,v_{0})$ for the system (\[eq:RequationFinal\])–(\[eq:OutgoingVlasovFinal\]) satisfying the reflecting gauge condition at $r=r_{0},+\infty$, according to Definition \[def:TypeII\], such that the quantities $\frac{\text{\textgreek{W}}_{/}^{2}}{1-\frac{1}{3}\Lambda r_{/}^{2}},r_{/}^{2}(T_{vv})_{/}$ and $\tan^{-1}r_{/}$ extend smoothly on $v=v_{0}$ and moreover $$\|\|(r_{/},\text{\textgreek{W}}_{/}^{2},\bar{f}_{in/},\bar{f}_{out/})\|\|_{\mathcal{C}\mathcal{S}}<\text{\textgreek{e}}\label{eq:SmallnessForCauchyStability}$$ for some $0<\text{\textgreek{e}}\le\text{\textgreek{e}}_{0}$, then the maximal development $(\mathcal{U};r,\text{\textgreek{W}}^{2},\bar{f}_{in},\bar{f}_{out})$ satisfies $$\mathcal{W}_{l_{}}\doteq\{0<u\le l_{}v_{0}\}\cap\{u<v<u+v_{0}\}\subset\mathcal{U}\label{eq:InclusionInMaximalDomain}$$ and $$\sqrt{-\Lambda}\sup_{\mathcal{W}_{l_{}}}\|\tilde{m}\|+\sup_{\mathcal{W}_{l_{}}}\log\Bigg(\frac{1-\frac{1}{3}\Lambda r^{2}}{1-\max\{\frac{2m}{r},0\}}\Bigg)+\sup_{\bar{u}}\int_{\{u=\bar{u}\}\cap\mathcal{W}_{l_{}}}\frac{rT_{vv}}{\partial_{v}r}\, dv+\sup_{\bar{v}}\int_{\{v=\bar{v}\}\cap\mathcal{W}_{l_{}}}\frac{rT_{uu}}{(-\partial_{u}r)}\, du<C_{l_{}}\text{\textgreek{e}}.\label{eq:SmallnessCauchyStability}$$ Proposition \[prop:CauchyStabilityOfAdS\] should be interpreted as a Cauchy stability statement for the pure AdS initial data set with respect to the topology defined by (\[eq:GeometricNormForCauchyStability\]) which is independent of the radius $r_{0}$ of the reflecting boundary. Considering the spherically symmetric Einstein–scalar field system (\[eq:EinsteinScalarFieldInDoubleNull\]) with an inner mirror placed at $\{r=r_{0}\}$, the analogue of the initial data norm (\[eq:GeometricNormForCauchyStability\]) (obtained using the substitution $(T_{vv})_{/}\rightarrow(\partial_{v}\text{\textgreek{f}})\|_{u=0}$) is rougher compared to the bounded variation norm of Christodoulou (see [@ChristodoulouBoundedVariation]). It is not known whether (\[eq:EinsteinScalarFieldInDoubleNull\]), restricted to the exterior of an inner mirror at $\{r=r_{0}\}$, satisfies a Cauchy stability estimate with respect to the analogue of the initial data norm (\[eq:GeometricNormForCauchyStability\]) which is independent of $r_{0}$ (although local existence and uniqueness follow trivially in this case for fixed $r_{0}$). In fact, Proposition \[prop:CauchyStabilityOfAdS\] is a special case of the following Cauchy stability estimate established in [@MoschidisMaximalDevelopment]: [3.2]{}\[Theorem 2 in [@MoschidisMaximalDevelopment]\]\[prop:CauchyStability\] For any $v_{1}<v_{2}$ and $0<r_{0}<(-\Lambda)^{-1/2}$, let $(r_{/i},\text{\textgreek{W}}_{/i}^{2},\bar{f}_{in/i},\bar{f}_{out/i})$, $i=1,2$, be two smooth asymptotically AdS boundary-characteristic initial data sets on $[v_{1},v_{2})$ for the system (\[eq:RequationFinal\])–(\[eq:OutgoingVlasovFinal\]) satisfying the reflective gauge condition at $r=r_{0},+\infty$, according to Definition \[def:TypeII\], such that the quantities $\frac{\text{\textgreek{W}}_{/i}^{2}}{1-\frac{1}{3}\Lambda r_{/i}^{2}},r_{/i}^{2}(T_{vv})_{/i}$ and $\tan^{-1}r_{/i}$ extend smoothly on $v=v_{2}$. Assume, also, the following conditions: 1. For some $u_{0}>0$, the maximal future development $(\mathcal{U}_{1};r_{1},\text{\textgreek{W}}_{1}^{2},\bar{f}_{in1},\bar{f}_{out1})$ of $(r_{/1},\text{\textgreek{W}}_{/1}^{2},\bar{f}_{in/1},\bar{f}_{out/1})$ satisfies $$\mathcal{W}_{u_{0}}\doteq\{0<u<u_{0}\}\cap\{u+v_{1}<v<u+v_{2}\}\subset\mathcal{U}_{1}$$ and $$\begin{aligned} \sup_{\mathcal{W}_{u_{0}}}\Bigg\{\Big\|\log\big(\frac{\text{\textgreek{W}}_{1}^{2}}{1-\frac{1}{3}\Lambda r_{1}^{2}}\big)\Big\|+\Big\|\log\Big(\frac{2\partial_{v}r_{1}}{1-\frac{2m_{1}}{r_{1}}}\Big)\Big\|+\Big\|\log\Big(\frac{1-\frac{2m_{1}}{r_{1}}}{1-\frac{1}{3}\Lambda r_{1}^{2}}\Big)\Big\|+\sqrt{-\Lambda}\|\tilde{m}_{1}\|\Bigg\}+\label{eq:UpperBoundNonTrappingForCauchyStability}\\ +\sup_{\bar{u}}\int_{\{u=\bar{u}\}\cap\mathcal{W}_{u_{0}}}r_{1}\frac{(T_{vv})_{1}}{\partial_{v}r_{1}}\, dv+\sup_{\bar{v}}\int_{\{v=\bar{v}\}\cap\mathcal{W}_{u_{0}}}r_{1}\frac{(T_{uu})_{1}}{-\partial_{u}r_{1}}\, du & =C_{0}<+\infty.\nonumber \end{aligned}$$ 2. The $(r_{/i},\text{\textgreek{W}}_{/i}^{2},\bar{f}_{in/i},\bar{f}_{out/i})$, $i=1,2$, are $\text{\textgreek{d}}$-close in the following sense: $$\begin{aligned} \sup_{v\in[v_{1},v_{2})}\Bigg\{\Big\|\log\big(\frac{\text{\textgreek{W}}_{/1}^{2}}{1-\frac{1}{3}\Lambda r_{/1}^{2}}\big)-\log\big(\frac{\text{\textgreek{W}}_{/2}^{2}}{1-\frac{1}{3}\Lambda r_{/2}^{2}}\big)\Big\|+\Big\|\log\Big(\frac{2\partial_{v}r_{/1}}{1-\frac{2m_{/1}}{r_{/1}}}\Big)-\log\Big(\frac{2\partial_{v}r_{/2}}{1-\frac{2m_{/2}}{r_{/2}}}\Big)\Big\|+\label{eq:GaugeDifferenceBoundCauchystability}\\ +\Big\|\log\Big(\frac{1-\frac{2m_{/_{1}}}{r_{/1}}}{1-\frac{1}{3}\Lambda r_{/1}^{2}}\Big)-\log\Big(\frac{1-\frac{2m_{/_{2}}}{r_{/2}}}{1-\frac{1}{3}\Lambda r_{/2}^{2}}\Big)\Big\|+\sqrt{-\Lambda}\|\tilde{m}_{/1}-\tilde{m}_{/2}\|\Bigg\}(v) & \le\text{\textgreek{d}}\nonumber \end{aligned}$$ and $$\sup_{v\in[v_{1},v_{2}]}(-\Lambda)\int_{v_{1}}^{v_{2}}\frac{\big\|r_{/1}^{2}\frac{(T_{vv})_{/1}}{\partial_{v}\text{\textgreek{r}}_{1}}(\bar{v})-r_{/2}^{2}\frac{(T_{vv})_{/2}}{\partial_{v}\text{\textgreek{r}}_{2}}(\bar{v})\big\|}{\|\text{\textgreek{r}}_{/}(v)-\text{\textgreek{r}}_{/}(\bar{v})\|+\text{\textgreek{r}}_{/}(v_{1})}\, d\bar{v}\le\text{\textgreek{d}},\label{eq:DifferenceBoundCauchyStability}$$ where $C_{1}$ is a large fixed absolute constant, $\text{\textgreek{d}}$ satisfies $$0\le\text{\textgreek{d}}\le\text{\textgreek{d}}_{0}\doteq\exp\big(-\exp\big(C_{1}(1+C_{0})\frac{u_{0}}{v_{2}-v_{1}}\big)\big)\label{eq:SmallnessDeltaForCauchyStability}$$ and $\text{\textgreek{r}}_{/}$ is defined by the relation $$\text{\textgreek{r}}_{/}(v)\doteq\tan^{-1}\big(\sqrt{-\frac{\Lambda}{3}}r_{/}(v)\big).$$ Then, the maximal development $(\mathcal{U}_{2};r_{2},\text{\textgreek{W}}_{2}^{2},\bar{f}_{in2},\bar{f}_{out2})$ of $(r_{/2},\text{\textgreek{W}}_{/2}^{2},\bar{f}_{in/2},\bar{f}_{out/2})$ satisfies $$\mathcal{W}_{u_{0}}\subset\mathcal{U}_{2}$$ and $$\begin{aligned} \sup_{\mathcal{W}_{u_{0}}}\Bigg\{\Big\|\log\big(\frac{\text{\textgreek{W}}_{1}^{2}}{1-\frac{1}{3}\Lambda r_{1}^{2}}\big)-\log\big(\frac{\text{\textgreek{W}}_{2}^{2}}{1-\frac{1}{3}\Lambda r_{2}^{2}}\big)\Big\|+\Big\|\log\Big(\frac{2\partial_{v}r_{1}}{1-\frac{2m_{1}}{r_{1}}}\Big)-\log\Big(\frac{2\partial_{v}r_{2}}{1-\frac{2m_{2}}{r_{2}}}\Big)\Big\|+\label{eq:UpperBoundNonTrappingForCauchyStability-1}\\ +\Big\|\log\Big(\frac{1-\frac{2m_{1}}{r_{1}}}{1-\frac{1}{3}\Lambda r_{1}^{2}}\Big)-\log\Big(\frac{1-\frac{2m_{2}}{r_{2}}}{1-\frac{1}{3}\Lambda r_{2}^{2}}\Big)\Big\|+\sqrt{-\Lambda}\|\tilde{m}_{1}-\tilde{m}_{2}\|\Bigg\}+\nonumber \\ +\sup_{\bar{u}}\int_{\{u=\bar{u}\}\cap\mathcal{W}_{u_{0}}}\big\|r_{1}(T_{vv})_{1}-r_{2}(T_{vv})_{2}\big\|\, dv+\sup_{\bar{v}}\int_{\{v=\bar{v}\}\cap\mathcal{W}_{u_{0}}}\big\|r_{1}(T_{uu})_{1}-r_{2}(T_{uu})_{2}\big\|\, du\nonumber \\ \le\exp\big(\exp\big( & C_{1}(1+C_{0})\big)\frac{u_{0}}{v_{2}-v_{1}}\big)\text{\textgreek{d}}.\nonumber \end{aligned}$$ By repeating the proof of Theorem \[prop:CauchyStability\], the Cauchy stability estimate (\[eq:UpperBoundNonTrappingForCauchyStability-1\]) also holds in the case when $(\mathcal{U}_{i};r_{i},\text{\textgreek{W}}_{i}^{2},\bar{f}_{in;i},\bar{f}_{out;i})$, $i=1,2$, are the maximal developments of $(r_{/i},\text{\textgreek{W}}_{/i}^{2},\bar{f}_{in/i},\bar{f}_{out/i})$, i.e. when $\mathcal{W}_{u_{0}}$ is replaced by $$\mathcal{W}_{u_{0}}^{(-)}\doteq\{-u_{0}\le u<0\}\cap\{u+v_{1}<v<u+v_{2}\}$$ and (\[eq:UpperBoundNonTrappingForCauchyStability\]) holds on $\mathcal{W}_{u_{0}}^{(-)}$ in place of $\mathcal{W}_{u_{0}}$. \[sec:The-main-result:Details\]Final statement of Theorem \[thm:TheoremDetailedIntro\]: the non-linear instability of AdS ========================================================================================================================= The main result of this paper is the following: [1]{}\[final version\]\[thm:TheTheorem\] For any $\text{\textgreek{e}}\in(0,1]$, there exist $r_{0\text{\textgreek{e}}}$, $v_{0\text{\textgreek{e}}}$ depending smoothly on $\text{\textgreek{e}}$ such that $$r_{0\text{\textgreek{e}}}\xrightarrow{\text{\textgreek{e}}\rightarrow0}0$$ and $$\sqrt{-\Lambda}v_{0\text{\textgreek{e}}}\xrightarrow{\text{\textgreek{e}}\rightarrow0}\frac{\pi}{\sqrt{3}},$$ as well as a family $(r_{/}^{(\text{\textgreek{e}})},(\text{\textgreek{W}}_{/}^{(\text{\textgreek{e}})})^{2},\bar{f}_{in/}^{(\text{\textgreek{e}})},\bar{f}_{out/}^{(\text{\textgreek{e}})})$ of smooth asymptotically AdS boundary-characteristic initial data sets for the system (\[eq:RequationFinal\])–(\[eq:OutgoingVlasovFinal\]) satisfying the reflecting gauge condition at $r=r_{0},+\infty$, such that the following hold: 1. The family $(r_{/}^{(\text{\textgreek{e}})},(\text{\textgreek{W}}_{/}^{(\text{\textgreek{e}})})^{2},\bar{f}_{in/}^{(\text{\textgreek{e}})},\bar{f}_{out/}^{(\text{\textgreek{e}})})$ satisfies $$\|\|(r_{/}^{(\text{\textgreek{e}})},(\text{\textgreek{W}}_{/}^{(\text{\textgreek{e}})})^{2},\bar{f}_{in/}^{(\text{\textgreek{e}})},\bar{f}_{out/}^{(\text{\textgreek{e}})})\|\|_{\mathcal{C}\mathcal{S}}\xrightarrow{\text{\textgreek{e}}\rightarrow0}0,\label{eq:DecayInInitialDataNorm}$$ where $\|\|\cdot\|\|_{CS}$ is the norm defined by (\[eq:GeometricNormForCauchyStability\]). 2. There exists a trapped sphere, i.e. point $(u_{\dagger},v_{\dagger})$ in the maximal future development $(\mathcal{U}_{\text{\textgreek{e}}};r_{\text{\textgreek{e}}},\text{\textgreek{W}}_{\text{\textgreek{e}}}^{2},\bar{f}_{in\text{\textgreek{e}}},\bar{f}_{out\text{\textgreek{e}}})$ of $(r_{/}^{(\text{\textgreek{e}})},(\text{\textgreek{W}}_{/}^{(\text{\textgreek{e}})})^{2},\bar{f}_{in/}^{(\text{\textgreek{e}})},\bar{f}_{out/}^{(\text{\textgreek{e}})})$ such that $$\frac{2m}{r}(u_{\dagger},v_{\dagger})>1.\label{eq:TrappedSurfaceOccurs}$$ In particular, in view of Theorem \[thm:maximalExtension\], $(\mathcal{U}_{\text{\textgreek{e}}};r_{\text{\textgreek{e}}},\text{\textgreek{W}}_{\text{\textgreek{e}}}^{2},\bar{f}_{in\text{\textgreek{e}}},\bar{f}_{out\text{\textgreek{e}}})$ has a non-empty future event horizon $\mathcal{H}^{+}$ (defined by (\[eq:FutureEventHorizon\])), satisfying the properties 4.a and 4.b of Theorem \[thm:maximalExtension\], and a complete conformal infinity $\mathcal{I}$ (satisfying (\[eq:CompleConformalInfinity\])). If $$\text{\textgreek{d}}(\text{\textgreek{e}})\doteq\|\|(r_{/}^{(\text{\textgreek{e}})},(\text{\textgreek{W}}_{/}^{(\text{\textgreek{e}})})^{2},\bar{f}_{in/}^{(\text{\textgreek{e}})},\bar{f}_{out/}^{(\text{\textgreek{e}})})\|\|_{\mathcal{C}\mathcal{S}},$$ the point $(u_{\dagger},v_{\dagger})$ satisfies the upper bound $$u_{\dagger}\le\exp\big(\exp(\text{\textgreek{d}}^{6}(\text{\textgreek{e}}))\big)v_{0}.\label{eq:UpperBoundTrappedSurface}$$ On the other hand, in view of Proposition \[prop:CauchyStabilityOfAdS\], we necessarily have $$u_{\dagger}\xrightarrow{\text{\textgreek{e}}\rightarrow0}+\infty.$$ In the simpler case when one is interested in a weaker instability statement, such as the existence of a point $(u_{\dagger},v_{\dagger})$ where $$\frac{2m}{r}\Big\|_{(u_{\dagger},v_{\dagger})}>\frac{1}{2}\label{eq:WeakerinstabilitySatatement}$$ (instead of the stronger bound (\[eq:TrappedSurfaceOccurs\])), the proof of Theorem \[thm:TheTheorem\] can be substantially simplified. In the case of (\[eq:WeakerinstabilitySatatement\]), the upper bound (\[eq:UpperBoundTrappedSurface\]) can be improved into a polynomial bound $$u_{\dagger}\le(\text{\textgreek{d}}(\text{\textgreek{e}}))^{-C_{1}}v_{0},$$ for some fixed $C_{1}>0$. \[sec:Preliminary-constructions\]Construction of the initial data and notation ============================================================================== As described already in Section \[sub:Sketch-of-the-proof\] of the introduction, the initial data family in Theorem\[thm:TheTheorem\] will be such that their development consists of a large number of initially ingoing Vlasov beams. In this section, we will construct such a family $(r_{/\text{\textgreek{e}}},\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{2},\bar{f}_{in/\text{\textgreek{e}}},\bar{f}_{out/\text{\textgreek{e}}})$ of asymptotically AdS boundary-characteristic initial data for (\[eq:RequationFinal\])–(\[eq:OutgoingVlasovFinal\]). The family $(r_{/}^{(\text{\textgreek{e}})},(\text{\textgreek{W}}_{/}^{(\text{\textgreek{e}})})^{2},\bar{f}_{in/}^{(\text{\textgreek{e}})},\bar{f}_{out/}^{(\text{\textgreek{e}})})$ in the statement of Theorem\[thm:TheTheorem\] will be eventually obtained from $(r_{/\text{\textgreek{e}}},\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{2},\bar{f}_{in/\text{\textgreek{e}}},\bar{f}_{out/\text{\textgreek{e}}})$ after possibly adding a suitable perturbation (see Section \[sec:Proof\]). This section is organised as follows: In Section \[sub:Parameters-and-auxiliary\], we will introduce a certain hierarchy of parameters that will be necessary for the construction of $(r_{/\text{\textgreek{e}}},\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{2},\bar{f}_{in/\text{\textgreek{e}}},\bar{f}_{out/\text{\textgreek{e}}})$ in Section \[sub:Construction-of-the-initial-data\]. In Section \[sub:Notational-conventions-andNotational5\], we will introduce some basic notation related to the maximal future development $(\mathcal{U}_{\text{\textgreek{e}}};r_{\text{\textgreek{e}}},\text{\textgreek{W}}_{\text{\textgreek{e}}}^{2},\bar{f}_{in\text{\textgreek{e}}},\bar{f}_{out\text{\textgreek{e}}})$ of $(r_{/\text{\textgreek{e}}},\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{2},\bar{f}_{in/\text{\textgreek{e}}},\bar{f}_{out/\text{\textgreek{e}}})$. Finally, in Section \[sub:Some-geometric-constructions\], we will perform some basic geometric constructions on $(\mathcal{U}_{\text{\textgreek{e}}};r_{\text{\textgreek{e}}},\text{\textgreek{W}}_{\text{\textgreek{e}}}^{2},\bar{f}_{in\text{\textgreek{e}}},\bar{f}_{out\text{\textgreek{e}}})$, related to the separation of $\mathcal{U}_{\text{\textgreek{e}}}$ into various subregions by the Vlasov beams arising from the initial data. \[sub:Parameters-and-auxiliary\]Parameters and auxiliary functions ------------------------------------------------------------------ Let us fix some smooth and strictly increasing functions $h_{0},h_{1},h_{2}:(0,1)\rightarrow(0,1)$, so that $$\lim_{\text{\textgreek{e}}\rightarrow0^{+}}h_{0}(\text{\textgreek{e}})=\lim_{\text{\textgreek{e}}\rightarrow0^{+}}h_{1}(\text{\textgreek{e}})=\lim_{\text{\textgreek{e}}\rightarrow0^{+}}h_{2}(\text{\textgreek{e}})=0,$$ $$\lim_{\text{\textgreek{e}}\rightarrow0^{+}}\text{\textgreek{e}}\cdot\exp(\frac{1}{(h_{1}(\text{\textgreek{e}}))^{6}})=\lim_{\text{\textgreek{e}}\rightarrow0^{+}}h_{1}(\text{\textgreek{e}})\cdot\exp\Big(\exp(\frac{1}{(h_{0}(\text{\textgreek{e}}))^{6}})\Big)=0\label{eq:h_1_h_0_definition}$$ and $$\lim_{\text{\textgreek{e}}\rightarrow0^{+}}h_{2}(\text{\textgreek{e}})\cdot\exp(\text{\textgreek{e}}^{-2})=0.\label{eq:h_2definition}$$ In particular, the following relations hold for $\text{\textgreek{e}}\ll1$: $$h_{2}(\text{\textgreek{e}})\ll\text{\textgreek{e}}\ll h_{1}(\text{\textgreek{e}})\ll h_{0}(\text{\textgreek{e}})\ll1.\label{eq:SchematicRelationParameters}$$ Let $\text{\textgreek{q}}:\mathbb{R}\rightarrow[0,1]$ be a smooth cut-off function, satisfying $\text{\textgreek{q}}\|_{[-1,1]}=1$, $\text{\textgreek{q}}\|_{\mathbb{R}\backslash[-2,2]}=0$ and $$\text{\textgreek{q}}\|_{(-2,2)}>0,\label{eq:SupportedCutOff}$$ and let $\text{\textgreek{e}}_{0}\ll1$ be a small enough absolute constant. For any $0<\text{\textgreek{e}}<\text{\textgreek{e}}_{0}$, any $r_{0}>0$ satisfying $$1-\exp\big(-2(h_{0}(\text{\textgreek{e}}))^{-4}\big)<\frac{r_{0}}{\frac{2}{\sqrt{-\Lambda}}\text{\textgreek{e}}-\frac{1}{3}\Lambda r_{0}^{3}}<1-\frac{1}{2}\exp\big(-2(h_{0}(\text{\textgreek{e}}))^{-4}\big)\label{eq:BoundMirror}$$ (note that (\[eq:BoundMirror\]) implies that $\frac{r_{0}\sqrt{-\Lambda}}{2\text{\textgreek{e}}}=1+O\big(\exp\big(-2(h_{0}(\text{\textgreek{e}}))^{-4}\big)\big)$ as $\text{\textgreek{e}}\rightarrow0$), we will define the following function on $[0,+\infty)\times(0,+\infty)$:$$\bar{f}_{\text{\textgreek{e}}}(v,p^{u})\doteq C_{\text{\textgreek{e}}r_{0}}\sum_{j=0}^{\lceil1/h_{1}(\text{\textgreek{e}})\rceil}\text{\textgreek{q}}\Big(p^{u}-3\Big)\cdot\frac{1}{h_{2}(\text{\textgreek{e}})}\text{\textgreek{q}}\Big(\frac{(v-v^{(j)})\sqrt{-\Lambda}-2h_{2}(\text{\textgreek{e}})}{h_{2}(\text{\textgreek{e}})}\Big)\cdot h_{(j)}(\text{\textgreek{e}})\cdot\text{\textgreek{e}},\label{eq:TheIngoingVlasovInitially}$$ for some constant $C_{\text{\textgreek{e}}r_{0}}$ to be specified in terms of $\text{\textgreek{e}},r_{0}$ later, where $\lceil1/h_{1}(\text{\textgreek{e}})\rceil$ denotes the least integer greater than or equal to $1/h_{1}(\text{\textgreek{e}})$, $$v^{(j)}=\frac{\pi}{\sqrt{-\Lambda}}-j\frac{\text{\textgreek{e}}}{h_{1}(\text{\textgreek{e}})\sqrt{-\Lambda}}\label{eq:DefinitionV_j}$$ for any $0\le j\le\lceil1/h_{1}(\text{\textgreek{e}})\rceil$, $$h_{(0)}=h_{0},\label{eq:TopBeamWeight}$$ and $$h_{(j)}=h_{1}$$ for all $1\le j\le\lceil1/h_{1}(\text{\textgreek{e}})\rceil$. \[sub:Construction-of-the-initial-data\]Construction of the initial data family ------------------------------------------------------------------------------- For any $0<\text{\textgreek{e}}<\text{\textgreek{e}}_{0}$, any $r_{0}$ satisfying (\[eq:BoundMirror\]), we will define the following asymptotically AdS boundary-characteristic initial data set according to Definition \[def:TypeII\]: \[def:ThefamilyOfInitialData\] For any $0<\text{\textgreek{e}}<\text{\textgreek{e}}_{0}$, any $r_{0}$ satisfying (\[eq:BoundMirror\]), we define $v_{0}=v_{0}(r_{0},\text{\textgreek{e}})>0$ and the set of smooth functions $r_{/\text{\textgreek{e}}}:[0,v_{0})\rightarrow[r_{0},+\infty)$, $\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{2}:[0,v_{0})\rightarrow(0,+\infty)$, $\bar{f}_{in/\text{\textgreek{e}}}:[0,v_{0})\times(0,+\infty)\rightarrow[0,+\infty)$ and $\bar{f}_{out/\text{\textgreek{e}}}:[0,v_{0})\times(0,+\infty)\rightarrow[0,+\infty)$ by the requirement that $(r_{/\text{\textgreek{e}}},\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{2},\bar{f}_{in/\text{\textgreek{e}}},\bar{f}_{out/\text{\textgreek{e}}})$ is an asymptotically AdS boundary-characteristic initial data set on $[0,v_{0})$ for the system (\[eq:RequationFinal\])–(\[eq:OutgoingVlasovFinal\]) satisfying the reflecting gauge condition at $r=r_{0},+\infty$ so that $$\frac{\partial_{v}r_{/\text{\textgreek{e}}}}{1-\frac{2m_{/\text{\textgreek{e}}}}{r_{/_{\text{\textgreek{e}}}}}}=\frac{1}{2}\label{eq:ConditionOnDvRInitiallyFamily}$$ (where $m_{/\text{\textgreek{e}}}$ is defined in terms of $r_{/\text{\textgreek{e}}},\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{2}$ by (\[eq:DefinitionHawkingMassCharacteristic\])), $$\bar{f}_{out/\text{\textgreek{e}}}=0\label{eq:OutgoingInitialdataFamily}$$ and $$\bar{f}_{in/\text{\textgreek{e}}}(v,p^{u})=\bar{f}_{\text{\textgreek{e}}}(v,p^{u})\label{eq:IngoingInitialDataFamily}$$ for all $0\le v\le v_{0}$ and $p^{u}>0$. The constant $C_{\text{\textgreek{e}}r_{0}}$ in (\[eq:TheIngoingVlasovInitially\]) is fixed in terms of $\text{\textgreek{e}},r_{0}$ by the requirement that $$\lim_{v\rightarrow_{v_{0}}}\tilde{m}_{/\text{\textgreek{e}}}=\frac{\text{\textgreek{e}}}{\sqrt{-\Lambda}}\label{eq:MassAtinfinityFixedInitially}$$ (in particular, there exists some fixed (large) $C_{0}>1$, independent of $\text{\textgreek{e}},r_{0}$, so that $C_{\text{\textgreek{e}}r_{0}}\in[C_{0}^{-1},C_{0}]$ for any $0<\text{\textgreek{e}}<\text{\textgreek{e}}_{0}$, any $r_{0}$ satisfying (\[eq:BoundMirror\]).[^14] The conditions (\[eq:ConditionOnDvRInitiallyFamily\])–(\[eq:IngoingInitialDataFamily\]) determine $v_{0}$ and $r_{/\text{\textgreek{e}}},\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{2}$ uniquely in terms of $\text{\textgreek{e}},r_{0}$. While $(r_{/\text{\textgreek{e}}},\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{2},\bar{f}_{in/\text{\textgreek{e}}},\bar{f}_{out/\text{\textgreek{e}}})$ depend on both $\text{\textgreek{e}}$ and $r_{0}$, we will only use the subscript $\text{\textgreek{e}}$ in their notation, since most of the estimates that we will later establish for their maximal development will depend only on $\text{\textgreek{e}}$. The initial data $(r_{/}^{(\text{\textgreek{e}})},(\text{\textgreek{W}}_{/}^{(\text{\textgreek{e}})})^{2},\bar{f}_{in/}^{(\text{\textgreek{e}})},\bar{f}_{out/}^{(\text{\textgreek{e}})})$ in the statement of Theorem \[thm:TheTheorem\] will eventually be chosen to be small perturbations of $(r_{/\text{\textgreek{e}}},\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{2},\bar{f}_{in/\text{\textgreek{e}}},\bar{f}_{out/\text{\textgreek{e}}})$ (see Section \[sec:Proof\]). \[sub:Notational-conventions-andNotational5\]Notational conventions and basic computations ------------------------------------------------------------------------------------------ For any $0<\text{\textgreek{e}}<\text{\textgreek{e}}_{0}$ and any $r_{0}$ satisfying (\[eq:BoundMirror\]), let $(r_{/\text{\textgreek{e}}},\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{2},\bar{f}_{in/\text{\textgreek{e}}},\bar{f}_{out/\text{\textgreek{e}}})$ be the initial data set defined by Definition \[def:ThefamilyOfInitialData\]. Assuming that $\text{\textgreek{e}}_{0}$ is fixed small enough, for any $0<\text{\textgreek{e}}<\text{\textgreek{e}}_{0}$ and any $r_{0}$ satisfying (\[eq:BoundMirror\]), the initial data set $(r_{/\text{\textgreek{e}}},\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{2},\bar{f}_{in/\text{\textgreek{e}}},\bar{f}_{out/\text{\textgreek{e}}})$ satisfies the following estimate depending only on $\text{\textgreek{e}}$: $$\|\|(r_{/\text{\textgreek{e}}},\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{2},\bar{f}_{in/\text{\textgreek{e}}},\bar{f}_{out/\text{\textgreek{e}}})\|\|_{\mathcal{C}\mathcal{S}}\le Ch_{0}(\text{\textgreek{e}}),\label{eq:CSnormFamily}$$ where $\|\|\cdot\|\|_{\mathcal{C}\mathcal{S}}$ is defined by (\[eq:GeometricNormForCauchyStability\]) and $C>0$ is a fixed constant. Let $(\mathcal{U}_{\text{\textgreek{e}}};r_{\text{\textgreek{e}}},\text{\textgreek{W}}_{\text{\textgreek{e}}}^{2},\bar{f}_{in\text{\textgreek{e}}},\bar{f}_{out\text{\textgreek{e}}})$ be the maximal future development of $(r_{/\text{\textgreek{e}}},\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{2},\bar{f}_{in/\text{\textgreek{e}}},\bar{f}_{out/\text{\textgreek{e}}})$ (see Theorem \[thm:maximalExtension\]). In view of Proposition \[prop:CauchyStabilityOfAdS\], the bound (\[eq:CSnormFamily\]) implies that, for any fixed $u_{}>0$ and any $\text{\textgreek{d}}>0$, there exists an $\text{\textgreek{e}}_{\text{\textgreek{d}}u_{}}>0$ sufficiently small depending only on $\text{\textgreek{d}}$ and $u_{}$ so that, for any $0\le\text{\textgreek{e}}<\text{\textgreek{e}}_{\text{\textgreek{d}}u_{}}$: $$\mathcal{W}_{u_{}}\doteq\{0<u<u_{}\}\cap\{u<v<u+v_{0}\}\subset\mathcal{U}\label{eq:InclusionInMaximalDomain-1}$$ and $$\sqrt{-\Lambda}\sup_{\mathcal{W}_{u_{}}}\|\tilde{m}_{\text{\textgreek{e}}}\|+\sup_{\mathcal{W}_{u_{}}}\log\Bigg(\frac{1-\frac{1}{3}\Lambda r_{\text{\textgreek{e}}}^{2}}{1-\max\{\frac{2m_{\text{\textgreek{e}}}}{r_{\text{\textgreek{e}}}},0\}}\Bigg)+\sup_{\bar{u}}\int_{\{u=\bar{u}\}\cap\mathcal{W}_{u_{}}}\frac{r_{\text{\textgreek{e}}}(T_{vv})_{\text{\textgreek{e}}}}{\partial_{v}r_{\text{\textgreek{e}}}}\, dv+\sup_{\bar{v}}\int_{\{v=\bar{v}\}\cap\mathcal{W}_{u_{}}}\frac{r_{\text{\textgreek{e}}}(T_{uu})_{\text{\textgreek{e}}}}{(-\partial_{u}r_{\text{\textgreek{e}}})}\, du<\text{\textgreek{d}},\label{eq:SmallnessCauchyStability-1}$$ where $m_{\text{\textgreek{e}}},\tilde{m}_{\text{\textgreek{e}}},(T_{uu})_{\text{\textgreek{e}}},(T_{vv})_{\text{\textgreek{e}}}$ are defined in terms of $r_{\text{\textgreek{e}}},\text{\textgreek{W}}_{\text{\textgreek{e}}}^{2}\bar{f}_{in\text{\textgreek{e}}},\bar{f}_{out\text{\textgreek{e}}}$ by (\[eq:DefinitionHawkingMass\]), (\[eq:RenormalisedHawkingMass\]), (\[eq:T\_uuComponent\]) and (\[eq:T\_vvComponent\]). In particular, if $$g_{AdS}=-\text{\textgreek{W}}_{AdS,r_{0},v_{0}}^{2}dudv+r_{AdS,r_{0},v_{0}}g_{\mathbb{S}^{2}}$$ is the pure AdS metric in a spherically symmetric coordinate chart $(u,v)$ such that $r_{AdS,r_{0},v_{0}}=r_{0}$ on $\{u=v\}$ and $r_{AdS,r_{0},v_{0}}=+\infty$ on $\{u=v-v_{0}\}$,[^15] then $(\mathcal{U}_{\text{\textgreek{e}}};r_{\text{\textgreek{e}}},\text{\textgreek{W}}_{\text{\textgreek{e}}}^{2},\bar{f}_{in\text{\textgreek{e}}},\bar{f}_{out\text{\textgreek{e}}})$, when restricted on $\mathcal{W}_{u_{}}$, is $\text{\textgreek{d}}$-close to $(\mathcal{W}_{u_{}};r_{AdS,r_{0},v_{0}},\text{\textgreek{W}}_{AdS,r_{0},v_{0}}^{2},0,0)$ with respect to the (gauge invariant) distance defined by (\[eq:GeometricNormForCauchyStability\]). Notice also that (\[eq:SmallnessCauchyStability-1\]) implies that, provided $\text{\textgreek{d}}$ is small enough, the spacetime $(\mathcal{W}_{u_{}}\times\mathbb{S}^{2},g_{\text{\textgreek{e}}})$ does not contain any trapped surface, where $$g_{\text{\textgreek{e}}}=-\text{\textgreek{W}}_{\text{\textgreek{e}}}^{2}dudv+r_{\text{\textgreek{e}}}^{2}g_{\mathbb{S}^{2}}.$$ Notice that, in view of the conservation of $\tilde{m}$ on $\text{\textgreek{g}}_{0}$ and $\mathcal{I}$ (see (\[eq:ConstantMassMirror\]) and (\[eq:ConstantMassInfinity\])), we have: $$\tilde{m}_{\text{\textgreek{e}}}\|_{\text{\textgreek{g}}_{0}}=0\label{eq:MassAxis}$$ and $$\tilde{m}_{\text{\textgreek{e}}}\|_{\mathcal{I}}=\lim_{v\rightarrow v_{0}}\tilde{m}_{/\text{\textgreek{e}}}(v)=\frac{\text{\textgreek{e}}}{\sqrt{-\Lambda}}.\label{eq:MassInfinity}$$ For each $0\le j\le\lceil1/h_{1}(\text{\textgreek{e}})\rceil$, we can associate to the beam centered at $v=v^{(j)}+\frac{2}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})$ the mass difference $$\mathfrak{D}\tilde{m}_{/}^{(j)}\doteq\tilde{m}_{/\text{\textgreek{e}}}\big(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})\big)-\tilde{m}_{/\text{\textgreek{e}}}\big(v^{(j)}\big).\label{eq:InitialMassDifferenceBeams}$$ Notice that $$\mathcal{D}\tilde{m}_{/}^{(0)}\simeq\frac{h_{0}(\text{\textgreek{e}})\text{\textgreek{e}}}{\sqrt{-\Lambda}}$$ and, for all $1\le j\le\lceil1/h_{1}(\text{\textgreek{e}})\rceil$: $$\mathcal{D}\tilde{m}_{/}^{(j)}\simeq\frac{h_{1}(\text{\textgreek{e}})\text{\textgreek{e}}}{\sqrt{-\Lambda}}.$$ Furthermore: $$\sum_{j=0}^{\lceil1/h_{1}(\text{\textgreek{e}})\rceil}\mathcal{D}\tilde{m}_{/}^{(j)}=\lim_{v\rightarrow v_{0}}\tilde{m}_{/\text{\textgreek{e}}}(v)=\frac{\text{\textgreek{e}}}{\sqrt{-\Lambda}}.$$ \[sub:Some-geometric-constructions\]Some geometric constructions on $\mathcal{U}_{\epsilon}$ -------------------------------------------------------------------------------------------- For any $0<\text{\textgreek{e}}<\text{\textgreek{e}}_{0}$ and any $r_{0}$ satisfying (\[eq:BoundMirror\]), we will define some special subsets of the domain $\mathcal{U}_{\text{\textgreek{e}}}$ of the maximal future development $(\mathcal{U}_{\text{\textgreek{e}}};r_{\text{\textgreek{e}}},\text{\textgreek{W}}_{\text{\textgreek{e}}}^{2},\bar{f}_{in\text{\textgreek{e}}},\bar{f}_{out\text{\textgreek{e}}})$ of the initial data set $(r_{/\text{\textgreek{e}}},\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{2},\bar{f}_{in/\text{\textgreek{e}}},\bar{f}_{out/\text{\textgreek{e}}})$. In the rest of this section, we will adopt the convention that the boundary $\partial A$ of a subset $A\subseteq\mathcal{U}_{\text{\textgreek{e}}}$ is the boundary of $A$ as a subset of $\mathbb{R}^{2}$ (with respect to the ambient topology of $\mathbb{R}^{2}$) Let us define the domain of outer communications $\mathcal{D}_{\text{\textgreek{e}}}$ of $\mathcal{U}_{\text{\textgreek{e}}}$ as $$\mathcal{D}_{\text{\textgreek{e}}}\doteq J^{-}(\mathcal{I})\cap\mathcal{U}_{\text{\textgreek{e}}},\label{eq:DomainOfOuterCommunications}$$ where $J^{-}(\mathcal{I})$ is the causal past of $\mathcal{I}$ with respect to the reference metric (\[eq:ComparisonUVMetric\]) (see (\[eq:PastOfInfinity\])). In accordance with Theorem \[thm:maximalExtension\], we will also define the future event horizon $\mathcal{H}_{\text{\textgreek{e}}}^{+}$ of $\mathcal{U}_{\text{\textgreek{e}}}$ as $$\mathcal{H}_{\text{\textgreek{e}}}^{+}\doteq\partial\mathcal{D}_{\text{\textgreek{e}}}\cap\mathcal{U}_{\text{\textgreek{e}}}.\label{eq:FutureEventHorizon}$$ Note that we allow $\mathcal{H}_{\text{\textgreek{e}}}^{+}$ to be empty. In view of Theorem \[thm:maximalExtension\], in the case when $\mathcal{H}_{\text{\textgreek{e}}}^{+}$ is non-empty, it is necessarily of the form $$\mathcal{H}_{\text{\textgreek{e}}}^{+}=\{u=u_{\mathcal{H}_{\text{\textgreek{e}}}^{+}}\}\cap\mathcal{U}_{\text{\textgreek{e}}}$$ and has infinite affine length. We will also define $$\mathcal{J}_{\text{\textgreek{e}}}\doteq J^{-}(\text{\textgreek{g}}_{0})\cap\mathcal{U}_{\text{\textgreek{e}}}.$$ Notice that, as a consequence of Theorem \[thm:maximalExtension\], on $\mathcal{J}_{\text{\textgreek{e}}}\cup\mathcal{D}_{\text{\textgreek{e}}}$ we have $$1-\frac{2m}{r}>0,$$ i.e. trapped spheres can only appear in the region $\mathcal{U}_{\text{\textgreek{e}}}\backslash(\mathcal{J}_{\text{\textgreek{e}}}\cup\mathcal{D}_{\text{\textgreek{e}}})$. In the case $\mathcal{H}_{\text{\textgreek{e}}}^{+}\neq\emptyset$, Theorem \[thm:maximalExtension\] also implies that $\mathcal{J}_{\text{\textgreek{e}}}\backslash\mathcal{D}_{\text{\textgreek{e}}}\neq\emptyset$. For any $v_{}\in[0,v_{0}]$ and any integer $n\ge1$, we will define $$U_{n}(v_{})\doteq v_{}+(n-1)v_{0}$$ and $$V_{n}(v_{})=v_{}+nv_{0}.$$ We will also set $$V_{0}(v_{})\doteq v_{}.$$ Notice that the segment $\{u=U_{n}(v_{})\}\cap\mathcal{U}_{\text{\textgreek{e}}}$ is the image of the ingoing null geodesic of $\mathcal{U}_{\text{\textgreek{e}}}$ emanating from the point $(0,v_{})$ after $n$ reflections off $\text{\textgreek{g}}_{0}$ and $n-1$ reflections off $\mathcal{I}$, while the segment $\{v=V_{n}(v_{})\}\cap\mathcal{U}_{\text{\textgreek{e}}}$ is the image of the same null geodesic after $n$ reflections off $\text{\textgreek{g}}_{0}$ and $n$ reflections off $\mathcal{I}$. Let us define the domains $\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j)}\subset\mathcal{U}_{\text{\textgreek{e}}}$ for any $n\in\mathbb{N}$, $0\le i\le\lceil1/h_{1}(\text{\textgreek{e}})\rceil$ and $i\le j\le\lceil1/h_{1}(\text{\textgreek{e}})\rceil+i+1$ by the relation $$\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j)}=\Big\{ U_{n}\big(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})\big)<u<U_{n}\big(v^{(i-1)}\big)\Big\}\cap\Big\{ V_{n}\big(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})\big)<v<V_{n}\big(v^{(j-1)}\big)\Big\}\cap\mathcal{U}_{\text{\textgreek{e}}},\label{eq:Domains_D}$$ where we have used the following conventions in the expression (\[eq:Domains\_D\]): 1. $U_{n}\big(v^{(-1)}\big)\doteq U_{n+1}\big(v^{(\lceil1/h_{1}(\text{\textgreek{e}})\rceil)}\big)$. 2. $V_{n}\big(v^{(\lceil1/h_{1}(\text{\textgreek{e}})\rceil+l)}+c\big)\doteq V_{n-1}\big(v^{(l-1)}+c\big)$ for any integer $1\le l\le\lceil1/h_{1}(\text{\textgreek{e}})\rceil$ and any $c\ge0$. The boundary of the domains $\mathcal{R}_{\text{\textgreek{e}}n}^{(i,i)}$, $0\le i\le\lceil1/h_{1}(\text{\textgreek{e}})\rceil$, contains a segment $\mathcal{I}$, while the boundary of the domains $\mathcal{R}_{\text{\textgreek{e}}n}^{(i,\lceil1/h_{1}(\text{\textgreek{e}})\rceil+1+i)}$, $0\le i\le\lceil1/h_{1}(\text{\textgreek{e}})\rceil$, contains a segment in $\text{\textgreek{g}}_{0}$. Notice that $T_{uu}=T_{vv}=0$ in $\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j)}$. In particular, all the domains $(\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j)}\times\mathbb{S}^{2},g_{\text{\textgreek{e}}})$ are isometric to a region of a member of the Schwarzschild-AdS family (or to a region of pure AdS spacetime), and the renormalised mass function $\tilde{m}_{\text{\textgreek{e}}}$ is constant on them. We will define for any $0\le i\le\lceil1/h_{1}(\text{\textgreek{e}})\rceil$, $i\le j\le\lceil1/h_{1}(\text{\textgreek{e}})\rceil+i+1$ and $n\in\mathbb{N}$ such that $\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j)}\neq\emptyset$: $$\tilde{m}_{\text{\textgreek{e}}n}^{(i,j)}\doteq\tilde{m}_{\text{\textgreek{e}}}\|_{\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j)}}.\label{eq:Renormalised_Mass_D_n}$$ In view of (\[eq:MassAxis\]) and (\[eq:MassInfinity\]), we immediately calculate that for all $0\le i\le\lceil1/h_{1}(\text{\textgreek{e}})\rceil$ and all $n\in\mathbb{N}$ such that $\mathcal{R}_{\text{\textgreek{e}}n}^{(i,i)},\mathcal{R}_{\text{\textgreek{e}}n}^{(i,\lceil1/h_{1}(\text{\textgreek{e}})\rceil+1+i)}\neq\emptyset$: $$\tilde{m}_{\text{\textgreek{e}}n}^{(i,\lceil1/h_{1}(\text{\textgreek{e}})\rceil+1+i)}=0\label{eq:ZeroMassNearAxis}$$ and $$\tilde{m}_{\text{\textgreek{e}}n}^{(i,i)}=\tilde{m}_{\text{\textgreek{e}}0}^{(i,i)}=\frac{\text{\textgreek{e}}}{\sqrt{-\Lambda}}.\label{eq:MassAtInfinity}$$ For any $n\in\mathbb{N}$, $0\le i\le\lceil1/h_{1}(\text{\textgreek{e}})\rceil$ and $i+1\le j\le\lceil1/h_{1}(\text{\textgreek{e}})\rceil+i$, we will define the interaction regions: $$\mathcal{N}_{\text{\textgreek{e}}n}^{(i,j)}\doteq\Big\{ U_{n}\big(v^{(i)}\big)\le u\le U_{n}\big(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})\big)\Big\}\cap\Big\{ V_{n}\big(v^{(j)}\big)\le u\le V_{n}\big(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})\big)\Big\}\cap\mathcal{U}_{\text{\textgreek{e}}},\label{eq:InteractionRegion}$$ where the conventions stated below (\[eq:Domains\_D\]) hold regarding indices smaller than $0$ or larger than $\lceil1/h_{1}(\text{\textgreek{e}})\rceil$. Let us define for any $0\le i\le\lceil1/h_{1}(\text{\textgreek{e}})\rceil$, $i\le j\le\lceil1/h_{1}(\text{\textgreek{e}})\rceil+i+1$ and $n\in\mathbb{N}$ such that $\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j)}\neq\emptyset$: $$r_{\text{\textgreek{e}}n}^{(i,j)}\doteq r_{\text{\textgreek{e}}}\big(U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\big).\label{eq:r_n}$$ Note that $r_{\text{\textgreek{e}}n}^{(i,i)}=+\infty$ and $r_{\text{\textgreek{e}}n}^{(i,\lceil1/h_{1}(\text{\textgreek{e}})\rceil+i+1)}=r_{0\text{\textgreek{e}}}$. Finally, let us remark that, in view of property (\[eq:SupportOfCutOff\]) of the cut-off used in the construction of the initial data and equations (\[eq:ConservationT\_vv\])–(\[eq:ConservationT\_uu\]), for any $1\le n\le n_{f}$, $0\le i\le\lceil1/h_{1}(\text{\textgreek{e}})\rceil$, $i\le j\le\lceil1/h_{1}(\text{\textgreek{e}})\rceil+i+1$, we have $$T_{uu}>0\mbox{ on }\Big\{ U_{n}(v^{(i)})<u<U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\Big\}\label{eq:TuuSupport}$$ and $$T_{vv}>0\mbox{ on }\Big\{ V_{n}(v^{(j)})<u<V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\Big\}.\label{eq:TvvSupport}$$ \[sec:Proof\]Proof of Theorem \[thm:TheTheorem\] ================================================ In this Section, we will prove Theorem \[thm:TheTheorem\]. In order to simplify our notation, from now on, we will often drop the subscripts $\text{\textgreek{e}}$ in notations related to the maximal future development $(\mathcal{U}_{\text{\textgreek{e}}};r_{\text{\textgreek{e}}},\text{\textgreek{W}}_{\text{\textgreek{e}}}^{2},\bar{f}_{in\text{\textgreek{e}}},\bar{f}_{out\text{\textgreek{e}}})$ of the initial data $(r_{/\text{\textgreek{e}}},\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{2},\bar{f}_{in/\text{\textgreek{e}}},\bar{f}_{out/\text{\textgreek{e}}})$ (see Definition \[def:ThefamilyOfInitialData\]). For any $0<\text{\textgreek{e}}<\text{\textgreek{e}}_{0}$ (provided $\text{\textgreek{e}}_{0}$ is fixed sufficiently small), any $r_{0}>0$ satisfying (\[eq:BoundMirror\]), let $(\mathcal{U}_{\text{\textgreek{e}}};r,\text{\textgreek{W}}^{2},\bar{f}_{in},\bar{f}_{out})$ be the maximal future development of $(r_{/\text{\textgreek{e}}},\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{2},\bar{f}_{in/\text{\textgreek{e}}},\bar{f}_{out/\text{\textgreek{e}}})$, and let us define $$u_{+}\doteq\sup\big\{ u_{}>0:\mbox{ }1-\frac{2m}{r}>h_{3}(\text{\textgreek{e}})\mbox{ on }\mathcal{U}_{\text{\textgreek{e}}}\cap\{u<u_{}\}\big\}\label{eq:UpperUNonTrapping}$$ and $$\mathcal{U}_{\text{\textgreek{e}}}^{+}=\mathcal{U}_{\text{\textgreek{e}}}\cap\big\{ u<\min\{u_{+},(h_{1}(\text{\textgreek{e}}))^{-2}v_{0\text{\textgreek{e}}}\}\big\},\label{eq:DefinitionUntrappedRegion}$$ where $$h_{3}(\text{\textgreek{e}})=\exp\Big\{-\exp\Big((h_{1}(\text{\textgreek{e}}))^{-5}\exp\big(-2(h_{0}(\text{\textgreek{e}}))^{-4}\big)\Big)\Big\}.\label{eq:h_3definition}$$ Let us also set $$k\doteq\lceil1/h_{1}(\text{\textgreek{e}})\rceil\label{eq:kappa}$$ and $$n_{f}\doteq\lfloor(u_{+}-v^{(0)})/v_{0}\rfloor,\label{eq:defNf}$$ where $\lceil x\rceil$ denotes the least integer greater than or equal to $x$, while $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$. The proof of Theorem \[thm:TheTheorem\] will follow in two steps: First, in Section \[sub:NearlyTrapped\], we will show that: $$\sup_{\mathcal{U}_{\text{\textgreek{e}}}^{+}}\big(1-\frac{2m}{r}\big)=h_{3}(\text{\textgreek{e}}),\label{eq:NearTrappingIsAchieved}$$ i.e. that $\mathcal{U}_{\text{\textgreek{e}}}^{+}$ contains a nearly-trapped sphere. Then, in Section \[sub:FinalStep\], we will show that, at the final step of the evolution, either a trapped sphere is formed, or there exists a small perturbation of the initial data $(r_{/\text{\textgreek{e}}},\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{2},\bar{f}_{in/\text{\textgreek{e}}},\bar{f}_{out/\text{\textgreek{e}}})$ giving rise to a trapped sphere. Before proving (\[eq:NearTrappingIsAchieved\]), we will need to establish some necessary bounds for the evolution of $(r,\text{\textgreek{W}}^{2},\bar{f}_{in},\bar{f}_{out})$ in the region $\mathcal{U}_{\text{\textgreek{e}}}^{+}$. These bounds, which will be obtained in Section \[sub:Inductive-bounds\], will be used both in Section \[sub:NearlyTrapped\] and in Section \[sub:FinalStep\]. \[sub:Inductive-bounds\]Inductive bounds for the evolution in the region $\mathcal{U}_{\text{\textgreek{e}}}^{+}$ ----------------------------------------------------------------------------------------------------------------- In this Section, we will establish a number of useful bounds for $(\mathcal{U}_{\text{\textgreek{e}}}^{+};r,\text{\textgreek{W}}^{2},\bar{f}_{in},\bar{f}_{out})$. These bounds will include a number of inductive bounds for the quantities $\tilde{m}_{n}^{(1,k+1)}$, $r_{n}^{(k,k+1)}$ and $r_{n}^{(1,k+1)}$ (with $k$ defined by (\[eq:kappa\])), that will be of fundamental significance in the proof of Theorem \[thm:TheTheorem\]. In particular, we will prove the following result: \[prop:TheMainBootstrapBeforeTrapping\] For any $0<\text{\textgreek{e}}<\text{\textgreek{e}}_{0}$, the following bounds hold for $(\mathcal{U}_{\text{\textgreek{e}}}^{+};r,\text{\textgreek{W}}^{2},\bar{f}_{in},\bar{f}_{out})$: 1. On $\mathcal{U}_{\text{\textgreek{e}}}^{+}$, we can estimate: $$\Big\|\log\Big(\frac{-\partial_{u}r}{1-\frac{1}{3}\Lambda r^{2}}\Big)\Big\|+\Big\|\log\Big(\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big)\Big\|\le\big(h_{1}(\text{\textgreek{e}})\big)^{-4}\log\big((h_{3}(\text{\textgreek{e}}))^{-1}\big).\label{eq:RoughBoundGeometry}$$ 2. For any $1\le n\le n_{f}$: $$r_{n}^{(0,k)}\ge\frac{\text{\textgreek{e}}^{-\frac{1}{2}}}{\sqrt{-\Lambda}},\label{eq:BoundForRAwayInteractionProp}$$ $$r_{n}^{(k,k+1)}\le\frac{\text{\textgreek{e}}^{\frac{1}{2}}}{\sqrt{-\Lambda}},\label{eq:UpperBoundForAxisInteractionProp}$$ $$\frac{2(\tilde{m}\|_{\mathcal{I}}-\tilde{m}_{n}^{(1,k+1)})}{r_{0}}\ge\exp\big(-2(h_{0}(\text{\textgreek{e}}))^{-4}\big),\label{eq:EnoughMassBehind}$$ $$\frac{2(\tilde{m}\|_{\mathcal{I}}-\tilde{m}_{n}^{(1,k+1)})}{r_{0}}\le1-\frac{1}{C_{0}}h_{0}(\text{\textgreek{e}})\label{eq:NotEnoughMassBehind}$$ and $$\frac{r_{n}^{(1,k+1)}}{r_{0}}-1\ge\exp\Big(-(h_{0}(\text{\textgreek{e}}))^{-4}\Big),\label{eq:BoundSecondBeamchanged}$$ where $C_{0}>1$ is a large fixed constant (independent of all the parameters). 3. For any $2\le n\le n_{f}$: $$\frac{\tilde{m}_{n}^{(1,k+1)}}{\tilde{m}_{n-1}^{(1,k+1)}}\ge1+\frac{1}{4}\exp\big(-2(h_{0}(\text{\textgreek{e}}))^{-4}\big)\frac{r_{0}}{r_{n}^{(k,k+1)}}\label{eq:BoundForMassIncrease}$$ and $$\frac{r_{n}^{(k,k+1)}-r_{0}}{r_{n-1}^{(k,k+1)}-r_{0}}\le1+2C_{0}\frac{r_{0}}{r_{n-1}^{(k,k+1)}}\Big(\Big\|\log\big(1-\frac{2\tilde{m}_{n-1}^{(1,k+1)}}{r_{0}}\big)\Big\|+(h_{0}(\text{\textgreek{e}}))^{-4}\Big).\label{eq:BoundForMaxBeamSeparation}$$ Before presenting the proof of Proposition \[prop:TheMainBootstrapBeforeTrapping\] (in Section \[sub:Proof-of-Proposition\]), we will briefly comment on the nature of the bounds (\[eq:RoughBoundGeometry\])–(\[eq:BoundForMaxBeamSeparation\]) and their relation with the specific choice of the parameters (\[eq:h\_1\_h\_0\_definition\])–(\[eq:h\_2definition\]). ### \[sub:Remark-on-Proposition\]Remarks on Proposition \[prop:TheMainBootstrapBeforeTrapping\] The bounds (\[eq:RoughBoundGeometry\])–(\[eq:BoundForMaxBeamSeparation\]) in Proposition \[prop:TheMainBootstrapBeforeTrapping\] lie at the heart of the proof of Theorem \[thm:TheTheorem\]. The precise form of the initial data (\[eq:TheIngoingVlasovInitially\]), the range (\[eq:BoundMirror\]) for the mirror radius $r_{0}$ and the asymptotic bounds (\[eq:h\_1\_h\_0\_definition\])–(\[eq:h\_2definition\]) on the parameters $h_{0},h_{1},h_{2}$ were carefuly chosen so that (\[eq:RoughBoundGeometry\])–(\[eq:BoundForMaxBeamSeparation\]) can be obtained. We will now proceed to briefly comment on the role of the bounds (\[eq:RoughBoundGeometry\])–(\[eq:BoundForMaxBeamSeparation\]) in the proof of Theorem \[thm:TheTheorem\]. The reader is advised to review first the sketch of the proof in Section \[sub:Sketch-of-the-proof\] of the introduction. Let us remark that, in the notation of Section \[sub:Sketch-of-the-proof\], $$\mathcal{E}_{\text{\textgreek{z}}_{0};n}=\tilde{m}_{n}^{(1,k+1)},$$ $$r_{\text{\textgreek{g}}_{0};n}=r_{n}^{(k,k+1)}$$ and $$r_{\text{\textgreek{g}}_{0};n}^{(1)}=r_{n}^{(1,k+1)}.$$ The bound (\[eq:RoughBoundGeometry\]) is a “trivial” bound controlling quantities related to the chosen gauge. The right hand side of (\[eq:RoughBoundGeometry\]), upon integration across any specific beam (in a direction transversal to the beam), will yield a small quantity, in view of the fact that the width of the null beams emanating from $u=0$, $v\sim v^{(j)}$ was chosen to be $\sim h_{2}(\text{\textgreek{e}})$ and, moreover, $h_{2}(\text{\textgreek{e}})$ was chosen in (\[eq:h\_2definition\]) to be small compared to the right hand side of (\[eq:RoughBoundGeometry\]). This fact will prove convenient for the proof of Proposition \[prop:TheMainBootstrapBeforeTrapping\] and Theorem \[thm:TheTheorem\], as it will enable us to “ignore” the variation of certain quantities across the width of any specific beam. That is to say, the bound (\[eq:RoughBoundGeometry\]) will enable us to frequently treat the null beams as line segments having negligible width. The bounds (\[eq:BoundForRAwayInteractionProp\])–(\[eq:UpperBoundForAxisInteractionProp\]) are quantitative expressions of the fact that the set of interactions of the beams splits into two portions, one close to $r=r_{0}$ and one close to $\mathcal{I}$. The lower bound (\[eq:EnoughMassBehind\]) is necessary in order to establish (\[eq:BoundForMassIncrease\]). In order to obtain (\[eq:EnoughMassBehind\]), it is necessary that $r_{0}$ satisfies the upper bound of (\[eq:BoundMirror\]). The upper bound (\[eq:NotEnoughMassBehind\]) implies that a trapped sphere (i.e. a sphere where $\frac{2m}{r}>1$) can not be formed at $\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j)}$ for any $j>k+1$, since one can also show that $\tilde{m}\le\tilde{m}\|_{\mathcal{I}}-\tilde{m}_{n}^{(1,k+1)}$ in those regions. In order to obtain (\[eq:NotEnoughMassBehind\]), it is necessary that the mirror radius $r_{0}$ satisfies the lower bound of (\[eq:BoundMirror\]). In the language of Section \[sub:Sketch-of-the-proof\] of the introduction, the bound (\[eq:BoundSecondBeamchanged\]) states that, when $\text{\textgreek{z}}_{0}$ reaches $\{r=r_{0}\}$ for the $n$-th time, the $r$-distance of the top beam $\text{\textgreek{z}}_{0}$ from the second-to-top beam $\text{\textgreek{z}}_{1}$, i.e. $r_{n}^{(1,k+1)}-r_{0}$, can be bounded from below by a small multiple of $r_{0}$ which is large compared to $h_{1}(\text{\textgreek{e}})r_{0}$. As a consequence of (\[eq:BoundSecondBeamchanged\]) and the bound (\[eq:BoundMirror\]) for $r_{0}$, for any $i\neq1$, $\mathcal{R}_{\text{\textgreek{e}}n}^{(i,k+1)}$ does not contain a trapped sphere. As a result, combining (\[eq:NotEnoughMassBehind\]), (\[eq:BoundForRAwayInteractionProp\]) and (\[eq:BoundSecondBeamchanged\]), we infer that, among all regions $\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j)}$, a trapped sphere can only appear for $i=1$, $j=k+1$. This fact serves to simplify the proof of Theorem \[thm:TheTheorem\], by avoiding considering multiple scenarios of trapped surface formation. Furthermore, it is crucial in obtaining (\[eq:BoundForMaxBeamSeparation\]). Establishing (\[eq:BoundSecondBeamchanged\]) is the most demanding part of the proof of Proposition \[prop:TheMainBootstrapBeforeTrapping\]. It requires obtaining a lower bound in the rate of decrease of $r_{n}^{(1,k+1)}$ in terms of the rate of increase of $\tilde{m}_{n}^{(1,k+1)}$, using also the fact that $\tilde{m}_{n}^{(1,k+1)}\lesssim r_{0}$ before a trapped sphere is formed (see the relations (\[eq:UsefulBoundAlmostThere\]) and (\[eq:ControlInMassration\]) in the next section). The bound (\[eq:BoundForMassIncrease\]) is a technical version of the bound (\[eq:InductiveEnIntro\]), and its proof follows from the ideas outlined in Section \[sub:Sketch-of-the-proof\]. In obtaining (\[eq:BoundForMassIncrease\]), the lower bound of (\[eq:EnoughMassBehind\]) is necessary. Finally, the bound (\[eq:BoundForMaxBeamSeparation\]) is a technical version of the bound (\[eq:InductiveBoundRinIntro\]) in Section \[sub:Sketch-of-the-proof\] and provides an estimate for the decrease of the multiplicative factor in the right hand side of (\[eq:BoundForMassIncrease\]). In obtaining (\[eq:BoundForMaxBeamSeparation\]) when $\frac{2m}{r}\simeq1$, the fact that $\frac{2m}{r}$ is bounded away from $1$ everywhere but on $\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j)}$ is crucially used (in particular, the bound (\[eq:BoundSecondBeamchanged\]) is necessary for (\[eq:BoundForMaxBeamSeparation\])). As is evident from the above discussion, most of the technical difficulties in the proof of Proposition \[prop:TheMainBootstrapBeforeTrapping\] are associated to issues related with the near-trapped regime $\frac{2m}{r}\simeq1$. In the case when, instead of the stronger bound (\[eq:TrappedSurfaceOccurs\]), one is merely interested in establishing the weaker instability estimate (\[eq:WeakerinstabilitySatatement\]), the proof of Proposition \[prop:TheMainBootstrapBeforeTrapping\] simplifies substantially: In that case, it is not necessary to demand that the worst instability scenario takes place in $\mathcal{R}_{\text{\textgreek{e}}n}^{(1,k+1)}$. In particular, the bounds (\[eq:NotEnoughMassBehind\]) and (\[eq:BoundSecondBeamchanged\]) can be omitted from the proof. Moreover, the lower bound for $r_{0}$ in (\[eq:BoundMirror\]) can be relaxed, and the exponentials in the relations (\[eq:h\_1\_h\_0\_definition\]) between $h_{0}(\text{\textgreek{e}}),h_{1}(\text{\textgreek{e}})$ can be replaced by polynomial functions. ### \[sub:Proof-of-Proposition\]Proof of Proposition \[prop:TheMainBootstrapBeforeTrapping\] In this section, we will make use of the $O(\cdot)$ convention: For any pair of functions $\mathcal{F},\mathcal{G}$ defined on the same domain, with $\mathcal{G}\ge0$, the notation $$\mathcal{F}=O(\mathcal{G})$$ will imply that $$\|\mathcal{F}\|\le C\cdot\mathcal{G}$$ for some universal constant $C>0$ which is independent of all the parameters in the statement of Theorem \[thm:TheTheorem\]. We should also remark that, throughout this proof, we will adopt the convention on the indices stated under (\[eq:Domains\_D\]), i.e.: 1. $U_{n}\big(v^{(-1)}\big)\doteq U_{n+1}\big(v^{(k)}\big)$. 2. $V_{n}\big(v^{(k+l)}+c\big)\doteq V_{n-1}\big(v^{(l-1)}+c\big)$ for any integer $1\le l\le k$ and any $c\ge0$. In view of (\[eq:UpperUNonTrapping\]), on $\mathcal{U}_{\text{\textgreek{e}}}^{+}$ we have $$\partial_{u}r<0<\partial_{v}r\label{eq:NonTrappingQualitativ}$$ and $$\partial_{u}\tilde{m}\le0\le\partial_{v}\tilde{m}.\label{eq:NonTrappingMassSign}$$ We will split the proof of Theorem \[thm:TheTheorem\] into two parts: In the first (and shortest) part, we will establish the bound (\[eq:RoughBoundGeometry\]) through a standard continuity argument. The proof of (\[eq:RoughBoundGeometry\]) will also yield (\[eq:BoundForRAwayInteractionProp\]) and (\[eq:UpperBoundForAxisInteractionProp\]). In the second (and more extended) part, we will establish the bounds (\[eq:EnoughMassBehind\])–(\[eq:BoundForMaxBeamSeparation\]) by induction on $n$. ### Part I: Proof of (\[eq:RoughBoundGeometry\])–(\[eq:UpperBoundForAxisInteractionProp\]) {#part-i-proof-of-eqroughboundgeometryequpperboundforaxisinteractionprop .unnumbered} Let $u_{}>0$ be such, so that on $$\mathcal{U}_{\text{\textgreek{e}}}^{}\doteq\mathcal{U}_{\text{\textgreek{e}}}^{+}\cap\{u<u_{}\},$$ we can bound $$\Big\|\log\Big(\frac{-\partial_{u}r}{1-\frac{1}{3}\Lambda r^{2}}\Big)\Big\|+\Big\|\log\Big(\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big)\Big\|\le2(h_{1}(\text{\textgreek{e}}))^{-4}\log\big((h_{3}(\text{\textgreek{e}}))^{-1}\big).\label{eq:RoughBoundGeometryBootstrap}$$ By showing that (\[eq:RoughBoundGeometry\]) holds on $\mathcal{U}_{\text{\textgreek{e}}}^{}$, it will follow (by applying a standard continuity argument) that (\[eq:RoughBoundGeometry\]) holds on the whole of $\mathcal{U}_{\text{\textgreek{e}}}^{+}$. #### Inductive formulas for $\partial_{u}r$ and $\partial_{v}r$ and proof of (\[eq:RoughBoundGeometry\]). * {#inductive-formulas-for-partial_ur-and-partial_vr-and-proof-of-eqroughboundgeometry. .unnumbered} From equation (\[eq:EquationRForProof\]), we can readily derive the following renormalised equation: $$\partial_{v}\partial_{u}\Big\{\sqrt{-\frac{3}{\Lambda}}\tan^{-1}\Big(\sqrt{-\frac{\Lambda}{3}}r\Big)\Big\}=-2\frac{\tilde{m}}{r^{2}}\frac{(1-\Lambda r^{2})}{(1-\frac{1}{3}\Lambda r^{2})}\Big(\frac{-\partial_{u}r}{1-\frac{1}{3}\Lambda r^{2}}\Big)\Big(\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big)\label{eq:RenormalisedREquation}$$ Let $n\ge1$, $0\le i\le k$, $i\le j\le k+i$, $\bar{u}<u_{}$ and $v_{b}$ be such that $$U_{n}\big(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})\big)\le\bar{u}\le U_{n}\big(v^{(i-1)}\big)$$ and $$V_{n}\big(v^{(j)}\big)\le v_{b}\le V_{n}\big(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})\big).$$ Integrating equation (\[eq:RenormalisedREquation\]) for $u=\bar{u}$ from $v=V_{n}\big(v^{(j)}\big)$ up to $v=v_{b}$, using also the fact that $$\partial_{u}\Big\{\sqrt{-\frac{3}{\Lambda}}\tan^{-1}\Big(\sqrt{-\frac{\Lambda}{3}}r\Big)\Big\}=\frac{\partial_{u}r}{1-\frac{1}{3}\Lambda r^{2}},$$ we obtain: $$\begin{aligned} \frac{-\partial_{u}r}{1-\frac{1}{3}\Lambda r^{2}}\Bigg\|_{(\bar{u},V_{n}(v^{(j)}))}=\frac{-\partial_{u}r}{1-\frac{1}{3}\Lambda r^{2}} & \Bigg\|_{(\bar{u},v_{b})}+\label{eq:SimpleIntegrationInVOverBeam}\\ & +O\Big(\sup_{\{u=\bar{u}\}\cap\{V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\le v\le V_{n}(v^{(j-1)})\}}\Big\|\frac{\tilde{m}}{r^{2}}\Big(\frac{-\partial_{u}r}{1-\frac{1}{3}\Lambda r^{2}}\Big)\Big(\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big)\Big\|h_{2}(\text{\textgreek{e}})\Big).\nonumber \end{aligned}$$ Using the bootstrap bound (\[eq:RoughBoundGeometryBootstrap\]), combined with the trivial bounds $$\tilde{m}\le\tilde{m}\|_{\mathcal{I}}=\frac{\text{\textgreek{e}}}{\sqrt{-\Lambda}}$$ and $$r\ge r_{0}$$ (following from (\[eq:DerivativeTildeUMass\]), (\[eq:DerivativeTildeVMass\]) and (\[eq:NonTrappingQualitativ\])), as well as the relation (\[eq:h\_2definition\]) for $h_{2}(\text{\textgreek{e}})$, the relation (\[eq:SimpleIntegrationInVOverBeam\]) yields: $$\frac{-\partial_{u}r}{1-\frac{1}{3}\Lambda r^{2}}\Bigg\|_{(\bar{u},V_{n}(v^{(j)}))}=\frac{-\partial_{u}r}{1-\frac{1}{3}\Lambda r^{2}}\Bigg\|_{(\bar{u},v_{b})}+O((h_{2}(\text{\textgreek{e}}))^{1/2}).\label{eq:D_uRdoesntchangemuchOverBeam}$$ Similarly, integrating (\[eq:RenormalisedREquation\]) for $v=\bar{v}$ from $u=U_{n}\big(v^{(i)}\big)$ up to $u=u_{b}$ for any $V_{n}\big(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})\big)\le\bar{v}\le V_{n}\big(v^{(j-1)}\big)$ and any $U_{n}\big(v^{(i)}\big)\le u_{b}\le U_{n}\big(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})\big)$ (assuming that $u_{b}<u_{}$), we infer: $$\frac{\partial_{v}r}{1-\frac{1}{3}\Lambda r^{2}}\Bigg\|_{(U_{n}(v^{(i)}),\bar{v})}=\frac{\partial_{v}r}{1-\frac{1}{3}\Lambda r^{2}}\Bigg\|_{(u_{b},\bar{v})}+O((h_{2}(\text{\textgreek{e}}))^{1/2}).\label{eq:D_vRdoesntchangemuchOverBeam}$$ By multiplying and dividing each factor with $1-\frac{2m}{r}=1-\frac{2\tilde{m}}{r}-\frac{1}{3}\Lambda r^{2}$, the relations (\[eq:D\_uRdoesntchangemuchOverBeam\]) and (\[eq:D\_vRdoesntchangemuchOverBeam\]) are equivalent to $$\begin{aligned} \frac{-\partial_{u}r}{1-\frac{2m}{r}}\Bigg\|_{(\bar{u},V_{n}(v^{(j)}))}= & \frac{-\partial_{u}r}{1-\frac{2m}{r}}\Bigg\|_{(\bar{u},v_{b})}\cdot\Bigg(\frac{1-\frac{2\tilde{m}}{r(1-\frac{1}{3}\Lambda r^{2})}\Big\|_{(\bar{u},v_{b})}}{1-\frac{2\tilde{m}}{r(1-\frac{1}{3}\Lambda r^{2})}\Big\|_{(\bar{u},V_{n}(v^{(j)}))}}+O((h_{2}(\text{\textgreek{e}}))^{1/2})\Bigg)\label{eq:KappaBarChangeOverBeam}\end{aligned}$$ and $$\begin{aligned} \frac{\partial_{v}r}{1-\frac{2m}{r}}\Bigg\|_{(U_{n}(v^{(i)}),\bar{v})}= & \frac{\partial_{v}r}{1-\frac{2m}{r}}\Bigg\|_{(u_{b},\bar{v})}\cdot\Bigg(\frac{1-\frac{2\tilde{m}}{r(1-\frac{1}{3}\Lambda r^{2})}\Big\|_{(u_{b},\bar{v})}}{1-\frac{2\tilde{m}}{r(1-\frac{1}{3}\Lambda r^{2})}\Big\|_{(U_{n}(v^{(i)}),\bar{v})}}+O((h_{2}(\text{\textgreek{e}}))^{1/2})\Bigg).\label{eq:KappaChangeOverBeam}\end{aligned}$$ In the vacuum case, where $\tilde{m}$ is contant, the factors in the right hand side of (\[eq:KappaBarChangeOverBeam\]) and (\[eq:KappaChangeOverBeam\]) become identically $1$. In our case, however, where matter is present, by relaxing our definition of $h_{2}$ and considering the limit $h_{2}\rightarrow0$ for fixed $\text{\textgreek{e}}$, the dominant terms in the factors in the right hand side of (\[eq:KappaBarChangeOverBeam\]) and (\[eq:KappaChangeOverBeam\]), i.e. the first summands, do not converge to $1$. This is because, in this limit, while the function $r$ remains $C^{1}$, the renormalised Hawking mass $\tilde{m}$ has a jump discontinuity across the beam. Since $T_{uu}=T_{vv}=0$ on $\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j)}$ for any $n\ge1$, any $0\le i\le k$ and any $i\le j\le k+i+1$, the relations (\[eq:DerivativeInUDirectionKappa\])–(\[eq:DerivativeInVDirectionKappaBar\]) imply that: $$\partial_{v}\Big(\frac{-\partial_{u}r}{1-\frac{2m}{r}}\Big)\Bigg\|_{\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j)}}=\partial_{u}\Big(\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big)\Bigg\|_{\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j)}}=0.$$ In particular, along lines of the form $\{u=\bar{u}\}$, the quantity $\frac{-\partial_{u}r}{1-\frac{2m}{r}}$ remains constant on $\{u=\bar{u}\}\cap\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j)}$ for any $\bar{u}<u_{}$ such that $\{u=\bar{u}\}\cap\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j)}$ is non-trivial. In view of (\[eq:KappaBarChangeOverBeam\]), the quantities $\frac{-\partial_{u}r}{1-\frac{2m}{r}}\Bigg\|_{\{u=\bar{u}\}\cap\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j+1)}}$ and $\frac{-\partial_{u}r}{1-\frac{2m}{r}}\Bigg\|_{\{u=\bar{u}\}\cap\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j)}}$ (for any $\bar{u}<u_{}$ such that $\{u=\bar{u}\}\cap\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j)}$ is non-trivial) are related by $$\frac{-\partial_{u}r}{1-\frac{2m}{r}}\Bigg\|_{\{u=\bar{u}\}\cap\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j)}}=\frac{-\partial_{u}r}{1-\frac{2m}{r}}\Bigg\|_{\{u=\bar{u}\}\cap\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j+1)}}\cdot\Bigg(\frac{1-\frac{2\tilde{m}}{r(1-\frac{1}{3}\Lambda r^{2})}\Big\|_{(\bar{u},V_{n}(v^{(j)}))}}{1-\frac{2\tilde{m}}{r(1-\frac{1}{3}\Lambda r^{2})}\Big\|_{(\bar{u},V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})))}}+O((h_{2}(\text{\textgreek{e}}))^{1/2})\Bigg).\label{eq:KappaBarChangeAdjacentDomains}$$ Similarly, the quantity $\frac{\partial_{v}r}{1-\frac{2m}{r}}$ remains constant along segments of the form $\{v=\bar{v}\}\cap\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j)}$, and $\frac{\partial_{v}r}{1-\frac{2m}{r}}\Bigg\|_{\{v=\bar{v}\}\cap\mathcal{R}_{\text{\textgreek{e}}n}^{(i+1,j)}}$ and $\frac{\partial_{v}r}{1-\frac{2m}{r}}\Bigg\|_{\{v=\bar{v}\}\cap\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j)}}$ are related (in view of (\[eq:KappaChangeOverBeam\])) by $$\begin{aligned} \frac{\partial_{v}r}{1-\frac{2m}{r}}\Bigg\|_{\{v=\bar{v}\}\cap\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j)}}= & \frac{\partial_{v}r}{1-\frac{2m}{r}}\Bigg\|_{\{v=\bar{v}\}\cap\mathcal{R}_{\text{\textgreek{e}}n}^{(i+1,j)}}\cdot\Bigg(\frac{1-\frac{2\tilde{m}}{r(1-\frac{1}{3}\Lambda r^{2})}\Big\|_{(U_{n}(v^{(i)}),\bar{v})}}{1-\frac{2\tilde{m}}{r(1-\frac{1}{3}\Lambda r^{2})}\Big\|_{(U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),\bar{v})}}+O((h_{2}(\text{\textgreek{e}}))^{1/2})\Bigg).\label{eq:KappaChangeAdjacentRegions}\end{aligned}$$ We infer, therefore, that for any point $(\bar{u},\bar{v})\in\mathcal{R}_{\text{\textgreek{e}},n}^{(i,j)}$ for some $n\ge2$, $0\le i\le k$ and $i\le j\le k+i+1$ such that $\bar{u}<u_{}$, the following relations hold between $(\bar{u},\bar{v})$ and $(\bar{u}-v_{0},\bar{v}-v_{0})\in\mathcal{R}_{\text{\textgreek{e}},n-1}^{(i,j)}$: $$\begin{aligned} \frac{-\partial_{u}r}{1-\frac{2m}{r}}\Bigg\|_{(\bar{u},\bar{v})}=\frac{-\partial_{u}r}{1-\frac{2m}{r}}\Bigg\|_{(\bar{u}-v_{0},\bar{v}-v_{0})} & \times\prod_{\bar{j}=j}^{k+i}\Bigg(\frac{1-\frac{2\tilde{m}}{r(1-\frac{1}{3}\Lambda r^{2})}\Big\|_{(\bar{u},V_{n}(v^{(\bar{j})}))}}{1-\frac{2\tilde{m}}{r(1-\frac{1}{3}\Lambda r^{2})}\Big\|_{(\bar{u},V_{n}(v^{(\bar{j})}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})))}}+O((h_{2}(\text{\textgreek{e}}))^{1/2})\Bigg)\times\label{eq:TotalChangeKappaBarEachIteration}\\ & \times\prod_{\bar{i}=i}^{k+i}\Bigg(\frac{1-\frac{2\tilde{m}}{r(1-\frac{1}{3}\Lambda r^{2})}\Big\|_{(U_{n}(v^{(\bar{i})}),\bar{u})}}{1-\frac{2\tilde{m}}{r(1-\frac{1}{3}\Lambda r^{2})}\Big\|_{(U_{n}(v^{(\bar{i})}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),\bar{u})}}+O((h_{2}(\text{\textgreek{e}}))^{1/2})\Bigg)\times\nonumber \\ & \times\prod_{\bar{j}=k+i+1}^{k+j}\Bigg(\frac{1-\frac{2\tilde{m}}{r(1-\frac{1}{3}\Lambda r^{2})}\Big\|_{(\bar{u}-v_{0},V_{n}(v^{(\bar{j})}))}}{1-\frac{2\tilde{m}}{r(1-\frac{1}{3}\Lambda r^{2})}\Big\|_{(\bar{u}-v_{0},V_{n}(v^{(\bar{j})}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})))}}+O((h_{2}(\text{\textgreek{e}}))^{1/2})\Bigg)\nonumber \end{aligned}$$ and $$\begin{aligned} \frac{\partial_{v}r}{1-\frac{2m}{r}}\Bigg\|_{(\bar{u},\bar{v})}=\frac{\partial_{v}r}{1-\frac{2m}{r}}\Bigg\|_{(\bar{u}-v_{0},\bar{v}-v_{0})} & \times\prod_{\bar{i}=i}^{j-1}\Bigg(\frac{1-\frac{2\tilde{m}}{r(1-\frac{1}{3}\Lambda r^{2})}\Big\|_{(U_{n}(v^{(\bar{i})}),\bar{v})}}{1-\frac{2\tilde{m}}{r(1-\frac{1}{3}\Lambda r^{2})}\Big\|_{(U_{n}(v^{(\bar{i})}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),\bar{v})}}+O((h_{2}(\text{\textgreek{e}}))^{1/2})\Bigg)\times\label{eq:TotalChangeKappaEachIteration}\\ & \times\prod_{\bar{j}=j}^{k+j}\Bigg(\frac{1-\frac{2\tilde{m}}{r(1-\frac{1}{3}\Lambda r^{2})}\Big\|_{(\bar{u}-v_{0},V_{n}(v^{(\bar{j})}))}}{1-\frac{2\tilde{m}}{r(1-\frac{1}{3}\Lambda r^{2})}\Big\|_{(\bar{u}-v_{0},V_{n}(v^{(\bar{j})}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})))}}+O((h_{2}(\text{\textgreek{e}}))^{1/2})\Bigg)\times\nonumber \\ & \times\prod_{\bar{i}=j}^{k+i}\Bigg(\frac{1-\frac{2\tilde{m}}{r(1-\frac{1}{3}\Lambda r^{2})}\Big\|_{(U_{n}(v^{(\bar{i})}),\bar{v}-v_{0})}}{1-\frac{2\tilde{m}}{r(1-\frac{1}{3}\Lambda r^{2})}\Big\|_{(U_{n}(v^{(\bar{i})}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),\bar{v}-v_{0})}}+O((h_{2}(\text{\textgreek{e}}))^{1/2})\Bigg).\nonumber \end{aligned}$$ The relation (\[eq:TotalChangeKappaBarEachIteration\]) is obtained as follows (see also Figure 6.1): First, (\[eq:KappaBarChangeAdjacentDomains\]) determines the evolution of $\frac{-\partial_{u}r}{1-\frac{2m}{r}}$ (according to (\[eq:KappaBarChangeAdjacentDomains\])) along the line $\{u=\bar{u}\}$ in the past direction, from $(\bar{u},\bar{v})$ up to $\text{\textgreek{g}}_{0}$. Then, using the boundary relation $$\frac{-\partial_{u}r}{1-\frac{2m}{r}}\Big\|_{\text{\textgreek{g}}_{0}}=\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big\|_{\text{\textgreek{g}}_{0}},$$ one repeats the same procedure for $\frac{\partial_{v}r}{1-\frac{2m}{r}}$ along $\{v=\bar{u}\}$ from $\text{\textgreek{g}}_{0}$ up to $\mathcal{I}$. Finally, using $$\frac{-\partial_{u}r}{1-\frac{2m}{r}}\Big\|_{\mathcal{I}}=\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big\|_{\mathcal{I}},$$ and following the evolution of $\frac{-\partial_{u}r}{1-\frac{2m}{r}}$ along $\{u=\bar{u}-v_{0}\}$ from $\mathcal{I}$ up to $(\bar{u}-v_{0},\bar{v}-v_{0})$, one arrives at (\[eq:TotalChangeKappaBarEachIteration\]). The relation (\[eq:TotalChangeKappaEachIteration\]) is similarly obtained by following the same procedure along the lines $\{v=\bar{v}\}$ (up to $\mathcal{I}$), $\{u=\bar{v}-v_{0}\}$ (from $\mathcal{I}$ up to $\text{\textgreek{g}}_{0}$) and $\{v=\bar{v}-v_{0}\}$ (from $\mathcal{I}$ up to $(\bar{u}-v_{0},\bar{v}-v_{0})$). In view of the bound $$1-\frac{2m}{r}\ge h_{3}(\text{\textgreek{e}})\label{eq:LowerBoundTrappingParameter}$$ on $\mathcal{U}_{\text{\textgreek{e}}}^{+}$ (see (\[eq:UpperUNonTrapping\])), we can estimate in the region $\{r\le\text{\textgreek{e}}^{1/2}(-\Lambda)^{1/2}\}\cap\mathcal{U}_{\text{\textgreek{e}}}^{+}$: $$1-\frac{2\tilde{m}}{r(1-\frac{1}{3}\Lambda r^{2})}=\frac{1-\frac{2m}{r}}{1-\frac{1}{3}\Lambda r^{2}}\ge\frac{1}{2}h_{1}(\text{\textgreek{e}}).\label{eq:FirstTrivial}$$ On the other hand, in the region $\{r\ge\text{\textgreek{e}}^{1/2}(-\Lambda)^{1/2}\}\cap\mathcal{U}_{\text{\textgreek{e}}}^{+}$, using (\[eq:MassInfinity\]) to bound $\tilde{m}$ we can trivially estimate (in view also of (\[eq:h\_1\_h\_0\_definition\])): $$1-\frac{2\tilde{m}}{r(1-\frac{1}{3}\Lambda r^{2})}\ge1-\frac{2\text{\textgreek{e}}}{\text{\textgreek{e}}^{\frac{1}{2}}}\ge h_{1}(\text{\textgreek{e}}).\label{eq:SecondTrivial}$$ Combining (\[eq:FirstTrivial\]) and (\[eq:SecondTrivial\]), using also the fact that $\tilde{m}\ge\tilde{m}\|_{\text{\textgreek{g}}_{0}}=0$ on $\mathcal{U}_{\text{\textgreek{e}}}^{+}$, we can bound $1-\frac{2\tilde{m}}{r(1-\frac{1}{3}\Lambda r^{2})}$ from above and below everywhere on $\mathcal{U}_{\text{\textgreek{e}}}^{+}$ as: $$\frac{1}{2}h_{3}(\text{\textgreek{e}})\le1-\frac{2\tilde{m}}{r(1-\frac{1}{3}\Lambda r^{2})}\le1.\label{eq:TrivialFactor}$$ Thus, by considering the logarithm of the relations (\[eq:TotalChangeKappaBarEachIteration\])–(\[eq:TotalChangeKappaEachIteration\]) and noting that the resulting right hand side contains $\sim k=\lceil1/h_{1}(\text{\textgreek{e}})\rceil$ summands, each controlled with the help of (\[eq:TrivialFactor\]), we readily obtain for any $n\ge2$, $0\le i\le k$ and $i\le j\le k+i+1$ and any point $(\bar{u},\bar{v})\in\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j)}$ with $\bar{u}<u_{}$: $$\Bigg\|\log\Big(\frac{-\partial_{u}r}{1-\frac{2m}{r}}\Big)\Big\|_{(\bar{u},\bar{v})}-\log\Big(\frac{-\partial_{u}r}{1-\frac{2m}{r}}\Big)\Big\|_{(\bar{u}-v_{0},\bar{v}-v_{0})}\Bigg\|\le\frac{C}{h_{1}(\text{\textgreek{e}})}\log\big((h_{3}(\text{\textgreek{e}}))^{-1}\big)\label{eq:RoughBoundForIteration}$$ and $$\Bigg\|\log\Big(\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big)\Big\|_{(\bar{u},\bar{v})}-\log\Big(\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big)\Big\|_{(\bar{u}-v_{0},\bar{v}-v_{0})}\Bigg\|\le\frac{C}{h_{1}(\text{\textgreek{e}})}\log\big((h_{3}(\text{\textgreek{e}}))^{-1}\big).\label{eq:RoughBoundForIteration-1}$$ In view of (\[eq:KappaBarChangeOverBeam\])–(\[eq:KappaChangeOverBeam\]), the bounds (\[eq:RoughBoundForIteration\]) and (\[eq:RoughBoundForIteration-1\]) (stated in the case when $(\bar{u},\bar{v})$ belongs to a vacuum region $\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j)}$) also hold when $(\bar{u},\bar{v})$ belongs to a beam, i.e. when $U_{n}\big(v^{(i)}\big)\le\bar{u}\le U_{n}\big(v^{(i-1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})\big)$ or $V_{n}\big(v^{(j)}\big)\le\bar{v}\le V_{n}\big(v^{(i-1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})\big)$ for some $n\ge2$, $0\le i\le k$ and $i\le j\le k+i+1$. Therefore, for any $n\ge2$, the bounds (\[eq:KappaBarChangeOverBeam\])–(\[eq:KappaChangeOverBeam\]) hold on the whole of $$\mathcal{U}_{\text{\textgreek{e}};n}^{}\doteq\{U_{n}(v^{(k)})\le u\le U_{n+1}(v^{(k)})\}\cap\mathcal{U}_{\text{\textgreek{e}}}^{}.$$ From (\[eq:DefinitionUntrappedRegion\]) and the definition (\[eq:defNf\]), it follows that $$n_{f}\le(h_{1}(\text{\textgreek{e}}))^{-2}.\label{eq:UpperBoundNf}$$ Since $n\le n_{f}$ (because $\mathcal{U}_{\text{\textgreek{e}}}^{}\subset\mathcal{U}_{\text{\textgreek{e}}}^{+}$), by substituting $(\bar{u},\bar{v})\rightarrow(\bar{u}-v_{0},\bar{v}-v_{0})$ in (\[eq:RoughBoundForIteration\])–(\[eq:RoughBoundForIteration-1\]) $n-2$ times and using (\[eq:UpperBoundNf\]), (\[eq:ConditionOnDvRInitiallyFamily\]), (\[eq:TrivialFactor\]) as well as the Cauchy stability estimate of Proposition \[prop:CauchyStabilityOfAdS\] for the region $\{0\le u\le2v_{0}\}$, we readily obtain $$\sup_{\mathcal{U}_{\text{\textgreek{e}}}^{}}\Bigg\{\Big\|\log\Big(\frac{-\partial_{u}r}{1-\frac{1}{3}\Lambda r^{2}}\Big)\Big\|+\Big\|\log\Big(\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big)\Big\|\Bigg\}\le\frac{C}{(h_{1}(\text{\textgreek{e}}))^{3}}\log\big((h_{3}(\text{\textgreek{e}}))^{-1}\big).\label{eq:ImprovedRoughBoundBootstrap}$$ Thus, (\[eq:RoughBoundGeometry\]) holds on $\mathcal{U}_{\text{\textgreek{e}}}^{}$ in view of the relation (\[eq:h\_1\_h\_0\_definition\]) for the parameter $h_{1}(\text{\textgreek{e}})$ (provided $\text{\textgreek{e}}_{0}$ is small enough). Therefore (as explained in the beginning of the proof), a standard continuity argument yields that (\[eq:RoughBoundGeometry\]) actually holds on the whole of $\mathcal{U}_{\text{\textgreek{e}}}^{+}$. #### Proof of (\[eq:BoundForRAwayInteractionProp\]) and (\[eq:UpperBoundForAxisInteractionProp\]). * {#proof-of-eqboundforrawayinteractionprop-and-equpperboundforaxisinteractionprop. .unnumbered} For any $1\le n\le n_{f}$, we can bound in view of the definition (\[eq:DefinitionV\_j\]) of $v^{(j)}$ and the bound (\[eq:RoughBoundGeometry\]): $$\begin{aligned} \Bigg\|\tan^{-1}\Big(\sqrt{-\frac{\Lambda}{3}}r\Big)\Big\|_{\big(U_{n}(v^{(0)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),V_{n}(v^{(k)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\big)}- & \tan^{-1}\Big(\sqrt{-\frac{\Lambda}{3}}r\Big)\Big\|_{\mathcal{I}\cap\{v=V_{n}(v^{(k)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\}}\Bigg\|\label{eq:BoundInteractionRegionAway}\\ = & \sqrt{-\frac{\Lambda}{3}}\int_{U_{n}(v^{(k)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))}^{U_{n}(v^{(0)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))}\frac{-\partial_{u}r}{1-\frac{1}{3}\Lambda r^{2}}\Big\|_{\big(u,V_{n}(v^{(k)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\big)}\, du\nonumber \\ \le & \frac{C\sqrt{-\Lambda}}{(h_{1}(\text{\textgreek{e}}))^{4}}\log\big((h_{3}(\text{\textgreek{e}}))^{-1}\big)\big\|v^{(k)}-v^{(0)}\big\|\nonumber \\ \le & \frac{C\text{\textgreek{e}}}{(h_{1}(\text{\textgreek{e}}))^{6}}\log\big((h_{3}(\text{\textgreek{e}}))^{-1}\big)\nonumber \end{aligned}$$ and $$\begin{aligned} \Bigg\|\tan^{-1}\Big(\sqrt{-\frac{\Lambda}{3}}r\Big)\Big\|_{\big(U_{n}(v^{(k)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),V_{n}(v^{(k+1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\big)}- & \tan^{-1}\Big(\sqrt{-\frac{\Lambda}{3}}r\Big)\Big\|_{\text{\textgreek{g}}_{0}\cap\{v=V_{n}(v^{(k+1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\}}\Bigg\|\label{eq:BoundInteractionRegionNear}\\ = & \sqrt{-\frac{\Lambda}{3}}\int_{U_{n}(v^{(k)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))}^{U_{n}(v^{(0)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))}\frac{-\partial_{u}r}{1-\frac{1}{3}\Lambda r^{2}}\Big\|_{\big(u,V_{n}(v^{(k+1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\big)}\, du\nonumber \\ \le & \frac{C\sqrt{-\Lambda}}{(h_{1}(\text{\textgreek{e}}))^{4}}\log\big((h_{3}(\text{\textgreek{e}}))^{-1}\big)\big\|v^{(k)}-v^{(0)}\big\|\nonumber \\ \le & \frac{C\text{\textgreek{e}}}{(h_{1}(\text{\textgreek{e}}))^{6}}\log\big((h_{3}(\text{\textgreek{e}}))^{-1}\big).\nonumber \end{aligned}$$ From (\[eq:BoundInteractionRegionAway\]) and (\[eq:BoundInteractionRegionNear\]) we readily obtain (\[eq:BoundForRAwayInteractionProp\]) and (\[eq:UpperBoundForAxisInteractionProp\]), respectively, in view of the relations (\[eq:h\_1\_h\_0\_definition\]) and (\[eq:h\_3definition\]) for $h_{1},h_{3}$, respectively, and the fact that $r\|_{\text{\textgreek{g}}_{0}}=r_{0}$, $r\|_{\mathcal{I}}=+\infty$. ### Part II: Proof of (\[eq:EnoughMassBehind\])–(\[eq:BoundForMaxBeamSeparation\]) {#part-ii-proof-of-eqenoughmassbehindeqboundformaxbeamseparation .unnumbered} We will now proceed to establish the bounds (\[eq:EnoughMassBehind\])–(\[eq:BoundForMaxBeamSeparation\]). To this end, we will first derive some useful estimates for the differences of the renormalied masses $\tilde{m}_{n}^{(i,j)}$ associated to the vacuum regions around each interaction region $\mathcal{N}_{\text{\textgreek{e}}n}^{(i,j)}$.[^16] #### Relations for the change in the mass difference of the beams. {#relations-for-the-change-in-the-mass-difference-of-the-beams. .unnumbered} Let us introduce the notion of the mass difference for the beams $\{U_{n}(v^{(i)})\le u\le U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\}$ and $\{V_{n}(v^{(j)})\le v\le V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\}$ around their interaction region $\mathcal{N}_{\text{\textgreek{e}}n}^{(i,j)}$: For any $1\le n\le n_{f}$, $0\le i\le k$ and $i+1\le j\le k+i$, we define the initial mass differences $$\begin{aligned} (\mathfrak{D}_{-}\tilde{m})_{n}^{(i,j)} & \doteq\tilde{m}_{n}^{(i+1,j+1)}-\tilde{m}_{n}^{(i,j+1)}\label{eq:DefinitionIncomingEnergy}\\ (\overline{\mathfrak{D}}_{-}\tilde{m})_{n}^{(i,j)} & \doteq\tilde{m}_{n}^{(i+1,j)}-\tilde{m}_{n}^{(i+1,j+1)}\nonumber \end{aligned}$$ and the final mass differences $$\begin{aligned} (\mathfrak{D}_{+}\tilde{m})_{n}^{(i,j)} & \doteq\tilde{m}_{n}^{(i+1,j)}-\tilde{m}_{n}^{(i,j)}\label{eq:DefinitionOutcomingEnergy}\\ (\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(i,j)} & \doteq\tilde{m}_{n}^{(i,j)}-\tilde{m}_{n}^{(i,j+1)}.\nonumber \end{aligned}$$ Note that $(\mathfrak{D}_{-}\tilde{m})_{n}^{(i,j)}$ and $(\mathfrak{D}_{+}\tilde{m})_{n}^{(i,j)}$ are the mass differences around the outgoing beam $\{U_{n}(v^{(i)})\le u\le U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\}$ before and after crossing the region $\mathcal{N}_{\text{\textgreek{e}}n}^{(i,j)}$, respectively, while $(\overline{\mathfrak{D}}_{-}\tilde{m})_{n}^{(i,j)}$ and $(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(i,j)}$ are the mass differences around the ingoing beam $\{V_{n}(v^{(j)})\le v\le U_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\}$ before and after crosssing the region $\mathcal{N}_{\text{\textgreek{e}}n}^{(i,j)}$. Note the trivial identity $$(\mathfrak{D}_{-}\tilde{m})_{n}^{(i,j)}+(\overline{\mathfrak{D}}_{-}\tilde{m})_{n}^{(i,j)}=(\mathfrak{D}_{+}\tilde{m})_{n}^{(i,j)}+(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(i,j)}.\label{eq:ConservationOfMassDifference}$$ We will establish the following bounds for any $1\le n\le n_{f}$, $1\le i\le k$ and $i+1\le j\le k+i$: $$(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(i,j)}=(\overline{\mathfrak{D}}_{-}\tilde{m})_{n}^{(i,j)}\cdot\exp\Bigg(\frac{2}{\bar{r}_{n}^{(i,j)}}\frac{(\mathfrak{D}_{-}\tilde{m})_{n}^{(i,j)}}{1-\frac{2\tilde{m}_{n}^{(i+1,j)}}{\bar{r}_{n}^{(i,j)}}-\frac{1}{3}\Lambda(\bar{r}_{n}^{(i,j)})^{2}}\big(1-\mathfrak{Err}_{1,n}^{(i,j)}\big)\big(1-\mathfrak{Err}_{\backslash n}^{(i,j)}\big)\Bigg)\label{eq:MassDifferenceIncreaseInUDirection}$$ and: $$(\mathfrak{D}_{+}\tilde{m})_{n}^{(i,j)}=(\mathfrak{D}_{-}\tilde{m})_{n}^{(i,j)}\cdot\exp\Bigg(-\frac{2}{\bar{r}_{n}^{(i,j)}}\frac{(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(i,j)}}{1-\frac{2\tilde{m}_{n}^{(i+1,j)}}{\bar{r}_{n}^{(i,j)}}-\frac{1}{3}\Lambda(\bar{r}_{n}^{(i,j)})^{2}}\big(1-\mathfrak{Err}_{1,n}^{(i,j)}\big)\big(1-\mathfrak{Err}_{/n}^{(i,j)}\big)\Bigg),\label{eq:MassDifferenceDecreaseInVDirection}$$ where the terms $\mathfrak{Err}_{1,n}^{(i,j)}$ in (\[eq:MassDifferenceIncreaseInUDirection\]) and (\[eq:MassDifferenceDecreaseInVDirection\]) are allowed to be different from each other, but they both satisfy the bound $$0\le\mathfrak{Err}_{1,n}^{(i,j)}\le1-\frac{\bar{r}_{n}^{(i,j)}-2\tilde{m}_{n}^{(i+1,j)}-\frac{1}{3}\Lambda(\bar{r}_{n}^{(i,j)})^{3}}{r\big\|_{\big(U_{n}(v^{(i)}),V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\big)}-2\tilde{m}_{n}^{(i,j+1)}-\frac{1}{3}\Lambda r^{3}\big\|_{\big(U_{n}(v^{(i)}),V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\big)}}\label{eq:BoundErrorTermForInteraction}$$ and $\mathfrak{Err}_{\backslash n}^{(i,j)}$, $\mathfrak{Err}_{/n}^{(i,j)}$ satisfy the bounds $$0\le\mathfrak{Err}_{\backslash n}^{(i,j)}\le1-\frac{(\mathfrak{D}_{+}\tilde{m})_{n}^{(i,j)}}{(\mathfrak{D}_{-}\tilde{m})_{n}^{(i,j)}}\label{eq:BoundErrorTermIngoingInteraction}$$ and $$0\le\mathfrak{Err}_{/n}^{(i,j)}\le1-\frac{(\overline{\mathfrak{D}}_{-}\tilde{m})_{n}^{(i,j)}}{(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(i,j)}}.\label{eq:BoundErrorTermOugoingInteraction}$$ Moreover, the following estimate will be useful in the proof of (\[eq:BoundSecondBeamchanged\]): For any $1\le n\le n_{f}$, $1\le i\le k$ and $k+1\le j\le k+i$, $$(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(i,j)}\ge(\overline{\mathfrak{D}}_{-}\tilde{m})_{n}^{(i,j)}\cdot\exp\Bigg(\frac{1}{5C_{0}}\frac{(\mathfrak{D}_{-}\tilde{m})_{n}^{(i,j)}}{\bar{r}_{n}^{(i,j)}}\Bigg).\label{eq:UsefulEstimate}$$ Notice that, as a consequence of (\[eq:MassDifferenceIncreaseInUDirection\]) and (\[eq:MassDifferenceDecreaseInVDirection\]), during the interaction of the two beams at $\mathcal{N}_{\text{\textgreek{e}}n}^{(i,j)}$, the mass difference $\overline{\mathfrak{D}}\tilde{m}$ of the ingoing beam increases, while the mass difference $\mathfrak{D}\tilde{m}$ of the outgoing beam decrases. Proof of (\[eq:MassDifferenceIncreaseInUDirection\]) and (\[eq:MassDifferenceDecreaseInVDirection\]). By differentiating (\[eq:DerivativeTildeVMass\]) in $u$ and using (\[eq:DerivativeInUDirectionKappa\]) and (\[eq:ConservationT\_vv\]), we readily obtain the following wave-type equation for $\tilde{m}$: $$\partial_{u}\partial_{v}\tilde{m}=-F(r,\tilde{m})\partial_{u}\tilde{m}\partial_{v}\tilde{m},\label{eq:WaveEquationMass}$$ where $$F(r,\tilde{m})\doteq\frac{2}{r-2\tilde{m}-\frac{1}{3}\Lambda r^{3}}.\label{eq:OriginalFMassEquation}$$ Note that, formally, equation (\[eq:WaveEquationMass\]) can be rewritten as $$\partial_{v}\log(-\partial_{u}\tilde{m})=-F(r,\tilde{m})\partial_{v}\tilde{m}\label{eq:OutgoingEquationMass}$$ or $$\partial_{u}\log(\partial_{v}\tilde{m})=F(r,\tilde{m})(-\partial_{u}\tilde{m})\label{eq:IngoingEquationMass}$$ (note, however, that $\log(-\partial_{u}\tilde{m})$, $\log(\partial_{v}\tilde{m})$ will not be well defined when $\partial_{u}\tilde{m}=0$ or $\partial_{v}\tilde{m}=0$). For any $1\le n\le n_{f}$, $1\le i\le k$ and $i+1\le j\le k+i$, integrating equation (\[eq:WaveEquationMass\]) first in $u$, for $U_{n}(v^{(i)}))\le u\le U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))$, and then in $v$, for $V_{n}(v^{(j)})\le v\le V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))$, we obtain: $$\tilde{m}_{n}^{(i,j)}-\tilde{m}_{n}^{(i,j+1)}=\int_{V_{n}(v^{(j)})}^{V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))}\partial_{v}\tilde{m}\|_{(U_{n}(v^{(i)})),v)}\cdot\exp\Big(2\int_{U_{n}(v^{(i)})}^{U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))}\frac{-\partial_{u}\tilde{m}}{r-2m}\Big\|_{(u,v)}\, du\Big)\, dv.\label{eq:BeforeIngoingDifference}$$ Note that, at the formal level, the derivation of (\[eq:BeforeIngoingDifference\]) is easiest seen by integrating equation (\[eq:IngoingEquationMass\]) first in $u$, then exponentiating, and then integrating in $v$. This procedure can actually be done rigorously, since $\partial_{u}\tilde{m}<0<\partial_{v}\tilde{m}$ in the interior of $\mathcal{N}_{\text{\textgreek{e}}n}^{(i,j)}$, in view of (\[eq:DerivativeTildeUMass\]), (\[eq:DerivativeTildeVMass\]) and (\[eq:TuuSupport\])–(\[eq:TvvSupport\]). In view of (\[eq:NonTrappingQualitativ\])–(\[eq:NonTrappingMassSign\]), we can bound for any $U_{n}(v^{(i)})\le u\le U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))$ and any $V_{n}(v^{(j)})\le v\le V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))$: $$\begin{aligned} \bar{r}_{n}^{(i,j)}-2\tilde{m}_{n}^{(i+1,j)}-\frac{1}{3}\Lambda(\bar{r}_{n}^{(i,j)})^{3}\le\big(r- & 2m\big)\Big\|_{(u,v)}\le\label{eq:TrivialBoundR}\\ & \le r\big\|_{\big(U_{n}(v^{(i)}),V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\big)}-2\tilde{m}_{n}^{(i,j+1)}-\frac{1}{3}\Lambda r^{3}\big\|_{\big(U_{n}(v^{(i)}),V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\big)},\nonumber \end{aligned}$$ where $$\bar{r}_{n}^{(i,j)}\doteq r\|_{(U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),V_{n}(v^{(j)}))}.$$ Therefore, using (\[eq:TrivialBoundR\]) to estimate $\frac{1}{r-2m}$, from (\[eq:BeforeIngoingDifference\]) we readily infer that: $$\tilde{m}_{n}^{(i,j)}-\tilde{m}_{n}^{(i,j+1)}=\int_{V_{n}(v^{(j)})}^{V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))}\partial_{v}\tilde{m}\|_{(U_{n}(v^{(i)})),v)}\cdot\exp\Bigg(\frac{2}{\bar{r}_{n}^{(i,j)}}\frac{\tilde{m}\|_{(U_{n}(v^{(i)}),v)}-\tilde{m}\|_{(U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),v)}}{1-\frac{2\tilde{m}_{n}^{(i+1,j)}}{\bar{r}_{n}^{(i,j)}}-\frac{1}{3}\Lambda(\bar{r}_{n}^{(i,j)})^{2}}\big(1-\mathfrak{Err}_{1,n}^{(i,j)}(v)\big)\Bigg)\, dv,\label{eq:bla}$$ where, for any $V_{n}(v^{(j)})\le v\le V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))$, $\mathfrak{Err}_{1,n}^{(i,j)}(v)$ satisfies the bound (\[eq:BoundErrorTermForInteraction\]). Equations (\[eq:DerivativeTildeUMass\]), (\[eq:DerivativeInVDirectionKappaBar\]) and (\[eq:EquationT\_uu\]) imply that, for any $V_{n}(v^{(j)})\le v\le V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))$, $$\partial_{v}\big(\tilde{m}\|_{(U_{n}(v^{(i)}),v)}-\tilde{m}\|_{(U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),v)}\big)\le0$$ and, therefore, for any $V_{n}(v^{(j)})\le v\le V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))$: $$(\mathfrak{D}_{+}\tilde{m})_{n}^{(i,j)}\le\tilde{m}\|_{(U_{n}(v^{(i)}),v)}-\tilde{m}\|_{(U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),v)}\le(\mathfrak{D}_{-}\tilde{m})_{n}^{(i,j)}.\label{eq:MonotonicitymassDifference}$$ The bound (\[eq:MonotonicitymassDifference\]) implies that (\[eq:bla\]) can be expressed as $$\tilde{m}_{n}^{(i,j)}-\tilde{m}_{n}^{(i,j+1)}=(\tilde{m}_{n}^{(i+1,j)}-\tilde{m}_{n}^{(i+1,j+1)})\cdot\exp\Bigg(\frac{2}{\bar{r}_{n}^{(i,j)}}\frac{(\mathfrak{D}_{-}\tilde{m})_{n}^{(i,j)}}{1-\frac{2\tilde{m}_{n}^{(i+1,j)}}{\bar{r}_{n}^{(i,j)}}-\frac{1}{3}\Lambda(\bar{r}_{n}^{(i,j)})^{2}}\big(1-\mathfrak{Err}_{1,n}^{(i,j)}\big)\big(1-\mathfrak{Err}_{\backslash n}^{(i,j)}\big)\Bigg)\label{eq:bla-1}$$ where $\mathfrak{Err}_{1,n}^{(i,j)}$ satisfies the bound (\[eq:BoundErrorTermForInteraction\]) and $\mathfrak{Err}_{\backslash n}^{(i,j)}$ satisfies the bound (\[eq:BoundErrorTermIngoingInteraction\]). In view of (\[eq:DefinitionIncomingEnergy\]) and (\[eq:DefinitionOutcomingEnergy\]), (\[eq:bla-1\]) is equivalent to (\[eq:MassDifferenceIncreaseInUDirection\]). Similarly, integrating equation (\[eq:WaveEquationMass\]) first in $v$, for $V_{n}(v^{(j)})\le v\le V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))$, and then in $u$, for $U_{n}(v^{(i)}))\le u\le U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))$ (see also (\[eq:OutgoingEquationMass\])), we obtain (\[eq:MassDifferenceDecreaseInVDirection\]). Proof of (\[eq:UsefulEstimate\]). Recall $F$ defined by (\[eq:OriginalFMassEquation\]) and let us define the function $\bar{F}:\mathcal{D}_{\bar{F}}\rightarrow(0,+\infty)$, where $$\mathcal{D}_{\bar{F}}=\big\{(x,y)\in\mathbb{R}^{2}\mbox{ }x>0\mbox{ and }x-y-\frac{2}{3}\Lambda x^{2}>0\big\},$$ by the relation $$\bar{F}(x,y)\doteq\frac{2}{x-y-\frac{2}{3}\Lambda x^{2}}.\label{eq:ModifiedF}$$ Note that, in view of (\[eq:RoughBoundGeometry\]), (\[eq:UpperBoundForAxisInteractionProp\]), (\[eq:BoundMirror\]) and (\[eq:h\_2definition\]), for any $\text{\textgreek{m}}\ge0$ for which $$\inf_{(u,v)\in\mathcal{N}_{\text{\textgreek{e}}n}^{(i,j)}}\big\{ r(u,v)-2\text{\textgreek{m}}-\frac{1}{3}\Lambda r^{2}(u,v)\big\}>h_{3}(\text{\textgreek{e}}),\label{eq:ConditionFWellDefined}$$ we can readily bound: $$\max_{(u,v)\in\mathcal{N}_{\text{\textgreek{e}}n}^{(i,j)}}\bar{F}(r(u,v),\text{\textgreek{m}})<\min_{(u,v)\in\mathcal{N}_{\text{\textgreek{e}}n}^{(i,j)}}F(r(u,v),\text{\textgreek{m}})\label{eq:BoundForLemma}$$ and $$\partial_{\text{\textgreek{m}}}\bar{F}(r(u,v),\text{\textgreek{m}}),\mbox{ }\partial_{\text{\textgreek{m}}}F(r(u,v),\text{\textgreek{m}})>0\label{eq:IncreasingForLemma}$$ (note that $F(r\|_{\mathcal{N}_{\text{\textgreek{e}}n}^{(i,j)}},\text{\textgreek{m}})$ and $\bar{F}(r\|_{\mathcal{N}_{\text{\textgreek{e}}n}^{(i,j)}},\text{\textgreek{m}})$ are well-defined and positive under the condition (\[eq:ConditionFWellDefined\])). For any $1\le n\le n_{f}$, $1\le i\le k$ and $k+1\le j\le k+i$, let us consider the following characteristic initial value problem on $\mathcal{N}_{\text{\textgreek{e}}n}^{(i,j)}$: $$\begin{cases} \partial_{u}\partial_{v}\bar{m}=-\bar{F}(r,\bar{m})\partial_{u}\bar{m}\partial_{v}\bar{m} & \mbox{on }\mathcal{N}_{\text{\textgreek{e}}n}^{(i,j)},\\ \bar{m}=\tilde{m} & \mbox{on }[U_{n}(v^{(i)})),U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))]\times\{V_{n}(v^{(j)}))\}\cup\\ & \hphantom{\mbox{on }\cup}\cup\{U_{n}(v^{(i)}))\}\times[V_{n}(v^{(j)})),V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))]. \end{cases}\label{eq:ModifiedInitialValueProblem}$$ Note that $\tilde{m}$ satisfies the same characteristic initial value problem with $F(r,\tilde{m})$ in place of $\bar{F}(r,\bar{m})$. Notice also that, in view of (\[eq:DerivativeTildeUMass\])–(\[eq:DerivativeTildeVMass\]) and (\[eq:TuuSupport\])–(\[eq:TvvSupport\]), the initial data for $\tilde{m}$ and $\bar{m}$ satisfy: $$\partial_{u}\tilde{m}<0\mbox{ on}\Big\{ U_{n}(v^{(i)})<u<U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\Big\}\label{eq:TuuSupport-1}$$ and $$\partial_{v}\tilde{m}>0\mbox{ on}\Big\{ V_{n}(v^{(j)})<u<V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\Big\}.\label{eq:TvvSupport-1}$$ Therefore, in view of (\[eq:BoundForLemma\]), (\[eq:IncreasingForLemma\]) and (\[eq:TuuSupport-1\])–(\[eq:TvvSupport-1\]), an application of Lemma \[lem:HyperbolicMaximumPrinciple\] (see Section \[sub:Auxiliary-lemmas\]) with $\tilde{m},\bar{m}$ in place of $z_{2},z_{1}$, respectively, yields the following a priori bounds for a solution $\bar{m}$ of (\[eq:ModifiedInitialValueProblem\]): $$\bar{m}\le\tilde{m}\mbox{ on }\mathcal{N}_{\text{\textgreek{e}}n}^{(i,j)}\label{eq:AprioriBoundMtilde'}$$ and $$\partial_{u}\bar{m}<0<\partial_{v}\bar{m}\mbox{ in the interior of }\mathcal{N}_{\text{\textgreek{e}}n}^{(i,j)}.\label{eq:MonotonicityFromLemma}$$ Notice that the a priori bound (\[eq:AprioriBoundMtilde’\]) and the initial data in (\[eq:ModifiedInitialValueProblem\]) imply that $\bar{m}\ge0$ and that (\[eq:ConditionFWellDefined\]) holds for $\text{\textgreek{m}}=\tilde{m}$ and $\text{\textgreek{m}}=\bar{m}$; in particular, $\bar{F}(r,\bar{m})$ is well defined and positive on $\mathcal{N}_{\text{\textgreek{e}}n}^{(i,j)}$. Thus, it readily follows (using standard arguments) that (\[eq:ModifiedInitialValueProblem\]) indeed has a unique smooth solution $\bar{m}$ satisfying (\[eq:AprioriBoundMtilde’\]). With $\bar{m}$ defined on $\mathcal{N}_{\text{\textgreek{e}}n}^{(i,j)}$ as above for any $1\le n\le n_{f}$, $1\le i\le k$ and $k+1\le j\le k+i$, we will define the following modified versions of (\[eq:DefinitionIncomingEnergy\]) and (\[eq:DefinitionOutcomingEnergy\]): $$\begin{aligned} (\mathfrak{D}_{-}\bar{m})_{n}^{(i,j)} & \doteq\bar{m}\|_{(U_{n}(v^{(i)}),V_{n}(v^{(j)}))}-\bar{m}\|_{(U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),V_{n}(v^{(j)}))}\label{eq:DefinitionIncomingEnergyModified}\\ (\overline{\mathfrak{D}}_{-}\bar{m})_{n}^{(i,j)} & \doteq\bar{m}\|_{(U_{n}(v^{(i)}),V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})))}-\bar{m}\|_{(U_{n}(v^{(i)}),V_{n}(v^{(j)}))}\nonumber \end{aligned}$$ and $$\begin{aligned} (\mathfrak{D}_{+}\bar{m})_{n}^{(i,j)} & \doteq\bar{m}\|_{(U_{n}(v^{(i)}),V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})))}-\bar{m}\|_{(U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})))}\label{eq:DefinitionOutcomingEnergyModified}\\ (\overline{\mathfrak{D}}_{+}\bar{m})_{n}^{(i,j)} & \doteq\bar{m}\|_{(U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})))}-\bar{m}\|_{(U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),V_{n}(v^{(j)}))}.\nonumber \end{aligned}$$ Note that, in view of the initial data for (\[eq:ModifiedInitialValueProblem\]): $$\begin{aligned} (\mathfrak{D}_{-}\bar{m})_{n}^{(i,j)} & =(\mathfrak{D}_{-}\tilde{m})_{n}^{(i,j)}\label{eq:EqualityTwoEnergiesInitially}\\ (\overline{\mathfrak{D}}_{-}\bar{m})_{n}^{(i,j)} & =(\overline{\mathfrak{D}}_{-}\tilde{m})_{n}^{(i,j)},\nonumber \end{aligned}$$ while, in view of the bound (\[eq:AprioriBoundMtilde’\]) (and the initial data for (\[eq:ModifiedInitialValueProblem\])): $$\begin{aligned} (\mathfrak{D}_{+}\bar{m})_{n}^{(i,j)} & \ge(\mathfrak{D}_{+}\tilde{m})_{n}^{(i,j)}\label{eq:InequalityOutcomingEnergies}\\ (\overline{\mathfrak{D}}_{+}\bar{m})_{n}^{(i,j)} & \le(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(i,j)}.\nonumber \end{aligned}$$ By repeating exactly the same steps that led to (\[eq:MassDifferenceIncreaseInUDirection\]) and (\[eq:MassDifferenceDecreaseInVDirection\]) but using (\[eq:ModifiedInitialValueProblem\]) instead of (\[eq:WaveEquationMass\]), we obtain for any $1\le n\le n_{f}$, $1\le i\le k$ and $k+1\le j\le k+i$: $$(\overline{\mathfrak{D}}_{+}\bar{m})_{n}^{(i,j)}=(\overline{\mathfrak{D}}_{-}\bar{m})_{n}^{(i,j)}\cdot\exp\Bigg(\frac{2}{\bar{r}_{n}^{(i,j)}}\frac{(\mathfrak{D}_{-}\bar{m})_{n}^{(i,j)}}{1-\frac{\tilde{m}_{n}^{(i+1,j)}}{\bar{r}_{n}^{(i,j)}}-\frac{2}{3}\Lambda(\bar{r}_{n}^{(i,j)})^{2}}\big(1-\mathfrak{\overline{Err}}_{1,n}^{(i,j)}\big)\big(1-\mathfrak{\overline{Err}}_{\backslash n}^{(i,j)}\big)\Bigg)\label{eq:MassDifferenceIncreaseInUDirectionModified}$$ and $$(\mathfrak{D}_{+}\bar{m})_{n}^{(i,j)}=(\mathfrak{D}_{-}\bar{m})_{n}^{(i,j)}\cdot\exp\Bigg(-\frac{2}{\bar{r}_{n}^{(i,j)}}\frac{(\overline{\mathfrak{D}}_{+}\bar{m})_{n}^{(i,j)}}{1-\frac{\tilde{m}_{n}^{(i+1,j)}}{\bar{r}_{n}^{(i,j)}}-\frac{2}{3}\Lambda(\bar{r}_{n}^{(i,j)})^{2}}\big(1-\mathfrak{\overline{Err}}_{1,n}^{(i,j)}\big)\big(1-\mathfrak{\overline{Err}}_{/n}^{(i,j)}\big)\Bigg),\label{eq:MassDifferenceDecreaseInVDirectionModified}$$ where $$0\le\mathfrak{\overline{Err}}_{1,n}^{(i,j)}\le1-\frac{\bar{r}_{n}^{(i,j)}-\tilde{m}_{n}^{(i+1,j)}-\frac{2}{3}\Lambda(\bar{r}_{n}^{(i,j)})^{3}}{r\big\|_{\big(U_{n}(v^{(i)}),V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\big)}-\tilde{m}_{n}^{(i,j+1)}-\frac{2}{3}\Lambda r^{3}\big\|_{\big(U_{n}(v^{(i)}),V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\big)}},\label{eq:BoundErrorTermForInteraction-1}$$ $$0\le\mathfrak{\overline{Err}}_{\backslash n}^{(i,j)}\le1-\frac{(\mathfrak{D}_{+}\bar{m})_{n}^{(i,j)}}{(\mathfrak{D}_{-}\bar{m})_{n}^{(i,j)}}\label{eq:BoundErrorTermIngoingInteraction-1-1}$$ and $$0\le\mathfrak{\overline{Err}}_{/n}^{(i,j)}\le1-\frac{(\overline{\mathfrak{D}}_{-}\bar{m})_{n}^{(i,j)}}{(\overline{\mathfrak{D}}_{+}\bar{m})_{n}^{(i,j)}}\label{eq:BoundErrorTermOugoingInteraction-1}$$ (and, as before, we allow the terms $\mathfrak{\overline{Err}}_{1,n}^{(i,j)}$ in (\[eq:MassDifferenceIncreaseInUDirectionModified\]) and (\[eq:MassDifferenceDecreaseInVDirectionModified\]) to be different). Because $$\tilde{m}_{n}^{(i+1,j)}\le\tilde{m}\|_{\mathcal{I}}\le\frac{2}{3}r_{0}\le\frac{2}{3}\bar{r}_{n}^{(i,j)}\label{eq:UpperBoundMassFromR}$$ (in view of (\[eq:MassInfinity\]), (\[eq:BoundMirror\]), (\[eq:NonTrappingQualitativ\]) and (\[eq:NonTrappingMassSign\])), from (\[eq:MassDifferenceDecreaseInVDirectionModified\]) (using also (\[eq:UpperBoundForAxisInteractionProp\]) and the fact that $\mathfrak{\overline{Err}}{}_{1,n}^{(i,j)},\mathfrak{\overline{Err}}{}_{/n}^{(i,j)}\ge0$) we can estimate for any $1\le n\le n_{f}$, $1\le i\le k$ and $k+1\le j\le k+i$: $$\begin{aligned} (\mathfrak{D}_{+}\bar{m})_{n}^{(i,j)} & =(\mathfrak{D}_{-}\bar{m})_{n}^{(i,j)}\cdot\exp\Bigg(-\frac{2}{\bar{r}_{n}^{(i,j)}}\frac{(\overline{\mathfrak{D}}_{+}\bar{m})_{n}^{(i,j)}}{1-\frac{\tilde{m}_{n}^{(i+1,j)}}{\bar{r}_{n}^{(i,j)}}-\frac{2}{3}\Lambda(\bar{r}_{n}^{(i,j)})^{2}}\big(1-\mathfrak{\overline{Err}}{}_{1,n}^{(i,j)}\big)\big(1-\mathfrak{\overline{Err}}{}_{/n}^{(i,j)}\big)\Bigg)\label{eq:OneMoreSillyEstimate}\\ & \ge(\mathfrak{D}_{-}\bar{m})_{n}^{(i,j)}\cdot\exp\Bigg(-\frac{8(\overline{\mathfrak{D}}_{+}\bar{m})_{n}^{(i,j)}}{\bar{r}_{n}^{(i,j)}}\Bigg)\nonumber \end{aligned}$$ In view of the fact that $$(\overline{\mathfrak{D}}_{+}\bar{m})_{n}^{(i,j)}\le(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(i,j)}=\tilde{m}_{n}^{(i,j)}-\tilde{m}_{n}^{(i,j+1)}\le\tilde{m}\|_{\mathcal{I}}-0\le\frac{2}{3}r_{0}\le\frac{2}{3}\bar{r}_{n}^{(i,j)}$$ (following from (\[eq:InequalityOutcomingEnergies\])), (\[eq:OneMoreSillyEstimate\]) yields $$(\mathfrak{D}_{+}\bar{m})_{n}^{(i,j)}\ge e^{-\frac{16}{3}}(\mathfrak{D}_{-}\bar{m})_{n}^{(i,j)}.\label{eq:DonAlready}$$ In view of (\[eq:BoundErrorTermIngoingInteraction-1-1\]), (\[eq:DonAlready\]) implies that $$1-\mathfrak{\overline{Err}}{}_{\backslash n}^{(i,j)}\ge\frac{1}{C_{0}}.\label{eq:BoundForErrorInteraction}$$ Using (\[eq:BoundForErrorInteraction\]) in (\[eq:MassDifferenceIncreaseInUDirectionModified\]), we obtain: $$(\overline{\mathfrak{D}}_{+}\bar{m})_{n}^{(i,j)}\ge(\overline{\mathfrak{D}}_{-}\bar{m})_{n}^{(i,j)}\cdot\exp\Bigg(\frac{2}{C_{0}\bar{r}_{n}^{(i,j)}}\frac{(\mathfrak{D}_{-}\bar{m})_{n}^{(i,j)}}{1-\frac{\tilde{m}_{n}^{(i+1,j)}}{\bar{r}_{n}^{(i,j)}}-\frac{2}{3}\Lambda(\bar{r}_{n}^{(i,j)})^{2}}\big(1-\mathfrak{\overline{Err}}_{1,n}^{(i,j)}\big)\Bigg).\label{eq:MassDifferenceIncreaseInUDirectionModified-1}$$ In view of (\[eq:UpperBoundMassFromR\]) and (\[eq:UpperBoundForAxisInteractionProp\]), we can also estimate $$\frac{1-\mathfrak{\overline{Err}}{}_{1,n}^{(i,j)}}{1-\frac{\tilde{m}_{n}^{(i+1,j)}}{\bar{r}_{n}^{(i,j)}}-\frac{2}{3}\Lambda(\bar{r}_{n}^{(i,j)})^{2}}\ge\frac{1}{10}$$ and, thus, (\[eq:MassDifferenceIncreaseInUDirectionModified-1\]) yields: $$(\overline{\mathfrak{D}}_{+}\bar{m})_{n}^{(i,j)}\ge(\overline{\mathfrak{D}}_{-}\bar{m})_{n}^{(i,j)}\cdot\exp\Bigg(\frac{1}{5C_{0}}\frac{(\mathfrak{D}_{-}\bar{m})_{n}^{(i,j)}}{\bar{r}_{n}^{(i,j)}}\Bigg).\label{eq:MassDifferenceIncreaseInUDirectionModified-1-1}$$ From (\[eq:MassDifferenceIncreaseInUDirectionModified-1-1\]) and the relations (\[eq:EqualityTwoEnergiesInitially\]) and (\[eq:InequalityOutcomingEnergies\]), we readily obtain (\[eq:UsefulEstimate\]). #### Proof of (\[eq:NotEnoughMassBehind\]). {#proof-of-eqnotenoughmassbehind. .unnumbered} For any $1\le n\le n_{f}$, from (\[eq:MassDifferenceIncreaseInUDirection\]) we readily obtain that, for any $1\le i\le k$: $$(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(i,k+1)}\ge(\overline{\mathfrak{D}}_{-}\tilde{m})_{n}^{(i,k+1)}.\label{eq:Increasemassingoing}$$ Applying (\[eq:Increasemassingoing\]) successively for $i=1,2,\ldots k$, using also the identity $$(\overline{\mathfrak{D}}_{-}\tilde{m})_{n}^{(i,j)}=(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(i+1,j)}\label{eq:EqualMassDifferenceWhennoIntraction}$$ (which follows from the fact that $\tilde{m}$ is constant over each $\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j)}$), we thus infer that, for any $1\le i\le k$: $$(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(i,k+1)}\ge(\overline{\mathfrak{D}}_{-}\tilde{m})_{n}^{(k,k+1)}=(\tilde{m}_{n-1}^{(0,0)}-\tilde{m}_{n-1}^{(0,1)}).\label{eq:TotalWeakBoundIngoing}$$ Since $$\tilde{m}_{n-1}^{(0,0)}=\tilde{m}_{n-1}^{(1,1)}=\tilde{m}\|_{\mathcal{I}}$$ and $$(\mathfrak{D}_{+}\tilde{m})_{n-1}^{(0,1)}=\tilde{m}_{n-1}^{(1,1)}-\tilde{m}_{n-1}^{(0,1)}=(\overline{\mathfrak{D}}_{-}\tilde{m})_{n}^{(k,k+1)},\label{eq:EqualMassDifferenceSfterReflection}$$ from (\[eq:TotalWeakBoundIngoing\]) we infer that, for any $1\le i\le k$: $$(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(i,k+1)}\ge(\mathfrak{D}_{+}\tilde{m})_{n-1}^{(0,1)}.\label{eq:WeakBoundIngoingFinalOutgoing}$$ Similarly as for the derivation of (\[eq:Increasemassingoing\]), applying the relation (\[eq:MassDifferenceDecreaseInVDirection\]) successively for $i=0$ and $j=1,2,\ldots k$ (with $n-1$ in place of $n$), we infer: $$(\mathfrak{D}_{+}\tilde{m})_{n-1}^{(0,1)}=(\mathfrak{D}_{-}\tilde{m})_{n-1}^{(0,k)}\cdot\exp\Bigg(-\sum_{j=1}^{k}\frac{2}{\bar{r}_{n-1}^{(0,j)}}\frac{(\overline{\mathfrak{D}}_{-}\tilde{m})_{n-1}^{(0,j)}}{1-\frac{2\tilde{m}_{n-1}^{(0,j)}}{\bar{r}_{n-1}^{(0,j)}}-\frac{1}{3}\Lambda(\bar{r}_{n-1}^{(0,j)})^{2}}\big(1-\mathfrak{Err}_{1,n-1}^{(0,j)}\big)\big(1-\mathfrak{Err}_{/n-1}^{(0,j)}\big)\Bigg).\label{eq:MassDifferenceDecreaseInVDirection-1-1}$$ In view of the bound (\[eq:MassInfinity\]) for the total mass $\tilde{m}\|_{\mathcal{I}}$, the lower bound (\[eq:BoundForRAwayInteractionProp\]) for $r_{n}^{(0,k)}$ and the fact that $$\bar{r}_{n-1}^{(0,j)}\le r_{n-1}^{(0,k)}\le\bar{r}_{n-1}^{(0,j)}\big(1+(h_{2}(\text{\textgreek{e}}))^{1/2}\big)$$ for $1\le j\le k$ (following from (\[eq:RoughBoundGeometry\]), and (\[eq:h\_2definition\])), we can estimate $$\sum_{j=1}^{k}\frac{2}{\bar{r}_{n-1}^{(0,j)}}\frac{(\overline{\mathfrak{D}}_{-}\tilde{m})_{n-1}^{(0,j)}}{1-\frac{2\tilde{m}_{n-1}^{(0,j)}}{\bar{r}_{n-1}^{(0,j)}}-\frac{1}{3}\Lambda(\bar{r}_{n-1}^{(0,j)})^{2}}\big(1-\mathfrak{Err}_{1,n-1}^{(0,j)}\big)\big(1-\mathfrak{Err}_{/n-1}^{(0,j)}\big)\le\text{\textgreek{e}}^{\frac{3}{2}}.\label{eq:BoundForDecrease}$$ Therefore, (\[eq:MassDifferenceDecreaseInVDirection-1-1\]) yields: $$\log\frac{(\mathfrak{D}_{+}\tilde{m})_{n-1}^{(0,1)}}{(\mathfrak{D}_{-}\tilde{m})_{n-1}^{(0,k)}}\ge-\text{\textgreek{e}}^{3/2}.\label{eq:TotalMassDecreaseTopInteraction-1}$$ From (\[eq:WeakBoundIngoingFinalOutgoing\]) and (\[eq:TotalMassDecreaseTopInteraction-1\]) we thus infer that, for any $1\le i\le k$ and any $2\le n\le n_{f}$: $$\log\frac{(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(i,k+1)}}{\tilde{m}_{n-1}^{(1,k+1)}}\ge-\text{\textgreek{e}}^{3/2}.\label{eq:LowerBoundGeneralInteraction}$$ From (\[eq:LowerBoundGeneralInteraction\]) for $i=1$ and the fact that, for any $1\le n\le n_{f}$: $$(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(1,k+1)}=(\mathfrak{D}_{-}\tilde{m})_{n}^{(0,k)}=\tilde{m}_{n}^{(1,k+1)},$$ we thus infer that, for all $2\le n\le n_{f}$: $$\log\frac{\tilde{m}_{n}^{(1,k+1)}}{\tilde{m}_{n-1}^{(1,k+1)}}\ge-\text{\textgreek{e}}^{3/2}.\label{eq:LowerboundQuotientMass}$$ Applying (\[eq:LowerboundQuotientMass\]) successively $n-1$ times, we thus infer for any $2\le n\le n_{f}$: $$\log\frac{\tilde{m}_{n}^{(1,k+1)}}{\tilde{m}_{1}^{(1,k+1)}}\ge-\text{\textgreek{e}}^{3/2}(n-1).\label{eq:FinalLowerBoundTopMass}$$ The bound (\[eq:BoundMirror\]) for $r_{0}$ and the form (\[eq:TheIngoingVlasovInitially\]) of the initial data imply that $$\frac{2(\tilde{m}_{/}(v^{(0)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))-\tilde{m}_{/}(v^{(0)}))}{r_{0}}\ge\frac{4}{C_{0}}h_{0}(\text{\textgreek{e}}).\label{eq:BoundForTheMirror-1}$$ Therefore, from (\[eq:TotalWeakBoundIngoing\]) and (\[eq:BoundForTheMirror-1\]) we infer that, for all $1\le i\le1$: $$\frac{2(\overline{\mathfrak{D}}_{+}\tilde{m})_{1}^{(i,k+1)}}{r_{0}}\ge\frac{4}{C_{0}}h_{0}(\text{\textgreek{e}}).\label{eq:TrivialBoundMassFirst}$$ From (\[eq:FinalLowerBoundTopMass\]) and (\[eq:TrivialBoundMassFirst\]) for $i=1$ (when $(\overline{\mathfrak{D}}_{+}\tilde{m})_{1}^{(1,k+1)}=\tilde{m}_{1}^{(1,k+1)}$), using also the fact that $n_{f}\le(h_{1}(\text{\textgreek{e}}))^{-2}$, we thus deduce that, for all $1\le n\le n_{f}$: $$\frac{2\tilde{m}_{n}^{(1,k+1)}}{r_{0}}\ge\frac{2}{C_{0}}h_{0}(\text{\textgreek{e}}).\label{eq:AllSoTrivialBounds}$$ The relations (\[eq:MassInfinity\]) and (\[eq:AllSoTrivialBounds\]) readily yield (\[eq:NotEnoughMassBehind\]). #### Proof of (\[eq:EnoughMassBehind\]). {#proof-of-eqenoughmassbehind. .unnumbered} In view of the bound (\[eq:LowerBoundTrappingParameter\]), we infer that, for any $1\le n\le n_{f}$: $$1-\frac{2\tilde{m}_{n}^{(1,k+1)}}{r\|_{\big(U_{n}(v^{(1)}),V_{n}(v^{(k+1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\big)}}-\frac{1}{3}\Lambda r^{2}\|_{\big(U_{n}(v^{(1)}),V_{n}(v^{(k+1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\big)}\ge h_{3}(\text{\textgreek{e}}).\label{eq:FromTrivialBoundForTrapping}$$ Using the bounds $$\frac{r\|_{\big(U_{n}(v^{(1)}),V_{n}(v^{(k+1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\big)}}{r_{0}}\le1+(h_{2}(\text{\textgreek{e}}))^{1/2}$$ (derived from (\[eq:RoughBoundGeometry\]), (\[eq:UpperBoundForAxisInteractionProp\]) and (\[eq:h\_2definition\])) and $$\frac{r_{0}}{\frac{2}{\sqrt{-\Lambda}}\text{\textgreek{e}}-\frac{1}{3}\Lambda r_{0}^{3}}<1-\frac{1}{2}\exp\big(-2(h_{0}(\text{\textgreek{e}}))^{-4}\big)\label{eq:UpperboundMirror}$$ (from (\[eq:BoundMirror\])), as well as the relation (\[eq:MassInfinity\]) for $\tilde{m}\|_{\mathcal{I}}$, we can readily derive from (\[eq:FromTrivialBoundForTrapping\]) that: $$\frac{2(\tilde{m}\|_{\mathcal{I}}-\tilde{m}_{n}^{(1,k+1)})}{r_{0}}\ge2\big(1+(h_{2}(\text{\textgreek{e}}))^{1/2}\big)^{-1}\exp\big(-2(h_{0}(\text{\textgreek{e}}))^{-4}\big)+\frac{1}{3}\Lambda r_{0}^{2}\big(1+(h_{2}(\text{\textgreek{e}}))^{1/2}\big).\label{eq:AlmostThereTrivial}$$ The bound (\[eq:EnoughMassBehind\]) follows readily from (\[eq:UpperboundMirror\]) and (\[eq:AlmostThereTrivial\]). #### Proof of (\[eq:BoundForMassIncrease\]). {#proof-of-eqboundformassincrease. .unnumbered} For any $2\le n\le n_{f}$, applying the relation (\[eq:MassDifferenceIncreaseInUDirection\]) successively for $j=k+1$ and $i=1,2,\ldots k$, using also the identity (\[eq:EqualMassDifferenceWhennoIntraction\]) and the trivial bound $$(\mathfrak{D}_{-}\tilde{m})_{n}^{(i,j)}\big(1-\mathfrak{Err}_{\backslash n}^{(i,k+1)}\big)\ge(\mathfrak{D}_{+}\tilde{m})_{n}^{(i,k+1)}$$ (following directly from (\[eq:BoundErrorTermIngoingInteraction\])), we obtain: $$\begin{aligned} (\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(1,k+1)} & =(\tilde{m}_{n-1}^{(0,0)}-\tilde{m}_{n-1}^{(0,1)})\cdot\exp\Bigg(\sum_{i=1}^{k}\frac{2}{\bar{r}_{n}^{(i,k+1)}}\frac{(\mathfrak{D}_{-}\tilde{m})_{n}^{(i,k+1)}}{1-\frac{2\tilde{m}_{n}^{(i+1,k+1)}}{\bar{r}_{n}^{(i,j)}}-\frac{1}{3}\Lambda(\bar{r}_{n}^{(i,k+1)})^{2}}\big(1-\mathfrak{Err}_{1,n}^{(i,k+1)}\big)\big(1-\mathfrak{Err}_{\backslash n}^{(i,k+1)}\big)\Bigg)\label{eq:MassDifferenceIncreaseInUDirection-1}\\ & \ge(\tilde{m}_{n-1}^{(0,0)}-\tilde{m}_{n-1}^{(0,1)})\cdot\exp\Bigg(\sum_{i=1}^{k}\frac{2}{\bar{r}_{n}^{(i,k+1)}}\frac{(\mathfrak{D}_{+}\tilde{m})_{n}^{(i,k+1)}}{1-\frac{2\tilde{m}_{n}^{(i+1,k+1)}}{\bar{r}_{n}^{(i,j)}}-\frac{1}{3}\Lambda(\bar{r}_{n}^{(i,k+1)})^{2}}\big(1-\mathfrak{Err}_{1,n}^{(i,k+1)}\big)\Bigg).\nonumber \end{aligned}$$ In view of (\[eq:RoughBoundGeometry\]), (\[eq:h\_2definition\]), (\[eq:BoundErrorTermForInteraction\]), (\[eq:UpperBoundForAxisInteractionProp\]) and the fact that $$r_{0}\le\bar{r}_{n}^{(i,k+1)}\le\bar{r}_{n}^{(k,k+1)},$$ for $1\le i\le k$ (following from (\[eq:NonTrappingQualitativ\])), we can bound for any $1\le i\le k$: $$\begin{gathered} \frac{2}{\bar{r}_{n}^{(i,k+1)}}\frac{1}{1-\frac{2\tilde{m}_{n}^{(i+1,k+1)}}{\bar{r}_{n}^{(i,j)}}-\frac{1}{3}\Lambda(\bar{r}_{n}^{(i,k+1)})^{2}}\big(1-\mathfrak{Err}_{1,n}^{(i,k+1)}\big)\\ \ge2\min\Bigg\{\frac{1}{r\big\|_{\big(U_{n}(v^{(i)}),V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\big)}-2\tilde{m}_{n}^{(i,j+1)}-\frac{1}{3}\Lambda r^{3}\big\|_{\big(U_{n}(v^{(i)}),V_{n}(v^{(j)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\big)}},\frac{1}{\bar{r}_{n}^{(i,j)}-2\tilde{m}_{n}^{(i+1,k+1)}-\frac{1}{3}\Lambda(\bar{r}_{n}^{(i,k+1)})^{3}}\Bigg\}\\ \ge\frac{2-O(\text{\textgreek{e}})}{r_{n}^{(k,k+1)}}.\label{eq:LowerBoundNearRInteraction}\end{gathered}$$ Furthermore, $$\sum_{i=1}^{k}(\mathfrak{D}_{+}\tilde{m})_{n}^{(i,k+1)}=\sum_{i=1}^{k}(\tilde{m}_{n}^{(i+1,k+1)}-\tilde{m}_{n}^{(i,k+1)})=\tilde{m}\|_{\mathcal{I}}-\tilde{m}_{n}^{(1,k+1)}.\label{eq:TotalMassDifferenceTopInteraction}$$ Therefore, in view of (\[eq:EnoughMassBehind\]), (\[eq:LowerBoundNearRInteraction\]), (\[eq:TotalMassDifferenceTopInteraction\]) and the fact that $$(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(1,k+1)}=\tilde{m}_{n}^{(1,k+1)}-\tilde{m}_{n}^{(1,k+2)}=\tilde{m}_{n}^{(1,k+1)},$$ the bound (\[eq:MassDifferenceIncreaseInUDirection-1\]) yields $$\begin{aligned} \tilde{m}_{n}^{(1,k+1)} & \ge(\tilde{m}_{n-1}^{(0,0)}-\tilde{m}_{n-1}^{(0,1)})\cdot\exp\Big(\frac{2-O(\text{\textgreek{e}})}{r_{n}^{(k,k+1)}}\big(\tilde{m}\|_{\mathcal{I}}-\tilde{m}_{n}^{(1,k+1)}\big)\Big)\label{eq:TotalMassIncreaseTopInteraction}\\ & \ge(\tilde{m}_{n-1}^{(0,0)}-\tilde{m}_{n-1}^{(0,1)})\cdot\exp\Big(\frac{r_{0}}{2r_{n}^{(k,k+1)}}\exp\big(-2(h_{0}(\text{\textgreek{e}}))^{-4}\big)\Big).\nonumber \end{aligned}$$ Using the bound (\[eq:TotalMassDecreaseTopInteraction-1\]) and the fact that $$(\mathfrak{D}_{+}\tilde{m})_{n-1}^{(0,1)}=\tilde{m}_{n-1}^{(1,1)}-\tilde{m}_{n-1}^{(0,1)}=\tilde{m}_{n-1}^{(0,0)}-\tilde{m}_{n-1}^{(0,1)}$$ and $$(\mathfrak{D}_{-}\tilde{m})_{n-1}^{(0,k)}=\tilde{m}_{n-1}^{(1,k+1)},$$ we can estimate: $$(\tilde{m}_{n-1}^{(0,0)}-\tilde{m}_{n-1}^{(0,1)})\ge e^{-\text{\textgreek{e}}^{3/2}}\tilde{m}_{n-1}^{(1,k+1)}.\label{eq:TotalDecreaseOnceMore}$$ From (\[eq:TotalMassIncreaseTopInteraction\]) and (\[eq:TotalDecreaseOnceMore\]) we thus obtain (in view also of (\[eq:UpperBoundForAxisInteractionProp\]) and the properties (\[eq:h\_1\_h\_0\_definition\]) of $h_{0}(\text{\textgreek{e}})$): $$\tilde{m}_{n}^{(1,k+1)}\ge\tilde{m}_{n-1}^{(1,k+1)}\exp\Big(\frac{r_{0}}{4r_{n}^{(k,k+1)}}\exp\big(-2(h_{0}(\text{\textgreek{e}}))^{-4}\big)\Big).\label{eq:BoundToShowForMass}$$ In particular, (\[eq:BoundForMassIncrease\]) holds for all $2\le n\le n_{f}$. #### Proof of (\[eq:BoundSecondBeamchanged\]). {#proof-of-eqboundsecondbeamchanged. .unnumbered} Combining (\[eq:LowerBoundGeneralInteraction\]) and (\[eq:NotEnoughMassBehind\]) (using also (\[eq:TrivialBoundMassFirst\]) in the case $n=1$, as well as (\[eq:MassInfinity\]) and (\[eq:BoundMirror\]) for $\tilde{m}\|_{\mathcal{I}}$, $r_{0}$), we can readily estimate for any $1\le n\le n_{f}$ and any $1\le i\le k$: $$\frac{2(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(i,k+1)}}{r_{0}}\ge\frac{1}{C_{0}}h_{0}(\text{\textgreek{e}}).\label{eq:BoundForEnergyRemainingForTheOtherBemas}$$ Similarly, in view of (\[eq:EqualMassDifferenceWhennoIntraction\]), (\[eq:EqualMassDifferenceSfterReflection\]) and (\[eq:TotalMassDecreaseTopInteraction-1\]), we can bound for any $1\le n\le n_{f}$ and any $1\le i\le k$: $$\frac{2(\overline{\mathfrak{D}}_{-}\tilde{m})_{n}^{(i,k+1)}}{r_{0}}\ge\frac{1}{C_{0}}h_{0}(\text{\textgreek{e}}).\label{eq:BoundForInitialMassesInteraction}$$ Using the relation $$(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(i,k+1)}=\tilde{m}_{n}^{(i,k+1)}-\tilde{m}_{n}^{(i,k+2)}$$ and the trivial bounds $$\tilde{m}_{n}^{(i,k+1)}\le\tilde{m}\|_{\mathcal{I}}$$ and $$\max_{k+2\le j\le k+i+1}\tilde{m}_{n}^{(i,j)}=\tilde{m}_{n}^{(i,k+2)}$$ (following from the monotonicity properties (\[eq:NonTrappingMassSign\]) of $\tilde{m}$), from (\[eq:BoundForEnergyRemainingForTheOtherBemas\]) we obtain for any $1\le n\le n_{f}$: $$\min_{\substack{1\le i\le k,\\ k+2\le j\le k+i+1 } }\frac{2(\tilde{m}\|_{\mathcal{I}}-\tilde{m}_{n}^{(i,j)})}{r_{0}}\ge\frac{1}{C_{0}}h_{0}(\text{\textgreek{e}}).\label{eq:ActualNotEnoughEnergy}$$ In view of (\[eq:MassInfinity\]) and (\[eq:BoundMirror\]) and the properties (\[eq:h\_1\_h\_0\_definition\]) of $h_{0}(\text{\textgreek{e}})$, from (\[eq:ActualNotEnoughEnergy\]) we infer that, for any $1\le n\le n_{f}$: $$\max_{\substack{1\le i\le k,\\ k+2\le j\le k+i+1 } }\frac{2\tilde{m}_{n}^{(i,j)}}{r_{0}}\le1-\frac{1}{3C_{0}}h_{0}(\text{\textgreek{e}}).\label{eq:AwayFromTrappingBehindTheFirstBeam}$$ In particular, for any $1\le n\le n_{f}$, (\[eq:AwayFromTrappingBehindTheFirstBeam\]) implies that: $$\sup_{\{u\le v\le V_{n}(v^{(k+1)})\}\cap\{u\ge U_{n}(v^{(k)})\}}\frac{1}{1-\frac{2m}{r}}\le4C_{0}(h_{0}(\text{\textgreek{e}}))^{-1}.\label{eq:UpperBoundForTrappingBetween TwoBeams}$$ The main estimate that will be used in the proof of (\[eq:BoundSecondBeamchanged\]) is the following bound: For any $1\le n_{1}<n_{2}\le n_{f}$ and any $V_{n_{2}}(v^{(k+2)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\le v\le V_{n_{2}}(v^{(k+1)})$: $$\begin{aligned} \partial_{v}r\big\|_{(U_{n_{2}}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),v)}\ge & \partial_{v}r\big\|_{(U_{n_{1}-1}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),v-(n_{2}-n_{1}-1)v_{0})}\times\label{eq:UsefulBoundAlmostThere}\\ & \hphantom{\partial_{v}r}\times\exp\Big(-C_{0}^{5}(h_{0}(\text{\textgreek{e}}))^{-3}\max_{n_{1}\le n\le n_{2}}\Big\{\frac{r_{n}^{(1,k+1)}}{r_{0}}\Big\}\log\Big(\frac{\tilde{m}{}_{n_{2}}^{(1,k+1)}}{\tilde{m}{}_{n_{1}}^{(1,k+1)}}\Big)-2\text{\textgreek{e}}^{1/2}\Big).\nonumber \end{aligned}$$ Proof of (\[eq:UsefulBoundAlmostThere\]). For any $1\le n\le n_{f}$, $1\le i\le k$ and $k+1\le j\le k+i$, integrating (\[eq:DerivativeInUDirectionKappa\]) from $u=U_{n}(v^{(i)})$ up to $U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))$ (and using (\[eq:DerivativeTildeUMass\])), we infer that, for all $V_{n}(v^{(j+1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\le\bar{v}\le V_{n}(v^{(j)})$: $$\begin{aligned} \log\Big(\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big)\Big\|_{(U_{n}(v^{(i)}),\bar{v})}-\log\Big(\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big) & \Big\|_{(U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),\bar{v})}\label{eq:FormulaForComparisonMassDiffAndRChange-1}\\ = & 4\pi\int_{U_{n}(v^{(i)})}^{U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))}\frac{rT_{uu}}{-\partial_{u}r}\Big\|_{(u,\bar{v})}\, du\nonumber \\ = & 2\int_{U_{n}(v^{(i)})}^{U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))}\frac{-\partial_{u}\tilde{m}}{r(1-\frac{2m}{r})}\Big\|_{(u,\bar{v})}\, du.\nonumber \end{aligned}$$ In view of the monotonicity properties (\[eq:NonTrappingQualitativ\]), (\[eq:NonTrappingMassSign\]) of $r$ and $\tilde{m}$, we can estimate: $$\begin{aligned} 2\int_{U_{n}(v^{(i)})}^{U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))}\frac{-\partial_{u}\tilde{m}}{r(1-\frac{2m}{r})}\Big\|_{(u,\bar{v})}\, du & \le\sup_{U_{n}(v^{(i)})\le\bar{u}\le U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))}\Big(\frac{2}{r(1-\frac{2m}{r})}\Big)\Big\|_{(u,\bar{v})}\int_{U_{n}(v^{(i)})}^{U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))}(-\partial_{u}\tilde{m})\Big\|_{(u,\bar{v})}\, du\label{eq:OneMoreTrivialBound}\\ & \le\frac{2}{r(U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),\bar{v})}\sup_{U_{n}(v^{(i)})\le\bar{u}\le U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))}\Bigg(\frac{1}{1-\frac{2m}{r}\big\|_{(\bar{u},\bar{v})}}\Bigg)(\mathfrak{D}_{-}\tilde{m})_{n}^{(i,j)}.\nonumber \end{aligned}$$ From (\[eq:FormulaForComparisonMassDiffAndRChange-1\]), (\[eq:OneMoreTrivialBound\]) and the bound (\[eq:UpperBoundForTrappingBetween TwoBeams\]) for $1-\frac{2m}{r}$, we infer that $$\begin{aligned} \log\Big(\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big)\Big\|_{(U_{n}(v^{(i)}),\bar{v})}- & \log\Big(\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big)\Big\|_{(U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),\bar{v})}\label{eq:FormulaForComparisonMassDiffAndRChange-1-1}\\ \le & 8C_{0}(h_{0}(\text{\textgreek{e}}))^{-1}\frac{\bar{r}_{n}^{(i,j)}}{r(U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),\bar{v})}\cdot\frac{(\mathfrak{D}_{-}\tilde{m})_{n}^{(i,j)}}{\bar{r}_{n}^{(i,j)}}.\nonumber \end{aligned}$$ Notice that, in view of the bound (\[eq:UsefulEstimate\]), we can estimate: $$\frac{(\mathfrak{D}_{-}\tilde{m})_{n}^{(i,j)}}{\bar{r}_{n}^{(i,j)}}\le5C_{0}\log\Big(\frac{(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(i,j)}}{(\overline{\mathfrak{D}}_{-}\tilde{m})_{n}^{(i,j)}}\Big).\label{eq:ComeOn!!}$$ Thus, from (\[eq:FormulaForComparisonMassDiffAndRChange-1-1\]) and (\[eq:ComeOn!!\]), we deduce that, for any $1\le n\le n_{f}$, $1\le i\le k$ and $k+1\le j\le k+i$ and any $V_{n}(v^{(j+1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\le\bar{v}\le V_{n}(v^{(j)})$: $$\begin{aligned} \log\Big(\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big)\Big\|_{(U_{n}(v^{(i)}),\bar{v})}- & \log\Big(\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big)\Big\|_{(U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),\bar{v})}\le\label{eq:FormulaForComparisonMassDiffAndRChange}\\ \le & 40C_{0}^{2}(h_{0}(\text{\textgreek{e}}))^{-1}\frac{\bar{r}_{n}^{(i,j)}}{r(U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),\bar{v})}\log\Big(\frac{(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(i,j)}}{(\overline{\mathfrak{D}}_{-}\tilde{m})_{n}^{(i,j)}}\Big).\nonumber \end{aligned}$$ Applying the relation (\[eq:FormulaForComparisonMassDiffAndRChange\]) successively for $i=1,\ldots,k$ and $\bar{v}=v$, using also the fact that $\partial_{u}\Big(\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big)=0$ on each $\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j)}$, we obtain: $$\begin{aligned} \frac{\partial_{v}r}{1-\frac{2m}{r}}\Big\|_{(U_{n}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),\bar{v})}\ge & \frac{\partial_{v}r}{1-\frac{2m}{r}}\Big\|_{(U_{n-1}(v^{(0)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),\bar{v})}\times\label{eq:SecondRelationForRDifference}\\ & \hphantom{\int_{B}^{B}}\times\exp\Bigg(-40C_{0}^{2}(h_{0}(\text{\textgreek{e}}))^{-1}\sum_{i=1}^{k}\frac{\bar{r}_{n}^{(i,k+1)}}{r(U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),\bar{v})}\log\Big(\frac{(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(i,k+1)}}{(\overline{\mathfrak{D}}_{-}\tilde{m})_{n}^{(i,k+1)}}\Big)\Bigg).\nonumber \end{aligned}$$ For any $i=1,\ldots,k$, integrating (\[eq:DerivativeInUDirectionKappa\]) in $u$ from $u=U_{n}(v^{(i)})$ up to $U_{n}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))$ for $v=v_{}\in[\bar{v},V_{n}(v^{(j)})]$ and using (\[eq:DerivativeTildeUMass\]) and the fact that $\partial_{u}\tilde{m}=0$ on $\mathcal{R}_{\text{\textgreek{e}}n}^{(i,j)}$, we infer: $$\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big\|_{(U_{n}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),v_{})}=\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big\|_{(U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),v_{})}\exp\Bigg(-2\int_{U_{n}(v^{(i)})}^{U_{n}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))}\frac{-\partial_{u}\tilde{m}}{r-2\tilde{m}-\frac{1}{3}\Lambda r^{3}}\Big\|_{(u,v_{})}\, du\Bigg).\label{eq:BoundForDvRFirst}$$ Using the fact that $r\ge r_{0}$, from (\[eq:BoundForDvRFirst\]) we infer (in view of the monotonicity property (\[eq:NonTrappingMassSign\]) for $\tilde{m}$) that $$\begin{aligned} \frac{\partial_{v}r}{1-\frac{2m}{r}}\Big\|_{(U_{n}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),v_{})}\ge & \frac{\partial_{v}r}{1-\frac{2m}{r}}\Big\|_{(U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),v_{})}\exp\Bigg(-2\int_{U_{n}(v^{(i)})}^{U_{n}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))}\frac{-\partial_{u}\tilde{m}\|_{(u,v_{})}}{r_{0}-2\tilde{m}\|_{(u,v_{})}-\frac{1}{3}\Lambda r_{0}^{3}}\, du\Bigg)\label{eq:BoundForDvRSecond}\\ & =\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big\|_{(U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),v_{})}\exp\Bigg(-\int_{U_{n}(v^{(i)})}^{U_{n}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))}\partial_{u}\Big(\log\big(1-\frac{2\tilde{m}\|_{(u,v_{})}}{r_{0}}-\frac{1}{3}\Lambda r_{0}^{2}\big)\Big)\, du\Bigg)\nonumber \\ & =\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big\|_{(U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),v_{})}\cdot\Bigg(\frac{1-\frac{2\tilde{m}\|_{(U_{n}(v^{(i)}),v_{})}}{r_{0}}-\frac{1}{3}\Lambda r_{0}^{2}}{1-\frac{2\tilde{m}\|_{(U_{n}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),v_{})}}{r_{0}}-\frac{1}{3}\Lambda r_{0}^{2}}\Bigg).\nonumber \end{aligned}$$ In view of the bounds (\[eq:UpperBoundForAxisInteractionProp\]) and (\[eq:AwayFromTrappingBehindTheFirstBeam\]), (\[eq:BoundForDvRSecond\]) yields: $$\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big\|_{(U_{n}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),v_{})}\ge\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big\|_{(U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),v_{})}\frac{h_{0}(\text{\textgreek{e}})}{4C_{0}}.\label{eq:AtLast!!}$$ Integrating (\[eq:AtLast!!\]) in $v_{}\in[\bar{v},V_{n}(v^{(j)})]$ and using (\[eq:UpperBoundForTrappingBetween TwoBeams\]) (and (\[eq:UpperBoundForAxisInteractionProp\])) for the $\frac{1}{1-\frac{2m}{r}}$ factors, we thus obtain: $$\bar{r}_{n}^{(1,k+1)}-r_{0}\ge\Big(\bar{r}_{n}^{(i,k+1)}-r(U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),\bar{v})\Big)\frac{(h_{0}(\text{\textgreek{e}}))^{2}}{16C_{0}^{2}}$$ and, thus (in view of (\[eq:RoughBoundGeometry\]), (\[eq:h\_2definition\]) and the fact that $r_{0}\le\min\{r_{n}^{(1,k+1)},r(U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),\bar{v})\}$): $$\frac{\bar{r}_{n}^{(i,k+1)}}{r(U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),\bar{v})}\le16C_{0}^{2}(h_{0}(\text{\textgreek{e}}))^{-2}\frac{r_{n}^{(1,k+1)}}{r_{0}}.\label{eq:ForQuotientOfR}$$ From (\[eq:ForQuotientOfR\]) and (\[eq:EqualMassDifferenceWhennoIntraction\]), it follows that: $$\begin{aligned} \sum_{i=1}^{k}\frac{\bar{r}_{n}^{(i,k+1)}}{r(U_{n}(v^{(i)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),v)} & \log\Big(\frac{(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(i,k+1)}}{(\overline{\mathfrak{D}}_{-}\tilde{m})_{n}^{(i,k+1)}}\Big)\label{eq:EstimateForTheExponentialInChange}\\ \le & 16C_{0}^{2}(h_{0}(\text{\textgreek{e}}))^{-2}\frac{r_{n}^{(1,k+1)}}{r_{0}}\sum_{i=1}^{k}\log\Big(\frac{(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(i,k+1)}}{(\overline{\mathfrak{D}}_{-}\tilde{m})_{n}^{(i,k+1)}}\Big)\nonumber \\ = & 16C_{0}^{2}(h_{0}(\text{\textgreek{e}}))^{-2}\frac{r_{n}^{(1,k+1)}}{r_{0}}\Big\{\sum_{i=1}^{k-1}\log\Big(\frac{(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(i,k+1)}}{(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(i+1,k+1)}}\Big)+\log\Big(\frac{(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(k,k+1)}}{\tilde{m}\|_{\mathcal{I}}-\tilde{m}_{n-1}^{(0,1)}}\Big)\Big\}\nonumber \\ = & 16C_{0}^{2}(h_{0}(\text{\textgreek{e}}))^{-2}\frac{r_{n}^{(1,k+1)}}{r_{0}}\log\Big(\frac{(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(1,k+1)}}{\tilde{m}\|_{\mathcal{I}}-\tilde{m}_{n-1}^{(0,1)}}\Big)\nonumber \\ = & 16C_{0}^{2}(h_{0}(\text{\textgreek{e}}))^{-2}\frac{r_{n}^{(1,k+1)}}{r_{0}}\log\Big(\frac{(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(1,k+1)}}{(\mathfrak{D}_{+}\tilde{m})_{n-1}^{(0,1)}}\Big)\nonumber \\ \le & 16C_{0}^{2}(h_{0}(\text{\textgreek{e}}))^{-2}\frac{r_{n}^{(1,k+1)}}{r_{0}}\Big\{\log\Big(\frac{(\overline{\mathfrak{D}}_{+}\tilde{m})_{n}^{(1,k+1)}}{(\mathfrak{D}_{-}\tilde{m})_{n-1}^{(0,k)}}\Big)+\text{\textgreek{e}}^{3/2}\Big\}\nonumber \\ = & 16C_{0}^{2}(h_{0}(\text{\textgreek{e}}))^{-2}\frac{r_{n}^{(1,k+1)}}{r_{0}}\Big\{\log\Big(\frac{\tilde{m}{}_{n}^{(1,k+1)}}{\tilde{m}{}_{n-1}^{(1,k+1)}}\Big)+\text{\textgreek{e}}^{3/2}\Big\}\nonumber \end{aligned}$$ (where the inequality at the sixth line of (\[eq:EstimateForTheExponentialInChange\]) follows from (\[eq:TotalMassDecreaseTopInteraction-1\])). Therefore, (\[eq:SecondRelationForRDifference\]) and (\[eq:EstimateForTheExponentialInChange\]) yield for any $1\le n\le n_{f}$ and any $V_{n}(v^{(k+2)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\le v\le V_{n}(v^{(k+1)})$: $$\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big\|_{(U_{n}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),v)}\ge\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big\|_{(U_{n-1}(v^{(0)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),v)}\exp\Big(-C_{0}^{5}(h_{0}(\text{\textgreek{e}}))^{-3}\frac{r_{n}^{(1,k+1)}}{r_{0}}\Big\{\log\Big(\frac{\tilde{m}{}_{n}^{(1,k+1)}}{\tilde{m}{}_{n-1}^{(1,k+1)}}\Big)+\text{\textgreek{e}}^{3/2}\Big\}\Big).\label{eq:LowerBoundDvRBetweenBeams}$$ In view of (\[eq:DerivativeInVDirectionKappaBar\]), we can bound for any $U_{n-1}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\le u\le U_{n-1}(v^{(0)})$: $$\frac{-\partial_{u}r}{1-\frac{2m}{r}}\Big\|_{(u,V_{n-1}(v^{(1)}))}\ge\frac{-\partial_{u}r}{1-\frac{2m}{r}}\Big\|_{(u,V_{n-1}(v^{(k+1)})}.\label{eq:LowerBoundDuRBetweenBeams}$$ Hence, using the fact that - [ From (\[eq:ZeroMassNearAxis\]), (\[eq:UpperBoundForAxisInteractionProp\]): $$\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big\|_{(U_{n}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),v)}=\frac{\partial_{v}r}{1-\frac{1}{3}\Lambda r^{2}}\Big\|_{(U_{n}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),v)}=(1+O(\text{\textgreek{e}}))\partial_{v}r\|_{(U_{n}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),v)},$$ ]{} - [ From (\[eq:EquationRForProof\]), (\[eq:MassInfinity\]), (\[eq:BoundForRAwayInteractionProp\]) and (\[eq:RoughBoundGeometry\]) $$\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big\|_{(U_{n-1}(v^{(0)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),v)}=\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big\|_{(U_{n-1}(v^{(0)}),v)}(1+O(\text{\textgreek{e}})),$$ ]{} - [ From (\[eq:DerivativeInVDirectionKappaBar\]), (\[eq:DerivativeInUDirectionKappa\]) and the gauge condition (\[eq:GaugeInfinityMaximal\]): $$\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big\|_{\mathcal{R}_{n-1}^{(1,1)}\cap\{v=\bar{v}\}}=\frac{-\partial_{u}r}{1-\frac{2m}{r}}\Big\|_{\mathcal{R}_{n-1}^{(1,1)}\cap\{u=\bar{v}-v_{0}\}},$$ ]{} the bounds (\[eq:LowerBoundDvRBetweenBeams\]) and (\[eq:LowerBoundDuRBetweenBeams\]) yield for any $V_{n}(v^{(k+2)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\le v\le V_{n}(v^{(k+1)})$: $$\partial_{v}r\big\|_{(U_{n}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),v)}\ge\frac{-\partial_{u}r}{1-\frac{2m}{r}}\Big\|_{(v-v_{0},V_{n-1}(v^{(k+1)})}\exp\Big(-C_{0}^{5}(h_{0}(\text{\textgreek{e}}))^{-3}\frac{r_{n}^{(1,k+1)}}{r_{0}}\log\Big(\frac{\tilde{m}{}_{n}^{(1,k+1)}}{\tilde{m}{}_{n-1}^{(1,k+1)}}\Big)-\text{\textgreek{e}}^{1/2}\Big).\label{eq:AlmostUsefulBoundForBeamSeperation}$$ In view of (\[eq:ZeroMassNearAxis\]), (\[eq:UpperBoundForAxisInteractionProp\]) and the fact that $$\frac{-\partial_{u}r}{1-\frac{2m}{r}}\Big\|_{\mathcal{R}_{n-1}^{(1,k+2)}\cap\{u=\bar{u}\}}=\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big\|_{\mathcal{R}_{n-1}^{(1,k+2)}\cap\{v=\bar{u}\}}$$ (following from the gauge condition (\[eq:GaugeMirrorMaximal\])), from (\[eq:AlmostUsefulBoundForBeamSeperation\]) we infer for any $V_{n}(v^{(k+2)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\le v\le V_{n}(v^{(k+1)})$: $$\partial_{v}r\big\|_{(U_{n}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),v)}\ge\partial_{v}r\big\|_{(U_{n-1}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),v-v_{0})}\exp\Big(-C_{0}^{5}(h_{0}(\text{\textgreek{e}}))^{-3}\frac{r_{n}^{(1,k+1)}}{r_{0}}\log\Big(\frac{\tilde{m}{}_{n}^{(1,k+1)}}{\tilde{m}{}_{n-1}^{(1,k+1)}}\Big)-2\text{\textgreek{e}}^{1/2}\Big).\label{eq:UsefulInductiveBoundForBeamSeperation}$$ Iterating (\[eq:UsefulInductiveBoundForBeamSeperation\]) for $n_{1}<n\le n_{2}$, we thus obtain (\[eq:UsefulBoundAlmostThere\]). Let $2\le n_{1}\le n_{f}$ be such so that $$r_{n_{1}}^{(1,k+1)}\le2r_{0}$$ and $$r_{n_{1}-1}^{(1,k+1)}>2r_{0}.\label{eq:PreviousN1}$$ Note that if no such $n_{1}$ exists, then (\[eq:BoundSecondBeamchanged\]) is automatically true ((\[eq:PreviousN1\]) holds for $r_{1}^{(1,k+1)}$ as a corollary of the Cauchy stability estimates of Proposition \[prop:CauchyStabilityOfAdS\] and the choice of the initial data). Let us also define $$n_{2}=\max\big\{ n_{1}\le n\le n_{f}:\, r_{l}^{(1,k+1)}\le2r_{0}\mbox{ for all }n_{1}\le l\le n\big\}.$$ In order to establish (\[eq:BoundSecondBeamchanged\]), it suffices to establish that, for all $n_{1}\le n\le n_{2}$: $$\frac{r_{n}^{(1,k+1)}}{r_{0}}-1\ge\exp\Big(-C_{0}^{7}(h_{0}(\text{\textgreek{e}}))^{-3}\log\big((h_{0}(\text{\textgreek{e}}))^{-1}\big)\Big).\label{eq:BoundToShowSecondBeam}$$ In view of the fact that $$\sup_{1\le l_{1}<l_{2}\le n_{}}\frac{\tilde{m}_{l_{2}}^{(1,k+1)}}{\tilde{m}_{l_{1}}^{(1,k+1)}}\le C_{0}(h_{0}(\text{\textgreek{e}}))^{-1}\label{eq:ControlInMassration}$$ (following from (\[eq:TheIngoingVlasovInitially\]), (\[eq:UpperBoundMassFromR\]) and the fact that the sequence $\tilde{m}_{n}^{(1,k+1)}$ is increasing in $n$ as a consequence of (\[eq:BoundToShowForMass\])), from (\[eq:UsefulBoundAlmostThere\]) we infer that, for any $n_{1}\le n\le n_{2}$ and any $V_{n_{2}}(v^{(k+2)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\le v\le V_{n_{2}}(v^{(k+1)})$: $$\partial_{v}r\big\|_{(U_{n}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),v)}\ge\partial_{v}r\big\|_{(U_{n_{1}-1}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),v-(n-n_{1})v_{0})}\exp\Big(-C_{0}^{6}(h_{0}(\text{\textgreek{e}}))^{-3}\log\big((h_{0}(\text{\textgreek{e}}))^{-1}\big)\Big).\label{eq:UsefulBoundAlmostThere-1}$$ Thus, integrating (\[eq:UsefulBoundAlmostThere-1\]) from $v=V_{n}(v^{(k+2)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))$ up to $v=V_{n}(v^{(k+1)})$ and using (\[eq:PreviousN1\]), we immediately infer (\[eq:BoundToShowSecondBeam\]). #### Proof of (\[eq:BoundForMaxBeamSeparation\]). * {#proof-of-eqboundformaxbeamseparation. .unnumbered} In view of (\[eq:h\_2definition\]), (\[eq:KappaChangeAdjacentRegions\]), (\[eq:KappaBarChangeAdjacentDomains\]), as well as the boundary condition (\[eq:GaugeInfinityMaximal\]) and the bounds (\[eq:BoundForRAwayInteractionProp\]) and (\[eq:UpperBoundForAxisInteractionProp\]), the following one-sided bound holds for all $2\le n\le n_{f}$: $$\begin{aligned} r_{n}^{(k,k+1)}-r_{0} & =\int_{V_{n}(v^{(2k)})}^{V_{n}(v^{(k+1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))}\partial_{v}r\Big\|_{(U_{n}(v^{(k)}),v)}\, dv\label{eq:IntegralForChangeOfTotalBeamWidth}\\ & \le\int_{V_{n}(v^{(2k)})}^{V_{n}(v^{(k+1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))}\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big\|_{(U_{n}(v^{(k)}),v)}\Big(1+O(\text{\textgreek{e}})\Big)\, dv\nonumber \\ & =\int_{U_{n-1}(v^{(k)})}^{U_{n-1}(v^{(0)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))}\frac{-\partial_{u}r}{1-\frac{2m}{r}}\Big\|_{(u,V_{n-1}(v^{(k+1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})))}\Big(1+O(\text{\textgreek{e}})\Big)\, du.\nonumber \end{aligned}$$ We can readily compute (using also (\[eq:h\_2definition\]), (\[eq:UpperBoundForAxisInteractionProp\]) and (\[eq:ImprovedRoughBoundBootstrap\])): $$\begin{aligned} \int_{U_{n-1}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))}^{U_{n-1}(v^{(0)})}\frac{-\partial_{u}r}{1-\frac{2m}{r}} & \Big\|_{(u,V_{n-1}(v^{(k+1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})))}\, du\label{eq:TermThatWillGiveTheLogarithm}\\ = & \int_{U_{n-1}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))}^{U_{n-1}(v^{(0)})}\Big(1-\frac{2\tilde{m}_{n-1}^{(1,k+1)}}{r\|_{(u,V_{n-1}(v^{(k+1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})))}}-\frac{1}{3}\Lambda r^{2}\|_{(u,V_{n-1}(v^{(k+1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})))}\Big)^{-1}\times\nonumber \\ & \hphantom{\int_{U_{n-1}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))}^{U_{n-1}(v^{(0)})}\Big(1-\frac{2\tilde{m}_{n-1}^{(1,k+1)}}{r\|_{(u,V_{n-1}(v^{(k+1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})))}}-\frac{1}{3}\Lambda}\times(-\partial_{u}r)\|_{(u,V_{n-1}(v^{(k+1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})))}\, du\nonumber \\ = & \int_{r_{0}+O((h_{2}(\text{\textgreek{e}}))^{1/2})}^{r_{n-1}^{(1,k+1)}}\Big(1-\frac{2\tilde{m}_{n-1}^{(1,k+1)}}{r}-\frac{1}{3}\Lambda r^{2}\Big)^{-1}\, dr\nonumber \\ \le & r_{n-1}^{(1,k+1)}-r_{0}+C_{0}\tilde{m}_{n-1}^{(1,k+1)}\Big\|\log\big(1-\frac{2\tilde{m}_{n-1}^{(1,k+1)}}{r_{0}}\big)\Big\|.\nonumber \end{aligned}$$ From (\[eq:BoundMirror\]) and (\[eq:BoundSecondBeamchanged\]) we can similarly estimate: $$\begin{aligned} \int_{U_{n-1}(v^{(k)})}^{U_{n-1}(v^{(1)})}\frac{-\partial_{u}r}{1-\frac{2m}{r}} & \Big\|_{(u,V_{n-1}(v^{(k+1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})))}\, du\le\label{eq:TermThatWillGiveTheLogarithm-1}\\ \le & \int_{r_{n-1}^{(1,k+1)}+O((h_{2}(\text{\textgreek{e}}))^{1/2})}^{r_{n-1}^{(k,k+1)}}\Big(1-\frac{2\tilde{m}\|_{\mathcal{I}}}{r}+O(\text{\textgreek{e}})\Big)^{-1}\, dr\le\nonumber \\ \le & r_{n-1}^{(k,k+1)}+C_{0}\tilde{m}\|_{\mathcal{I}}\Big\|\log\big(\exp\big(h_{0}(\text{\textgreek{e}})\big)^{-4}\big)+1\Big\|\le\nonumber \\ \le & r_{n-1}^{(k,k+1)}-r_{n-1}^{(1,k+1)}+C_{0}\tilde{m}\|_{\mathcal{I}}\big(h_{0}(\text{\textgreek{e}})\big)^{-4}.\nonumber \end{aligned}$$ From (\[eq:h\_2definition\]), (\[eq:ImprovedRoughBoundBootstrap\]), (\[eq:IntegralForChangeOfTotalBeamWidth\]), (\[eq:TermThatWillGiveTheLogarithm\]) and (\[eq:TermThatWillGiveTheLogarithm-1\]) one readily obtains the bound (\[eq:BoundForMaxBeamSeparation\]). \[sub:NearlyTrapped\]Formation of a nearly-trapped sphere --------------------------------------------------------- In this section, we will establish (\[eq:NearTrappingIsAchieved\]), using the bounds (\[eq:RoughBoundGeometry\])–(\[eq:BoundForMaxBeamSeparation\]) of Proposition \[prop:TheMainBootstrapBeforeTrapping\]. Let us set $$n_{max}\doteq\max\big\{ n_{}\in\mathbb{N}:\,\mathcal{R}_{n}^{(1,k+1)}\subset\mathcal{U}_{\text{\textgreek{e}}}^{+}\mbox{ for all }n\le n_{}\big\}.\label{eq:DefinitionNmax}$$ Note that, in view of (\[eq:DefinitionUntrappedRegion\]) and (\[eq:defNf\]), $n_{max}$ satisfies $$n_{f}\le n_{max}\le n_{f}+1.\label{eq:doubleBoundNmax}$$ Thus, (\[eq:UpperUNonTrapping\]) implies that $$n_{max}\le(h_{1}(\text{\textgreek{e}}))^{-2}.\label{eq:InitialUpperBoundForNmax}$$ Notice that (\[eq:doubleBoundNmax\]) and the definition (\[eq:DefinitionUntrappedRegion\]) imply that, if $$n_{max}<\frac{1}{2}(h_{1}(\text{\textgreek{e}}))^{-2},\label{eq:NonTrivalBoundNmax}$$ then, necessarily, (\[eq:NearTrappingIsAchieved\]) holds. Thus, in order to establish (\[eq:NearTrappingIsAchieved\]), it suffices to show (\[eq:NonTrivalBoundNmax\]). We will show (\[eq:NonTrivalBoundNmax\]) by applying Lemma \[lem:ForMassIncrease\] (see Section \[sub:Auxiliary-lemmas\]). In particular, setting for any $1\le n\le n_{max}+1$ $$\begin{aligned} \text{\textgreek{m}}_{n} & \doteq\frac{2\tilde{m}_{n-1}^{(1,k+1)}}{r_{0}},\label{eq:mu_n}\\ \text{\textgreek{r}}_{n} & \doteq\frac{r_{n-1}^{(k,k+1)}}{r_{0}},\label{eq:rho_n}\end{aligned}$$ the inductive bounds (\[eq:BoundForMassIncrease\]) and (\[eq:BoundForMaxBeamSeparation\]) imply that $\text{\textgreek{m}}_{n},\text{\textgreek{r}}_{n}$ satisfy $$\begin{aligned} \text{\textgreek{r}}_{n+1} & \le\text{\textgreek{r}}_{n}+C_{1}\log\big((1-\text{\textgreek{m}}_{n})^{-1}+1\big),\label{eq:InductiveRelationSequence-1}\\ \text{\textgreek{m}}_{n+1} & \ge\text{\textgreek{m}}_{n}\exp\big(\frac{c_{1}}{\text{\textgreek{r}}_{n+1}}\big),\nonumber \end{aligned}$$ for any $1\le n\le n_{max}+1$, with $$C_{1}=(h_{0}(\text{\textgreek{e}}))^{-4}$$ and $$c_{1}=\frac{1}{16}\exp\big(-2(h_{0}(\text{\textgreek{e}}))^{-4}\big).$$ Furthermore, setting $$\text{\textgreek{d}}=h_{3}(\text{\textgreek{e}})$$ (where $h_{3}(\cdot)$ is defined by (\[eq:h\_3definition\])), the definition (\[eq:DefinitionNmax\]) immediately implies that $$\max_{0\le n\le n_{max}}\text{\textgreek{m}}_{n}<1-\text{\textgreek{d}}.\label{eq:NerTrappedMu-1}$$ Note that, in view of Definition \[def:ThefamilyOfInitialData\] and Proposition \[prop:CauchyStabilityOfAdS\], we have $$\text{\textgreek{m}}_{0}\sim h_{0}(\text{\textgreek{e}})$$ and $$\text{\textgreek{r}}_{0}\sim(h_{1}(\text{\textgreek{e}}))^{-1}.$$ Hence, as a consequence of (\[eq:h\_1\_h\_0\_definition\]) and (\[eq:h\_3definition\]), $$\Big(\frac{C_{1}}{c_{1}}\Big)^{8}\ll\frac{\text{\textgreek{r}}_{0}}{\text{\textgreek{m}}_{0}}\label{eq:LargenessRho0-1}$$ and $$\text{\textgreek{d}}<\big(\frac{\text{\textgreek{m}}_{0}}{\text{\textgreek{r}}_{0}}\big)^{(C_{1}/c_{1})^{4}}.\label{eq:SmallnessDelta-1}$$ The relations (\[eq:InductiveRelationSequence-1\])–(\[eq:SmallnessDelta-1\]) allow us to apply Lemma \[lem:ForMassIncrease\] (see Section \[sub:Auxiliary-lemmas\]) with $n_{}=n_{max}+1$ for the sequence $\text{\textgreek{r}}_{n},\text{\textgreek{m}}_{n}$. Thus, in view of Lemma \[lem:ForMassIncrease\], we obtain the following upper bound for $n_{max}$: $$n_{max}+1\le\exp\Big(\exp\big(2(h_{0}(\text{\textgreek{e}}))^{-4}\big)\Big)(h_{1}(\text{\textgreek{e}}))^{-1}.\label{eq:UpperBoundNmax}$$ In particular, (\[eq:NonTrivalBoundNmax\]) (and, thus, (\[eq:NearTrappingIsAchieved\])) holds. \[sub:FinalStep\]The final step of the evolution ------------------------------------------------ In this section, we will complete the proof of Theorem \[thm:TheTheorem\], using the near-trapping bound (\[eq:NearTrappingIsAchieved\]), the bounds (\[eq:RoughBoundGeometry\])–(\[eq:BoundForMaxBeamSeparation\]) of Proposition \[prop:TheMainBootstrapBeforeTrapping\], as well a backwards-in-time Cauchy stability estimate (see Lemma \[lem:PerturbationInitialData\] in Section \[sub:Cauchy-Stability-Backwards\]). The bound (\[eq:NearTrappingIsAchieved\]), combined with the estimates (\[eq:NotEnoughMassBehind\]) and (\[eq:BoundSecondBeamchanged\]) of Proposition \[prop:TheMainBootstrapBeforeTrapping\], imply that, necessarily (in view also of (\[eq:h\_1\_h\_0\_definition\]), (\[eq:h\_2definition\]), (\[eq:RoughBoundGeometry\]), (\[eq:BoundForMassIncrease\]) and (\[eq:DefinitionNmax\])): $$\frac{2\tilde{m}_{n_{max}+1}^{(1,k+1)}}{r_{0}}\ge1-2h_{3}(\text{\textgreek{e}}).\label{eq:NearTrappedLastIteration}$$ Therefore, applying again Lemma \[lem:ForMassIncrease\] for $\text{\textgreek{m}}_{n},\text{\textgreek{r}}_{n}$ (defined by (\[eq:mu\_n\]), (\[eq:rho\_n\])) and $n_{}=n_{max}+1$ yields, in view of (\[eq:NearTrappedLastIteration\]), that, either $$\text{\textgreek{m}}_{n_{max}+1}>1+h_{3}(\text{\textgreek{e}}),\label{eq:TheCaseOfTrappedSurface}$$ or $$1-2h_{3}(\text{\textgreek{e}})\le\text{\textgreek{m}}_{n_{max}+1}\le1+h_{3}(\text{\textgreek{e}})\label{eq:TheCaseOfNearTrappedSurface}$$ and $$\begin{aligned} \text{\textgreek{m}}_{n_{max}} & \le1-\exp\Big(-\exp\big(2(h_{0}(\text{\textgreek{e}}))^{-4}\big)\Big)(h_{1}(\text{\textgreek{e}}))^{2}\label{eq:AlmostTrappedLastStep}\\ \max\{\text{\textgreek{r}}_{n_{max}+1},\text{\textgreek{r}}_{n_{max}}\} & \le\exp\Big(\exp\big(2(h_{0}(\text{\textgreek{e}}))^{-4}\big)\Big)(h_{1}(\text{\textgreek{e}}))^{-1}\log\big((h_{1}(\text{\textgreek{e}}))^{-1}\big).\label{eq:MaxSeperationLastStep}\end{aligned}$$ Let us set $$\bar{v}_{}\doteq V_{n_{max}+1}\big(v^{(k+1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})\big)\label{eq:vbar}$$ (recall that (\[eq:vbar\\]) equals $V_{n_{max}}\big(v^{(0)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})\big)$, in view of our conventions on the indices). The proof of Theorem \[thm:TheTheorem\] will follow by showing that - Either $$\inf_{\mathcal{U}_{\text{\textgreek{e}}}\cap\{v=\bar{v}_{}\}}\big(1-\frac{2m}{r}\big)<0\label{eq:TrappedSurfaceFormed}$$ (in which case $(r_{/}^{(\text{\textgreek{e}})},(\text{\textgreek{W}}_{/}^{(\text{\textgreek{e}})})^{2},\bar{f}_{in/}^{(\text{\textgreek{e}})},\bar{f}_{out/}^{(\text{\textgreek{e}})})=(r_{/\text{\textgreek{e}}},\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{2},\bar{f}_{in/\text{\textgreek{e}}},\bar{f}_{out/\text{\textgreek{e}}})$ in the statement of Theorem \[thm:TheTheorem\]). - Or $$\inf_{\mathcal{U}_{\text{\textgreek{e}}}\cap\{v=\bar{v}_{}\}}\big(1-\frac{2m'}{r'}\big)<0,\label{eq:PerturbedTrappedSurfaceFormed}$$ where $(r',(\text{\textgreek{W}}')^{2},\bar{f}_{in}^{\prime},\bar{f}_{out}^{\prime})$ is a (possibly different) smooth solution to the system (\[eq:RequationFinal\])–(\[eq:OutgoingVlasovFinal\]) arising as a future development of an asymptotically AdS boundary-characteristic initial data set $(r_{/\text{\textgreek{e}}}^{\prime},(\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{\prime})^{2},\bar{f}_{in/\text{\textgreek{e}}}^{\prime},\bar{f}_{out/\text{\textgreek{e}}}^{\prime})$ on $\{u=0\}\cap\{0\le v\le v_{0\text{\textgreek{e}}}\}$ (satisfying the reflecting gauge condition at $r=r_{0},+\infty$) which is $(h_{1}(\text{\textgreek{e}}))^{2}$ close to $(r_{/\text{\textgreek{e}}},\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{2},\bar{f}_{in/\text{\textgreek{e}}},\bar{f}_{out/\text{\textgreek{e}}})$ with respect to the norm (\[eq:GeometricNormForCauchyStability\]), i.e. satisfies, in particular, (\[eq:GaugeDifferenceBoundCauchystability-1\]) and (\[eq:DifferenceBoundCauchyStability-1\]) (in which case $(r_{/}^{(\text{\textgreek{e}})},(\text{\textgreek{W}}_{/}^{(\text{\textgreek{e}})})^{2},\bar{f}_{in/}^{(\text{\textgreek{e}})},\bar{f}_{out/}^{(\text{\textgreek{e}})})=(r_{/\text{\textgreek{e}}}^{\prime},(\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{\prime})^{2},\bar{f}_{in/\text{\textgreek{e}}}^{\prime},\bar{f}_{out/\text{\textgreek{e}}}^{\prime})$ in the statement of Theorem \[thm:TheTheorem\]). Notice that, in both cases, (\[eq:DecayInInitialDataNorm\]) follows readily from (\[eq:CSnormFamily\]) and (\[eq:h\_1\_h\_0\_definition\]). To this end, we will proceed to treat the cases (\[eq:TheCaseOfTrappedSurface\]) and (\[eq:TheCaseOfNearTrappedSurface\]) separately. Case I.* Assume that (\[eq:TheCaseOfTrappedSurface\]) holds. Then, we will show that (\[eq:TrappedSurfaceFormed\]) also holds. We will argue by contradiction, assuming that $$\inf_{\mathcal{U}_{\text{\textgreek{e}}}\cap\{v=\bar{v}_{}\}}\big(1-\frac{2m}{r}\big)\ge0.\label{eq:UntrappedForContradiction}$$ Let us set $$\mathcal{C}_{}\doteq\{U_{n_{max}+1}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\le u<U_{n_{max}+1}(v^{(0)})\}\cap\{v=\bar{v}_{}\}\cap\mathcal{U}_{\text{\textgreek{e}}}.$$ The renormalised mass $\tilde{m}$ is constant on $\mathcal{C}_{}$, satisfying in particular $$\tilde{m}\|_{\mathcal{C}_{}}=\tilde{m}_{n_{max}+1}^{(1,k+1)}.\label{eq:MassOnIngoingFinalLine}$$ Since $\partial_{u}r<0$ on $\mathcal{U}_{\text{\textgreek{e}}}$ (see (\[eq:NegativeDerivativeRMaximal\])), from (\[eq:UntrappedForContradiction\]) and the fact that $\mathcal{C}_{}$ does not contain its future endpoint, we infer the following stronger bound: $$1-\frac{2m}{r}\big\|_{\mathcal{C}_{}}>0.\label{eq:CompletelyUntrappedForContradiction}$$ Thus, we also have $$\partial_{v}r\|_{\mathcal{C}_{}}>0.\label{eq:PositiveDvrForLater}$$ We will now show that the future endpoint of $\mathcal{C}_{}$ is exactly $(U_{n_{max}+1}(v^{(0)}),\bar{v}_{})$. If there existed some $(u_{b},\bar{v}_{})\in(\partial\mathcal{U}_{\text{\textgreek{e}}}\backslash\mathcal{I})$ such that $U_{n_{max}+1}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\le u_{b}<U_{n_{max}+1}(v^{(0)})$, then Theorem \[thm:maximalExtension\] on the strucure of the maximal future development would imply that $r$ extends continuously on $(u_{b},\bar{v}_{})$ with $$r(u_{b},\bar{v}_{})=r_{0\text{\textgreek{e}}}.\label{eq:R0OnFutureBoundary}$$ However, in that case, (\[eq:TheCaseOfTrappedSurface\]), (\[eq:MassOnIngoingFinalLine\]) and (\[eq:R0OnFutureBoundary\]) would imply that, for some $u_{b}$ close enough to $u_{b}$ $$1-\frac{2m}{r}\big\|_{(u_{b},\bar{v}_{})}<0,$$ which is a contradiction in view of (\[eq:UntrappedForContradiction\]). Therefore, $$\{U_{n_{max}+1}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\le u<U_{n_{max}+1}(v^{(0)})\}\cap\{v=\bar{v}_{}\}\cap(\partial\mathcal{U}_{\text{\textgreek{e}}}\backslash\mathcal{I})=\emptyset,$$ and, thus $$\mathcal{C}_{}=\{U_{n_{max}+1}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\le u<U_{n_{max}+1}(v^{(0)})\}\cap\{v=\bar{v}_{}\}.\label{eq:TheWholeSegmentInTheDevelopment}$$ In order to complete the proof in the case when (\[eq:TheCaseOfTrappedSurface\]) holds, it suffices to establish that $$\limsup_{\bar{u}\rightarrow U_{n_{max}+1}(v^{(0)})}\frac{r\|_{(\bar{u},\bar{v}_{})}}{r_{0}}\le1+O\big((h_{2}(\text{\textgreek{e}}))^{1/2}\big).\label{eq:AlmostOnMirror}$$ Assuming that (\[eq:AlmostOnMirror\]) holds, from (\[eq:TheCaseOfTrappedSurface\]), (\[eq:MassOnIngoingFinalLine\]) and (\[eq:AlmostOnMirror\]) (in view also of (\[eq:h\_2definition\]), (\[eq:h\_3definition\])) we readily obtain $$\liminf_{\bar{u}\rightarrow U_{n_{max}+1}(v^{(0)})}\Big(1-\frac{2m}{r}\Big)\Big\|_{(\bar{u},\bar{v}_{})}<-\frac{1}{2}h_{3}(\text{\textgreek{e}})<0,$$ which is a contradiction in view of (\[eq:UntrappedForContradiction\]). Let us set $$\begin{aligned} \mathcal{B}_{}\doteq\{U_{n_{max}+1}(v^{(1)}+ & \frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\le u<U_{n_{max}+1}(v^{(0)})\}\cap\{V_{n_{max}+1}(v^{(k)})\le v\le\bar{v}_{}\}.\end{aligned}$$ From (\[eq:TheWholeSegmentInTheDevelopment\]) and the structure of the maximal future development of general initial data sets for (\[eq:RequationFinal\])–(\[eq:OutgoingVlasovFinal\]) (see Theorem \[thm:maximalExtension\]), we infer that $$\mathcal{B}_{}\subset\mathcal{U}_{\text{\textgreek{e}}}.$$ Furthermore, in view of (\[eq:ConstrainVFinal\]) and (\[eq:PositiveDvrForLater\]), we infer that $$\partial_{v}r\big\|_{\mathcal{B}_{}}>0$$ and, thus (in view of (\[eq:NegativeDerivativeRMaximal\])): $$1-\frac{2m}{r}\big\|_{\mathcal{B}_{}}>0.\label{eq:UntrappedInRectangle}$$ In view of (\[eq:DefinitionNmax\]) and the bounds (\[eq:NotEnoughMassBehind\]) and (\[eq:BoundSecondBeamchanged\]), we have $$\big\{ u\le U_{n_{max}+1}(v^{(1)})\big\}\cap\mathcal{U}_{\text{\textgreek{e}}}\subset\mathcal{U}_{\text{\textgreek{e}}}^{+}.$$ Therefore, as a consequence of (\[eq:RoughBoundGeometry\]), we can estimate $$\log\Big(\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big)\Big\|_{\big\{ u=U_{n_{max}+1}(v^{(1)})\big\}\cap\mathcal{U}_{\text{\textgreek{e}}}}\le\big(h_{1}(\text{\textgreek{e}})\big)^{-4}\log\big((h_{3}(\text{\textgreek{e}}))^{-1}\big).\label{eq:FromRoughBound}$$ Since (\[eq:DerivativeInUDirectionKappa\]) implies that $$\partial_{u}\log\Big(\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big)\le0,$$ from (\[eq:FromRoughBound\]) (and (\[eq:UpperBoundForAxisInteractionProp\]) ) we infer the one sided bound: $$\partial_{v}r\big\|_{\mathcal{B}_{}}\le2\big(h_{1}(\text{\textgreek{e}})\big)^{-4}\log\big((h_{3}(\text{\textgreek{e}}))^{-1}\big).\label{eq:BoundForRseperationAlreadyTrapped}$$ Integrating (\[eq:BoundForRseperationAlreadyTrapped\]) from $v=V_{n_{max}+1}(v^{(k)})$ up to $V_{n_{max}+1}(v^{(k+1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))$ using (\[eq:h\_2definition\]), we finally obtain (\[eq:AlmostOnMirror\]). Thus, the proof in the case when (\[eq:TheCaseOfTrappedSurface\]) holds is complete. Case II.* Assume that (\[eq:TheCaseOfNearTrappedSurface\]) holds. Then, (\[eq:AlmostTrappedLastStep\]) and (\[eq:MaxSeperationLastStep\]) also hold. As a consequence of (\[eq:NotEnoughMassBehind\]), (\[eq:BoundSecondBeamchanged\]) and (\[eq:BoundForMassIncrease\]), the bound (\[eq:AlmostTrappedLastStep\]) implies that $$\inf_{\{u\le U_{n_{max}}(v^{(0)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\}\cap\mathcal{U}_{\text{\textgreek{e}}}}\Big(1-\frac{2\tilde{m}}{r}\Big)\ge\frac{1}{2}\exp\Big(-\exp\big(2(h_{0}(\text{\textgreek{e}}))^{-4}\big)\Big)(h_{1}(\text{\textgreek{e}}))^{2}.$$ Therefore, using (\[eq:NotEnoughMassBehind\]), (\[eq:BoundSecondBeamchanged\]) and (\[eq:BoundForMassIncrease\]) to estimate $(1-\frac{2m}{r})$ in the region $$\{U_{n_{max}}(v^{(0)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\le u\le U_{n_{max}+1}(v^{(0)})\}\backslash\mathcal{R}_{n_{max}+1}^{(1,k+1)},$$ we infer that $$\inf_{\{u\le U_{n_{max}+1}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\}\cap\mathcal{U}_{\text{\textgreek{e}}}}\Big(1-\frac{2\tilde{m}}{r}\Big)\ge\frac{1}{2}\exp\Big(-\exp\big(2(h_{0}(\text{\textgreek{e}}))^{-4}\big)\Big)(h_{1}(\text{\textgreek{e}}))^{2}.\label{eq:AwayFromTrappedBeforePerurbation}$$ Notice that, while $1-\frac{2\tilde{m}}{r}$ becomes $\sim h_{3}(\text{\textgreek{e}})$ in $\{u\le U_{n_{max}+1}(v^{(0)})\}\cap\mathcal{U}_{\text{\textgreek{e}}}$ (in view of (\[eq:TheCaseOfNearTrappedSurface\])), when restricting to the subregion $\{u\le U_{n_{max}+1}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\}\cap\mathcal{U}_{\text{\textgreek{e}}}$, the improved bound (\[eq:AwayFromTrappedBeforePerurbation\]) holds. Let us set $$u_{}\doteq U_{n_{max}+1}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),\label{eq:u_}$$ noticing that $$supp(r^{2}T_{vv})\cap\{u=u_{}\}\subset\{r\le\text{\textgreek{e}}^{1/2}\}\label{eq:SupportNotAtInfinity}$$ as a consequence of (\[eq:BoundForRAwayInteractionProp\]). Let us also fix a smooth cut-off function $\text{\textgreek{q}}_{\text{\textgreek{e}}}:[u_{},u_{}+v_{0\text{\textgreek{e}}})\rightarrow[0,1]$ such that $$\text{\textgreek{q}}_{\text{\textgreek{e}}}(v)=1\mbox{ for }v\in[V_{n_{max}+1}(v^{(k+1)}),V_{n_{max}+1}(v^{(k+1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))]\label{eq:SupportOfCutOff}$$ and $$\text{\textgreek{q}}_{\text{\textgreek{e}}}(v)=0\mbox{ for }v\in[u_{},u_{}+v_{0\text{\textgreek{e}}})\backslash[V_{n_{max}+1}(v^{(k+1)}-\frac{1}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}})),V_{n_{max}+1}(v^{(k+1)}+\frac{5}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))].$$ We will then define the function $\widetilde{T}_{vv}:[u_{},u_{}+v_{0\text{\textgreek{e}}})\rightarrow\mathbb{R}$ by the relation $$\widetilde{T}_{vv}(v)\doteq\exp\big(-2C_{\text{\textgreek{e}}}^{2}\frac{u_{}}{v_{0}}\big)(h_{1}(\text{\textgreek{e}}))^{2}\text{\textgreek{q}}_{\text{\textgreek{e}}}(v)T_{vv}(u_{},v),\label{eq:WidetildeT}$$ where $C_{\text{\textgreek{e}}}$ is defined by (\[eq:Cepsilon\]). Notice that, since $$2\pi\int_{V_{n_{max}+1}(v^{(k+1)})}^{V_{n_{max}+1}(v^{(k+1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))}\frac{(1-\frac{2m}{r})}{\partial_{v}r}r^{2}T_{vv}\Big\|_{(u_{},v)}\, dv=\tilde{m}_{n_{max}+1}^{(1,k+1)}-\tilde{m}_{n_{max}+1}^{(1,k+2)}=\tilde{m}_{n_{max}+1}^{(1,k+1)},\label{eq:MassDifferenceForPerturbation}$$ we can readily bound in view of (\[eq:WidetildeT\]), (\[eq:BoundSecondBeamchanged\]), (\[eq:TheCaseOfNearTrappedSurface\]) and (\[eq:MassDifferenceForPerturbation\]): $$\sup_{u_{}\le\bar{v}\le u_{}+v_{0\text{\textgreek{e}}}}(-\Lambda)\int_{u_{}}^{u_{}+v_{0\text{\textgreek{e}}}}\frac{r^{2}(u_{},v)\frac{\|\tilde{T}_{vv}(v)\|}{\partial_{v}\text{\textgreek{r}}(u_{},v)}}{\|\text{\textgreek{r}}(u_{},v)-\text{\textgreek{r}}(u_{},\bar{v})\|+\text{\textgreek{r}}(u_{},u_{})}dv\le\exp\big(-C_{\text{\textgreek{e}}}^{2}\frac{u_{}}{v_{0\text{\textgreek{e}}}}\big)(h_{1}(\text{\textgreek{e}}))^{2},\label{eq:SmallnessPerturbation-1}$$ where $\text{\textgreek{r}}$ is defined in terms of $r$ by the relation $$\text{\textgreek{r}}\doteq\tan^{-1}\Big(\sqrt{-\frac{\Lambda}{3}}r\Big).$$ Applying the backwards-in-time Cauchy stability lemma \[lem:PerturbationInitialData\] (see Section \[sub:Auxiliary-lemmas\]), for $u_{}$ given by (\[eq:u\_\\]) and $\widetilde{T}_{vv}$ given by (\[eq:WidetildeT\]) (in view of (\[eq:AwayFromTrappedBeforePerurbation\]), (\[eq:SupportNotAtInfinity\]) and (\[eq:SmallnessPerturbation-1\])), we infer that there exists an smooth asymptotically AdS boundary-characteristic initial data set $(r_{/\text{\textgreek{e}}}^{\prime},(\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{\prime})^{2},\bar{f}_{in/\text{\textgreek{e}}}^{\prime},\bar{f}_{out/\text{\textgreek{e}}}^{\prime})$ on $\{u=0\}$ for the system (\[eq:RequationFinal\])–(\[eq:OutgoingVlasovFinal\]) satisfying the reflecting gauge condition at $r=r_{0\text{\textgreek{e}}},+\infty$ with the following properties: 1. The initial data sets $(r_{/\text{\textgreek{e}}},\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{2},\bar{f}_{in/\text{\textgreek{e}}},\bar{f}_{out/\text{\textgreek{e}}})$ and $(r_{/\text{\textgreek{e}}}^{\prime},(\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{\prime})^{2},\bar{f}_{in/\text{\textgreek{e}}}^{\prime},\bar{f}_{out/\text{\textgreek{e}}}^{\prime})$ satisfy (\[eq:GaugeDifferenceBoundCauchystability-1\]) and (\[eq:DifferenceBoundCauchyStability-1\]). 2. The maximal development $(\mathcal{U}_{\text{\textgreek{e}}}^{\prime};r^{\prime},(\text{\textgreek{W}}^{\prime})^{2},\bar{f}_{in}^{\prime},\bar{f}_{out}^{\prime})$ of $(r_{/\text{\textgreek{e}}},\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{2},\bar{f}_{in/\text{\textgreek{e}}},\bar{f}_{out/\text{\textgreek{e}}})$ satisfies (\[eq:ComparableDevelopments\]), (\[eq:EqualROnTheSupport\]) and (\[eq:NewIngoingEnergyMomentum\]). Using primes to denote quantities associated to $(r^{\prime},(\text{\textgreek{W}}^{\prime})^{2},\bar{f}_{in}^{\prime},\bar{f}_{out}^{\prime})$, we can readily estimate in view of (\[eq:ComparableDevelopments\]), (\[eq:EqualROnTheSupport\]), (\[eq:NewIngoingEnergyMomentum\]) and (\[eq:WidetildeT\]): $$\begin{aligned} \tilde{m}^{\prime}\big\|_{(u_{},\bar{v}_{})} & =\int_{V_{n_{max}+1}(v^{(k+1)})}^{V_{n_{max}+1}(v^{(k+1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))}\frac{(1-\frac{2m^{\prime}}{r^{\prime}})}{\partial_{v}r^{\prime}}(r^{\prime})^{2}T_{vv}^{\prime}\Big\|_{(u_{},v)}\, dv\label{eq:NewMass}\\ & =\int_{V_{n_{max}+1}(v^{(k+1)})}^{V_{n_{max}+1}(v^{(k+1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))}\frac{(1-\frac{2m^{\prime}}{r})}{\partial_{v}r}r^{2}(T_{vv}+\widetilde{T}_{vv})\Big\|_{(u_{},v)}\, dv\nonumber \\ & \ge(1+\exp\big(-2C_{\text{\textgreek{e}}}^{2}\frac{u_{}}{v_{0}}\big)(h_{1}(\text{\textgreek{e}}))^{2})\tilde{m}\big\|_{(u_{},\bar{v}_{})}.\nonumber \end{aligned}$$ Therefore, since $\tilde{m}\big\|_{(u_{},\bar{v}_{})}=\tilde{m}_{n_{max}+1}^{(1,k+1)}$, the bound (\[eq:TheCaseOfNearTrappedSurface\]) (in view also of (\[eq:h\_3definition\]) and (\[eq:Cepsilon\])) implies that $$\frac{2\tilde{m}^{\prime}\big\|_{(u_{},\bar{v}_{})}}{r_{0}}\ge(1+\exp\big(-2C_{\text{\textgreek{e}}}^{2}\frac{u_{}}{v_{0}}\big)(h_{1}(\text{\textgreek{e}}))^{2})(1-2h_{3}(\text{\textgreek{e}}))\ge1+h_{3}(\text{\textgreek{e}}).\label{eq:PerturbationIsTrapped}$$ Since $\tilde{m}\big\|_{(u_{},\bar{v}_{})}$ is constant on $$\mathcal{C}_{}^{\prime}\doteq\{U_{n_{max}+1}(v^{(1)}+\frac{4}{\sqrt{-\Lambda}}h_{2}(\text{\textgreek{e}}))\le u<U_{n_{max}+1}(v^{(0)})\}\cap\{v=\bar{v}_{}\}\cap\mathcal{U}_{\text{\textgreek{e}}}^{\prime}$$ and satisfies (\[eq:PerturbationIsTrapped\]), we can now repeat the same arguments as in Case I (i.e. the case when (\[eq:TheCaseOfTrappedSurface\]) holds) in order to infer that (\[eq:PerturbedTrappedSurfaceFormed\]) holds. Thus, the proof of Theorem \[thm:TheTheorem\] is complete. \[sub:Auxiliary-lemmas\]Some auxiliary lemmas --------------------------------------------- In this section, we will prove some lemmas necessary for the proof of Proposition \[prop:TheMainBootstrapBeforeTrapping\] and Theorem \[thm:TheTheorem\]. ### \[sub:MaximumPrinciple\] A maximum principle for $1+1$ wave-type equations The following lemma provides a comparison inequality for certain $1+1$ equations of wave type, and is used in the proof of Proposition \[prop:TheMainBootstrapBeforeTrapping\]. \[lem:HyperbolicMaximumPrinciple\]For any $u_{0}<u_{1}$, $v_{0}<v_{1}$ and $a\in\mathbb{R}$, let $F_{1},F_{2}:[u_{0},u_{1}]\times[v_{0},v_{1}]\times(-\infty,a]\rightarrow(0,+\infty)$ be smooth functions so that $$\max_{(u,v)\in[u_{0}u_{1}]\times[v_{0},v_{1}]}F_{1}(u,v,z)<\min_{(u,v)\in[u_{0}u_{1}]\times[v_{0},v_{1}]}F_{2}(u,v,z)\label{eq:RelationF_1F_2}$$ for any $z\in(-\infty,a]$ and $$\partial_{z}F_{1}(u,v,z),\partial_{z}F_{2}(u,v,z)\ge0\label{eq:IncreasingInLastVariable}$$ for any $(u,v,z)\in[u_{0},u_{1}]\times[v_{0},v_{1}]\times(-\infty,a]$. Suppose also $z_{1},z_{2}:[u_{0},u_{1}]\times[v_{0},v_{1}]\rightarrow(-\infty,a]$ are smooth solutions to the equations $$\partial_{v}\partial_{u}z_{1}=-F_{1}(u,v,z_{1})\partial_{u}z_{1}\partial_{v}z_{1}\label{eq:Equation1}$$ and $$\partial_{v}\partial_{u}z_{2}=-F_{2}(u,v,z_{2})\partial_{u}z_{2}\partial_{v}z_{2},\label{eq:Equation2}$$ satisfying the same characteristic initial data $$z_{1}(u,v_{0})=z_{2}(u,v_{0})=z_{\backslash}(u),\label{eq:InData1}$$ $$z_{1}(u_{0},v)=z_{2}(u_{0},v)=z_{/}(v).\label{eq:InData2}$$ where $z_{/}:[v_{0},v_{1}]\rightarrow(-\infty,a)$ and $z_{\backslash}:[u_{0},u_{1}]\rightarrow(-\infty,a)$ are smooth functions so that $$z_{/}(v_{0})=z_{\backslash}(v_{1}),$$ $$\partial_{v}z_{/}\|_{(v_{0},v_{1})}>0\label{eq:IncreasingRight}$$ and $$\partial_{u}z_{\backslash}\|_{(u_{0},u_{1})}<0.\label{eq:DecreasingLeft}$$ Then, the functions $z_{1},z_{2}$ satisfy $$\partial_{u}z_{i}<0<\partial_{v}z_{i},\mbox{ }i=1,2\label{eq:Monotonicity}$$ in $(u_{0},u_{1})\times(v_{0},v_{1})$ and $$z_{1}\le z_{2}\label{eq:InequalityZ1,2}$$ everywhere on $[u_{0},u_{1}]\times[v_{0},v_{1}]$. We will first establish (\[eq:Monotonicity\]). By applying a standard continuity argument, rewriting equation (\[eq:Equation1\]) as $$\partial_{v}\log(-\partial_{u}z_{1})=-\partial_{v}z_{1}F_{1}(u,v,z_{1})\label{eq:DvDerivativeZ1}$$ and integrating in $v$, using also the property (\[eq:DecreasingLeft\]) of the initial data, we obtain that $$\partial_{u}z_{1}<0$$ everywhere on $(u_{0},u_{1})\times(v_{0},v_{1})$. Similarly, rewriting (\[eq:Equation1\]) as $$\partial_{u}\log(\partial_{v}z_{1})=-\partial_{u}z_{1}F_{1}(u,v,z_{1})$$ and integrating in $u$, using (\[eq:DecreasingLeft\]), and then repeating the same procedure for $z_{2}$, we finally obtain (\[eq:Monotonicity\]). In order to establish (\[eq:InequalityZ1,2\]), we will argue by continuity: Let $u_{}\in[u_{0},u_{1})$ be such that (\[eq:InequalityZ1,2\]), $$\partial_{v}z_{1}\le\partial_{v}z_{2}\label{eq:ContinuityDv}$$ and $$\partial_{u}z_{1}\le\partial_{u}z_{2}\label{eq:ContinuityDu}$$ hold on $[u_{0},u_{}]\times[v_{0},v_{1}]$. Note that $u_{}=u_{0}$ satisfies this condition: In this case, (\[eq:InequalityZ1,2\]) and (\[eq:ContinuityDv\]) follow directly from (\[eq:InData2\]), while (\[eq:ContinuityDu\]) follows by integrating (\[eq:DvDerivativeZ1\]) (and its analogue for $z_{2}$) and using (\[eq:RelationF\_1F\_2\]). We will show that there exists a $\text{\textgreek{d}}>0$, such that (\[eq:InequalityZ1,2\]), (\[eq:ContinuityDv\]) and (\[eq:ContinuityDu\]) hold on $[u_{0},u_{}+\text{\textgreek{d}})\times[v_{0},v_{1}]$. For any $\bar{v}\in(v_{0},v_{1}]$, integrating (\[eq:Equation1\]) and (\[eq:Equation2\]) in $v$ along $\{u_{}\}\times[v_{0},\bar{v}]$, we obtain: $$\log(-\partial_{u}z_{1})(u_{},\bar{v})=-\int_{v_{0}}^{\bar{v}}F_{1}(u_{},v,z_{1})\partial_{v}z_{1}\, dv+\log(-\partial_{u}z_{\backslash})(u_{})\label{eq:LogDuZ1}$$ and $$\log(-\partial_{u}z_{2})(u_{},\bar{v})=-\int_{v_{0}}^{\bar{v}}F_{2}(u_{},v,z_{2})\partial_{v}z_{2}\, dv+\log(-\partial_{u}z_{\backslash})(u_{}).\label{eq:LogDuZ2}$$ Let us define the auxiliary functions $F_{1;u_{}\bar{v}},F_{2;u_{}\bar{v}}:(-\infty,a]\rightarrow(0,+\infty)$ by the relations $$F_{1;u_{}\bar{v}}(z)=\max_{v\in[v_{0},\bar{v}]}F_{1}(u_{},v,z)\label{eq:Aux1}$$ and $$F_{2;u_{}\bar{v}}(z)=\min_{v\in[v_{0},\bar{v}]}F_{2}(u_{},v,z).\label{eq:Aux2}$$ In view of (\[eq:RelationF\_1F\_2\]), (\[eq:IncreasingInLastVariable\]) and the fact that (\[eq:InequalityZ1,2\]) holds on $\{u_{}\}\times[v_{0},\bar{v}]$,[^17] we can bound for any $v\in[v_{0},\bar{v}]$: $$F_{1;u_{}\bar{v}}(z_{1}(u_{},v))<F_{2;u_{}\bar{v}}(z_{1}(u_{},v))\le F_{2;u_{}\bar{v}}(z_{2}(u_{},v)).\label{eq:InequalityAuxiliary}$$ Thus, subtracting (\[eq:LogDuZ1\]) and (\[eq:LogDuZ2\]) and using (\[eq:InequalityAuxiliary\]) and (\[eq:ContinuityDv\]) (and the fact that $\partial_{v}z_{2}>0$, $\bar{v}>v_{0}$), we readily infer that $$\begin{aligned} \log(-\partial_{u}z_{1}) & (u_{},\bar{v})-\log(-\partial_{u}z_{2})(u_{},\bar{v})\label{eq:ComparisonDu}\\ & \ge\int_{v_{0}}^{\bar{v}}F_{2;u_{}\bar{v}}(z_{2}(u_{},v))\partial_{v}z_{2}(u_{},v)\, dv-\int_{v_{0}}^{\bar{v}}F_{1;u_{}\bar{v}}(z_{1}(u_{},v))\partial_{v}z_{1}(u_{},v)\, dv\nonumber \\ & >0.\nonumber \end{aligned}$$ From (\[eq:ComparisonDu\]) we thus infer that, for any $v_{0}<\bar{v}\le v_{1}$: $$\partial_{u}z_{1}(u_{},\bar{v})<\partial_{u}z_{2}(u_{},\bar{v}).\label{eq:AlmostThereComparison}$$ Therefore, since $z_{1},z_{2}$ are smooth, there exists a continuous function $\text{\textgreek{d}}_{u}:[v_{0},v_{1}]\rightarrow[0,1)$ with $\text{\textgreek{d}}_{u}\|_{(v_{0},v_{1}]}>0$, such that $$\partial_{u}z_{1}(u,v)\le\partial_{u}z_{2}(u,v)\mbox{ for }\{v_{0}\le v\le v_{1}\}\cap\{u_{}\le u\le u_{}+\text{\textgreek{d}}_{u}(v))\}.\label{eq:AlmostThereComparison-2}$$ Similarly, by integrating equations (\[eq:Equation1\]) and (\[eq:Equation2\]) in $u$ along $[u_{0},u_{1}]\times\{v_{0}\}$ and repeating a similar procedure (using (\[eq:InData1\])), we also obtain that there exists a continuous function $\text{\textgreek{d}}_{v}:[u_{0},u_{1}]\rightarrow[0,1)$ with $\text{\textgreek{d}}_{v}\|_{(u_{0},u_{1}]}>0$, such that $$\partial_{v}z_{1}(\bar{u},v_{0})\le\partial_{v}z_{2}(\bar{u},v_{0})\mbox{ for }\{u_{0}\le u\le u_{1}\}\cap\{v_{0}\le v\le v_{0}+\text{\textgreek{d}}_{v}(u))\}.\label{eq:AlmostThereComparison-1}$$ From (\[eq:AlmostThereComparison\]) and (\[eq:AlmostThereComparison-1\]), we infer that there exists some $\text{\textgreek{d}}>0$, such that $$z_{1}\le z_{2}\mbox{ on }(u_{},u_{}+\text{\textgreek{d}})\times[v_{0},v_{1}].\label{eq:DoneFirstInequality}$$ In particular, (\[eq:InequalityZ1,2\]) holds on $[u_{0},u_{}+\text{\textgreek{d}})\times[v_{0},v_{1}]$. Furthermore, for any $\bar{u}\in(u_{},u_{}+\text{\textgreek{d}})$ and any $\bar{v}\in(v_{0},v_{0}+\text{\textgreek{d}}_{v}(\bar{u}))$, repeating the procedure leading to (\[eq:ComparisonDu\]) with $\bar{u}$ in place of $u_{}$ and using (\[eq:AlmostThereComparison-1\]) and (\[eq:DoneFirstInequality\]) in place of (\[eq:ContinuityDv\]) and (\[eq:InequalityZ1,2\]), respectively, we infer that: $$\partial_{u}z_{1}(\bar{u},\bar{v})\le\partial_{u}z_{2}(\bar{u},\bar{v}).\label{eq:OneMoreComparison}$$ Thus, combining (\[eq:AlmostThereComparison-2\]) and (\[eq:OneMoreComparison\]), we infer that (\[eq:ContinuityDu\]) holds on $[u_{0},u_{}+\text{\textgreek{d}}')\times[v_{0},v_{1}]$, for some $0<\text{\textgreek{d}}'\le\text{\textgreek{d}}$. Finally, the bound (\[eq:ContinuityDv\]) on $[u_{0},u_{}+\text{\textgreek{d}}')\times[v_{0},v_{1}]$ follows in a similar way as the proof of (\[eq:ComparisonDu\]), by integrating equations (\[eq:Equation1\]) and (\[eq:Equation2\]) in $u\in[u_{0},u_{}+\text{\textgreek{d}}')$ for any $\bar{v}\in(v_{0},v_{1})$ and using (\[eq:RelationF\_1F\_2\]), (\[eq:InequalityZ1,2\]) and (\[eq:ContinuityDu\]) (which we have shown that they hold on $[u_{0},u_{}+\text{\textgreek{d}}')\times[v_{0},v_{1}]$). We will omit the details. ### \[sub:TheInductionLemma\]A lemma for a system of inductive inequalities The following lemma is used to show that the inductive bounds (\[eq:BoundForMassIncrease\]) and (\[eq:BoundForMaxBeamSeparation\]) for $\tilde{m}_{n}^{(1,k+1)}$ and $r_{n}^{(k,k+1)}$ indeed lead to the formation of an almost-trapped surface. \[lem:ForMassIncrease\]Let $0<c_{1}\ll1\ll C_{1}$, and $0<\text{\textgreek{m}}_{0}\ll1\ll\text{\textgreek{r}}_{0}$, $0<\text{\textgreek{d}}\ll1$ be given variables, such that $$\Big(\frac{C_{1}}{c_{1}}\Big)^{8}\ll\frac{\text{\textgreek{r}}_{0}}{\text{\textgreek{m}}_{0}}\label{eq:LargenessRho0}$$ and $$\text{\textgreek{d}}<\big(\frac{\text{\textgreek{m}}_{0}}{\text{\textgreek{r}}_{0}}\big)^{(C_{1}/c_{1})^{4}}.\label{eq:SmallnessDelta}$$ Let also $\text{\textgreek{m}}_{n},\text{\textgreek{r}}_{n}>0$, be sequences of positive numbers, with $\text{\textgreek{m}}_{n}$ increasing in $n$, such that for $0\le n\le n_{}$ they satisfy $$\begin{aligned} \text{\textgreek{r}}_{n+1} & \le\text{\textgreek{r}}_{n}+C_{1}\log\big((1-\text{\textgreek{m}}_{n})^{-1}+1\big),\label{eq:InductiveRelationRho}\\ \text{\textgreek{m}}_{n+1} & \ge\text{\textgreek{m}}_{n}\exp\big(\frac{c_{1}}{\text{\textgreek{r}}_{n+1}}\big),\label{eq:InductiveRelationMu}\end{aligned}$$ and $$\max_{0\le n\le n_{}-1}\text{\textgreek{m}}_{n}<1-\text{\textgreek{d}}.\label{eq:NerTrappedMu}$$ Then, $$n_{}\le\big(\frac{C_{1}}{c_{1}}\big)^{3}\text{\textgreek{r}}_{0}\text{\textgreek{m}}_{0}^{-(C_{1}/c_{1})^{2}}.\label{eq:UpperBoundN0}$$ Furthermore, if $1-\text{\textgreek{d}}\le\text{\textgreek{m}}_{n_{}}\le1+\text{\textgreek{d}}$, we can bound: $$\text{\textgreek{m}}_{n_{}-1}\le1-\big(\frac{c_{1}}{C_{1}}\big)^{3}\text{\textgreek{r}}_{0}^{-2}\text{\textgreek{m}}_{0}^{2(C_{1}/c_{1})^{2}}\label{eq:UpperBoundMassN_}$$ and $$\max\{\text{\textgreek{r}}_{n_{}},\text{\textgreek{r}}_{n_{}-1}\}\le\big(\frac{C_{1}}{c_{1}}\big)^{4}\frac{\text{\textgreek{r}}_{0}}{\text{\textgreek{m}}_{0}^{(C_{1}/c_{1})^{2}}}\log\big(\frac{\text{\textgreek{r}}_{0}}{\text{\textgreek{m}}_{0}}\big).\label{eq:UpperBoundRhoN_}$$ Notice that the right hand side of (\[eq:UpperBoundN0\]) is independent of $\text{\textgreek{d}}$. Let us define for any integer $k\ge1$ $$n_{k}=\max\Big\{0\le n\le n_{}:\mbox{ }\text{\textgreek{m}}_{l}\le1-\frac{1}{2^{k}}\mbox{ for all }0\le l\le n\Big\},\label{eq:DefinitionNk}$$ using the convention $$n_{0}=0.$$ Notice that, in view of the fact that the sequence $\text{\textgreek{m}}_{n}$ is increasing, for all $k\ge1$ and all $n_{k-1}<n\le n_{k}$ we can estimate: $$1-\frac{1}{2^{k-1}}\le\text{\textgreek{m}}_{n}\le1-\frac{1}{2^{k}}\label{eq:TrivialBoundMuN}$$ (note that, in the case $n_{k-1}=n_{k}$, there is no $n$ satisfying $n_{k-1}<n\le n_{k}$ and (\[eq:TrivialBoundMuN\])). Using (\[eq:TrivialBoundMuN\]), from (\[eq:InductiveRelationRho\]) we can bound for any $k\ge1$ such that $n_{k-1}<n_{k}$ and any $n_{k-1}<n\le n_{k}$: $$\text{\textgreek{r}}_{n}\le\text{\textgreek{r}}_{n_{k-1}}+2C_{1}(\log2)k(n-n_{k-1})\label{eq:InductiveRelationSeperation}$$ and, therefore, for any $0\le n\le n_{k}$ we have: $$\text{\textgreek{r}}_{n}\le2C_{1}(\log2)\Big(\sum_{l=1}^{k-1}l(n_{l}-n_{l-1})+k(n-n_{k-1})\Big)+\text{\textgreek{r}}_{0}\label{eq:BestEstimateForRhoN}$$ (note that $\eqref{eq:BestEstimateForRhoN}$ holds for all $0\le n\le n_{k}$, while the bounds (\[eq:TrivialBoundMuN\]) and (\[eq:InductiveRelationSeperation\]) are non-trivial only for those values of $k$ for which $n_{k}>n_{k-1}$). Let us set $$k_{1}\doteq32\lceil\log\frac{C_{1}}{c_{_{1}}}\rceil.\label{eq:k_1}$$ Then, (\[eq:InductiveRelationSeperation\]) implies that, for all $0\le n\le n_{k_{1}}$ $$\text{\textgreek{r}}_{n}\le\text{\textgreek{r}}_{0}+2C_{1}(\log2)k_{1}n\label{eq:BoundRhoBeforeN_k1}$$ and, thus, (\[eq:InductiveRelationMu\]) implies that $$\log\big(\frac{\text{\textgreek{m}}_{n_{k_{1}}}}{\text{\textgreek{m}}_{0}}\big)\ge c_{1}\sum_{n=1}^{n_{k_{1}}}\text{\textgreek{r}}_{n}^{-1}\ge c_{1}\sum_{n=1}^{n_{k_{1}}}\frac{1}{\text{\textgreek{r}}_{0}+2C_{1}(\log2)k_{1}n}\ge\frac{c_{1}\log\Big(\frac{\text{\textgreek{r}}_{0}+2C_{1}(\log2)k_{1}n_{k_{1}}}{\text{\textgreek{r}}_{0}+2C_{1}(\log2)k_{1}}\Big)}{4C_{1}(\log2)k_{1}}.\label{eq:AlmostBoundNk_1}$$ From (\[eq:DefinitionNk\]) and (\[eq:AlmostBoundNk\_1\]) we readily infer that $$\frac{c_{1}\log\Big(\frac{\text{\textgreek{r}}_{0}+2C_{1}(\log2)k_{1}n_{k_{1}}}{\text{\textgreek{r}}_{0}+2C_{1}(\log2)k_{1}}\Big)}{4C_{1}(\log2)k_{1}}\le-\log(\text{\textgreek{m}}_{0})$$ and, therefore (using also (\[eq:LargenessRho0\])): $$n_{k_{1}}\le\frac{\text{\textgreek{r}}_{0}}{\text{\textgreek{m}}_{0}^{(C_{1}/c_{1})^{2}}}.\label{eq:FinalBoundNk_1}$$ For any $k\ge2$ such that $n_{k}>n_{k-1}+1$, from (\[eq:InductiveRelationMu\]), (\[eq:DefinitionNk\]), (\[eq:TrivialBoundMuN\]) and (\[eq:BestEstimateForRhoN\]) we readily infer: $$\frac{1}{2^{k-2}}\ge\log\frac{\text{\textgreek{m}}_{n_{k}}}{\text{\textgreek{m}}_{n_{k-1}+1}}\ge c_{1}\sum_{n=n_{k-1}+2}^{n_{k}}\text{\textgreek{r}}_{n}^{-1}\ge\frac{c_{1}}{4C_{1}(\log2)}\frac{1}{(k-1)+\sum_{l=2}^{k-1}(l-1)\frac{(n_{l}-n_{l-1})}{n_{k}-n_{k-1}-1}}$$ and, hence: $$\begin{aligned} n_{k}-n_{k-1}-1 & \le\frac{\sum_{l=2}^{k-1}(l-1)(n_{l}-n_{l-1})}{\frac{c_{1}}{4C_{1}(\log2)}2^{k-2}-(k-1)}\label{eq:InductiveRelationNk}\\ & \le\frac{k(k-1)}{2}\frac{\max_{2\le l\le k-1}(n_{l}-n_{l-1})}{\frac{c_{1}}{4C_{1}(\log2)}2^{k-2}-(k-1)}\nonumber \\ & \le\frac{k(k-1)}{2}\frac{1}{\frac{c_{1}}{4C_{1}(\log2)}2^{k-2}-(k-1)}n_{k-1}.\nonumber \end{aligned}$$ The relation (\[eq:InductiveRelationNk\]) also holds (trivially) when $n_{k}\le n_{k-1}+1$. Thus, for any $k\ge k_{1}$, the bound (\[eq:InductiveRelationNk\]) yields: $$n_{k}\le\big(1+\frac{C_{1}}{c_{1}}2^{-\frac{k-2}{4}}\big)n_{k-1}+1$$ and, therefore, for any $k\ge2$: $$n_{k}\le16\frac{C_{1}}{c_{1}}\big(n_{k_{1}}+\max\{k-k_{1},0\}\big).\label{eq:FirstTotalUpperBoundNk}$$ In view of (\[eq:FinalBoundNk\_1\]), we thus obtain for any $k\ge2$: $$n_{k}\le16\frac{C_{1}}{c_{1}}\big(\frac{\text{\textgreek{r}}_{0}}{\text{\textgreek{m}}_{0}^{(C_{1}/c_{1})^{2}}}+\max\{k-k_{1},0\}\big).\label{eq:RoughGrowthRateNk}$$ Let us set $$k_{2}\doteq4k_{1}+2\frac{\log\big(\text{\textgreek{r}}_{0}/\text{\textgreek{m}}_{0}^{(C_{1}/c_{1})^{2}}\big)}{\log2}.\label{eq:DefinitionK2}$$ Note that (\[eq:SmallnessDelta\]), (\[eq:NerTrappedMu\]) and (\[eq:DefinitionNk\]) implies that $$n_{k_{2}}\le n_{}-1.$$ In view of (\[eq:RoughGrowthRateNk\]), we have $$n_{k_{2}}\le\big(\frac{C_{1}}{c_{1}}\big)^{3}\frac{\text{\textgreek{r}}_{0}}{\text{\textgreek{m}}_{0}^{(C_{1}/c_{1})^{2}}}\label{eq:UpperBoundNk_2}$$ and, for all $k\ge k_{2}$ (in view of (\[eq:InductiveRelationNk\]) and (\[eq:RoughGrowthRateNk\])): $$n_{k}-n_{k-1}\le1.$$ Furthermore, (\[eq:BestEstimateForRhoN\]) implies (in view of (\[eq:DefinitionK2\]) and (\[eq:UpperBoundNk\_2\])) that $$\max\{\text{\textgreek{r}}_{n_{k_{2}}+1},\text{\textgreek{r}}_{n_{k_{2}}}\}\le\big(\frac{C_{1}}{c_{1}}\big)^{3}\frac{\text{\textgreek{r}}_{0}}{\text{\textgreek{m}}_{0}^{(C_{1}/c_{1})^{2}}}\log\big(\frac{\text{\textgreek{r}}_{0}}{\text{\textgreek{m}}_{0}}\big).\label{eq:UpperBoundRhok_2}$$ In view of (\[eq:DefinitionNk\]), we have $$\text{\textgreek{m}}_{n_{k_{2}}}\le1-2^{-k_{2}}<\text{\textgreek{m}}_{n_{k_{2}}+1}.\label{eq:SeperationMn_k_2}$$ We will consider two cases, depending on whether $\text{\textgreek{m}}_{n_{k_{2}}+1}$ is larger than $1-\text{\textgreek{d}}$ or not. 1. In the case $\text{\textgreek{m}}_{n_{k_{2}}+1}\ge1-\text{\textgreek{d}}$, (\[eq:NerTrappedMu\]) implies that $n_{k_{2}}+1=n_{}$. Thus, (\[eq:UpperBoundN0\]) follows from (\[eq:UpperBoundNk\_2\]). Furthermore, (\[eq:UpperBoundRhoN\_\\]) follows from (\[eq:UpperBoundRhok\_2\]), while (\[eq:UpperBoundMassN\_\\]) follows from (\[eq:SeperationMn\_k\_2\]). 2. In the case $\text{\textgreek{m}}_{n_{k_{2}}+1}<1-\text{\textgreek{d}}$, we can assume without loss of generality that $n_{k_{2}}\le n_{}-2$ (otherwise, (\[eq:UpperBoundN0\]) follows from (\[eq:UpperBoundNk\_2\])). From (\[eq:InductiveRelationRho\]), (\[eq:UpperBoundRhok\_2\]) and (\[eq:SeperationMn\_k\_2\]), we thus infer that $$\text{\textgreek{r}}_{n_{k_{2}}+2}\le\big(\frac{C_{1}}{c_{1}}\big)^{3}\frac{\text{\textgreek{r}}_{0}}{\text{\textgreek{m}}_{0}^{(C_{1}/c_{1})^{2}}}\log\big(\frac{\text{\textgreek{r}}_{0}}{\text{\textgreek{m}}_{0}}\big)+C_{1}\log\big((1-\text{\textgreek{m}}_{n_{k_{2}}+1})^{-1}\big)\label{eq:UpperBoundNextRho}$$ Hence, setting $$M\doteq\big(\frac{C_{1}}{c_{1}}\big)^{3}\frac{\text{\textgreek{r}}_{0}}{\text{\textgreek{m}}_{0}^{(C_{1}/c_{1})^{2}}}\log\big(\frac{\text{\textgreek{r}}_{0}}{\text{\textgreek{m}}_{0}}\big),\label{eq:DefinitionM}$$ from (\[eq:InductiveRelationMu\]) and (\[eq:SeperationMn\_k\_2\]) we calculate: $$\begin{aligned} \text{\textgreek{m}}_{n_{k_{2}}+2} & \ge\text{\textgreek{m}}_{n_{k_{2}}+1}\exp\Big(\frac{c_{1}}{\text{\textgreek{r}}_{n_{k_{2}}+2}}\Big)\label{eq:LowerBoundMnBiggerThan1}\\ & \ge\begin{cases} (1-2^{-k_{2}})e^{\frac{c_{1}}{2M}}, & \mbox{if }\log\big((1-\text{\textgreek{m}}_{n_{k_{2}}+1})^{-1}\big)\le\frac{M}{C_{1}}\\ \text{\textgreek{m}}_{n_{k_{2}}+1}\Big(1+\frac{c_{1}}{C_{1}\log\big((1-\text{\textgreek{m}}_{n_{k_{2}}+1})^{-1}+1\big)}\Big), & \mbox{if }\log\big((1-\text{\textgreek{m}}_{n_{k_{2}}+1})^{-1}\big)>\frac{M}{C_{1}}. \end{cases}\nonumber \end{aligned}$$ If $\log\big((1-\text{\textgreek{m}}_{n_{k_{2}}+1})^{-1}\big)\le\frac{M}{C_{1}}$, in view of (\[eq:DefinitionK2\]) and (\[eq:DefinitionM\]) we can bound (using also (\[eq:LargenessRho0\])) $$(1-2^{-k_{2}})e^{\frac{c_{1}}{2M}}\ge\Big(1-\frac{\text{\textgreek{m}}_{0}^{2(C_{1}/c_{1})^{2}}}{\text{\textgreek{r}}_{0}^{2}}\Big)\Big(1+\frac{c_{1}}{2}\big(\frac{c_{1}}{C_{1}}\big)^{3}\frac{\text{\textgreek{m}}_{0}^{(C_{1}/c_{1})^{2}}}{\text{\textgreek{r}}_{0}\log\big(\frac{\text{\textgreek{r}}_{0}}{\text{\textgreek{m}}_{0}}\big)}\Big)>1+\text{\textgreek{d}}.$$ If $\log\big((1-\text{\textgreek{m}}_{n_{k_{2}}+1})^{-1}\big)>\frac{M}{C_{1}}\gg(C_{1}/c_{1})^{3}$, we can also estimate: $$\text{\textgreek{m}}_{n_{k_{2}}+1}\Big(1+\frac{c_{1}}{C_{1}\log\big((1-\text{\textgreek{m}}_{n_{k_{2}}+1})^{-1}+1\big)}\Big)\ge\Big(1-e^{-\log\big((1-\text{\textgreek{m}}_{n_{k_{2}}+1})^{-1}\big)}\Big)\Big(1+\frac{c_{1}}{C_{1}\log\big((1-\text{\textgreek{m}}_{n_{k_{2}}+1})^{-1}+1\big)}\Big)>1+\text{\textgreek{d}}.$$ Therefore, (\[eq:LowerBoundMnBiggerThan1\]) implies that $$\text{\textgreek{m}}_{n_{k_{2}}+2}>1+\text{\textgreek{d}}\label{eq:MuBiggerThan1SecondCase}$$ and, hence, $n_{k_{2}+2}=n_{}$. Thus, (\[eq:UpperBoundN0\]) follows again from (\[eq:UpperBoundNk\_2\]). ### \[sub:Cauchy-Stability-Backwards\]Cauchy stability backwards in time The following lemma, which is essentially a backwards-in-time Cauchy stability estimate for late time perturbations of $(\mathcal{U}_{\text{\textgreek{e}}};r,\text{\textgreek{W}}^{2},\bar{f}_{in},\bar{f}_{out})$, is an easy corollary of Theorem \[prop:CauchyStability\]. \[lem:PerturbationInitialData\] For any $0<\text{\textgreek{e}}<\text{\textgreek{e}}_{0}$ (provided $\text{\textgreek{e}}_{0}$ is sufficiently small), any $r_{0}>0$ satisfying (\[eq:BoundMirror\]), let $(\mathcal{U}_{\text{\textgreek{e}}};r,\text{\textgreek{W}}^{2},\bar{f}_{in},\bar{f}_{out})$ be the maximal future development of $(r_{/\text{\textgreek{e}}},\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{2},\bar{f}_{in/\text{\textgreek{e}}},\bar{f}_{out/\text{\textgreek{e}}})$, and let us set $$C_{\text{\textgreek{e}}}\doteq\exp\Big(\exp\big(-2(h_{0}(\text{\textgreek{e}}))^{-4}\big)(h_{1}(\text{\textgreek{e}}))^{-4}\Big).\label{eq:Cepsilon}$$ Then, for any $0\le u_{}\le(h_{1}(\text{\textgreek{e}}))^{-2}v_{0\text{\textgreek{e}}}$ such that $$\mathcal{W}_{u_{}}\doteq\{0<u\le u_{}\}\cap\{u<v<u+v_{0\text{\textgreek{e}}}\}\subset\mathcal{U}_{\text{\textgreek{e}}},$$ $$\sup_{\mathcal{W}_{u_{}}}\big(1-\frac{2\tilde{m}}{r}\big)^{-1}\le C_{\text{\textgreek{e}}}\label{eq:UpperBoundTrappingInTheRegionOfPerturbation}$$ and $$u_{}+v_{0\text{\textgreek{e}}}\notin supp\big(r^{2}T_{vv}(u_{},\cdot)\big),\label{eq:EbdsNotInSupport}$$ and for any $\tilde{T}_{vv}:(u_{},u_{}+v_{0\text{\textgreek{e}}})\rightarrow\mathbb{R}$ smooth and compactly supported satisfying $\tilde{T}_{vv}(\cdot)\ge-T_{vv}(u_{},\cdot)$ and $$\sup_{u_{}\le\bar{v}\le u_{}+v_{0\text{\textgreek{e}}}}(-\Lambda)\int_{u_{}}^{u_{}+v_{0\text{\textgreek{e}}}}\frac{r^{2}(u_{},v)\frac{\|\tilde{T}_{vv}(v)\|}{\partial_{v}\text{\textgreek{r}}(u_{},v)}}{\|\text{\textgreek{r}}(u_{},v)-\text{\textgreek{r}}(u_{},\bar{v})\|+\text{\textgreek{r}}(u_{},u_{})}dv\le\exp\big(-C_{\text{\textgreek{e}}}^{2}\frac{u_{}}{v_{0\text{\textgreek{e}}}}\big)(h_{1}(\text{\textgreek{e}}))^{2}\label{eq:SmallnessPerturbation}$$ with $$\text{\textgreek{r}}(u,v)\doteq\tan^{-1}\Big(\sqrt{-\frac{\Lambda}{3}}r\Big),\label{eq:RhoFunctionAuxiliary}$$ the following statement holds: There exists a smooth asymptotically AdS boundary-characteristic initial data set $(r_{/\text{\textgreek{e}}}^{\prime},(\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{\prime})^{2},\bar{f}_{in/\text{\textgreek{e}}}^{\prime},\bar{f}_{out/\text{\textgreek{e}}}^{\prime})$ on $\{u=0\}$ for the system (\[eq:RequationFinal\])–(\[eq:OutgoingVlasovFinal\]) satisfying the reflecting gauge condition at $r=r_{0\text{\textgreek{e}}},+\infty$ with the following properties: 1. The initial data sets $(r_{/\text{\textgreek{e}}},\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{2},\bar{f}_{in/\text{\textgreek{e}}},\bar{f}_{out/\text{\textgreek{e}}})$ and $(r_{/\text{\textgreek{e}}}^{\prime},(\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{\prime})^{2},\bar{f}_{in/\text{\textgreek{e}}}^{\prime},\bar{f}_{out/\text{\textgreek{e}}}^{\prime})$ are $(h_{1}(\text{\textgreek{e}}))^{2}$ close in the (\[eq:GeometricNormForCauchyStability\]) norm, and in particular: $$\begin{aligned} \sup_{v\in[0,v_{0\text{\textgreek{e}}})}\Bigg\{\Big\|\log\big(\frac{\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{2}}{1-\frac{1}{3}\Lambda r_{/\text{\textgreek{e}}}^{2}}\big)-\log\big(\frac{(\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{\prime})^{2}}{1-\frac{1}{3}\Lambda(r_{/\text{\textgreek{e}}}^{\prime})^{2}}\big)\Big\|+\Big\|\log\Big(\frac{2\partial_{v}r_{/\text{\textgreek{e}}}}{1-\frac{2m_{/\text{\textgreek{e}}}}{r_{/\text{\textgreek{e}}}}}\Big)-\log\Big(\frac{2\partial_{v}r_{/\text{\textgreek{e}}}^{\prime}}{1-\frac{2m_{/\text{\textgreek{e}}}^{\prime}}{r_{/\text{\textgreek{e}}}^{\prime}}}\Big)\Big\|+\label{eq:GaugeDifferenceBoundCauchystability-1}\\ +\Big\|\log\Big(\frac{1-\frac{2m_{/_{\text{\textgreek{e}}}}}{r_{/\text{\textgreek{e}}}}}{1-\frac{1}{3}\Lambda r_{/\text{\textgreek{e}}}^{2}}\Big)-\log\Big(\frac{1-\frac{2m_{/\text{\textgreek{e}}}^{\prime}}{r_{/\text{\textgreek{e}}}^{\prime}}}{1-\frac{1}{3}\Lambda(r_{/\text{\textgreek{e}}}^{\prime})}\Big)\Big\|+\sqrt{-\Lambda}\|\tilde{m}_{/\text{\textgreek{e}}}-\tilde{m}_{/\text{\textgreek{e}}}^{\prime}\|\Bigg\}(v) & \le(h_{1}(\text{\textgreek{e}}))^{2}\nonumber \end{aligned}$$ and $$\sup_{\bar{v}\in[0,v_{0\text{\textgreek{e}}}]}\int_{0}^{v_{0\text{\textgreek{e}}}}\frac{\big\|r_{/\text{\textgreek{e}}}^{2}\frac{(T_{vv})_{/\text{\textgreek{e}}}}{\partial_{v}\text{\textgreek{r}}_{/\text{\textgreek{e}}}}(v)-(r_{/\text{\textgreek{e}}}^{\prime})^{2}\frac{(T_{vv})_{/\text{\textgreek{e}}}^{\prime}}{\partial_{v}\text{\textgreek{r}}_{/\text{\textgreek{e}}}^{\prime}}(v)\big\|}{\|\text{\textgreek{r}}_{/\text{\textgreek{e}}}(\bar{v})-\text{\textgreek{r}}_{/\text{\textgreek{e}}}(v)\|+\text{\textgreek{r}}_{/\text{\textgreek{e}}}(0)}\, dv\le(h_{1}(\text{\textgreek{e}}))^{2}.\label{eq:DifferenceBoundCauchyStability-1}$$ 2. The maximal future development $(\mathcal{U}_{\text{\textgreek{e}}}^{\prime};r^{\prime},(\text{\textgreek{W}}^{\prime})^{2},\bar{f}_{in}^{\prime},\bar{f}_{out}^{\prime})$ of $(r_{/\text{\textgreek{e}}}^{\prime},(\text{\textgreek{W}}_{/\text{\textgreek{e}}}^{\prime})^{2},\bar{f}_{in/\text{\textgreek{e}}}^{\prime},\bar{f}_{out/\text{\textgreek{e}}}^{\prime})$ satisfies $$\mathcal{W}_{u_{}}\subset\mathcal{U}_{\text{\textgreek{e}}}^{\prime},\label{eq:ComparableDevelopments}$$ $$r^{\prime}\|_{\{u=u_{}\}\cap supp(T_{vv})}=r\|_{\{u=u_{}\}\cap supp(T_{vv})}\label{eq:EqualROnTheSupport}$$ and $$T_{vv}^{\prime}\|_{\{u=u_{}\}}=T_{vv}\|_{\{u=u_{}\}}+\tilde{T}_{vv}.\label{eq:NewIngoingEnergyMomentum}$$ In view of (\[eq:BoundMirror\]), (\[eq:MassInfinity\]), (\[eq:NotEnoughMassBehind\]) and (\[eq:BoundSecondBeamchanged\]), we can readily estimate $$\sup_{\mathcal{W}_{u_{}}\backslash\cup_{n}\mathcal{R}_{n}^{(1,k+1)}}\Big(1-\frac{2\tilde{m}}{r}\Big)^{-1}\le2\exp\Big((h_{0}(\text{\textgreek{e}}))^{-4}\Big).\label{eq:TotalDistanceFromTrappingAway}$$ Therefore, using (\[eq:UpperBoundTrappingInTheRegionOfPerturbation\]) for $\cup_{n}\mathcal{R}_{n}^{(1,k+1)}$ and (\[eq:TotalDistanceFromTrappingAway\]) for $\mathcal{W}_{u_{}}\backslash\cup_{n}\mathcal{R}_{n}^{(1,k+1)}$, the relations (\[eq:TotalChangeKappaBarEachIteration\]) and (\[eq:TotalChangeKappaEachIteration\]) imply (in view of (\[eq:h\_2definition\]), (\[eq:BoundForRAwayInteractionProp\]) and the fact that $u_{}\le(h_{1}(\text{\textgreek{e}}))^{-2}v_{0\text{\textgreek{e}}}$) that $$\sup_{\mathcal{W}_{u_{}}}\Bigg(\Big\|\log\Big(\frac{-\partial_{u}r}{1-\frac{2m}{r}}\Big)\Big\|+\Big\|\log\Big(\frac{\partial_{v}r}{1-\frac{2m}{r}}\Big)\Big\|\Bigg)\le(h_{1}(\text{\textgreek{e}}))^{-3}\exp\big((h_{0}(\text{\textgreek{e}}))^{-4}\big).\label{eq:BoundDuRDvRAwayFromSevereTrapping}$$ Similarly, equations (\[eq:DerivativeInUDirectionKappa\]) and (\[eq:DerivativeInVDirectionKappaBar\]), in view of the relations (\[eq:TotalChangeKappaBarEachIteration\]), (\[eq:TotalChangeKappaEachIteration\]) (using again the bounds (\[eq:UpperBoundTrappingInTheRegionOfPerturbation\]) (\[eq:TotalDistanceFromTrappingAway\])) imply that $$\sup_{\bar{u}}\int_{\{u=\bar{u}\}\cap\mathcal{W}_{u_{}}}rT_{vv}\, dv+\sup_{\bar{v}}\int_{\{v=\bar{v}\}\cap\mathcal{W}_{u_{}}}rT_{uu}\, du\le(h_{1}(\text{\textgreek{e}}))^{-1}\exp\big((h_{0}(\text{\textgreek{e}}))^{-4}\big).\label{eq:BoundrTAwayFromSevereTrapping}$$ Let us fix a set of smooth functions $r_{/}^{},(\text{\textgreek{W}}_{/}^{})^{2}:[u_{},u_{}+v_{0\text{\textgreek{e}}})\rightarrow(0,+\infty)$ and $\bar{f}_{in/}^{},\bar{f}_{out/}^{}:[u_{},u_{}+v_{0\text{\textgreek{e}}})\times(0,+\infty)\rightarrow[0,+\infty)$ satisfying the following requirements: 1. $(r_{/}^{},(\text{\textgreek{W}}_{/}^{})^{2},\bar{f}_{in/}^{},\bar{f}_{out/}^{})$ is a smooth asymptotically AdS boundary-characteristic initial data set for for the system (\[eq:RequationFinal\])–(\[eq:OutgoingVlasovFinal\]) on $\{u_{}\}\times[u_{},u_{}+v_{0\text{\textgreek{e}}})$ satisfying the reflecting gauge condition at $r^{}=r_{0\text{\textgreek{e}}},+\infty$. 2. The function $r_{/}^{}$ satisfies for any $v$ such that $(u_{},v)\in supp(T_{vv})$ $$r_{/}^{}(v)=r\|_{supp(T_{vv})\cap\{u=u_{}\}}.\label{eq:SameRLateTime}$$ 3. The function $\bar{f}_{in/}^{}$ satisfies for all $v\in[u_{},u_{}+v_{0\text{\textgreek{e}}})$: $$\int_{0}^{+\infty}\big((\text{\textgreek{W}}_{/}^{})^{2}(v)p^{v}\big)^{2}\bar{f}_{in/}^{}(v;p^{v})\,(r_{/}^{})^{2}(v)\frac{dp^{v}}{p^{v}}=T_{vv}(u_{},v)+\tilde{T}_{vv}(v).\label{eq:IngoingEnergyMomentumDeformed}$$ 4. The function $\bar{f}_{out/}^{}$ satisfies for all $(v,p^{v})\in[u_{},u_{}+v_{0\text{\textgreek{e}}})\times(0,+\infty)$: $$\bar{f}_{out/}^{}(v;p^{v})=\bar{f}_{out}(u_{},v;p^{v}).\label{eq:SameOutgoingF}$$ 5. The initial data sets $(r,\text{\textgreek{W}}^{2},\bar{f}_{in},\bar{f}_{out})\|_{u=u_{}}$ and $(r_{/}^{},(\text{\textgreek{W}}_{/}^{})^{2},\bar{f}_{in/}^{},\bar{f}_{out/}^{})$ satisfy $$\begin{aligned} \sup_{v\in[u_{},u_{}+v_{0\text{\textgreek{e}}})}\Bigg\{\Big\|\log\big(\frac{\text{\textgreek{W}}^{2}}{1-\frac{1}{3}\Lambda r^{2}}\big)\Big\|_{u=u_{}}-\log\big(\frac{(\text{\textgreek{W}}_{/}^{})^{2}}{1-\frac{1}{3}\Lambda(r_{/}^{})^{2}}\big)\Big\|+\Big\|\log\Big(\frac{2\partial_{v}r}{1-\frac{2m}{r}}\Big)\Big\|_{u=u_{}}-\log & \Big(\frac{2\partial_{v}r_{/}^{}}{1-\frac{2m_{/}^{}}{r_{/}^{}}}\Big)\Big\|+\label{eq:GaugeDifferenceBoundCauchystability-2}\\ +\Big\|\log\Big(\frac{1-\frac{2m}{r}}{1-\frac{1}{3}\Lambda r^{2}}\Big)\Big\|_{u=u_{}}-\log\Big(\frac{1-\frac{2m_{/}^{}}{r_{/}^{}}}{1-\frac{1}{3}\Lambda(r_{/}^{})}\Big)\Big\|+\sqrt{-\Lambda}\big\|\tilde{m}\|_{u=u_{}}-\tilde{m}_{/}^{}\big\|\Bigg\}(v) & \le\exp\big(-\frac{1}{2}C_{\text{\textgreek{e}}}^{2}\frac{u_{}}{v_{0}}\big)(h_{1}(\text{\textgreek{e}}))^{2}\nonumber \end{aligned}$$ and $$\sup_{v\in[v_{1},v_{2}]}\int_{v_{1}}^{v_{2}}\frac{\big\|r^{2}T_{vv}(u_{},\bar{v})-(r_{/}^{})^{2}(T_{vv})_{/}^{}(\bar{v})\big\|}{\|\text{\textgreek{r}}_{/}(v)-\text{\textgreek{r}}_{/}(\bar{v})\|+\text{\textgreek{r}}_{/}(v_{1})}\, d\bar{v}\le\exp\big(-\frac{1}{2}C_{\text{\textgreek{e}}}^{2}\frac{u_{}}{v_{0}}\big)(h_{1}(\text{\textgreek{e}}))^{2}.\label{eq:DifferenceBoundCauchyStability-2}$$ Remark.* As a consequence of (\[eq:EbdsNotInSupport\]), by suitably deforming $r_{/}^{}$ near $v=u_{}+v_{0\text{\textgreek{e}}}$, we can always arrange that (\[eq:GaugeInfinityInitialData\]) and (\[eq:SameRLateTime\]) are satisfied simultaneously. Furthermore, since $\tilde{T}_{vv}$ is compactly supported in $(u_{},u_{}+v_{0\text{\textgreek{e}}})$, we can always choose $\bar{f}_{in/}^{}=f_{in}\|_{\{u=u_{}\}}$ in a neighborhood of $v=u_{},u_{}+v_{0\text{\textgreek{e}}}$, so that (\[eq:LeftBoundaryConditionInitialData\]) and (\[eq:RightBoundaryConditionInitialData\]) are satisfied. Finally, $(r_{/}^{},(\text{\textgreek{W}}_{/}^{})^{2},\bar{f}_{in/}^{},\bar{f}_{out/}^{})$ can be chosen so that (\[eq:GaugeDifferenceBoundCauchystability-2\]) and (\[eq:DifferenceBoundCauchyStability-2\]) are satisfied because of (\[eq:SmallnessPerturbation\]) and the relations (\[eq:RelationHawkingMass\]), (\[eq:DerivativeInVDirectionKappaBar\]) and (\[eq:DerivativeTildeVMass\]). Let us now consider the two sets of initial data $(r,\text{\textgreek{W}}^{2},\bar{f}_{in},\bar{f}_{out})\|_{u=u_{}}$ and $(r_{/}^{},(\text{\textgreek{W}}_{/}^{})^{2},\bar{f}_{in/}^{},\bar{f}_{out/}^{})$ on $\{u=u_{}\}\cap\{u_{}\le v<u_{}+v_{0\text{\textgreek{e}}}\}$. The maximal development of $(r,\text{\textgreek{W}}^{2},\bar{f}_{in},\bar{f}_{out})\|_{u=u_{}}$ (see the remark below Theorem \[thm:maximalExtension\]) coincides with $(\mathcal{W}_{u_{}};r,\text{\textgreek{W}}^{2},\bar{f}_{in},\bar{f}_{out})$ when restricted on $\{u\ge0\}$ and, in view of (\[eq:BoundDuRDvRAwayFromSevereTrapping\]) and (\[eq:BoundrTAwayFromSevereTrapping\]), satisfies $$\begin{aligned} \sup_{\mathcal{W}_{u_{}}}\Bigg\{\Big\|\log\big(\frac{\text{\textgreek{W}}^{2}}{1-\frac{1}{3}\Lambda r^{2}}\big)\Big\|+\Big\|\log\Big(\frac{2\partial_{v}r}{1-\frac{2m}{r}}\Big)\Big\|+\Big\|\log\Big(\frac{1-\frac{2m}{r}}{1-\frac{1}{3}\Lambda r^{2}}\Big)\Big\|+\sqrt{-\Lambda}\|\tilde{m}\|\Bigg\}+\label{eq:UpperBoundNonTrappingForCauchyStability-2}\\ +\sup_{\bar{u}}\int_{\{u=\bar{u}\}\cap\mathcal{W}_{u_{}}}rT_{vv}\, dv+\sup_{\bar{v}}\int_{\{v=\bar{v}\}\cap\mathcal{W}_{u_{}}}rT_{uu}\, du & \le4(h_{1}(\text{\textgreek{e}}))^{-3}\exp\big(-(h_{0}(\text{\textgreek{e}}))^{-4}\big).\nonumber \end{aligned}$$ Therefore, in view of (\[eq:UpperBoundNonTrappingForCauchyStability-2\]), (\[eq:GaugeDifferenceBoundCauchystability-2\]) and (\[eq:DifferenceBoundCauchyStability-2\]), Theorem \[prop:CauchyStability\] applied for the past development of $(r,\text{\textgreek{W}}^{2},\bar{f}_{in},\bar{f}_{out})\|_{u=u_{}}$ on $\mathcal{W}_{u_{}}$ (see the remark below Theorem \[prop:CauchyStability\]) implies that the maximal past development $(\mathcal{U}^{};r^{},(\text{\textgreek{W}}^{})^{2},\bar{f}_{in}^{},\bar{f}_{out}^{})$ of $(r_{/}^{},(\text{\textgreek{W}}_{/}^{})^{2},\bar{f}_{in/}^{},\bar{f}_{out/}^{})$ satsfies $$\mathcal{W}_{u_{}}\subset\mathcal{U}^{}$$ and $$\begin{aligned} \sup_{\mathcal{W}_{u_{}}}\Bigg\{\Big\|\log\big(\frac{\text{\textgreek{W}}^{2}}{1-\frac{1}{3}\Lambda r^{2}}\big)-\log\big(\frac{(\text{\textgreek{W}}^{})^{2}}{1-\frac{1}{3}\Lambda(r^{})^{2}}\big)\Big\|+\Big\|\log\Big(\frac{2\partial_{v}r}{1-\frac{2m}{r}}\Big)-\log\Big(\frac{2\partial_{v}r^{}}{1-\frac{2m^{}}{r^{}}}\Big)\Big\|+\label{eq:UpperBoundNonTrappingForCauchyStability-1-1}\\ +\Big\|\log\Big(\frac{1-\frac{2m}{r}}{1-\frac{1}{3}\Lambda r^{2}}\Big)-\log\Big(\frac{1-\frac{2m^{}}{r^{}}}{1-\frac{1}{3}\Lambda(r^{})^{2}}\Big)\Big\|+\sqrt{-\Lambda}\|\tilde{m}-\tilde{m}^{}\|\Bigg\}+\nonumber \\ +\sup_{\bar{u}}\int_{\{u=\bar{u}\}\cap\mathcal{W}_{u_{}}}\big\|rT_{vv}-r^{}(T_{vv})^{}\big\|\, dv+\sup_{\bar{v}}\int_{\{v=\bar{v}\}\cap\mathcal{W}_{u_{}}}\big\|rT_{uu}-r^{}(T_{uu})^{}\big\|\, du & \le(h_{1}(\text{\textgreek{e}}))^{3}.\nonumber \end{aligned}$$ Thus, the proof of the lemma concludes by setting $$(r_{/}^{\prime},(\text{\textgreek{W}}_{/}^{\prime})^{2},\bar{f}_{in/}^{\prime},\bar{f}_{out/}^{\prime})\doteq(r^{},(\text{\textgreek{W}}^{})^{2},\bar{f}_{in}^{},\bar{f}_{out}^{})\|_{u=0}.$$ [^1]: Here, Cauchy stability of $(\mathcal{M}_{AdS},g_{AdS})$ refers to Cauchy stability of the conformal compactification of $(\mathcal{M}_{AdS},g_{AdS})$ (including, therefore, the timelike boundary $\mathcal{I}$); see the discussion in the next section. [^2]: In fact, well-posedness for the smooth initial value problem for the Einstein–Vlasov system also holds outside spherical symmetry, see [@ChoquetBruhat1971]. In the case $\Lambda=0$, the stability of Minkowski spacetime for the Einstein–massless Vlasov system without any symmetry assumptions was recently established by Taylor [@Taylor2017]. [^3]: This problem is circumvented in 4+1 dimensions by the biaxial Bianchi IX symmetry class referred to earlier (see [@BizonChmajSchmidt2005]). [^4]: A similar result can presumably also be deduced for the vacuum Einstein equations (\[eq:VacuumEinsteinEquations\]) reduced under the biaxial Bianchi IX symmetry in $4+1$ dimensions, following by an amalgamation of the proofs of [@Holzegel5d2010] and [@HolSmul2013]. [^5]: The result of [@Christodoulou1991] was restricted to the case $\Lambda=0$, but the proof can be readily modified to include the case $\Lambda<0$. [^6]: We should remark that (\[eq:TrappedSurface\]) follows from (\[eq:IndicatorBlackHole\]) under the assumption that $\partial_{u}r<0$ (which always holds provided, initially, $\partial_{u}r\|_{u=0}<0$; see [@ChristodoulouBoundedVariation]). [^7]: where time is measured with respect to the (dimensionless) coordinate function $\bar{t}=\sqrt{-\Lambda}(u+v)$. [^8]: Note that the renormalised Hawking mass $\tilde{m}^{(\text{\textgreek{e}})}$ is constant on $\mathcal{I}$ when imposing a reflecting boundary condition. [^9]: This is possible in view of the fact that, for solutions $(r,\text{\textgreek{W}}^{2},\text{\textgreek{t}},\bar{\text{\textgreek{t}}})$ to (\[eq:EinsteinNullDust\]), $r$ remains uniformly continuous in the limit when $\text{\textgreek{t}},\bar{\text{\textgreek{t}}}$ tend to $\text{\textgreek{d}}$-functions in the $u,v$ variables, respectively. [^10]: Note that, necessarily, $\overline{\mathfrak{D}}_{+}\tilde{m}-\overline{\mathfrak{D}}_{-}\tilde{m}=-(\mathfrak{D}_{+}\tilde{m}-\mathfrak{D}_{-}\tilde{m})$. [^11]: We should remark that, once a trapped surface $\mathcal{S}$ has formed, $\{r=r_{0\text{\textgreek{e}}}\}\cap J^{+}(\mathcal{S})$ (where $J^{+}(\mathcal{S})$ is the future of $\mathcal{S}$) will be spacelike and we will not study the evolution of the spacetime beyond $\{r=r_{0\text{\textgreek{e}}}\}\cap J^{+}(\mathcal{S})$. In particular, no more reflections of the beams will occur in the future of $\mathcal{S}$. See Theorem \[thm:MaximalDevelopmentIntro\]. [^12]: The example of two outgoing null rays in the exterior of Schwarzschild–AdS, with mass $M\ll(-\Lambda)^{-\frac{1}{2}}$, serves to illustrate this phenomenon: The $r$-separation of two rays emanating from the region close to the future event horizon $\mathcal{H}^{+}$, where $\frac{2m}{r}\sim1$, increases dramatically by the time they reach the region $r\sim(-\Lambda)^{-\frac{1}{2}}$. This is, of course, nothing other than the celebrated red-shift effect. [^13]: In other words, there is no point in $\text{\textgreek{g}}$ which lies on the curve $\{v=v_{\mathcal{I}}\}$, where $(u_{\mathcal{I}},v_{\mathcal{I}})$ is the future limit point of $\mathcal{I}$. [^14]: In fact, it suffices to choose $C_{0}=50$. [^15]: Note that such a coordinate chart is not unique. [^16]: A relation for the change the mass differences of two intersecting, infinitely thin null dust beams was also obtained in [@PoissonIsrael1990]. [^17]: Note that we can immediately restrict from $[u_{0},u_{1}]\times[v_{0},v_{1}]$ to $\{u_{*}\}\times[v_{0},\bar{v}]$ in (\[eq:RelationF\_1F\_2\]).
--- abstract: 'We investigate the changes in spin and orbital patterns induced by magnetic transition metal ions without an orbital degree of freedom doped in a strongly correlated insulator with spin-orbital order. In this context we study the $3d$ ion substitution in $4d$ transition metal oxides in the case of $3d^3$ doping at either $3d^2$ or $4d^4$ sites which realizes orbital dilution in a Mott insulator. Although we concentrate on this doping case as it is known experimentally and more challenging than other oxides due to finite spin-orbit coupling, the conclusions are more general. We derive the effective $3d-4d$ (or $3d-3d$) superexchange in a Mott insulator with different ionic valencies, underlining the emerging structure of the spin-orbital coupling between the impurity and the host sites and demonstrate that it is qualitatively different from that encountered in the host itself. This derivation shows that the interaction between the host and the impurity depends in a crucial way on the type of doubly occupied $t_{2g}$ orbital. One finds that in some cases, due to the quench of the orbital degree of freedom at the $3d$ impurity, the spin and orbital order within the host is drastically modified by doping. The impurity acts either as a spin defect accompanied by an orbital vacancy in the spin-orbital structure when the host-impurity coupling is weak, or it favors doubly occupied active orbitals (orbital polarons) along the $3d-4d$ bond leading to antiferromagnetic or ferromagnetic spin coupling. This competition between different magnetic couplings leads to quite different ground states. In particular, for the case of a finite and periodic $3d$ atom substitution, it leads to striped patterns either with alternating ferromagnetic/antiferromagnetic domains or with islands of saturated ferromagnetic order. We find that magnetic frustration and spin degeneracy can be lifted by the quantum orbital flips of the host but they are robust in special regions of the incommensurate phase diagram. Orbital quantum fluctuations modify quantitatively spin-orbital order imposed by superexchange. In contrast, the spin-orbit coupling can lead to anisotropic spin and orbital patterns along the symmetry directions and cause a radical modification of the order imposed by the spin-orbital superexchange. Our findings are expected to be of importance for future theoretical understanding of experimental results for $4d$ transition metal oxides doped with $3d^3$ ions. We suggest how the local or global changes of the spin-orbital order induced by such impurities could be detected experimentally.' author: - Wojciech Brzezicki - 'Andrzej M. Oleś' - Mario Cuoco date: 24 December 2014 title: \| Spin-Orbital Order Modified by Orbital Dilution in Transition Metal Oxides:\ From Spin Defects to Frustrated Spins Polarizing Host Orbitals --- Introduction ============ The studies of strongly correlated electrons in transition metal oxides (TMOs) focus traditionally on $3d$ materials [@Ima98], mainly because of high-temperature superconductivity discovered in cuprates and more recently in iron-pnictides, and because of colossal magnetoresistance manganites. The competition of different and complex types of order is ubiquitous in strongly correlated TMOs mainly due to coupled spin-charge-orbital where frustrated exchange competes with the kinetic energy of charge carriers. The best known example is spin-charge competition in cuprates, where spin, charge and superconducting orders intertwine [@Ber11] and stripe order emerges in the normal phase as a compromise between the magnetic and kinetic energy [@Lee06; @Voj09]. Remarkable evolution of the stripe order under increasing doping is observed [@Yam98] and could be reproduced by the theory based on the extended Hubbard model [@Fle01]. Hole doping in cuprates corresponds to the removal of the spin degree of freedom. Similarly, hole doping in a simplest system with the orbital order in $d^1$ configuration removes locally orbital degrees of freedom and generates stripe phases which involve orbital polarons [@Wro10]. It was predicted recently that orbital domain walls in bilayer manganites should be partially charged as a result of competition between orbital-induced strain and Coulomb repulsion [@Li14], which opens a new route towards charge-orbital physics in TMOs. We will show below that the stripe-like order may also occur in doped spin-orbital systems. These systems are very challenging and their doping leads to very complex and yet unexplored spin-orbital-charge phenomena [@Zaa93]. A prerequisite to the phenomena with spin-orbital-charge coupled degrees of freedom is the understanding of undoped systems [@Tok00], where the low-energy physics and spin-orbital order are dictated by effective spin-orbital superexchange [@Kug82; @Ole05; @Woh11] and compete with spin-orbital quantum fluctuations [@Fei97; @Kha00; @Kha05]. Although ordered states occur in many cases, the most intriguing are quantum phases such as spin [@Bal10] or orbital [@Fei05] liquids. Recent experiments on a copper oxide Ba$_3$CuSb$_2$O$_9$ [@Nak12; @Qui12] have triggered renewed efforts in a fundamental search for a quantum spin-orbital liquid [@Nor08; @Karlo; @Nas12; @Mil14], where spin-orbital order is absent and electron spins are randomly choosing orbitals which they occupy. A signature of strong quantum effects in a spin-orbital system is a disordered state which persists down to very low temperatures. A good example of such a disordered spin-orbital liquid state is as well FeSc$_2$S$_4$ which does not order in spite of finite Curie-Weiss temperature $\Theta_{\rm CW}=-45$ K [@Fri04], but shows instead signatures of quantum criticality [@Che09; @Mit14]. Spin-orbital interactions may be even more challenging — for instance previous attempts to find a spin-orbital liquid in the Kugel-Khomskii model [@Fei97] or in LiNiO$_2$ [@Ver04] turned out to be unsuccessful. In fact, in the former case certain types of exotic spin order arise as a consequence of frustrated and entangled spin-orbital interactions [@Brz12; @Brz13], and a spin-orbital entangled resonating valence bond state was recently shown to be a quantum superposition of strped spin-singlet covering on a square lattice [@Cza15]. In contrast, spin and orbital superexchange have different energy scales and orbital interactions in LiNiO$_2$ are much stronger and dominated by frustration [@Rey01]. Hence the reasons behind the absence of magnetic long range order are more subtle [@Rei05]. In all these cases orbital fluctuations play a prominent role and spin-orbital entanglement [@Ole12] determines the ground state. The role of charge carriers in spin-orbital systems is under very active investigation at present. In doped La$_{1-x}$(Sr,Ca)$_x$MnO$_3$ manganites several different types of magnetic order compete with one another and occur at increasing hole doping [@Dag01; @Dag05; @Tok06]. Undoped LaMnO$_3$ is an antiferromagnetic (AF) Mott insulator, with large $S=2$ spins for $3d^4$ ionic configurations of Mn$^{3+}$ ions stabilized by Hund’s exchange, coupled via the spin-orbital superexchange due to $e_g$ and $t_{2g}$ electron excitations [@Fei99]. The orbital $e_g$ degree of freedom is removed by hole doping when Mn$^{3+}$ ions are generated, and this requires careful modeling in the theory that takes into account both $3d^4$ and $3d^3$ electronic configurations of Mn$^{3+}$ and Mn$^{4+}$ ions [@Kil99; @Cuo02; @Feh04; @Dag04; @Gec05; @Kha11]. In fact, the orbital order changes radically with increasing doping in La$_{1-x}$(Sr,Ca)$_x$MnO$_3$ systems at the magnetic phase transitions between different types of magnetic order [@Tok06], as weel as at La$_{0.7}$Ca$_{0.3}$MnO$_3$/BiFeO$_3$ heterostructures, where it offers a new route to enhancing multiferroic functionality [@She14]. The double exchange mechanism [@deG60] triggers ferromagnetic (FM) metallic phase at sufficient doping; in this phase the spin and orbital degrees of freedom decouple and spin excitations are explained by the orbital liquid [@KhaKi; @Ole02]. Due to distinct magnetic and kinetic energy scales, even low doping may suffice for a drastic change in the magnetic order, as observed in electron-doped manganites [@Sak10]. A rather unique example of a spin-orbital system with strongly fluctuating orbitals, as predicted in the theory [@Kha01; @Kha04; @Hor08] and seen experimentally [@Ulr03; @Zho07; @Ree11], are the perovskite vanadates with competing spin-orbital order [@Fuj10]. In these $t_{2g}$ systems $xy$ orbitals are filled by one electron and orbital order of active $\{yz,zx\}$ orbitals is strongly influenced by doping with Ca (Sr) ions which replace Y (La) ones in YVO$_3$ (LaVO$_3$). In this case finite spin-orbit coupling modifies the spin-orbital phase diagram [@Hor03]. In addition, the AF order switches easily from the $G$-type AF ($G$-AF) to $C$-type AF ($C$-AF) order in the presence of charge defects in Y$_{1-x}$Ca$_x$VO$_3$. Already at low $x\simeq 0.02$ doping the spin-orbital order changes and spectral weight is generated within the Mott-Hubbard gap [@Fuj08]. Although one might imagine that the orbital degree of freedom is thereby removed, a closer inspection shows that this is not the case as the orbitals are polarized by charge defects [@Hor11] and readjust near them [@Ave13]. Removing the orbital degree of freedom in vanadates would be only possible by electron doping generating instead $d^3$ ionic configurations, but such a doping by charge defects would be very different from the doping by transition metal ions of the same valence considered below. Also in $4d$ materials spin-orbital physics plays a role [@Hot06], as for instance in Ca$_{2-x}$Sr$_x$RuO$_4$ systems with Ru$^{4+}$ ions in $4d^4$ configuration [@Miz01; @Lee02; @Kog04; @Fan05; @Sug13]. Recently it has been shown that unconventional magnetism is possible for Ru$^{4+}$ and similar ions where spin-orbit coupling plaus a role [@Kha13; @Kha14]. Surprisingly, these systems are not similar to manganites but to vanadates where one finds as well ions with active $t_{2g}$ orbitals. In the case of ruthenates the $t_{2g}^4$ Ru$^{4+}$ ions have low $S=1$ spin as the splitting between the $t_{2g}$ and $e_g$ levels is large. Thus the undoped Ca$_2$RuO$_4$ is a hole analogue of a vanadate [@Kha01; @Kha04], with $t_{2g}$ orbital degree of freedom and $S=1$ spin per site in both cases. This gives new opportunities to investigate spin-orbital entangled states in $t_{2g}$ system, observed recently by angle resolved photoemission [@Vee14]. Here we focus on a novel and very different doping from all those considered above, namely on a substitutional doping by other magnetic ions in a plane built by transition metal and oxygen ions, for instance in the $(a,b)$ plane of a monolayer or in perovskite ruthenates or vanadates. In this study we are interested primarily in doping of a TMO with $t_{2g}$ orbital degrees of freedom, where doped magnetic ions have no orbital degree of freedom and realize orbital dilution. In addition, we deal with the simpler case of $3d$ doped ions where we can neglect spin-orbit interaction which should not be ignored for $4d$ ions. We emphasize that in contrast to manganites where holes within $e_g$ orbitals participate in transport and are responsible for the colossal magnetoresitance, such doped hole are immobile due to the ionic potential at $3d$ sites and form defects in spin-orbital order of a Mott insulator. We encounter here a different situation from the dilution effects in the 2D $e_g$ orbital system considered so far [@Tan07] as we deel with magnetic ions at doped sites. It is challenging to investigate how such impurities modify locally or globally spin-orbital order of the host. The doping which realizes this paradigm is by either Mn$^{4+}$ or Cr$^{3+}$ ions with large $S=3/2$ spins stabilized by Hund’s exchange, and orbital dilution occurs either in a TMO with $d^2$ ionic configuration as in the vanadium perovskites, or in $4d$ Mott insulators as in ruthenates. It has been shown that dilute Cr doping for Ru reduces the temperature of the orthorhombic distortion and induces FM behavior in Ca$_2$Ru$_{1-x}$Cr$_x$O$_4$ (with $0<x<0.13$) [@Qi10]. It also induces surprising negative volume thermal expansion via spin-orbital order. Such defects, on one hand, can weaken the spin-orbital coupling in the host, but on the other hand, may open a new channel of interaction between the spin and orbital degree of freedom through the host-impurity exchange, see Fig. \[fig:bond\_schem\]. Consequences of such doping are yet unexplored and are expected to open a new route in the research on strongly correlated oxides. ![ (a) Schematic view of the orbital dilution when the $3d^3$ ion with no orbital degree of freedom and spin $S=3/2$ substitutes $4d^4$ one with spin $S=1$ on a bond having specific spin and orbital character in the host (gray arrows). Spins are shown by red arrows and doubly occupied $t_{2g}$ orbitals (doublons) are shown by green symbols for $a$ and $c$ orbitals, respectively. (b) If an inactive orbital along the bond is removed by doping, the total spin exchange is AF. (c) On the contrary, active orbitals at the host site can lead to either FM (top) or AF (bottom) exchange coupling, depending on the energy levels mismatch and difference in the Coulomb couplings between the impurity and the host. We show the case when the host site is unchanged in the doping process. \[fig:bond\_schem\]](fig2_new){width=".92\columnwidth"} The physical example for the present theory are the insulating phases of $3d-4d$ hybrid structures, where doping happens at $d^4$ transition metal sites, and the value of the spin is locally changed from $S=1$ to $S=3/2$. As a demonstration of the highly nontrivial physics emerging in $3d-4d$ oxides, remarkable effects have already been observed, for instance, when Ru ions are replaced by Mn, Ti, Cr or other $3d$ elements. The role of Mn doping in the SrRuO Ruddlesden-Popper series is strongly linked to the dimensionality through the number $n$ of RuO$_2$ layers in the unit cell. The Mn doping of the SrRuO$_3$ cubic member drives the system from the itinerant FM state to an insulating AF configuration in a continuous way via a possible unconventional quantum phase transition [@Cao05]. Doping by Mn ions in Sr$_{3}$Ru$_2$O$_{7}$ leads to a metal-to-insulator transition and AF long-range order for more than $5\%$ Mn concentration [@Hos12]. Subtle orbital rearrangement can occur at the Mn site, as for instance the inversion of the crystal field in the $e_g$ sector observed via x-ray absorption spectroscopy [@Hos08]. Neutron scattering studies indicate the occurrence of an unusual $E$-type antiferromagnetism in doped systems (planar order with FM zigzag chains with AF order between them) with moments aligned along the $c$ axis within a single bilayer [@Mes12]. Furthermore, the more extended $4d$ orbitals would a priori suggest a weaker correlation than in $3d$ TMOs due to a reduced ratio between the intraatomic Coulomb interaction and the electron bandwidth. Nevertheless, the (effective) $d$-bandwidth is reduced by the changes in the $3d$-$2p$-$3d$ bond angles in distorted structures which typically arise in these materials. This brings these systems on the verge of a metal-insulator transition [@Kim09], or even into the Mott insulating state with spin-orbital order, see Fig. \[fig:host\]. Hence, not only $4d$ materials share common features with $3d$ systems, but are also richer due to their sensitivity to the lattice structure and to relativistic effects due to larger spin-orbit [@Ann06] or other magneto-crystalline couplings. ![ Schematic view of $C$-AF spin order coexisting with $G$-AO orbital order in the $(a,b)$ plane of an undoped Mott insulator with $4d^4$ ionic configurations. Spins are shown by arrows while doubly occupied $xy$ and $yz$ orbitals ($c$ and $a$ doublons, see text) form a checkerboard pattern. Equivalent spin-orbital order is realized for V$^{3+}$ ions in $(b,c)$ planes of LaVO$_3$ [@Fuj10], with orbitals standing for empty orbitals (holes).[]{data-label="fig:host"}](caf_host){width=".72\columnwidth"} To simplify the analysis we assume that onsite Coulomb interactions are so strong that charge degrees of freedom are projected out, and only virtual charge transfer can occur between $3d$ and $4d$ ions via the oxygen ligands. For convenience, we define the orbital degree of freedom as a doublon (double occupancy) in the $t_{2g}^4$ configuration. The above $3d$ doping leads then effectively to the removal of a doublon in one of $t_{2g}$ orbitals which we label as $\{a,b,c\}$ (this notation is introduced in Ref. [@Kha05] and explained below) and to replacing it by a $t_{2g}^3$ ion. To our knowledge, this is the only example of removing the orbital degree of freedom in $t_{2g}$ manifold realized so far and below we investigate possible consequences of this phenomenon. Another possibility of orbital dilution which awaits experimental realization would occur when a $t_{2g}$ degree of freedom is removed by replacing a $d^2$ ion by a $d^3$ one, as for instance by Cr$^{3+}$ doping in a vanadate — here a doublon is an empty $t_{2g}$ orbital, i.e., filled by two holes. Before presenting the details of the quantitative analysis, let us concentrate of the main idea of the superexchange modified by doping in a spin-orbital system. The $d^3$ ions have singly occupied all three $t_{2g}$ orbitals and $S=3/2$ spins due to Hund’s exchange. While a pair of $d^3$ ions, e.g. in SrMnO$_3$, is coupled by AF superexchange [@Ole02], the superexchange for the $d^3-d^4$ bond has a rather rich structure and may also be FM. The spin exchange depends then on whether the orbital degree of freedom is active and participates in charge excitations along a considered bond or electrons of the doublon cannot move along this bond due to the symmetry of $t_{2g}$ orbital, as explained in Fig. \[fig:bond\_schem\]. This qualitative difference to systems without active orbital degrees of freedom is investigated in detail in Sec. \[sec:model\]. The main outcomes of our analysis are: (i) the determination of the effective spin-orbital exchange Hamiltonian describing the low-energy sector for the $3d-4d$ hybrid structure, (ii) establishing that a $3d^3$ impurity without an orbital degree of freedom modifies the orbital order in the $4d^4$ host, (iii) providing the detailed way how the microscopic spin-orbital order within the $4d^4$ host is modified around the $3d^3$ impurity, and (iv) suggesting possible spin-orbital patterns that arise due to periodic and finite substitution (doping) of $4d$ atoms in the host by $3d$ ones. The emerging physical scenario is that the $3d$ impurity acts as an orbital vacancy when the host-impurity coupling is weak and as an orbital polarizer of the bonds active $t_{2g}$ doublon configurations when it is strongly coupled to the host. The tendency to polarize the host orbitals around the impurity turns out to be robust and independent of spin configuration. Otherwise, it is the resulting orbital arrangement around the impurity and the strength of Hund’s coupling at the impurity that set the character of the host-impurity magnetic exchange. The remaining of the paper is organized as follows. In Sec. \[sec:model\] we introduce the effective model describing the spin-orbital superexchange at the $3d-4d$ bonds which serves to investigate the changes of spin and orbital order around individual impurities and at finite doping. We arrive at a rather general formulation which emphasizes the impurity orbital degree of freedom, being a doublon, and present some technical details of the derivation in Appendix A. The strategy we adopt is to analyze first the ground state properties of a single $3d^3$ impurity surrounded by $4d^4$ atoms by investigating how the spin-orbital pattern in the host may be modified at the nearest neighbor (NN) sites to the $3d$ atom. This study is performed for different spin-orbital patterns of the $4d$ host with special emphasis on the alternating FM chains ($C$-AF order) which coexist with $G$-type alternating orbital ($G$-AO) order, see Fig. \[fig:host\]. We address the impurity problem within the classical approximation in Sec. \[sec:mfa\]. As explained in Sec. \[sec:two\], there are two nonequivalent cases which depend on the precise modification of the orbital order by the $3d$ impurity, doped either to replace a doublon in $a$ orbital (Sec. \[sec:impa\]) or the one in $c$ orbital (Sec. \[sec:impc\]). Starting from the single impurity solution we next address periodic arrangements of $3d$ atoms at different concentrations. We demonstrate that the spin-orbital order in the host can be radically changed by the presence of impurities, leading to striped patterns with alternating FM/AF domains and islands of fully FM states. In Sec. \[sec:dopg\] we consider the modifications of spin-orbital order which arise at periodic doping with macroscopic concentration. Here we limit ourselves to two representative cases: (i) commensurate $x=1/8$ doping in Sec. \[sec:dop8\], and (ii) two doping levels $x=1/5$ and $x=1/9$ being incommensurate with underlying two-sublattice order (Fig. \[fig:host\]) which implies simultaneous doping at two sublattices, i.e., at both $a$ and $c$ doublon sites, as presented in Secs. \[sec:dop5\] and \[sec:dop9\]. Finally, in Sec. \[sec:orb\] we investigate the modifications of the classical phase diagram induced by quantum fluctuations, and in Secs. \[sec:soc\] and \[sec:JH\] we discuss representative results obtained for finite spin-orbit coupling (calculation details of the treatment of spin-orbit interaction are presented in Appendix B). The paper is concluded by a general discussion of possible emerging scenarios for the $3d^3$ impurities in $4d^4$ host, a summary of the main results and perspective of future experimental investigations of orbital dilution in Sec. \[sec:sum\]. The spin-orbital model {#sec:model} ====================== In this Section we consider a $3d$ impurity in a strongly correlated $4d$ TMO and derive the effective $3d^3-4d^4$ spin-orbital superexchange. It follows from the coupling between $3d$ and $4d$ orbitals via oxygen $2p$ orbitals due to the $p-d$ hybridization. In a strongly correlated system it suffices to concentrate on a pair of atoms forming a bond $\langle ij\rangle$, as the effective interactions are generated by charge excitations $d^4_id^4_j\leftrightharpoons d^5_id^3_j$ along a single bond [@Ole05]. In the reference $4d$ host both atoms on the bond $\langle ij\rangle$ are equivalent and one considers, $$H(i,j)=H_t(i,j)+H_{\rm int}(i)+H_{\rm int}(j). \label{host}$$ The Coulomb interaction $H_{\rm int}(i)$ is local at site $i$ and we describe it by the degenerate Hubbard model [@Ole83], see below. We implement a strict rule that the hopping within the $t_{2g}$ sector is allowed in a TMO only between two neighboring orbitals of the same symmetry which are active along the bond direction [@Kha00; @Har03; @Dag08], and neglect the interorbital processes originating from the octahedral distortions such as rotation or tilting. Indeed, in ideal undistorted (perovskite or square lattice) geometry the orbital flavor is conserved as long as the spin-orbit coupling may be neglected. The interorbital hopping elements are smaller by at least one order of magnitude and may be treated as corrections in cases where distortions play a role to the overall scenario established below. The kinetic energy for a representative $3d$-$2p$-$4d$ bond, i.e., after projecting out the oxygen degrees of freedom, is given by the hopping in the host $\propto t_h$ between sites $i$ and $j$, $$H_t(i,j)=-t_h\sum_{\mu(\gamma),\sigma} \left(d_{i\mu\sigma}^{\dagger}d_{j\mu\sigma}^{} +d_{j\mu\sigma}^{\dagger}d_{i\mu\sigma}^{}\right). \label{eq:hop4d}$$ Here $d_{i\mu\sigma}^{\dagger}$ are the electron creation operators at site $i$ in the spin-orbital state $(\mu\sigma)$. The bond $\langle ij\rangle$ points along one of the two crystallographic directions, $\gamma=a,b$, in the two-dimensional (2D) square lattice. Without distortions, only two out of three $t_{2g}$ orbitals are active along each bond $\langle 12\rangle$ and contribute to $H_t(i,j)$, while the third orbital lies in the plane perpendicular to the $\gamma$ axis and thus the hopping via oxygen is forbidden by symmetry. This motivates a convenient notation as follows [@Kha00], $$\left\|a\right\rangle\equiv\left\|yz\right\rangle, \quad \left\|b\right\rangle\equiv\left\|xz\right\rangle, \quad \left\|c\right\rangle\equiv\left\|xy\right\rangle, \label{eq:or_defs}$$ with the $t_{2g}$ orbital inactive along a given direction $\gamma\in\{a,b,c\}$ labeled by the index $\gamma$. We consider a 2D square lattice with transition metal ions connected via oxygen orbitals as in a RuO$_2$ $(a,b)$ plane of Ca$_2$RuO$_4$ (SrRuO$_3$). In this case $\|a\rangle$ ($\|b\rangle$) orbitals are active along the $b$ ($a$) axis, while $\|c\rangle$ orbitals are active along both $a,b$ axes. To derive the superexchange in a Mott insulator, it is sufficient to consider a bond which connects nearest neighbor sites, $\langle ij\rangle\equiv\langle 12\rangle$. Below we consider a bond between an impurity site $i=1$ occupied by a $3d$ ion and a neighboring host $4d$ ion at site $j=2$. The Hamiltonian for this bond can be then expressed in the following form, $$H(1,2)=H_t(1,2)+H_{\rm int}(1)+H_{\rm int}(2)+H_{\rm ion}(2). \label{Hub}$$ The total Hamiltonian contains the kinetic energy term $H_t(1,2)$ describing the electron charge transfer via oxygen orbitals, the onsite interaction terms $H_{\rm int}(m)$ for the $3d$ $(4d)$ ion at site $m=1,2$, and the local potential of the $4d$ atom, $H_{ion}(2)$, which takes into account the mismatch of the energy level structure between the two ($4d$ and $3d$) atomic species and prevents valence fluctuations when the host is doped, even in the absence of local Coulomb interaction. The kinetic energy in Eq. (\[Hub\]) is given by, $$H_t(1,2)=-t\sum_{\mu(\gamma),\sigma} \left(d_{1\mu\sigma}^{\dagger}d_{2\mu\sigma}^{} +d_{2\mu\sigma}^{\dagger}d_{1\mu\sigma}^{}\right), \label{eq:hop}$$ where $d_{m\mu\sigma}^{\dagger}$ is the electron creation operator at site $m=1,2$ in the spin-orbital state $(\mu\sigma)$. The bond $\langle 12\rangle$ points along one of the two crystallographic directions, $\gamma=a,b$, and again the orbital flavor is conserved [@Kha00; @Har03; @Dag08]. The Coulomb interaction on an atom at site $m=1,2$ depends on two parameters [@Ole83]: (i) intraorbital Coulomb repulsion $U_m$, and (ii) Hund’s exchange $J_m^H$. The label $m$ stands for the ion and distinguishes between these terms at the $3d$ and $4d$ ion, respectively. The interaction is expressed in the form, $$\begin{aligned} H_{\rm int}(m)&=& U_m\sum_{\mu}n_{m\mu\uparrow}n_{m\mu\downarrow} - 2J_m^{H}\sum_{\mu<\nu}\vec{S}_{m\mu}\!\cdot\!\vec{S}_{m\nu} \nonumber \\ &+&\left(U_m-\frac{5}{2}J_m^{H}\right) \sum_{{\mu<\nu\atop \sigma\sigma'}}n_{m\mu\sigma}n_{m\nu\sigma'} \nonumber \\ &+& J_m^{H}\sum_{\mu\not=\nu}d_{m\mu\uparrow}^{\dagger} d_{m\mu\downarrow}^{\dagger}d_{m\nu\downarrow}^{}d_{m\nu\uparrow}^{}. \label{Hint}\end{aligned}$$ The terms standing in the first line of Eq. (\[Hint\]) contribute to the magnetic instabilities in degenerate Hubbard model [@Ole83] and decide about spin order, both in an itinerant system and in a Mott insulator. The remaining terms contribute to the multiplet structure and are of importance for the correct derivation of the superexchange which follows from charge excitations, see below. Finally, we include a local potential on the $4d$ atom which encodes the energy mismatch between the host and the impurity orbitals close to the Fermi level and prevents valence fluctuations on the $4d$ ion due to the $3d$ doping. This term has the following general structure, $$H_{\rm ion}(2)=I_2^e\left(4-\sum_{\mu,\sigma}n_{2\mu\sigma}\right)^2, \label{eq:ion2}$$ with $\mu=a,b,c$. The effective Hamiltonian for the low energy processes is derived from $H(1,2)$ (\[Hub\]) by a second order expansion for charge excitations generated by $H_t(1,2)$, and treating the remaining part of $H(1,2)$ as an unperturbed Hamiltonian. We are basically interested in virtual charge excitations in the manifold of degenerate ground states of a pair of $3d$ and $4d$ atoms on a bond, see Fig. \[fig:3d4d\]. These quantum states are labeled as $\left\{e_1^k\right\}$ with $k=1,\dots,4$ and $\left\{e_{2}^p\right\}$ with $p=1,\dots,9$ and their number follows from the solution of the onsite quantum problem for the Hamiltonian $H_{\rm int}(i)$. For the $3d$ atom the relevant states can be classified according to the four components of the total spin $S_1=3/2$ for the $3d$ impurity atom at site $m=1$, three components of $S_2=1$ spin for the $4d$ host atom at site $m=2$ and for the three different positions of the double occupied orbital (doublon). Thus, the effective Hamiltonian will contain spin products $(\vec{S}_1\!\cdot\!\vec{S}_2)$ between spin operators defined as, $$\vec{S}_{m}=\frac{1}{2}\sum_{\gamma} d_{m\gamma\alpha}^{\dagger} \vec{\sigma}_{\alpha\beta}^{}d_{m\gamma\beta}^{}, \label{eq:spin_op}$$ for $m=1,2$ sites and the operator of the doublon position at site $m=2$, $$D_2^{(\gamma)}=\left( d_{2\gamma\uparrow}^{\dagger}d_{2\gamma\uparrow}^{}\right) \left(d_{2\gamma\downarrow}^{\dagger}d_{2\gamma\downarrow}^{}\right). \label{eq:dub}$$ The doublon operator identifies the orbital $\gamma$ within the $t_{2g}$ manifold of the $4d$ ion with a double occupancy (occupied by the doublon) and stands in what follows for the orbital degree of freedom. It is worth noting that the hopping (\[eq:hop\]) does not change the orbital flavor thus we expect that the resulting Hamiltonian is diagonal in the orbital degrees of freedom with only $D_2^{(\gamma)}$ operators. ![ Schematic representation of one configuration belonging to the manifold of 36 degenerate ground states for a representative $3d-4d$ bond $\langle 12\rangle$ as given by the local Coulomb Hamiltonian $H_{\rm int}(m)$ (\[Hint\]) with $m=1,2$. The dominant exchange processes considered here are those that move one of the four electrons on the $4d$ atom to the $3d$ neighbor and back. The stability of the $3d^{3}$-$4d^{4}$ charge configurations is provided by the local potential energy $I_{2}^{e}$, see Eq. (\[eq:ion2\]). []{data-label="fig:3d4d"}](3d4d){width="0.6\columnwidth"} Following the standard second order perturbation expansion for spin-orbital systems [@Ole05], we can write the matrix elements of the low energy exchange Hamiltonian, ${\cal H}_J^{(\gamma)}(i,j)$, for a bond $\langle 12\rangle\parallel\gamma$ along the $\gamma$ axis as follows, $$\begin{aligned} &&\!\big\langle e_{1}^{k},e_{2}^{l}\big\|{\cal H}_J^{(\gamma)}(1,2) \big\|e_{1}^{k'},e_{2}^{l'}\big\rangle = -\sum_{n_{1},n_{2}}\frac{1}{\varepsilon_{n1}+\varepsilon_{n2}} \nonumber \\ \!\times&\!\big\langle& e_{1}^{k},e_{2}^{l}\big\|H_t(1,2)\big\| n_{1},n_{2}\big\rangle\! \times \!\big\langle n_{1},n_{2}\big\|H_t(1,2)\big\| e_{1}^{k'}\!,e_{2}^{l'}\!\big\rangle, \label{eq:pert_exp}\end{aligned}$$ with $\varepsilon_{nm}=E_{n,m}-E_{0,m}$ being the excitation energies for atoms at site $m=1,2$ with respect to the unperturbed ground state. The superexchange Hamiltonian ${\cal H}_J^{(\gamma)}(1,2)$ for a bond along $\gamma$ can be expressed in a matrix form by a $36\times 36$ matrix, with dependence on $U_m$, $J_m^H$, and $I_{e}$ elements. There are two types of charge excitations: (i) $d^3_1d^4_2\leftrightharpoons d^4_1d^3_2$ one which creates a doublon at the $3d$ impurity, and (ii) $d^3_1d^4_2\leftrightharpoons d^2_1d^5_2$ one which adds another doublon at the $4d$ host site in the intermediate state. The second type of excitations involves more doubly occupied orbitals and has much larger excitation energy. It is therefore only a small correction to the leading term (i), as we discuss in Appendix A. Similar as in the case of doped manganites [@Ole02], the dominant contribution to the effective low-energy spin-orbital Hamiltonian for the $3d-4d$ bond stems from the $d^3_1d^4_2\leftrightharpoons d^4_1d^3_2$ charge excitations, as they do not involve an extra double occupancy and the Coulomb energy $U_2$. The $3d^3_14d^4_2\leftrightharpoons 3d^4_14d^3_2$ charge excitations can be analyzed in a similar way as the $3d^3_i3d^4_j\leftrightharpoons 3d^4_i3d^3_j$ ones for an $\langle ij\rangle$ bond in doped manganites [@Ole02]. In both cases the total number of doubly occupied orbitals does not change, so the main contributions come due to Hund’s exchange. In the present case, one more parameter plays a role, $$\Delta=I_{e}+3(U_{1}-U_{2})-4(J_{1}^{H}-J_{2}^{H}), \label{Delta}$$ which stands for the mismatch potential energy (\[eq:ion2\]) renormalized by the onsite Coulomb interactions $\{U_m\}$ and by Hund’s exchange $\{J_m^H\}$. On a general ground we expect $\Delta$ to be a positive quantity, since the repulsion $U_m$ should be larger for smaller $3d$ shells than for the $4d$ ones and $U_m$ is the largest energy scale in the problem. Let us have a closer view on this dominant contribution of the effective low-energy spin-orbital Hamiltonian for the $3d-4d$ bond, given by Eq. (\[eq:s-ex\]). For the analysis performed below and the clarity of our presentation it is convenient to introduce some scaled parameters related to the interactions within the host and between the host and the impurity. For this purpose we employ the exchange couplings $J_{\rm imp}$ and $J_{\rm host}$, $$\begin{aligned} \label{eq:bothJ} J_{\rm imp}&=&\frac{t^{2}}{4\Delta}, \\ J\label{eq:hostJ} _{\rm host}&=&\frac{4t_h^{2}}{U_2},\end{aligned}$$ which follow from the virtual charge excitations generated by the kinetic energy, see Eqs. (\[eq:hop4d\]) and (\[eq:hop\]). We use their ratio to investigate the influence of the impurity on the spin-orbital order in the host. Here $t_h$ is the hopping amplitude between two $t_{2g}$ orbitals at NN $4d$ atoms, $J_{2}^{H}$ and $U_{2}$ refer to the host, and $\Delta$ (\[Delta\]) is the renormalized ionization energy of the $3d-4d$ bonds. The results depend as well on Hund’s exchange element for the impurity and on the one at host atoms, $$\begin{aligned} \label{eq:etai} \eta_{\rm imp} &=&\frac{J_{1}^{H}}{\Delta},\\ \label{eq:etah} \eta_{\rm host}&=&\frac{J_2^{H}}{U_2},\end{aligned}$$ Note that the ratio introduced for the impurity, $\eta_{\rm imp}$ (\[eq:etai\]), has here a different meaning from Hund’s exchange used here for the host, $\eta_{\rm host}$ (\[eq:etah\]), which cannot be too large by construction, i.e., $\eta_{\rm host}<1/3$. With the parametrization introduced above, the dominant term in the impurity-host Hamiltonian for the impurity spin $\vec{S}_i$ interacting with the neighboring host spins $\{\vec{S}_j\}$ at $j\in{\cal N}(i)$, deduced from ${\cal H}_{3d-4d}^{(\gamma)}(1,2)$ Eq. (\[eq:s-ex\]), can be written in a rather compact form as follows $${\cal H}_{3d-4d}(i)\simeq\sum_{\gamma,j\in{\cal N}(i)} \left\{J_{S}(D_j^{(\gamma)})(\vec{S}_i\!\cdot\!\vec{S}_j) + E_D D_j^{(\gamma)}\right\}, \label{eq:H123}$$ where the orbital (doublon) dependent spin couplings $J_S(D_j^{(\gamma)})$ and the doublon energy $E_D$ depend on $\eta_{\rm imp}$. The evolution of the exchange couplings are shown in Fig. \[fig:JSEd\]. We note that the dominant energy scale is $E_D^{\gamma}$, so for a single $3d-4d$ bond the doublon will avoid occupying the inactive ($\gamma$) orbital and the spins will couple with $J_{S}(D_j^{(\gamma)}=0)$ which can be either AF if $\eta_{\rm imp}\lesssim0.43$ or FM if $\eta_{\rm imp}>0.43$. Thus the spins at $\eta_{\rm imp}=\eta_{\rm imp}^c\simeq 0.43$ will decouple according to the ${\cal H}_J^{(\gamma)}(i,j)$ exchange. ![ Evolution of the spin exchange $J_S(D_2^{(\gamma)})$ and the doublon energy $E_D$, both given in Eq. (\[eq:H123\]) for increasing Hund’s exchange $\eta_{\rm imp}$ at the impurity. \[fig:JSEd\]](3d4d_alt){width="0.9\columnwidth"} Let us conclude this Section by writing the complete superexchange Hamiltonian, $${\cal H}={\cal H}_{3d-4d}+{\cal H}_{4d-4d}+{\cal H}_{so}, \label{fullH}$$ where ${\cal H}_{3d-4d}\equiv\sum_i{\cal H}_{3d-4d}(i)$ includes all the $3d-4d$ bonds around impurities, ${\cal H}_{4d-4d}$ stands for the the effective spin-orbital Hamiltonian for the $4d$ host bonds, and ${\cal H}_{so}$ is the spin-orbit interaction in the host. The former term we explain below, while the latter one is defined in Sec. \[sec:soc\], where we analyze the quantum corrections and the consequences of spin-orbit interaction. The superexchange in the host for the bonds $\langle ij\rangle$ along the $\gamma=a,b$ axes [@Cuo06], $${\cal H}_{4d-4d}=J_{\rm host}\sum_{\langle ij\rangle\parallel\gamma} \left\{J_{ij}^{(\gamma)}(\vec{S}_{i}\!\cdot\!\vec{S}_{j}+1) +K_{ij}^{(\gamma)}\right\}, \label{eq:Hhost}$$ depends on $J_{ij}^{(\gamma)}$ and $K_{ij}^{(\gamma)}$ operators acting only in the orbital space. They are expressed in terms of the pseudospin operators defined in the orbital subspace spanned by the two orbital flavors active along a given direction $\gamma$, i.e., $$\begin{aligned} J_{ij}^{(\gamma)} & = & \frac{1}{2}\left(2r_{1}+1\right) \left(\vec{\tau}_{i}^{}\!\cdot\!\vec{\tau}_{j}^{}\right)^{(\gamma)} -\frac{1}{2}r_2\left(\tau_i^z\tau_j^z\right)^{(\gamma)} \nonumber \\ & + & \frac{1}{8}\left(n_{i}^{}n_{j}^{}\right)^{(\gamma)} \left(2r_{1}\!-\!r_{2}\!+\!1\right)-\frac{1}{4}r_1 \left(n_{i}^{}+n_{j}^{}\right)^{(\gamma)}, \\ K_{ij}^{(\gamma)} & = & r_1\left(\vec{\tau}_{i}\!\cdot\!\vec{\tau}_{j}\right)^{(\gamma)} +\!r_2\left(\tau_i^z\tau_j^z\right)^{(\gamma)} +\!\frac14\left(r_1+r_2\right)\left(n_{i}^{}n_{j}^{}\right)^{(\gamma)} \nonumber \\ & - & \frac14\left(r_1+1\right)\left(n_i^{}+n_j^{}\right)^{(\gamma)}.\end{aligned}$$ with $$r_1=\frac{\eta_{\rm host}}{1-3\eta_{\rm host}}, \hskip .5cm r_2=\frac{\eta_{\rm host}}{1+2\eta_{\rm host}}, \label{rr}$$ standing for the multiplet structure in charge excitations, and the orbital operators $\{\vec{\tau}_i^{\,(\gamma)},n_i^{(\gamma)}\}$ that for the $\gamma=c$ axis take the form: $$\begin{aligned} \vec{\tau}_{i}^{\,(c)}&=&\frac{1}{2}\big(\begin{array}[t]{cc} a_{i}^{\dagger} & b_{i}^{\dagger}\end{array}\big)\cdot\vec{\sigma}\cdot \big(\begin{array}[t]{cc}a_i^{} & b_i^{}\end{array}\big)^{\intercal}, \\ n_{i}^{(c)}&=&a_{i}^{\dagger}a_{i}^{}+b_{i}^{\dagger}b_{i}^{}.\end{aligned}$$ For the directions $\gamma=a,b$ in the considered $(a,b)$ plane one finds equivalent expressions by cyclic permutation of the axis labels $\{a,b,c\}$ in the above formulas. This problem is isomorphic with the spin-orbital superexchange in the vanadium perovskites [@Kha01; @Kha04], where a hole in the $\{a,b\}$ doublet plays an equivalent role to the doublon in the present case. The operators $\{a_{i}^{\dagger},b_{i}^{\dagger},c_{i}^{\dagger}\}$ are the doublon (hard core boson) creation operators in the orbital $\gamma=a,b,c$, respectively, and they satisfy the local constraint, $$a_{i}^{\dagger}a_i^{}+b_{i}^{\dagger}b_i^{}+c_{i}^{\dagger}c_i^{}=1,$$ meaning that exactly one doublon (\[eq:dub\]) occupies one of the three $t_{2g}$ orbitals at each site $i$. These bosonic occupation operators coincide with the previously used doublon occupation operators $D^{(\gamma)}_j$, i.e., $D^{(\gamma)}_j=\gamma^{\dagger}_j\gamma_j$ with $\gamma=a,b,c$. Below we follow first the classical procedure to determine the ground states of single impurities in Sec. \[sec:imp\], and at macroscopic doping in Sec \[sec:dop\]. Single 3 impurity in 4 host {#sec:imp} =========================== Classical treatment of the impurity problem {#sec:mfa} ------------------------------------------- In this Section we describe the methodology that we applied for the determination of the phase diagrams for a single impurity reported below in Secs. \[sec:impa\], and next at macroscopic doping, as presented in Sec. \[sec:dop\]. Let us consider first the case of a single $3d$ impurity in the $4d$ host. Since the interactions in the model Hamiltonian are only effective ones between NN sites, it is sufficient to study the modification of the spin-orbital order around the impurity for a given spin-orbital configuration of the host by investigating a cluster of $L=13$ sites shown in Fig. \[fig:setup\]. We assume the $C$-AF spin order (FM chains coupled antiferromagnetically) accompanied by $G$-AO order within the host which is the spin-orbital order occurring for the realistic parameters of a RuO$_2$ plane [@Cuo06], see Fig. \[fig:host\]. Such a spin-orbital pattern turns out to be the most relevant one when considering the competition between the host and the impurity as due to the AO order within the $(a,b)$ plane. Other possible configurations with uniform orbital order and AF spin pattern, e.g. $G$-AF order, will also be considered in the discussion throughout the manuscript. The sites $i=1,2,3,4$ inside the cluster in Fig. \[fig:setup\] have active spin and orbital degrees of freedom while the impurity at site $i=0$ has only spin degree of freedom. At the remaining sites the spin-orbital configuration is assigned, following the order in the host, and it does not change along the computation. ![Schematic top view of the cluster used to obtain the phase diagrams of the $3d$ impurity within the $4d$ host in an $(a,b)$ plane. The impurity is at the central site $i=0$ which belongs to the $c$ orbital host sublattice. For the outer sites in this cluster the spin-orbital configuration is fixed and determined by the undoped $4d$ host (with spins and $c$ orbitals shown here) having $C$-AF/$G$-AO order, see Fig. \[fig:host\]. For the central $i=0$ site the spin state and for the host sites $i=1,\dots,4$ the spin-orbital configurations are determined by minimizing the energy of the cluster. []{data-label="fig:setup"}](setup){width="0.7\columnwidth"} To determine the ground state we assume that the spin-orbital degrees of freedom are treated as classical variables. This implies that for the bonds between atoms in the host we use the Hamiltonian (\[eq:Hhost\]) and neglect quantum fluctuations, i.e., in the spin sector we keep only the $z$th (Ising) spin components and in the orbital one only the terms which are proportional to the doublon occupation numbers (\[eq:dub\]) and to the identity operators. Similarly, for the impurity-host bonds we use the Hamiltonian Eq. (\[eq:H123\]) by keeping only the $z$th projections of spin operators. Since we do neglect the fluctuation of the spin amplitude it is enough to consider only the maximal and minimal values of $\langle S^z_i\rangle$ for spin $S=3/2$ at the impurity sites and $S=1$ at the host atoms. With these assumptions we can construct all the possible configurations by varying the spin and orbital configurations at the sites from $i=1$ to $i=4$ in the cluster shown in Fig. \[fig:setup\]. Note that the outer ions in the cluster belong all to the same sublattice, so two distinct cases have to be considered to probe all the configurations. Since physically it is unlikely that a single impurity will change the orbital order of the host globally thus we will not compare the energies from these two cases and analyze two classes of solutions separately, see Sec. \[sec:two\]. Then, the lowest energy configuration in each class provides the optimal spin-orbital pattern for the NNs around the $3d$ impurity. In the case of degenerate classical states, the spin-orbital order is established by including quantum fluctuations. In the case of a periodic doping analyzed in Sec. \[sec:dop\], we use a similar strategy in the computation. Taking the most general formulation, we employ larger clusters having both size and shape that depend on the impurity distribution and on the spin-orbital order in the host. For this purpose, the most natural choice is to search for the minimum energy configuration in the elementary unit cell that can reproduce the full lattice by a suitable choice of the translation vectors. This is computationally expensive but doable for a periodic distribution of the impurities that is commensurate to the lattice because it yields a unit cell of relatively small size for doping around $x=0.1$. Otherwise, for the incommensurate doping the size of the unit cell can lead to a configuration space of a dimension that impedes finding of the ground state. This problem is computationally more demanding and to avoid the comparison of all the energy configurations, we have employed the Metropolis algorithm at low temperature to achieve the optimal configuration iteratively along the convergence process. Note that this approach is fully classical, meaning that the spins of the host and impurity are treated as Ising variables and the orbital fluctuations in the host’s Hamiltonian Eq. (\[eq:Hhost\]) are omitted. They will be addressed in Sec. \[sec:orb\]. Two nonequivalent $3d$ doping cases {#sec:two} ----------------------------------- The single impurity problem is the key case to start with because it shows how the short-range spin-orbital correlations are modified around the $3d$ atom due to the host and host-impurity interactions in Eq. (\[fullH\]). The analysis is performed by fixing the strength between Hund’s exchange and Coulomb interaction within the host (\[Hint\]) at $\eta_{\rm host}=0.1$, and by allowing for a variation of the ratio between the host-impurity interaction (\[eq:H123\]) and the Coulomb coupling at the impurity site. The choice of $\eta_{\rm host}=0.1$ is made here because this value is within the physically relevant range for the case of the ruthenium materials. Small variations of $\eta_{\rm host}$ do not affect the obtained results qualitatively. As we have already discussed in the model derivation, the sign of the magnetic exchange between the impurity and the host depends on the orbital occupation of the $4d$ doublon around the $3d$ impurity. The main aspect that controls the resulting magnetic configuration is then given by the character of the doublon orbitals around the impurity, depending on whether they are active or inactive along the considered $3d-4d$ bond. To explore such a competition quantitatively we investigate $G$-AO order for the host with alternation of $a$ and $c$ doublon configurations accompanied by the $C$-AF spin pattern, see Fig. \[fig:host\]. Note that the $a$ orbitals are active only along the $b$ axis, while the $c$ orbitals are active along the both axes: $a$ and $b$ [@Dag08]. This state has the lowest energy for the host in a wide range of parameters for Hund’s exchange, Coulomb element and crystal-field potential [@Cuo06]. Due to the specific orbital pattern of Fig. \[fig:host\], the $3d$ impurity can substitute one of two distinct $4d$ sites which are considered separately below, either with $a$ or with $c$ orbital occupied by the doublon. Since the two $4d$ atoms have nonequivalent surrounding orbitals, not always active along the $3d-4d$ bond, we expect that the resulting ground state will have a modified spin-orbital order. Indeed, if the $3d$ atom replaces the $4d$ one with the doublon in the $a$ orbital, then all the $4d$ neighboring sites have active doublons along the connecting $3d-4d$ bonds because they are in the $c$ orbitals. On the contrary, the substitution at the $4d$ site with $c$ orbital doublon configuration leads to an impurity state with its neighbors having both active and inactive doublons. Therefore, we do expect a more intricate competition for the latter case of an impurity occupying the $4d$ site with $c$ orbital configuration. Indeed, this leads to frustrated host-impurity interactions, as we show in Sec. \[sec:impc\]. Doping removing a doublon in $a$ orbital {#sec:impa} ---------------------------------------- We start by considering the physical situation where the $3d$ impurity replaces a $4d$ ion with the doublon within the $a$ orbital. The ground state phase diagram and the schematic view of the spin-orbital pattern are reported in Fig. \[fig:pd\_caf\_a\] in terms of the ratio $J_{\rm imp}/J_{\rm host}$ (\[eq:etai\]) and the strength of Hund’s exchange coupling $\eta_{\rm imp}$ (\[eq:bothJ\]) at the $3d$ site. There are three different ground states that appear in the phase diagram. Taking into account the structure of the $3d-4d$ spin-orbital exchange (\[eq:H123\]) we expect that, in the regime where the host-impurity interaction is greater than that in the host, the $4d$ neighbors to the impurity tend to favor the spin-orbital configuration set by the $3d-4d$ exchange. In this case, since the orbitals surrounding the $3d$ site already minimize the $3d-4d$ Hamiltonian, we expect that the optimal spin configuration corresponds to the $4d$ spins aligned either antiferromagnetically or ferromagnetically with respect to the impurity $3d$ spin. ![ Top — Phase diagram of the $3d$ impurity in the $(a,b)$ plane with the $C$-AF/$G$-AO order in $4d$ host for the impurity doped at the sublattice with an $a$-orbital doublon. Different colors refer to local spin order around the impurity, AF and FM, while FS indicates the intermediate regime of frustrated impurity spin. Bottom — Schematic view of spin-orbital patterns for the ground state configurations shown in the top panel. The $3d$ atom is at the central site, the dotted frame highlights the $4d$ sites where the impurity induces a a spin reversal. In the FS$a$ phase the question mark stands for that the frustrated impurity spin within the classical approach but frustration is released by the quantum fluctuations of the NN $c$ orbitals in the $a$ direction resulting in small AF couplings along the $a$ axis, and spins obey the $C$-AF order (small arrow). The labels FM$a$ and AF$a$ refer to the local spin order around the $3d$ impurity site with respect to the host — these states differ by spin inversion at the $3d$ atom site. \[fig:pd\_caf\_a\]](caf_asub){width="1\columnwidth"} The neighbor spins are AF to the $3d$ spin impurity in the AF$a$ phase, while the FM$a$ phase is just obtained from AF$a$ by reversing the spin at the impurity, and having all the $3d-4d$ bonds FM. It is interesting to note that due to the host-impurity interaction the $C$-AF spin pattern of the host is modified in both the AF$a$ and the FM$a$ ground states. Another intermediate configuration which emerges when the host-impurity exchange is weak in the intermediate FS$a$ phase where the impurity spin is undetermined and its configuration in the initial $C$-AF phase is degenerate with the one obtained after the spin-inversion operation. This is a singular physical situation because the impurity does not select a specific direction even if the surrounding host has a given spin-orbital configuration. Such a degeneracy is clearly verified at the critical point $\eta_{\rm imp}^c\simeq 0.43$ where the amplitude of the $3d-4d$ coupling vanishes when the doublon occupies the active orbital. Interestingly, such a degenerate configuration is also obtained at $J_{\rm imp}/J_{\rm host}<1$ when the host dominates and the spin configuration at the $4d$ sites around the impurity are basically determined by $J_{\rm host}$. In this case, due to the $C$-AF spin order, always two bonds are FM and other two have AF order, independently of the spin orientation at the $3d$ impurity. This implies that both FM or AF couplings along the $3d-4d$ bonds perfectly balance each other which results in the degenerate FS$a$ phase. It is worth pointing out that there is a quite large region of the phase diagram where the FS$a$ state is stabilized and the spin-orbital order of the host is not affected by doping with the possibility of having large degeneracy in the spin configuration of the impurities. On the other hand, by inspecting the $c$ orbitals around the impurity (Fig. \[fig:pd\_caf\_a\]) from the point of view of the full host’s Hamiltonian Eq. (\[eq:Hhost\]) with orbital flips included, $(\tau_{i}^{\gamma +}\tau_{j}^{\gamma -}+{\rm H.c.})$, one can easily find out that the frustration of the impurity spin can be released by quantum orbital fluctuations. Note that the $c$ orbitals around the impurity in the $a$ ($b$) direction have quite different surroundings. The ones along the $a$ axis are connected by two active bonds along the $b$ axis with orbitals $a$, as in Fig. \[fig:host\_flips\](a), while the ones along $b$ are connected with only one active $a$ orbital along the same $b$ axis. This means that in the perturbative expansion the orbital flips will contribute only along the $b$ bonds (for the present $G$-AO order) and admix the $a$ orbital character to $c$ orbitals along them, while such processes will be blocked for the bonds along the $a$ axis, as also for $b$ orbitals along the $b$ axis, see Fig. \[fig:host\_flips\](b). ![ Schematic view of the two types of orbital bonds found in the $4d$ host: (a) an active bond with respect to orbital flips, $(\tau_{i}^{\gamma +}\tau_{j}^{\gamma -}+{\rm H.c.})$, and (b) an inactive bond, where orbital fluctuations are blocked by the orbital symmetry — here the orbitals are static and only Ising terms contribute to the ground state energy.[]{data-label="fig:host_flips"}](host_flips){width="0.9\columnwidth"} This fundamental difference can be easily included in the host-impurity bond in the mean-field manner by setting $\langle D_{i\pm b,\gamma}\rangle=0$ for the bonds along the $b$ axis and $0<\langle D_{i\pm a,\gamma}\rangle<1$ for the bonds along the $a$ axis. Then one can easily check that for the impurity spin pointing downwards we get the energy contribution from the spin-spin bonds which is given by $E_\downarrow=\alpha(\eta_{\rm host})\langle D_{i\pm a,\gamma}\rangle$, and for the impurity spin pointing upwards we have $E_\uparrow=-\alpha(\eta_{\rm host})\langle D_{i\pm a,\gamma}\rangle$, with $\alpha(\eta_{\rm host})>0$. Thus, it is clear that any admixture of the virtual orbital flips in the host’s wave function polarize the impurity spin upwards so that the $C$-AF order of the host will be restored. Doping removing a doublon in $c$ orbital {#sec:impc} ---------------------------------------- Let us move to the case of the $3d$ atom replacing the doublon at $c$ orbital. As anticipated above, this configuration is more intricate because the orbitals surrounding the impurity, as originated by the $C$-AF/AO order within the host, lead to nonequivalent $3d-4d$ bonds. There are two bonds with the doublon occupying an inactive orbital (and has no hybridization with the $t_{2g}$ orbitals at $3$d atom), and two remaining bonds with doublons in active $t_{2g}$ orbitals. Since the $3d-4d$ spin-orbital exchange depends on the orbital polarization of $4d$ sites we do expect a competition which may modify significantly the spin-orbital correlations in the host. Indeed, one observes that three configurations compete, denoted as AF1$c$, AF2$c$ and FM$c$, see Fig. \[fig:pd\_caf\_c\]. In the regime where the host-impurity exchange dominates the system tends to minimize the energy due to the $3d-4d$ spin-orbital coupling and, thus, the orbitals become polarized in the active configurations compatible with the $C$-AF/$G$-AO pattern and the host-impurity spin coupling is AF for $\eta_{\rm imp}\leq 0.43$, while it is FM otherwise. This region resembles orbital polarons in doped manganites [@Kil99; @Dag04]. Also in this case, the orbital polarons arise because they minimize the double exchange energy [@deG60]. ![ Top — Phase diagram of the $3d$ impurity in the $4d$ host with $C$-AF/$G$-AO order and the impurity doped at the $c$ doublon sublattice. Different colors refer to local spin order around the impurity: AF$c1$, AF$c2$, and FM. Bottom — Schematic view of spin-orbital patterns for the two AF ground state configurations shown in the top panel; the FM$c$ phase differs from the AF$c2$ one only by spin inversion at the $3d$ atom. The $3d$ atom is at the central site and has no doublon orbital, the frames highlight the spin-orbital defects caused by the impurity. As in Fig. \[fig:pd\_caf\_a\], the labels AF and FM refer to the impurity spin orientation with respect to the neighboring $4d$ sites. []{data-label="fig:pd_caf_c"}](caf_csub){width="1\columnwidth"} On the contrary, for weak spin-orbital coupling between the impurity and the host there is an interesting cooperation between the $3d$ and $4d$ atoms. Since the strength of the impurity-host coupling is not sufficient to polarize the orbitals at the $4d$ sites, it is preferable to have an orbital rearrangement to the configuration with inactive orbitals on $3d-4d$ bonds and spin flips at $4d$ sites. In this way the spin-orbital exchange is optimized in the host and also on the $3d-4d$ bonds. The resulting state has an AF coupling between the host and the impurity as it should when all the orbitals surrounding the $3d$ atoms are inactive with respect to the bond direction. This modification of the orbital configuration induces the change in spin orientation. The double exchange bonds (with inactive doublon orbitals) along the $b$ axis are then blocked and the total energy is lowered, in spite of the frustrated spin-orbital exchange in the host. As a result, the AF1$c$ state the spins surrounding the impurity are aligned and antiparallel to the spin at the $3d$ site. Concerning the host $C$-AF/$G$-AO order we note that it is modified only along the direction where the FM correlations develop and spin defects occur within the chain doped by the $3d$ atom. The FM order is locally disturbed by the $3d$ defect antiferromagnetically coupled spins surrounding it. Note that this phase is driven by the orbital vacancy as the host develops more favorable orbital bonds to gain the energy in the absence of the orbital degrees of freedom at the impurity. At the same time the impurity-host bonds do not generate too big energy losses as: (i) either $\eta_{\rm imp}$ is so small that the loss due to $E_D$ is compensated by the gain from the superexchange $\propto J_{S}(D_j^{(\gamma)}=1)$ (all these bonds are AF), see Fig. \[fig:JSEd\], or (ii) $J_{\rm imp}/J_{\rm host}$ is small meaning that the overall energy scale of the impurity-host exchange remains small. Interestingly, if we compare the AF1$c$ with the AF2$c$ ground states we observe that the disruption of the $C$-AF/$G$-AO order is anisotropic and occurs either along the FM chains in the AF1$c$ phase or perpendicular to the FM chains in the AF2$c$ phase. No spin frustration is found here, in contrast to the FS$a$ phase in the case of $a$ doublon doping, see Fig. \[fig:pd\_caf\_a\]. Finally, we point out that a very similar phase diagram can be obtained assuming that the host has the FM/$G$-AO order with $a$ and $b$ orbitals alternating from site to site. Such configuration can be stabilized by a distortion that favors the out-of-plane orbitals. In this case there is no difference in doping at one or the other sublattice. The main difference is found in energy scales — for the $G$-AF/$C$-AO order the diagram is similar to the one of Fig. \[fig:pd\_caf\_c\] if we rescale $J_{\rm imp}$ by half, which means that the $G$-AF order is softer than the $C$-AF one. Note also that in the peculiar AF1$c$ phase the impurity does not induce any changes in the host for the FM/AO ordered host. Thus we can safely conclude that the observed change in the orbital order for the $C$-AF host in the AF1$c$ phase is due to the presence of the $c$ orbitals which are not directional in the $(a,b)$ plane. Summarizing, we have shown the complexity of local spin-orbital order around $t_{2g}^3$ impurities in a $4t_{2g}^4$ host. It is remarkable that such impurity spins not only modify the spin-orbital order around them in a broad regime of parameters, but also are frequently frustrated. This highlights the importance of quantum effects beyond the present classical approach which release frustration as we show in Sec. \[sec:orb\]. Periodic 3 doping in 4 host {#sec:dop} =========================== General remarks on finite doping {#sec:dopg} -------------------------------- In this Section we analyze the spin-orbital patterns due to a finite concentration $x$ of $3d$ impurities within the $4d$ host with $C$-AF/$G$-AO order, assuming that the $3d$ impurities are distributed in a periodic way. The study is performed for three representative doping distributions — the first one $x=1/8$ is commensurate with the underlying spin-orbital order and the other two are incommensurate with respect to it, meaning that in such cases doping at both $a$ and $c$ doublon sites is imposed simultaneously. As the impurities lead to local energy gains due to $3d-4d$ bonds surrounding them, we expect that the most favorable situation is when they are isolated and have maximal distances between one another. Therefore, we selected the largest possible distances for the three doping levels used in our study: $x=1/8$, $x=1/5$, and $x=1/9$. This choice allows us to cover different regimes of competition between the spin-orbital coupling within the host and the $3d-4d$ coupling. While single impurities may only change spin-orbital order locally, we use here a high enough doping to investigate possible global changes in spin-orbital order, i.e., whether they can occur in the respective parameter regime. The analysis is performed as for a single impurity, by assuming the classical spin and orbital variables and by determining the configuration with the lowest energy. For this analysis we set the spatial distribution of the $3d$ atoms and we determine the spin and orbital profile that minimizes the energy. $C$-AF phase with $x=1/8$ doping {#sec:dop8} -------------------------------- We begin with the phase diagram obtained at $x=1/8$ $3d$ doping, see Fig. \[fig:1to8\]. In the regime of strong impurity-host coupling the $3d-4d$ spin-orbital exchange determines the orbital and spin configuration of the $4d$ atoms around the impurity. The most favorable state is when the doublon occupies $c$ orbitals at the NN sites to the impurity. The spin correlations between the impurity and the host are AF (FM), if the amplitude of $\eta_{\rm imp}$ is below (above) $\eta_{\rm imp}^c$, leading to the AF$a$ and the FM$a$ states, see Fig. \[fig:1to8\]. The AF$a$ state has a striped-like profile with AF chains alternated by FM domains (consisting of three chains) along the diagonal of the square lattice. Even if the coupling between the impurity and the host is AF for all the bonds in the AF$a$ state, the overall configuration has a residual magnetic moment originating by the uncompensated spins and by the cooperation between the spin-orbital exchange in the $4d$ host and that for the $3d-4d$ bonds. Interestingly, at the point where the dominant $3d-4d$ exchange tends to zero (i.e., for $\eta_{\rm imp}\simeq\eta_{\rm imp}^c$), one finds a region of the FS$a$ phase which is analogous to the FS$a$ phase found in Sec. \[sec:impa\] for a single impurity, see Fig. \[fig:pd\_caf\_a\]. Again the impurity spin is frustrated in purely classical approach but this frustration is easily released by the orbital fluctuations in the host so that the $C$-AF order of the host can be restored. This state is stable for the amplitude of $\eta_{\rm imp}$ being close to $\eta_{\rm imp}^c$. ![ Top panel — Ground state diagram obtained for periodic $3d$ doping $x=1/8$. Different colors refer to local spin order around the impurity: AF$a$, AF$c$, FS$a$, and FM$a$. Bottom panel — Schematic view of the ground state configurations within the four 8-site unit cells (indicated by blue dashed lines) for the phases shown in the phase diagram. The question marks in FS$a$ phase indicate frustrated impurity spins within the classical approach — the spin direction (small arrows) is fixed only by quantum fluctuations. The $3d$ atoms are placed at the sites where orbitals are absent.[]{data-label="fig:1to8"}](1to8_diag){width="1\columnwidth"} The regime of small $J_{\rm imp}/J_{\rm host}$ ratio is qualitatively different — an orbital rearrangement around the impurity takes place, with a preference to move the doublons into the inactive orbitals along the $3d-4d$ bonds. Such orbital configurations favor the AF spin coupling at all the $3d-4d$ bonds which is stabilized by the $4d-4d$ superexchange [@Fei99]. This configuration is peculiar because it generally breaks inversion and does not have any plane of mirror symmetry. It is worth pointing out that the original order in the $4d$ host is completely modified by the small concentration of $3d$ ions and one finds that the AF coupling between the $3d$ impurity and the $4d$ host generally leads to patterns such as the AF$c$ phase where FM chains alternate with AF ones in the $(a,b)$ plane. Another relevant issue is that the cooperation between the host and impurity can lead to a fully polarized FM$a$ state. This implies that doping can release the orbital frustration which was present in the host with the $C$-AF/$G$-AO order. Phase diagram for periodic $x=1/5$ doping {#sec:dop5} ----------------------------------------- ![image](1to5_diag){width="90.00000%"} ![ Isotropic surrounding of the degenerate impurity spins in the FS$v$ and FS$p$ phases in the case of $x=1/5$ periodic doping (Fig. \[fig:1to5\]). Frames mark the clusters which are not connected with orbitally active bonds. \[fig:FSv\]](surr_PM){width=".96\columnwidth"} Next we consider doping $x=1/5$ with a given periodic spatial profile which concerns both doublon sublattices. We investigate the $3d$ spin impurities separated by the translation vectors $\vec{u}=(i,j)$ and $\vec{v}=(2,-1)$ (one can show that for general periodic doping $x$, $\|\vec{u}\|^2=x^{-1}$) so there is a mismatch between the impurity periodicity and the two-sublattice $G$-AO order in the host. One finds that the present case, see Fig. \[fig:1to5\], has similar general structure of the phase diagram to the case of $x=1/8$ (Fig. \[fig:1to8\]), with AF correlations dominating for $\eta_{\rm imp}$ lower than $\eta_{\rm imp}^c$ and FM ones otherwise. Due to the specific doping distribution there are more phases appearing in the ground state phase diagram. For $\eta_{\rm imp}<\eta_{\rm imp}^c$ the most stable spin configuration is with the impurity coupled antiferromagnetically to the host. This happens both in the AF vacancy (AF$v$) and the AF polaronic (AF$p$) ground states. The difference between the two AF states arises due to the orbital arrangement around the impurity. For weak ratio of the impurity to the host spin-orbital exchange, $J_{\rm imp}/J_{\rm host}$, the orbitals around the impurity are all inactive ones. On the contrary, in the strong impurity-host coupling regime all the orbitals are polarized to be in active (polaronic) states around the impurity. Both states have been found as AF1$c$ and AF2$c$ phase in the single impurity problem (Fig. \[fig:pd\_caf\_c\]). More generally, for all phases the boundary given by an approximate hyperbolic relation $\eta_{\rm imp}\propto J_{\rm imp}^{-1}$ separates the phases where the orbitals around impurities in the $c$-orbital sublattice are all inactive (small $\eta_{\rm imp}$) from those where all the orbitals are active (large $\eta_{\rm imp}$). The inactive orbital around the impurity stabilize always the AF coupling between the impurity spin and host spins whereas the active orbitals can give either AF or FM exchange depending on $\eta_{\rm imp}$ (hence $\eta_{\rm imp}^c$, see Fig. \[fig:JSEd\]). Since the doping does not match the size of the elementary unit cell, the resulting ground states do not exhibit specific symmetries in the spin-orbital pattern. They are generally FM due to the uncompensated magnetic moments and the impurity feels screening by the presence of the surrounding it host spins being antiparallel to the impurity spin. ![image](1to9_diag){width="92.00000%"} By increasing Hund’s exchange coupling at the $3d$ ion the system develops fully FM state in a large region of the ground state diagram due to the possibility of suitable orbital polarization around the impurity. On the other hand, in the limit where the impurity-host bonds are weak, so either for $\eta_{\rm imp}\simeq\eta_{\rm imp}^c$ and large enough $J_{\rm imp}/J_{\rm host}$ so that all orbitals around the impurity are active, or just for small $J_{\rm imp}/J_{\rm host}$ we get the FS phases where the impurity spin at the $a$-orbital sublattice is undetermined in the present classical approach. This is a similar situation to the one found in the FS$a$ phase of a single impurity problem and at $x=1/8$ periodic doping, see Figs. \[fig:pd\_caf\_a\] and \[fig:1to8\], but there it was still possible to identify the favored impurity spin polarization by considering the orbital flips in the host around the impurity. However, the situation here is different as the host’s order is completely altered by doping and has became isotropic, in contrast to the initial $C$-AF order (Fig. \[fig:host\]) which breaks the planar symmetry between the $a$ and $b$ direction. It was precisely this symmetry breaking that favored one impurity spin polarization over the other one. Here this mechanism is absent — one can easily check that the neighborhood of the $c$ orbitals surrounding impurities is completely equivalent in both directions (see Fig. \[fig:FSv\] for the view of these surroundings) so that the orbital flip argument is no longer applicable. This is a peculiar situation in the classical approach and we indicate frustration in spin direction by question marks in Fig. \[fig:1to5\]. In Fig. \[fig:FSv\] we can see that both in FS vacancy (FS$v$) and FS polaronic (FS$p$) phase the orbitals are grouped in $3\times 3$ clusters and $2\times 2$ plaquettes, respectively, that encircle the degenerate impurity spins. For the FS$v$ phase we can distinguish between two kind of plaquettes with non-zero spin polarization differing by a global spin inversion. In the case of FS$p$ phases we observe four plaquettes with zero spin polarization arranged in two pairs related by a point reflection with respect to the impurity site. It is worthwhile to realize that these plaquettes are completely disconnected in the orbital sector, i.e., there are no orbitally active bonds connecting them (see Fig. \[fig:host\_flips\] for the pictorial definition of orbitally active bonds). This means that quantum effects of purely orbital nature can appear only at the short range, i.e., inside the plaquettes. However, one can expect that if for some reason the two degenerate spins in a single elementary cell will polarize then they will also polarize in the same way in all the other cells to favor long-range quantum fluctuations in the spin sector related to the translational invariance of the system. Phase diagram for periodic $x=1/9$ doping {#sec:dop9} ----------------------------------------- Finally we investigate low doping $x=1/9$ with a given periodic spatial profile, see Fig. \[fig:1to9\]. Here the impurities are separated by the translation vectors $\vec{u}=(0,3)$ and $\vec{v}=(3,0)$. Once again there is a mismatch between the periodic distribution of impurities and the host’s two-sublattice AO order, so we again call this doping incommensurate as it also imposes doping at both doublon sublattices. The ground state diagram presents gradually increasing tendency towards FM $3d-4d$ bonds with increasing $\eta_{\rm imp}$, see Fig. \[fig:1to9\]. These polaronic bonds polarize as well the $4d-4d$ bonds and one finds an almost FM order in the FM$p$ state. Altogether, we have found the same phases as at the higher doping of $x=1/5$, see Fig. \[fig:1to5\], i.e., AF$v$ and AF$p$ at low values $\eta_{\rm imp}$, FM$v$ and FM$p$ in the regime of high $\eta_{\rm imp}$, separated by the regime of frustrated impurity spins which occur within the phases: FS$v$, FS1$p$, and FS2$p$. The difference between the two AF (FM) states in Fig. \[fig:1to9\] is due to the orbital arrangement around the impurity. As for the other doping levels considered so far, $x=1/8$ and $x=1/5$, we find neutral (inactive) orbitals around $3d$ impurities in the regime of low $J_{\rm imp}/J_{\rm host}$ in AF$v$ and FM$v$ phases which lead to spin defects within the 1D FM chains in the $C$-AF spin order. A similar behavior was reported for single impurities in the low doping regime in Sec. \[sec:imp\]. This changes radically above the orbital transition for both types of local magnetic order, where the orbitals reorient into the active ones. One finds that spin orientations are then the same as those of their neighboring $4d$ atoms, with some similarities to those found at $x=1/5$, see Fig. \[fig:1to5\]. Frustrated impurity spins occur in the crossover regime between the AF and FM local order around impurities. This follows from the local configurations around them, which include two $\uparrow$-spins and two $\downarrow$-spins accompanied by $c$ orbitals at the NN $4d$ sites. This frustration is easily removed by quantum fluctuations and we suggest that this happens again in the same way as for $x=1/8$ doping, as indicated by small arrows in the respective FS phases shown in Fig. \[fig:1to9\]. Quantum effects beyond the classical approach {#sec:qua} ============================================= Spin-orbital quantum fluctuations {#sec:orb} --------------------------------- So far, we analyzed the ground states of $3d$ impurities in the $(a,b)$ plane of a $4d$ system using the classical approach. Here we show that this classical picture may be used as a guideline and is only quantitatively changed by quantum fluctuations if the spin-orbit coupling is weak. We start the analysis by considering the quantum problem in the absence of spin-orbit coupling (at $\lambda=0$). The orbital doublon densities, $$N_{\gamma}\equiv\sum_{i\in\rm host}\langle n_{i\gamma}\rangle, \label{doc}$$ with $\gamma=a,b,c$, and total $S^z$ are conserved quantities and thus good quantum numbers for a numerical simulation. To determine the ground state configurations in the parameters space and the relevant correlation functions we diagonalize exactly the Hamiltonian matrix (\[fullH\]) for the cluster of $L=8$ sites by means of the Lanczos algorithm. In Fig. \[fig:diag\](a) we report the resulting quantum phase diagram for an 8-site cluster having one impurity and assuming periodic boundary conditions, see Fig. \[fig:diag\](b). This appears to be an optimal cluster configuration because it contains a number of sites and connectivities that allows us to analyze separately the interplay between the host-host and the host-impurity interactions and to simulate a physical situation when the interactions within the host dominate over those between the host and the impurity. Such a problem is a quantum analogue of the single unit cell presented in Fig. \[fig:1to8\] for $x=1/8$ periodic doping. As a general feature that resembles the classical phase diagram, we observe that there is a prevalent tendency to have AF-like (FM-like) spin correlations between the impurity and the host sites in the region of $\eta_{\rm imp}$ below (above) the critical point at $\eta_{\rm imp}^c\simeq 0.43$ which separates these two regimes, with intermediate configurations having frustrated magnetic exchange. As we shall discuss below it is the orbital degree of freedom that turns out to be more affected by the quantum effects. Following the notation used for the classical case, we distinguish various quantum AF (QAF) ground states, i.e., QAF$cn$ ($n=1,2$) and QAF$an$ ($n=1,2$), as well as a uniform quantum FM (QFM) configuration, i.e., QFM$a$, and quantum frustrated one labeled as QFS$a$. ![image](diag2){width=".99\textwidth"} In order to visualize the main spin-orbital patterns contributing to the quantum ground state it is convenient to adopt a representation with arrows for the spin and ellipsoids for the orbital sector at any given host site. The arrows stand for the on-site spin projection $\langle S_i^z\rangle$, with the length being proportional to the amplitude. The length scale for the arrows is the same for all the configurations. Moreover, in order to describe the orbital character of the ground state we employed a graphical representation that makes use of an ellipsoid whose semi-axes $\{a,b,c\}$ length are given by the average amplitude of the squared angular momentum components $\{(L^x_i)^2,(L^y_i)^2,(L^z_i)^2\}$, or equivalently by the doublon occupation Eq. (\[eq:dub\]). For instance, for a completely flat circle (degenerate ellipsoid) lying in the plane perpendicular to the $\gamma$ axis only the corresponding $\gamma$ orbital is occupied. On the other hand, if the ellipsoid develops in all three directions $\{a,b,c\}$ it implies that more than one orbital is occupied and the distribution can be anisotropic in general. If all the orbitals contribute equally, one finds an isotropic spherical ellipsoid. Due to the symmetry of the Hamiltonian, the phases shown in the phase diagram of Fig. \[fig:diag\](a) can be characterized by the quantum numbers for the $z$-th spin projection, $S^z$, and the doublon orbital occupation $N_{\alpha}$ (\[doc\]), $\left(S^{z},N_a,N_b,N_c\right)$: QAF$c1$ $\left(-3.5,2,2,3\right)$, QFS$a2$ $\left(-1.5,3,1,3\right)$, QAF$a1$ $\left(-5.5,1,3,3\right)$, QAF$a2$ and QAF$a2$ $\left(-5.5,2,2,3\right)$, QFS$a1$ $\left(-0.5,3,0,4\right)$, and QFM$a$ $\left(-8.5,2,1,4\right)$. Despite the irregular shape of the cluster \[Fig. \[fig:diag\](b)\] there is also symmetry between the $a$ and $b$ directions. For this reason, the phases with $N_{a}\neq N_{b}$ can be equivalently described either by the set $\left(S^z,N_a,N_b,N_c\right)$ or $\left(S^z,N_b,N_a,N_c\right)$. The outcome of the quantum analysis indicates that the spin patterns are quite robust as the spin configurations of the phases QAF$a$, QAF$c$, QFS$a$ and QFM$a$ are the analogues of the classical ones. The effects of quantum fluctuations are more evident in the orbital sector where mixed orbital patterns occur if compared to the classical case. In particular, orbital inactive states around the impurity are softened by quantum fluctuations and on some bonds we find an orbital configuration with a superposition of active and inactive states. The unique AF states where the classical inactive scenario is recovered corresponds to the QAF$c1$ and QAF$c2$ ones in the regime of small $\eta_{\rm imp}$. A small hybridization of active and inactive orbitals along both the AF and FM bonds is also observed around the impurity for the QFS$a$ phases as one can note by the shape of the ellipsoid at host sites. Moreover, in the range of large $\eta_{\rm imp}$ where the FM state is stabilized, the orbital pattern around the impurity is again like in the classical case. A significant orbital rearrangement is also obtained within the host. We generally obtain an orbital pattern that is slightly modified from the pure AO configuration assumed in the classical case. The effect is dramatically different in the regime of strong impurity-host coupling (i.e., for large $J_{\rm{\rm imp}}$) with AF exchange (QAF$a2$) with the formation of an orbital liquid around the impurity and within the host, with doublon occupation represented by an almost isotropic shaped ellipsoid. Interestingly, though with a different orbital arrangement, the QFS$a1$ and the QFS$a2$ states are the only ones where the $C$-AF order of the host is recovered. For all the other phases shown in the diagram of Fig. \[fig:diag\] the coupling between the host and the impurity is generally leading to a uniform spin polarization with FM or AF coupling between the host and the impurity depending on the strength of the host-impurity coupling. Altogether, we conclude that the classical spin patterns are only quantitatively modified and are robust with respect to quantum fluctuations. Finite spin-orbit coupling {#sec:soc} -------------------------- In this Section we analyze the quantum effects in the spin and orbital order around the impurity in the presence of the spin-orbit coupling at the host $d^4$ sites. For the $t_{2g}^{4}$ configuration the strong spin-orbit regime has been considered recently by performing an expansion around the atomic limit where the angular $\vec{L}_i$ and spin $\vec{S}_i$ momenta form a spin-orbit singlet for the amplitude of the total angular momentum, $\vec{J}_i=\vec{L}_i+\vec{S}_i$ (i.e., $J=0$) [@Kha13]. The instability towards an AF state starting from the $J=0$ liquid has been obtained within the spin-wave theory [@Kha14] for the low energy excitations emerging from the spin-orbital exchange. In the analysis presented here we proceed from the limit of zero spin-orbit to investigate how the spin and orbital order are gradually suppressed when approaching the $J=0$ spin-orbit singlet state. This issue is addressed by solving the full quantum Hamiltonian (\[fullH\]) exactly on a cluster of $L=8$ sites including the spin-orbital exchange for the host and that one derived for the host-impurity coupling (\[fullH\]) as well as the spin-orbit term, $${\cal H}_{so}=\lambda \sum_{i\in\rm host}\vec{L}_i\cdot\vec{S}_i. \label{Hso}$$ where the sum includes the ions of the $4d$ host and we use the spin $S=1$ and the angular momentum $L=1$, as in the ionic $4d^4$ configurations. Here $\lambda$ is the spin-orbit coupling constant at $4d$ host ions, and the components of the orbital momentum $\vec{L}_i\equiv\{L^x_i,L^y_i,L^z_i\}$ are defined as follows: $$\begin{aligned} L^x_i&=&i \sum_{\sigma}(d^{\dagger}_{i,xy\sigma}d_{i,xz\sigma} -d^{\dagger}_{i,xz\sigma}d_{i,xy\sigma}), \nonumber \\ L^y_i&=&i \sum_{\sigma}(d^{\dagger}_{i,xy\sigma}d_{i,yz\sigma} -d^{\dagger}_{i,yz\sigma}d_{i,xy\sigma}), \nonumber \\ L^z_i&=&i \sum_{\sigma}(d^{\dagger}_{i,xz\sigma}d_{i,yz\sigma} -d^{\dagger}_{i,yz\sigma}d_{i,xz\sigma}).\end{aligned}$$ To determine the ground state and the relevant correlation functions we use again the Lanczos algorithm for the cluster of $L=8$ sites. Such an approach allows us to study the competition between the spin-orbital exchange and the spin-orbit coupling on equal footing without any simplifying approximation. Moreover, the cluster calculation permits to include the impurity in the host and deal with the numerous degrees of freedom without making approximations that would constrain the interplay of the impurity-host versus host-host interactions. Finite spin-orbit coupling significantly modifies the symmetry properties of the problem. Instead of the SU(2) spin invariance one has to deal with the rotational invariance related to the total angular momentum per site $\vec{J}_i=\vec{L}_i+\vec{S}_i$. Though the $\vec{L}_i\cdot\vec{S}_i$ term in Eq. (\[Hso\]) commutes with both total $\vec{J}^2$ and $J^z$, the full Hamiltonian for the host with impurities Eq. (\[fullH\]) has a reduced symmetry because the spin sector is now linearly coupled to the orbital which has only the cubic symmetry. Thus the remaining symmetry is a cyclic permutation of the $\{x,y,z\}$ axes. Moreover, $J^z$ is not a conserved quantity due to the orbital anisotropy of the spin-orbital exchange in the host and the orbital character of the impurity-host coupling. There one has a $\mathbb{Z}_2$ symmetry associated with the parity operator (-1)$^{J_{z}}$. Hence, the ground state can be classified as even or odd with respect to the value of $J^z$. This symmetry aspect can introduce a constraint on the character of the ground state and on the impurity-host coupling since the $J^z$ value for the impurity is only due to the spin projection while in the host it is due to the combination of the orbital and spin projection. A direct consequence is that the parity constraint together with the unbalance between the spin at the host and the impurity sites leads to a nonvanishing total projection of the spin and angular momentum with respect to a symmetry axis, e.g. the $z^{th}$ axis. It is worth to note that a fixed parity for the impurity spin means that it prefers to point in one direction rather than the other one which is not the case for the host’s spin and angular momentum. Thus the presence on the impurity for a fixed ${\cal P}$ will give a nonzero polarization along the quantization axis for every site of the system. Such a property holds for any single impurity with a half-integer spin. Another important consequence of the spin-orbit coupling is that it introduces local quantum fluctuations in the orbital sector even at the sites close to the impurity where the orbital pattern is disturbed. The spin-orbit term makes the on-site problem around the impurity effectively analogous to the Ising model in a transverse field for the orbital sector, with nontrivial spin-orbital entanglement [@Ole12] extending over the impurity neighborhood. ![image](ls_4conf){width="92.00000%"} In Figs. \[fig:4conf\] and \[fig:2conf\] we report the schematic evolution of the ground state configurations for the cluster of $L=8$ sites, with one-impurity and periodic boundary conditions as a function of increasing spin-orbit coupling. These patterns have been determined by taking into account the sign and the amplitude of the relevant spatial dependent spin and orbital correlation functions. The arrows associated to the spin degree of freedom can lie in $xy$ plane or out-of-plane (along $z$, chosen to be parallel to the $c$ axis) to indicate the anisotropic spin pattern. The out-of-plane arrow length is given by the on-site expectation value of $\langle S^z_i\rangle$ while the in-plane arrow length is obtained by computing the square root of the second moment, i.e., $\sqrt{\langle(S^x_i)^2}\rangle$ and $\sqrt{\langle(S^y_i)^2}\rangle$ of the $x$ and $y$ spin components corresponding to the arrows along $a$ and $b$, respectively. Moreover, the in-plane arrow orientation for a given direction is determined by the sign of the corresponding spin-spin correlation function assuming as a reference the orientation of the impurity spin. The ellipsoid is constructed in the same way as for the zero spin-orbit case above, with the addition of a color map that indicates the strength of the average $\vec{L}_i\cdot\vec{S}_i$ (i.e., red, yellow, green, blue, violet correspond with a growing amplitude of the local spin-orbit correlation function). The scale for the spin-orbit amplitude is set to be in the interval $0<\lambda<J_{\rm host}$. The selected values for the ground state evolution are given by the relation (with $m=1,2,\dots,10$), $$\lambda_m=\left[\,0.04+0.96\,\frac{(m-1)}{9}\,\right]J_{\rm host}. \label{lambda}$$ The scale is set such that $\lambda_1=0.04J_{\rm host}$ and $\lambda_{10}=J_{\rm host}$. This range of values allows us to explore the relevant physical regimes when moving from $3d$ to $4d$ and $5d$ materials with corresponding $\lambda$ being much smaller that $J_{\rm host}$, $\lambda\sim J_{\rm host}/2$ and $\lambda>J_{\rm host}$, respectively. For the performed analysis the selected values of $\lambda$ (\[lambda\]) are also representative of the most interesting regimes of the ground state as induced by the spin-orbit coupling. Let us start with the quantum AF phases QAF$c1$, QAF$c2$, QFS$a1$, QFS$a2$, QAF$a1$, and QAF$a2$. As one can observe the switching on of the spin-orbit coupling (i.e., $\lambda_1$ in Fig. \[fig:4conf\]) leads to anisotropic spin patterns with unequal moments for the in-plane and out-of-plane components. From weak to strong spin-orbit coupling, the character of the spin correlations keeps being AF between the impurity and the neighboring host sites in all the spin directions. The main change for the spin sector occurs for the planar components. For weak spin-orbit coupling the in-plane spin pattern is generally AF for the whole system in all the spatial directions (i.e., $G$-AF order). Further increase of the spin-orbit does not modify qualitatively the character of the spin pattern for the out-of-plane components as long as we do not go to maximal values of $\lambda\sim J_{\rm host}$ where local $\langle S_i^z\rangle$ moments are strongly suppressed. In this limit the dominant tendency of the system is towards formation of the spin-orbital singlets and the spin patterns shown in Fig. \[fig:4conf\] are the effect of the virtual singlet-triplet excitations [@Kha13]. Concerning the orbital sector, only for weak spin-orbit coupling around the impurity one can still observe a reminiscence of inactive orbitals as related to the orbital vacancy role at the impurity site in the AF phase. Such an orbital configuration is quickly modified by increasing the spin-orbit interaction and it evolves into a uniform pattern with almost degenerate orbital occupations in all the directions, and with preferential superpositions of $c$ and $(a,b)$ states associated with dominating $L^x$ and $L^y$ orbital angular components (flattened ellipsoids along the $c$ direction). An exception is the QAF$c2$ phase with the orbital inactive polaron that is stable up to large spin-orbit coupling of the order of $J_{\rm host}$. ![ Evolution of the ground state configurations for the QFS$a1$ and QFS$a2$ phases for selected increasing values of spin-orbit coupling $\lambda_m$, see Eq. (\[lambda\]). Arrows and ellipsoids indicate the spin-orbital state at a given site $i$. Color map indicates the strength of the average spin-orbit, $\langle{\vec L}_i\cdot{\vec S}_i\rangle$, i.e., red, yellow, green, blue, violet correspond to the growing amplitude of the above local correlation function.[]{data-label="fig:2conf"}](ls_2conf){width="0.95\columnwidth"} When considering the quantum FM configurations QFM$a1$ in Fig. \[fig:4conf\], we observe similar trends in the evolution of the spin correlation functions as obtained for the AF states. Indeed, the QFM$a$ exhibits a tendency to form FM chains with AF coupling for the in-plane components at weak spin-orbit that evolve into more dominant AF correlations in all the spatial directions within the host. Interestingly, the spin exchange between the impurity and the neighboring host sites shows a changeover from AF to FM for the range of intermediate-to-strong spin-orbit amplitudes. A peculiar response to the spin-orbit coupling is obtained for the QFS$a1$ phase, see Fig. \[fig:2conf\], which showed a frustrated spin pattern around the impurity already in the classical regime, with FM and AF bonds. It is remarkable that due to the close proximity with uniform FM and the AF states, the spin-orbit interaction can lead to a dramatic rearrangement of the spin and orbital correlations for such a configuration. At weak spin-orbit coupling (i.e., $\lambda\simeq\lambda_1$) the spin-pattern is $C$-AF and the increased coupling ($\lambda\simeq\lambda_2)$) keeps the $C$-AF order only for the in-plane components with the exception of the impurity site. It also modulates the spin moment distribution around the impurity along the $z$ direction. Further increase of $\lambda$ leads to complete spin polarization along the $z$ direction in the host, with antiparallel orientation with respect to the impurity spin. This pattern is guided by the proximity to the FM phase. The in-plane components develop a mixed FM-AF pattern with a strong $xy$ anisotropy most probably related to the different bond exchange between the impurity and the host. When approaching the regime of a spin-orbit coupling that is comparable to $J_{\rm{host}}$, the out-of-plane spin components dominate and the only out-of-plane spin polarization is observed at the impurity site. Such a behavior is unique and occurs only in the QFS$a$ phases. The cooperation between the strong spin-orbit coupling and the frustrated host-impurity spin-orbital exchange leads to an effective decoupling in the spin sector at the impurity with a resulting maximal polarization. On the other hand, as for the AF states, the most favorable configuration for strong spin-orbit has AF in-plane spin correlations. The orbital pattern for the QFS$a$ states evolves similarly to the AF cases with a suppression of the active-inactive interplay around the impurity and the setting of a uniform-like orbital configuration with unquenched angular momentum on site and predominant in-plane components. The response of the FM state is different in this respect as the orbital active states around the impurity are hardly affected by the spin-orbit while the host sites far from the impurity the local spin-orbit coupling is more pronounced. Finally, to understand the peculiar evolution of the spin configuration it is useful to consider the lowest order terms in the spin-orbital exchange that couple directly the orbital angular momentum with the spin. Taking into account the expression of the spin-orbital exchange in the host (\[Hso\]) and the expression of $\vec{L}_i$ one can show that the low energy terms on a bond that get more relevant in the Hamiltonian when the spin-orbit coupling makes a non-vanishing local angular momentum. As a result, the corresponding expressions are: $$\begin{aligned} H_{\rm host}^{a(b)}(i,j)&\approx & J_{{\rm host}}\left\{ a_1\vec{S}_{i}\!\cdot\!\vec{S}_{j}+ b_1S^{z}_{i} S^{z}_{j}L_{i}^{y(x)}L_{j}^{y(x)}\right\} \nonumber \\ &+&\lambda\left\{ \vec{L}_{i}\!\cdot\!\vec{S}_{i} +\vec{L}_{j}\!\cdot\!\vec{S}_{j}\right\}, \label{ls}\end{aligned}$$ with positive coefficients $a_1$ and $b_1$ that depend on $r_1$ and $r_2$ (\[rr\]). A definite sign for the spin exchange in the limit of vanishing spin-orbit coupling is given by the terms which go beyond Eq. (\[ls\]). Then, if the ground state has isotropic FM correlations (e.g. QFM$a$) at $\lambda=0$, the term $S^z_iS^z_jL_i^{y(x)}L_j^{y(x)}$ would tend to favor AF-like configurations for the in-plane orbital angular components when the spin-orbit interaction is switched on. This opposite tendency between the $z$ and $\{x,y\}$ components is counteracted by the local spin-orbit coupling that prevents to have coexisting FM and AF spin-orbital correlations. Such patterns would not allow to optimize the $\vec{L}_i\cdot\vec{S}_i$ amplitudes. One way out is to reduce the $z^{th}$ spin projection and to get planar AF correlations in the spin and in the host. A similar reasoning applies to the AF states where the negative sign of the $S^{z}_iS^{z}_j$ correlations favors FO alignment of the angular momentum components. As for the previous case, the opposite trend of in- and out-of-plane spin-orbital components is suppressed by the spin-orbit coupling and the in-plane FO correlations for the $\{L^x,L^y\}$ components leads to FM patterns for the in-plane spin part as well. Summarizing, by close inspection of Figs. \[fig:4conf\] and \[fig:2conf\] one finds an interesting evolution of the spin patterns in the quantum phases:\ (i) For the QAF states (Fig. \[fig:4conf\]), a spin canting develops at the host sites (i.e., the relative angle is between 0 and $\pi$) while the spins on impurity-host bonds are always AF. The canting in the host evolves, sometime in an inhomogeneous way, to become reduced in the strong spin-orbit coupling regime where ferro-like correlations tend to dominate. In this respect, when the impurity is coupled antiferromagnetically to the host it does not follow the tendency to form spin canting.\ (ii) In the QFM states (Fig. \[fig:2conf\]), at weak spin-orbit one observes spin-canting in the host and for the host-impurity coupling that persists only in the host whereas the spin-orbit interaction is increasing. Spin-orbit coupling versus Hund’s exchange {#sec:JH} ------------------------------------------ To probe the phase diagram of the system in presence of the spin-orbit coupling ($\lambda>0$) we solved the same cluster of $L=8$ sites as before along three different cuts in the phase diagram of Fig. \[fig:diag\](a) for three values of $\lambda$, i.e., small $\lambda=0.1J_{\rm host}$, intermediate $\lambda=0.5J_{\rm host}$, and large $\lambda=J_{\rm host}$. Each cut contained ten points, the cuts were parameterized as follows: (i) $J_{\rm imp}=0.7J_{\rm host}$ and $0\leq\eta_{\rm imp}\leq0.7$, (ii) $J_{\rm imp}=1.3J_{\rm host}$ and $0\leq\eta_{\rm imp}\leq0.7$, and (iii) $\eta_{\rm imp}=\eta_{\rm imp}^{c}\simeq0.43$ and $0\leq J_{\rm imp}\leq1.5J_{\rm host}$. In Fig. \[fig:JH\](a) we show the representative spin-orbital configurations obtained for $\lambda=0.5J_{\rm host}$ along the first cut shown in Fig. \[fig:JH\](b). Values of $\eta_{\rm imp}$ are chosen as $$\eta_{\rm imp}=\eta_{m}\equiv 0.7\,\frac{(m-1)}{9}, \label{cut}$$ with $m=1,\dots,10$ but not all the points are shown in Fig. \[fig:JH\](a) — only the ones for which the spin-orbital configuration changes substantially. The cut starts in the QAF$c2$ phase, according to the phase diagram of Fig. \[fig:JH\](b), and indeed we find a similar configuration to the one shown in Fig. \[fig:4conf\] for QAF$c2$ phase at $\lambda=\lambda_5$. Moving up in the phase diagram from $\eta_1$ to $\eta_2$ we see that the configuration evolves smoothly to the one which we have found in the QAF$a1$ phase at $\lambda=\lambda_5$ (not shown in Fig. \[fig:4conf\]). The evolution of spins is such that the out-of-plane moments are suppressed while in-plane ones are slightly enhanced. The orbitals become more spherical and the local spin-orbit average, $\langle \vec{L}_i\cdot\vec{S}_i\rangle$, becomes larger and more uniform, however for the apical site $i=7$ in the cluster \[Fig. \[fig:diag\](b)\] the trend is opposite — initially large value of spin-orbit coupling drops towards the uniform value. The points between $\eta_3$ and $\eta_7$ we skip as the evolution is smooth and the trend is clear, however the impurity out-of-plane moment begins to grow above $\eta_5$, indicating proximity to the QFS$a1$ phase. For this phase at intermediate and high $\lambda$ the impurity moment is much larger than all the others (see Fig. \[fig:2conf\]). ![(a) Evolution of the ground state configurations as for increasing $\eta_{\rm imp}$ and for a fixed value of spin-orbit coupling $\lambda=0.5J_{\rm host}$ along a cut in the phase diagram shown in panel (b), i.e., for $J_{\rm imp}=0.7J_{\rm host}$ and $0\leq\eta_{\rm imp}\leq 0.7$. Arrows and ellipsoids indicate the spin-orbital state at a given site $i$. Color map indicates the strength of the average spin-orbit, $\langle\vec{L}_i\cdot\vec{S}_i\rangle$, i.e., red, yellow, green, blue, violet correspond to the growing amplitude of the above correlation function.[]{data-label="fig:JH"}](cut_JH){width=".99\columnwidth"} For $\eta_{\rm imp}=\eta_7$ the orbital pattern clearly shows that we are in the QFS$a1$ phase at $\lambda=\lambda_5$ which agrees with the position of the $\eta_7$ point in the phase diagram, see Fig. \[fig:JH\](b). On the other hand, moving to the next $\eta_{\rm imp}$ point upward along the cut Eq. (\[cut\]) we already observe a configuration which is very typical for the QFM$a$ phase at intermediate $\lambda$ (here $\lambda=\lambda_7$ shown in Fig. \[fig:4conf\] but also $\lambda_6$, not shown). This indicates that the QFS$a1$ phase can be still distinguished at $\lambda=0.5J_{\rm host}$ and its position in the phase diagram is similar as in the $\lambda=0$ case, i.e., as an intermediate phase between the QAF$a1(2)$ and QFM$a$ one. Finally, we have found that also the two other cuts which were not shown here, i.e., for $J_{\rm imp}=1.3J_{\rm host}$ and increasing $\eta_{\rm imp}$ and for $\eta_{\rm imp}=\eta_{\rm imp}^{c}\simeq0.43$ and increasing $J_{\rm imp}$ confirm that the overall character of the phase diagram of Fig. \[fig:diag\](a) is preserved at this value of spin-orbit coupling, however firstly, the transitions between the phases are smooth and secondly, the subtle differences between the two QFS$a$, QAF$a$ and QAF$c$ phases are no longer present. This also refers to the smaller value of $\lambda$, i.e., $\lambda=0.1J_{\rm host}$, but already for $\lambda=J_{\rm host}$ the out-of-plane moments are so strongly suppressed (except for the impurity moment in the QFS$a1$ phase) and the orbital polarization is so weak (i.e., almost spherical ellipsoids) that typically the only distinction between the phases can be made by looking at the in-plane spin correlations and the average spin-orbit, $\langle\vec{L}_i\cdot\vec{S}_i\rangle$. In this limit we conclude that the phase diagram is (partially) melted by large spin-orbit coupling but for lower values of $\lambda$ it is still valid. Summary and conclusions {#sec:sum} ======================= We have derived the spin-orbital superexchange model for $3d^3$ impurities replacing $4d^4$ (or $3d^2$) ions in the $4d$ ($3d$) host in the regime of Mott insulating phase. Although the impurity has no orbital degree of freedom, we have shown that it contributes to the spin-orbital physics and influences strongly the orbital order. In fact, it tends to project out the inactive orbitals at the impurity-host bonds to maximize the energy gain from virtual charge fluctuations. In this case the interaction along the superexchange bond can be either antiferromagnetic or ferromagnetic, depending on the ratio of Hund’s exchange coupling at impurity ($J_1^H$) and host ($J_2^H$) ions and on the mismatch $\Delta$ between the $3d$ and $4d$ atomic energies, modified by the difference in Hubbard $U$’s and Hund’s exchange $J^H$’s at both atoms. This ratio, denoted $\eta_{\rm imp}$ (\[eq:etai\]), replaces here the conventional parameter $\eta=J_H/U$ often found in the spin-orbital superexchange models of undoped compounds (e.g., in the Kugel-Khomskii model for KCuF$_3$ [@Fei97]) where it quantifies the proximity to ferromagnetism. On the other hand, if the overall coupling between the host and impurity is weak in the sense of the total superexchange, $J_{\rm imp}$, with respect to the host value, $J_{\rm host}$, the orbitals being next to the impurity may be forced to stay inactive which modifies the magnetic properties — in such cases the impurity-host bond is always antiferromagnetic. As we have seen in the case of a single impurity, the above two mechanisms can have a nontrivial effect on the host, especially if the host itself is characterized by frustrated interactions, as it happens in the parameter regime where the $C$-AF phase is stable. For this reason we have focused mostly on the latter phase of the host and we have presented the phase diagrams of a single impurity configuration in the case when the impurity is doped on the sublattice where the orbitals form a checkerboard pattern with alternating $c$ and $a$ orbitals occupied by doublons. The diagram for the $c$-sublattice doping shows that in some sense the impurity is never weak, because even for a very small value of $J_{\rm imp}/J_{\rm host}$ it can release the host’s frustration around the impurity site acting as an orbital vacancy. On the other hand, for the $a$-sublattice doping when the impurity-host coupling is weak, i.e., either $J_{\rm imp}/J_{\rm host}$ is weak or $\eta_{\rm imp}$ is close to $\eta_{\rm imp}^c$, we have identified an interesting quantum mechanism releasing frustration of the impurity spin (that cannot be avoided in the purely classical approach). It turned out that in such situations the orbital flips in the host make the impurity spin polarize in such a way that the $C$-AF order of the host is completely restored. The cases of the periodic doping studied in this paper show that the host’s order can be completely altered already for rather low doping of $x=1/8$, even if the $J_{\rm imp}/J_{\rm host}$ is small. In this case we can stabilize a ferrimagnetic type of phase with a four-site unit cell having magnetization $\langle S_i^z\rangle=3/2$, reduced further by quantum fluctuations. We have established that the only parameter range where the host’s order remains unchanged is when $\eta_{\rm imp}$ is close to $\eta_{\rm imp}^c$ and $J_{\rm imp}/J_{\rm host}\gtrsim 1$. The latter value is very surprising as it means that the impurity-host coupling must be large enough to keep the host’s order unchanged — this is another manifestation of the orbital vacancy mechanism that we have already observed for a single impurity. Also in this case the impurity spins are fixed with the help of orbital flips in the host that lift the degeneracy which arises in the classical approach. We would like to point out that the quantum mechanism that lifts the ground state degeneracy mentioned above and the role of quantum fluctuations are of particular interest for the periodically doped checkerboard systems with $x=1/2$ doping which is a challenging problem for future research. From the point of view of generic, i.e., non-periodic doping, the most representative cases are those of a doping which is incommensurate with the two-sublattice spin-orbital pattern. To uncover the generic rules in such cases, we have studied periodic $x=1/5$ and $x=1/9$ doping. One finds that when the period of the impurity positions does not match the period of 2 for both the spin and orbital order of the host, interesting novel types of order emerge. In such cases the elementary cell must be doubled in both lattice directions which clearly gives a chance of realizing more phases than in the case of commensurate doping. Our results show that indeed, the number of phases increases from 4 to 7 and the host’s order is altered in each of them. Quite surprisingly, the overall character of the phase diagram remained unchanged with respect to the one for $x=1/8$ doping and, if we ignore the differences in configuration, it seems that only some of the phases got divided into two versions differing either by the spin bond’s polarizations around impurities (phases around $\eta_{\rm imp}^c$), or by the character of the orbitals around the impurities (phases with inactive orbitals in the limit of small enough product $\eta_{\rm imp}J_{\rm imp}$, versus phases with active orbitals in the opposite limit). Orbital polarization in this latter region resembles orbital polarons in doped manganites [@Dag04; @Gec05] — also here such states are stabilized by the double exchange [@deG60]. A closer inspection of underlying phases reveals however a very interesting degeneracy of the impurity spins at $x=1/5$ that arises again from the classical approach but this time it cannot be released by short-range orbital flips. This happens because the host’s order is already so strongly altered that it is no longer anisotropic (as it was the case of the $C$-AF phase) and there is no way to restore the orbital anisotropy around the impurities that could lead to spin-bonds imbalance and polarize the spin. In the case of lower $x=1/9$ doping such an effect is absent and the impurity spins are always polarized, as it happens for $x=1/8$. It shows that this is rather a peculiarity of the $x=1/5$ periodic doping. Indeed, one can easily notice that for $x=1/5$ every atom of the host is a nearest neighbor of some impurity. In contrast, for $x=1/8$ we can find three host’s atoms per unit cell which do not neighbor any impurity and for $x=1/9$ there are sixteen of them. For this reason the impurity effects are amplified for $x=1/5$ which is not unexpected although one may find somewhat surprising that the ground state diagrams for the lowest and the highest doping considered here are very similar. This suggests that the cooperative effects of multiple impurities are indeed not very strong in the low-doping regime, so the diagram obtained for $x=1/9$ can be regarded as generic for the dilute doping regime with uniform spatial profile. For the representative case of $x=1/8$ doping, we have presented the consequences of quantum effects beyond the classical approach. Spin fluctuations are rather weak for the considered case of large $S=1$ and $S=3/2$ spins, and we have shown that orbital fluctuations on superexchange bonds are more important. They are strongest in the regime of antiferromagnetic impurity-host coupling (which suggests importance of entangled states [@Ole12]) and enhance the tendency towards frustrated impurity spin configurations but do not destroy other generic trends observed when the parameters $\eta_{\rm imp}$ and $J_{\rm imp}/J_{\rm host}$ increase. Increasing spin-orbit coupling leads to qualitative changes in the spin-orbital order. When Hund’s exchange is small at the impurity sites, the antiferromagnetic bonds around it have reduced values of spin-orbit coupling term, but the magnetic moments reorient and survive in the $(a,b)$ planes, with some similarity to the phenomena occurring in the perovskite vanadates [@Hor03]. This quenches the magnetic moments at $3d$ impurities and leads to almost uniform orbital occupancies at the host sites. In contrast, frustration of impurity spins is removed and the impurity magnetization along the $c$ axis survives for large spin-orbit coupling. We would like to emphasize that the orbital dilution considered here influences directly the orbital degrees of freedom in the host around the impurities. The synthesis of hybrid compounds having both $3d$ and $4d$ transition metal ions will likely open a novel route for unconventional effects in complex materials. There are several reasons for expecting new scenarios in mixed $3d-4d$ spin-orbital-lattice materials, and we pointed out only some of them. On the experimental side, the changes of local order could be captured using inelastic neutron scattering or resonant inelastic x-ray scattering (RIXS). In fact, using RIXS can also bring an additional advantage: RIXS, besides being a perfect probe of both spin and orbital excitations, can also (indirectly) detect the nature of orbital ground state (supposedly also including the nature of impurities in the crystal) [@Woh12]. Unfortunately, there are no such experiments yet but we believe that they will be available soon. Short range order around impurities could be investigated by the excitation spectra at the resonant edges of the substituting atoms. Taking them both at finite energy and momentum can dive insights into the nature of the short range order around the impurity and then unveil information of the order within the host as well. Even if there are no elastic superlattice extra peaks one can expect that the spin-orbital correlations will emerge in the integrated RIXS spectra providing information of the impurity-host coupling and of the short range order around the impurity. Even more interesting is the case where the substituting atom forms a periodic array with small deviation from the perfect superlattice when one expects the emergence of extra elastic peaks which will clearly indicate the spin-orbital reconstruction. In our case an active orbital diluted site cannot participate coherently in the host spin-orbital order but rather may to restructure the host ordering [@Hos13]. At dilute impurity concentration we may expect broad peaks emerging at finite momenta in the Brillouin zone, indicating the formation of coherent islands with short range order around impurities. We also note that local susceptibility can be suitably measured by making use of resonant spectroscopies (e.g. nuclear magnetic resonance (NMR), electron spin resonance (ESR), nuclear quadrupole resonance (NQR), muon spin resonance ($\mu$SR), etcetera) that exploit the different magnetic or electric character of the atomic nuclei for the impurity and the host in the hybrid system. Finally, the random implantation of the muons in the sample can provide information of the relaxation time in different domains with unequal dopant concentration which may be nonuniform. For the given problem the differences in the resonant response can give relevant information about the distribution of the local fields, the occurrence of local order and provide access to the dynamical response within doped domains. The use of local spectroscopic resonance methods has been widely demonstrated to be successful when probing the nature and the evolution of the ground state in the presence of spin vacancies both for ordered and disordered magnetic configurations [@Lim02; @Bob09; @Sen11; @Bon12]. In summary, this study highlights the role of spin defects which lead to orbital dilution in spin-orbital systems. Using an example of $3d^3$ impurities in a $4d^4$ (or $3d^2$) host we have shown that impurities change radically the spin-orbital order around them, independently of the parameter regime. As a general feature we have found that doped $3d^3$ ions within the host with spin-orbital order have frustrated spins and polarize the orbitals of the host when the impurity-host exchange as well as Hund’s exchange at the impurity are both sufficiently large. This remarkable trend is independent of doping and is expected to lead to global changes of spin-orbital order in doped materials. While the latter effect is robust, we argue that the long-range spin fluctuations resulting from the translational invariance of the system will likely prevent the ground state from being macroscopically degenerate, so if the impurity spins in one unit cell happens to choose its polarization then the others will follow. On the contrary, in the regime of weak Hund’s exchange $3d^3$ ions act not only as spin defects which order antiferromagnetically with respect to their neighbors, but also induce doublons in inactive orbitals. Finally, we remark that this behavior with switching between inactive and active orbitals by an orbitally neutral impurity may lead to multiple interesting phenomena at macroscopic doping when global modifications of the spin-orbital order are expected to occur. Most of the results were obtained in the classical approximation but we have shown that modifications due to spin-orbit coupling do not change the main conclusion. We note that this generic treatment and the general questions addressed here, such as the release of frustration for competing spin structures due to periodic impurities, are relevant to double perovskites [@Pau13]. While the local orbital polarization should be similar, it is challenging to investigate disordered impurities, both theoretically and in experiment, to find out whether their influence on the global spin-orbital order in the host is equally strong. We thank Maria Daghofer and Krzysztof Wohlfeld for insightful discussions. W. B. and A. M. O. kindly acknowledge support by the Polish National Science Center (NCN) under Project No. 2012/04/A/ST3/00331. W. B. was also supported by the Foundation for Polish Science (FNP) within the START program. M. C. acknowledges funding from the EU — FP7/2007-2013 under Grant Agreement No. 264098 — MAMA. Derivation of superexchange =========================== Here we present the details of the derivation of the low energy spin-orbital Hamiltonian for the $3d^3-4d^4$ bonds around the impurity at site $i$. ${\cal H}_{3d-4d}(i)$, which follows from the perfurbation theory, as given in Eq. (\[eq:pert\_exp\]). Here we consider a single $3d^3-4d^4$ bond $\langle ij\rangle$. Two contributions to the effective Hamiltonian follow from charge excitations: (i) ${\cal H}_{J,43}^{(\gamma)}(i,j)$ due to $d^3_id^4_j\leftrightharpoons d^4_id^3_j$, and (ii) ${\cal H}_{J,25}^{(\gamma)}(i,j)$ due to $d^3_id^4_j\leftrightharpoons d^2_id^5_j$. Therefore the low energy Hamiltonian is, $${\cal H}_J^{(\gamma)}(i,j)= {\cal H}_{J,43}^{(\gamma)}(i,j)+{\cal H}_{J,25}^{(\gamma)}(i,j). \label{eq:HJ}$$ Consider first the processes which conserve the number of doubly occupied orbitals, $d^3_id^4_j\leftrightharpoons d^4_id^3_j$. Then by means of spin and orbital projectors, it is possible to express ${\cal H}_{J,43}^{(\gamma)}(i,j)$ for $i=1$ and $j=2$ as $$\begin{aligned} & &{\cal H}_{J,43}^{(\gamma)}(1,2)= \nonumber \\ &&- \left(\vec{S}_{1}\!\cdot\!\vec{S}_{2}\right)\frac{t^{2}}{18} \left\{ \frac{4}{\Delta}-\frac{7}{\Delta+3J_{2}^{H}} -\frac{3}{\Delta+5J_{2}^{H}}\right\} \nonumber \\ &&+ D_2^{(\gamma)}\left(\vec{S}_1\!\cdot\!\vec{S}_2\right) \frac{t^{2}}{18}\left\{ \frac{4}{\Delta}-\frac{1}{\Delta+3J_{2}^{H}} +\frac{3}{\Delta+5J_{2}^{H}}\right\} \nonumber \\ &&+ \left(D_2^{(\gamma)}\!-1\right)\frac{t^{2}}{12} \left\{ \frac{8}{\Delta}+\frac{1}{\Delta+3J_{2}^{H}} -\frac{3}{\Delta+5J_{2}^{H}}\right\}, \label{eq:s-ex}\end{aligned}$$ with the excitation energy $\Delta$ defined in Eq. (\[Delta\]). The resulting effective $3d-4d$ exchange in Eq. (\[eq:s-ex\]) consists of three terms: (i) The first one does not depend on the orbital configuration of the $4d$ atom and it can be FM or AF depending on the values $\Delta$ and the Hund’s exchange on the $3d$ ion. In particular, if $\Delta$ is the largest or the smallest energy scale, the coupling will be either AF or FM, respectively. (ii) The second term has an explicit dependence on the occupation of the doublon on the $4d$ atom via the projecting operator $D_2^{(\gamma)}$. This implies that a magnetic exchange is possible only if the doublon occupies the inactive orbital for a bond along a given direction $\gamma$. Unlike in the first term, the sign of this interaction is always positive favoring an AF configuration at any strength of $\Delta$ and $J_{1}^{H}$. (iii) Finally, the last term describes the effective processes which do not depend on the spin states on the $3d$ and $4d$ atoms. This contribution is of pure orbital nature, as it originates from the hopping between $3d$ and $4d$ atoms without affecting their spin configuration, and for this reason favors the occupation of active $t_{2g}$ orbitals along the bond by the doublon. Within the same scheme, we have derived the effective spin-orbital exchange that originates from the charge transfer processes of the type $3d_1^3 4d_2^4\leftrightharpoons 3d_i^24d_j^5$, ${\cal H}_{J,25}^{(\gamma)}(1,2)$. The effective low-energy contribution to the Hamiltonian for $i=1$ and $j=2$ reads $$\begin{aligned} \label{eq:s-ex2} &&{\cal H}_{J,25}^{(\gamma)}(1,2)= \frac{t^{2}}{U_{1}+U_{2}-\left(\Delta+3J_{2}^{H}-2J_{1}^{H}\right)} \nonumber\\ &\!\times\!&\left\{\frac{1}{3}D_2^{(\gamma)}\! \left(\vec{S}_{1}\!\cdot\!\vec{S}_{2}\right) \!+\frac{1}{3}\!\left(\vec{S}_{1}\!\cdot\!\vec{S}_{2}\right) \!-\frac{1}{2}\!\left(D_2^{(\gamma)}\!+1\right)\!\right\}.\end{aligned}$$ By inspection of the spin structure involved in the elemental processes that generate ${\cal H}_{J,25}^{(\gamma)}(1,2)$, one can note that it is always AF independently of the orbital configuration on the $4d$ atom exhibiting with a larger spin-exchange and an orbital energy gain if the doublon is occupying the inactive orbital along a given bond. We have verified that the amplitude of the exchange terms in ${\cal H}_{J,25}^{(\gamma)}(1,2)$ is much smaller than the ones which enter in ${\cal H}_{J,43}^{(\gamma)}(1,2)$ which justifies that one may simplify Eq. (\[eq:HJ\]) for $i=1$ and $j=2$ to $${\cal H}_J^{(\gamma)}(1,2)\simeq {\cal H}_{J,43}^{(\gamma)}(1,2), \label{eq:HJapp}$$ and neglect ${\cal H}_{J,25}^{(\gamma)}(1,2)$ terms altogether. This approximation is used in Sec. \[sec:model\]. Orbital operators in the L-basis ================================ The starting point to express the orbital operators appearing in the spin-orbital superexchange model (\[fullH\]) is the relation between quenched $\left\|a\right\rangle_i$, $\left\|b\right\rangle_i$, and $\left\|c\right\rangle_i$ orbitals at site $i$ and the eigenvectors $\left\|1\right\rangle_i$, $\left\|0\right\rangle_i$, and $\left\|-1\right\rangle_i$ of the angular momentum operator $L^z_i$. These are known to be $$\begin{aligned} \left\|a\right\rangle_i& = & \frac{1}{\sqrt{2}} \left(\left\|1\right\rangle_i+\left\|-1\right\rangle_i\right), \nonumber \\ \left\|b\right\rangle_i & = & \frac{-i}{\sqrt{2}} \left(\left\|1\right\rangle_i-\left\|-1\right\rangle_i\right), \nonumber \\ \left\|c\right\rangle_i & = & \left\|0\right\rangle_i .\end{aligned}$$ From this we can immediately get the occupation number operators for the doublon, $$\begin{aligned} D_i^{(a)} & =a^{\dagger}_ia_i^{}=\left\|a\right\rangle_i\left\langle a\right\|_i= & 1-\left(L^{x}_i\right)^{2},\nonumber \\ D_i^{(b)} & =b^{\dagger}_ib_i^{}=\left\|b\right\rangle_i\left\langle b\right\|_i= & 1-\left(L^{y}_i\right)^{2},\nonumber \\ D_i^{(c)} & =c^{\dagger}_ic_i^{}=\left\|c\right\rangle_i\left\langle c\right\|_i= & 1-\left(L^{z}_i\right)^{2},\end{aligned}$$ and the related $\{n_i^{(\gamma)}\}$ operators, $$\begin{aligned} n_i^{(a)} & =b^{\dagger}_ib_i^{}+c^{\dagger}_ic_i^{}= & \left(L^{x}_i\right)^{2},\nonumber \\ n_i^{(b)} & =c^{\dagger}_ic_i^{}+a^{\dagger}_ia_i^{}= & \left(L^{y}_i\right)^{2},\nonumber \\ n_i^{(c)} & =a^{\dagger}_ia_i^{}+b^{\dagger}_ib_i^{}= & \left(L^{z}_i\right)^{2}.\end{aligned}$$ The doublon hopping operators have a slightly different structure that reflects their noncommutivity, i.e., $$\begin{aligned} a^{\dagger}_ib_i^{}& = & \left\|a\right\rangle_i\left\langle b\right\|_i=iL^{y}_iL^{x}_i, \nonumber \\ b^{\dagger}_ic_i^{}& = & \left\|b\right\rangle_i\left\langle c\right\|_i=iL^{z}_iL^{y}_i, \nonumber \\ c^{\dagger}_ia_i^{}& = & \left\|c\right\rangle_i\left\langle a\right\|_i=iL^{x}_iL^{z}_i.\end{aligned}$$ These relations are sufficient to write the superexchange Hamiltonian for the host-host and impurity-host bonds in the $\left\{ L^x_i,L^y_i,L^z_i\right\}$ operator basis for the orbital part. However, in practice it is more convenient to work with real operators $\left\{L^+_i,L^-_i,L^z_i\right\}$ rather than with the original ones, $\left\{ L^x_i,L^y_i,L^z_i\right\}$. Thus we write the final relations which we used for the numerical calculations in terms of these operators, $$\begin{aligned} D_i^{(a)} & = &-\frac{1}{4}\left[\left(L^+_i\right)^{2} +\left(L^-_i\right)^{2}\right]+\frac12\left(L^z_i\right)^2,\nonumber \\ D_i^{(b)} & = &\hskip .2cm \frac{1}{4}\left[\left(L^+_i\right)^{2} +\left(L^-_i\right)^{2}\right]+\frac12\left(L^z_i\right)^2,\nonumber \\ D_i^{(c)} & = & 1-\left(L^z_i\right)^{2},\end{aligned}$$ for the doublon occupation numbers and going directly to the orbital $\vec{\tau}_i$ operators we find that, $$\begin{aligned} \tau^{+(a)}_i& = & \frac12\left(L^-_i-L^+_i\right)L^{z}_i, \nonumber \\ \tau^{+(b)}_i& = &\frac{-i}{2}\,L^z_i\left(L^+_i+L^{-}_i\right), \nonumber \\ \tau^{+(c)}_i& = & \frac{i}{4}\left[\left(L^{+}_i\right)^{2} -\left(L^{-}_i\right)^{2}\right]-\frac{i}{2}L^{z}_i,\end{aligned}$$ for the off-diagonal part and $$\begin{aligned} \tau^{z(a)}_i & = & \frac{1}{8}\left[\left(L^{+}_i\right)^{2} +\left(L^{-}_i\right)^2\right]+\frac{3}{4}\left(L^{z}_i\right)^2 -\frac{1}{2},\nonumber \\ \tau^{z(b)}_i & = & \frac{1}{8}\left[\left(L^{+}_i\right)^{2} +\left(L^{-}_i\right)^2\right]-\frac{3}{4}\left(L^{z}_i\right)^2 +\frac{1}{2},\nonumber \\ \tau^{z(c)}_i & = & -\frac{1}{4}\left[\left(L^{+}_i\right)^{2} +\left(L^{-}_i\right)^{2}\right],\end{aligned}$$ for the diagonal one. 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--- author: - 'P. Adamson' - 'I. Anghel' - 'N. Ashby' - 'A. Aurisano' - 'G. Barr' - 'M. Bishai' - 'A. Blake' - 'G. J. Bock' - 'D. Bogert' - 'R. Bumgarner' - 'S. V. Cao' - 'C. M. Castromonte' - 'S. Childress' - 'J. A. B. Coelho' - 'L. Corwin' - 'D. Cronin-Hennessy' - 'J. K. de Jong' - 'A. V. Devan' - 'N. E. Devenish' - 'M. V. Diwan' - 'C. O. Escobar' - 'J. J. Evans' - 'E. Falk' - 'G. J. Feldman' - 'B. Fonville' - 'M. V. Frohne' - 'H. R. Gallagher' - 'R. A. Gomes' - 'M. C. Goodman' - 'P. Gouffon' - 'N. Graf' - 'R. Gran' - 'K. Grzelak' - 'A. Habig' - 'S. R. Hahn' - 'J. Hartnell' - 'R. Hatcher' - 'J. Hirschauer' - 'A. Holin' - 'J. Huang' - 'J. Hylen' - 'G. M. Irwin' - 'Z. Isvan' - 'C. James' - 'S. R. Jefferts' - 'D. Jensen' - 'T. Kafka' - 'S. M. S. Kasahara' - 'G. Koizumi' - 'M. Kordosky' - 'A. Kreymer' - 'K. Lang' - 'J. Ling' - 'P. J. Litchfield' - 'P. Lucas' - 'W. A. Mann' - 'M. L. Marshak' - 'D. Matsakis' - 'N. Mayer' - 'A. McKinley' - 'C. McGivern' - 'M. M. Medeiros' - 'R. Mehdiyev' - 'J. R. Meier' - 'M. D. Messier' - 'W. H. Miller' - 'S. R. Mishra' - 'S. Mitchell' - 'S. Moed Sher' - 'C. D. Moore' - 'L. Mualem' - 'J. Musser' - 'D. Naples' - 'J. K. Nelson' - 'H. B. Newman' - 'R. J. Nichol' - 'J. A. Nowak' - 'J. O’Connor' - 'M. Orchanian' - 'R. B. Pahlka' - 'J. Paley' - 'T. E. Parker' - 'R. B. Patterson' - 'G. Pawloski' - 'A. Perch' - 'S. Phan-Budd' - 'R. K. Plunkett' - 'N. Poonthottathil' - 'E. Powers' - 'X. Qiu' - 'A. Radovic' - 'B. Rebel' - 'K. Ridl' - 'S. Römisch' - 'C. Rosenfeld' - 'H. A. Rubin' - 'M. C. Sanchez' - 'J. Schneps' - 'A. Schreckenberger' - 'P. Schreiner' - 'R. Sharma' - 'A. Sousa' - 'N. Tagg' - 'R. L. Talaga' - 'J. Thomas' - 'M. A. Thomson' - 'X. Tian' - 'A. Timmons' - 'S. C. Tognini' - 'R. Toner' - 'D. Torretta' - 'J. Urheim' - 'P. Vahle' - 'B. Viren' - 'A. Weber' - 'R. C. Webb' - 'C. White' - 'L. Whitehead' - 'L. H. Whitehead' - 'S. G. Wojcicki' - 'J. Wright' - 'V. Zhang' - 'R. Zwaska' ---
--- abstract: 'Parametric Subharmonic Instability (PSI) is one of the most important mechanisms that transfer energy from tidally-generated long internal waves to short steep waves. Breaking of these short waves results in diapycnal mixing through which oceanic abyssal stratification is maintained. It has long been believed that PSI is strongest between a primary internal wave and perturbative waves of half the frequency of the primary wave. Here, we rigorously show that this is not the case. Specifically, we show that neither the initial growth rate nor the maximum long-term amplification occur at the half frequency, and demonstrate that the dominant subharmonic waves have much longer wavelengths than previously thought.' author: - 'Y. Liang' - 'L.-A. Couston' - 'Q. C. Guo' - 'M.-R. Alam[^1]' title: Dominant Resonance in Parametric Subharmonic Instability of Internal Waves --- Introduction ============ Internal gravity waves are ubiquitous in world’s density-stratified oceans. They mainly arise from barotropic tides flowing over topographic features, or wind disturbing the upper ocean’s mixed layer [@wunsch2004]. Internal waves transport energy over long distances in oceans and when break result in considerable mixing which contributes to, e.g., oceanic circulations through lifting cold water from the ocean basin [@Garrett2003a], and the lives of a wide range of ocean creatures by redistributing nutrients [@Boyd2007; @Harris1986]. The underlying mechanism(s) that lead to the breaking of internal waves is not yet fully understood despite significant recent progresses made in our understanding of such waves. There is a nearly general consensus that low mode internal waves, such as those generated by tides, need to somehow transfer their energy to shorter waves which are steeper and hence more prone to breaking. Several mechanisms for such transfer of energy from long to short waves have been put forward, among them interaction with topographic features [@Lamb2014; @Sarkar2017; @Swinney2014] and parametric subharmonic instability [e.g. @Hasselmann1967; @Davis1967; @Dauxois2013] are the most important ones. Parametric Subharmonic Instability (PSI) is the instability of a primary internal wave to two lower-frequency internal waves with (initially) infinitesimal amplitudes thus disturbances. This happens if the primary and the two disturbance waves’ frequencies and wavenumber vectors (respectively $\omega_{0,1,2}$ and $\k_{0,1,2}$) satisfy the triad resonance condition [e.g. @Lombard1996; @Karimi2014; @Koudella2006] \[007\] \_0=\_1+\_2, \_0=\_1+\_2. Through this instability, energy is transferred from the primary wave to the two disturbance waves. The flow of energy may continue until the amplitudes of the disturbance waves, initially infinitesimal, become of the same order as the amplitude of the primary wave or even higher, and that is why the disturbance waves are said to be resonated. This resonance is not through an explicit external force but instead is caused by what appears to be parameters in the governing equations, hence it is called parametric resonance or instability[^2]. Resonated subharmonic waves obtained through PSI usually have smaller vertical and horizontal scales than the primary wave. Thus, PSI constitutes a pathway for energy transfer to steeper waves which are more prone to breaking. PSI was first studied [in 1960s, e.g. @Hasselmann1967; @Davis1967] as a subset of the general wave resonance theory [@Phillip1981; @Marcus2009], and has since been reported in several field studies [e.g. @Alford2008b; @Chinn2012a]. The current understanding of PSI is based on a linear stability theory [established in the 1970s, e.g. @Martin1972; @McEwan1971]. Assuming that the amplitude of the primary wave is constant, and that amplitudes of disturbance waves are much smaller than the primary wave, the linear stability theory predicts an exponential growth rate for a pair of perturbing waves that satisfy the resonance conditions [see e.g. @Martin1972; @Koudella2006; @Bourget2013]. An internal wave, however, can undergo parametric subharmonic instability simultaneously with a countably infinite pairs of subharmonic waves. In order to accurately determine the role of PSI on the evolution of oceanic internal wave spectrum as well as to answer how efficiently an internal wave can transfer its energy to smaller scales, it is critical to know which specific pair (or pairs) of disturbance waves draw the most energy from the primary internal wave. In other words, it is critical to know which triads are resonated the strongest out of all the different triad resonance possibilities. The classical linear stability theory assumes that these infinite resonance possibilities are independent (or decoupled) and predicts that the pair of subharmonic waves with half of the frequency of the primary wave has the largest growth rates and thus is expected to dominate in the process of PSI [e.g. @Staquet2002; @Bourget2013]. However, laboratory and numerical experiments on PSI do not support the predominance of half frequency resonant waves. For example, in a series of experiments on PSI in a wave tank $21$m long, $1.2$m deep and filled with linearly stratified fluid, Martin et al. [@Martin1972] obtained multiple subharmonic waves generated from PSI of a mode-three internal wave, but none of them were at the half frequency. In another attempt, Joubaud et al. [@Joubaud2012] sends a horizontally propagating mode-one internal wave at frequency $0.95N$ ($N$: Brunt-V[ä]{}is[ä]{}l[ä]{} frequency) but observes two subharmonic waves not at the half frequency but at frequencies $0.38N$ and $0.57N$. Also, direct simulation of internal beams generated by a tidal flow (frequency $\omega_0$) over bottom topography results in strongest subharmonic waves at frequencies $0.4\omega_0$ and $0.6\omega_0$ [@Korobov2008]. To address this discrepancy, here we consider the fully-coupled governing interaction equations that account for all triads satisfying PSI resonance condition . We solve this governing equation through multiple-scale analysis that obtains a uniformly-valid nonlinear instability solution. Our analysis determines that, contrary to linear stability theory prediction, it is a pair of subharmonic waves with frequencies different from $\omega_0/2$ that grow the largest in the PSI. In fact, in cases pairs of frequency $\omega_0/2$ receive the least amount of energy compared to other pairs in the pool of interactions. Furthermore, internal waves with frequencies near $\omega_0/2$, as can be derived from equation , have very large vertical wavenumbers and therefore are very short. In fact, at exactly $\omega_0/2$, the vertical wavenumber is infinite. Strongest resonant waves of PSI, as predicted by the nonlinear stability theory, have frequencies far from $\omega_0$ hence have finite wavenumbers. Therefore, nonlinear stability theory shows that the actual strongest resonant waves of PSI have much larger scales than what linear stability theory predicts. While it is not unexpected that nonlinear theory results in a long-term growth which is different from the linear theory predictions, it is obvious that both linear and nonlinear theories must give the same initial growth rates. But, to our surprise, our initial growth rate from nonlinear stability theory did not match the prevalent linear growth rate reported in the literature. After some scrutiny we realized that in the classical theory the effect of the second exponential term has been mistakenly neglected, resulting in an incorrect reported initial growth rate. Specifically, in linear stability analysis, amplitude growth in a resonance is usually obtained in the form $A=a\e^{bt}+c\e^{-bt}$, and therefore the growth rate is $(1/A)(\d A/\d t)$ = $ b(a\e^{bt}-c\e^{-bt})/(a\e^{bt}+c\e^{-bt})$. While at large times (at which linear stability analysis is usually not valid) the growth rate tends to $b$, the initial growth rate is $b[1-2c/(a+c)]$, that depends on $a,c$ and can potentially be very much different from $b$. In what follows, we present our nonlinear stability analysis and discuss both initial growth rate and overall long-time growth of each resonance triads. We validate our analytical solution with direct numerical simulation results obtained from non-hydrostatic Navier-Stokes solver MITgcm.\ Interaction Equations for Parametric Subharmonic Instability ============================================================ Consider an inviscid, incompressible and stably-stratified fluid bounded by a top rigid lid and a flat seafloor at the bottom. In a Cartesian coordinates system with $x,y$-axes on the rigid lid and $z$-axis positive upward, the governing equations read &\_0 /t=-p-g z, -h<z<0,\[g101\]\ &/t=0, -h<z<0,\[g1031\]\ &=0, -h<z<0,\[g1032\]\ &w=0, z=0, \[g1051\]\ &w=0, z=-h,\[g1052\] where $\b{u} = \{u, v, w\}$ is the velocity vector, $\rho$ is the density, $p$ is the pressure, and $g$ is the gravitational acceleration. A linear density profile is considered such that the background density is given by $\bar{\rho}(z)/\rho_0=1-a z$, with $\rho_0=\bar{\rho}(z)\|_{z=0}$. Equation is the momentum equation, represents conservation of salt (assuming that the density only depends on salinity in the equation of state and diffusion of salt is negligible), is conservation of mass, and equations , are kinematic boundary conditions on the rigid lid and the sea bottom respectively. ![(a) Geometric construction of several of possible triad resonances (c.f. ) between a primary internal wave of $\omega$=0.8 (red circle) and pairs of lower harmonic waves for $ah$=0.05. Square and triangles correspond to the higher mode propagating respectively in the same/opposite direction of the primary wave, and diamonds show a special case in which both subharmonic waves travel in the same direction. Six triads are highlighted by color solid lines. Black dashed lines show branches of the dispersion relation $\omega=\|k/\sqrt{k^2+m^2}\|$. (b) Variation of the normalized exponent ($\sigma/\sigma_{max}$) of linear stability analysis as a function of normalized frequency $\omega/\omega_0$. Clearly subharmonic waves with frequencies close to $\omega/2$ have the highest exponents [c.f. figure 11b of @Bourget2013]. (c) Normalized interaction coefficient $\|s_i/s_{max}\|$ as a function of normalized frequency $\omega/\omega_0$. The maximum initial growth rate occurs at the frequency $\omega/\omega_0$=0.83 although $\sigma$ is maximum at $\omega/\omega_0$=0.5.[]{data-label="fig11"}](11_1 "fig:"){width="9cm"} (-245,82)[(a)]{}\ ![(a) Geometric construction of several of possible triad resonances (c.f. ) between a primary internal wave of $\omega$=0.8 (red circle) and pairs of lower harmonic waves for $ah$=0.05. Square and triangles correspond to the higher mode propagating respectively in the same/opposite direction of the primary wave, and diamonds show a special case in which both subharmonic waves travel in the same direction. Six triads are highlighted by color solid lines. Black dashed lines show branches of the dispersion relation $\omega=\|k/\sqrt{k^2+m^2}\|$. (b) Variation of the normalized exponent ($\sigma/\sigma_{max}$) of linear stability analysis as a function of normalized frequency $\omega/\omega_0$. Clearly subharmonic waves with frequencies close to $\omega/2$ have the highest exponents [c.f. figure 11b of @Bourget2013]. (c) Normalized interaction coefficient $\|s_i/s_{max}\|$ as a function of normalized frequency $\omega/\omega_0$. The maximum initial growth rate occurs at the frequency $\omega/\omega_0$=0.83 although $\sigma$ is maximum at $\omega/\omega_0$=0.5.[]{data-label="fig11"}](12 "fig:"){width="9cm"} (-245,82)[(b)]{}\ ![(a) Geometric construction of several of possible triad resonances (c.f. ) between a primary internal wave of $\omega$=0.8 (red circle) and pairs of lower harmonic waves for $ah$=0.05. Square and triangles correspond to the higher mode propagating respectively in the same/opposite direction of the primary wave, and diamonds show a special case in which both subharmonic waves travel in the same direction. Six triads are highlighted by color solid lines. Black dashed lines show branches of the dispersion relation $\omega=\|k/\sqrt{k^2+m^2}\|$. (b) Variation of the normalized exponent ($\sigma/\sigma_{max}$) of linear stability analysis as a function of normalized frequency $\omega/\omega_0$. Clearly subharmonic waves with frequencies close to $\omega/2$ have the highest exponents [c.f. figure 11b of @Bourget2013]. (c) Normalized interaction coefficient $\|s_i/s_{max}\|$ as a function of normalized frequency $\omega/\omega_0$. The maximum initial growth rate occurs at the frequency $\omega/\omega_0$=0.83 although $\sigma$ is maximum at $\omega/\omega_0$=0.5.[]{data-label="fig11"}](13_1 "fig:"){width="9cm"} (-245,82)[(c)]{}\ System of equations admits, among other solutions, a propagating internal-wave solution. Considering that a primary internal wave ($\k_0,\omega_0$) with a finite initial amplitude co-exists with two perturbation waves ($\k_1,\omega_1$) and ($\k_2,\omega_2$) the vertical velocity to the leading order can be written in the form \[131\] w=\_[j=0,1,2]{}A\_j m\_j(z+h)e\^[ i(k\_j x-\_j t)]{}+ where $\k_j=k_j\hat i + m_j \hat z$, and denotes the complex conjugates. If triad resonance condition is satisfied between the three waves, then amplitudes $A_j$ will slowly change with time. Mathematically this is expressed by $A_j=A_j(\epsilon t)$, that is, amplitudes are functions of slow time ($\epsilon$ is a small parameter and a measure of the waves’ steepness). Let’s first define the following dimensionless variables &A\^\\_j=, t\^\=, \^\\_j=, k\^\\_j=k\_jh, m\^\*\_j=m\_jh. where $A_{00}=A_0(t)\|_{t=0}$. Through multiple-scale perturbation analysis, the differential equation that governs the evolution of a wave triad can be obtained, dropping all asterisks, as A\_0(t)/t=s\_0 A\_1(t)A\_2(t)\ A\_1(t)/t=s\_1 A\_0(t)\|[A]{}\_2(t)\ A\_2(t)/t=s\_2 A\_0(t)\|[A]{}\_1(t) where $\bar{A}_{1,2}(t)$ are complex conjugates of $A_{1,2}(t)$, and[^3] \[146\] s= - + +\_0\^2. The interaction coefficients for the two perturbing waves $s_1$ and $s_2$ can be obtained by simply swapping the physical parameters in the expression for $s_0$ in [c.f. e.g. @Mcewan2006; @Bourget2013]. Dominant Subharmonic Waves ========================== The subharmonic waves generated in the process of PSI must satisfy the resonant condition and the dispersion relation $\omega=\|k/\sqrt{k^2+m^2}\|$. As an example of the wave triads satisfying the resonance conditions, for the choice of $ah=5\times10^{-2}$ and $\omega_0=0.8$, we present in figure \[fig11\]a several of possible subharmonic wave triads that can be excited by PSI of the primary wave (red circle). The total number of triad possibilities is (countably) infinite. Note that there are two distinct branches of triads in figure \[fig11\]a, marked by triangles and squares. Clearly, as the wavenumber of perturbation waves involved in the triad increases the frequency of perturbation waves asymptotically tend to half the frequency of the primary wave ($\omega_0/2$). In other words, perturbation waves with frequencies near $\omega_0/2$ have large horizontal and vertical wavenumbers, i.e., $\|k_1,m_1\|\approx\|k_2,m_2\|\gg\|k_0,m_0\|$ [c.f. e.g. @Staquet2002]. Based on the linearized instability theory, i.e. assuming that the amplitude of the primary wave is constant (of course this only applies for the very initial period of the resonance), then becomes, d A\_1(t)/dt=s\_1 A\_0\|[A]{}\_2(t),\ d A\_2(t)/dt=s\_2 A\_0 \|[A]{}\_1(t), where $A_0$ is a constant. If the initial amplitudes of the two perturbing waves are respectively $A_1\|_{t=0}=\delta_1$ and $A_2\|_{t=0}=\delta_2$, the solution to $A_1(t)$ is, \[1452\] A\_1(t)=&1/2(+\_1) (t)\ &-1/2(-\_1) (-t) where $\sigma=\sqrt{s_1s_2} \|A_0\|$. A similar expression is obtained for $A_2(t)$ with subscripts 1,2 in swapped. In classical theory of PSI, $\sigma$ has been considered as the measure of growth of perturbation waves. We show in figure \[fig11\]b the plot of $\sigma/\sigma_{max}$ (i.e. $\sigma$ normalized by the maximum $\sigma$ found from all possible PSI triads) as a function of $\omega/\omega_0$, where $\omega$ refers to the frequency of perturbation waves. Colors of markers in figure \[fig11\]b are associated with the colors of triads depicted geometrically in figure \[fig11\]a. In each resonance, two perturbation waves are involved that are shown with a pair of same-color markers. For instance, triad of green color occurs between perturbation waves of frequencies $\omega_{1,2}/\omega_0$= 0.34, 0.66, and results in $\sigma/\sigma_{max}$=0.93. The two arc-shaped branches formed in this plot correspond to the two branches of triad possibilities in figure \[fig11\]a (triangles and squares, as discussed before). Behavior of $\sigma$ presented in figure \[fig11\]b matches exactly figure 11b of [@Bourget2013] except that in our case because of two horizontal top and bottom boundaries we get discrete modes only. ![Evolution of amplitudes of (a) the primary internal wave and (b) the six subharmonic pairs undergoing PSI corresponding to the case represented in figure \[fig11\]. Initial amplitudes of perturbation waves chosen as $A_{i1/i2}\|_{t=0}$=0.001 and up to mode 20th (which results in 38 PSI pairs) are considered in the numerical integration of . All waves undergo a modulation in time, and the maximum growth is obtained for perturbation wave of $\omega/\omega_0$=0.76. []{data-label="fig21"}](21_1 "fig:"){width="9cm"} (-260,70)[(a)]{}\ ![Evolution of amplitudes of (a) the primary internal wave and (b) the six subharmonic pairs undergoing PSI corresponding to the case represented in figure \[fig11\]. Initial amplitudes of perturbation waves chosen as $A_{i1/i2}\|_{t=0}$=0.001 and up to mode 20th (which results in 38 PSI pairs) are considered in the numerical integration of . All waves undergo a modulation in time, and the maximum growth is obtained for perturbation wave of $\omega/\omega_0$=0.76. []{data-label="fig21"}](21_2 "fig:"){width="8cm"} (-240,75)[(b)]{} Clearly figure \[fig11\]b suggests that the maximum of $\sigma$ is at $\omega/\omega_0$=0.5. But it is to be noted that this does not imply that the initial growth rate nor the long-time growth is highest at $\omega/\omega_0$=0.5. Specifically, at large $t$, it is in fact expected that the exponential term dominates the behavior. However, the linear stability analysis is not valid for large $t$, and a nonlinear analysis must be called. The linear theory is only applicable at initial times, and the initial growth rate is given by \[500\] \_[t=0]{}=s\_[1,2]{}A\_0\|\_[2,1]{}. Effect of $\sigma$ in the exponent, as can be seen from , is canceled by the coefficients in front, and therefore, the initial growth rate , is determined by $s_{1,2}$, and not $\sigma$. Behavior of normalized correct initial growth rate $s_i/s_{max}$ plotted against $\omega/\omega_0$ (figure \[fig11\]c) shows almost an opposite behavior when compared with that of $\sigma$. Most importantly, the highest value of correct initial growth rate $s_{max}$, does not occur at $\omega/\omega_0$=0.5, but at $\omega/\omega_0$=0.85. In fact, perturbation waves with frequencies near $\omega/\omega_0$=0.5 have almost the smallest initial growth rates[^4]. To determine the long term growth of waves involved in PSI, nonlinear stability analysis must be conducted. Since all pairs of perturbation waves (that satisfy resonance condition) simultaneously interact with the primary wave, a correct from of that takes into account the coupling between waves read &A\_0(t)/t=\_[i=1]{}\^[N]{}s\_[i0]{} A\_[i1]{}(t)A\_[i2]{}(t)\ &A\_[i1]{}(t)/t=s\_[i1]{} A\_0(t)\|[A]{}\_[i2]{}(t)\ &A\_[i2]{}(t)/t=s\_[i2]{} A\_0(t)\|[A]{}\_[i1]{}(t) where subscript $i$ denotes the $i$th subharmonic wave pair. From , it can be rigorously shown that \[1551\] { +\_[i=1]{}\^[N]{}} =0, which shows that our governing equation conserves energy. In other words, although energy is exchanged between a large number of waves, the total energy of the system is conserved and does not change with the time. As a case study, consider a primary internal wave of $\omega_0=0.8$ in a stratified water of $ah=5\times10^{-2}$ which is correspond to figure \[fig11\]. If we consider wave modes up to the mode 20th, that is 20 branches of the dispersion relation plot (dashed lines) in figure \[fig11\]a, then 38 pair of perturbation waves can be found to form resonance with our primary wave (6 of them are shown in figure \[fig11\]a). We assume all these perturbation waves have the same initial amplitude $A_i$=$1\times 10^{-3}$, which corresponds to a white-noise-like distribution of perturbation waves in the environment. Results of long time evolution of the primary wave, as well as few important perturbation waves, are shown in figure \[fig21\]a,b: at the stage $t<$6.8, amplitudes of some of perturbation waves increase, in cases substantially and even by orders of magnitudes, at the expense of a decrease in the energy of primary wave. Once the entire energy of the primary wave is depleted, then the process reverses and now energy flows back from (initially) perturbation waves to the primary wave. This modulation continues with a period of $T_i\sim$ 15 for primary wave and with the period of $T_p\sim 2T_i$ for perturbation waves. The most important feature of long-time evolution plots (figure \[fig21\]b) is the fact that the largest growth is observed at frequency $\omega/\omega_0=$0.76 ($A_i=$0.18 at t=21.7). The second and third largest growth are respectively at frequency $\omega/\omega_0=$0.83 ($A_i=$0.13 at t=21.7) and $\omega/\omega_0=$0.67 ($A_i=$0.12 at t=21.7); none at the frequency $\omega/\omega_0=$0.5. The perturbation wave associated with $\omega/\omega_0=$0.55 (which is closest to 0.5 in our database) gains a maximum value of $A_i=0.093$ which is barely 50% of the highest growth. ![Time-evolution of a primary internal wave of frequency $\omega=0.8$ with two perturbation waves of frequencies $\omega_{1,2}/\omega_0$=0.24,0.76. Direct simulations results of MITgcm (blue solid line) matches well with our analytical results using nonlinear stability theory though numerical integration of (red dashed line). For comparison, growth rate prediction by linear stability theory (black dash-dotted line) are also shown. Physical parameters for MITgcm simulations are $a=5\times10^{-5}$m$^{-1}$, $h=1000$m and initial amplitude is $1$m for the primary wave and $0.01$m for each of the two subharmonic waves.[]{data-label="fig6"}](cp_1 "fig:"){width="9cm"} ![Time-evolution of a primary internal wave of frequency $\omega=0.8$ with two perturbation waves of frequencies $\omega_{1,2}/\omega_0$=0.24,0.76. Direct simulations results of MITgcm (blue solid line) matches well with our analytical results using nonlinear stability theory though numerical integration of (red dashed line). For comparison, growth rate prediction by linear stability theory (black dash-dotted line) are also shown. Physical parameters for MITgcm simulations are $a=5\times10^{-5}$m$^{-1}$, $h=1000$m and initial amplitude is $1$m for the primary wave and $0.01$m for each of the two subharmonic waves.[]{data-label="fig6"}](cp_2 "fig:"){width="9cm"} ![Time-evolution of a primary internal wave of frequency $\omega=0.8$ with two perturbation waves of frequencies $\omega_{1,2}/\omega_0$=0.24,0.76. Direct simulations results of MITgcm (blue solid line) matches well with our analytical results using nonlinear stability theory though numerical integration of (red dashed line). For comparison, growth rate prediction by linear stability theory (black dash-dotted line) are also shown. Physical parameters for MITgcm simulations are $a=5\times10^{-5}$m$^{-1}$, $h=1000$m and initial amplitude is $1$m for the primary wave and $0.01$m for each of the two subharmonic waves.[]{data-label="fig6"}](cp_3 "fig:"){width="9cm"} We validate our analytical predictions with direct simulation run by MITgcm [@Marshall1997]. MITgcm is a finite-volume based open-source non-hydrostatic Navier-Stokes solver that is widely used for modeling stratified mediums [e.g. @Engqvist2004; @Klymak2010; @Lim2010; @Churaev2015; @Alford2015b]. We consider a rectangular domain with a periodic boundary condition on both ends in the x-direction, a free-slip wall at the bottom, and a free-slip rigid lid at the top. We use physical parameters $a=5\times10^{-5}$m$^{-1}$, $h=1000$m and consider a primary wave of mode one with $\omega_0=0.806$, $A_0\|_{t=0}$=1m, that resonates two subharmonic waves of frequencies $\omega_{1,2}/\omega_0=$0.24, 0.76, $A_{1,2}\|_{t=0}$=$0.01$m[^5]. The evolution of amplitude of the three waves over time obtained from the direct simulation of MITgcm (blue solid lines in figure \[fig6\]) compares well with our analytical results obtained from (red dashed curves). For comparison, we also show the results of linear stability analysis (black dash-dotted lines) on top of the other two curves. Clearly, at the initial stage of resonance and as long as the change in the amplitude of $A_0$ is small, linear stability analysis estimates the growth of perturbations with a satisfactory tolerance. But for later stages, linear theory over-predicts the growth. Conclusion ========== In summary, we showed, contrary to widely-accepted results drawn from linearized instability theory, that Parametric Subharmonic Instability (PSI) is strongest among a primary internal waves and perturbation waves at frequencies different from half the frequency of internal wave. Specifically, we proved that both the initial growth rate and the maximum amplitude reached by the (initially) perturbation waves are highest for subharmonic waves of low vertical modes with frequencies higher than $\omega_0/2$. It is straightforward to show that similar conclusion holds also in the spatial (or boundary value) problem, i.e. as waves propagate away from a source (e.g. a wavemaker) and interaction develops over the distance [c.f. e.g. @alam2009bragg]. Our finding suggests that the efficiency of converting internal wave energy from large scales to small scales through PSI may have been overestimated by previous studies, and dominant resonant waves may have been missed if sought at the half frequency. [10]{} Carl Wunsch and Raffaele Ferrari. Vertical mixing, energy, and the general circulation of the oceans. , 36:281–314, 2004. Chris Garrett. . , 301(5641):1858–9, 2003. Philip W. Boyd. . , 446(7139):989–991, 2007. Graham P. Harris. Chapman and Hall, 1986. Kevin G Lamb. . , 46:231–54, 2014. S. Sarkar and A. Scotti. From topographic internal gravity waves to turbulence. , 49(1):195–220, 2017. Likun Zhang and Harry L. Swinney. 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[^1]: reza.alam@berkeley.edu [^2]: Some literature [e.g. @Dauxois2018], to be precise, call this resonance a “Parametric Subharmonic instability or PSI” only if the disturbance waves have a frequency half of the frequency of primary wave. If the disturbance waves have a different frequency, then it is called “Triadic Resonant Instability or TRI”. [^3]: Our equation is in fact the same as (3.26) presented in [@Bourget2013] except for the pre-factor $1/4$ which is due to [@Bourget2013] considering a vertically unbounded domain whereas in our study we have vertical wall boundaries. Therefore, in our case, unlike [@Bourget2013], we must have standing waves in the vertical direction and also can only have discrete vertical wavenumbers $m$. [^4]: We would like to note that $s_i$ for some of the perturbation waves is negative and therefore the amplitude will first decrease. Usually in such cases, since the amplitude of perturbation waves are initially small, the amplitude quickly decreases to zero and then starts to grow on the negative side. The negative amplitude means that the wave gains a $\pi$-radian phase difference. [^5]: This specific pair of subharmonics are intentionally chosen since their wavenumber are $k_{1,2}/k_0$=-1/7,8/7 allowing us to use a periodic domain in the x-direction.
--- abstract: \| We investigate how different learning restrictions reduce learning power and how the different restrictions relate to one another. We give a complete map for nine different restrictions both for the cases of complete information learning and set-driven learning. This completes the picture for these well-studied delayable learning restrictions. A further insight is gained by different characterizations of conservative learning in terms of variants of cautious learning. Our analyses greatly benefit from general theorems we give, for example showing that learners with exclusively delayable restrictions can always be assumed total. author: - Timo Kötzing - Raphaela Palenta title: \| A Map of Update Constraints\ in Inductive Inference --- Introduction ============ \[sec:Introduction\] This paper is set in the framework of inductive inference, a branch of (algorithmic) learning theory. This branch analyzes the problem of algorithmically learning a description for a formal language (a computably enumerable subset of the set of natural numbers) when presented successively all and only the elements of that language. For example, a learner $h$ might be presented more and more even numbers. After each new number, $h$ outputs a description for a language as its conjecture. The learner $h$ might decide to output a program for the set of all multiples of $4$, as long as all numbers presented are divisible by $4$. Later, when $h$ sees an even number not divisible by $4$, it might change this guess to a program for the set of all multiples of $2$. Many criteria for deciding whether a learner $h$ is successful on a language $L$ have been proposed in the literature. Gold, in his seminal paper [@Gol:j:67], gave a first, simple learning criterion, ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learning[^1], where a learner is successful iff, on every text for $L$ (listing of all and only the elements of $L$) it eventually stops changing its conjectures, and its final conjecture is a correct description for the input sequence. Trivially, each single, describable language $L$ has a suitable constant function as a ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learner (this learner constantly outputs a description for $L$). Thus, we are interested in analyzing for which classes of languages ${\mathcal{L}}$ there is a single learner $h$ learning each member of ${\mathcal{L}}$. This framework is also sometimes known as language learning in the limit and has been studied extensively, using a wide range of learning criteria similar to ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learning (see, for example, the textbook [@Jai-Osh-Roy-Sha:b:99:stl2]). A wealth of learning criteria can be derived from ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learning by adding restrictions on the intermediate conjectures and how they should relate to each other and the data. For example, one could require that a conjecture which is consistent with the data must not be changed; this is known as conservative learning and known to restrict what classes of languages can be learned ([@Ang:j:80:lang-pos-data], we use ${\mathbf{Conv}}$ to denote the restriction of conservative learning). Additionally to conservative learning, the following learning restrictions are considered in this paper (see Section \[sec:LearningCriteria\] for a formal definition of learning criteria including these learning restrictions). In cautious learning (${\mathbf{Caut}}$, [@Osh-Sto-Wei:j:82:strategies]) the learner is not allowed to ever give a conjecture for a strict subset of a previously conjectured set. In non-U-shaped learning (${\mathbf{NU}}$, [@Bal-Cas-Mer-Ste-Wie:j:08]) a learner may never semantically abandon a correct conjecture; in strongly non-U-shaped learning (${\mathbf{SNU}}$, [@Cas-Moe:j:11:optLan]) not even syntactic changes are allowed after giving a correct conjecture. In decisive learning (${\mathbf{Dec}}$, [@Osh-Sto-Wei:j:82:strategies]), a learner may never (semantically) return to a semantically abandoned conjecture; in strongly decisive learning (${\mathbf{SDec}}$, [@Koe:c:14:stacs]) the learner may not even (semantically) return to syntactically abandoned conjectures. Finally, a number of monotonicity requirements are studied ([@Jan:j:91; @Wie:c:91; @Lan-Zeu:c:93]): in strongly monotone learning (${\mathbf{SMon}}$) the conjectured sets may only grow; in monotone learning (${\mathbf{Mon}}$) only incorrect data may be removed; and in weakly monotone learning (${\mathbf{WMon}}$) the conjectured set may only grow while it is consistent. The main question is now whether and how these different restrictions reduce learning power. For example, non-U-shaped learning is known not to restrict the learning power [@Bal-Cas-Mer-Ste-Wie:j:08], and the same for strongly non-U-shaped learning [@Cas-Moe:j:11:optLan]; on the other hand, decisive learning is restrictive [@Bal-Cas-Mer-Ste-Wie:j:08]. The relations of the different monotone learning restriction were given in [@Lan-Zeu:c:93]. Conservativeness is long known to restrict learning power [@Ang:j:80:lang-pos-data], but also known to be equivalent to weakly monotone learning [@Kin-Ste:j:95:mon; @Jai-Sha:j:98]. Cautious learning was shown to be a restriction but not when added to conservativeness in [@Osh-Sto-Wei:j:82:strategies; @Osh-Sto-Wei:b:86:stl], similarly the relationship between decisive and conservative learning was given. In Exercise 4.5.4B of [@Osh-Sto-Wei:b:86:stl] it is claimed (without proof) that cautious learners cannot be made conservative; we claim the opposite in Theorem \[thm:CautVarConv\]. This list of previously known results leaves a number of relations between the learning criteria open, even when adding trivial inclusion results (we call an inclusion trivial iff it follows straight from the definition of the restriction without considering the learning model, for example strongly decisive learning is included in decisive learning; formally, trivial inclusion is inclusion on the level of learning restrictions as predicates, see Section \[sec:LearningCriteria\]). With this paper we now give the complete picture of these learning restrictions. The result is shown as a map in Figure \[fig:GoldRelations\]. A solid black line indicates a trivial inclusion (the lower criterion is included in the higher); a dashed black line indicates inclusion (which is not trivial). A gray box around criteria indicates equality of (learning of) these criteria. A different way of depicting the same results is given in Figure \[fig:partialorderGold\] (where solid lines indicate any kind of inclusion). Results involving monotone learning can be found in Section \[sec:Monotone\], all others in Section \[sec:Caution\]. at (-5,-1) [${\mathbf{G}}$]{}; (nothing) at (0,0) [T ${\mathbf{NU}}$ ${\mathbf{SNU}}$]{}; (dec) at (0,-1.5) [${\mathbf{Dec}}$]{}; (sdec) at (0,-3) [${\mathbf{SDec}}$]{}; (mon) at (2.5,-4.5) [${\mathbf{Mon}}$]{}; (wmon) at (-2.5,-4.5) [${\mathbf{Caut}}$ ${\mathbf{WMon}}$ ${\mathbf{Conv}}$]{}; (smon) at (0,-6) [${\mathbf{SMon}}$]{}; (nothing) – (dec); (dec) – (sdec); (sdec) – (wmon); (sdec) – (mon); (mon) – (smon); (wmon) – (smon); For the important restriction of conservative learning we give the characterization of being equivalent to cautious learning. Furthermore, we show that even two weak versions of cautiousness are equivalent to conservative learning. Recall that cautiousness forbids to return to a strict subset of a previously conjectured set. If we now weaken this restriction to forbid to return to finite subsets of a previously conjectured set we get a restriction still equivalent to conservative learning. If we forbid to go down to a correct conjecture, effectively forbidding to ever conjecture a superset of the target language, we also obtain a restriction equivalent to conservative learning. On the other hand, if we weaken it so as to only forbid going to infinite subsets of previously conjectured sets, we obtain a restriction equivalent to no restriction. These results can be found in Section \[sec:Caution\]. In set-driven learning [@Wex-Cul:b:80] the learner does not get the full information about what data has been presented in what order and multiplicity; instead, the learner only gets the set of data presented so far. For this learning model it is known that, surprisingly, conservative learning is no restriction [@Kin-Ste:j:95:mon]! We complete the picture for set driven learning by showing that set-driven learners can always be assumed conservative, strongly decisive and cautious, and by showing that the hierarchy of monotone and strongly monotone learning also holds for set-driven learning. The situation is depicted in Figure \[fig:hierarchySetdriven\]. These results can be found in Section \[sec:SetDriven\]. at (-5,-1) [${\mathbf{Sd}}$]{}; (nothing) at (0,0) ; (mon) at (0,-1.5) [${\mathbf{Mon}}$]{}; (smon) at (0,-3) [${\mathbf{SMon}}$]{}; (nothing) – (mon); (mon) – (smon); Techniques ---------- A major emphasis of this paper is on the techniques used to get our results. These techniques include specific techniques for specific problems, as well as general theorems which are applicable in many different settings. The general techniques are given in Section \[sec:techniques\], one main general result is as follows. It is well-known that any ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learner $h$ learning a language $L$ has a locking sequence, a sequence $\sigma$ of data from $L$ such that, for any further data from $L$, the conjecture does not change and is correct. However, there might be texts such that no initial sequence of the text is a locking sequence. We call a learner such that any text for a target language contains a locking sequence strongly locking, a property which is very handy to have in many proofs. Fulk [@Ful:j:90:prudence] showed that, without loss of generality, a ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learner can be assumed strongly locking, as well as having many other useful properties (we call this the Fulk normal form, see Definition \[defn:FulkNormalForm\]). For many learning criteria considered in this paper it might be too much to hope for that all of them allow for learning by a learner in Fulk normal form. However, we show in Corollary \[cor:SinkLocking\] that we can always assume our learners to be strongly locking, total, and what we call syntactically decisive, never syntactically returning to syntactically abandoned hypotheses. The main technique we use to show that something is decisively learnable, for example in Theorem \[thm:NatnumSDec\], is what we call poisoning of conjectures. In the proof of Theorem \[thm:NatnumSDec\] we show that a class of languages is decisively learnable by simulating a given monotone learner $h$, but changing conjectures as follows. Given a conjecture $e$ made by $h$, if there is no mind change in the future with data from conjecture $e$, the new conjecture is equivalent to $e$; otherwise it is suitably changed, poisoned, to make sure that the resulting learner is decisive. This technique was also used in [@Cas-Koe:c:10:colt] to show strongly non-U-shaped learnability. Finally, for showing classes of languages to be not (strongly) decisively learnable, we adapt a technique known in computability theory as a “priority argument” (note, though, that we do not deal with oracle computations). We use this technique to reprove that decisiveness is a restriction to ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learning (as shown in [@Bal-Cas-Mer-Ste-Wie:j:08]), and then use a variation of the proof to show that strongly decisive learning is a restriction to decisive learning. Mathematical Preliminaries ========================== \[sec:MathPrelim\] Unintroduced notation follows [@Rog:b:87], a textbook on computability theory. ${\mathbb{N}}$ denotes the set of natural numbers, $\{0,1,2,\ldots\}$. The symbols $\subseteq$, $\subset$, $\supseteq$, $\supset$ respectively denote the subset, proper subset, superset and proper superset relation between sets; $\setminus$ denotes set difference. $\emptyset$ and ${\lambda}$ denote the empty set and the empty sequence, respectively. The quantifier $\forall^\infty x$ means “for all but finitely many $x$”. With ${\mathrm{dom}}$ and ${\mathrm{range}}$ we denote, respectively, domain and range of a given function. Whenever we consider tuples of natural numbers as input to a function, it is understood that the general coding function $\langle \cdot, \cdot \rangle$ is used to code the tuples into a single natural number. We similarly fix a coding for finite sets and sequences, so that we can use those as input as well. For finite sequences, we suppose that for any $\sigma \subseteq \tau$ we have that the code number of $\sigma$ is at most the code number of $\tau$. We let ${{\mathbb{S}\mathrm{eq}}}$ denote the set of all (finite) sequences, and ${{\mathbb{S}\mathrm{eq}}}_{\leq t}$ the (finite) set of all sequences of length at most $t$ using only elements $\leq t$. If a function $f$ is not defined for some argument $x$, then we denote this fact by $f(x){\mathclose{\hbox{$\uparrow$}}}$, and we say that $f$ on $x$ diverges; the opposite is denoted by $f(x){\mathclose{\hbox{$\downarrow$}}}$, and we say that $f$ on $x$ converges. If $f$ on $x$ converges to $p$, then we denote this fact by $f(x){\mathclose{\hbox{$\downarrow$}}}= p$. We let ${\mathfrak{P}}$ denote the set of all partial functions ${\mathbb{N}}\rightarrow {\mathbb{N}}$ and ${\mathfrak{R}}$ the set of all total such functions. ${\mathcal{P}}$ and ${\mathcal{R}}$ denote, respectively, the set of all partial computable and the set of all total computable functions (mapping ${\mathbb{N}}\rightarrow {\mathbb{N}}$). We let $\varphi$ be any fixed acceptable programming system for ${\mathcal{P}}$ (an acceptable programming system could, for example, be based on a natural programming language such as C or Java, or on Turing machines). Further, we let $\varphi_p$ denote the partial computable function computed by the $\varphi$-program with code number $p$. A set $L \subseteq {\mathbb{N}}$ is computably enumerable ([`ce`]{}) iff it is the domain of a computable function. Let ${\mathcal{E}}$ denote the set of all [`ce`]{} sets. We let $W$ be the mapping such that $\forall e: W(e) = {\mathrm{dom}}(\varphi_e)$. For each $e$, we write $W_e$ instead of $W(e)$. $W$ is, then, a mapping from ${\mathbb{N}}$ onto ${\mathcal{E}}$. We say that $e$ is an index, or program, (in $W$) for $W_e$. We let $\Phi$ be a Blum complexity measure associated with $\varphi$ (for example, for each $e$ and $x$, $\Phi_e(x)$ could denote the number of steps that program $e$ takes on input $x$ before terminating). For all $e$ and $t$ we let $W_e^t = {\{x \leq t \; \| \; \Phi_e(x) \leq t\}}$ (note that a complete description for the finite set $W_e^t$ is computable from $e$ and $t$). The symbol $\#$ is pronounced pause and is used to symbolize “no new input data” in a text. For each (possibly infinite) sequence $q$ with its range contained in ${\mathbb{N}}\cup \{\#\}$, let ${\mathrm{content}}(q) = ({\mathrm{range}}(q) \setminus \{\#\}$). By using an appropriate coding, we assume that $?$ and $\#$ can be handled by computable functions. For any function $T$ and all $i$, we use $T[i]$ to denote the sequence $T(0)$, …, $T(i-1)$ (the empty sequence if $i=0$ and undefined, if any of these values is undefined). We will use Case’s Operator Recursion Theorem ([ORT]{}), providing infinitary self-and-other program reference [@Cas:j:74; @Cas:j:94:self; @Jai-Osh-Roy-Sha:b:99:stl2]. [ORT]{} itself states that, for all operators $\Theta$ there are $f$ with $\forall z: \Theta(\varphi_z) = \varphi_{f(z)}$ and $e \in {\mathcal{R}}$, $$\label{eq:ORT} \forall a,b: \varphi_{e(a)}(b) = \Theta(e)(a,b).$$ Learning Criteria ----------------- \[sec:LearningCriteria\] In this section we formally introduce our setting of learning in the limit and associated learning criteria. We follow [@Koe:th:09] in its “building-blocks” approach for defining learning criteria. A learner is a partial computable function $h \in {\mathcal{P}}$. A language is a [`ce`]{} set $L \subseteq {\mathbb{N}}$. Any total function $T: {\mathbb{N}}\rightarrow {\mathbb{N}}\cup \{\#\}$ is called a text. For any given language $L$, a text for $L$ is a text $T$ such that ${\mathrm{content}}(T) = L$. Initial parts of this kind of text is what learners usually get as information. An interaction operator is an operator $\beta$ taking as arguments a function $h$ (the learner) and a text $T$, and that outputs a function $p$. We call $p$ the learning sequence (or sequence of hypotheses) of $h$ given $T$. Intuitively, $\beta$ defines how a learner can interact with a given text to produce a sequence of conjectures. We define the interaction operators ${\mathbf{G}}$, ${\mathbf{Psd}}$ (partially set-driven learning, [@Sch:th:84]) and ${\mathbf{Sd}}$ (set-driven learning, [@Wex-Cul:b:80]) as follows. For all learners $h$, texts $T$ and all $i$, $$\begin{aligned} {\mathbf{G}}(h,T)(i) & = & h(T[i]);\\ {\mathbf{Psd}}(h,T)(i) & = & h({\mathrm{content}}(T[i]),i);\\ {\mathbf{Sd}}(h,T)(i) & = & h({\mathrm{content}}(T[i])).\end{aligned}$$ Thus, in set-driven learning, the learner has access to the set of all previous data, but not to the sequence as in ${\mathbf{G}}$-learning. In partially set-driven learning, the learner has the set of data and the current iteration number. Successful learning requires the learner to observe certain restrictions, for example convergence to a correct index. These restrictions are formalized in our next definition. A learning restriction is a predicate $\delta$ on a learning sequence and a text. We give the important example of explanatory learning (${\mathbf{Ex}}$, [@Gol:j:67]) defined such that, for all sequences of hypotheses $p$ and all texts $T$, $$\begin{aligned} {\mathbf{Ex}}(p,T) \Leftrightarrow & \; p \mbox{ total }\wedge [\exists n_0 \forall n \geq n_0: p(n) = p(n_0) \wedge W_{p(n_0)} = {\mathrm{content}}(T)].\end{aligned}$$ Furthermore, we formally define the restrictions discussed in Section \[sec:Introduction\] in Figure \[fig:DefinitionsOfLearningRestrictions\] (where we implicitly require the learning sequence $p$ to be total, as in ${\mathbf{Ex}}$-learning; note that this is a technicality without major importance). $$\begin{aligned} {\mathbf{Conv}}(p,T) \Leftrightarrow & \; [\forall i: {\mathrm{content}}(T[i+1]) \subseteq W_{p(i)} \Rightarrow p(i) = p(i+1)];\\ {\mathbf{Caut}}(p,T) \Leftrightarrow & \; [\forall i,j: W_{p(i)} \subset W_{p(j)} \Rightarrow i < j];\\ {\mathbf{NU}}(p,T) \Leftrightarrow & \; [\forall i,j,k: i \leq j \leq k \; \wedge \; W_{p(i)} = W_{p(k)} = {\mathrm{content}}(T) \Rightarrow W_{p(j)} = W_{p(i)}];\\ {\mathbf{Dec}}(p,T) \Leftrightarrow & \; [\forall i,j,k: i \leq j \leq k \; \wedge \; W_{p(i)} = W_{p(k)} \Rightarrow W_{p(j)} = W_{p(i)}];\\ {\mathbf{SNU}}(p,T) \Leftrightarrow & \; [\forall i,j,k: i \leq j \leq k \; \wedge \; W_{p(i)} = W_{p(k)} = {\mathrm{content}}(T) \Rightarrow p(j) = p(i)];\\ {\mathbf{SDec}}(p,T) \Leftrightarrow & \; [\forall i,j,k: i \leq j \leq k \; \wedge \; W_{p(i)} = W_{p(k)} \Rightarrow p(j) = p(i)];\\ {\mathbf{SMon}}(p,T) \Leftrightarrow & \; [\forall i,j: i< j \Rightarrow W_{p(i)} \subseteq W_{p(j)}];\\ {\mathbf{Mon}}(p,T) \Leftrightarrow & \; [\forall i,j: i< j \Rightarrow W_{p(i)} \cap{\mathrm{content}}(T) \subseteq W_{p(j)}\cap{\mathrm{content}}(T)];\\ {\mathbf{WMon}}(p,T) \Leftrightarrow & \; [\forall i,j: i < j \wedge {\mathrm{content}}(T[j]) \subseteq W_{p(i)} \Rightarrow W_{p(i)} \subseteq W_{p(j)}].\end{aligned}$$ A variant on decisiveness is syntactic decisiveness, ${\mathbf{SynDec}}$, a technically useful property defined as follows. $${\mathbf{SynDec}}(p,T) \Leftrightarrow [\forall i,j,k: i \leq j \leq k \; \wedge \; p(i) = p(k) \Rightarrow p(j) = p(i)].$$ We combine any two sequence acceptance criteria $\delta$ and $\delta'$ by intersecting them; we denote this by juxtaposition (for example, all the restrictions given in Figure \[fig:DefinitionsOfLearningRestrictions\] are meant to be always used together with ${\mathbf{Ex}}$). With $\mathbf{T}$ we denote the always true sequence acceptance criterion (no restriction on learning). A learning criterion is a tuple $({\mathcal{C}},\beta,\delta)$, where ${\mathcal{C}}$ is a set of learners (the admissible learners), $\beta$ is an interaction operator and $\delta$ is a learning restriction; we usually write ${\mathcal{C}}{\mathbf{Txt}^{}}\beta\delta$ to denote the learning criterion, omitting ${\mathcal{C}}$ in case of ${\mathcal{C}}= {\mathcal{P}}$. We say that a learner $h \in {\mathcal{C}}$ ${\mathcal{C}}{\mathbf{Txt}^{}}\beta\delta$-learns a language $L$ iff, for all texts $T$ for $L$, $\delta(\beta(h,T),T)$. The set of languages ${\mathcal{C}}{\mathbf{Txt}^{}}\beta\delta$-learned by $h \in {\mathcal{C}}$ is denoted by ${\mathcal{C}}{\mathbf{Txt}^{}}\beta\delta(h)$. We write $[{\mathcal{C}}{\mathbf{Txt}^{}}\beta\delta]$ to denote the set of all ${\mathcal{C}}{\mathbf{Txt}^{}}\beta\delta$-learnable classes (learnable by some learner in ${\mathcal{C}}$). Delayable Learning Restrictions =============================== \[sec:techniques\] In this section we present technically useful results which show that learners can always be assumed to be in some normal form. We will later always assume our learners to be in the normal form established by Corollary \[cor:SinkLocking\], the main result of this section. We start with the definition of delayable. Intuitively, a learning criterion $\delta$ is delayable iff the output of a hypothesis can be arbitrarily (but not indefinitely) delayed. \[defn:Delayable\] Let $\vec{R}$ be the set of all non-decreasing $r: {\mathbb{N}}\rightarrow {\mathbb{N}}$ with infinite limit inferior, i.e. for all $m$ we have $\forall^\infty n: r(n) \geq m$. A learning restriction $\delta$ is delayable iff, for all texts $T$ and $T'$ with ${\mathrm{content}}(T) = {\mathrm{content}}(T')$, all $p$ and all $r \in \vec{R}$, if $(p,T) \in \delta$ and $\forall n: {\mathrm{content}}(T[r(n)]) \subseteq {\mathrm{content}}(T'[n])$, then $(p \circ r,T') \in \delta$. Intuitively, as long as the learner has at least as much data as was used for a given conjecture, then the conjecture is permissible. Note that this condition holds for $T = T'$ if $\forall n: r(n) \leq n$. Note that the intersection of two delayable learning criteria is again delayable and that all learning restrictions considered in this paper are delayable. As the name suggests, we can apply delaying tricks (tricks which delay updates of the conjecture) in order to achieve fast computation times in each iteration (but of course in the limit we still spend an infinite amount of time). This gives us equally powerful but total learners, as shown in the next theorem. While it is well-known that, for many learning criteria, the learner can be assumed total, this theorem explicitly formalizes conditions under which totality can be assumed (note that there are also natural learning criteria where totality cannot be assumed, such as consistent learning [@Jai-Osh-Roy-Sha:b:99:stl2]). \[thm:Total\] For any delayable learning restriction $\delta$, we have \[${\mathbf{Txt}^{}}{\mathbf{G}}\delta$\] = \[${\mathcal{R}}{\mathbf{Txt}^{}}{\mathbf{G}}\delta$\]. Let $h$ be a ${\mathbf{Txt}^{}}{\mathbf{G}}\delta$-learner and $e$ such that $\varphi_e = h$. We define a function $M$ such that, for all $\sigma$, $$M(\sigma) = \{\sigma' \subseteq \sigma\ \|\ \Phi_e(\sigma') \leq \|\sigma\|\} \cup \{{\lambda}\}.$$ We let $h'$ be the learner such that, for all $\sigma$, $$h'(\sigma) = h(\max(M(\sigma)).$$ As $h$ is required to have only total learning sequences, we have that $h({\lambda}){\mathclose{\hbox{$\downarrow$}}}$; thus, $h'$ is total computable using that $M$ is total computable. Let ${\mathcal{L}}= {\mathbf{Txt}^{}}{\mathbf{G}}\delta(h)$, $L \in {\mathcal{L}}$ and let $T$ be a text for $L$. Let $r(n) = \|\max(M(T[n]))\|$. Then we have, for all $n$, $h'(T[n]) = h(T[r(n)])$. Thus, if we show that $r \in \vec{R}$ we get that $h'$ ${\mathbf{Txt}^{}}{\mathbf{G}}\delta$-learns $L$ from $T$ using $\delta$ delayable. From the definition of $M$ we get that $r$ is non-decreasing and, for all $n$, $r(n) \leq n$. For any given $m$ there are $n,n'$ with $n' \geq n \geq m$ such that $\Phi_e(T[n]) \leq n'$. Thus, we have $r(n') \geq m$ and, as $r$ is non-decreasing, we get $\forall^\infty n : r(n) \geq m$ as desired. Next we define another useful property, which can always be assumed for delayable learning restrictions. \[defn:StronglyLocking\] A locking sequence for a learner $h$ on a language $L$ is any finite sequence $\sigma$ of elements from $L$ such that $h(\sigma)$ is a correct hypothesis for $L$ and, for sequences $\tau$ with elements from $L$, $h(\sigma \diamond \tau) = h(\sigma)$[@Blu-Blu:j:75]. It is well known that every learner $h$ learning a language $L$ has a locking sequence on $L$. We say that a learning criterion $I$ allows for strongly locking learning iff, for each $I$-learnable class of languages ${\mathcal{L}}$ there is a learner $h$ such that $h$ $I$-learns ${\mathcal{L}}$ and, for each $L \in {\mathcal{L}}$ and any text $T$ for $L$, there is an $n$ such that $T[n]$ is a locking sequence of $h$ on $L$ (we call such a learner $h$ strongly locking). With this definition we can give the following theorem. \[thm:DelayStronglyLocking\] Let $\delta$ be a delayable learning criterion. Then ${\mathcal{R}}{\mathbf{Txt}^{}}{\mathbf{G}}\delta{\mathbf{Ex}}$ allows for strongly locking learning. Let ${\mathcal{L}}$ and $h \in {\mathcal{R}}$ be such that $h$ ${\mathcal{R}}{\mathbf{Txt}^{}}{\mathbf{G}}\delta{\mathbf{Ex}}$-learns ${\mathcal{L}}$. We define a set $M(\rho, \sigma)$, for all $\rho$ and $\sigma$ such that $$M(\rho,\sigma) = {\{\tau \; \| \; \|\tau\| \leq \|\sigma\| \wedge {\mathrm{content}}(\tau) \subseteq {\mathrm{content}}(\sigma) \wedge h(\rho \diamond \tau) \neq h(\rho)\}}.$$ Thus, $M$ contains sequences with elements from ${\mathrm{content}}(\sigma)$ such that $h$ makes a mind change on $\sigma$ extended with such a sequence. Additionally, we define a function $f$ recursively such that, for all $\sigma, x$ and $T$, $$\begin{aligned} f(\emptyset) & = & \emptyset; \\ f(\sigma \diamond x) & = & \begin{cases} f(\sigma), &\mbox{if }M(f(\sigma),\sigma \diamond x) = \emptyset;\\ f(\sigma) \diamond \min(M(f(\sigma),\sigma \diamond x)) \diamond \sigma, &\mbox{otherwise;} \end{cases}\\ f(T) & = & \lim\limits_{n \rightarrow \infty}{f(T[n])}.\end{aligned}$$ Intuitively, $f$ searches for longer and longer sequences which are not locking sequences. We let $h'$ be the learner such that, for all $\sigma$, $$h'(\sigma) = h(f(\sigma)).$$ Note that $f$ is total (as $h$ is total), and thus $h'$ is total. Let $L \in {\mathcal{L}}$ and $T$ be a text for $L$. We will show now that $f(T)$ converges to a finite sequence. We have that $f(T)$ is finite. By way of contradiction, suppose that $f(T)$ is infinite, and let $T' = f(T)$. As $f(T)$ is infinite we get, for every $n$, an $m > n$ such that $f(T[m]) \neq f(T[n])$. Then we have $${\mathrm{content}}(T[n]) \subseteq {\mathrm{content}}(f(T[m])).$$ As this holds for every $n$, we get ${\mathrm{content}}(T) \subseteq {\mathrm{content}}(f(T))$. From the construction of $f$ we know that ${\mathrm{content}}(f(T)) \subseteq {\mathrm{content}}(T)$. Thus, $f(T)$ is a text for $L$. From the construction of $M$ we get that $h$ does not ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learns $L$ from $T'$ as $h$ changes infinitely often its mind, a contradiction. Next, we will show that $h'$ converges on $T$ and $h'$ is strongly locking. As $f(T)$ is finite, there is $n_0$ such that, for all $n \geq n_0$, $$\begin{aligned} f(T[n]) = f(T[n_0]).\end{aligned}$$ As $f(T)$ converges to $f(T[n_0])$, we get from the construction of $M$ that $f(T[n_0])$ is a locking sequence of $h$ on $L$. Therefore we get that, for all $\tau \in {{\mathbb{S}\mathrm{eq}}}(L)$, $$f(T[n_0]) = f(T[n_0] \diamond \tau)$$ and therefore $$h'(T[n_0]) = h'(T[n_0] \diamond \tau).$$ Thus, $h'$ is strongly locking and converges on $T$. To show that $h'$ fulfills the $\delta$-restriction, we let $T' = f(T[n_0]) \diamond T$ be a text for $L$ starting with $f(T[n_0])$. Let $r$ be such that $$r(n) = \begin{cases} \|f(T[n])\|, & \text{if } n \leq n_0; \\ r(n_0) + n - n_0, & \text{otherwise.} \end{cases}$$We now show $$h(T'[r(n)]) = h'(T[n]).$$ Case 1: $n \leq n_0$. Then we get $$\begin{aligned} h(T'[r(n)]) &= h(T'[\|f(T[n])\|]) \\ &= h(f(T[n])) &\text{as $T' = f(T[n_0]) \diamond T$} \\ &= h'(T[n]).\end{aligned}$$ Case 2: $n > n_0$. Then we get $$\begin{aligned} h(T'[r(n)]) &= h(T'[r(n_0) + n - n_0])\\ & = h(T'[\|f(T[n_0])\| + n - n_0]) \\ &= h(f(T[n_0])\diamond T[n-n_0]) &\text{as $T' = f(T[n_0]) \diamond T$}\\ &= h(f(T[n_0])) &\text{\hspace{-10mm}as $f(T[n_0])$ is a locking sequence of $h$} \\ &= h'(T[n]).\end{aligned}$$ Thus, all that remains to be shown is that $r \in \vec{R}$. Obviously, $r$ is non-decreasing. Especially, we have that $r$ is strongly monotone increasing for all $n > n_0$. Thus we have, for all $m$, $\forall^\infty n : r(n) \geq m$. Finally we show that ${\mathrm{content}}(T'[r(n)]) \subseteq {\mathrm{content}}(T[n])$. From the construction of $f$ we have, for all $n \leq n_0$, ${\mathrm{content}}(T'[\|f(T[n])\|]) \subseteq {\mathrm{content}}(T[n])$. From the construction of $r$ and $T'$ we get that, for all $n > n_0$, $T'(r(n)) = T(n)$. Thus we get, for all $n$, ${\mathrm{content}}(T'[r(n)]) \subseteq {\mathrm{content}}(T[n])$. Next we define semantic and pseudo-semantic restrictions introduced in [@Koe:c:14:stacs]. Intuitively, semantic restrictions allow for replacing hypotheses by equivalent ones; pseudo-sematic restrictions allow the same, as long as no new mind changes are introduced. \[defn:SemanticRestriction\] For all total functions $p \in {\mathfrak{P}}$, we let $$\begin{aligned} {\mathrm{Sem}}(p) & = & {\{ p' \in {\mathfrak{P}}\; \| \; \forall i: W_{p(i)} = W_{p'(i)}\}};\\ {\mathrm{Mc}}(p) & = & {\{ p' \in {\mathfrak{P}}\; \| \; \forall i: p'(i) \neq p'(i+1) \Rightarrow p(i) \neq p(i+1)\}}.\end{aligned}$$ A sequence acceptance criterion $\delta$ is said to be a semantic restriction iff, for all $(p,q) \in \delta$ and $p' \in {\mathrm{Sem}}(p)$, $(p',q) \in \delta$. A sequence acceptance criterion $\delta$ is said to be a pseudo-semantic restriction iff, for all $(p,q) \in \delta$ and $p' \in {\mathrm{Sem}}(p) \cap {\mathrm{Mc}}(p)$, $(p',q) \in \delta$. We note that the intersection of two (pseudo-) semantic learning restrictions is again (pseudo-) semantic. All learning restrictions considered in this paper are pseudo-semantic, and all except ${\mathbf{Conv}}$, ${\mathbf{SNU}}$, ${\mathbf{SDec}}$ and ${\mathbf{Ex}}$ are semantic. The next lemma shows that, for every pseudo-semantic learning restriction, learning can be done syntactically decisively. \[lem:SynDec\] Let $\delta$ be a pseudo-semantic learning criterion. Then we have $$[{\mathcal{R}}{\mathbf{Txt}^{}}{\mathbf{G}}\delta] = [{\mathcal{R}}{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{SynDec}}\delta].$$ Let a ${\mathbf{Txt}^{}}{\mathbf{G}}\delta$-learner $h \in {\mathcal{R}}$ be given. We define a learner $h' \in {\mathcal{R}}$ such that, for all $\sigma$, $$h'(\sigma) = \begin{cases} {\mathrm{pad}}(h(\sigma),\sigma), & \text{if } \sigma = \emptyset \text{ or } h(\sigma) \neq h(\sigma^-); \\ h'(\sigma^-), & \text{otherwise.} \end{cases}$$ The correctness of this construction is straightforward to check. As ${\mathbf{SynDec}}$ is a delayable learning criterion, we get the following corollary by taking Theorems \[thm:Total\] and \[thm:DelayStronglyLocking\] and Lemma \[lem:SynDec\] together. We will always assume our learners to be in this normal form in this paper. \[cor:SinkLocking\] Let $\delta$ be pseudo-semantic and delayable. Then ${\mathbf{Txt}^{}}{\mathbf{G}}\delta{\mathbf{Ex}}$ allows for strongly locking learning by a syntactically decisive total learner. Fulk showed that any ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learner can be (effectively) turned into an equivalent learner with many useful properties, including strongly locking learning [@Ful:j:90:prudence]. One of the properties is called order-independence, meaning that on any two texts for a target language the learner converges to the same hypothesis. Another property is called rearrangement-independence, where a learner $h$ is rearrangement-independent if there is a function $f$ such that, for all sequences $\sigma$, $h(\sigma) = f({\mathrm{content}}(\sigma),\|\sigma\|)$ (intuitively, rearrangement independence is equivalent to the existence of a partially set-driven learner for the same language). We define the collection of all the properties which Fulk showed a learner can have to be the Fulk normal form as follows. \[defn:FulkNormalForm\] We say a ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learner $h$ is in Fulk normal form if $(1) - (5)$ hold. 1. $h$ is order-independent. 2. $h$ is rearrangement-independent. 3. If $h$ ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learns a language $L$ from some text, then $h$ ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learns $L$. 4. If there is a locking sequence of $h$ for some $L$, then $h$ ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learns $L$. 5. For all ${\mathcal{L}}\in {\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}(h)$, $h$ is strongly locking on ${\mathcal{L}}$. The following theorem is somewhat weaker than what Fulk states himself. \[thm:FulkNormalForm\] Every ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learnable set of languages has a ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learner in Fulk normal form. Full-Information Learning ========================= \[sec:Caution\] In this section we consider various versions of cautious learning and show that all of our variants are either no restriction to learning, or equivalent to conservative learning as is shown in Figure \[fig:GoldCautiousV\]. Additionally, we will show that every cautious ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learnable language is conservative ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learnable which implies that $[{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Conv}}{\mathbf{Ex}}]$, $[{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{WMon}}{\mathbf{Ex}}]$ and $[{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Caut}}{\mathbf{Ex}}]$ are equivalent. Last, we will separate these three learning criteria from strongly decisive ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learning and show that $[{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{SDec}}{\mathbf{Ex}}]$ is a proper superset. \[thm:ConvInSDec\] We have that any conservative learner can be assumed cautious and strongly decisive, i.e. $$[\textbf{TxtGConvEx}] = [\textbf{TxtGConvSDecCautEx}].$$ Let $h \in {\mathcal{R}}$ and $\mathcal{L}$ be such that $h$ TxtGConvEx-learns $\mathcal{L}$. We define, for all $\sigma$, a set $M(\sigma)$ as follows $$M(\sigma) = \{\tau\ \|\ \tau \subseteq \sigma\ \land\ \forall x \in {\mathrm{content}}(\tau) : \Phi_{h(\tau)}(x) \leq \left\|\sigma\right\| \}.$$ We let $$\forall \sigma : h'(\sigma) = h(\max(M(\sigma))).$$ Let $T$ be a text for a language $L \in \mathcal{L}$. We first show that $h'$ TxtGEx-learns $L$ from the text $T$. As $h$ TxtGConvEx-learns $L$, there are $n$ and $e$ such that $\forall n' \geq n : h(T[n]) = h(T[n']) = e$ and $W_e = L$. Thus, there is $m \geq n$ such that $\forall x \in {\mathrm{content}}(T[n]) : \Phi_{h(T[n])}(x) \leq m$ and therefore $\forall m' \geq m : h'(T[m])=h'(T[m'])=e$. Next we show that $h'$ is strongly decisive and conservative; for that we show that, with every mind change, there is a new element of the target included in the conjecture which is currently not included but is included in all future conjectures; it is easy to see that this property implies both caution and strong decisiveness. Let $i$ and $i'$ be such that $\max(M(T[i'])) = T[i]$. This implies that $${\mathrm{content}}(T[i]) \subseteq W_{h'(T[i'])}.$$ Let $j' > i'$ such that $h'(T[i']) \neq h'(T[j'])$. Then there is $j > i$ such that $\max(M(T[j'])) = T[j]$ and therefore $${\mathrm{content}}(T[j]) \subseteq W_{h'(T[j'])}.$$ Note that in the following diagram $j$ could also be between $i$ and $i'$. (left) at (-1,0)\[\]; (right) at (12,0)\[\]; (labelil) at (4,-1)\[\][$h'(T[i']) = h(T[i])$]{}; (labeljl) at (10,-1)\[\][$h'(T[j']) = h(T[j])$]{}; at (4,-1.5) [${\mathrm{content}}(T[i]) \subseteq W_{h(T[i])}$]{}; at (10,-1.5) [${\mathrm{content}}(T[j]) \subseteq W_{h(T[j])}$]{}; (left) – (right); (1,1pt) – (1,-1pt) node\[anchor=south\][$i$]{} node\[anchor=north\][mind change $h$]{}; (4,1pt) – (4,-1pt) node\[anchor=south\][$i'$]{} node\[anchor=north\][mind change $h'$]{}; (7,1pt) – (7,-1pt) node\[anchor=south\][$j$]{} node\[anchor=north\][mind change $h$]{}; (10,1pt) – (10,-1pt) node\[anchor=south\][$j'$]{} node\[anchor=north\][mind change $h'$]{}; (4,0.5) – (10,0.5) node\[midway, above,yshift=12pt,\][no mind change $h'$]{}; As $h$ is conservative and ${\mathrm{content}}(T[i]) \subseteq W_{h(T[i])}$, there exists $\ell$ such that $i < \ell < j$ and $T(\ell) \notin W_{h(T[i])}$. Then we have $\forall n \geq j' : T(\ell) \in W_{h'(T[n])}$ as $T(\ell) \in W_{h'(T[j'])}$. Obviously $h'$ is conservative as it only outputs (delayed) hypotheses of $h$ (and maybe skip some) and $h$ is conservative. In the following we consider three new learning restrictions. The learning restriction ${\mathbf{Caut}}_\mathbf{Fin}$ means that the learner never returns a hypothesis for a finite set that is a proper subset of a previous hypothesis. ${\mathbf{Caut}}_\infty$ is the same restriction for infinite hypotheses. With ${\mathbf{Caut}}_\mathbf{Tar}$ the learner is not allowed to ever output a hypothesis that is a proper superset of the target language that is learned. \[defn:CautVariations\] $$\begin{aligned} {\mathbf{Caut}}_{\mathbf{Fin}}(p,T) &\Leftrightarrow [\forall i < j: W_{p(j)} \subset W_{p(i)} \Rightarrow W_{p(j)} \text{ is infinite}]\\ {\mathbf{Caut}}_{\infty}(p,T) &\Leftrightarrow [\forall i < j: W_{p(j)} \subset W_{p(i)} \Rightarrow W_{p(j)} \text{ is finite}]\\ {\mathbf{Caut}}_{\mathbf{Tar}}(p,T) &\Leftrightarrow [\forall i: \neg( {\mathrm{content}}(T) \subset W_{p(i)})] \end{aligned}$$ (nothing) at (1,0) [T]{}; (caut) at (1,-3.5) [${\mathbf{Caut}}$]{}; (cautinf) at (-1.5,-1) [${\mathbf{Caut}}_\infty$]{}; (cauttar) at (1,-2) [${\mathbf{Caut}}_\textbf{Tar}$]{}; (cautfin) at (3.5,-2) [${\mathbf{Caut}}_\textbf{Fin}$]{}; (caut) – (cautfin); (caut) – (cauttar); (caut) – (cautinf); (cautinf) – (nothing); (cauttar) – (nothing); (cautfin) – (nothing); (-3,-1.2) – (5,-1.2); The proof of the following theorem is essentially the same as given in [@Osh-Sto-Wei:b:86:stl] to show that cautious learning is a proper restriction of ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learning, we now extend it to strongly decisive learning. Note that a different extension was given in [@Bal-Cas-Mer-Ste-Wie:j:08] (with an elegant proof exploiting the undecidability of the halting problem), pertaining to behaviorally correct learning. The proof in [@Bal-Cas-Mer-Ste-Wie:j:08] as well as our proof would also carry over to the combination of these two extensions. \[thm:ConvWMonInT\] There is a class of languages that is ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{SDec}}{\mathbf{Mon}}{\mathbf{Ex}}$-learnable, but not ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Caut}}{\mathbf{Ex}}$-learnable. Let $h$ be a ${\mathbf{Psd}}$-learner as follows, $$\forall D, t : h(D,t) = \varphi_{\max(D)}(t),$$ and ${\mathcal{L}}= {\mathbf{Txt}^{}}{\mathbf{Psd}}{\mathbf{SDec}}{\mathbf{Mon}}{\mathbf{Ex}}(h)$. Suppose ${\mathcal{L}}$ is ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Caut}}{\mathbf{Ex}}$-learnable through learner $h' \in {\mathcal{R}}$. We define, for all $\sigma$ and $t$, the total computable predicate $Q(\sigma, t)$ as $$Q(\sigma,t) \Leftrightarrow {\mathrm{content}}(\sigma) \subset W_{h'(\sigma)}^t.$$ We let ${\mathrm{ind}}$ such that, for every set $D$, $W_{{\mathrm{ind}}(D)} = D$. Using ${\textbf{ORT}\xspace}$ we define $p$ and $e \in {\mathcal{R}}$ strongly monotone increasing such that for all $n$ and $t$, $$\begin{aligned} W_p &= {\mathrm{range}}(e);\\ \varphi_{e(n)} &= \begin{cases} {\mathrm{ind}}({\mathrm{content}}(e[n+1])), & \text{if } Q(e[n+1],t); \\ p, & \text{otherwise.} \end{cases}\end{aligned}$$ Case 1: For all $n$ and $t$, $Q(e[n+1],t)$ does not hold. Then we have $\varphi_{e(n)}(t) = p$ for all $n,t$. Thus $W_p \in {\mathcal{L}}$ as for any $D \subseteq W_p$, $h(D,t) = \varphi_{\max(D)}(t) = p$. But $h'$ does not ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Caut}}{\mathbf{Ex}}$-learns $W_p$ from text $e$ as for all $n$ and $t$, ${\mathrm{content}}(e[n])$ is not a proper subset of $W_{h'(e[n])}$ in $t$ steps although $W_p$ is infinite. Case 2: There are $n$ and $t$ such that $Q(e[n+1],t)$ holds. Then we have ${\mathrm{content}}(e[n+1]) \in {\mathcal{L}}$ as we will show now. Let $T$ be a text for ${\mathrm{content}}(e[n+1])$. As $e$ is monotone increasing we have that $e(n)$ is the maximal element in ${\mathrm{content}}(e[n+1])$. Additionally, for all $t' \geq t$, we have $\varphi_{e(n)}(t') = \varphi_{e(n)}(t) = {\mathrm{ind}}({\mathrm{content}}(e[n+1]))$. As $h$ makes only one mind change the strongly decisive and monotone conditions hold. Thus, there is $n_0$ such that, for all $n \geq n_0$, $h({\mathrm{content}}(T[n]),n) = h({\mathrm{content}}(T[n_0]),n_0) = {\mathrm{ind}}({\mathrm{content}}(e[n+1]))$, i.e. ${\mathrm{content}}(e[n+1]) \in {\mathcal{L}}$. The learner $h'$ does not ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Caut}}{\mathbf{Ex}}$-learn ${\mathrm{content}}(e[n+1])$ as we know from the predicate $Q$ that ${\mathrm{content}}(e[n+1]) \subset W_{h'({\mathrm{content}}(e[n+1]))}$ and the cautious learner $h'$ must not change to a proper subset of a previous hypothesis. The following theorem contradicts a theorem given as an exercise in [@Osh-Sto-Wei:b:86:stl] (Exercise 4.5.4B). \[thm:CautVarConv\] For $\delta \in \{{\mathbf{Caut}}, {\mathbf{Caut}}_\mathbf{Tar}, {\mathbf{Caut}}_\mathbf{Fin}\}$ we have $$[{\mathbf{Txt}^{}}{\mathbf{G}}\delta{\mathbf{Ex}}] = [{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Conv}}{\mathbf{Ex}}].$$ We get the inclusion \[TxtGConvEx\] $\subseteq$ \[TxtGCautEx\] as a direct consequence from Theorem \[thm:ConvInSDec\]. Obviously we have $[{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Caut}}{\mathbf{Ex}}] \subseteq [{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Caut}}_\mathbf{Tar}{\mathbf{Ex}}]$ and $[{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Caut}}{\mathbf{Ex}}] \subseteq [{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Caut}}_\mathbf{Fin}{\mathbf{Ex}}]$. Thus, it suffices to show $[{\mathbf{Txt}^{}}{\mathbf{G}}\delta{\mathbf{Ex}}] \subseteq [{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Conv}}{\mathbf{Ex}}]$. Let $\mathcal{L}$ be ${\mathbf{Txt}^{}}{\mathbf{G}}\delta{\mathbf{Ex}}$-learnable by a syntactically decisive learner $h \in \mathcal{R}$ (see Corollary \[cor:SinkLocking\]). Using the S-m-n Theorem we get a function $p \in \mathcal{R}$ such that $$\forall \sigma: W_{p(\sigma)} = \bigcup_{t \in \mathbb{N}} \begin{cases} W_{h(\sigma)}^t, &\mbox{if }\forall \rho \in (W_{h(\sigma)}^t)^, \|\sigma \diamond \rho\| \leq t: h(\sigma \diamond \rho) = h(\sigma);\\ \emptyset, &\mbox{otherwise.} \end{cases}$$ We let $Q$ be the following computable predicate. $$Q(\hat{\sigma},\sigma) \Leftrightarrow h(\hat{\sigma}) \neq h(\sigma)\ \land\ {\mathrm{content}}(\sigma) \not\subseteq W_{h(\hat{\sigma})}^{\|\sigma\|-1}.$$ For given sequences $\sigma$ and $\tau$ we say $\tau \preceq \sigma$ if $${\mathrm{content}}(\tau) \subseteq {\mathrm{content}}(\sigma)\ \land\ \|\tau\| \leq \|\sigma\|.$$ This means that, for every $\sigma$, the set of all $\tau$ such that $\tau \preceq \sigma$ is finite and computable. We define a learner $h'$ such that $h'(\sigma) = p(\hat{\sigma})$ where $\hat{\sigma} \preceq \sigma$ using recursion. For a given sequence $\sigma \neq \emptyset$ let $\hat{\sigma}$ be such that $h'(\sigma^-)=p(\hat{\sigma})$. $$\forall \sigma: h'(\sigma) = \begin{cases} p(\emptyset), &\mbox{if }\sigma = \emptyset;\\ p(\tau \diamond \sigma), &\mbox{else, if }\exists \tau, \hat{\sigma} \subseteq \tau \preceq \sigma: Q(\hat{\sigma},\tau);\footnotemark\\ h'(\sigma^-), &\mbox{otherwise.} \end{cases}\footnotetext{We choose the least such $\tau$, if existent.}$$ This means $h'$ only changes its hypothesis if $Q$ ensures that $h$ made a mind change and that the previous hypothesis does not contain something of the new input data. We first show that $h'$ is conservative. Let $\sigma$ and $\hat{\sigma}$ be such that $h'(\sigma^-) = p(\hat{\sigma})$ and let $\tau \preceq \sigma$ be such that $Q(\hat{\sigma},\tau)$. Then we have, for all $t \geq \|\tau\|$ with ${\mathrm{content}}(\tau) \subseteq W_{h(\sigma)}^t$, $$\begin{aligned} \neg [&\forall \rho \in (W_{h(\hat{\sigma})}^t)^, \|\hat{\sigma} \diamond \rho\| \leq t: h(\hat{\sigma} \diamond \rho) = h(\hat{\sigma})], \text{ which is equivalent to} \\ &\exists \rho \in (W_{h(\hat{\sigma})}^t)^, \|\hat{\sigma} \diamond \rho\| \leq t: h(\hat{\sigma} \diamond \rho) \neq h(\hat{\sigma});\end{aligned}$$ as there is $\rho$ such that $\hat{\sigma} \diamond \rho = \tau$. Therefore, we get ${\mathrm{content}}(\tau) \nsubseteq W_{p(\hat{\sigma})}$, as $W_{h(\hat{\sigma})}^t$ is monotone non-decreasing in $t$. Thus, $h'$ is conservative. Second, we will show that $h'$ converges on any text $T$ for a language $L \in \mathcal{L}$. Let $L \in \mathcal{L}$ and $T$ be a text for $L$. Thus, $h$ converges on $T$. Suppose $h'$ does not converge on $T$. Let $(p(\sigma_i))_{i \in \mathbb{N}}$ the corresponding sequence of hypotheses. Then $T' = \bigcup_{i \in \mathbb{N}} \sigma_i$ is a text for $L$ as for every $i \in \mathbb{N}$, $T(i) \in {\mathrm{content}}(\sigma_{i+1})$. As $h'$ infinitely often changes its mind, we have that, for infinitely many $\sigma_i$, there is, for each $i$, $\tau_i$ such that $\sigma_i \subseteq \tau_i \subseteq \sigma_{i+1}$ with $Q(\sigma_i,\tau_i)$ holds. As $Q(\sigma_i,\tau_i)$ means that $h(\sigma_i) \neq h(\tau_i)$, $h$ diverges on $T'$, a contradiction. Third we will show that $h'$ converges to a correct hypothesis. Let $\sigma$ be such that $h'$ converges to $p(\sigma)$ on $T$. In the following we consider two cases for this $\sigma$. Case 1:* If $\sigma$ is a locking sequence of $h$ on $L$ we have, for all $\tau \in {{\mathbb{S}\mathrm{eq}}}(L)$, $h(\sigma \diamond \tau) = h(\sigma)$ and especially for all $\rho \in (W_{h(\sigma)}^t)^$ with $\|\sigma \diamond \rho\| \leq t$, $h(\sigma \diamond \rho) = h(\sigma)$. Thus, $W_{p(\sigma)} = W_{h(\sigma)} = L$. Case 2:* Suppose $\sigma$ is not a locking sequence. As ${\mathrm{content}}(T) = L$ and $h'$ converges, we have for all $n$ and $\tau$ with $\sigma \subseteq \tau \preceq T[n]$, $\neg Q(\sigma, \tau)$. This means that, for all $\tau$ with elements of $L$ and $\sigma \subseteq \tau,$ $\neg Q(\sigma,\tau)$, i.e. $$\label{eq:NotQSigmaTau} \forall \tau \in {{\mathbb{S}\mathrm{eq}}}(L) : h(\sigma) = h(\tau)\ \lor\ {\mathrm{content}}(\tau) \subseteq W_{h(\sigma)}^{\|\tau\|-1}.$$ We now show $L \subseteq W_{h(\sigma)}$. If we have, for all $\tau \in {{\mathbb{S}\mathrm{eq}}}(L)$, $h(\sigma) = h(\tau)$, we get this directly from Equation (\[eq:NotQSigmaTau\]). Otherwise, let $\tau$ be such that $h(\sigma) \neq h(\sigma \diamond \tau)$. Let $x \in L$. Thus, $h(\sigma) \neq h(\sigma \diamond \tau \diamond x)$, as $h$ is syntactically decisive. From $\neg Q(\sigma, \sigma \diamond \tau \diamond x)$ we can conclude that ${\mathrm{content}}(\sigma \diamond \tau \diamond x) \subseteq W_{h(\sigma)}^{\|\sigma \diamond \tau \diamond x\|}$. Therefore we have, for all $x \in L$, $x \in W_{h(\sigma)}$ and thus ${\mathrm{content}}(T) = L \subseteq W_{h(\sigma)}$. Additionally we will show now that $W_{h(\sigma)} = W_{p(\sigma)}$. Obviously we have $W_{p(\sigma)} \subseteq W_{h(\sigma)}.$ To show that $W_{h(\sigma)} \subseteq W_{p(\sigma)}$ suppose there is $x \in W_{h(\sigma)}$ such that $x \notin W_{p(\sigma)}$. Then there is a minimal $t$ such that $x \in W_{h(\sigma)}^t$ but there is $\rho \in (W_{h(\sigma)}^t)^, \|\sigma \diamond \rho\| \leq t$ such that $h(\sigma \diamond \rho) \neq h(\sigma)$ and therefore $h(\sigma \diamond \rho \diamond x) \neq h(\sigma \diamond \rho).$ As we have $\neg Q(\sigma, \sigma \diamond \rho \diamond x)$ which is equivalent to $h(\sigma) = h(\sigma \diamond \rho \diamond x)\ \lor\ {\mathrm{content}}(\sigma \diamond \rho \diamond x) \subseteq W_{h(\sigma)}^{\|\sigma \diamond \rho \diamond x\| - 1}$ and we supposed that $h(\sigma \diamond \rho \diamond x) \neq h(\sigma)$ it follows that ${\mathrm{content}}(\sigma \diamond \rho \diamond x) \subseteq W_{h(\sigma)}^{\|\sigma \diamond \rho \diamond x\| - 1}.$ This is a contradiction as $\|\sigma \diamond \rho \diamond x\| -1 \leq t.$ Thus, for all $x \in L$ we have $x \in W_{p(\sigma)}$ and from $L \subseteq W_{h(\sigma)}$ we get $W_{h(\sigma)} \subseteq W_{p(\sigma)}$. \(a) $\delta = {\mathbf{Caut}}.$ We have that the learner must not change to a proper subset of a previous hypothesis and this means that $W_{h(\sigma)} = L$. \(b) $\delta = {\mathbf{Caut}}_\mathbf{Tar}.$ The learner $h$ never returns a hypothesis which is a proper superset of the language that is learned. Thus $W_{h(\sigma)} = L$. \(c) $\delta = {\mathbf{Caut}}_\mathbf{Fin}.$ As $h$ must not change to a finite subset of a previous hypothesis, we suppose that $W_{h(\sigma)} \supset L$ and both $W_{h(\sigma)}$ and $L$ are infinite. This means there is a sequence $\rho \in {{\mathbb{S}\mathrm{eq}}}(L) \subseteq {{\mathbb{S}\mathrm{eq}}}(W_{h(\sigma)})$ such that $h(\sigma) \neq h(\sigma \diamond \rho)$. Thus, $W_{p(\sigma)}$ is finite, a contradiction to $W_{h(\sigma)}$ being infinite. Therefore we have $W_{h(\sigma)} = L$. From the definitions of the learning criteria we have $[{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Conv}}{\mathbf{Ex}}] \subseteq [{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{WMon}}{\mathbf{Ex}}]$. Using Theorem \[thm:CautVarConv\] and the equivalence of weakly monotone and conservative learning (using ${\mathbf{G}}$) [@Kin-Ste:j:95:mon; @Jai-Sha:j:98], we get the following. \[cor:ConvWMonCaut\] We have $$[{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Conv}}{\mathbf{Ex}}] = [{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{WMon}}{\mathbf{Ex}}] = [{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Caut}}{\mathbf{Ex}}].$$ Using Corollary \[cor:ConvWMonCaut\] and Theorem \[thm:ConvInSDec\] we get that weakly monotone ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learning is included in strongly decisive ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learning. Theorem \[thm:ConvWMonInT\] shows that this inclusion is proper. \[cor:WMonInSDec\] We have $$[{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{WMon}}{\mathbf{Ex}}] \subset [{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{SDec}}{\mathbf{Ex}}].$$ The next theorem is the last theorem of this section and shows that forbidding to go down to strict infinite* subsets of previously conjectures sets is no restriction. \[thm:CautInfT\] We have $$[{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Caut}}_\infty{\mathbf{Ex}}] = [{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}].$$ Obviously we have $[{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Caut}}_{\infty}{\mathbf{Ex}}] \subseteq [{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}]$. Thus, we have to show that $[{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}] \subseteq [{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Caut}}_\mathbf{\infty}{\mathbf{Ex}}]$. Let ${\mathcal{L}}$ be a set of languages and $h$ be a learner such that $h$ ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learns ${\mathcal{L}}$ and $h$ is strongly locking on ${\mathcal{L}}$ (see Corollary \[cor:SinkLocking\]). We define, for all $\sigma$ and $t$, the set $M_{\sigma}^t$ such that $$M_{\sigma}^t = \{\tau\ \|\ \tau \in {{\mathbb{S}\mathrm{eq}}}(W_{h(\sigma)}^t \cup {\mathrm{content}}(\sigma))\ \land\ \|\tau \diamond \sigma\| \leq t\}.$$ Using the S-m-n Theorem we get a function $p \in \mathcal{R}$ such that $$\forall \sigma: W_{p(\sigma)} = {\mathrm{content}}(\sigma) \bigcup_{t \in \mathbb{N}} \begin{cases} W_{h(\sigma)}^t, &\mbox{if }\forall \rho \in M_{\sigma}^t: h(\sigma \diamond \rho) = h(\sigma);\\ \emptyset, &\mbox{otherwise.} \end{cases}$$ We define a learner $h'$ as $$\forall \sigma : h'(\sigma) = \begin{cases} p(\sigma), & \text{if } h(\sigma) \neq h(\sigma^-); \\ h'(\sigma^-), & \text{otherwise.} \end{cases}$$ We will show now that the learner $h'$ ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Caut}}_{\infty}{\mathbf{Ex}}$-learns ${\mathcal{L}}$. Let an $L \in {\mathcal{L}}$ and a text $T$ for $L$ be given. As $h$ is strongly locking there is $n_0$ such that for all $\tau \in {{\mathbb{S}\mathrm{eq}}}(L)$, $h(T[n_0] \diamond \tau) = h(T[n_0])$ and $W_{h(T[n_0])} = L$. Thus we have, for all $n \geq n_0$, $h'(T[n]) = h'(T[n_0])$ and $W_{h'(T[n_0])} = W_{p(T[n_0])} = W_{h(T[n_0])} = L$. To show that the learning restriction ${\mathbf{Caut}}_{\infty}$ holds, we assume that there are $i < j$ such that $W_{h'(T[j])} \subset W_{h'(T[i])}$ and $W_{h'(T[j])}$ is infinite. W.l.o.g. $j$ is the first time that $h'$ returns the hypothesis $W_{h'(T[j])}$. Let $\tau$ be such that $T[i] \diamond \tau = T[j]$. From the definition of the function $p$ we get that ${\mathrm{content}}(T[j]) \subseteq W_{h'(T[j])} \subseteq W_{h'(T[i])}$. Thus, ${\mathrm{content}}(\tau) \subseteq W_{h'(T[i])} = W_{p(T[i])}$ and therefore $W_{p(T[i])}$ is finite, a contradiction to the assumption that $W_{h'(T[j])}$ is infinite. Decisiveness ============ \[sec:Decisiveness\] In this section the goal is to show that decisive and strongly decisive learning separate (see Theorem \[thm:StronglyDecisiveLearning\]). For this proof we adapt a technique known in computability theory as a “priority argument” (note, though, that we are not dealing with oracle computations). In order to illustrate the proof with a simpler version, we first reprove that decisiveness is a restriction to ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learning (as shown in [@Bal-Cas-Mer-Ste-Wie:j:08]). For both proofs we need the following lemma, a variant of which is given in [@Bal-Cas-Mer-Ste-Wie:j:08] for the case of decisive learning; it is easy to see that the proof from [@Bal-Cas-Mer-Ste-Wie:j:08] also works for the cases we consider here. \[lem:NotNatnum\] Let ${\mathcal{L}}$ be such that ${\mathbb{N}}\not\in {\mathcal{L}}$ and, for each finite set $D$, there are only finitely many $L \in {\mathcal{L}}$ with $D \not\subseteq L$. Let $\delta \in \{{\mathbf{Dec}},{\mathbf{SDec}}\}$. Then, if ${\mathcal{L}}$ is ${\mathbf{Txt}^{}}{\mathbf{G}}\delta{\mathbf{Ex}}$-learnable, it is so learnable by a learner which never outputs an index for ${\mathbb{N}}$. Now we get to the theorem regarding decisiveness. Its proof is an adaptation of the proof given in [@Bal-Cas-Mer-Ste-Wie:j:08], rephrased as a priority argument. This rephrased version will be modified later to prove the separation of decisive and strongly decisive learning. \[thm:DecisiveLearning\] We have $$[{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Dec}}{\mathbf{Ex}}] \subset [{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}].$$ For this proof we will employ a technique from computability theory known as priority argument. For this technique, one has a set of requirements (we will have one for each $e \in {\mathbb{N}}$) and a priority on requirements (we will prioritize smaller $e$ over larger). One then tries to fulfill requirements one after the other in an iterative manner (fulfilling the unfulfilled requirement of highest priority without violating requirements of higher priority) so that, in the limit, the entire infinite list of requirements will be fulfilled. We apply this technique in order to construct a learner $h \in {\mathcal{P}}$ (and a corresponding set of learned sets ${\mathcal{L}}= {\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}(h)$). Thus, we will give requirements which will depend on the $h$ to be constructed. In particular, we will use a list of requirement $(R_e)_{e \in {\mathbb{N}}}$, where lower $e$ have higher priority. For each $e$, $R_e$ will correspond to the fact that learner $\varphi_e$ is not a suitable decisive learner for ${\mathcal{L}}$. We proceed with the formal argument. For each $e$, let Requirement $R_e$ be the disjunction of the following three predicates depending on the $h$ to be constructed. (i) $\exists x$: $\forall \sigma \in {{\mathbb{S}\mathrm{eq}}}({\mathbb{N}}\setminus \{x\}): \varphi_e(\sigma){\mathclose{\hbox{$\uparrow$}}}\vee W_{\varphi_e(\sigma)} \neq {\mathbb{N}}\setminus \{x\}$ and $h$ learns ${\mathbb{N}}\setminus \{x\}$. (ii) $\exists \sigma \in {{\mathbb{S}\mathrm{eq}}}{}: {\mathrm{content}}(\sigma) \subset W_{\varphi_e(\sigma)}$ and $h$ learns $W_{\varphi_e(\sigma)}$ and some $D$ with ${\mathrm{content}}(\sigma) \subseteq D \subset W_{\varphi_e(\sigma)}$. (iii) $\exists \sigma \in {{\mathbb{S}\mathrm{eq}}}: W_{\varphi_e(\sigma)} = {\mathbb{N}}$. If all $(R_e)_{e \in {\mathbb{N}}}$ hold, then every learner which never outputs an index for ${\mathbb{N}}$ fails to learn ${\mathcal{L}}$ decisively as follows. For each learner $\varphi_e$ which never outputs an index for ${\mathbb{N}}$, either (i) of $R_e$ holds, implying that some co-singleton is learned by $h$ but not by $\varphi_e$. Or (ii) holds, then there is a $\sigma$ on which $\varphi_e$ generalizes, but will later have to abandon this correct conjecture $p = \varphi_e(\sigma)$ in order to learn some finite set $D$; as, after the change to a hypothesis for $D$, the text can still be extended to a text for $W_p$, the learner is not decisive.[^2] Thus, all that remains is to construct $h$ in a way that all of $(R_e)_{e \in {\mathbb{N}}}$ are fulfilled. In order to coordinate the different requirements when constructing $h$ on different inputs, we will divide the set of all possible input sequences into infinitely many segments, of which every requirement can “claim” up to two at any point of the algorithm defining $h$; the chosen segments can change over the course of the construction, and requirements of higher priority might “take away” segments from requirements with lower priority (but not vice versa). We follow [@Bal-Cas-Mer-Ste-Wie:j:08] with the division of segments: For any set $A \subset {\mathbb{N}}$ we let ${\mathrm{id}}(A) = \min({\mathbb{N}}\setminus A)$ be the ID of $A$; for ease of notation, for each finite sequence $\sigma$, we let ${\mathrm{id}}(\sigma) = {\mathrm{id}}({\mathrm{content}}(\sigma))$. For each $s$, the $s$th segment contains all $\sigma$ with ${\mathrm{id}}(\sigma) = s$. We note that ${\mathrm{id}}$ is monotone, i.e. $$\label{eq:IDMonotone} \forall A,B \subset {\mathbb{N}}: A \subseteq B \Rightarrow {\mathrm{id}}(A) \leq {\mathrm{id}}(B).$$ The first way of ensuring some requirement $R_e$ is via (i); as this part itself is not decidable, we will check a “bounded” version thereof. We define, for all $e,t,s$, $$P_{e,t}(s) \Leftrightarrow (\forall \sigma \in {{\mathbb{S}\mathrm{eq}}}_{\leq t} \mid {\mathrm{id}}(\sigma) = s) \; \Phi_e(\sigma) > t \vee {\mathrm{content}}(\sigma) \not\subset W_{\varphi_e(\sigma)}^t.$$ For any $e$, if we can find an $s$ such that, for all $t$, we have $P_{e,t}(s)$, then it suffices to make $h$ learn ${\mathbb{N}}\setminus \{s\}$ in order to fulfill $R_e$ via part (i); this requires control over segment $s$ in defining $h$. Note that, if we ever cannot take control over some segment because some requirement with higher priority is already in control, then we will try out different $s$ (only finitely many are blocked). If we ever find a $t$ such that $\neg P_{e,t}(s)$, then we can work on fulfilling $R_e$ via (ii), as we directly get a $\sigma$ where $\varphi_e$ over the content generalizes. In order to fulfill $R_e$ via (ii) we have to choose a finite set $D$ with ${\mathrm{content}}(\sigma) \subseteq D \subset W_{\varphi_e(\sigma)}$. We will then take control over the segments corresponding to ${\mathrm{id}}(D)$ and ${\mathrm{id}}(W_{\varphi_e(\sigma)}^t)$ (for growing $t$), but not necessarily over segment $s$, and thus establish $R_e$ via (ii). Note that, again, the segments we desire might be blocked; but only finitely many are blocked, and we require control over ${\mathrm{id}}(D)$ and ${\mathrm{id}}(W_{\varphi_e(\sigma)}^t)$, both of which are at least $s$ (this follows from ${\mathrm{id}}$ being monotone, see Equation (\[eq:IDMonotone\]), and from ${\mathrm{content}}(\sigma) \subseteq D \subset W_{\varphi_e(\sigma)}^t$); thus, we can always find an $s$ for which we can either follow our strategy for (i) or for (ii) as just described. It is tempting to choose simply $D = {\mathrm{content}}(\sigma)$, this fulfills all desired properties. The main danger now comes from the possibility of $\varphi_e(\sigma)$ being an index for ${\mathbb{N}}$: this will imply that, for growing $t$, $y = {\mathrm{id}}(W_{\varphi_e(\sigma)}^t)$ will also be growing indefinitely. Of course, there is no problem with satisfying $R_e$, it now holds via (iii); but as soon as at least two requirements will take control over segments $y$ for indefinitely growing $y$, they might start blocking each other (more precisely, the requirement of higher priority will block the one of lower priority). We now need to know something about our later analysis: we will want to make sure that every requirement $R_e$ either (a) converges in which segments to control or (b) for all $n$, there is a time $t$ in the definition of $h$ after which $R_e$ will never have control over any segment corresponding to IDs $\leq n$; in fact, we will show this later by induction (see Claim \[claim:InductionProof\]). Any requirement which takes control over segments $y$ for indefinitely growing $y$ might be blocked infinitely often, and thus forced to try out different $s$ for fulfilling $R_e$, including returning to $s$ that were abandoned previously because of (back then) being blocked by a requirement of higher priority. Thus, such a requirement would fulfill neither (a) nor (b) from above. We will avoid this problem by not choosing $D = {\mathrm{content}}(\sigma)$, but instead choosing a $D$ which grows in ID along with the corresponding $W_{\varphi_e(\sigma)}^t$. The idea is to start with $D = {\mathrm{content}}(\sigma)$ and then, as $W_{\varphi_e(\sigma)}^t$ grows, add more elements. For this we make some definitions as follows. For a finite sequence $\sigma$ we let ${\mathrm{id}}'(\sigma)$ be the least element not in ${\mathrm{content}}(\sigma)$ which is larger than all elements of ${\mathrm{content}}(\sigma)$. For any finite sequence $\sigma$ and $e,t \geq 0$ we let $D^t_{e,\sigma}$ be such that $$D^t_{e,\sigma} = \begin{cases} {\mathrm{content}}(\sigma), &\mbox{if }{\mathrm{id}}(W_{\varphi_e(\sigma)}^t) \leq {\mathrm{id}}'(\sigma);\\ \{0,\ldots, {\mathrm{id}}(W_{\varphi_e(\sigma)}^t)-2\}, &\mbox{otherwise.} \end{cases}$$ For all $e,t$ and $\sigma$ with ${\mathrm{content}}(\sigma) \subset W_{\varphi_e(\sigma)}$ we have $$\label{eq:defDTSigma} {\mathrm{content}}(\sigma) \subseteq D^t_{e,\sigma} \subset W_{\varphi_e(\sigma)}.$$ Thus, we will use the sets $D^t_{e,\sigma}$ to satisfy (ii) of $R_e$ (in place of $D$). We now have all parts that are required to start giving the construction for $h$. In that construction we will make use of a subroutine which takes as inputs a set $B$ of blocked indices, a requirement $e$ and a time bound $t$, and which finds triples $(x,y,\sigma)$ with $x,y \not\in B$ such that $$\label{eq:defnTWitness} P_{e,t}(x) \mbox{ or } \big[{\mathrm{content}}(\sigma) \subset W_{\varphi_e(\sigma)}^t \wedge {\mathrm{id}}(D^t_{e,\sigma}) = x \wedge {\mathrm{id}}(W_{\varphi_e(\sigma)}^t) = y\big].$$ We call $(x,y,\sigma)$ fulfilling Equation (\[eq:defnTWitness\]) for given $t$ and $e$ a $t$-witness for $R_e$. The subroutine is called [`findWitness`]{} and is given in Algorithm \[alg:priorityArgumentDecSubroutine\]. `error` We now formally show termination and correctness of our subroutine. Let $e,t$ and a finite set $B$ be given. The algorithm [`findWitness`]{} on $(B,e,t)$ terminates and returns a $t$-witness $(x,y,\sigma)$ for $R_e$ such that $x,y \not\in B$. From the condition in line \[line:PCondition\] we see that the search in line \[line:sigmaSearch\] is necessarily successful, showing termination. Using the monotonicity of ${\mathrm{id}}$ from Equation (\[eq:IDMonotone\]) on Equation (\[eq:defDTSigma\]) we have that the subroutine [`findWitness`]{} cannot return `error` on any arguments $(B,e,t)$: for $s=\max(B)+1$, we either have $P_{e,t}(s)$ or the $x$ and $y$ chosen are larger than ${\mathrm{id}}(\sigma) = s > \max(B)$. With the subroutine given above, we now turn to the priority construction for defining $h$ detailed in Algorithm \[alg:priorityArgumentDec\]. This algorithm assigns witness tuples to more and more requirements, trying to make sure that they are $t$-witnesses, for larger and larger $t$. For each $e$, $w_e(t)$ will be the witness tuple associated with $R_e$ after $t$ iterations (defined for all $t \geq e$). We say that a requirement $R_e$ blocks an ID $n$ iff $n \in \{x,y\}$ for the witness tuple $w_e(t) = (x,y,\sigma)$ currently associated with $R_e$. We say that a tuple $(x,y,\sigma)$ is $(e,t)$-legal iff it is a $t$-witness for $R_e$ and $x$ and $y$ are not blocked by any $R_{e'}$ with $e' < e$. Clearly, it is decidable whether a triple is $(e,t)$-legal. In order to define the learner $h$ we will need some functions giving us indices for the languages to be learned. To that end, let $p,q \in {\mathcal{R}}$ (using the S-m-n Theorem) be such that $$\begin{aligned} \forall n: W_{q(n)} & = & {\mathbb{N}}\setminus \{n\};\\ \forall e,t,\sigma: W_{p(e,t,\sigma)} & = & D^t_{e,\sigma}.\end{aligned}$$ To increase readability, we allow assignments to values of $h$ for arguments on which $h$ was already defined previously; in this case, the new assignment has no effect. Regarding Algorithm \[alg:priorityArgumentDec\], note that lines 3–8 make sure that we have an appropriate witness tuple. We will later show that the sequence of assigned witness tuples will converge (for learners never giving a conjecture for ${\mathbb{N}}$). Lines 9–11 will try to establish the requirement $R_e$ via (i), once this fails it will be established in lines 12–16 via (ii). After this construction of $h$, we let ${\mathcal{L}}= {\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}(h)$ be the target to be learned. First note that the IDs blocked by different requirements are always disjoint (at the end of an iteration of $t$). As the major part of the analysis, we show the following claim by induction, showing that, for each $e$, either the triple associated with $R_e$ converges or it grows arbitrarily in both its $x$ and $y$ value (this is what we earlier had to carefully choose the $D$ for). \[claim:InductionProof\] For all $e$ we have $R_e$ and, for all $n$, there is $t_0$ such that either $$\forall t \geq t_0: R_e \mbox{ does not block any ID }\leq n$$ or $$\forall t \geq t_0: w_e(t) = w_e(t_0).$$ As our induction hypothesis, let $e$ be given such that the claim holds for all $e' < e$. Case 1: There is $t_0$ such that $\forall t \geq t_0: w_e(t) = w_e(t_0)$.\ Then, for all $t$, $(x,y,\sigma) = w_e(t_0)$ is a $t$-witness for $R_e$; in the case of $\forall t: P_{e,t}(x)$, we have that, for all but finitely many $\tau$ with ${\mathrm{id}}(\tau) = x$, $h(\tau) = q(x)$, and index for ${\mathbb{N}}\setminus \{x\}$; this implies ${\mathbb{N}}\setminus \{x\} \in {\mathcal{L}}$, which shows $R_e$. Otherwise we have, for all $t \geq t_0$, $D^t_{e,\sigma} = D^{t_0}_{e,\sigma}$. Furthermore we get, for all but finitely many $\tau$ with ${\mathrm{content}}(\tau) = D^{t_0}_{e,\sigma}$, $h(\tau) = p(e,t,\sigma)$, and index for $D^{t_0}_{e,\sigma}$; this implies $D^{t_0}_{e,\sigma} \in {\mathcal{L}}$. Consider now all those $\tau$ with ${\mathrm{id}}(\tau) = y$. If ${\mathrm{id}}(D^{t_0}_{e,\sigma}) = y$, then $h$ is already be defined on infinitely many such $\tau$, namely in case of ${\mathrm{content}}(\tau) = D^{t_0}_{e,\sigma}$. However, we have that $D^{t_0}_{e,\sigma}$ is a proper subset of $W_{\varphi_e(\sigma)}$, which shows that, on any text for $W_{\varphi_e(\sigma)}$, $h$ will eventually only output $\varphi_e(\sigma)$, which gives $W_{\varphi_e(\sigma)} \in {\mathcal{L}}$ as desired and, thus, $R_e$. Case 2: Otherwise.\ For each ID $s$ there exists at most finitely many $\sigma$ with ${\mathrm{id}}(\sigma) = s$ and $\sigma$ is used in the witness triple for $R_e$; this follows from the choice of $\sigma$ in the subroutine [`findWitness`]{} as a minimum, where, for larger $t$, all previously considered $\sigma$ are still considered (so that the chosen minimum might be smaller for larger $t$, but never go up, which shows convergence). A triple is only abandoned if it is not legal any more; this means it is either blocked or it is not a $t$-witness triple for some $t$. Using the induction hypothesis, the first can only happen finitely many times for any given tuple; the second implies the desired increase in both the $x$ and the $y$ value of the witness tuple. For this we also use our specific choice of $D$ as growing along with the ID of the associated $W_{\varphi_e(\sigma)}^t$ and we use that any witness tuple with a $\sigma$ with ${\mathrm{id}}(\sigma) = s$ has $x$ and $y$ value of at least $s$, due to the monotonicity of ${\mathrm{id}}$. To show $R_e$ (we will show (3)), let $t_1$ be the maximum over all $t_0$ existing for the converging $e' < e$ by the induction hypothesis and $e$. Let $(x,y,\sigma) = w_e(t_1)$ be the $t_1$-witness triple chosen for $R_e$ in iteration $t_1$. Suppose, by way of contradiction, that $\varphi_{e}(\sigma)$ is not an index for ${\mathbb{N}}$; let $n = {\mathrm{id}}(W_{\varphi_{e}(\sigma)})$. Let $t_2$ be the maximum over all $t_0$ found by the induction hypothesis for all $e' < e$ with the chosen $n$. Since the triple $(x,y,\sigma)$ is $(e,t)$-legal for all $t \geq t_2$, we get a contradiction to the unbounded growth of the witness triple. This shows that $\varphi_{e}(\sigma)$ is an index for ${\mathbb{N}}$, and thus we have $R_e$. With the last claim we now see that all requirement are satisfied. This implies that ${\mathcal{L}}$ cannot be ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Dec}}{\mathbf{Ex}}$-learned by a learner never using an index for ${\mathbb{N}}$ as conjecture. We have that ${\mathbb{N}}\not\in {\mathcal{L}}$. Furthermore, for any ID $s$, there are only finitely many sets in ${\mathcal{L}}$ with that ID; this implies that, for every finite set $D$, there are only finitely many elements $L \in {\mathcal{L}}$ with $D \not\subseteq L$. Thus, using Lemma \[lem:NotNatnum\], ${\mathcal{L}}$ is not decisively learnable at all. While the previous theorem showed that decisiveness poses a restriction on ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learning, the next theorem shows that the requirement of strong decisiveness is even more restrictive. The proof follows the proof of Theorem \[thm:DecisiveLearning\], with some modifications. \[thm:StronglyDecisiveLearning\] We have $$[{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{SDec}}{\mathbf{Ex}}] \subset [{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Dec}}{\mathbf{Ex}}].$$ We use the same language and definitions as in the proof of Theorem \[thm:DecisiveLearning\]. The idea of this proof is as follows. We build a set ${\mathcal{L}}$ with a priority construction just as in the proof of Theorem \[thm:DecisiveLearning\], the only essential change being in the definition of the hypothesis $p(e,t,\sigma)$: the change from $\varphi_e(\sigma)$ to $p(e,t,\sigma)$ and back to $\varphi_e(\sigma)$ on texts for $W_{\varphi_e(\sigma)}$ is what made ${\mathcal{L}}$ not decisively learnable. Thus, we will change $p(e,t,\sigma)$ to be a hypothesis for $W_{\varphi_e(\sigma)}$ as well – as soon as $\varphi_e$ changed its hypothesis on an extension of $\sigma$, and otherwise it is a hypothesis for $D_{e,\sigma}^t$ as before. This will make $h$ decisive on texts for $W_{\varphi_e(\sigma)}$, but $\varphi_e(\sigma)$ will not be strongly decisive. Furthermore, we will make sure that for sequences with ID $s$, only conjectures for sets with ID $s$ are used, so that indecisiveness can only possibly happen within a segment. Now the last source of ${\mathcal{L}}$ not being decisively learnable is as follows. When different requirements take turns with being in control over the segment, they might introduce returns to abandoned conjectures. To counteract this, we make sure that any conjecture which is ever abandoned on a segment of ID $s$ is for ${\mathbb{N}}\setminus \{s\}$, which will give decisiveness. We first define an alternative $p'$ for the function $p$ from that proof with the S-m-n Theorem such that, for all $e,t,\sigma$, $$W_{p'(e,t,\sigma)} = \begin{cases} W_{\varphi_e(\sigma)}, &\mbox{if }\exists \tau \mbox{ with }{\mathrm{content}}(\tau) \subseteq D_{e,\sigma}^t: \varphi_e(\sigma \diamond \tau) {\mathclose{\hbox{$\downarrow$}}}\neq \varphi_e(\sigma);\\ D_{e,\sigma}^t, &\mbox{otherwise.} \end{cases}$$ As we have $D_{e,\sigma}^t \subseteq W_{\varphi_e(\sigma)}$, this is a valid application of the S-m-n Theorem. We also want to replace the output of $h$ according to line \[line:folowE\] of Algorithm \[alg:priorityArgumentDec\]. To that end, let $g \in {\mathcal{R}}$ be as given by the S-m-n Theorem such that, for all $e$ and $\sigma$, $$W_{g(e,\sigma,y)} = W_{\varphi_e(\sigma)} \setminus \{y\}.$$ We construct now a learner $h$ again according to a priority construction, as given in Algorithm \[alg:priorityArgumentSDec\]. Note that lines 1–\[line:ElseLine\] are identical with the construction from Algorithm \[alg:priorityArgumentDec\] and lines 3–8 again make sure that we have an appropriate witness tuple and lines 9–11 try to establish the requirement $R_e$ via (i). The main difference lies in the way that $R_e$ is established once this fails in lines 12–18 via (ii): Here we need to check for a mind change and adjust what language $h$ should learn accordingly. It is easy to check that $h$, on any sequence $\sigma$, gives conjectures for languages of the same ID as that of $\sigma$. Thus, indecisiveness of $h$ can only occur within a segment. Next we will modify $h$ to avoid indecisiveness from different requirements taking turns controlling the same segment. With the S-m-n Theorem we let $f \in {\mathcal{R}}$ be such that, for all $\sigma$, $$W_{f(\sigma)} = \begin{cases} {\mathbb{N}}\setminus \{{\mathrm{id}}(\sigma)\}, &\mbox{if }\exists \tau \mbox{ with }{\mathrm{id}}(\sigma) \not\in {\mathrm{content}}(\tau): h(\sigma) \neq h(\sigma \diamond \tau);\\ W_{h(\sigma)}, &\mbox{otherwise.} \end{cases}$$ Let $h'$ be such that, for all $\sigma$, $$h'(\sigma) = \begin{cases} h'(\sigma^-), &\mbox{if }\sigma \neq \emptyset \mbox{ and } h(\sigma) = h(\sigma^-);\\ f(\sigma), &\mbox{otherwise.} \end{cases}$$ We now let ${\mathcal{L}}= {\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Dec}}{\mathbf{Ex}}(h')$. It is easy to see that $h'$ is decisive on all texts where it always makes an output, since indecisiveness can again only happen within a segment, and $f$ poisons any possible non-final conjectures within a segment. Let a strongly decisive learner $\overline{h}$ for ${\mathcal{L}}$ be given which never makes a conjecture for ${\mathbb{N}}$ (we are reasoning with Lemma \[lem:NotNatnum\] again). Let $e$ be such that $\varphi_e = \overline{h}$. Reasoning as in the proof of Theorem \[thm:DecisiveLearning\], we see that there is a triple $(x,y,\sigma)$ such that $w_e$ converges to that triple in the construction of $h'$. If, for all $t$, $P_{e,t}(x)$, then we have that ${\mathbb{N}}\setminus \{x\} \in {\mathcal{L}}$ (on any sequences with ID $x$, $h'$ gives an output for ${\mathbb{N}}\setminus \{x\}$, and it converges). Assume now that there is $t_0$ such that, for all $t \geq t_0$, we have $\neg P_{e,t}(x)$. Case 1: There is $\tau$ with ${\mathrm{content}}(\tau) \subseteq D^t_{e,\sigma}$ such that $\varphi_e(\sigma \diamond \tau) \neq \varphi_e(\sigma)$.\ Let $T$ be a text for $L = W_{\varphi_e(\sigma)}$. Then $h'$ on $T$ converges to an index for $L$, giving $L \in {\mathcal{L}}$. But this shows that $\overline{h} = \varphi_e$ was not strongly decisive on any text for $L$ starting with $\sigma \diamond \tau$, a contradiction. Case 2: Otherwise.\ Let $T$ be a text for $L = D^t_{e,\sigma}$. Then $h'$ on $T$ converges to an index for $L$, giving $L \in {\mathcal{L}}$. But $\overline{h} = \varphi_e$ converges on any text for $L$ starting with $\sigma$ to $\varphi_e(\sigma)$, a contradiction to $D^t_{e,\sigma} \subset W_{\varphi_e(\sigma)}$ (so the convergence is not to a correct hypothesis). In both cases we get the desired contradiction. Set-driven Learning =================== \[sec:SetDriven\] In this section we give theorems regarding set-driven learning. For this we build on the result that set-driven learning can always be done conservatively [@Kin-Ste:j:95:mon]. First we show that any conservative set-driven learner can be assumed to be cautious and syntactically decisive, an important technical lemma. \[thm:SdSyntDec\] We have $$[{\mathbf{Txt}^{}}{\mathbf{Sd}}{\mathbf{Ex}}] = [{\mathbf{Txt}^{}}{\mathbf{Sd}}{\mathbf{Conv}}{\mathbf{SynDec}}{\mathbf{Ex}}].$$ In other words, every set-driven learner can be assumed syntactically decisive. Let a set-driven learner $h$ be given. Following [@Kin-Ste:j:95:mon] we can $h$ assume to be conservative. We define a learner $h'$ such that, for all finite sets $C$, $$\begin{aligned} h'(C) = \begin{cases} \text{pad}(h(C),0), & \text{if } \forall D \subseteq C : h(D) = h(C) \rightarrow\\ &\;\;\; \forall D', D \subseteq D' \subseteq C : h(D') = h(D); \\ {\mathrm{pad}}(h(C),\|C\|+1), & \text{otherwise.} \end{cases}\end{aligned}$$ Let ${\mathcal{L}}= {\mathbf{Txt}^{}}{\mathbf{Sd}}{\mathbf{Conv}}{\mathbf{Ex}}(h)$. We will show that $h'$ is syntactically decisive and ${\mathbf{Txt}^{}}{\mathbf{Sd}}{\mathbf{Conv}}{\mathbf{Ex}}$-learns ${\mathcal{L}}$. Let $L \in {\mathcal{L}}$ be given and let $T$ be a text for $L$. First, we show that $h'$ ${\mathbf{Txt}^{}}{\mathbf{Ex}}$-learns $L$ from $T$. As $h$ is a set driven learner there is $n_0$ such that $\forall n \geq n_0 : h({\mathrm{content}}(T[n_0])) = h({\mathrm{content}}(T[n]))$ and $W_{h({\mathrm{content}}(T[n_0]))} = L$. We will show that, for all $T[n]$ with $n \geq n_0$, the first condition in the definition of $h'$ holds. Let $n \geq n_0$ and suppose there are $D$ and $D'$ with $$\begin{aligned} D &\subseteq {\mathrm{content}}(T[n]), \\ h(D) &= h({\mathrm{content}}(T[n])) = h({\mathrm{content}}(T[n_0]))\end{aligned}$$ and $$\begin{aligned} D &\subseteq D' \subseteq {\mathrm{content}}(T[n]), \\ h(D) &\neq h(D').\end{aligned}$$ As $W_{h(D)} = L$ and $h$ is conservative, $h$ must not change its hypothesis. Thus, for all $D'$ with $D \subseteq D' \subseteq L$ we get $h(D') = h(D)$, a contradiction. Thus we have, for all $n \geq n_0$, $$\begin{aligned} h'({\mathrm{content}}(T[n])) &= h'({\mathrm{content}}(T[n_0])) \\ &= {\mathrm{pad}}(h({\mathrm{content}}(T[n_0])),0)\end{aligned}$$ and $W_{h'({\mathrm{content}}(T[n_0]))} = W_{{\mathrm{pad}}(h({\mathrm{content}}(T[n_0])),0)} = L$, i.e. $h'$ ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learns $L$. Second, we will show that $h'$ is conservative. Whenever $h$ makes a mind change, $h'$ will also make a mind change; as, for all $n$, $W_{h({\mathrm{content}}(T[n]))} = W_{h'({\mathrm{content}}(T[n]))}$, we have that $h'$ is conservative in these cases. Thus, we have to show that $h'$ is conservative whenever it changes its mind because the first condition in the definition does not hold. Let $n$ such that $$h'({\mathrm{content}}(T[n])) \neq h'({\mathrm{content}}(T[n-1]))$$ because the first condition in the definition of $h'$ is violated. Let $C = {\mathrm{content}}(T[n])$. Thus, there are $D$ and $D'$ with $D \subseteq D' \subseteq C$ such that $h(D) = h(C)$ and $h(D') \neq h(C)$. We consider the case that $h(T[n]) = h(T[n-1])$ as otherwise $h'$ is obviously conservative. As $h$ is conservative we can conclude that there is $x \in D'$ such that $x \notin W_{h(D)}$. If not we could construct a text $T'$ with elements of $D$ on which $h$ would not be conservative. Thus there is $x \in D' \subseteq C$ such that $$x \notin W_{h(C)} = W_{h(T[n])} = W_{h(T[n-1])} = W_{h'(T[n-1])}$$ and therefore $h'$ is still conservative if it changes its mind. To show that $h'$ is syntactically decisive let $C \subseteq D \subseteq E$ such that $h'(C) \neq h'(D)$ and $h'(C) = h'(E)$. This implies that $C \subset E$. Thus $0 \neq \|C\| + 1 \neq \|E\|+1$ and therefore the second component in ${\mathrm{pad}}$ is different for $C$ and $E$. This implies that $h'(C) \neq h'(E)$ as ${\mathrm{pad}}$ is injective. The following Theorem is the main result of this section, showing that set-driven learning can be done not just conservatively, but also strongly decisively and cautiously at the same time. \[thm:SdConvCautSDec\] We have $$[{\mathbf{Txt}^{}}{\mathbf{Sd}}{\mathbf{Ex}}] = [{\mathbf{Txt}^{}}{\mathbf{Sd}}{\mathbf{Conv}}{\mathbf{SDec}}{\mathbf{Caut}}{\mathbf{Ex}}].$$ Following [@Kin-Ste:j:95:mon] we can assume a set-driven learner to be conservative. Let $h$ and $\mathcal{L}$ be such that $h$ TxtSdConvEx-learns $\mathcal{L}$ and suppose that $h$ is syntactically decisive using Lemma \[thm:SdSyntDec\]. We define a function $p$ using the S-m-n Theorem such that, for every set $D$ and $e$, $$W_{p(D,e)} = D \bigcup_{t \in {\mathbb{N}}} \begin{cases} W_e^t, & \text{if } h(D \cup W_e^t) = e; \\ \emptyset, & \text{otherwise.} \end{cases}$$ We define a function $N$ such that, for any finite set $D$, $$\begin{aligned} N(D) = \{ D' \subseteq D\ \|\ &h(D) = h(D')\}.\end{aligned}$$ We define $h'$, for all finite sets $D$, as $$h'(D) = p(\min(N(D)), h(D))$$ Let $L \in \mathcal{L}$ be given and let $T$ be a text for $L$. We first show that $h'$ ${\mathbf{Txt}^{}}{\mathbf{Sd}}{\mathbf{Ex}}$-learns $L$ from $T$. As $h$ TxtSdEx-learns $L$ we know that $h$ is strongly locking on $T$ (this was shown in [@Cas-Koe:c:10:colt]). Thus there is $n_0$ such that $T[n_0]$ is a locking sequence. Let $D' \subseteq {\mathrm{content}}(T[n_0])$ be minimal with $h(D') = h({\mathrm{content}}(T[n_0]))$. Thus we have, for all $n \geq n_0$, $\min(N({\mathrm{content}}(T[n]))) = D'$. From the construction of $p$ and $h$ syntactically decisive we get $$W_{p(D',h(D'))} = W_{h(D')}.$$ This shows that $h'$ ${\mathbf{Txt}^{}}{\mathbf{Sd}}{\mathbf{Ex}}$-learns $L$. Next we show the following claim. \[claim:SynDecConcl\] $\forall D\ (\forall D' \subseteq D\ \|\ D' \notin N(D))\ \forall C \in N(D) : C\backslash W_{h'(D')} \neq \emptyset.$ As $h$ is syntactically decisive we have that, for all $D''$ with $D' \subseteq D'' \subseteq D$, $h(D') = h(D'') = h(D).$ Therefore we get $$h(D') \neq h(D' \cup C).$$ Suppose, by way of contradiction, $C \subseteq W_{h'(D')}$. This implies that there is $t$ such that $C \subseteq D' \cup W_{h(D')}^t$ with $h(D' \cup W_{h(D')}^t) = h(D')$, according to the definitions of $h'$ and $p$. But, as $D' \subseteq D' \cup C \subseteq D' \cup W_{h(D')}^t$, this is a contradiction to $h$ being syntactically decisive. Let $i \leq j$ be such that $h'({\mathrm{content}}(T[i])) \neq h'({\mathrm{content}}(T[j]))$. To increase readability we let $D_0 = {\mathrm{content}}(T[i])$ and $D_1 = {\mathrm{content}}(T[j])$. As $h$ is syntactically decisive, $h'$ only changes its mind if $h$ changed its mind before. Thus we have $h(D_0) \neq h(D_1).$ As $D_0 \subseteq D_1$ and $D_0 \notin N(D_1)$ we get from Claim \[claim:SynDecConcl\] (with $C= D = D_1$ and $D' = D_0$) that $$D_1 \backslash W_{h'(D_0)} \neq \emptyset.$$ This shows that $h'$ is conservative. We will now show that $$W_{h'(D_1)} \nsubseteq W_{h'(D_0)},$$ as this implies that $h'$ is cautious and strongly decisive. From the construction of $h'$ we get that there is $B \subseteq D_1$ with $h(B) = h(D_1)$ such that $h'$ is consistent on $B$, i.e. $B \subseteq W_{h'(D_1)}.$ Using Claim \[claim:SynDecConcl\] again (this time with $C = B$, $D = D_1$ and $D' = D_0$), we see that there is $$x \in B \backslash W_{h'(D_0)} \subseteq W_{h'(D_1)} \backslash W_{h'(D_0)},$$ which shows that $W_{h'(D_0)} \not\subseteq W_{h'(D_1)}$. Monotone Learning ================= \[sec:Monotone\] In this section we show the hierarchies regarding monotone and strongly monotone learning, simultaneously for the settings of ${\mathbf{G}}$ and ${\mathbf{Sd}}$ in Theorems \[thm:SMon\] and \[thm:WMonNotMon\]. With Theorems \[thm:NatnumSDec\] and \[thm:MonInSDec\] we establish that monotone learnabilty implies strongly decisive learnability. \[thm:SMon\] There is a language ${\mathcal{L}}$ that is ${\mathbf{Txt}^{}}{\mathbf{Sd}}{\mathbf{Mon}}{\mathbf{WMon}}{\mathbf{Ex}}$-learnable but not ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{SMon}}{\mathbf{Ex}}$-learnable, i.e. $$[{\mathbf{Txt}^{}}{\mathbf{Sd}}{\mathbf{Mon}}{\mathbf{WMon}}{\mathbf{Ex}}] \backslash [{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{SMon}}{\mathbf{Ex}}] \neq \emptyset.$$ This is a standard proof which we include for completeness. Let $L_k = \{0, 2, 4, \dots, 2k, 2k+1\}$ and ${\mathcal{L}}= \{2{\mathbb{N}}\}\cup\{L_k\ \|\ k \in {\mathbb{N}}\}$. Let $e$ such that $W_e = 2{\mathbb{N}}$ and $p$ using the S-m-n Theorem such that, for all $k$, $$W_{p(k)} = L_k.$$ We first show that ${\mathcal{L}}$ is ${\mathbf{Txt}^{}}{\mathbf{Sd}}{\mathbf{Mon}}{\mathbf{WMon}}{\mathbf{Ex}}$-learnable. We let a learner $h$ such that, for all $\sigma$, $$h({\mathrm{content}}(\sigma)) = \begin{cases} e, & \text{if every } x \in {\mathrm{content}}(\sigma) \text{ is even;} \\ p(y), & \text{if } y \text{ is the least odd datum in } {\mathrm{content}}(\sigma). \end{cases}$$ Let $L_k \in {\mathcal{L}}$ and $T$ be a text for $L_k$. Thus, there is $n_0$ such that $T(n_0-1) = 2k+1$ and any element in ${\mathrm{content}}(T[n_0-1])$ is even. Then, we have, for all $n \geq n_0$, $h({\mathrm{content}}(T[n_0])) = h({\mathrm{content}}(T[n]))$ and $W_{h(t[n_0])} = W_{p(k)} = L_k$. It is easy to see that $h$ makes exactly one mind change on $T$ and this is at $n_0$. We have $W_e \cap {\mathrm{content}}(T)$ is a subset of $W_{p(k)} \cap {\mathrm{content}}(T)$ as $\{0,2, \dots, 2k\} \subseteq L_k$. Thus $h$ is monotone. Additionally $h$ is weakly monotone as it change its mind only if the first time a odd element is presented in the text and the previous hypotheses are $2{\mathbb{N}}$. Now, suppose that there is $h' \in {\mathcal{R}}$ and $h'$ ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{SMon}}{\mathbf{Ex}}$-learns ${\mathcal{L}}$. Let $\sigma$ be a locking sequence of $h'$ on $2{\mathbb{N}}$ and $k$ such that, for all $x \in {\mathrm{content}}(\sigma), x \leq 2k+1$. We let $T$ be a text for $L_k$ starting with $\sigma$. As $2{\mathbb{N}}\nsubseteq L_k$ we have that $h'$ is not strongly monotone on $T$ or $h$ does not ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learns $L_k$ from $T$. \[thm:WMonNotMon\] There is ${\mathcal{L}}$ such that ${\mathcal{L}}$ is ${\mathbf{Txt}^{}}{\mathbf{Sd}}{\mathbf{WMon}}{\mathbf{Ex}}$-learnable but not ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Mon}}{\mathbf{Ex}}$-learnable. This is a standard proof which we include for completeness. Let $L_k = \{x\ \|\ x \leq 2k+1\}$ and ${\mathcal{L}}= \{2{\mathbb{N}}\} \cup \{L_k\ \|\ k \in {\mathbb{N}}\}$. Let $e$ such that $W_e = 2{\mathbb{N}}$ and $p$ using the S-m-n Theorem such that, for all $k$, $$W_{p(k)} = L_k.$$ We define, for all $\sigma$, a learner $h$ such that $$h({\mathrm{content}}(\sigma)) = \begin{cases} e, & \text{if every element in } {\mathrm{content}}(\sigma) \text{ is even;} \\ p(y), &\text{else, } y \text{ is the maximal odd element in } {\mathrm{content}}(\sigma). \end{cases}$$ Let $L_k \in {\mathcal{L}}$ and a $T$ be a text for $L_k$. Then, there is $n_0$ such that $2k+1 \in {\mathrm{content}}(T[n_0])$ for the first time. Thus we have that for all $n \geq n_0, h({\mathrm{content}}(T[n_0])) = h({\mathrm{content}}(T[n]))$ and $W_{h({\mathrm{content}}(T[n_0]))} = W_{p(k)} = L_k$. Obviously $h$ learns $L_k$ weakly mononote as the learner only change its mind if a greater odd element appears in the text. Suppose now there is a learner $h' \in {\mathcal{R}}$ such that $h'$ ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Mon}}{\mathbf{Ex}}$-learns ${\mathcal{L}}$. Let $\sigma$ be a locking sequence of $h'$ on $2{\mathbb{N}}$ and $k$ such that, for all $x \in {\mathrm{content}}(\sigma)$, $x \leq 2k+1$. Let $\sigma' \supseteq \sigma$ a locking sequence of $h'$ on $L_{k}$ and $T$ be a text for $L_{k+1}$ starting with $\sigma'$. Let $\sigma'' \supseteq \sigma'$ be a locking sequence of $h'$ on $L_{k+1}$. Then, we have $$\begin{aligned} W_{h'(\sigma)} &= 2{\mathbb{N}}; \\ W_{h'(\sigma')} &= L_k ; \\ W_{h'(\sigma'')} &= L_{k+1}.\end{aligned}$$ As the datum $2k+2$ is in $2{\mathbb{N}}$ and in $L_{k+1}$ but not in $L_k$, $h'$ is not monotone on the text $T$ for $L_{k+1}$. The following theorem is an extension of a theorem from [@Bal-Cas-Mer-Ste-Wie:j:08], where the theorem has been shown for decisive learning instead of strongly decisive learning. \[thm:NatnumSDec\] Let ${\mathbb{N}}\in {\mathcal{L}}$ and ${\mathcal{L}}$ be ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learnable. Then, we have ${\mathcal{L}}$ is ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{SDec}}{\mathbf{Ex}}$-learnable. Let $h$ be a learner in Fulk normal form such that $h$ ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learns ${\mathcal{L}}$ with ${\mathbb{N}}\in {\mathcal{L}}$. As $h$ is strongly locking on ${\mathcal{L}}$ there is a locking sequence of $h$ on ${\mathbb{N}}$. Using this locking sequence we get an uniformly enumerable sequence $(L_i)_{i\in {\mathbb{N}}}$ of languages such that, 1. for $i \neq j$ and $L \supseteq L_i$, $L' \supseteq L_j$ with $L_i =^* L$, $L_j =^* L'$, $L \neq L'$; 2. for all $L \supseteq L_i$ with $L_i =^* L$, $L \notin {\mathcal{L}}$. We define a set $N(\sigma)$ such that, for every $\sigma$, $$N(\sigma) = L_{\|\sigma\|} \cup {\mathrm{content}}(\sigma).$$ We define, for all $\sigma$, a set $M(\sigma)$ such that $$M(\sigma) = \{\lambda \} \cup \{\tau\ \|\ \tau \subseteq \sigma\ \land\ h(\tau) \neq h(\tau^-)\ \land\ \forall x \in {\mathrm{content}}(\tau) : \Phi_{h(\tau)}(x) \leq \left\|\sigma\right\| \}.$$ Using the S-m-n Theorem we get a function $p \in {\mathcal{R}}$ such that, for all $\sigma$, $$W_{p(\sigma)} = \bigcup_{t \in \mathbb{N}} \begin{cases} W_{h(\sigma)}^t, & \text{if } \forall \rho \in W_{h(\sigma)}^t : h(\sigma) = h(\sigma \diamond \rho); \\ N(\sigma), & \text{otherwise.} \end{cases}$$ We will use the $p(\sigma)$ as hypotheses. Note that any hypothesis $p(\sigma)$ is either semantically equivalent to $h(\sigma)$ or, if $\sigma$ is not a locking sequence of $h$ for any language, $p(\sigma)$ is an index for a finite superset of $L_{\sigma}$. In the latter case we call the hypothesis $p(\sigma)$ poisoned. We define a learner $h'$ such that, for all $\sigma$, $$h'(\sigma) = p(\max(M(\sigma))).$$ Let $L \in {\mathcal{L}}$ and $T$ be a text for $L$. As $h$ is strongly locking and $h$ ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learns ${\mathcal{L}}$ there is $n_0$ such that, for all $\sigma \in {{\mathbb{S}\mathrm{eq}}}(L)$, $h(T[n_0]) = h(T[n_0] \diamond \sigma)$ and $W_{h(T[n_0])} = L$. Thus, there is $n_1 > n_0$ such that, for all $x \in {\mathrm{content}}(T[n_0])$, $\Phi_{h(T[n_0])}(x) \leq n_1$. This implies that, for all $n \geq n_1$, $h'(T[n_1]) = h'(T[n])$ and $$W_{h'(T[n_1])} = W_{p(\max(M(T[n_1])))} = \bigcup_{t \in \mathbb{N}} W_{h(T[n_0])}^t = L.$$ Next, we will show that $h'$ is strongly decisive. Suppose there are $i \leq j \leq k$ such that $W_{h'(T[i])} = W_{h'(T[k])}$ and $h'(T[i]) \neq h'(T[j])$. From the construction of the learner $h'$ we get $h(T[i]) \neq h(T[j])$. Case 1: $h'(T[i])$ is not a poisoned hypothesis. Independently of whether $h'(T[k])$ is poisoned or not, there is $\sigma \subseteq T[k]$ such that ${\mathrm{content}}(\sigma) \subseteq W_{h'(T[k])}$. ($T[k]$ if the hypothesis is poisend, $\max(M(T[k]))$ otherwise.) As $h'(T[i])$ is not poisened and $h(T[i]) \neq h(T[k])$ we get through the construction of $p$ that ${\mathrm{content}}(\sigma) \nsubseteq W_{h'(T[i])}$. Thus, we have $W_{h'(T[i])} \neq W_{h'(T[k])}$, a contradiction. Case 2: $h'(T[i])$ is poisoned. Thus, we have $T[i] \subseteq W_{h'(T[i])}$. Case 2.1: $h'(T[k])$ is not poisoned. Thus, $T[k]$ is a locking sequence on $h$ for a language $L \in {\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}(h)$ and $W_{h'(T[k])} \in {\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}(h)$. As $h'(T[i])$ is poisoned we have $W_{h'(T[i])} \notin {\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}(h)$. Thus, we get $W_{h'(T[i])} \neq W_{h'(T[k])}$, a contradiction. Case 2.2: $h'(T[k])$ is poisoned. As $T[i] \subset T[k]$ and $N(T[i]) =^* W_{h'(T[i])}$ and $N(T[k]) =^* W_{h'(T[k])}$ we have $W_{h'(T[i])} \neq W_{h'(T[k])}$. \[thm:MonInSDec\] We have that any monotone ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learnable class of languages is strongly decisive learnable, while the converse does not hold, i.e. $$[{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Mon}}{\mathbf{Ex}}] \subset [{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{SDec}}{\mathbf{Ex}}].$$ Let $h \in {\mathcal{R}}$ be a learner and ${\mathcal{L}}= {\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Mon}}{\mathbf{Ex}}(h)$. We distinguish the following two cases. We call ${\mathcal{L}}$ dense iff it contains a superset of every finite set. Case 1: ${\mathcal{L}}$ is dense. We will show now that $h$ ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{SMon}}{\mathbf{Ex}}$-learns the class ${\mathcal{L}}$. Let $L \in {\mathcal{L}}$ and $T$ be a text for $L$. Suppose there are $i$ and $j$ with $i < j$ such that $W_{h(T[i])} \nsubseteq W_{h(T[j])}$. Thus, we have $W_{h(T[i])}\backslash W_{h(T[j])} \neq \emptyset$. Let $x \in W_{h(T[i])}\backslash W_{h(T[j])}$. As ${\mathcal{L}}$ is dense there is a language $L' \in {\mathcal{L}}$ such that ${\mathrm{content}}(T[j]) \cup \{x\} \in L'$. Let $T'$ be a text for $L'$ and $T''$ be such that $T'' = T[j] \diamond T'$. Obviously, $T''$ is a text for $L'$. We have that $x \in W_{h(T''[i])}$ but $x \notin W_{h(T''[j])}$ which is a contradiction as $h$ is monotone. Thus, $h$ ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{SMon}}{\mathbf{Ex}}$-learns ${\mathcal{L}}$, which implies that $h$ ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{WMon}}{\mathbf{Ex}}$-learns ${\mathcal{L}}$. Using Corollary \[cor:WMonInSDec\] we get that ${\mathcal{L}}$ is ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{SDec}}{\mathbf{Ex}}$-learnable. Case 2: ${\mathcal{L}}$ is not dense. Thus, ${\mathcal{L}}' = {\mathcal{L}}\cup {\mathbb{N}}$ is ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$-learnable. Using Theorem \[thm:NatnumSDec\] ${\mathcal{L}}'$ is ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{SDec}}{\mathbf{Ex}}$-learnable and therefore so is ${\mathcal{L}}$. Note that $[{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{SDec}}{\mathbf{Ex}}] \subseteq [{\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Mon}}{\mathbf{Ex}}]$ does not hold as in Case 1 with Corollary \[cor:WMonInSDec\] a proper subset relation is used. [BCM[[$^{+}$]{}]{}08]{} D. Angluin. Inductive inference of formal languages from positive data. , 45:117–135, 1980. L. Blum and M. Blum. Toward a mathematical theory of inductive inference. , 28:125–155, 1975. G. Baliga, J. Case, W. Merkle, F. Stephan, and W. Wiehagen. When unlearning helps. , 206:694–709, 2008. J. Case and T. K[ö]{}tzing. Strongly non-[U]{}-shaped learning results by general techniques. In [Proc. of COLT (Conference on Learning Theory)]{}, pages 181–193, 2010. J. Case and S. Moelius. Optimal language learning from positive data. , 209:1293–1311, 2011. M. Fulk. 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Language learning from texts: Mind changes, limited memory and monotonicity. , 123:224–241, 1995. S. Lange and T. Zeugmann. Monotonic versus non-monotonic language learning. In [Proc. of Nonmonotonic and Inductive Logic]{}, pages 254–269, 1993. D. Osherson, M. Stob, and S. Weinstein. Learning strategies. , 53:32–51, 1982. D. Osherson, M. Stob, and S. Weinstein. . MIT Press, Cambridge, Mass., 1986. H. Rogers. . McGraw Hill, New York, 1967. Reprinted by MIT Press, Cambridge, Massachusetts, 1987. G. Schäfer-Richter. . PhD thesis, RWTH Aachen, 1984. K. Wexler and P. Culicover. . MIT Press, Cambridge, Massachusetts, 1980. R. Wiehagen. A thesis in inductive inference. In [Proc. of Nonmonotonic and Inductive Logic]{}, pages 184–207, 1991. [^1]: ${\mathbf{Txt}^{}}$ stands for learning from a text of positive examples; ${\mathbf{G}}$ stands for Gold, who introduced this model, and is used to to indicate full-information learning; ${\mathbf{Ex}}$ stands for explanatory. [^2]: One might wonder why the U-shape can be achieved on a language which is to be learned: after all, those can be avoided, according to the theorem that non-U-shaped learning is not a restriction to ${\mathbf{Txt}^{}}{\mathbf{G}}{\mathbf{Ex}}$ [@Bal-Cas-Mer-Ste-Wie:j:08]. However, the price for avoiding it is to output a conjecture for ${\mathbb{N}}$.
--- abstract: \| It has been pointed out that supersymmetric extensions of the Standard Model can induce significant changes to the theoretical prediction of the ratio $\Gamma\left(K\rightarrow e\nu\right)/\Gamma \left(K\rightarrow\mu\nu\right)\equiv R_{K}$, through lepton flavour violating couplings. In this work we carry out a full computation of all one-loop corrections to the relevant $\nu\ell H^{+}$ vertex, and discuss the new contributions to $R_{K}$ arising in the context of different constrained (minimal supergravity inspired) models which succeed in accounting for neutrino data, further considering the possibility of accommodating a near future observation of a $\mu\to e\gamma$ transition. We also re-evaluate the prospects for $R_{K}$ in the framework of unconstrained supersymmetric models. In all cases, we address the question of whether it is possible to saturate the current experimental sensitivity on $R_{K}$ while in agreement with the recent limits on $B$-meson decay observables (in particular BR($B_{s}\to\mu^+\mu^-$) and BR($B_{u}\to\tau\nu$)), as well as BR($\tau\to e \gamma$) and available collider constraints. Our findings reveal that in view of the recent bounds, and even when enhanced by effective sources of flavour violation in the right-handed $\tilde{e}-\tilde{\tau}$ sector, constrained supersymmetric (seesaw) models typically provide excessively small contributions to $R_{K}$. Larger contributions can be found in more general settings, where the charged Higgs mass can be effectively lowered, and even further enhanced in the unconstrained MSSM. However, our analysis clearly shows that even in this last case SUSY contributions to $R_{K}$ are still unable to saturate the current experimental bounds on this observable, especially due to a strong tension with the $B_{u}\to\tau\nu$ bound. --- CFTP/12-004\ PCCF RI 12-03\ Revisiting the $\Gamma\left(K\rightarrow e\nu\right)/ \Gamma\left(K\rightarrow\mu\nu\right)$ ratio in supersymmetric unified models\ Renato M. Fonseca$^{a}$, J. C. Romão$^{a}$ and A. M. Teixeira$^{b}$ $^{a}$ Centro de Física Teórica de Partículas, CFTP, Instituto Superior Técnico,\ Universidade Técnica de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal $^{b}$ Laboratoire de Physique Corpusculaire, CNRS/IN2P3 – UMR 6533,\ Campus des Cézeaux, 24 Av. des Landais, F-63171 Aubière Cedex, France Introduction ============ Neutrino oscillations have provided the first experimental manifestation of flavour violation in the lepton sector, fuelling the need to consider extensions of the Standard Model (SM) that succeed in explaining the smallness of neutrino masses and the observed pattern of their mixings [@Fogli:2011qn; @Schwetz:2011zk; @theta13]. In addition to the many facilities dedicated to study neutral leptons, there is currently a great experimental effort to search for signals of flavour violation in the charged lepton sector (cLFV), since such an observation would provide clear evidence for the existence of new physics beyond the SM (trivially extended to accommodate massive neutrinos). The quest for the origin of the underlying mechanism of flavour violation in the lepton sector has been actively pursued in recent years, becoming even more challenging as the MEG experiment is continually improving the sensitivity to $\mu\to e\gamma$ decays [@arXiv:1107.5547], thus opening the door for a possible measurement (observation) in the very near future. The current bounds on other radiative decays (i.e. $\ell_{i}\to\ell_{j}\gamma$), or three-body decays ($\ell_{i}\to3\ell_{j}$) are already impressive [@PDG], and are expected to be further improved in the future. Supersymmetric (SUSY) extensions of the SM offer new sources of CP and flavour violation, in both quark and lepton sectors. Given the strong experimental constraints, especially on CP and flavour violating observables involving the strongly interacting sector, phenomenological analyses in general favour the so-called “flavour-blind” mechanisms of SUSY breaking, where universality of the soft breaking terms is assumed at some high energy scale: in these constrained scenarios, the only sources of flavour violation (FV) are the quark and charged lepton Yukawa couplings. In order to accommodate current neutrino data, mechanisms of neutrino mass generation, such as the seesaw (in its different realisations - for a review of the latter, see for instance [@Abada:2007ux; @Abada:2011rg]), are often implemented in the framework of (constrained) SUSY models: in the case of the so-called “SUSY-seesaw”, radiatively induced flavour violation in the slepton sector [@Borzumati:1986qx] can provide sizable contributions to cLFV observables. The latter have been extensively studied, both at high- and low-energies, over the past years (see e.g. [@Raidal:2008jk]). Flavour violation can be also incorporated in a more phenomenological approach, where at low-energies new sources of FV are present in the soft SUSY breaking terms. However, these are severely constrained by a large number of observables (see, e.g. [@Antonelli:2009ws] and references therein). In addition to the above mentioned rare lepton decays, leptonic and semi-leptonic meson decays also offer a rich testing ground for cLFV. Here we will be particularly interested in leptonic $K$ decays, which (as is also the case of leptonic $\pi$ decays) constitute very good probes of violation of lepton universality. The potential of these observables, especially regarding SUSY extensions of the SM, was firstly noticed in [@Masiero:2005wr], and later investigated in greater detail in [@Masiero:2008cb; @Ellis:2008st; @Girrbach:2012km]. By themselves, these decays are heavily hampered by hadronic uncertainties and, in order to reduce the latter (and render these decays an efficient probe of new physics), one usually considers the ratio $$R_{K}\,\equiv\,\frac{\Gamma\left(K^{+}\rightarrow e^{+}\nu \left[\gamma\right]\right)}{\Gamma\left(K^{+}\rightarrow\mu^{+}\nu \left[\gamma\right]\right)}\,,\label{eq:rk:def}$$ since in this case the hadronic uncertainties cancel to a very good approximation. As a consequence, the SM prediction can be computed with high precision [@Marciano:1993sh; @Finkemeier:1995gi; @Cirigliano:2007xi]. The most recent analysis has provided the following value [@Cirigliano:2007xi]: $$R_{K}^{\textrm{SM}}=(2.477\pm0.001)\times10^{-5}\,.\label{eq:Cirigliano:2007xi}$$ On the experimental side, the NA62 collaboration has recently obtained very stringent bounds [@Goudzowski:2011tc]: $$R_{K}^{\textrm{exp}}\,=\,(2.488\pm0.010)\,\times10^{-5}\,,\label{eq:rk:NA62}$$ which should be compared with the SM prediction (Eq. (\[eq:Cirigliano:2007xi\])). In order to do so, it is often useful to introduce the following parametrisation, $$R_{K}^{\textrm{exp}}\,=\, R_{K}^{\textrm{SM}}\left(1+\Delta r\right)\,, \quad\quad\Delta r\equiv\nicefrac{R_{K}}{R_{K}^{\textrm{SM}}}-1\,, \label{eq:deltark}$$ where $\Delta r$ is a quantity denoting potential contributions arising from scenarios of new physics (NP). Comparing the theoretical SM prediction to the current bounds (i.e., Eqs. (\[eq:Cirigliano:2007xi\], \[eq:rk:NA62\])), one verifies that observation is compatible with the SM (at 1$\sigma$) for $$\Delta r\,=\,\left(4\pm4\right)\times10^{-3}\,.\label{eq:deltarexp}$$ Previous analyses have investigated supersymmetric contributions to $R_{K}$ in different frameworks, as for instance low-energy SUSY extensions of the SM (i.e. the unconstrained Minimal Supersymmetric Standard Model (MSSM)) [@Masiero:2005wr; @Masiero:2008cb; @Girrbach:2012km], or non-minimal grand unified models (where higher dimensional terms contribute to fermion masses) [@Ellis:2008st]. These studies have also considered the interplay of $R_{K}$ with other important low-energy flavour observables, magnetic and electric lepton moments and potential implications for leptonic CP violation. Distinct computations, based on an approximate parametrisation of flavour violating effects - the Mass Insertion Approximation (MIA) [@Hall:1985dx] - allowed to establish that SUSY LFV contributions can induce large contributions to the breaking of lepton universality, as parametrised by $\Delta r$. The dominant FV contributions are in general associated to charged-Higgs mediated processes, being enhanced due to non-holomorphic effects - the so-called “HRS” mechanism [@Hall:1993gn] -, and require flavour violation in the $RR$ block of the charged slepton mass matrix. It is important to notice that these Higgs contributions have been known to have an impact on numerous observables, and can become especially relevant for the large $\tan\beta$ regime [@Hou:1992sy; @Hall:1993gn; @Chankowski:1994ds; @Babu:1999hn; @Carena:1999py; @Babu:2002et; @Brignole:2003iv; @Brignole:2004ah; @Arganda:2004bz; @Paradisi:2005tk; @Paradisi:2006jp; @RamseyMusolf:2007yb]. In the present work, we re-evaluate the potential of a broad class of supersymmetric extensions of the SM to saturate the current measurement of $R_{K}$. Contrary to previous studies, we conduct a full computation of the one-loop corrections to the $\nu\ell H^{+}$ vertex, taking into account the important contributions from non-holormophic effective Higgs-mediated interactions. When possible we establish a bridge between our results and approximate analytical expressions in the literature, and we stress the potential enhancements to the total SUSY contributions. In our numerical analysis we re-investigate the prospects regarding $R_{K}$ of a constrained MSSM onto which several seesaw realisations are embedded (type I [@seesaw:I] and II [@seesaw:II], as well as the inverse seesaw [@Mohapatra:1986bd]), also briefly addressing $L$–$R$ symmetric models [@LRmodels1; @LRmodels2]. We then consider more relaxed scenarios, such as non-universal Higgs mass (NUHM) models at high-scale (which are known to enhance this class of observables [@Ellis:2008st] due to potentially lighter charged Higgs boson masses), and discuss the general prospects of unconstrained low-energy SUSY models. In all cases, we revisit the $R_{K}$ observable in the light of new experimental data: in addition to LHC bounds[^1] on the sparticle spectrum [@LHC:2011] and a number of low-energy flavour-related bounds [@PDG; @arXiv:1107.5547], we implement the very recent LHCb results concerning the BR($B_{s}\to\mu^{+}\mu^{-}$) [@Aaij:2012ac]. As we discuss here, the increasing tension with low-energy observables, in particular with $B_{u}\to\tau\nu$, precludes sizable SUSY contributions to $R_{K}$ even in the context of otherwise favoured candidate models as is the case of semi-constrained and unconstrained SUSY models. This document is organised as follows. Section \[sec:RK:formulae\] is devoted to the computation of the 1-loop MSSM prediction for $R_{K}$. We compare our (full) result to the approximations in the literature by means of the mass insertion approximation (among other simplifications), and discuss the dominant sources of flavour violation, and the implications to other observables. Our results for a number of models are collected in Section \[sec:res\]. Further discussion and concluding remarks are given in Section \[sec:concs\]. In the Appendices, we detail the computation of the renormalised charged lepton - neutrino - charged Higgs vertex, and summarise the key features of two supersymmetric seesaw realisations (types I and II) used in the numerical analysis. Supersymmetric contributions to $R_{K}$ {#sec:RK:formulae} ======================================= In the SM, the decay widths of pseudoscalar mesons into light leptons are given by $$\Gamma^{\text{SM}}(P^{\pm}\!\!\to\ell^{\pm}\nu)= \frac{G_{F}^{2}m_{P}m_{\ell}^{2}}{8\pi} \left(\!1-\!\frac{m_{\ell}^{2}}{m_{P}^{2}}\right)^{2}\!\! f_{P}^{2} \|V_{qq^{\prime}}\|^{2} , \label{eq:SM:Pdecays}$$ where $P$ denotes $\pi,K,D$ or $B$ mesons, with mass $m_{P}$ and decay constant $f_{P}$, and where $G_{F}$ is the Fermi constant, $m_{\ell}$ the lepton mass and $V_{qq^{\prime}}$ the corresponding Cabibbo-Kobayashi-Maskawa (CKM) matrix element. These decays are helicity suppressed (as can be seen from the factor $m_{\ell}^{2}$ in Eq. (\[eq:SM:Pdecays\])), and the prediction for their amplitude is thus hampered by the hadronic uncertainties in the meson decay constants. As mentioned in the Introduction, ratios of these amplitudes are independent of $f_{P}$ to a very good approximation, and the SM prediction can then be computed very precisely. Concerning the kaon decay ratio $R_{K}$, the SM prediction (inclusive of internal bremsstrahlung radiation) is [@Cirigliano:2007xi] $$R_{K}^{\text{SM}}\,=\,\left(\frac{m_{e}}{m_{\mu}}\right)^{2}\, \left(\frac{m_{K}^{2}-m_{e}^{2}}{m_{K}^{2}-m_{\mu}^{2}}\right)^{2}\, \left(1+\delta R_{\text{QED}}\right)\,,$$ where $\delta R_{\text{QED}}$ is a small electromagnetic correction accounting for internal bremsstrahlung and structure-dependent effects ($\delta R_{\text{QED}}=(-3.60\pm0.04)\%$ [@Cirigliano:2007xi]). -------------------------------------------------------------------------------------------------------------------------------------------- ![Tree level contributions to $R_{K}$ - SM and charged Higgs.[]{data-label="fig:RK:SM:Higgs"}](Diagram1.pdf "fig:"){width="50mm"} \[+2mm\] ![Tree level contributions to $R_{K}$ - SM and charged Higgs.[]{data-label="fig:RK:SM:Higgs"}](Diagram2.pdf "fig:"){width="50mm"} -------------------------------------------------------------------------------------------------------------------------------------------- In supersymmetric models, the extended Higgs sector can play an important rôle in lepton flavour violating transitions and decays (see [@Hou:1992sy; @Hall:1993gn; @Chankowski:1994ds; @Babu:1999hn; @Carena:1999py; @Babu:2002et; @Brignole:2003iv; @Brignole:2004ah; @Arganda:2004bz; @Paradisi:2005tk; @Paradisi:2006jp; @RamseyMusolf:2007yb]). The effects of the additional Higgs are also sizable in meson decays through a charged Higgs boson, as schematically depicted in Fig. \[fig:RK:SM:Higgs\]. In particular, for kaons, one finds [@Hou:1992sy] $$\begin{aligned} \Gamma(K^{\pm}\to\ell^{\pm}\nu)\,=&\,\Gamma^{\text{SM}}(K^{\pm} \to\ell^{\pm}\nu) \nonumber\\ &\times \left(1-\tan^{2}\beta\,\frac{m_{K}^{2}}{m_{H^+}^{2}}\, \frac{m_{s}}{m_{s}+m_{u}}\right)^{2};\label{eq:kaon:gamma:smsusy}\end{aligned}$$ however, despite this new tree-level contribution, $R_{K}$ is unaffected, as the extra factor does not depend on the (flavoured) leptonic part of the process. New contributions to $R_{K}$ only emerge at higher order: at one-loop level, there are box and vertex contributions, wave function renormalisation, which can be both lepton flavour conserving (LFC) and lepton flavour violating. Flavour conserving contributions arise from loop corrections to the $W^{\pm}$ propagator, through heavy Higgs exchange (neutral or charged) as well as from chargino/neutralino-sleptons (in the latter case stemming from non-universal slepton masses, i.e., a selectron-smuon mass splitting). As concluded in [@Masiero:2005wr], in the framework of SUSY models where lepton flavour is conserved, the new contributions to $\Delta r^{\text{SUSY}}$ are too small to be within experimental reach. On the other hand, Higgs mediated LFV processes are capable of providing an important contribution when the kaon decays into a electron plus a tau-neutrino. For such LFV Higgs couplings to arise, the leptonic doublet ($L$) must couple to more than one Higgs doublet. However, at tree level in the MSSM, $L$ can only couple to $H_{1}$, and therefore such LFV Higgs couplings arise only at loop level, due to the generation of an effective non-holomorphic coupling between $L$ and $H_{2}^{}$ - the HRS mechanism [@Hall:1993gn] - which is a crucial ingredient in enhancing the Higgs contributions to LFV observables. In what follows, we address the impact of these non-holomorphic terms for $R_{K}$. LFV Higgs mediated contributions to $R_{K}$ ------------------------------------------- We consider as starting point the MSSM, defined by its superpotential and soft-SUSY breaking Lagrangian. We detail below the relevant terms for our discussion: $$\mathcal{W}=\hat{U}^{c} Y^{u} \hat{Q} \hat{H}_{2} -\hat{D}^{c} Y^{d} \hat{Q} \hat{H}_{1} - \hat{E}^{c} Y^{l} \hat{L} \hat{H}_{1} -\mu \hat{H}_{1} \hat{H}_{2} , \label{eq:Wlepton0:def}$$ $$\begin{aligned} \mathcal{V}_{\text{soft}}=&-\mathcal{L}_{\text{soft}}\, =(M_{\alpha}\,\psi_{\alpha}\,\psi_{\alpha}+\text{h.c.})+ m_{H_{i}}^{2}\, H_{i}^{}\, H_{i} \nonumber \\ &+(B\, H_{1}\, H_{2}+\text{h.c.}) +\tilde{\ell}_{L}^{}\, m_{\tilde{L}}^{2}\, \tilde{\ell}_{L}\,+\tilde{\ell}_{R}^{}\, m_{\tilde{R}}^{2}\tilde{\ell}_{R}\, \nonumber \\ & +\,(H_{1}\,\tilde{\ell}_{R}^{}\, A^{l}\,\tilde{\ell}_{L}\,+\text{h.c.})+...\,,\end{aligned}$$ where $M_{\alpha}$ denotes the soft-gaugino mass terms, “...” stand for the squark terms, and we have omitted flavour indices. For the SU(2) superfield products, we adopt the convention $\hat{H}_{1}\,\hat{H}_{2}\equiv\hat{H}_{1}^{1}\, \hat{H}_{2}^{2}-\hat{H}_{1}^{2}\,\hat{H}_{2}^{1}$ (and likewise for similar cases). From an effective theory approach, the HRS mechanism can be accounted for by additional terms, corresponding to the higher-order corrections to the Higgs-neutrino-charged lepton interaction (schematically depicted in Fig. \[fig:RK:Hvertex\]). ![Corrections to the $\nu\ell H^{+}$ vertex, as discussed in the text.[]{data-label="fig:RK:Hvertex"}](Diagram4.pdf){width="85mm"} At tree-level, the Lagrangian describing the $\nu\ell H^{\pm}$ interaction is given by $$\begin{aligned} \mathcal{L}_{0}^{H^{\pm}}\,=&\,\overline{\nu}_{L}\, Y^{l\dagger}\,\ell_{R}\, H_{1}^{-}+\textrm{h.c.}\, \nonumber\\ =&\,\left(2^{3/4}\, G_{F}^{1/2}\right)\,\tan\beta\, \overline{\nu}_{L}\, M^{l}\,\ell_{R}\, H^{+}+\textrm{h.c.}\,, \label{eq:L:nulH0}\end{aligned}$$ with $M^{l}=\textrm{diag}\left(m_{e},m_{\mu},m_{\tau}\right)$. At loop level, two new terms are generated: $\overline{\nu}_{L}\Delta^{+}\ell_{R}H_{2}^{+}- \overline{\ell}_{L}\Delta^{0}\ell_{R}H_{2}^{0}+\textrm{h.c.}$. The second one, with $\Delta^{0}$, forces a redefinition of the charged lepton Yukawa couplings, $Y^{l\dagger}=\frac{M^{l}}{v_{1}}$ $\to$ $Y^{l\dagger}\approx\frac{M^{l}}{v_{1}}-\Delta^{0}\tan\beta$, which in turn implies a redefinition of the charged lepton propagator; the term with $\Delta^{+}$ corrects the Higgs-neutrino-charged lepton vertex[^2]. Once these terms are taken into account, the interaction Lagrangian, Eq. (\[eq:L:nulH0\]), becomes $$\begin{aligned} \mathcal{L}^{H^{\pm}} & \,=\,\left(2^{3/4}\, G_{F}^{1/2}\right)\, \tan\beta\,\,\overline{\nu}_{L}\, M^{l}\,\ell_{R}\, H^{+}\, \nonumber\\ &+\,\cos\beta\,\,\overline{\nu}_{L}\,\left(\Delta^{+} -\Delta^{0}\,\tan^{2}\beta\right)\,\ell_{R}\, H^{+}+\textrm{h.c.}\,. \label{eq:L:nulH}\end{aligned}$$ Since in the $\textrm{SU(2)}_{L}$-preserving limit $\Delta^{+}=\Delta^{0}$, it is reasonable to assume that, after electroweak (EW) symmetry breaking, both terms remain approximately of the same order of magnitude. Hence, it is clear that the contribution associated with $\Delta^{0}$ (the loop contribution to the charged lepton mass term) will be enhanced by a a factor of $\tan^{2}\beta$ when compared to the one associated with $\Delta^{+}$. This simple discussion allows to understand the origin of the dominant SUSY contribution[^3] to $R_{K}$. As we proceed to discuss, a quantitative assessment of the corrections to $\Delta^{+}$ and $\Delta^{0}$ requires considering the higher-order effects on the vertex $\overline{\nu}_{L}\, Z^{H}\,\ell_{R}\, H^{+}$ (see also [@Bellazzini:2010gn]). The $Z^{H}$ matrix depends on the following (loop-induced) quantities: - $\eta_{L}^{\ell}$ and $\eta_{L}^{\nu}$ (corrections to the kinetic terms of $\ell_{L}$ and $\nu_{L}$); - $\eta_{m}^{\ell}$ (correction to the charged lepton mass term); - $\eta^{H}$ (correction to the $\nu\ell H$ vertex). The expressions for the distinct $\eta$-parameters can be found in Appendix \[sec:app1\]. Instead of $Z^{H}$, which includes both tree and loop level effects, it proves to be more convenient to use the following combination, $$-\frac{\tan\beta}{2^{3/4}G_{F}^{1/2}}\, \left(\frac{m_{K}}{m_{H^+}}\right)^{2}\,\frac{m_{s}}{m_{s}+m_{u}}\, Z^{H}\,\left(M^{l}\right)^{-1}\,\equiv\,\epsilon\,\mathbf{1}+ \Delta\,,\label{eq:epsilondelta}$$ where $$\begin{aligned} \epsilon & =-\tan^{2}\beta\, \left(\frac{m_{K}}{m_{H^+}}\right)^{2}\frac{m_{s}}{m_{s}+m_{u}}\,, \label{eq:epsilon:def}\\ \Delta & =\epsilon\,\left[\frac{\eta_{L}^{\ell}}{2}- \frac{\eta_{L}^{\nu}}{2}+\left(\frac{\eta^{H}}{2^{3/4}\, G_{F}^{1/2} \tan\beta}-\eta_{m}^{\ell}\right) \left(M^{l}\right)^{-1}\right] .\label{eq:delta:def}\end{aligned}$$ In the above, $\epsilon$ encodes the tree level Higgs mediated amplitude (which does not change the SM prediction for $R_{K}$), while $\Delta$, a matrix in lepton flavour space, encodes the 1-loop effects. The main contribution is expected to arise from $\eta_{m}^{\ell}$. The $\Delta r$ observable is then related to $\epsilon$ and $\Delta$ as follows: $$\Delta r\equiv\frac{R_{K}}{R_{K}^{\text{SM}}}-1\, =\,\frac{\left[\left(\mathbf{1}+ \frac{\Delta^{\dagger}}{1+\epsilon}\right) \left(\mathbf{1}+\frac{\Delta}{1+\epsilon}\right)\right]_{ee}} {\left[\left(\mathbf{1}+\frac{\Delta^{\dagger}}{1+\epsilon}\right) \left(\mathbf{1}+\frac{\Delta}{1+\epsilon}\right)\right]_{\mu\mu}}-1\,. \label{eq:deltar:epsilondelta}$$ If the slepton mixing is sufficiently large, this expression can be approximated as $$\Delta r\,\approx\,2\,\operatorname{Re}(\Delta_{ee})\, +\,(\Delta^{\dagger}\Delta)_{ee}\,.$$ In the above, the first (linear) term on the right hand-side is due to an interference with the SM process, and is thus lepton flavour conserving. As shown in [@Masiero:2005wr], this contribution can be enhanced through both large $RR$ and $LL$ slepton mixing. On the other hand, the quadratic term $(\Delta^{\dagger}\Delta)_{ee}$ can be augmented mainly through a large LFV contribution from $\Delta_{\tau e}$, which can only be obtained in the presence of significant $RR$ slepton mixing. Generating $\Delta r$: sources of flavour violation and experimental constraints -------------------------------------------------------------------------------- In order to understand the dependence of $\Delta r$ on the SUSY parameters, and the origin of the dominant contributions to this observable, an approximate expression for $\Delta$ is required. Firstly, we notice that the previous discussion, leading to Eq. (\[eq:L:nulH\]), suggests that the $\eta_{m}^{\ell}$ term is responsible for the dominant contributions to $\Delta r$. Thus, in what follows, and for the purpose of obtaining simple analytical expressions, we shall neglect the contributions of the other terms (although these are included in the numerical analysis of Section \[sec:res\]). A fairly simple analytical insight can be obtained when working in the limit in which the virtual particles in the loops (sleptons and gauginos) are assumed to have similar masses, so that their relative mass splittings are indeed small. In this limit, one can Taylor-expand the loop functions entering $\eta_{m}^{\ell}$ (see Appendix \[sec:app1\]); working to third order in this expansion, and keeping only the terms enhanced by a factor of $m_{\tau}\,\tan\beta\,\frac{m_{\text{SUSY}}}{m_{\text{EW}}}$ (where $m_{\text{SUSY}},\, m_{\text{EW}}$ denote the SUSY breaking scale and EW scale, respectively), we obtain $$\begin{aligned} \Delta r\,\sim\, & \left[1+X\left(1- \frac{9}{10}\frac{\delta}{\overline{m}_{\widetilde{\ell},\chi^{0}}^{2}}\right) \left(m_{\tilde{L}}^{2}\right)_{e\tau}\right]^{2}-1 \nonumber\\ &+X^{2}\left[-\mu^{2} +\delta\left(3-\frac{3}{10}\frac{\mu^{2}+2M_{1}^{2}} {\overline{m}_{\widetilde{\ell},\chi^{0}}^{2}}\right)\right]^{2}, \label{eq:deltar:approx:long}\end{aligned}$$ where $\mu$, $M_{1}$ and $(m_{\tilde{L}}^{2})_{e\tau}$ denote the low-energy values of the Higgs bilinear term, bino soft-breaking mass, and off-diagonal entry of the soft-breaking left-handed slepton mass matrix, respectively. We have also introduced $\overline{m}_{\widetilde{\ell},\chi^{0}}^{2}= \frac{1}{2}(\langle{m}_{\widetilde{\ell}}^{2}\rangle+ \langle{m}_{\chi^{0}}^{2}\rangle)$, the average mass squared of sleptons and neutralinos ($\approx m^2_{\text{SUSY}}$), and $\delta=\frac{1}{2}(\langle{m}_{\widetilde{\ell}}^{2}\rangle- \langle{m}_{\chi^{0}}^{2}\rangle)$, the corresponding splitting. The quantity $X$ is given by $$X\,\equiv\,\frac{1}{192\pi^{2}}\, m_{K}^{2}\, g'^{2}\,\mu\, M_{1}\,\frac{\tan^{3}\beta}{m_{H^+}^{2}}\, \frac{m_{\tau}}{m_{e}}\, \frac{\left(m_{\tilde{R}}^{2}\right)_{\tau e}} {(\overline{m}_{\widetilde{\ell},\chi^{0}}^{2})^{3}}\,, \label{eq:deltar:approx:X}$$ and it illustrates in a transparent (albeit approximate) way the origin of the terms contributing to the enhancement of $R_{K}$: in addition to the factor ${\tan^{3}\beta}/{m_{H^+}^{2}}$, usually associated with Higgs exchanges, the crucial flavour violating source emerges from the off-diagonal $(\tau e)$ entry of the right-handed slepton soft-breaking mass matrix. Using the above analytical approximation, one easily recovers the results in the literature, usually obtained using the MIA. For instance, Eq. (11) of Ref. [@Masiero:2005wr] amounts to $$\Delta r\,\sim\,2X\,\left(m_{\tilde{L}}^{2}\right)_{e\tau}\, +\, X^{2}\,\left(m_{\tilde{L}}^{2}\right)_{e\tau}^{2}\, +\, X^{2}\,\delta^{2}\,, \label{eq:deltar:approx}$$ which stems from having kept the dominant (crucial) second and third order contributions in the expansion: $X^{2}\delta^{2}$ and $2X\left(m_{\tilde{L}}^{2}\right)_{e\tau}+ X^{2}\left(m_{\tilde{L}}^{2}\right)_{e\tau}^{2}$, respectively. Regardless of the approximation considered, it is thus clear that the LFV effects on kaon decays into a $e\nu$ or $\mu\nu$ pair can be enhanced in the large $\tan\beta$ regime (especially in the presence of low values of $m_{H^+}$), and via a large $RR$ slepton mixing $\left(m_{\tilde{R}}^{2}\right)_{\tau e}$. Although the latter is indeed the privileged source, notice that, as can be seen from Eq. (\[eq:deltar:approx\]), a strong enhancement can be obtained from sizable flavour violating entries of the left-handed slepton soft-breaking mass, $\left(m_{\tilde{L}}^{2}\right)_{e\tau}$. This is in fact a globally flavour conserving effect (which can also account for negative contributions to $R_{K}$). Previous experimental measurements of $R_{K}$ appeared to favour values smaller than the SM theoretical estimation, thus motivating the study of regimes leading to negative values of $\Delta r$ [@Masiero:2005wr], but these regimes have now become disfavoured in view of the present bounds, Eq. (\[eq:deltarexp\]). Clearly, these Higgs mediated exchanges, as well as the FV terms at the origin of the strong enhancement to $R_{K}$, will have an impact on a number of other low-energy observables, as can be easily inferred from the structure of Eqs. (\[eq:deltar:approx:long\]-\[eq:deltar:approx\]). This has been extensively addressed in the literature [@Masiero:2005wr; @Masiero:2008cb; @Ellis:2008st; @Girrbach:2012km], and here we will only briefly discuss the most relevant observables: electroweak precision data on the anomalous electric and magnetic moments of the electron, as well as the naturalness of the electron mass, directly constrain the $\eta_{m}^{\ell}$ corrections (and $\eta_{L}^{\ell}$, $\eta^{H}$); low-energy cLFV observables, such as $\tau\to\ell\gamma$ and $\tau\to3\ell$ decays are also extremely sensitive probes of Higgs mediated exchanges, and in the case of $\tau-e$ transitions, depend on the same flavour violating entries. It has been suggested that positive and negative values of $\Delta r$ can be of the order of 1%, still in agreement with data on the electron’s electric dipole moment and on $\tau\rightarrow\ell\gamma$ [@Masiero:2005wr; @Masiero:2008cb; @Ellis:2008st]. Finally, other meson decays, such as $B\to\ell\ell$ (and $B\to\ell\nu$), exhibit a similar dependence on $\tan\beta$, $\tan^{n}\beta/{m_{H^+}}^{4}$ [@BToLL] ($n$ ranging from 2 to 6, depending on the other SUSY parameters), and may also lead to indirect bounds on $\Delta r$. In particular, the strict bounds on BR($B_{u}\to\tau\nu$) [@PDG] and the very recent limits on BR($B_{s}\to\mu^{+}\mu^{-}$) [@Aaij:2012ac] might severely constrain the allowed regions in SUSY parameter space for large $\tan\beta$. Although we will come to this issue in greater detail when discussing the numerical results, it is clear that the similar nature of the $K^{+}\to\ell\nu$ and $B_{u}\to\tau\nu$ processes (easily inferred from a generalization of Eq. (\[eq:kaon:gamma:smsusy\]), see e.g. [@Hou:1992sy; @Isidori:2006pk]) will lead to a tension when light charged Higgs masses are considered to saturate the bounds on $R_{K}$. Supersymmetric models of neutrino mass generation (such as the SUSY seesaw) naturally induce sizable cLFV contributions, via radiatively generated off-diagonal terms in the $LL$ (and to a lesser extent $LR$) slepton soft-breaking mass matrices [@Borzumati:1986qx]. In addition to explaining neutrino masses and mixings, such models can also easily account for values of BR($\mu\to e\gamma$), within the reach of the MEG experiment. In view of the recent confirmation of a large value for the Chooz angle ($\theta_{13}\sim8.8^{\circ}$) [@theta13] and on the impact it might have on $(m_{\tilde{L}}^{2})_{e\tau}$, in the numerical analysis of the following section we will also consider different realisations of the SUSY seesaw (type I [@seesaw:I], II [@seesaw:II] and inverse [@Mohapatra:1986bd]), embedded in the framework of constrained SUSY models. We will also revisit semi-constrained scenarios allowing for light values of $m_{H^+}$, re-evaluating the predictions for $R_{K}$ under a full, one loop-computation, and in view of recent experimental data. Finally, we confront these (semi-)constrained scenarios with general, low-energy realisations, of the MSSM. Prospects for $R_{K}$: unified vs unconstrained SUSY models {#sec:res} =========================================================== In this section we evaluate the SUSY contributions to $R_{K}$, with the results obtained via the full expressions for $\Delta r$, as described in Section \[sec:RK:formulae\]. These were implemented into the SPheno public code [@Porod:2003um], which was accordingly modified to allow the different studies. It is important to stress that although some approximations have still been done (as previously discussed), the results based on the present computation strongly improve upon those so far reported in the literature (mostly obtained using the MIA). Although the different contributions cannot be easily disentangled due to having carried a full computation, our results automatically include all one-loop lepton flavour violating and lepton flavour conserving contributions (in association with charged Higgs mediation, see footnote \[footpag5\]). As mentioned before, we evaluate $R_{K}$ in the framework of constrained, semi-constrained (NUHM) and unconstrained SUSY models. Concerning the first two, we assume some flavour blind mechanism of SUSY breaking (for instance minimal supergravity (mSUGRA) inspired), so that the soft breaking parameters obey universality conditions at some high-energy scale, which we choose to be the gauge coupling unification scale $M_{\text{GUT}}\sim10^{16}$ GeV, $$\begin{aligned} \left(m_{\tilde{Q}}\right)_{ij}^{2}\, =&\,\left(m_{\tilde{U}}\right)_{ij}^{2}\, =\,\left(m_{\tilde{D}}\right)_{ij}^{2}\, =\,\left(m_{\tilde{L}}\right)_{ij}^{2}\, =\, \left(m_{\tilde{R}}\right)_{ij}^{2} \nonumber \\ =&\, m_{0}^{2}\,\delta_{ij}\,,\,\,\nonumber \\[+2mm] \left(A^{l}\right)_{ij}\,=&\, A_{0}\,(Y^{l})_{ij}\,. \label{eq:msugra.univ}\end{aligned}$$ In the above, $m_{0}$ and $A_{0}$ are the universal scalar soft-breaking mass and trilinear couplings of the cMSSM, and $i,j$ denote lepton flavour indices ($i,j=1,2,3$). In the latter case, the gaugino masses are also assumed to be universal, their common value being denoted by $M_{1/2}$. We will also consider the supersymmetrisation of several mechanisms for neutrino mass generation. More specifically, we have considered the type I and type II SUSY seesaw (as detailed in Appendix \[app:seesaw\]). We briefly comment on the inverse SUSY seesaw, and discuss a $L-R$ model. The strict universality boundary conditions of Eqs. (\[eq:msugra.univ\]) will be relaxed for the Higgs sector when we address NUHM scenarios, so that in the latter case we will have $$\begin{aligned} & m_{H_{1}}^{2}\,\neq m_{H_{2}}^{2}\,\neq\, m_{0}^{2}\,.\label{eq:msugra.nuhm}\end{aligned}$$ All the above universality hypothesis will be further relaxed when, for completeness, and to allow a final comparison with previous analyses, we address the low-energy unconstrained MSSM. In our numerical analysis, we took into account LHC bounds on the SUSY spectrum [@LHC:2011], as well as the constraints from low-energy flavour dedicated experiments [@PDG], and neutrino data [@Fogli:2011qn; @Schwetz:2011zk]. In particular, concerning lepton flavour violation, we have considered [@PDG; @arXiv:1107.5547]: $$\begin{aligned} & \text{BR}(\tau\to e\gamma)\,<3.3\times10^{-8}\, \quad(90\%\text{C.L.})\,,\label{eq:cLFVbounds1}\\ & \text{BR}(\tau\to3\, e)\,<2.7\times10^{-8}\, \quad(90\%\text{C.L.})\,,\label{eq:cLFVbounds2}\\ & \text{BR}(\mu\to e\gamma)\,<2.4\times10^{-12}\, \quad(90\%\text{C.L.})\,,\\ & \text{BR}(B_{u}\to\tau\nu)\,>9.7\times10^{-5}\, \quad(2\,\sigma)\,. \label{eq:cLFVbounds3}\end{aligned}$$ Also relevant are the recent LHCb bounds [@Aaij:2012ac] $$\begin{aligned} & \text{BR}(B_{s}\to\mu^{+}\mu^{-})\,<4.5\times10^{-9}\, \quad(95\%\text{C.L.})\,,\label{eq:Bbounds1}\\ & \text{BR}(B\to\mu^{+}\mu^{-})\,<1.03\times10^{-9}\, \quad(95\%\text{C.L.})\,.\label{eq:Bbounds2}\end{aligned}$$ When addressing models for neutrino mass generation, we take the following (best-fit) values for the neutrino mixing angles [@Schwetz:2011zk] (where $\theta_{13}$ is already in good agreement with the most recent results from [@theta13]), $$\begin{aligned} & \sin^{2}\theta_{12}\,=\ 0.312_{-0.015}^{+0.017},\quad\sin^{2}\theta_{23}\,=\ 0.52_{-0.07}^{+0.06}, \nonumber \\[+2mm] &\sin^{2}\theta_{13}\approx0.013_{-0.005}^{+0.007}\,, \label{eq:mixingangles:data}\\[+2mm] & \Delta\, m_{\text{12}}^{2}\,=\,(7.59_{-0.18}^{+0.20})\,\times10^{-5}\,\,\text{eV}^{2}\,, \\[+2mm] &\Delta\, m_{\text{13}}^{2}\,=(2.50_{-0.16}^{+0.09})\,\times10^{-3}\,\,\text{eV}^{2}\,. \end{aligned}$$ Regarding the leptonic mixing matrix ($U_{\text{MNS}}$) we adopt the standard parametrisation. In the present analysis, all CP violating phases are set to zero[^4]. mSUGRA inspired scenarios: cMSSM and the SUSY seesaw ---------------------------------------------------- We begin by re-evaluating, through a full computation of the one-loop corrections, the maximal amount of supersymmetric contributions to $R_{K}$ in constrained SUSY scenarios. For a first evaluation of $R_{K}$, we consider different cMSSM (mSUGRA-like) points, defined in Table \[table:mSUGRA:points\]. Among them are several cMSSM benchmark points from [@AbdusSalam:2011fc], representative of low and large $\tan\beta$ regimes, as well as some variations. Notice that, as mentioned before, these choices are compatible with having a Higgs boson mass above 118 GeV but will be excluded once we require $m_{h}$ to lie close to 125 GeV as suggested by LHC results [@LHC:Higgs:2012]. $m_{0}$ (GeV) $M_{1/2}$ (GeV) $\tan\beta$ $A_{0}$ (GeV) sign($\mu$) -------- --------------- ----------------- ------------- --------------- ------------- 10.3.1 300 450 10 0 1 P20 330 500 20 -500 1 P30 330 500 30 -500 1 40.1.1 330 500 40 -500 1 40.3.1 1000 350 40 -500 1 As could be expected from Eqs. (\[eq:deltar:approx:long\]-\[eq:deltar:approx\]), in a strict cMSSM scenario (in agreement with the experimental bounds above referred to) the SUSY contributions to $R_{K}$ are extremely small; motivated by the need to accommodate neutrino data, and at the same time accounting for values of BR($\mu\to e\gamma$) within MEG reach, we implement type I and type II seesaws in mSUGRA-inspired models (see Appendices \[app:seesawI\] and \[app:seesawII\]). Regarding the heavy-scale mediators, we considered degenerate right-handed neutrinos, as well as degenerate scalar triplets. We set the seesaw scale aiming at maximising the (low-energy) non-diagonal entries of the soft-breaking slepton mass matrices, while still in agreement with the current low-energy bounds (see Eqs. (\[eq:cLFVbounds1\]-\[eq:Bbounds2\])). In particular, we have tried to maximise the $LL$ contributions to $\Delta r$, i.e., $(m_{\tilde{L}}^{2})_{e\tau}$, and to obtain BR($\mu\to e\gamma$) within MEG reach (i.e. $10^{-13}\lesssim$ BR($\mu\to e\gamma$)$\lesssim2.4\times10^{-12}$). However, and due to the fact that both seesaw realisations fail to account for radiatively induced LFV in the right-handed slepton sector, one finds values $\|\Delta r\|\lesssim2\times10^{-8}$. It is worth emphasising that if one further requires $m_{h}$ to lie close to 125 GeV (as suggested by recent findings [@LHC:Higgs:2012]), then one is led to regions in mSUGRA parameter space where, due to the much heavier sparticle masses and typically lower values of $\tan\beta$, the SUSY contributions to $R_{K}$ become even further suppressed. Thus, and even under a full computation of the corrections to the $\nu\ell H^{+}$ vertex, we nevertheless confirm that, as firstly put forward in the analyses of [@Masiero:2005wr; @Masiero:2008cb] strictly constrained SUSY and SUSY seesaw models indeed fail to account for values of $R_{K}$ close to the present limits. Clearly, new sources of flavour violation, associated to the right-handed sector are required: in what follows, we maintain universality of soft-breaking terms allowing, at the grand unified (GUT) scale, for a single $\tau-e$ flavour violating entry in $m_{\tilde{R}}^{2}$. This approach is somewhat closer to the lines of [@Masiero:2005wr; @Masiero:2008cb; @Ellis:2008st; @Girrbach:2012km], although in our computation we will still conduct a full evaluation of the distinct contributions to $\Delta r$, and we consider otherwise universal soft-breaking terms. Without invoking a specific framework/scenario of SUSY breaking that would account for such a pattern, we thus set $$\delta_{31}^{RR}\,=\,\frac{(m_{\tilde{R}}^{2})_{\tau e}}{m_{0}^{2}}\, \neq0\,. \label{eq:mia:sleptons}$$ As discussed above, low-energy constraints on LFV observables (especially $\tau\rightarrow e\gamma$), severely constrain this entry. In Fig. \[fig:cMSSM:seesaw\], we present our results for $\Delta r$ scanning the $m_{0}-M_{1/2}$ plane for a regime of large $\tan\beta$. We have set $\delta_{31}^{RR}=0.1$, $\tan\beta=40$, and taken $A_{0}=-500$ GeV. The surveys displayed in the panels correspond to having embedded a type I (left) or type II (right) seesaw onto this near-mSUGRA framework. --------------------------------------------- --------------------------------------------- ![image](GUT81.pdf){width="0.42\linewidth"} ![image](GUT83.pdf){width="0.42\linewidth"} --------------------------------------------- --------------------------------------------- As can be readily seen from Fig. \[fig:cMSSM:seesaw\], once the constraints from low-energy observables have been applied, in the type I SUSY seesaw, the maximum values for $\Delta r$ are $\mathcal{O}(10^{-7})$, associated to the region with a lighter SUSY spectra (which is in turn disfavoured by a “heavy” light Higgs). Even for the comparatively small non-universality, $\delta_{31}^{RR}=0.1$, a considerable region of the parameter space is excluded due to excessive contributions to BR($B_{u}\to\tau\nu$) and BR($\tau\to e\gamma$), thus precluding the possibility of large values of $\Delta r$. In a regime of large $\tan\beta$, the contributions to BR($B_{s}\to\mu^{+}\mu^{-}$) are also sizable, and the recent LHCb results seem to exclude the regions of the parameter space where one could still have $\Delta r\sim\mathcal{O}(10^{-6,-7})$. The excessive SUSY contributions to BR($B_{s}\to\mu^{+}\mu^{-}$) can be somewhat reduced by adjusting $A_0$ (in Fig. \[fig:cMSSM:seesaw\] we fixed $A_0=-500$ GeV) and the values of $\Delta r$ can be slightly augmented by increasing $\delta_{31}^{RR}$; in the latter case, the $\tau\to e\gamma$ bound proves to be the most constraining, and values of $\Delta r$ larger than $\mathcal{O}(10^{-6,-7})$ cannot be obtained in these constrained SUSY seesaw models. The situation is somewhat different for the type II case: firstly notice that a sizable region in the $m_{0}-M_{1/2}$ plane is associated to negative contributions to $R_{K}$, which are currently disfavoured. In the remaining (allowed) parameter space, the values of $\Delta r$ are slightly smaller than for the type I case: this is a consequence of a non trivial interplay between a smaller value for the splitting $\delta=\frac{1}{2}(\langle{m}_{\widetilde{\ell}}^{2}\rangle- \langle{m}_{\chi^{0}}^{2}\rangle)$ (induced by a lighter spectra), and a lighter charged Higgs boson. (We notice that accommodating light neutral Higgs with $m_{h}>118$ GeV is also comparatively more difficult in the type II SUSY seesaw.) Notice that in both SUSY seesaws it is fairly easy to accommodate a potential observation of BR($\mu\to e\gamma$) $\sim10^{-13}$ by MEG, taking for instance $M_{{\rm Seesaw}}\sim10^{12}$ GeV for the type I and II seesaw mechanisms. For both cases, larger values of $\delta_{31}^{RR}=0.5$ can be taken, but these typically lead to conflicting situations with low-energy observables; lowering $\tan\beta$ can ease the existing tension, at the expense of also reducing $\Delta r$. We summarise this on Table \[table:mSUGRA:results\], for simplicity in association with a type I SUSY seesaw. [lccccccc]{} & [$\delta_{31}^{RR}$ ]{} & [$\Delta r$ ]{} & [[@c@]{}]{} [$m_{H^+}$]{}\ [(GeV)]{}\ & [BR($\tau\to e\gamma$) ]{} & [[@c@]{}]{} [BR($B_{u}\to\tau\nu$)]{}\ [$(\times10^{-4})$]{}\ & [[@c@]{}]{} [BR($B_{s}\to\mu^{+}\mu^{-}$) ]{}\ [$(\times10^{-9})$]{}\ & [BR($\mu\to e\gamma$)]{}\ \ [10.3.1 - I ]{} & [0 ]{} & [7.2$\times10^{-11}$]{} & [715 ]{} & [2.5$\times10^{-16}$]{} & [1.17]{} & [ 4.0]{} & [7.2$\times10^{-14}$]{}\ [10.3.1 - I ]{} & [0.1 ]{} & [8.5$\times10^{-11}$]{} & [715]{} & [2.9$\times10^{-10}$]{} & [1.17]{} & [ 4.0]{} & [1.8$\times10^{-13}$ ]{}\ [10.3.1 - I ]{} & [0.5 ]{} & [5.1$\times10^{-9}$]{} & [715 ]{} & [8.5$\times10^{-9}$ ]{} & [1.12]{} & [ 4.0]{} & [9.7$\times10^{-15}$]{}\ \ [0.1mm P20 - I ]{} & [0.1 ]{} & [4.3$\times10^{-9}$]{} & [800]{} & [3.5$\times10^{-9}$ ]{} & [1.15]{} & [ 4.0]{} & [2.0$\times10^{-12}$]{}\ \ [0.1mm P30 - I ]{} & [0.1 ]{} & [1.2$\times10^{-7}$]{} & [725 ]{} & [1.4$\times10^{-8}$]{} & [1.11]{} & [ 4.3]{} & [1.7$\times10^{-14}$ ]{}\ \ [40.3.1 - I ]{} & [0 ]{} & [1.6$\times10^{-8}$]{} & [818]{} & [3.1$\times10^{-15}$ ]{} & 1.09 & [4.4]{} & [1.2$\times10^{-12}$]{}\ [40.3.1 - I ]{} & [0.1 ]{} & [6.0$\times10^{-8}$]{} & [818]{} & [2.9$\times10^{-10}$ ]{} & 1.09 & [4.4]{} & [1.2$\times10^{-12}$]{}\ [40.3.1 - I ]{} & [0.5 ]{} & [2.0$\times10^{-6}$]{} & [818]{} & [2.0$\times10^{-8}$ ]{} & 1.09 & [4.4]{} & [3.3$\times10^{-12}$]{}\ A few comments are in order regarding the summary of Table \[table:mSUGRA:results\]: even with a large value for $\delta_{31}^{RR}$, and in the large $\tan\beta$ regime, the maximum attainable values for $\Delta r$ are much below the current experimental sensitivity, at most $2 \times 10^{-6}$. As mentioned before, if we further take into account the recent discovery of a new boson at LHC [@LHC:Higgs:2012] with a mass around 125 GeV, and interpret it as the lightest neutral CP-even Higgs boson of the MSSM, the attainable values for $\Delta r$ will be extremely small. In order to conclude this part of the analysis we provide a comprehensive overview of the constrained MSSM prospects regarding $R_{K}$, presenting in Fig. \[fig:cMSSM:survey\] a survey of the (type I seesaw) mSUGRA parameter space, for two different regimes of $\delta_{31}^{RR}$, taking all present bounds (including the recent ones on $m_{h}$) into account. The panels of Fig. \[fig:cMSSM:survey\] allow to recover the information that could be expected from the discussion following Fig. \[fig:cMSSM:seesaw\]: for fixed values of $A_0$ and $\tan \beta$, increasing $\delta_{31}^{RR}$ indeed allows to augment the SUSY contributions to $\Delta r$ although, as can be seen from the right-panel, the constraints from BR($\tau\to e\gamma$) become increasingly harder to accommodate. (Notice that the latter could be avoided by increasing the SUSY scale (i.e. on regions of the parameter space with large $m_0$ and/or $M_{1/2}$) - however, and as visible from Fig. \[fig:cMSSM:survey\], in a constrained SUSY framework this would lead to heavier charged Higgs masses, and in turn to suppressed contributions to $\Delta r$.) --------------------------------------------------- --------------------------------------------------- ![image](mSUGRAv3_01.pdf){width="0.45\linewidth"} ![image](mSUGRAv3_07.pdf){width="0.45\linewidth"} --------------------------------------------------- --------------------------------------------------- Although we do not display an analogous plot here, the situation is very similar for the type II SUSY seesaw (slightly even more constrained due to the fact that accommodating $m_{h}\sim125$ GeV is more difficult in these models [@Hirsch:2012ti]). In view of the above discussion it is clear that even taking into account all 1-loop corrections to the $\nu\ell H^{+}$ vertex, values of $\Delta r$, large enough to saturate current observation, cannot be reached in the framework of constrained SUSY models (and its seesaw extensions accommodating neutrino data). In this sense, and even though we have followed a different approach, our results follow the conclusions of [@Ellis:2008st]. We also stress that recent experimental bounds (both from flavour observables and collider searches) add even more severe constraints to the maximal possible values of $\Delta r$. mSUGRA inspired scenarios: inverse seesaw and $L-R$ models ----------------------------------------------------------- We briefly comment here on the prospects of the inverse SUSY seesaw concerning $R_{K}$: recently, it was pointed out that some flavour violating observables can be enhanced by as much as two orders of magnitude in a model with the inverse seesaw mechanism [@Abada:2011hm]. Within such a framework, right-handed (s)neutrino masses can be relatively light, and as a consequence these ${\nu_{R}}$, ${\widetilde{\nu}_{R}}$ states do not decouple from the theory until the TeV scale, hence potentially providing important contributions to different low-energy processes. Nevertheless, the specific contributions to $\Delta r$ are suppressed by a factor $\frac{m_{e}^{2}}{m_{\tau}^{2}}$, with respect to those discussed above (see Eq. (\[eq:deltar:approx:X\])), so that we do not expect a significant enhancement of SUSY 1-loop effects to $R_{K}$ due to the inverse seesaw mechanism. For completeness (and although we do not provide specific details here), we have considered a specific $L-R$ seesaw model [@LRmodels2]. In this framework, non-vanishing values of $\delta_{31}^{RR}$ can be dynamically generated. We have numerically verified that typically one finds $\delta_{31}^{RR}\lesssim0.01$ (we do not dismiss that larger values might be found, although certainly requiring a considerable amount of fine-tuning in the input parameters). We have not done a dedicated $\Delta r$ calculation for this case, but taking into account that the effect scales with $(\delta_{31}^{RR})^{2}$, we also expect the typical range for $\Delta r$ to be far below the current experimental sensitivity. mSUGRA inspired scenarios: NUHM ------------------------------- As can be seen from the approximate expression for $\Delta r$ in Eqs. (\[eq:deltar:approx:X\], \[eq:deltar:approx\]), regimes associated with both large $\tan\beta$ and a light charged Higgs can greatly enhance this observable [@Ellis:2008st] ($\Delta r\propto\nicefrac{\tan^{6}\beta}{m_{H^+}^{4}}$). By relaxing the mSUGRA-inspired universality conditions for the Higgs sector, as occurs in NUHM scenarios, one can indeed have very low masses for the $H^{+}$ boson at low energies. This regime corresponds to a narrow strip in parameter space where $m_{H_{1}}^{2}\!\approx\! m_{H_{2}}^{2}$, in particular when both are close to $-(2.2\:\textrm{TeV})^{2}$. In addition to favouring electroweak symmetry breaking, since $m^2_{H^{+}}\sim\left\|m_{H_{1}}^{2}-m_{H_{2}}^{2}\right\|$ (even accounting for RG evolution of the parameters down to the weak scale), it is expected that the charged Higgs can be made very light with some fine tuning [@Ellis:2008st]. In order to explore the maximal possible values of $\Delta r$, a small scan was conducted around this region, where $m_{H^+}$ changes very rapidly (see Table \[tab:NUHM\]). [cccccc]{} & $m_{0}$ & $M_{1/2}$ & $m_{H_{1}}^{2}$, $m_{H_{2}}^{2}$ & $\tan\beta$ & $\delta_{31}^{RR}$\ & $\text{\footnotesize (GeV)}$ & $\text{\footnotesize (GeV)}$ & $\text{\footnotesize (GeV}^{2}\text{\footnotesize )}$ & &\ \ Min & 0 & 100 & $-5.2\times10^{6}$ & 40 & 0.1\ Max & 1500 & 1500 & $-4.6\times10^{6}$ & 40 & 0.7\ ------------------------------------------------------- -------------------------------------------------------- ![image](DeltaR-mHp-NUHM.pdf){width="0.49\linewidth"} ![image](BuTauNu-Mhp-NUHM.pdf){width="0.49\linewidth"} ------------------------------------------------------- -------------------------------------------------------- As can be verified from the left-hand panel of Fig. \[fig:NUHM:deltar\], one could in principle have semi-constrained regimes leading to sizable values of $R_{K}$, $\mathcal{O}(10^{-2})$. Once all (collider and low-energy) bounds have been imposed, one has at most $\Delta r\lesssim10^{-4}$ (in association with $m_{H^+}\gtrsim500$ GeV). Moreover, it is interesting to notice that SUSY contributions to BR($B_{u}\to\tau\nu$), which become non-negligible for lighter $H^{\pm}$, have a negative interference with those of the SM, lowering the latter BR to values below the current experimental bound. This can be seen on the right-hand panel of Fig. \[fig:NUHM:deltar\]. We will return to this topic in greater detail in the following subsection, when addressing the unconstrained MSSM. Unconstrained MSSM ------------------ To conclude the numerical discussion, and to allow for a better comparison between our approach and those usually followed in other recent analyses (for instance [@Masiero:2008cb; @Girrbach:2012km]), we conduct a final study of the unconstrained, low-energy MSSM. Thus, and in what follows, we make no hypothesis concerning the source of lepton flavour violation, nor on the underlying mechanism of SUSY breaking. Massive neutrinos are introduced by hand (no assumption being made on their nature), and although charged interactions do violate lepton flavour, as parametrised by the $U_{\text{MNS}}$ matrix, no sizable contributions to BR($\mu\to e\gamma$) should be expected, as these would be suppressed by the light neutrino masses. At low-energies, no constraints (other than the relevant experimental bounds) are imposed on the SUSY spectrum (for simplicity, we will assume a common value for all sfermion trilinear couplings at the low-scale, $A_i=A_0$). The soft-breaking slepton masses are allowed to be non-diagonal, so that a priori a non-negligible mixing in the slepton sector can occur. In order to better correlate the source of flavour violation at the origin of $\Delta r$ with the different experimental bounds, we again allow for a single FV entry in the slepton mass matrices: $\delta_{31}^{RR}\sim0.5$ (otherwise setting all other $\delta_{ij}^{XY}=0$). [cccccccccccc]{}\ & $\mu$ & $m_{A}$ & $M_{1}$, $M_{2}$ & $M_{3}$ & $A_{0}$ & $m_{L}$ & $m_{R}$ & $m_{Q},m_{U},m_{D}$ & $\tan\beta$ & $\delta_{31}^{RR}$ & other $\delta_{ij}^{XY}$\ \ Min & 100 & 50 & 100 & 1100 & -1000 & 100 & 100 & 1200 & 30 & 0.5 & 0\ Max & 3000 & 1500 & 2500 & 2500 & 1000 & 2200 & 2500 & 5000 & 60 & 0.5 & 0\ In our scan we have varied the input parameters in the ranges collected in Table \[tab:ranges\]. We have also applied all relevant constraints on the low-energy observables, Eq. (\[eq:cLFVbounds1\]-\[eq:Bbounds2\]), as well as the constraints on the SUSY spectrum [@PDG; @LHC:2011]. In particular we have assumed the conservative limits $$m_{\tilde{q}_{L,R}}>1000\,\text{GeV}\,,\quad\quad m_{\tilde{g}}>1000\,\text{GeV}\,.$$ Concerning the light Higgs boson mass, no constraint was explicitly imposed. We just notice here that values close to 125 GeV [@LHC:Higgs:2012], or even larger, are easily achievable due to the heavy squark masses. -------------------------------------------------- ------------------------------------------------ ![image](DeltaR-mh0.pdf){width="0.49\linewidth"} ![image](mHp-Atop.pdf){width="0.49\linewidth"} -------------------------------------------------- ------------------------------------------------ This can be observed from the left panel of Fig. \[fig:DeltaR-mh0\], where we display the output of the above scan, presenting the values of $\Delta r$ versus the associated light Higgs boson mass, $m_{h}$. As expected, no explicit correlation between $m_{h}$ and $\Delta r$ is manifest, nor with the other (relevant) flavour-related low-energy bounds. For completeness, and to better illustrate the following discussion, we present on the right-hand panel of Fig. \[fig:DeltaR-mh0\] the charged Higgs mass as a function of $A_0$, again under a colour scheme denoting the experimental bounds applied in each case. Identical to what was observed in Fig. \[fig:NUHM:deltar\] (notice that NUHM models correspond, at low-energies, to a subset of these general cases), regimes of very light charged Higgs are indeed present, in association with small to moderate (negative) regimes for $A_0$. Nevertheless, these regimes - which could potentially enhance $\Delta r$ - are likewise excluded due to a strong conflict with $\textrm{BR}\left(B_{u}\to\tau\nu\right)$. This can be further confirmed from the left panel of Fig. \[fig:DeltaRComparisonCuts\], where we display the possible range of variation for $\Delta r$ as a function of $m_{H^+}$, colour-coding the different applied bounds. -------------------------------------------------- ------------------------------------------------------------ ![image](DeltaR-mHp.pdf){width="0.49\linewidth"} ![image](DeltaRComparisonCuts.pdf){width="0.49\linewidth"} -------------------------------------------------- ------------------------------------------------------------ As can be seen from both panels of Fig. \[fig:DeltaRComparisonCuts\], values $\Delta r\approx\mathcal{O}(10^{-2},10^{-1})$ could be obtainable, in agreement with Refs [@Girrbach:2012km; @Masiero:2005wr; @Masiero:2008cb; @Ellis:2008st]. However, the situation is substantially altered when one takes into account the current experimental bounds on $B$ decays ($B_{u}\to\tau\nu$ and $B_{s}\to\mu^{+}\mu^{-}$) and $\tau\to e\gamma$. As is manifest from the left panel of Fig. \[fig:DeltaRComparisonCuts\], once experimental bounds - other than $B_{u}\to\tau\nu$ - are imposed, one could in principle have $\Delta r^{\text{max}}\approx\mathcal{O}(10^{-2})$; however, taking into account the limits from BR($B_{u}\to\tau\nu$), one is now led to $\Delta r\lesssim 10^{-3}$. A few comments are in order regarding the impact of the different low-energy bounds from radiative tau decays and $B$-physics observables. Firstly, let us consider the $\tau\to e\gamma$ decay: although directly depending on $\delta_{31}^{RR}$, its amplitude is (roughly) suppressed by the fourth power of the average SUSY scale, $m_{\text{SUSY}}$. As can be seen from Eqs. (\[eq:deltar:approx:X\], \[eq:deltar:approx\]), $\Delta r$ only depends on the charged Higgs mass - if the latter is assumed to be an EW scale parameter, $\Delta r$ will be thus independent of $m_{\text{SUSY}}$ in these unconstrained models. As such, it is possible to evade the $\tau\to e\gamma$ bound by increasing the soft SUSY masses, and this can indeed be seen from the right-hand panel of Fig. \[fig:DeltaRComparisonCuts\], where a number of “blue” points are found to lie below the BR($\tau\to e\gamma$) bound. Secondly, the $B_{s}\to\mu^{+}\mu^{-}$ decay can be a severe constraint regarding the SUSY contributions to $\Delta r$ in the case of constrained models (see, e.g., Figs. \[fig:cMSSM:seesaw\] and \[fig:cMSSM:survey\]). We notice that $B_{s}\to\mu^{+}\mu^{-}$ is approximately proportional to $A_{0}^{2}$ (see for instance [@Isidori:2006pk]) while $\Delta r$ shows no such dependence: thus a regime of small trilinear couplings easily allows evade the $B_{s}\to\mu^{+}\mu^{-}$ bounds. Finally, let us discuss the $B_{u}\to\tau\nu$ bounds. Notice that this is a process essentially identical to the charged kaon decays at the origin of the $R_{K}$ ratio (the only difference being that the $K^{+}$ meson is to be replaced by a $B_{u}$ and the $e$/$\mu$ in the decay products by a kinematically allowed $\tau$), and hence its tree-level decay width can be inferred from Eqs. (\[eq:SM:Pdecays\]) and (\[eq:kaon:gamma:smsusy\]). Due to a negative interference between the SM and the MSSM contributions, given by the term proportional to $\tan^2\beta/m^2_{H^\pm}$ in Eq. (\[eq:kaon:gamma:smsusy\]), regimes of low $m_{H^+}$ lead to excessively small values of $B_{u}\to\tau\nu$ (below the experimental bound), effectively setting a lower bound for for $m^2_{H^\pm}$ (see right panel of Fig. \[fig:NUHM:deltar\], in relation to the discussion of NUHM models). In turn, this excludes regimes of $m_{H^+}$ associated to sizable values of $\Delta r$, as is clear from the comparison of the “blue” and “green” regions of the left panel of Fig. \[fig:DeltaRComparisonCuts\]. In summary, we conclude that saturating the experimental bound on $R_{K}$ clearly proves to be extremely difficult (if not impossible), even in the unconstrained MSSM, especially in view of the stringent constraints from $B_{u}\to\tau\nu$. Conclusions {#sec:concs} =========== In this work we have revisited supersymmetric contributions to $R_{K}=\Gamma\left(K\rightarrow e\nu\right)$$/\Gamma\left(K\rightarrow\mu\nu\right)$, considering the potential of a broad class of constrained SUSY models to saturate the current measurement of $R_{K}$. We based our analysis in a full computation of the one-loop corrections to the $\nu\ell H^{+}$ vertex; we have also derived (when possible) illustrative analytical approximations, which in addition to offering a more transparent understanding of the rôle of the different parameters, also allow to establish a bridge between our results and previous ones in the literature. Our analysis further revisited the $R_{K}$ observable in the light of new experimental data, arising from flavour physics as well as from collider searches. We numerically evaluated the contributions to $R_{K}$ arising in the context of different minimal supergravity inspired models which account for observed neutrino data, further considering the possibility of accommodating a near future observation of a $\mu\to e\gamma$ decay. As expected from the (mostly) $LL$ nature of the radiatively induced charged lepton flavour violation, type I and II seesaw mechanisms implemented in the cMSSM provide minimal contributions to $R_{K}$, thus implying that such cMSSM SUSY seesaws cannot saturate the present value for $\Delta r$. We then considered unified models where the flavour-conserving hypothesis on the $RR$ slepton sector is relaxed by allowing a non-vanishing $\delta_{31}^{RR}$ ($e-\tau$ sector). In all models, special attention was given to experimental constraints, especially four observables which turn out to play a particularly relevant rôle: the recent interval for the lightest neutral Higgs boson mass provided by the CMS and ATLAS collaborations, BR($B_{s}\to\mu^{+}\mu^{-}$), BR($B_{u}\to\tau\nu$) and BR($\tau\to e\gamma$). These last two exhibit a dependence on $m_{H^+}$ ($B_{u}\to\tau\nu$) and on $\delta_{31}^{RR}$ ($\tau\to e\gamma$) similar to that of $\Delta r$. The SUSY contributions to $\Delta r$ are thus maximised in a regime in which $m_{H^+}$ and $\delta_{31}^{RR}$ are such that the experimental limits for $B_{u}\to\tau\nu$ and $\tau\to e\gamma$ are simultaneously saturated; in this regime one must then accommodate the bounds on other observables, such as $m_h$ and BR($B_{s}\to\mu^{+}\mu^{-}$). For a minimal deviation from a pure cMSSM scenario allowing for non-vanishing values of $\delta_{31}^{RR}$, we can have values for $\Delta r$ at most of the order of $10^{-6}$. In fact, the requirement of having a Higgs boson mass of 125-126 GeV is much more constraining on the cMSSM parameter space than, for instance $B_{s}\to\mu^{+}\mu^{-}$ (which is sub-dominant, and can be overcome by variations of the trilinear coupling, $A_0$). In order to have $\Delta r \sim \mathcal{O}(10^{-6})$, one must significantly increase $\delta_{31}^{RR}$ so to marginally overlap the regions of $m_h \sim 125$ GeV, while still in agreement with $\tau\to e\gamma$. Models where the charged Higgs mass can be significantly lowered, as is the case of NUHM models, allow to increase the SUSY contributions to $\Delta r$, which can be as large as $10^{-4}$ (larger values being precluded due to $B_{u}\to\tau\nu$ decay constraints). More general models, as the unconstrained MSSM realised at low-energies, offer more degrees of freedom, and the possibility to better accommodate/evade the different experimental constraints. In the unconstrained MSSM, one can find values of $\Delta r$ one order of magnitude larger, $\Delta r \sim \mathcal{O}(10^{-3})$. Again, any further augmentation is precluded due to incompatibility with the bounds on $B_{u}\to\tau\nu$. However $\Delta r \sim \mathcal{O}(10^{-3})$ still remains one order of magnitude shy of the current experimental sensitivity to $R_{K}$, and also substantially lower than some of the values previously found in the literature. As such, if SUSY is indeed discovered, and unless there is significant progress in the experimental sensitivity to $R_{K}$, it seems unlikely that the contributions to $R_{K}$ of the SUSY models studied here will be testable in the near future. On the other hand, any near-future measurement of $\Delta r$ larger than $\mathcal{O}(10^{-3})$ would unambiguously point towards a scenario different than those here addressed (mSUGRA-like seesaw, NUHM and the phenomenological MSSM). It should be kept in mind that the analysis presented here focused on the impact of LFV interactions. Should the discrepancy between the SM and experimental observations turn out to be much smaller than $10^{-4}$, a more detailed approach and evaluation will then be necessary. Acknowledgments {#acknowledgments .unnumbered} =============== R.M.F. is thankful for the hospitality of the LPC Clermont-Ferrand. The work of R.M.F has been supported by Fundação para a Ciência e a Tecnologia through the fellowship SFRH/BD/47795/2008. R.M.F. and J. C. R. also acknowledge the financial support from the EU Network grant UNILHC PITN-GA-2009-237920 and from Fundação para a Ciência e a Tecnologia grants CFTP-FCT UNIT 777, CERN/FP/83503/2008 and PTDC/FIS/102120/2008. A. M. T. acknowledges partial support from the European Union FP7 ITN-INVISIBLES (Marie Curie Actions, PITN- GA-2011-289442). Renormalisation of the $\nu\ell H^{+}$ vertex {#sec:app1} ============================================= In what follows we detail the computation leading to Eqs. (\[eq:epsilondelta\]-\[eq:delta:def\]), and we further refer to [@Bellazzini:2010gn] for a similar analysis. As expected, loop effects contribute to both kinetic and mass terms of charged leptons as well as to the ${\nu}\ell H^{+}$ vertex: $$\begin{aligned} \mathcal{L}_{0}^{H^{\pm}}\!\! = & i\,\overline{\ell}_{L} \left(\mathbf{1}+ \eta_{L}^{\ell}\right)\slashed{\partial}\ell_{L}+i\,\overline{\ell}_{R} \left(\mathbf{1}+\eta_{R}^{\ell}\right) \slashed{\partial}\ell_{R}\nonumber\\[+1mm] &\!\! +i\, \overline{\nu}_{L}\left(\mathbf{1}+\eta_{L}^{\nu}\right) \slashed{\partial}\nu_{L}- \left[\overline{\ell}_{L} \left(M^{l0}+\eta_{m}^{\ell}\right)\ell_{R}+\textrm{h.c.}\right]\nonumber \\[+1mm] &\!\! +\left[\overline{\nu}_{L}\left(2^{3/4}G_{F}^{1/2}\,\tan\beta\, M^{l0}+\eta^{H}\right)\ell_{R}H^{+}+\textrm{h.c.}\right]\,.\end{aligned}$$ Here $M^{l0}$ denotes the bare charged lepton mass and the $\eta$’s correspond to loop contributions to the various terms. The (new) kinetic terms can be recast into a canonical form by means of unitary rotations of the fields ($K_{L}^{\ell}$, $K_{R}^{\ell}$, $K_{L}^{\nu}$), which are then renormalised by diagonal transformations ($\hat{Z}_{L}^{\ell}$, $\hat{Z}_{R}^{\ell}$, $\hat{Z}_{L}^{\nu}$): $$\begin{aligned} \ell_{L}^{\text{old}} \! = & K_{L}^{\ell}\left(\hat{Z}_{L}^{\ell}\right)^{-\frac{1}{2}}\!\! \ell_{L}^{\text{new}} \ \textrm{;} \quad \hat{Z}_{L}^{\ell}={K_{L}^{\ell}}^{\dagger} \left(\mathbf{1}+\eta_{L}^{\ell} \right)K_{L}^{\ell}\,,\\ \ell_{R}^{\text{old}} \! = & K_{R}^{\ell} \left(\hat{Z}_{R}^{\ell}\right)^{-\frac{1}{2}}\!\! \ell_{R}^{\text{new}} \ \textrm{;}\quad \hat{Z}_{R}^{\ell}={K_{R}^{\ell}}^{\dagger} \left(\mathbf{1}+\eta_{R}^{\ell}\right)K_{R}^{\ell}\,,\\ \nu_{L}^{\text{old}} \! = & K_{L}^{\nu}\left(\hat{Z}_{L}^{\nu} \right)^{-\frac{1}{2}}\!\! \nu_{L}^{\text{new}} \ \textrm{;}\quad \hat{Z}_{L}^{\nu}={K_{L}^{\nu}}^{\dagger} \left(\mathbf{1}+\eta_{L}^{\nu}\right)K_{L}^{\nu}\,. \end{aligned}$$ Two unitary rotation matrices ($R_{L}^{\ell}$, $R_{R}^{\ell}$) are further required to diagonalise the charged lepton mass matrix, and one finally has $$\begin{aligned} \ell_{L}^{\text{old}} & = & K_{L}^{\ell}\left(\hat{Z}_{L}^{\ell}\right)^{-\frac{1}{2}}\, R_{L}^{\ell}\,\ell_{L}^{\text{new}}\,,\\ \ell_{R}^{\text{old}} & = & K_{R}^{\ell}\left(\hat{Z}_{R}^{\ell}\right)^{-\frac{1}{2}}\, R_{R}^{\ell}\,\ell_{R}^{\text{new}}\,,\\ \nu_{L}^{\text{old}} & = & K_{L}^{\nu}\left(\hat{Z}_{L}^{\nu}\right)^{-\frac{1}{2}}\, R_{L}^{\ell}\,\nu_{L}^{\text{new}}\,.\end{aligned}$$ In the new basis, the mass terms now read $$\begin{aligned} \mathcal{L}^{\textrm{mass}}\equiv&-\overline{\ell}_{L}\, M^{l}\,\ell_{R} +\textrm{h.c.} \nonumber \\ =&-\overline{\ell}_{L}\,{R_{L}^{\ell}}^{\dagger} \left[\left(\hat{Z}_{L}^{\ell}\right)^{-\frac{1}{2}} \,{K_{L}^{\ell}}^{\dagger}\left(M^{l0} +\eta_{m}^{\ell}\right) \right. \nonumber\\ &\left.\hskip 17mm K_{R}^{\ell}\left(\hat{Z}_{R}^{l} \right)^{-\frac{1}{2}}\right]\, R_{R}^{\ell}\,\ell_{R}+\textrm{h.c.}\,. \end{aligned}$$ The above equation relates the unknown parameter $M^{l0}$ with the physical mass matrix $M^{l}$. Using the latter to rewrite the ${\nu}\ell H^{+}$ vertex one finds $$\begin{aligned} \mathcal{L}^{H^{\pm}} & \equiv & \overline{\nu}_{L}\, Z^{H}\,\ell_{R}\, H^{+}+\textrm{h.c.}\,, \end{aligned}$$ where $$\begin{aligned} Z^{H} = & 2^{3/4}G_{F}^{1/2}\,\tan\beta\,{R_{L}^{\ell}}^{\dagger} \left(\hat{Z}_{L}^{\nu}\right)^{-\frac{1}{2}}{K_{L}^{\nu}}^{\dagger}\, K_{L}^{\ell}\left(\hat{Z}_{L}^{\ell}\right)^{\frac{1}{2}}\, R_{L}^{\ell} \, M^{l}\nonumber \\ & +{R_{L}^{\ell}}^{\dagger}\left(\hat{Z}_{L}^{\nu} \right)^{-\frac{1}{2}}\,{K_{L}^{\nu}}^{\dagger} \left(-2^{3/4}G_{F}^{1/2}\,\tan\beta\,\eta_{m}^{\ell} +\eta^{H}\right)\nonumber\\ & \hskip 5mm K_{R}^{\ell} \left(\hat{Z}_{R}^{\ell}\right)^{-\frac{1}{2}}\, K_{R}^{\ell}\,.\end{aligned}$$ To one-loop order, this exact expression simplifies to $$\begin{aligned} Z^{H} = & 2^{3/4}G_{F}^{1/2}\,\tan\beta\,\left[\left(\mathbf{1}+\frac{\eta_{L}^{\ell}}{2}-\frac{\eta_{L}^{\nu}}{2}\right)\, M^{l}-\eta_{m}^{\ell}\right]+\eta^{H}\,.\end{aligned}$$ The expressions for the $\eta$’s can be computed from the relevant Feynman diagrams (assuming zero external momenta): $$\begin{aligned} -\left(4\pi\right)^{2}\hskip -1mm &\left(\eta_{m}^{\ell}\right)_{ij}= \, N_{i\alpha\beta}^{R\left(\ell\right)}N_{j\alpha\beta}^{L\left(\ell\right)} m_{\chi_{\alpha}^{0}} B_{0}\left(0,m_{\chi_{\alpha}^{0}}^{2},m_{\widetilde{\ell}_{\beta}}^{2}\right) \nonumber\\ + & C_{i\alpha\beta}^{R\left(\ell\right)} C_{j\alpha\beta}^{L\left(\ell\right)}m_{\chi_{\alpha}^{\pm}} B_{0}\!\left(0,m_{\chi_{\alpha}^{\pm}}^{2},m_{\widetilde{\nu}_{\beta}}^{2}\right)\!, \label{eq:app:etaellm}\\[+1mm] -\left(4\pi\right)^{2}\hskip -1mm &\left(\eta_{R}^{\ell}\right)_{ij}= \, N_{i\alpha\beta}^{L\left(\ell\right)}N_{j\alpha\beta}^{L\left(\ell\right)} B_{1}\left(0,m_{\chi_{\alpha}^{0}}^{2},m_{\widetilde{\ell}_{\beta}}^{2}\right) \nonumber\\ +& C_{i\alpha\beta}^{L\left(\ell\right)}C_{j\alpha\beta}^{L\left(\ell\right)} B_{1}\left(0,m_{\chi_{\alpha}^{\pm}}^{2},m_{\widetilde{\nu}_{\beta}}^{2}\right), \label{eq:app:etaellR}\\[+1mm] -\left(4\pi\right)^{2}\hskip -1mm &\left(\eta_{L}^{\ell}\right)_{ij}= \, N_{i\alpha\beta}^{R\left(\ell\right)}N_{j\alpha\beta}^{R\left(\ell\right)} B_{1}\left(0,m_{\chi_{\alpha}^{0}}^{2},m_{\widetilde{\ell}_{\beta}}^{2}\right) \nonumber\\ + & C_{i\alpha\beta}^{R\left(\ell\right)}C_{j\alpha\beta}^{R\left(\ell\right)} B_{1}\left(0,m_{\chi_{\alpha}^{\pm}}^{2},m_{\widetilde{\nu}_{\beta}}^{2} \right)\!,\label{eq:app:etaellL}\\[+1mm] -\left(4\pi\right)^{2}\hskip -1mm &\left(\eta_{L}^{\nu}\right)_{ij}= \, N_{i\alpha\beta}^{R\left(\nu\right)}N_{j\alpha\beta}^{R\left(\nu\right)} B_{1}\left(0,m_{\chi_{\alpha}^{0}}^{2},m_{\widetilde{\nu}_{\beta}}^{2}\right) \nonumber\\ + & C_{i\alpha\beta}^{R\left(\nu\right)}C_{j\alpha\beta}^{R\left(\nu\right)} B_{1}\left(0,m_{\chi_{\alpha}^{\pm}}^{2},m_{\widetilde{\ell}_{\beta}}^{2}\right), \label{eq:app:etanuL}\\[+1mm] -\left(4\pi\right)^{2}\hskip -1mm &\left(\eta^{H}\right)_{ij}= \, C_{i\beta\gamma}^{R\left(\nu\right)}N_{j\alpha\gamma}^{L\left(\ell\right)} \left[D_{\beta\alpha2}^{L\left(S^{+}\right)}m_{\chi_{\alpha}^{0}} m_{\chi_{\beta}^{\pm}}\right.\nonumber\\ & \left.\hskip 25mm C_{0}\left(0,0,0,m_{\chi_{\alpha}^{0}}^{2},m_{\chi_{\beta}^{\pm}}^{2},m_{\widetilde{\ell}_{\gamma}}^{2}\right)\right.\nonumber \\ + & \left. D_{\beta\alpha2}^{R\left(S^{+}\right)} dC_{00}\left(0,0,0,m_{\chi_{\alpha}^{0}}^{2},m_{\chi_{\beta}^{\pm}}^{2},m_{\widetilde{\ell}_{\gamma}}^{2} \right)\right]\nonumber \\ & +\, N_{i\alpha\gamma}^{R\left(\nu\right)} C_{j\beta\gamma}^{L\left(\ell\right)}\left[D_{\beta\alpha2}^{L\left(S^{+}\right)} m_{\chi_{\alpha}^{0}}m_{\chi_{\beta}^{\pm}} \right.\nonumber\\ &\left.\hskip 30mm C_{0}\left(0,0,0,m_{\chi_{\alpha}^{0}}^{2},m_{\chi_{\beta}^{\pm}}^{2},m_{\widetilde{\nu}_{\gamma}}^{2}\right)\right.\nonumber \\ & \left.+\, D_{\beta\alpha2}^{R\left(S^{+}\right)} dC_{00}\left(0,0,0,m_{\chi_{\alpha}^{0}}^{2},m_{\chi_{\beta}^{\pm}}^{2},m_{\widetilde{\nu}_{\gamma}}^{2}\right)\right]\nonumber \\ & +\, N_{i\alpha\beta}^{R\left(\nu\right)} N_{j\alpha\gamma}^{L\left(\ell\right)} g_{2\gamma\beta}^{\left(S^{+}\widetilde{\ell}\widetilde{\nu}^{}\right)} m_{\chi_{\gamma}^{0}}\nonumber\\ &\hskip 20mm C_{0}\left(0,0,0,m_{\widetilde{\ell}_{\gamma}}^{2},m_{\widetilde{\nu}_{\beta}}^{2},m_{\chi_{\alpha}^{0}}^{2}\right)\,,\label{eq:app:etaH}\end{aligned}$$ with $B_{0,1},C_{0},C_{0,0}$ denoting the usual loop integral functions $$\begin{aligned} & B_{0}\left(0,x,y\right)=\,\Delta_{\varepsilon}+1 -\frac{x\log\frac{x}{\mu^{2}}-y\log\frac{y}{\mu^{2}}}{x-y}\,,\\ & B_{1}\left(0,x,y\right)=\,-\frac{1}{2}\left[\Delta_{\varepsilon} +\frac{3x-y}{2\left(x-y\right)}\right.\nonumber\\ &\left.\hskip 25mm -\log\frac{y}{\mu^{2}}+\left(\frac{x}{x-y}\right)^{2}\log\frac{y}{x}\right]\,,\label{eq:app:B1}\\ & C_{0}\left(0,0,0,x,y,z\right)=\,\frac{xy\log\frac{x}{y}+yz\log\frac{y}{z}+zx\log\frac{z}{x}}{\left(x-y\right)\left(y-z\right)\left(z-x\right)}\,,\\ & dC_{00}\left(0,0,0,x,y,z\right)=\,\Delta_{\varepsilon}+1\nonumber\\ &\hskip -2mm +\frac{x^{2}\left(y-z\right)\log\frac{x}{\mu^{2}}+y^{2}\left(z-x\right)\log\frac{y}{\mu^{2}}+z^{2}\left(x-y\right)\log\frac{z}{\mu^{2}}}{\left(x-y\right)\left(y-z\right)\left(z-x\right)}\,.\end{aligned}$$ Here $d=4-\varepsilon$, $\mu$ is the regularisation parameter and $\Delta_{\varepsilon}=\frac{2}{\varepsilon}-\gamma+\log4\pi$. For the couplings notation we followed [@romao:MSSM]. The comparison of the above expressions with the corresponding ones derived in Ref. [@Bellazzini:2010gn], reveals a fair agreement; we nevertheless notice that the neutralino and chargino masses are absent from the analogous of Eq. (\[eq:app:etaellm\]), and that the order of the arguments of $B_{1}$ in Eqs. (\[eq:app:etaellR\], \[eq:app:etaellL\], \[eq:app:etanuL\]) appears reversed. Moreover, we find small discrepancies (which cannot be accounted by the distinct notations used) in the expressions for $\eta_{m}^{\ell}$ and $\eta_{H}$, cf. Eq. (\[eq:app:etaellm\]) and Eq. (\[eq:app:etaH\]), respectively. SUSY seesaw models {#app:seesaw} ================== In its different realisations, the seesaw mechanism offers one of the most appealing explanations for the smallness of neutrino masses and the pattern of neutrino mixing angles. Moreover, when embedded in the framework of SUSY models - the so-called SUSY seesaw - the seesaw offers the interesting feature that flavour violation in the neutrino sector (encoded in non-diagonal neutrino Yukawa couplings) can radiatively induce flavour violation in the slepton sector at low-energies [@Borzumati:1986qx], leading to potentially sizable contributions to a large array of observables. In what follows we briefly summarise the most relevant features of different realisations of the seesaw mechanism. In particular, we will consider “high-scale” seesaws, i.e., where the additional states are assumed to be much heavier than the electroweak scale (in association with large values of the corresponding couplings). Type I SUSY seesaw {#app:seesawI} ------------------ In a type I SUSY seesaw, the MSSM superfield content is extended by three right-handed Majorana neutrino superfields. The lepton superpotential is thus extended as $$\mathcal{W}_{\text{I}}^{\text{lepton}}=\hat{N}^{c} Y^{\nu} \hat{L} \hat{H}_{2}+\hat{E}^{c} Y^{l} \hat{L} \hat{H}_{1} + \frac{1}{2} \hat{N}^{c} M_{N} \hat{N}^{c} ,\label{eq:WleptonI:def}$$ where, and without loss of generality, one can work in a basis where both $Y^{l}$ and $M_{N}$ are diagonal ($Y^{l}=\operatorname{diag}(Y^{e},Y^{\mu},Y^{\tau})$, $M_{N}=\operatorname{diag}(M_{N_{1}},M_{N_{2}},M_{N_{3}})$). The relevant slepton soft-breaking terms are now $$\begin{aligned} \mathcal{V}_{\text{soft\ I}}^{\text{slepton}}=& m_{\tilde{L}}^{2}\,\tilde{l}_{L}\,\tilde{l}_{L}^{} +m_{\tilde{R}}^{2}\,\tilde{l}_{R}\,\tilde{l}_{R}^{} +m_{\tilde{\nu}_{R}}^{2}\,\tilde{\nu}_{R}\,\tilde{\nu}_{R}^{} +\!\left(A^{l}\, H_{1}\,\tilde{l}_{L}\,\tilde{l}_{R}^{} \right.\nonumber\\ &\left. +A^{\nu}\, H_{2}\,\tilde{\nu}_{L}\,\tilde{\nu}_{R}^{} +B^{\nu}\,\tilde{\nu}_{R}\,\tilde{\nu}_{R}+\text{h.c.}\right).\end{aligned}$$ Should this be embedded into a cMSSM, then the additional soft breaking parameters would also obey universality conditions at the GUT scale, $(m_{\widetilde{\nu}_{R}})_{ij}^{2}=m_{0}^{2}$ and $(A^{\nu})_{ij}=A_{0}(Y^{\nu})_{ij}$. In this case, the light neutrino masses are given by $$m_{\nu}^{\text{I}}\,=\,-{m_{D}^{\nu}}^{T}M_{N}^{-1}m_{D}^{\nu}\,,\label{eq:seesawI:light}$$ with $m_{D}^{\nu}=Y^{\nu}\, v_{2}$ ($v_{i}$ being the vacuum expectation values (VEVs) of the neutral Higgs scalars, $v_{1(2)}=\, v\,\cos(\sin)\beta$, with $v=174$ GeV), and where $M_{N_{i}}$ corresponds to the masses of the heavy right-handed neutrino eigenstates. The light neutrino matrix $m_{\nu}$ is diagonalized by the $U_{\text{MNS}}$ as $m_{\nu}^{\text{diag}}={U_{\text{MNS}}}^{T}m_{\nu}U_{\text{MNS}}$. A convenient means of parametrising the neutrino Yukawa couplings, while at the same time allowing to accommodate the experimental data, is given by the Casas-Ibarra parametrisation [@Casas:2001sr], which reads at the seesaw scale, $M_{N}$, $$Y^{\nu}v_{2}=m_{D}^{\nu}\,=\, i\sqrt{M_{N}^{\text{diag}}}\, R\,\sqrt{m_{\nu}^{\text{diag}}}\,{U_{\text{MNS}}}^{\dagger}\,.\label{eq:seesaw:casas}$$ In the above, $R$ is a complex orthogonal $3\times3$ matrix that encodes the possible mixings involving the right-handed neutrinos, in addition to those of the low-energy sector (i.e. $U_{\text{MNS}}$) and which can be parametrised in terms of three complex angles $\theta_{i}$ $(i=1,2,3)$. In our analysis, we assumed degenerate right-handed neutrino masses and real parameters, so that the results are effectively independent of the choice of the $\theta_{i}$. Even under universality conditions at the GUT scale, the non-trivial flavour structure of $Y^{\nu}$ will induce (through the running from $M_{\text{GUT}}$ down to the seesaw scale, $M_{N}$) flavour mixing in the otherwise approximately flavour conserving soft-SUSY breaking terms. In particular, there will be radiatively induced flavour mixing in the slepton mass matrices, manifest in the $LL$ and $LR$ blocks of the $6\times6$ slepton mass matrix; an analytical estimation using the leading order (LLog) approximation leads to the following corrections to the slepton mass terms: $$\begin{aligned} \label{eq:LFV:LLog} (\Delta m_{\tilde{L}}^{2})_{_{ij}} & \,=\,-\frac{1}{8\,\pi^{2}}\,(3\, m_{0}^{2}+A_{0}^{2})\,({Y^{\nu}}^{\dagger}\, L\, Y^{\nu})_{ij}\,,\nonumber \\ (\Delta A^{l})_{_{ij}} & \,=\,-\frac{3}{16\,\pi^{2}}\, A_{0}\, Y_{ij}^{l}\,({Y^{\nu}}^{\dagger}\, L\, Y^{\nu})_{ij}\,,\nonumber \\ (\Delta m_{\tilde{R}}^{2})_{_{ij}} & \,\simeq\,0\,\,;\, L_{kl}\,\equiv\,\log\left(\frac{M_{\text{GUT}}}{M_{N_{k}}}\right)\,\delta_{kl}\,.\end{aligned}$$ The amount of flavour violation is encoded in the matrix elements $({Y^{\nu}}^{\dagger}LY^{\nu})_{ij}$ of Eq. (\[eq:LFV:LLog\]). Type II SUSY seesaw {#app:seesawII} ------------------- The implementation of a type II SUSY seesaw model requires the addition of at least two SU(2) triplet superfields [@Rossi:2002zb]. Should one aim at preserving gauge coupling unification, then complete SU(5) multiplets must be added to the MSSM content. Under the SM gauge group, the 15* decomposes as $\pmb{15}=S+T+Z$, where $S\sim(6,1,-2/3)$, $T\sim(1,3,1)$ and $Z\sim(3,2,1/6)$. In the SU(5) broken phase (below the GUT scale), the superpotential contains the following terms: $$\begin{aligned} \mathcal{W}_{\text{II}}\, &=\,\frac{1}{\sqrt{2}}\,\left(Y_{T}\,\hat{L}\,\hat{T}_{1}\, \hat{L}+Y_{S}\,\hat{D}\,\hat{S}\,\hat{D}\right)\, +\, Y_{Z}\,\hat{D}\,\hat{Z}\,\hat{L}\, \nonumber\\ & +\, Y^{d}\,\hat{D}^{c}\,\hat{Q}\,\hat{H}_{1}\, +\, Y^{u}\,\hat{U}^{c}\,\hat{Q}\,\hat{H}_{2}\, +\, Y^{l}\,\hat{E}^{c}\,\hat{L}\,\hat{H}_{1}\nonumber \\ \, & +\,\frac{1}{\sqrt{2}}\,\left(\lambda_{1} \hat{H}_{1}\,\hat{T}_{1}\,\hat{H}_{1}+\lambda_{2}\,\hat{H}_{2}\,\hat{T}_{2}\,\hat{H}_{2}\right)\, +\, M_{T}\,\hat{T}_{1}\,\hat{T}_{2}\nonumber\\ & +\, M_{Z}\,\hat{Z}_{1}\,\hat{Z}_{2}\, +\, M_{S}\,\hat{S}_{1}\,\hat{S}_{2}\,+\,\mu\,\hat{H}_{1}\,\hat{H}_{2}\,,\end{aligned}$$ where we have omitted flavour indices for simplicity (for shortness we will not detail the soft breaking Lagrangian here, see e.g. [@Rossi:2002zb]). 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This amounts to a small multiplicative effect on $\Delta r$ which we will not discuss here (see [@Girrbach:2012km] for details). [^4]: We will assume that we are in a strictly CP conserving framework, where all terms are taken to be real. This implies that there will be no contributions to observables such as electric dipole moments, or CP asymmetries.
--- author: - \| \ Department of Physics, Baylor University, Waco, TX 76798-7316,USA\ E-mail: - \| Walter Wilcox\ Department of Physics, Baylor University, Waco, TX 76798-7316, USA\ E-mail: - \| Ronald B. Morgan\ Department of Mathematics, Baylor University, Waco, TX 76798-7328, USA\ E-mail: bibliography: - 'sample.bib' title: New Noise Subtraction Methods in Lattice QCD --- Introduction ============ Lattice QCD is a set of numerical techniques which uses a finite space-time lattice to simulate the interactions between quarks and gluons. The evaluation of quark loop effects on a given lattice is essential but extremely computer time intensive and approximation techniques must be introduced[@Wilcox]. In this paper, we describe some new noise subtraction methods useful in evaluating quark operators. Perturbative subtraction[@Thron] is a standard method which we will be comparing to and attempting to improve upon. This paper will be focusing on eigenspectrum subtraction (deflation), polynomial subtraction, and combination methods. Methods ======= Many operators in lattice QCD simulations are flooded with noise. Several strategies could be applied to reduce the variance of these operators, which originates in the off-diagonal components of the associated quark matrix. One basic strategy is to mimic the off-diagonal elements of the inverse of the quark matrix with another traceless matrix, thereby maintaining the trace but with reduced statistical uncertainty. We have applied several new techniques to non-subtracted (NS) lattice data. These are termed eigenvalue subtraction (ES)[@Guerrero], Hermitian forced eigenvalue subtraction (HFES)[@Guerrero2] and polynomial subtraction (POLY)[@Liu]. In [@Guerrero2] we also introduced techniques which combine deflation with other subtraction methods. Here we will compare the various methods and show how effective the combination methods are. We work with the standard Wilson matrix in the quenched approximation. The size of the lattice we used is $24^{3}\times 32$, the number of noises is 200, the kappa value is 0.155, and we use $Z4$ noise. Linear equations are solved using GMRES-DR (generalized minimum residual algorithm-deflated and restarted) for the first noise and GMRES-Proj (similar algorithm projected over eigenvectors) for remaining noises[@Morgan] . In order to calculate the trace, the quark matrix $M$ is projected over a finite number of noises $\eta^{(n)}$, $Mx^{(n)}=\eta^{(n)}$, and solution vectors $x^{(n)}$ are extracted. All of our methods attempt to design a traceless matrix $\tilde M^{-1}$ in order to obtain off-diagonal elements as close to $M^{-1}$ as possible. We then use solution vectors formed from such a matrix. Unfortunately, this matrix is not completely traceless, so we will have to re-add the appropriate trace. For the trace of the inverse quark matrix one has $$Tr\left(M^{-1}\right)=\frac{1}{N}\sum_n^N{\left( \eta ^{(n)\dagger} \left( x^{(n)} -\tilde x^{(n)}\right) \right)} + Tr\left( \tilde M^{-1}\right),$$ for $N$ noises, where $x^{(n)}$ is the solution vector generated when implementing the GMRES algorithms and $\tilde x^{(n)}$ is given by $$\tilde x^{(n)}\equiv \tilde M^{-1}\eta^{(n)}.$$ For any operator $\Theta$, the appropriate trace becomes $$Tr\left( \Theta M^{-1}\right)=\frac{1}{N}\sum_n^N{\left( \eta ^{(n)\dagger} \Theta \left(x^{(n)} -\tilde x^{(n)}\right) \right)} + Tr\left( \Theta \tilde M^{-1}\right).$$ Note that adding $Tr\left(\Theta \tilde M^{-1}\right)$ has no influence on the noise error bar, which is what is studied here. Eigenvalue Subtraction (ES) --------------------------- The spectrum of low eigenvalues of matrices can limit the performance of iterative solvers. We have emphasized the role of deflation in accelerating the convergence of algorithms in Ref.[@Wilcox2]. Here we investigate deflation effects in statistical error reduction. Consider the vectors $e_R^{(q)}$ and $e_L^{(q)\dagger}$, which are defined as normalized right and left eigenvectors of the matrix $M$, as in $$Me_R^{(q)}=\lambda^{(q)} e_R^{(q)},$$ and $$e_L^{(q)\dagger} M=\lambda^{(q)} e_L^{(q)\dagger},$$ where $\lambda^{(q)}$ is the eigenvalue associated with both eigenvectors. With a full set $N$ of eigenvectors and eigenvalues the matrix $M$ can be fully formed as $$%M=∑_(q=1)^N▒〖e_R^((q)) λ^((q)) e_L^(〖(q)〗^† ) 〗 M=\sum_{q=1}^{N} {e_R^{(q)} \lambda^{(q)} e_{L}^{(q)\dagger}},$$ or $$M=V_R \Lambda V_L^{\dagger},$$ where $V_R$ contains the right eigenvectors and $V_L^{\dagger}$ contains the left eigenvectors. $\Lambda$ is a purely diagonal matrix made up of the eigenvalues of $M$ in the order they appear in both $V_R$ and $V_L$. Deflating out eigenvalues with the linear equation solver GMRES-DR can mimic the low eigenvalue structure of the inverse of matrix $M$ as $$%BS (M^' ) ̃_eig^(-1)≡(V^' ) ̌_R (Λ^' ) ̃^(-1) (V^' ) ̃_L^* \tilde{M}_{eig}^{-1} \equiv \tilde{V}_{R} \tilde{\Lambda}^{-1} \tilde{V}_{L}^{\dagger},$$ where $\tilde{V}_{R}$ and $\tilde{V}_{L}^{\dagger}$ are the computed right and left eigenvectors and $ \tilde{\Lambda}^{-1}$ is the inverse of eigenvalues. In the results we will see that if we näively subtract eigenvalues from a non-Hermitian matrix, we often end up expanding the size of error bars. This happens because many of the right handed eigenvectors of a non-Hermitian matrix can point in the same direction, a condition referred to as highly non-normal". The trace takes the following form in this method, $$Tr\left( \Theta M^{-1}\right)=\frac{1}{N}\sum_n^N{\left( \eta ^{(n)\dagger} \Theta \left( x^{(n)} -\tilde x^{(n)}_{eig}\right) \right)} + Tr\left( \Theta \tilde M^{-1}_{eig}\right),$$ where $\tilde x^{(n)}_{eig} = \tilde M_{eig}^{-1}\eta^{(n)}$. This last operation does not add matrix vector products. The generation of eigenmodes only requires the super convergence solution of a single right hand side with GMRES-DR. As pointed out previously[@Guerrero], there is a relation between the even-odd eigenvectors for the reduced system and the full eigenvectors. Other right hand sides are accelerated with GMRES-Proj using the eigenvalues generated. Hermitian Forced Eigenvalue Subtraction (HFES) ---------------------------------------------- To avoid the non-normal problem we force our matrix to be formulated in a Hermitian manner. The easiest way for us to do this with the Wilson matrix is to multiply by the Dirac $\gamma_5$ matrix. It is important for the algorithm to do the multiplication on the [right]{}, $M\gamma_5$, to avoid using cyclic properties which fail in finite noise space. We can then form the low eigenvalue structure of $M\gamma_5$ from these eigenvalues. We define $$M'\equiv M\gamma_5.$$ We can now form normalized eigenvectors ${e'}_R^{(n)}$, eigenvalues $\lambda '^{(n)}$ and solution vectors $\tilde{x}'{_{eig}^{(n)}}$ for this new Hermitian matrix and perform a calculation similar to the ES method, accounting properly for the extra $\gamma_5$ factors. The trace of any operator $\Theta$, for the HFES method takes the following form, $$Tr\left( \Theta M^{-1}\right)=\frac{1}{N}\sum_n^N{\left( \eta ^{(n)\dagger} \Theta \left( x^{(n)} -\tilde{x}'{_{eig}^{(n)}}\right) \right)} + Tr\left( \Theta \gamma_5\tilde {M}'{_{eig}^{-1}}\right),$$ where $$\tilde{x}'{_{eig}^{(n)}} \equiv\gamma_5\tilde{M}'{_{eig}^{-1}}\eta^{(n)} =\gamma_5 \sum_{q}^{Q}{\frac{1}{{\lambda ' }^{(q)}} { e'}_{R}^{(q)} \left({e'}_{R}^{(q)\dagger} \eta^{(n)} \right) }$$ and $$\tilde{M}'{_{eig}^{-1}} \equiv \tilde{V}_{R}' \tilde{\Lambda'}^{-1} \tilde{V}_{R}'^{\dagger}.$$ $\tilde{V}_{R}'$ is a matrix whose columns are the Q smallest right eigenvectors of $M'$. $\tilde{\Lambda '}^{-1}$ is the diagonal matrix of size Q that contains the inverse of eigenvalues $1/{{\lambda'}^{(q)}}$ as the diagonal elements. The price paid here, similar to the ES method, is a single extra super convergence on one right hand side for the non-reduced Hermitian system $M'$ with GMRES-DR to extract eigenvectors and eigenvalues. Polynomial Subtraction (POLY) ----------------------------- Our goal is to find more efficient methods than perturbative subtraction (PS), where to 6th order: $$\tilde{M}_{pert}^{-1} \equiv 1+\kappa P+(\kappa P)^2+(\kappa P)^3+(\kappa P)^4+(\kappa P)^5+(\kappa P)^6.$$ $P$ is the quark hopping matrix and $\kappa$ is the usual expansion parameter. The polynomial method is similar to that of perturbative subtraction. The only difference is that the coefficients are allowed to be different from one, $$\tilde{M}_{poly}^{-1} \equiv a_1+ a_2\kappa P+a_3(\kappa P)^2+a_4(\kappa P)^3+a_5(\kappa P)^4+a_6(\kappa P)^5+a_7(\kappa P)^6,$$ where the $a_i$’s are the coefficients obtained from min-res projection[@Liu]. The trace in this method takes the form, $$Tr\left( \Theta M^{-1}\right)=\frac{1}{N}\sum_n^N{\left( \eta ^{(n)\dagger} \cdot \Theta \left( x^{(n)} -\tilde x^{(n)}_{poly}\right) \right)} + Tr\left( \Theta \tilde M^{-1}_{poly}\right),$$ where $$\tilde x^{(n)}_{poly}\equiv\tilde M^{-1}_{poly}\eta^{(n)}.$$ Combination Methods (HFPOLY and HFPS) ------------------------------------- We have developed two methods which combine the error reduction techniques of HFES with POLY and PS, called HFPOLY and HFPS. Näively, for POLY we could think of this method as a subtracted combination: $\tilde M^{-1}_{poly}+\gamma_5 \tilde M'^{-1}_{eig}$. However, this presents a possible conflict since $\tilde M^{-1}_{poly}$ will overlap on the deflated Hermitian eigenvector space. In order to prevent this, we also remove low eigenmode information from $\tilde {M}^{-1}_{poly}$. Since $\tilde {M}_{poly}$ is not Hermitian, the procedure is to define $\tilde M'_{poly}=\tilde {M}_{poly}\gamma_5$ and remove its overlapping Hermitian eigenvalue information using the eigenvectors from $M'$. Following the idea in Ref.[@Guerrero], we define $${e'}_R^{(q)\dagger}\tilde M'^{-1}_{poly}{e'}_R^{(q)}\equiv\frac{1}{\xi'^{(q)}},$$ where ${e'}^{(q)}_{R}$ is the eigenmode of $M'$ generated within HFES method and $1/\xi'^{(q)}$ are the approximate eigenvalues of $\tilde M'^{-1}_{poly}$. The trace takes the following form, $$\begin{aligned} Tr\left( \Theta M^{-1}\right ) = \frac{1}{N} & \sum_n^N { \left( \eta^{(n)\dagger} \left[ \Theta x^{{(n)}} - \Theta\tilde{x}'{_{eig}^{(n)}} - \left( \Theta \tilde{x}_{poly}^{(n)} - \Theta \tilde{x}'{_{eigpoly}^{(n)}} \right) \right ] \right) } \nonumber\\+& Tr \left( \Theta \gamma_5 \tilde{M'}_{eig}^{-1}\right)+Tr\left( \Theta \tilde{M}_{poly}^{-1}- \Theta \gamma_5 \tilde{M'}_{eigpoly}^{{-1}} \right) ,\end{aligned}$$ where $$\tilde{x}'{_{eigpoly}^{(n)}} \equiv\gamma_5\tilde{M}'{_{eigpoly}^{-1}}\eta^{(n)} =\gamma_5 \sum_{q}^{Q}{\frac{1}{\xi'^{(q)}} { e'}_{R}^{(q)} \left({e'}_{R}^{(q)\dagger} \eta^{(n)} \right) }.$$ $\tilde{x}'{_{eig}^{(n)}}$ and $\tilde x^{(n)}_{poly}$ are defined in previous sections and $$\tilde{M}_{eigpoly}^{'^{-1}} \equiv \tilde{V}_{R}' \Xi^{-1} \tilde{V}_{R}'^{\dagger},$$ where $\tilde{V}_{R}'$ is defined above also and $\Xi^{-1}$ is the diagonal matrix of size Q that contains approximate inverse eigenvalues, $1/\xi'^{(q)}$. In the case of HFPS, $\tilde{M}_{poly}^{-1}$ is replaced by $\tilde{M}_{pert}^{-1}$ and all the calculations are repeated. Results ======== Figure 1 shows the calculated error bars for the nonlocal current operator in the one direction (other currents are similar) as a function of deflated eigenvectors. Similarly, Figure 2 shows error bars for the local current operator in the one direction. Figure 3 represents the error bar for the scalar operator. We deflated approximately 140 eigenvectors. The ES method does not decrease the error bars as the number of deflated eigenvectors is increased. This arises from the non-normal nature of the non-Hermitian eigenvectors[^1]. The HFES method reduces the error bars in all cases. However it does not outperform PS for the number of eigenvectors removed. The POLY method is better than PS for nonlocal and scalar operators but the difference is not that significant for local operators. The HFPOLY combo is the most efficient method. We define the relative efficiency, $RE$, of the two methods as $$RE\equiv \left( \frac {1}{\delta y^{2}}-1\right) \times 100,$$ where $\delta y$ is the relative error bar. The relative error bars for HFPOLY combo as compared to PS are approximately $0.77, 0.75$ and $0.74$ for scalar, local and nonlocal operators, respectively. That means the HFPOLY method is more efficient than the PS method by 68%, 77% and 81%, respectively. We expect efficiency to improve further as we move on towards lower quark masses. Note that similar results for deflation applied to hierarchical probing have been obtained in Ref.[@Gambhir]. ![Error bars for a local spatial vector as a function of deflated eigenvalues.[]{data-label="fig:2"}](nonlocal1.eps){width=".75\textwidth"} ![Error bars for a local spatial vector as a function of deflated eigenvalues.[]{data-label="fig:2"}](local1.eps){width=".75\textwidth"} ![Error bars for the scalar operator as a function of deflated eigenvalues.[]{data-label="fig3"}](scalar.eps){width=".75\textwidth"} Conclusions =========== Our polynomial and perturbative deflation combination methods produce very encouraging results for $\kappa=0.155$. Although our quark mass is not small in this investigation, we are hopeful that our methods will be effective at smaller quark mass. Acknowledgements ================ All of the numerical work was performed using the High Performance Cluster at Baylor University. This work is partially supported by URC grant funding, Baylor University. [99]{} [^1]: Note that the numerical ES results in Ref.[@Guerrero] were in error; see Ref.[@Guerrero2]
--- address: \| Division EP, CERN, CH-1211 Genève-23, Switzerland\ E-mail: Isabel.Trigger@cern.ch author: - Isabel Trigger title: 'Constraining CP-Violating TGCs and Measuring W-Polarization at OPAL' --- The Spin Density Matrix for ============================ Polarization properties of produced in collisions are summarized by the two-particle joint spin density matrix (SDM). The SDM is the product of the amplitudes for producing a and of respective helicities $\tau_+$, $\tau_-$: $$ \rho_{\tau_{-}{\tau^{\prime}}\!\!_{-}\tau_{+}{\tau^{\prime}}\!\!_{+}}(s,\cos\theta_{\rm W}) = \frac{\sum_{\lambda}F^{(\lambda)}_{\tau_{-}\tau_{+}}(F^{(\lambda)}_{\tau_{-}^{ \prime}\tau_{+}^{\prime}})^{}}{\sum_{\lambda\tau_{+}\tau_{-}}\|F^{(\lambda)}_{ \tau_{-}\tau_{+}}\|^{2}} \, . \label{eq:jsdm}$$ The diagonal elements are probabilities of producing with the corresponding helicity combinations, and are strictly real. The potentially complex off-diagonal terms represent interference between helicity states. The data considered in the analysis were collected with the OPAL detector in 1998, at a centre-of-mass energy of 189 GeV. They correspond to an integrated luminosity of $\sim 183\mbox{ pb}^{-1}$, with 1065 events identified as $\PWp\PWm\to \mathrm{q {\bar{q}}} \ell \nu$. Only semi-leptonic decays are used for this analysis, as fully leptonic and fully hadronic decays cannot be unambiguously reconstructed. Due to the restricted sample size, it is not possible to measure all 81 elements of the joint SDM; however, all elements of the nine-element single-particle SDM, obtained by summing over the helicity states of the , may be measured: $$ \rho^{W^{-}}_{\tau_{-}{\tau^{\prime}}\!\!_{-}}(s,\cos\theta_{\rm W}) = \sum_{\tau_{+}}\rho_{\tau_{ -}{\tau^{\prime}}\!\!_{-}\tau_{+}\tau_{+}}(s,\cos\theta_{\rm W}) \, . \label{eq:ssdm}$$ Its diagonal elements are the probabilities of producing a with helicity $+1, 0$ or $-1$. In order to use both $\mathrm{ q {\bar q}}\ell^-\bar{\nu}$ and $\mathrm{ q {\bar q}}\ell^+\nu$, CPT is assumed. The five kinematic variables used in the analysis are the cosine of the polar production angle of the in the laboratory frame ($\cos\theta_W$), and the decay angles given by the directions of the fermion and anti-fermion with respect to the and respectively in the $\mathrm{W}$ rest-frames ($\cos\theta_\ell^\ast,\phi_\ell^\ast,$ $\cos\theta_j^\ast, \phi_j^\ast)$, with an ambiguity of $\pi$ for both jet angles due to the difficulty of distinguishing the up-quark jet from the anti–down-quark jet. With the hadronic part of the event, it is therefore only possible to measure combinations of SDM elements which are symmetric under $\cos\theta^\ast \to -\cos\theta^\ast$ and $\phi^\ast \to \phi^\ast+\pi$. Fortunately, these include $\rho_{++}+\rho_{--}$ and $\rho_{00}$, so the full two-particle SDM element $\rho_{0000}$ and the combinations $\rho_{++++}+\rho_{++--}+\rho_{--++}+\rho_{----}$ and $\rho_{++00}+\rho_{--00}+\rho_{00++}+\rho_{00--}$ may be measured. SDM elements are measured by forming histograms of the $\cos\theta_W$ distribution obtained in the data, and weighting each event by a projection operator which is a function of $\cos\theta_\ell^\ast,\phi_\ell^\ast,\cos\theta_j^\ast, \phi_j^\ast$. Different operators project out each independent element of the SDM. Results are shown in figure \[fig:sdm\]. Constraints arise from imposing tree-level CPT-invariance: $$\begin{aligned} {\rm Re}(\rho^{W^{-}}_{\tau_{1}\tau_{2}}) &-&{\rm Re}(\rho^{W^{+}}_{-\tau_{1}-\tau_{2}}) = 0 \label{eq:cptreal}\\ {\rm Im}(\rho^{W^{-}}_{\tau_{1}\tau_{2}}) &+& {\rm Im}(\rho^{W^{+}}_{-\tau_{1}-\tau_{2}}) = 0 \, . \label{eq:cptim}\end{aligned}$$ Since CPT has already been assumed, any deviation from these must arise from loop effects. CP-invariance would further imply that: $${\rm Im}(\rho^{W^{-}}_{\tau_{1}\tau_{2}}) - {\rm Im}(\rho^{W^{+}}_{-\tau_{1}-\tau_{2}}) = 0 \, . \label{eq:cpim}$$ Combining equations \[eq:cptim\] and \[eq:cpim\] shows that all single-particle SDM coefficients must be strictly real in the absence of CP-violation. Helicity Cross-Sections {#sec:polar} ======================= Differential production cross-sections for $\mathrm{W}^\pm$ of particular helicities are: $$\frac{{\rm d}\sigma_h}{{\rm d}\!\cos\theta_{\rm W}} = f_h \frac{{\rm d}\sigma}{{\rm d}\!\cos\theta_{\rm W}} \, , \label{eq:crsec}$$ where $f_{\rm T}=\rho_{++}+\rho_{--}$, $f_{\rm L}=\rho_{00}$, $f_{\rm TT}=\rho_{++++}+\rho_{++--}+\rho_{--++}+\rho_{----}$, $f_{\rm LL}=\rho_{0000}$, $f_{\rm TL}=\rho_{++00}+\rho_{--00}+\rho_{00++}+\rho_{00--}$. Corresponding total cross-sections may be obtained by integrating over the full range of ${\rm d}\!\cos\theta_{\rm W}$. Results are given in table \[tab:pol\]. Correction factors are included for detector and reconstruction effects. [Data]{} [SM Exp.]{} ------------------------------- ------------------------------- ------------------- W$ \rightarrow \ell{\nu}$ 0.842 $\pm$ 0.048 $\pm$ 0.023 0.746 $\pm$ 0.006 W$\rightarrow {\rm q}{\rm q}$ 0.738 $\pm$ 0.045 $\pm$ 0.025 0.741 $\pm$ 0.006 All 0.790 $\pm$ 0.033 $\pm$ 0.016 0.743 $\pm$ 0.004 W$\rightarrow \ell{\nu}$ 0.158 $\pm$ 0.048 $\pm$ 0.023 0.254 $\pm$ 0.006 W$\rightarrow {\rm q}{\rm q}$ 0.262 $\pm$ 0.045 $\pm$ 0.025 0.259 $\pm$ 0.006 All 0.210 $\pm$ 0.033 $\pm$ 0.016 0.257 $\pm$ 0.004 : Fractions of $\mathrm{W}$ polarizations, and of pairs of each helicity combination. Expected values are from generator level EXCALIBUR Monte Carlo. \[tab:pol\] Measured Expected ------------------------------------------ ----------------------------------- ----------------------- $\sigma_{\rm TT}/\sigma_{\rm total}$ 0.781 $\pm$ 0.090 $\pm$ 0.033 0.572 $\pm$ 0.010 [$\sigma_{\rm LL}/\sigma_{\rm total}$]{} [0.201 $\pm$ 0.072 $\pm$ 0.018]{} [0.086 $\pm$ 0.008]{} $\sigma_{\rm TL}/\sigma_{\rm total}$ 0.018 $\pm$ 0.147 $\pm$ 0.038 0.342 $\pm$ 0.016 : Fractions of $\mathrm{W}$ polarizations, and of pairs of each helicity combination. Expected values are from generator level EXCALIBUR Monte Carlo. \[tab:pol\] The and polarizations are about 7% correlated. This effect is included in the systematic errors. Total cross-sections for the various helicity states are very strongly correlated. It should be noted that $\rho_{0000}$ can have no CP-violating contributions,[@rho0000] and $\frac{{\rm d}\sigma_{\rm LL}}{{\rm d}\!\cos\theta_{\rm W}}$ is thus completely insensitive to CP-violation. The helicity fractions for TT, LL and TL differ by about $2\sigma$ from Standard Model (SM) predictions, giving a $\chi^2$ probability of about 10% for SM compatibility. Triple Gauge Boson Couplings {#sec:tgc} ============================ Of nine possible helicity pairings, seven are allowed in the $s$-channel processes $\Pep\Pem\to {{\mathrm Z^} / \gamma^}\to \PWp\PWm$ ($+-$, $-+$ occur only in $t$-channel $\nu$-exchange, as $\|\tau_+ - \tau_-\|=2$). There are then seven free parameters in the Lagrangian describing each of the triple gauge boson couplings $\PWp\PWm\PZz$ and $\PWp\PWm\Pgg$, here called $\kappa_V, g_1^V, \lambda_V, g_5^V, \tilde\kappa_V, g_4^V, \tilde\lambda_V$ ($V=\PZz, \Pgg$). The first two are unity in the SM, and the others zero. The first three conserve C and P, $g_5^V$ violates C and P but conserves CP, and the last three violate CP. The real elements of the SDM are sensitive to all of the coupling parameters; the imaginary elements are also sensitive to CP-violating parameters. It is impossible to fit all fourteen parameters simultaneously with the limited data sample. Each is measured separately, with the others set to their SM values, except those related to the tested parameter through $SU(2)\times U(1)$ symmetry[@goun] ($\tilde\kappa_Z = -\tan^2\theta_W \tilde\kappa_\gamma$; $\tilde\lambda_Z = \tilde\lambda_\gamma$; $g_4^Z = g_4^\gamma$). This leaves three independent parameters to test, chosen to be $\tilde\kappa_Z, \tilde\lambda_Z, g_4^Z$. There are strong constraints on CP-violation in electromagnetic interactions from the constraints on the electric dipole moment of the neutron,[@neutron] which is why an alternative would be to ignore the gauge symmetry constraints and set couplings to zero. Coupling values from a fit to the SDM elements are given in table \[tab:tgc\]. Fit $\tilde{\kappa}_{\rm z}$ $g^{\rm z}_{4}$ $\tilde{\lambda}_{\rm z}$ ----------------------- ------------------------------------ ------------------------------------ ------------------------------------ SDM Elements $-0.19^{+0.08}_{-0.07}$ $\;\;\;$$0.00^{+0.21}_{-0.20}$ $-0.12^{+0.17}_{-0.16}$ $\cos\theta_{\rm W}$ $-0.19^{+0.46}_{-0.08}$ $\;\;\;$$0.7^{+0.4}_{-1.8}$ $-0.29^{+0.69}_{-0.11}$ Combined $-0.19^{+0.06}_{-0.05}$ $\;\;\;$$0.01^{+0.22}_{-0.22}$ $-0.19^{+0.18}_{-0.13}$ Expected Stat. Error $\;\;\;\pm$0.11 $\;\;\;\pm$0.19 $\;\;\;\pm$0.12 Final Fit Including Systematics \[0pt\][$-0.20^{+0.10}_{-0.07}$]{} \[0pt\][$-0.02^{+0.32}_{-0.33}$]{} \[0pt\][$-0.18^{+0.24}_{-0.16}$]{} : Measured values of CP-violating TGC parameters. Both the SDM elements and the $\cos\theta_{\rm W}$ production distribution are used in the calculation. Errors are statistical only except in the case of the final combined fit.[]{data-label="tab:tgc"} Table \[tab:tgc\] also includes results from a $\chi^2$ fit to the $\cos\theta_W$ distribution. The $\cos\theta_W$ distribution is the most sensitive to variations in CP-conserving couplings, but is relatively insensitive to CP-violating couplings. Figure \[fig:chi\] shows $\chi^2$ curves for TGCs measured from SDM elements and from $\cos\theta_W$ distributions. The real SDM elements and $\cos\theta_W$ are sensitive only to the magnitude of CP-violating couplings (their dependence on the couplings is quadratic), and so have a double minimum. The imaginary SDM elements depend linearly on the CP-violating couplings and can thus lift the degeneracy. The CP-conserving couplings measured from the SDM elements are fully compatible with the results of the OPAL optimal observable analysis.[@oo] Conclusions =========== The SDM method allows direct measurement of the fraction of $\mathrm{W}$ bosons produced with longitudinal polarization. This longitudinal component of the $\mathrm{W}$ is a result of the electroweak symmetry breaking mechanism. It also provides constraints on CP-violation in TGCs. All results are compatible with SM predictions. This analysis is more fully described elsewhere.[@sdm] [99]{} G. Gounaris, J. Layssac, G. Moultaka and F.M. Renard, [Int. Journal of Modern Physics]{} A[8]{}, 3285 (1993). G. Gounaris, C.G. Papadopoulos, [Eur. Phys. J.]{} C[2]{}, 365 (1998).\ C.G. Papadopoulos, [Comput. Phys. Commun.]{} [101]{}, 183 (1997). P.G. Harris [et al.]{}, [Phys. Rev. Lett.]{} [82]{}, 904 (1999).\ K.F. Smith [et al.]{}, [Phys. Lett.]{} B[234]{}, 191 (1990). OPAL Collaboration, G. Abbiendi [et al.]{}, “Measurement of Triple Gauge Boson Couplings from Production at LEP Energies up to 189 GeV,” CERN-EP-2000-114, [ Submitted to Eur. Phys. J. C]{}. OPAL Collaboration, G. Abbiendi [et al.]{}, “Measurement of W Boson Polarisations and CP-violating Triple Gauge Couplings from Production at LEP,” CERN-EP-2000-113, [ Submitted to Eur. Phys. J. C*]{}.
--- abstract: 'Recently, differentiable neural architecture search methods significantly reduce the search cost by constructing a super network and relax the architecture representation by assigning architecture weights to the candidate operations. All the existing methods determine the importance of each operation directly by architecture weights. However, architecture weights cannot accurately reflect the importance of each operation; that is, the operation with the highest weight might not related to the best performance. To alleviate this deficiency, we propose a simple yet effective solution to neural architecture search, termed as exploiting operation importance for effective neural architecture search (EoiNAS), in which a new indicator is proposed to fully exploit the operation importance and guide the model search. Based on this new indicator, we propose a gradual operation pruning strategy to further improve the search efficiency and accuracy. Experimental results have demonstrated the effectiveness of the proposed method. Specifically, we achieve an error rate of 2.50% on CIFAR-10, which significantly outperforms state-of-the-art methods. When transferred to ImageNet, it achieves the top-1 error of 25.6%, comparable to the state-of-the-art performance under the mobile setting.' author: - \| Xukai Xie\ Tianjin university\ [xkxie@tju.edu.cn]{} - \| Yuan Zhou^\^\ Tianjin university\ [zhouyuan@tju.edu.cn]{} - \| Sun-Yuan Kung\ Princeton University\ [kung@princeton.edu]{} bibliography: - 'egbib.bib' title: Exploiting Operation Importance for Differentiable Neural Architecture Search --- Introduction ============ Designing reasonable network architecture for specific problems is a challenging task. Better designed network architectures usually lead to significant performance improvement. In recent years, neural architecture search (NAS) [@1; @2; @3; @6; @13; @15; @19; @34] has demonstrated success in designing neural architectures automatically. Many architectures produced by NAS methods have achieved higher accuracy than those manually designed in tasks such as image classification [@1], super resolution [@22], semantic segmentation [@4; @7] and object detection [@11]. NAS methods not only boost the model performance, but also liberate human experts from the tedious architecture tweaking work. ![Correlation between stand-alone model and learned architecture weights. We replace the selected operation in the first edge of the first cell for the final architecture with all the other candidate operations both in the DARTS [@15] and GDAS [@26], and fully train them until converge.[]{data-label="fig:long"}](fig1){width="0.8\linewidth"} \[fig:onecol\] So far, there exist three basic frameworks that have gained a growing interest, i.e., evolutionary algorithm (EA)-based NAS [@14; @19; @24], reinforcement learning (RL)-based NAS [@1; @2; @18], and gradient-based NAS [@15; @25; @26]. In both EA-based and RL-based approaches, their searching procedures require the validation accuracy of numerous architecture candidates, which is computationally expensive. For example, the reinforcement learning method [@1; @2] trains and evaluates more than 20,000 neural networks across 500 GPUs over 4 days. These approaches use a large amount of computational resources, which is inefficient and unaffordable. ![image](fig2){width="1.0\linewidth"} \[fig:onecol\] To eliminate such deficiency, gradient-based NAS methods [@15; @25; @26; @33] such as DARTS [@15] and GDAS [@26] are recently presented. They construct a super network and relax the architecture representation by assigning continuous weights to the candidate operations. In DARTS, a computation cell is searched as the building block of the final architecture and each cell is represented as a directed acyclic graph (DAG) consisting of an ordered sequence of N* nodes. Then the concrete search space is relaxed into a continuous one, so that network and architecture parameters can be well-optimized by gradient descent. It achieved comparable performance to EA-based [@14] and RL-based [@1] methods while only requiring a search cost of a few GPU-days. In order to further accelerate the searching procedure, GDAS [@26] samples one sub-graph according to the architecture weights in a differentiable way at each training iteration. Existing methods select the candidate operations based on their architecture weights to derive the target architecture. Stand-alone models are constructed to generate weights for all possible architectures in the search space. However, architecture weights cannot accurately reflect the importance of each operation. To illustrate this issue, the obtained accuracy of stand-alone model is compared with the corresponding architecture weights. Their correlation is plotted in Figure 1. We can see that the operation with the highest architecture weight dose not achieves the best accuracy. Furthermore, the architecture weights of candidate operations are often close to each other; in this case, it is difficult to decide which candidate operation is the optimal one. Figure 2 illustrates the procedure of NAS. Given the limitation of architecture weights, it is natural to ask the question: will we be able to improve architecture search performance if we apply a more effective indicator to guide the model search? To this end, we propose a simple yet effective solution to neural architecture search, termed as exploiting operation importance for effective neural architecture search (EoiNAS). The main idea of our method has two parts: 1\) It is well-recognized that operation A is better than operation B if A has fewer training epochs and higher validation accuracy during the search process. According to this criterion, a new indicator is proposed to fully exploit the operation importance and guide the model search. Training iterations and validation accuracy for each operation can be recorded in the search space. 2\) Based on this new indicator, we propose a gradual operation pruning strategy to further improve the search efficiency and accuracy. We denote the training of every k epochs as a step. In each step, we prune the most inferior operation according to the new indicator. This process continues until only one operation remains; this operation can be regarded as the best operation to derive the final architecture. Owing to the gradual operation pruning strategy, our super network exhibits fast convergence. The effectiveness of EoiNAS is verified on the standard vision setting, i.e., searching on CIFAR-10, and evaluating on both CIFAR-10/100 and ImageNet datasets. We achieve state-of-the-art performance of 2.50% test error on CIFAR-10 using 3.4M parameters. When transferred to ImageNet, it achieves top-1/5 errors of 25.6%/8.3% respectively, comparable to the state-of-the-art performance under the mobile setting. The remainder of this paper is organized as follows: In Section 2, we review the related work of recent neural architecture search algorithms and describe our search method in Section 3. After experiments are shown in Section 4, we conclude this paper in Section 5. Related Work ============ With the rapid development of deep learning, significant gain in performance has been brought to a wide range of computer vision problems, most of which owed to manually designed network architectures [@8; @10; @12; @21; @23; @27]. Recently, a new research field named neural architecture search (NAS) [@1; @2; @3; @6; @13] has been attracting increasing attentions. The goal is to find automatic ways of designing neural architectures to replace conventional handcrafted ones. According to the heuristics to explore the large architecture space, existing NAS approaches can be roughly divided into three categories, namely, evolutionary algorithm-based approaches [@14; @19; @24], reinforcement learning-based approaches [@1; @2; @18] and gradient-based approaches [@15; @25; @26]. Reinforcement learning based NAS. A reinforcement learning based approach has been proposed by Zoph et al. [@1; @2] for neural architecture search. They use a recurrent network as a controller to generate the model description of a child neural network designated for a given task. The resulted architecture (NASNet) improved over the existing hand-crafted network models at its time. Evolutionary algorithm-based NAS. An alternative search technique has been proposed by Real et al. [@19] where an evolutionary (genetic) algorithm has been used to find a neural architecture tailored for a given task. The evolved neural network (AmoebaNet), further improved the performance over NASNet. Although these works achieved state-of-the-art results on various classification tasks, their main disadvantage is the large amount of computational resources they demand. Gradient-based NAS. Contrary to treating architecture search as a black-box optimization problem, gradient based neural architecture search methods [@15; @25; @26] utilized the gradient obtained in the training process to optimize neural architecture. DARTS [@15] relaxed the search space to be continuous, so that the architecture can be optimized with respect to its validation set performance by gradient descent. Therefore, gradient-based approaches successfully accelerate the architecture search procedure, only several GPU days are required. Because DARTS optimized the entire super network during the search process, it may suffer from discrepancy between the continuous architecture encoding and the derived discrete architecture. GDAS [@26] suggested an alternative method to alleviate this discrepancy. GDAS approaches the search problem as sampling from a distribution of architectures, where the distribution itself is learned in a continuous way. The distribution is expressed via slack softened one-hot variables that multiply the operations and make the sampling procedure differentiable. SNAS [@25] applied a similar technique to constrain the architecture parameters to be one-hot to tackle the inconsistency in optimizing objectives between search and evaluation scenarios. In order to bridge the depth gap between search and evaluation scenarios, PDARTS [@33] divide the search process into multiple stages and progressively increase the network depth at the end of each stage. In addition, MdeNAS [@32] propose a multinomial distribution learning method for extremely effective NAS, which considers the search space as a joint multinomial distribution and the distribution is optimized to have high expectation of the performance. ![ Search space. (a) A cell contains 7 nodes, two input nodes, four intermediate nodes that apply sampled operations on the input nodes and upper nodes, and an output node that concatenates the outputs of the four intermediate nodes. (b) The edge between two nodes denotes a possible operation sampled according to the discrete probability distribution $\Gamma$ in the search space.[]{data-label="fig:long"}](fig3){width="0.8\linewidth"} \[fig:onecol\] Methodology =========== We first describe our search space and continuous relaxation in general form in Section 3.1, where the computation procedure for an architecture is represented as a directed acyclic graph. We then propose a new indicator to fully exploit the importance of each operation in Section 3.2. Finally, we design an gradual operation pruning strategy to make the super network exhibit fast convergence and high training accuracy in Section 3.3. Search Space and Continuous Relaxation -------------------------------------- In this work, we leverage GDAS [@26] as our baseline framework. Our goal is to search a robust cell and apply it to a network of L cells. As shown in Figure 3, a cell is defined as a directed acyclic graph (DAG) of N nodes, $\{ x_0, x_2, \cdots, x_{N-1} \}$, where each node is a network layer, i.e., performing a specific mathematical function. We denote the operation space as $ O $ , in which each element represents a candidate operation $o(\cdot)$. An edge $f_{i,j}$ represents the information flow connecting node $x_{i}$ and $x_{j}$, which consists of a set of operations weighted by the architecture weights $\beta_{i,j}$, and is thus formulated as: $$\label{1} f_{i,j}(x_i)= \sum\limits_{o\in O} \beta_{i,j}^o o(x_i)$$ $$\label{2} \mathrm{s.t.}\hspace{2mm} \beta_{i,j}^o= \frac{exp(\alpha_{i,j}^o)}{\sum_{o^{'}\in O} exp(\alpha_{i,j}^{o^{'}})}$$ where $\alpha_{i,j}^o$ is the o-th element of an O-dimensional learnable vector $\alpha_{i,j}\in\mathbb{R}^{O}$, and $\beta_{i,j}$ encodes the sampling distribution of the function between node $x_{i}$ and $x_{j}$, as we will discuss below. Intuitively, a well learned $\beta=\{\beta_{i,j}^o\}$ could represent the relative importance of the operation o for transforming the feature map $x_{i}$. Similar to GDAS, between node $x_{i}$ and $x_{j}$, we sample one operation from $ O $ according to a discrete probability distribution $\Gamma_{i,j}$ which is characterized by Eq. (2). During the search, we calculate each node in a cell as: $$\label{3} x_{j}= \sum_{i\textless j} f_{i,j}(x_{i})$$ where $f_{i,j}$ is sampled from $\Gamma_{i,j}$. Since the operation $f_{i,j}$ is sampled from a discrete probability distribution, we cannot back-propagate gradients to optimize $\alpha_{i,j}$. To allow back-propagation, we use the Gumbel-Max trick [@27; @30] and softmax function [@28] to re-formulate Eq. (3) to Eq. (4), which provides an efficient way to draw samples from a discrete probability distribution in a differentiable way. $$\label{4} x_{j}= \sum_{i=1}^{j-1} \sum_{o=1}^O h_{i,j}^o f_{i,j}^o (x_{i};W_{i,j}^o)$$ $$\label{6} \mathrm{s.t.}\hspace{2mm} h_{i,j}^o = \frac{exp((\alpha_{i,j}^o+\tau_o)/T)}{\sum_{o^{'}=1}^O exp((\alpha_{i,j}^{o^{'}}+\tau_{o^{'}})/T)}$$ Here $\tau_o$ are i.i.d samples drawn from Gumbel (0,1); $f_{i,j}^o$ indicates the o-th function in O; $h_{i,j}^o$ is the o-th element of $h_{i,j}$; $W_{i,j}^o$ is the weight of $f_{i,j}^o$ for the transformation function between node $x_{i}$ and $x_{j}$; T is the temperature parameter [@27], which controlls the Gumbel-Softmax distribution. As the parameter T approaches zero, the Gumbel-Softmax distribution becomes equivalent to the discrete probability distribution. The temperature parameter is annealed from 5.0 to 0.0 during our search. Our candidate operation set $O$ contains the following 8 operations: (1) identity, (2) zero, (3) $3\times 3$ separable convolutions, (4) $3\times 3$ dilated separable convolutions, (5) $5\times 5$ separable convolutions, (6) $5\times 5$ dilated separable convolutions, (7) $3\times 3$ average pooling, (8) $3\times 3$ max pooling. We search for two kinds of cells, i.e., the normal cell and the reduction cell. When searching the normal cell, each operation in $O$ has the stride of 1. For the reduction cell, the stride of operations on 2 input nodes is 2. Once we discover the best normal cell and reduction cell, we stack copies of these best cells to make up a neural network. ![Ideal and reality cases of the weight deviation. The dark dashed curve indicates the ideal weight distribution, and the red solid curve denotes the weight distribution might occur in real cases. The deviation of architecture weights should be large enough, so that we can clearly judge which operation is more important.[]{data-label="fig:long"}](fig4){width="0.75\linewidth"} \[fig:onecol\] ![Distribution of architecture weights. The small circles, ’$\times$’ and solid curves denote each operation weight and Kernel Distribution Estimate (KDE) of the weight distribution respectively. Architecture weights are distributed too densely, which makes it difficult to distinguish the important operations from the others.[]{data-label="fig:long"}](fig5){width="0.72\linewidth"} \[fig:onecol\] Operation Importance Indicator ------------------------------ Architecture Weights Deviation. In previous algorithms, operation importance is ranked by the architecture weights $\beta$, which is supposed to represent the relative importance of a candidate operation verse the others. When the search process is over, they select the most important operation and prune other inferior operations according to the value of the architecture weights. However, architecture weights cannot accurately reflect the importance of each operation. As shown in Figure 4, the dark dashed curve and the red solid curve indicates the architecture distribution in ideal and real cases respectively. In ideal cases, the deviation of architecture weights is large enough, so that we can clearly judge which operation is more important. However, this requirement may not always hold and it might lead to unexpected results. In Figure 5, statistical information collected from the cell of DARTS and GDAS demonstrates this analysis. The small circles and ’$\times$’ show each observation in this weight distribution, and the solid curves denote the Kernel Distribution Estimate (KDE) [@38], which is a non-parametric way to estimate the probability density function of a random variable. As shown in Figure 5, there is a large quantity of operations whose architecture weights are distributed on a small interval, which makes it difficult to distinguish the important operations from the others. Figure 2 (c) also illustrates this issue: a change in ranking occurs between architecture weight ranking and the true one. The Proposed Indicator. It is well-recognized that operation A is better than operation B if A has fewer training epochs and higher validation accuracy during the search process. Therefore, For each operation, the ratio of training iterations and validation accuracy can be used to determine the operation importance. This ratio is represented by $$\label{8} C_{i,j}^o = \frac{C_{i,j}^{a^o}}{C_{i,j}^{e^o}}$$ where $C_{i,j}^{a^o}$ and $C_{i,j}^{e^o}$ is the validation accuracy and training iterations of each operation on each edge respectively. The value of accuracy parameters might also close to each other, which will affect importance judgement; in this case, we consider the operation with higher architecture weights will be more important. Therefore, we combine accuracy parameters with architecture weights to obtain an effective indicator I as Eq. (7), which can fully exploit the importance of each operation. $$\label{7} I_{i,j}^o = \beta_{i,j}^o + \lambda C_{i,j}^o$$ where $\beta_{i,j}^o$ is the architecture weights of the o-th operation between node $x_{i}$ and $x_{j}$, $\lambda$ is a parameter to control the balance between the two parts, which is set to 0.5 in this work. Compared to previous methods [@15; @26] that judge the operation importance directly by architecture weights, our proposed indicator can effectively reflect the operation importance, which can help to select the optimal operation, so as to achieve the highest accuracy. Apply this effective indicator can be able to improve architecture search performance significantly. Based on this new indicator I, gradual operation pruning strategy is proposed during the search process to further improve the search efficiency and accuracy, as we will discuss next. -- ------------------------------ ----- ----------- ------------ -------- --------- ---------------- Times Params (days) (million) C10(%) C100(%) ResNet + CutOut [@8] $-$ $-$ 1.7 4.61 22.10 manual DenseNet [@10] $-$ $-$ 25.6 3.46 17.18 manual SENet [@31] $-$ $-$ 11.2 4.05 $-$ manual MetaQNN [@36] 10 8-10 11.2 6.92 27.14 RL NAS [@1] 800 21-28 7.1 4.47 $-$ RL NASNet-A [@2] 450 3-4 3.3 3.41 $-$ RL NASNet-A + CutOut [@2] 450 3-4 3.3 2.65 $-$ RL ENAS [@18] 1 0.45 4.6 3.54 19.43 RL ENAS + CutOut [@18] 1 0.45 4.6 2.89 $-$ RL AmoebaNet-A + CutOut [@19] 450 7.0 3.2 3.34 18.93 evolution AmoebaNet-B + CutOut [@19] 450 7.0 2.8 2.55 $-$ evolution Hierarchical NAS [@14] 200 1.5 61.3 3.63 $-$ evolution Progressive NAS [@13] 100 1.5 3.2 3.63 19.53 SMBO DARTS (1st) + CutOut [@15] 1 0.38 3.3 3.00 17.76 gradient-based DARTS (2nd) + CutOut [@15] 1 1.0 3.4 2.82 17.54 gradient-based SNAS + CutOut [@25] 1 1.5 2.9 2.98 $-$ gradient-based GDAS [@26] 1 1.0 3.4 3.87 19.68 gradient-based GDAS + CutOut [@26] 1 1.0 3.4 2.93 18.38 gradient-based MdeNAS + CutOut [@32] 1 0.16 3.61 2.55 $-$ MDL Random Search + CutOut [@15] 1 4.0 3.2 3.29 $-$ random EoiNAS 1 0.6 3.4 3.42 18.4 gradient-based EoiNAS + CutOut 1 0.6 3.4 2.50 17.3 gradient-based -- ------------------------------ ----- ----------- ------------ -------- --------- ---------------- \[table:CIFAR\] Gradual Operation Pruning Strategy ---------------------------------- In existing methods, all candidate operations are always kept during the search process and unimportant operations are removed directly by the architecture weights until the search is over to derive the final architecture. However, for some unimportant operations, we do not need to waste time and computation resources to sample and train. Therefore, we propose a gradual operation pruning strategy to further improve the search efficiency and accuracy. We denote the training of every k epochs as a step. In a step, we prune the most inferior operation according to the new indicator. In the next step, we judge the most inferior operation in the remaining operations and prune it. This process continues until only one operation remains; this operation can be regarded as the best operation to derive the final architecture. Owing to the gradual operation pruning strategy, our super network exhibits fast convergence. Our definite searching algorithm is presented in Algorithm 1. At the initialization of the search process, we perform gradient-descent based optimization over the network parameters in the first 20 epochs. It helps obtaining balanced architecture weights between parameterized operations (e.g. convolution operation) and non-parameterized operations (e.g. skip-connect operation). Then, we perform a gradient-descent based optimization for the architecture parameters $\alpha$ and network parameters $w$ in an alternating manner. Specifically, we optimize the operation weights by descending $\bigtriangledown_wL_{train}(w,\alpha;T)$ on the training set, and optimize the architecture parameters by descending $\bigtriangledown_{\alpha}L_{val}(w,\alpha;T)$ on the validation set. An operation will be pruned after 20 epochs if its corresponding operation importance indicator $I_{i,j}^o$, which is updated along training iterations, is the lowest. When the search procedure is finished, we decode the discrete cell architecture by first retaining the two strongest predecessors for each node (with the strength from node $x_{i}$ and $x_{j}$, being $max_{o, O^o\neq zero}I_{i,j}^o$), and then choose the most likely operation by taking the argmax. Training set: $D_T$; Validation set: $D_V$;\ Operation set: $O$\ Network parameters: $w$; Architecture parameters: $\alpha$;\ Validation accuracy: $C^a$; Training iterations: $C^e$;\ Temperature parameter: $T$ Sample a sub-graph to train according to Eq. (5); Update $w$ by $\bigtriangledown_wL_{train}(w,\alpha;T)$ on $D_T$; Update $\alpha$ by $\bigtriangledown_{\alpha}L_{val}(w,\alpha;T)$ on $D_V$; Update $C^e$, $C^a$; Calculate operation importance $I$ by Eq. (7); $O=O\backslash\{O_{argmin_{o}I^o} \}$\ [end while]{} Derive the final architecture based on the indicator I; Optimize the architecture on the training set Experiments =========== Datasets -------- We conduct experiments on three popular image classification datasets, including CIFAR-10, CIFAR-100 [@29] and ImageNet [@20]. Architecture search is performed on CIFAR-10, and the discovered architectures are evaluated on all three datasets. Both CIFAR-10 and CIFAR-100 have 50K training and 10K testing RGB images with a fixed spatial resolution of $32\times 32$. These images are equally distributed over 10 classes and 100 classes in CIFAR-10 and CIFAR-100 respectively. In the architecture search scenario, the training set is equally split into two subsets, one for updating network parameters and the other for updating the architecture parameters. In the evaluation scenario, the standard training/testing split is used. We use ImageNet to test the transferability of the architectures discovered on CIFAR-10. Specificaly, we use a subset of ImageNet, namely ILSVRC2012, which contains 1,000 object categories and 1.28M training and 50K validation images. Following the conventions [@2; @15], we apply the mobile setting where the input image size is $224\times 224$. -- ----------------------- ----- ----------- ------------ ----------- ------- ------- ---------------- Times Params MAdds (days) (million) (million) Top-1 Top-5 Inception-v1 [@27] $-$ $-$ 6.6 1448 30.2 10.1 manual MobileNet-V2 [@16] $-$ $-$ 3.4 300 28.0 $-$ manual MobileNet-V3 [@17] $-$ $-$ 5.4 219 24.8 $-$ manual ShuffleNet [@9] $-$ $-$ 5.0 524 26.3 $-$ manual NASNet-A [@2] 450 3-4 5.3 564 26.0 8.4 RL NASNet-B [@2] 450 3-4 5.3 488 27.2 8.7 RL NASNet-C [@2] 450 3-4 4.9 558 27.5 9.0 RL AmoebaNet-A [@19] 450 7.0 5.1 555 25.5 8.0 evolution AmoebaNet-B [@19] 450 7.0 5.3 555 26.0 8.5 evolution AmoebaNet-C [@19] 450 7.0 6.4 570 24.3 7.6 evolution Progressive NAS [@13] 100 1.5 5.1 588 25.8 8.1 SMBO DARTS (2nd) [@15] 1 1.0 4.9 595 26.9 9.0 gradient-based SNAS [@25] 1 1.5 4.3 522 27.3 9.2 gradient-based GDAS [@26] 1 1.0 5.3 581 26.0 8.5 gradient-based MdeNAS [@32] 1 0.16 6.1 596 25.5 7.9 MDL EoiNAS 1 0.6 5.0 570 25.6 8.3 gradient-based -- ----------------------- ----- ----------- ------------ ----------- ------- ------- ---------------- \[table:ImageNet\] ![Detailed structure of the best cells discovered on CIFAR-10 by our EoiNAS. (a) Normal cell. (b) Reduction cell. The definition of the operations on the edges is in Section 3.1. In the normal cell, the stride of operations on 2 input nodes is 1 and the stride is 2 in the reduction cell.[]{data-label="fig:long"}](fig6){width="1.0\linewidth"} \[fig:onecol\] Implementation Details ---------------------- Following the pipeline in GDAS [@26], our experiments consist of three stages. First, EoiNAS is applied to search for the best normal/reduction cells on CIFAR-10. Then, a larger network is constructed by stacking the learned cells and retrained on both CIFAR-10 and CIFAR-100. The performance of EoiNAS is compared with other state-of-the-art NAS methods. Finally, we transfer the cells learned on CIFAR-10 to ImageNet to evaluate their performance on larger datasets. Network Configrations. The neural cells for CNN are searched on CIFAR-10 following [@15; @25; @26]. The candidate function set O has 8 different functions as introduced in Section 3.1. By default, we train a small network of 8 cells for 160 epochs in total and set the number of initial channels in the first convolution layer C as 16. Cells located at the 1/3 and 2/3 of the total depth of the network are reduction cells, in which all the operations adjacent to the input nodes are of stride two. Parameter Settings. For network parameters w, we use the SGD optimization. We start with a learning rate of 0.025 and anneal it down to 0.001 following a cosine schedule. We use the momentum of 0.9 and the weight decay of 0.0003. For architecture parameters $\alpha$ ,we use zero initialization which implies equal amount of attention over all possible operations. And we use the Adam optimization [@35] with the learning rate of 0.0003, momentum (0.5; 0.999) and the weight decay of 0.001. To control the temperature parameter T of the Gumbel Softmax in Eq. (5), we use an exponentially decaying schedule. The T is initialized as 5 and finally reduced to 0. Following [@26], we run EoiNAS 4 times with different random seeds and pick the best cell based on its validation performance. This procedure can reduce the high variance of the searched results. Our EoiNAS takes about 0.6 GPU days to finish the search procedure on a single NVIDIA 1080Ti GPU. The best cells searched by EoiNAS is shown in Figure 6. Results on CIFAR-10 and CIFAR-100 --------------------------------- For CIFAR, we built a network with 20 cells and 36 input channels, and trained it by 600 epochs with batch size 128. Cutout regularization [@5] of length 16, drop-path of probability 0.3 and auxiliary towers of weight 0.4 [@1] are applied. A standard SGD optimizer with a weight decay of 0.0003 and a momentum of 0.9 is used. The initial learning rate is 0.025, which is decayed to 0 following the cosine rule. Evaluation results and comparison with state-of-the-art approaches are summarized in Table 1. As demonstrated in Table 1, EoiNAS achieves test errors of 2.50% and 17.3% on CIFAR-10 and CIFAR-100, respectively, with a search cost of only 0.6 GPU-days. To obtain the same performance, AmoebaNet [@19] spent four orders of magnitude more computational resources (0.6 GPU-days vs 3150 GPU-days). Our EoiNAS also outperforms GDAS [@26] and SNAS [@25] by a large margin. Notably, architectures discovered by EoiNAS outperform MdeNAS [@32], the previously most efficient approach, while with fewer parameters. In addition, we compare our method to random search (RS) [@15], which is considered as a very strong baseline. Note that the accuracy of the model searched by EoiNAS is 0.7% higher than that of RS. Results on ImageNet ------------------- The ImageNet dataset is used to test the transferability of architectures discovered on CIFAR-10. We adopt the same network configurations as GDAS [@26], i.e., a network of 14 cells and 48 input channels. The network is trained by 250 epochs with batch size 128 on a single NVIDIA 1080Ti GPU, which takes 12 days with the PyTorch [@37] implementation. The network parameters are optimized using an SGD optimizer with an initial learning rate of 0.1 (decayed linearly after each epoch), a momentum of 0.9 and a weight decay of $3\times 10^{-5}$. Additional enhancements including label smoothing and auxiliary loss tower are applied during training. Evaluation results and comparison with state-of-the-art approaches are summarized in Table 2. Architecture discovered by EoiNAS outperforms that by GDAS by a large margin in terms of classification accuracy and model size. It demonstrates the transfer capability of the discovered architecture from small dataset to large dataset. Ablation Studies ---------------- In addition, we have conducted a series of ablation studies that validate the importance and effectiveness of the proposed operation importance indicator as well as gradual operation pruning strategy incorporated in the design of EoiNAS. In Table 3, we show ablation studies on CIFAR-10. GDAS [@26] is our baseline frame work. The GOP means the gradual operation pruning strategy, the OII means the proposed operation importance indicator. All architectures are trained by 600 epochs. As the results show, our super network exhibits fast convergence and high training accuracy owing to the gradual operation pruning strategy. The structure of the best cells discovered on CIFAR-10 is shown in Figure 7. Through prune inferior operations gradually during the search process, we achieve much improvement in performance while using less search times. Table 3 also demonstrated the effectiveness of the proposed operation importance indicator. The proposed indicator can better judge the importance of each operation and achieve higher accuracy. Such results reveal the necessity of the operation importance indicator. -------------- -------------- -------------- ----------- ------------ ----------- Times Params Error (days) (million) (%) Baseline $\times$ $\times$ 1.0 3.4 2.93 Baseline+GOP $\checkmark$ $\times$ 0.6 3.4 2.72 EoiNAS $\checkmark$ $\checkmark$ 0.6 3.4 2.50 -------------- -------------- -------------- ----------- ------------ ----------- : Ablation studies on CIFAR-10. The GDAS [@26] is our baseline. The GOP means the gradual operation pruning strategy, the OII means the proposed operation importance indicator. All architectures are trained by 600 epochs. \[table:WT2\] Searched Architecture Analysis ------------------------------ In differentiable NAS methods, architecture weights is not able to accurately reflect the importance of each operation as discussed in Section 1, because the accuracy of the fully trained stand-alone model and their corresponding architecture weights have low correlation. The proposed operation importance indicator can better decide which operation should be keep on each edge and which edges should be the input of each node, especially for the selection of skip-connect. The skip-connect operation plays an important role in cell structure. As well studied in [@39; @40], including a reasonable number and location of skip connections would make the gradient flows easier and optimization of deep neural network more stable. Compared the searched results in Figure 6 and Figure 7, architecture discovered by EoiNAS on CIFAR-10 tend to preserve the skip-connect operations in a hierarchical way, which can facilitate gradient back propagation and make the network have a better convergence ![Detailed structure of the best cells discovered on CIFAR-10 only by gradual operation pruning strategy. When pruning inferior operations and derive the final architecture during the search process, the operation importance are determined only by architecture weights.[]{data-label="fig:long"}](fig7){width="1.0\linewidth"} \[fig:onecol\] Besides, compared with Figure 6 and Figure 7, we can see that EoiNAS encourages connections in a cell to cascade more levels, in other words, there are more layers in the cell, making the evaluation network further deeper and achieving better classification performance. Finally, the combination of the operation importance indicator with the gradual operation pruning strategy can further enhance each other. The indicator is able to accurately represent the importance of operation and determine the remaining and pruning operations. Meanwhile, through gradually prune inferior operations, we can obtain more accurate indicator. Conclusion ========== In this paper, we presented EoiNAS, a simple yet efficient architecture search algorithm for convolutional networks, in which a new indicator was proposed to fully exploit the operation importance to guide the model search. A gradual operation pruning strategy was proposed during the search process to further improve the search efficiency. By gradually pruning the inferior operations based on the proposed operation importance indicator, EoiNAS drastically reduced the computation consumption while achieving excellent model accuracies on CIFAR-10/100 and ImageNet, which outperformed the human-designed networks and other state-of-the-art NAS methods.
--- abstract: 'We describe a method for generating entanglement between two spatially separated dipoles coupled to optical micro-cavities. The protocol works even when the dipoles have different resonant frequencies and radiative lifetimes. This method is particularly important for solid-state emitters, such as quantum dots, which suffer from large inhomogeneous broadening. We show that high fidelities can be obtained over a large dipole detuning range without significant loss of efficiency. We analyze the impact of higher order photon number states and cavity resonance mismatch on the performance of the protocol.' author: - Deepak Sridharan and Edo Waks title: Generating entanglement between quantum dots with different resonant frequencies based on Dipole Induced Transparency --- 1. Introduction =============== Generation of entanglement between qubits is an important operation for a large variety of applications in quantum information processing. Such states can be used to the realize schemes such as transmission of secret messages via quantum key distribution[@Ekert:1991; @Jennewein:2000] and teleportation of quantum information[@Bennett:1993; @Bouwmeester:1997; @Buttler:1998; @Mattle:1996]. The exchange of entanglement between two distant parties is also required for implementation of quantum repeaters [@Briegel:1998] which use a combination of entanglement swapping and entanglement purification[@Dur:1999] to achieve unconditional secure communication over arbitrarily long distances. To date, a variety of methods have been proposed for creating entanglement between spatially separated nodes. One of the most common methods is to transmit entangled photons generated by parametric down-conversion[@Kwiat:1995]. Entanglement protocols for atomic systems have also been proposed[@Jaksch:1999; @Duan:2001; @Childress:2006; @Barrett:2005; @Loock:2006]. Atom entanglement has the advantages that quantum information can be stored for long time periods, which is important for long distance quantum networking. Semiconductor based approaches to quantum information processing are currently an area of great interest because they offer the potential for a compact and scalable quantum information architecture. Furthermore, solid-state emitters such as semiconductor quantum dots (QDs), can be coupled to ultra-compact cavity waveguide systems to form highly integrated quantum systems [@Dirk:2007; @Dirk:2005]. A major challenge in using solid-state emitters is that they suffer from enormous inhomogeneous broadening, typically caused by emitter size variation and strain fields in the host material. The inhomogeneous broadening makes it difficult to find two emitters with identical emission wavelengths. Protocols to date for generating atom entanglement require the dipoles to emit indistinguishable photons, and are thus difficult to implement in semiconductor systems. In order to implement quantum networking in semiconductors, we need a protocol that works even when the dipoles emit photons that are distinguishable. In this paper, we describe a protocol for creating entanglement between two dipoles with different radiative properties, such as different emission wavelengths or radiative lifetimes. The proposed protocol uses Dipole Induced Transparency (DIT) to achieve the desired entanglement which occurs when a dipole is coupled to an optical cavity[@Waks:2006]. When the coupling is sufficiently strong, the dipole can switch a cavity from being highly transmitting to highly reflecting. The switching contrast is determined by the atomic cooperativity, which is the ratio of the lifetime of the uncoupled emitter to the modified lifetime of the cavity-coupled emitter. Enhancement of spontaneous emission has been observed in semiconductor emitters coupled to a variety of different micro-cavity architectures[@Dirk:2005; @Gerard:1998; @Moreau:2001; @Yoshie:2004; @Badolato:2005; @Reithmaier:2004]. Modification of cavity reflectivity by coupling a quantum dot to a photonic crystal nanocavity has been recently observed[@Englund:2007; @Kartik:2007]. In section 2, we describe the basic protocol under idealized assumptions that all fields can be expanded to first order in photon number and that the two cavities have the same resonant frequencies. Section 3 then considers the effect of higher order photon number states on the efficiency and fidelity of entanglement. The impact of non-linear behavior away from the weak excitation limit is also discussed. In section 4 we investigate the effect of cavity frequency mismatch on the entanglement. Finally, in section 5 we perform a precise numerical simulation of the entanglement generation protocol for one specific implementation using the exciton bi-exciton cascade of a single Indium Arsenide(InAs) quantum dot. Numerical results from recent experimental work is used to show that this may be a promising method for achieving entanglement between two QDs for the first time. 2. Protocol for entanglement generation ======================================= The schematic for generating entanglement between two spatially separated dipoles that emit distinguishable photons is shown in Fig 1. Each qubit consists of a dipole coupled to a double sided cavity. Each dipole is assumed to have three states: a ground state, a long lived metastable state and an excited state, which we refer to as ${\|g\rangle}$, ${\|m\rangle}$ and ${\|e\rangle}$ respectively. The states ${\|g\rangle}$ and ${\|m\rangle}$ represent the two qubit states of the dipole. The transition from the ground state to the excited state for dipole 1, may be detuned by $\delta_1$ from the resonant frequency $\omega_1$ of cavity 1. Similarly, the transition from the ground state to the excited state for dipole 2 may be detuned by $\delta_2$ from the resonant frequency $\omega_2$ of cavity 2. We assume that when the dipole is in state ${\|g\rangle}$ it couples to the cavity mode via an optical transition to state ${\|e\rangle}$, while when it is in state ${\|m\rangle}$ it does not optically couple to the cavity mode. State ${\|m\rangle}$ may be decoupled from the cavity due to either spectral detuning or selection rules. Although state ${\|m\rangle}$ is illustrated in the diagram as having an energy level that is in between states ${\|g\rangle}$ and ${\|e\rangle}$, this is not required. The only requirement is that when the dipole is in state ${\|m\rangle}$ it is decoupled from the cavity. This point will be analyzed in more detail when we consider using the exciton-biexciton transitions of an InAs QD to achieve entanglement. The desired level structure described above can be realized in a variety of solid-state material systems. In semiconductor quantum dots one can use the exciton and biexciton transitions[@Santori:2001], as well as the spin-based bright and dark exciton states[@Stevenson:2006]. In addition, three level structures can also be achieved using quantum-dot molecules[@Stinaff:2006], charged quantum dots[@Gammon:2007] and impurity bound excitons[@Fu:2005]. Similar qubits states could also be realized in other materials such as diamond using neutral and negatively charged nitrogen vacancy defects[@Santori:2006; @Pieter:2006]. The decay rates of the two dipoles is given by $\gamma_1$ and $\gamma_2$ respectively. To characterize the interaction between the dipoles and the cavity modes, we define the operators $\hat{\mathbf{\sigma_{1-}}}$ and $\hat{\mathbf{\sigma_{2-}}}$. They represent the dipole lowering operators for the dipoles in cavities 1 and 2 respectively. It should be noted that $\hat{\mathbf{\sigma_{1-}}}$ and $\hat{\mathbf{\sigma_{2-}}}$ represent the dipole lowering operators for the g-e transition. Although state ${\|m\rangle}$ is decoupled from the cavity, we still define the dipole operators $\hat{\mathbf{\sigma_{m1-}}}$ and $\hat{\mathbf{\sigma_{m2-}}}$ for the dipole in state ${\|m\rangle}$. These dipole are detuned by $\delta_{m1}$ and $\delta_{m2}$ from their respective cavities. We define $\hat{\textbf{a}}_{in}$ and $\hat{\textbf{c}}_{in}$ as the two input modes, $\hat{\textbf{a}}_{out}$ and $\hat{\textbf{c}}_{out}$ as the reflected modes, and $\hat{\textbf{b}}_{out}$ and $\hat{\textbf{d}}_{out}$ as the transmitted modes to the two cavities, as illustrated in Fig. 1. The energy decay rate of cavity 1 into the reflected and transmitted modes is given by $\kappa_{r1}$ and $\kappa_{t1}$ respectively. Similarly, the energy decay rates of cavity 2 into the reflected and transmitted modes is given by $\kappa_{r2}$ and $\kappa_{t2}$ respectively. There is also the decay rate $\kappa_{l1}$ and $\kappa_{l2}$ into the parasitic leaky modes that is due to losses such as material absorption and out of plane scattering. The field inside the cavities are represented by the cavity field operators $\hat{\textbf{f}}_1$ and $\hat{\textbf{f}}_2$. The protocol works as follows. Both the dipoles are initialized to be in an equal superposition of qubit states ${\|g\rangle}$ and ${\|m\rangle}$. This can be achieved by first driving the dipoles into the lowest energy state by either waiting several radiative lifetimes or optical pumping. The qubit state can then be rotated by either a direct $\pi/2$ transition, or a raman transition[@Stievater:2001]. The choice of method depends on the specifics of the dipole and material system. Once the initialization step is complete, the initialized state of the two dipole system is given by $1/2({\|gg\rangle}+{\|mm\rangle}+{\|gm\rangle}+{\|mg\rangle})$. After the initialization of the dipoles, a weak coherent field ${\|\alpha\rangle}$ with frequency $\omega$ is inserted at input $\hat{\textbf{a}}_{in}$. Simultaneously, another weak coherent field ${\|\beta\rangle}$ that is phase coherent with ${\|\alpha\rangle}$ (i.e. originates from a common laser source) is injected at $\hat{\textbf{c}}_{in}$. These input fields interact with the cavity-dipole system. The interaction between the input field $\hat{\textbf{a}}_{in}$ and cavity-dipole system 1 can be characterized by the Heisenbergs equations of motion for the cavity field operator $\hat{\textbf{f}}_1$ and the dipole lowering operator $\hat{\mathbf{\sigma_{1-}}}$ $$\begin{split} &\frac{d \hat{\textbf{f}_1}}{dt} = -(i\omega_0 + (\kappa_{r1}+\kappa_{t1}+\kappa_{l1})/2)\hat{\mathbf{f}_1}-\sqrt{\kappa_{r1}}\hat{\mathbf{a_{in}}}-ig\mathbf{\sigma_{1-}}-ig\mathbf{\sigma_{m1-}} \\ &\frac{d\hat{\mathbf{\sigma_{1-}}}}{dt}=(-i(\omega_0+\delta_1)+\gamma_1)\mathbf{\sigma_{1-}}+ig\sigma_{z1}\hat{\textbf{f}_1}\\ &\frac{d\hat{\mathbf{\sigma_{m1-}}}}{dt}=(-i(\omega_0+\delta_m1)+\gamma_1)\mathbf{\sigma_{m1-}}+ig\sigma_{zm1}\hat{\textbf{f}_1}\\ \end{split}$$ Similar equations can also be written for the interaction of the input field $\hat{\textbf{c}}_{in}$ with cavity-dipole system 2. The interaction between the input fields and the cavity-dipole systems results in part of the field being transmitted into the modes $\hat{\textbf{b}}^{\dagger}_{out}$ and $\hat{\textbf{d}}^{\dagger}_{out}$, while the remainder is reflected into the modes $\hat{\textbf{a}}^{\dagger}_{out}$ and $\hat{\textbf{c}}^{\dagger}_{out}$, or absorbed by the QD. The amount of light reflected and transmitted is given by the cavity reflection and transmission coefficients. Our analysis works in the weak excitation limit, where predominantly the quantum dots are populated in the ground state. In this limit, $\langle\sigma_{z1}(t)\rangle \approx -1$. This also implies that the population inversion for the dipole in state ${\|m\rangle}$ is close to $0$ i.e. $\langle\sigma_{zm1}(t)\rangle \approx 0$ Using this limit in Eq. 1, we can derive the reflection and transmission coefficients to be[@Walls:1994] $$\begin{split} &r_{1}(\omega) = \frac{(-i\Delta\omega_{1} + \frac{g^2}{-i(\Delta\omega_{1} - \delta_{1}) + \gamma_{1}})+ (\kappa_{r1}-\kappa_{t1}-\kappa_{l1})/2}{(-i\Delta\omega_{1} + {(\kappa_{r1}+\kappa_{t1}+\kappa_{l1})/2}+ \frac{g^2}{-i(\Delta\omega_{1} - \delta_{1,2}) + \gamma_{1}})}\\ &t_{1}(\omega) = \frac{\sqrt{\kappa_{r1}\kappa_{t1}}}{(-i\Delta\omega_{1} + (\kappa_{r1}+\kappa_{t1}+\kappa_{l1})/2 + \frac{g^2}{-i(\Delta\omega_{1} - \delta_{1}) + \gamma_{1}})} \end{split}$$ where $\Delta\omega_1 = \omega - \omega_1$. These equations are obtained for cavity-dipole system 1. The reflection and transmission coefficients for dipole 2 are identical to dipole 1 in form, and are obtained by substituting the dipole 2 parameters into Eq. 2. To get a better feel for Eq. 2, it is helpful to first consider the simplified case where $\Delta\omega_{1} = 0$. Assume first that the dipole is in state ${\|g\rangle}$ and that the g-e transition is resonant with the cavity such that $\delta_{1} = 0$. We see that maximum reflection and minimum transmission occurs for the case when $\kappa_{r1} = \kappa_{t1} + \kappa_{l1}$, called the critical coupling condition. This condition ensures that no light is reflected from the cavity when the incident field is directly on cavity resonance. We represent the decay rate $\kappa_{r1}$ at critical coupling as $\kappa_1$. Hence, the transmission and reflection coefficients simplify to t = 1/(1+C) and r = C/(1+C), where C=$g^2/\gamma_1\kappa_1$ is called the atomic cooperativity. If C$>>1$, which is the desired operation regime, then r=1 and all of the light is reflected. Now suppose the dipole is instead in state ${\|m\rangle}$ which is detuned from the cavity by $\delta_m$. We then have t = 1/(1+CL) and r = CL/(1+CL), where L=$\gamma_1/(\gamma_1+i\delta_m)$ is a Lorentzian function. If we assume that either state ${\|m\rangle}$ is highly detuned from the resonance of the g-e transition ($\delta_m>>g^2/\kappa_1$) or the transition is very weak due to selection rules (C$\approx 0$), then t=1 and now the light is completely transmitted. Thus, by changing the state of the dipole from ${\|g\rangle}$ to ${\|m\rangle}$ we can completely change the reflectivity of the cavity. In a realistic system we cannot assume that the two dipoles are resonant with their respective cavities, since in general they will have different resonant frequencies. Nor can we assume the reflection and transmission coefficients will reach their ideal limits because $\delta_m$ is not infinitely large and we usually don’t have perfect selection rules to cancel out the m-e transition. In this case we define for dipole 1, t$_{1}^g$ and r$_{1}^g$ as the transmission and reflection coefficients when the dipole is in state ${\|g\rangle}$, and t$_{1}^m$ and r$_{1}^m$ when the dipole is in state ${\|m\rangle}$. We define r$_2^g$,t$_2^g$,r$_2^m$, and t$_2^m$ analogously for dipole 2. These coefficients can be calculated by plugging in the appropriate values corresponding to the different transitions of the dipoles. It is important to emphasize that we do not assume that the $\delta$, g, $\gamma$, and $\kappa$ are the same for both dipoles. The protocol we describe works even if all of these parameters are different, which is why it is so useful in semiconductors. Before continuing, it is worth noting that in much of the literature the atomic cooperativity C is often interchanged with the Purcell factor, defined as the ratio of the lifetime of the dipole inside the cavity to that of a dipole in bulk or free space, which we denote $\gamma_{bulk}$. Although the atomic cooperativity is the correct parameter to use in the strictest sense, it is a very difficult parameter to measure. The Purcell factor in contrast is easier to measure and almost always gives a lower bound on the parameter C in any realistic system. The reason for this is that $\gamma_{bulk}$ is due both to radiative and non-radiative decay. In contrast the decay rate $\gamma$ is the decay rate into non-cavity modes and is mainly due to non-radiative processes, as radiation into modes other than the cavity mode is highly suppressed. Thus, outside of some atypical cases where the cavity has two modes resonant with the QD, we expect that $\gamma_{bulk}>\gamma$. Therefore, in virtually all cases C can be replaced by the Purcell factor to get a lower bound on the performance of the system. We first investigate the protocol under the assumptions that the resonant frequencies of both the cavities are the same($\omega_1=\omega_2$), and the input fields ${\|\alpha\rangle}$ and ${\|\beta\rangle}$ are sufficiently weak that we may expand them to first order in photon number. The initial state of the system(dipoles and fields) is given by ${\|\Psi_i\rangle}=1/2({\|gg\rangle}+{\|mm\rangle}+{\|gm\rangle}+{\|mg\rangle}) (\alpha\hat{\textbf{a}}^{\dagger}_{in}+\beta\hat{\textbf{c}}^{\dagger}_{in})$. The fields, after interacting with the cavities, are transformed according to cavity reflection and transmission coefficients. That is, if dipole 1 is in state ${\|g\rangle}$ then $\hat{\textbf{a}}_{in} \rightarrow r_1^g \hat{\textbf{a}}_{out} + t_1^g \hat{\textbf{b}}_{out}$, and if it is in state ${\|m\rangle}$ then $\hat{\textbf{a}}_{in} \rightarrow r_1^m \hat{\textbf{a}}_{out} + t_1^m \hat{\textbf{b}}_{out}$. The transformation for photon in $\hat{\textbf{c}}_{in}$ and dipole 2 is defined in a completely analogous way. The reflected fields from the two cavities are mixed on a 50/50 beamsplitter that applies the transformation: $\hat{\textbf{a}}^{\dagger}_{out}\rightarrow(\hat{\textbf{d}}_{1}+\hat{\textbf{d}}_{2})/\sqrt{2}, \hat{\textbf{c}}^{\dagger}_{out}\rightarrow(\hat{\textbf{d}}_{1}-\hat{\textbf{d}}_{2})/\sqrt{2}$. The final state of the QDs can be obtained by applying the cavity and beamsplitter transformations. If a detection event is observed in detector $\hat{\textbf{d$_2$}}$, then the state of the two QDs collapses to $$\begin{split} &{\|\Psi_f\rangle}= \frac{1}{N}[(\alpha r_1^g-\beta r_2^g){\|gg\rangle}+\alpha r_1^g{\|gm\rangle} -\beta r_2^g{\|mg\rangle}] \end{split}$$ where $N^2=\|\alpha r_1^g-\beta r_2^g\|^2+\|\alpha r_1^g\|^2+\|\beta r_2^g\|^2$. In general, r$_1^g \ne$ r$_2^g$ because the dipoles have different resonant frequencies. However, we can correct for this mismatch by properly adjusting the amplitudes of the fields. If the amplitudes are selected such that $$\alpha r_1^g=\beta r_2^g \label{match}$$ the state of the qubits is projected onto ${\|\Psi_-\rangle}=({\|gm\rangle}-{\|mg\rangle})/\sqrt{2}$ which is an ideal entangled state. Thus, by properly choosing the amplitude and phase of the input coherent fields ${\|\alpha\rangle}$ and ${\|\beta\rangle}$, we ensure that a detection at $\hat{\textbf{d$_2$}}$ creates an entangled state of dipoles. Note that the entanglement generation is accomplished despite the fact that the two dipoles may have completely different resonant frequencies or decay rates. 3. Higher order photon numbers ============================== The matching condition $\alpha r_1^g=\beta r_2^g$ as given in Eq \[match\] gives a relationship between $\alpha$ and $\beta$, but does not tell us how large to make $\alpha$. In general, we want to make $\|\alpha\|^2$ as large as possible to improve the chances of a detection event at $\hat{\textbf{d$_2$}}$. The probability of detecting a photon at detector $\hat{\textbf{d$_2$}}$ is defined as the efficiency $\eta$ of the protocol. When the fields ${\|\alpha\rangle}$ and ${\|\beta\rangle}$ are weak, efficiency of the protocol is proportional to the intensity of the field at $\hat{\textbf{d$_2$}}$ and can be derived to be $\|\alpha r_1^g\|^2/4$. The factor $1/4$ appears because $50\%$ of the field is transmitted into the modes $\hat{\textbf{b}}^{\dagger}_{out}$ and $\hat{\textbf{d}}^{\dagger}_{out}$ and another $50\%$ is lost when the beamsplitter splits the photons equally between $\hat{\textbf{d$_1$}}$ and $\hat{\textbf{d$_2$}}$. We see that we can achieve higher efficiencies by increasing input photon flux rate $\|\alpha\|^2$. However, if we make $\alpha$ too large we can no longer expand the fields to first order in photon number and higher order photon number contributions will become important. Higher order photon number contributions are undesirable because they serve as a decoherence mechanism. In the ideal case where only one photon is injected into the system, a detection event at $\hat{\textbf{d$_2$}}$ ensures that there are no other photons in the system which may carry “which path” information about the state of the dipole. Now suppose we consider the second order process of simultaneously injecting two photons into the input ports ${\|\alpha\rangle}$ and ${\|\beta\rangle}$. In the ideal case (both dipoles are on resonance with the cavities), if the state of the two dipoles is ${\|gm\rangle}$, cavity 1 will reflect its incident photon while cavity 2 will transmit the second photon. The transmitted photon in cavity 2 will always keep track of the fact that dipole 2 was in state ${\|m\rangle}$, and this information cannot be erased by the beamsplitter. Thus, we expect the state to be completely decohered when this happens. We will now consider not only this specific case, but full expansion of the coherent fields $\alpha$ and $\beta$ to all photon numbers to see how the final state of the system is affected. The initial state of the system is given by ${\|\Psi_i\rangle}=1/2({\|gg\rangle}+{\|mm\rangle}+{\|gm\rangle}+{\|mg\rangle}{\|\alpha\rangle}{\|\beta\rangle}$. The coherent states ${\|\alpha\rangle}$ and ${\|\beta\rangle}$ can also be written as ${\|\alpha\rangle}= D_1(\alpha){\|0\rangle}$ and ${\|\beta\rangle}= D_2(\beta){\|0\rangle}$. $D_1$ and $D_2$ are the displacement operators and are given by $$\begin{split} &D_1(\alpha)=e^{\alpha \hat{\textbf{a}}^{\dagger}_{in} - \alpha^* \hat{\textbf{a}}_{in}}\\ &D_2(\beta)=e^{\beta \hat{\textbf{c}}^{\dagger}_{in} - \beta^* \hat{\textbf{c}}_{in}}\\ \end{split}$$ The displacement operator provides as convenient way of writing the coherent states and includes all the higher order photon numbers contributions. The final state of system ${\|\Psi_f\rangle}$ can be obtained by applying the cavity and beamsplitter transformations to the initial state ${\|\Psi_i\rangle}$. After applying the transformations, the final state of the QDs is obtained by tracing out over the photon fields conditioned on a detection event at detector $\hat{\textbf{d$_2$}}$. The state of the dipoles is therefore given by the reduced density matrix $$\rho_{dipoles}=\frac{tr_{\mathrm{(fields)}}\{{\langle M\|\Psi_f\rangle}{\langle \Psi_f\|M\rangle}\}}{tr_{\mathrm{(dipoles \& fields)}}\{{\langle M\|\Psi_f\rangle}{\langle \Psi_f\|M\rangle}\}}$$ The matrix $M = \sum_{n=1}^\infty{\|n\rangle}_{d_2}{\langle n\|}$ is a positive projector that projects the state of the system onto a subspace containing at least one photon in $\hat{\textbf{d}}_{2}$. This projection models the measurement performed by the photon counter, which registers a detection event as long as there is at least one photon in the detection mode. Since the final state of QDs is mixed, we need a figure of merit to measure how well the QDs are entangled. In this paper we use the fidelity, which is defined as the overlap integral between the desired final state and the actual final state of the system. In our protocol, the desired final state is the maximally entangled Bell state ${\|\Psi_-\rangle}$. Thus, the expression for fidelity is ${\langle \Psi_-\|}{\rho_{dipoles}}{\|\Psi_-\rangle}$. If the actual final state is same as the desired final state, we have a perfect entangled state and a fidelity of 1. A fidelity of 0.5 implies that the state of the QDs is a random mixture of ${\|gm\rangle}$ and ${\|mg\rangle}$ and completely decohered. An analytical expression for the fidelity can be calculated by evaluating $\rho_{dipoles}$ and averaging over the state ${\|\Psi_-\rangle}$. We have carried out this calculation, but the expression for the fidelity is messy and the math is involved. The procedure for calculating the fidelity along with the final analytical expression are given in Appendix A. The expression in the appendix is used for subsequent calculations of fidelity. We also define the efficiency $\eta$ as the probability of getting a detection at detector $\hat{\textbf{d}}_{2}$. Mathematically this is given by the expression, $\eta = {tr_{\mathrm{(dipoles \& fields)}}\{{\langle M\|\Psi_f\rangle}{\langle \Psi_f\|M\rangle}\}}$. Using the matching condition $\alpha r_1^g=\beta r_2^g$, the expression can be simplified to $\eta = 0.5(1-e^{\|\alpha r_1^g\|^2/2})$. For the calculations in this paper, we use parameters that are appropriate for InAs quantum dots coupled to photonic crystal defect cavities. We represent the total decay rate out of each cavity $\kappa_{r1}+\kappa_{t1}+\kappa_{l1}$ and $\kappa_{r2}+\kappa_{t2}+\kappa_{l2}$ as $\kappa$ and set it to be equal to $100$ GHz. This corresponds to a cavity Q of 3300. We set g = 20 GHz for both the quantum dots. We estimate dipole decay rate $\gamma$ within the cavity to be 0.125 GHz, using reported data that measured the lifetime of several quantum dots that were placed inside a photonic crystal cavity, but heavily detuned [@Englund:2005]. Using these values we calculate C to be 32 and the cavity-dipole systems to be $96.7\%$ reflective on resonance. For the chosen values of g and $\kappa$, the cavity-dipole systems are in the weak coupling regime(g$<\kappa$/4). However, the analysis in this paper is completely general and is equally valid also for the strong coupling regime. In Fig \[fig 3\], we plot both fidelity and efficiency as a function of $\|\alpha r_1^g\|^2$. Fidelity is plotted for four values of $\delta_1/\kappa$, ranging from 0 to 1, with $\delta_2$ fixed at 0. Note that the efficiency is only a function of $\|\alpha r_1^g\|$, so the plot of efficiency is the same for all values of $\delta_1$. From Fig \[fig 3\], we see that there is a tradeoff between fidelity and efficiency as we increase $\alpha$. When $\|\alpha r_1^g\|^2<<1$ the fidelity is close to 1, indicating an ideal entangled state, which is consistent with our predictions in the weak field limit. In the region 0.1$<\|\alpha r_1^g\|^2<$1, the fidelity quickly drops due to the presence of higher photon number contributions. In the limit $\|\alpha r_1^g\|^2>>1$, the fidelity asymptotically approaches 0.5, indicating the higher photon number contributions have completely decohered the state. When $\|\alpha r_1^g\|^2<<1$, the fidelity curves for different values of $\delta_1$ nearly overlap. Fidelity stays close to 1 in this region. However, in the region 0.1$<\|\alpha\|^2<$1, the fidelity curves for different values of $\delta_1$ separate out. There is a drop in fidelity with increase in dipole detuning from $\delta_1=0$ to $\delta_1= \kappa$. Also, efficiency is a function of $\|\alpha r_1^g\|$ alone and does not change with $\delta_1$. This implies that fidelity decreases with increase in dipole detuning for a constant efficiency. In Fig \[fig 3\], we also plot a line of constant fidelity of 0.85. Note that for every value of $\delta_1$ there is a unique point on the plot corresponding to a fidelity of 0.85. As $\delta_1$ increases, this point shifts to lower values of $\|\alpha r_1^g\|^2$. Since, efficiency is a function of $\|\alpha r_1^g\|$, this in turn implies a decrease in efficiency. Thus, it is important to consider how the efficiency of the protocol changes for a fixed value of fidelity. To investigate this, we plot efficiency as a function of $\delta_1/\kappa$ for several values of $\delta_2$ for a constant fidelity of 0.85 in Fig \[fig 4\]. We see that even though there is a loss of efficiency, the change is gradual and there is only a 50$\%$ reduction over a cavity linewidth. Also, we would expect that if we added another detuning $\delta_2$, efficiency would decrease. However, this does not happen. From Fig \[fig 4\] we see that the effect of $\delta_2$ is to shift the efficiency curves by the detuning $\delta_2$ without altering the shape. So, the protocol can be used to obtain high efficiencies over a wide range of dipole detunings. 4. Validity of Weak Excitation Limit ===================================== In the protocol we describe, dipole detuning is compensated by adjusting the amplitude and phase of the input coherent fields until the matching condition $\alpha r_1^g = \beta r_2^g$ is satisfied. The more detuning we have, the larger the amplitude required by the coherent field in order to achieve the desired efficiency. It is possible that at some point, the amplitudes required by the coherent fields will be so large that the g-e transition of the QDs will be saturated leading to an optical nonlinearity and linewidth broadening[@Garnier:2007]. Because of this, the cavity reflection and transmission equations will depend on the pump power and Eq 2 needs to be modified accordingly. However, our protocol is intended to work in the linear regime wherein the QDs are unsaturated. This is possible only if the amplitude of the input fields is within a certain limit called the weak excitation limit. The weak excitation limit is defined as $\langle\sigma_z(t)\rangle \approx -1$, which is equivalent to the statement $\langle\sigma_+\sigma_-\rangle<<1$. and is necessary for Eq. 2 to be valid. This condition puts a constraint on the operation of the protocol. In order to investigate the implication of the weak excitation constraint, we start with the Heisenbergs equation of motion for the cavity field operator $\hat{\textbf{f}_1}$ and the dipole lowering operator $\hat{\mathbf{\sigma_1-}}$ given in Eq 1. We will consider cavity-dipole system 1. Similar equations are also applicable for cavity-dipole system 2. Eliminating $\hat{\textbf{b}}$ from Eq 1, we have $$[\frac{\kappa}{2}(i\delta_1+\gamma)-g^2]\hat{\mathbf{\sigma_-}}=-ig\sqrt{\kappa}\hat{\textbf{a}}^{\dagger}_{in}$$ Using the fact the cooperativity index C is $g^2/\gamma \kappa> 1$, the equation can be further simplified and multiplied with its conjugate to obtain $$\langle\sigma_+\sigma_-\rangle=\frac{g^2\kappa}{(g^4+\delta_1^2 \kappa^2/4)} \langle{\hat{\mathbf{a}}^{\dagger}_{in}}\hat{\mathbf{a}}_{in}\rangle \label{return}$$ The parameter $\langle\sigma_+\sigma_-\rangle$ represents the probability of the QD being in the excited state. In the weak excitation limit, $\langle\sigma_+\sigma_-\rangle << 1$. We also identify $\langle{\hat{\mathbf{a}}^{\dagger}_{in}}\hat{\mathbf{a}}_{in}\rangle$ as the total flux of photons in the input field ${\|\alpha\rangle}$. Using this in Eq \[return\], the weak excitation constraint thus puts a limit on $\|\alpha\|^2$ given by $$\label{weaklimit} \frac{\|\alpha\|^2}{\tau_p} << \frac{g^2}{\kappa} + \frac{\kappa\delta_1^2}{g^2}$$ where $\tau_p$ is the pulse width of the laser. From Eq \[weaklimit\], we see that when there is no detuning $\delta_1$, the flux of photons in the input field ${\|\alpha\rangle}$ should be less than the modified lifetime of the QD within the cavity $\frac{g^2}{\kappa}$. This is understandable because, if the first photon excites the QD and the second photon comes in before the QD has decayed, we will no longer be in the weak excitation limit. However, if the QD is off resonant from the cavity with detuning $\delta_1$, not all the light that comes in couples to the QD. Therefore, we will be able to pump the QDs with much more power before we exceed the weak excitation limit. This is given by the detuning dependent term $\frac{\kappa\delta_1^2}{g^2}$ in Eq \[weaklimit\]. Eq \[weaklimit\] conveys more than the weak excitation limit of ${\|\alpha\rangle}$. If we apply the matching condition $\alpha r_1^g = \beta r_2^g$ in Eq \[weaklimit\], we obtain a limit on the flux of photons in the input field ${\|\beta\rangle}$ given by $$\label{weaklimitb} \frac{\|\beta\|^2}{\tau_p} << \frac{g^2}{\kappa} + \frac{\kappa\delta_2^2}{g^2}$$ We recognize this as the weak excitation limit equation for the field ${\|\beta\rangle}$ which we would have obtained had we used the Heisenbergs equations of motion for cavity-dipole system 2. This implies that if we pick ${\|\alpha\rangle}$ such that it satisfies the weak excitation limit of cavity-dipole system 1, the matching condition automatically ensures that the flux of photons in ${\|\beta\rangle}$ is within the weak excitation limit of cavity-dipole 2. Note that by making $\tau_p$ sufficiently long, we can always ensure that the system is in the weak excitation limit and that nonlinearities do no contribute. However, because we are using longer pulses the entanglement rate is reduced. The rate of entanglement generation is proportional to the rate at which the cavity reflects photons, given by $ R = \|\alpha r_1^g\|^2/\tau_p$. Using the upper limit on $\|\alpha\|^2/\tau_p$ from Eq \[weaklimit\] and cavity reflectivity $r_1^g$ from Eq 2, we get $$\label{weakreflected} R << \frac{g^2}{\kappa}$$ The above equation implies that the system will remain in the linear weak excitation limit provided that the rate of reflected photons is less the 1 photon per modified lifetime of the dipole. Note that this result is true regardless of the detunings, and is therefore valid in all cases. It is instructive to compare the limits on the entanglement rate imposed by nonlinearities to the limits imposed by which-path information given in Section 3. The analysis of higher order photon numbers in the previous section showed that reflected photons $\|\alpha r_1^g\|<<1$ to have a high fidelity entangled state between the QDs. In contrast, the analysis of weak excitation limit in this section puts an upper bound on the rate of the input photons in ${\|\alpha\rangle}$ and ${\|\beta\rangle}$ given by $\|\alpha r_1^g\|^2/\tau_p << g^2/\kappa$ . Thus, the two analyses are fundamentally different in that one limits the total number of input photons and the other limits the rate of incoming photons. Although one might expect the nonlinear limit analyzed in section 4 to be important, it turns out that the analysis of section 3 is more restrictive, and is therefore the important limit to consider. To understand why, we first note that nonlinearities can always be suppressed by increasing the pulse duration $\tau_p$. No mater how many photons we inject into the system, if we make the pulses sufficiently long we will always be in the weak excitation limit. In contrast, which-path information does not depend on pulse duration, and therefore cannot be suppressed. Furthermore, in order to stay in the monochromatic limit (i.e. to use the single frequency approximation) it has been shown in previous work that the pulse duration must be longer than the modified spontaneous emission lifetime of the dipole[@Waks:2006]. If we combine this with the results of section 3, which state that the number of reflected photons $\|\alpha r_1^g\|^2<<1$, these two conditions already constrain us to work in the regime where the $\|\alpha r_1^g\|^2/\tau_p << g^2/\kappa$. Thus, we expect the entanglement to decohere due to which-path information before the nonlinear behavior in section 4 is observed. For this reason deviation from weak excitation does not pose any additional restrictions to the protocol that were not already present in the linear scattering regime. 5. Effects of Cavity detuning ============================= In previous sections, we considered the idealized case where both the cavities had identical resonant frequencies. However in realistic systems, this will not be the case. Fabrication imperfections may lead to slightly different resonances for the two cavities. Clearly, if even a small amount of mismatch between the cavities were to result in no entanglement, the usefulness of our protocol would be questionable. Thus, it is important to consider how sensitive the protocol is to cavity resonance mismatch. Now let’s consider the case where the two cavities do not have the same resonant frequency. The analysis of the protocol in the presence of cavity detuning becomes involved for two reasons. First, it is no longer clear which frequency we should use for the coherent fields ${\|\alpha\rangle}$ and ${\|\beta\rangle}$. We do not know whether to place it on resonance with one of the cavities or somewhere in between. This can depend on both the cavity separation $\Delta\omega_s$ and dipole detunings $\delta_1$ and $\delta_2$. Second, the matching condition used in the previous section $\alpha r_1^g=\beta r_2^g$, is not guaranteed to be optimal. If a detection event is observed in detector $\hat{\textbf{d$_2$}}$, then the state of the two QDs is $$\begin{split} &{\|\Psi_f\rangle}_{dipoles}= \frac{1}{N}[(\alpha r_1^g-\beta r_2^g){\|gg\rangle}+(\alpha r_1^m-\beta r_2^m){\|mm\rangle}\\ &+\alpha r_1^m{\|mg\rangle} -\beta r_2^m{\|gm\rangle}]\\ \end{split}$$where $N^2=\|\alpha r_1^m-\beta r_2^m\|^2+\|\alpha r_1^m\|^2+\|\beta r_2^m\|^2$. The matching condition $\alpha r_1^g=\beta r_2^g$ ensures that we do not have any detection at $\hat{\textbf{d$_2$}}$ if both the dipoles are in the state ${\|g\rangle}$. However, the field amplitude at $\hat{\textbf{d$_2$}}$ if both the dipoles are in the state ${\|m\rangle}$ i.e. ($\alpha r_1^m-\beta r_2^m)$ is not compensated. This results in imperfect destructive interference at detector $\hat{\textbf{d$_2$}}$. Thus, there is a small probability of detection at $\hat{\textbf{d$_2$}}$ when both the dipoles are in state ${\|m\rangle}$. This causes a loss of fidelity. In order to obtain the state that comes closest to the desired final entangled state, we must optimize the fidelity with respect to $\omega$, $\alpha$ and $\beta$. For calculating the effects of cavity detuning, we choose the frequency midway between the two cavity frequencies as the reference frequency $\Delta_{ref}$. Based on this reference frequency, $\omega_1 = -\Delta\omega_s/2$ and $\omega_2 = \Delta\omega_s/2$. Also, it will be easier if we define the dipole detunings in terms of the reference frequency rather than the cavity frequencies. We define $\Delta_1 = \delta_1 + \omega_1$ and $\Delta_2 = \delta_2 + \omega_2$, which are the dipole detunings of dipoles 1 and 2 with respect to the reference frequency located midway between the two cavities. These definitions ensure that when increasing the cavity separation $\Delta\omega_s$ we do not affect the QDs. This is important because we can obtain information about the effects of cavity detuning alone by making these definitions. Figure 4 plots the dependence of fidelity on the laser frequency for several different values of $\Delta_1$. The cavity separation $\Delta\omega_s = \omega_2-\omega_1$ is set to 50 GHz, and $\Delta_2=0.25\kappa$. The figure is optimized over the real and imaginary parts of $\frac{\alpha}{\beta}$. The value of the maximum fidelity for the three curves occurs at three different frequencies. The frequency at which we get maximum fidelity is the optimal frequency $\omega$. The fidelity at that frequency is the maximum fidelity that can be obtained for that particular configuration of $\Delta\omega_s$, $\Delta_1$ and $\Delta_2$. In Fig. 5, we plot optimized fidelity as a function of cavity detuning $\Delta\omega_s$ for different values of $\Delta_1$ with $\Delta_2=0$. When $\Delta\omega_s=0$, which represents the case when there is no cavity detuning, fidelity is 1. As the two cavities move apart, the spectra of the two cavities no longer overlap. Thus, there is a small probability of photon detection at $\hat{\textbf{d}}_{2}$ when the dipoles are in the state ${\|mm\rangle}$. This results in a loss of fidelity. Surprisingly, however, the fidelity does not continue to decrease, but instead increases back to 1 at some value of $\Delta \omega_s$. As we keep increasing $\Delta\omega_s$ further, for a certain value of the laser frequency $\omega$, both r$_1^g$ and r$_2^g$ are 0. If a detection event is observed in detector $\hat{\textbf{d$_2$}}$, then the state of the two QDs collapses to $$\begin{split} &{\|\Psi_f\rangle}_{dipoles}= \frac{1}{N}[(\alpha r_1^m-\beta r_2^m){\|mm\rangle}+\alpha r_1^m{\|mg\rangle} -\beta r_2^m{\|gm\rangle}] \end{split}$$where $N^2=\|\alpha r_1^m-\beta r_2^m\|^2+\|\alpha r_1^m\|^2+\|\beta r_2^m\|^2$. In this special case there is a second matching condition, given by $\alpha r_1^m = \beta r_2^m$, that again projects the two dipoles onto ${\|\Psi_-\rangle}=({\|gm\rangle}-{\|mg\rangle})/\sqrt{2}$. It is this second matching condition that results in the fidelity of 1 at the second peak. Our optimization algorithm naturally detects these two optimal regions, and gives us the best performance in the intermediate regime. Thus, given any set of operating conditions we have the ability to determine the best set of amplitudes and input frequencies. We note that in many cases fidelities exceeding 0.95 can be achieved even with an 60 GHz detuning, which is more than half a cavity linewidth. The fabrication of cavities with resonance frequencies that are repeatable within a linewidth is well within current technological capabilities. We can also consider what happens when we have both cavity detuning and dipole detuning. In Fig. 6, we plot optimized fidelity as a function of cavity detuning $\Delta\omega_s$, and dipole detuning $\Delta_1$. A maximum fidelity of 1 is obtained when $\Delta\omega_s = 0$, which represents the case when the two cavities have the same resonant frequency. For $\Delta\omega_s<g$ (where g=20GHz in the plot) the fidelity is largely independent of the detuning $\Delta_1$ and is only determined by cavity separation. When $\Delta\omega_s$ becomes larger, fidelity increases again due to the second matching condition and a dependence on the dipole detuning now becomes apparent. This dependence on detuning comes about from the fact that the second matching condition is a function of $\Delta_1$, as illustrated in Fig. 5. From Fig 6, fidelity is over 0.75 for a cavity linewidth separation(100 GHz) of the cavities even over a wide range of dipole detunings. Thus, we can use the protocol to obtain high fidelities even if the cavities and dipoles are detuned. 6. Exciton-Biexciton Implementation =================================== The protocol to generate entanglement between two QDs relies on considering the QD as a three level system. One of the ways to implement this three level system is by making use of the excitonic and biexcitonic transitions. A QD consists of three states: the ground state, an exciton state X consisting of a single electron-hole pair within the QD and a biexciton state XX which is formed when two electron-hole pairs are trapped inside the QD. The recombination of an electron-hole pair in the XX state generates the biexciton XX photon. Similarly, the recombination of an electron-hole pair in X state generates the X photon. The X and XX photons have different energies due to the coulomb and exchange interactions between the carriers. The typical energy separation between the two lines is 1mev[@Gonz:2007]. Thus, we can make use of this difference in energies to spectrally isolate the two lines. The schematic of the QD as a three level system is shown in Fig \[fig7\]. We identify the three states of the QD as the ground, X and XX states. We are free to assign these three states as ${\|g\rangle}$, ${\|m\rangle}$ and ${\|e\rangle}$ in a variety of different combinations. In fact, there are several ways to assign these levels, but probably the most convenient approach is given in the inset of Fig \[fig7\]. In the figure we have identified the ground state of the QD as state ${\|m\rangle}$, the single exciton state as state ${\|g\rangle}$, and the bi-exciton state as state ${\|e\rangle}$. This choice of the level configuration has a number of advantages. First, single qubit operations between ${\|g\rangle}$ and ${\|m\rangle}$ can be directly applied by pulses resonant with the single exciton transition. Second, by placing the bi-exciton transition on resonance with the cavity, we can enhance the exciton to bi-exciton transition to get DIT, while at the same time suppressing the single exciton lifetime in order to increase the coherence time of the qubit. This is illustrated in Fig \[fig7\]. We assume the biexciton transition to be on resonance with the cavity frequency. This is indicated in Fig \[fig7\] where the XX transition is in the middle of the cavity spectrum. The X transition line is detuned from the cavity by $\delta_X$. The vacuum Rabi frequencies of X and XX transitions are given by g$_X$ and g$_{XX}$ respectively. Similarly, the decay rates of the two transitions is given by $\gamma_X$ and $\gamma_{XX}$ respectively. In the exciton bi-exciton scheme the degree of cavity enhancement directly impacts our ability to create an entangled state. This is because both the exciton and bi-exciton are strongly radiative states, and the only way to enhance one while suppressing the other is to use cavity lifetime modification. In other qubit implementations, such as dark state excitons[@Stevenson:2006], this is not as much of a problem because selection rules make the dark exciton long lived regardless of cavity. To quantitatively address this issue, we first calculate the coherence time of the exciton state which is given by solving for the decay rate of $\sigma_-$ in Eq. 1. The coherence time of the qubit is given by $$\begin{split} &\Gamma_{X} = \frac{g_X^2\kappa}{\delta_X^2+\kappa^2}+\gamma_X + \frac{1}{T_2}\\ \end{split}$$ where we have added the dipole dephasing rate $1/T_2$ to the decay rate. From the above equation one can see that increasing $\delta_X$ decreases the decoherence rate until it finally saturates at a minimum value of $\gamma_X+1/T_2$. At this point, increasing the detuning of the exciton will not help as we are limited by non-radiative and dephasing processes. The coherence time of the dipole should be compared to an appropriate time scale in order to determine if entanglement can be generated. Although there are a number of different factors that should be considered in this comparison, the minimum requirement for generating entanglement is that the duration of the entangling pulse should be shorter than the coherence time of the qubit. If this is not the case, the qubit will begin to decohere before the entangling pulse has finished interacting with the cavity-dipole system, and there is no hope of generating high-fidelity entanglement. In previous work in Ref [@Waks:2006], it has been shown that when the pulse is resonant with the dipole, it must be much longer than the modified spontaneous emission lifetime of the dipole in order to be monochromatic. Thus, in the worst case when the dipole is resonant with the cavity we need $1/\tau_p <<g_{XX}^2/\kappa$. We thus argue that an important figure of merit is the ratio of the coherence time of the qubit to the entanglement pulse width, given by $$N_{ent} = \frac{g_{XX}^2}{\kappa\Gamma_X}$$ This ratio determines the maximum number of entanglements that can be performed before the system decoheres. If $N_{ent}>1$, there is enough time for the pulses to finish their interaction with the QDs before the system has decohered. Otherwise, the QDs will start to decohere before the pulses have finished their interaction and high fidelity entanglement will be impossible. For calculations, we choose experimental values taken from the paper of Hennessy et. al. [@Hennessy:2007] which investigates the coupling of an Indium Arsenide (InAs) quantum dot coupled to a photonic crystal cavity patterned in Gallium Arsenide (GaAs) by electron beam lithography. This experimental work reports g=20 GHz and cavity linewidth of 25 GHz which corresponds to a Q of 13300. However, the cavity linewidth is the bare cavity Q which corresponds to the decay into the leaky modes. In order to achieve critical coupling with the cavity, we need another in-plane mode with a decay rate equal to the bare cavity decay rate. This mode can be implemented in a photonic crystal as a waveguide coupled to the cavity. Thus, the total decay rate of the cavity is double that of the bare cavity decay rate. Hence, we use $\kappa=50$ GHz in our calculations. We use $g_X = g_{XX} = g$. For values of T$_2$ we use 2 ns, which are appropriate values for InAs QDs[@Langbein:2004]. For these values, $g_{XX}^2/\kappa$ is 8 GHz , $\Gamma_{x}$ is 0.93 GHz and $N_{ent}$ is 8.6. The fact that $N_{ent}>1$ ensures that we can complete an entanglement operation well before the QDs have decohered. For a cavity linewidth of $50$ GHz, the exciton line lies outside the cavity spectrum($\delta_{X} = 250$GHz). However, the exciton line still couples to the cavity and we cannot ignore the presence of the extra transition coupled to the cavity. So, we cannot substitute for g as 0 in Eq 2 in order to obtain the cavity reflection and transmission equations when the QD is in state ${\|m\rangle}$. We need to use the vacuum Rabi frequency as the value for g to obtain the values of r$^m_1$, t$^m_1$, r$^m_2$ and t$^m_2$. The changes in the transmission and reflection coefficients will modify the final state of the QDs and hence the fidelity of the system. In general we cannot assume that the XX transition is not detuned from the cavity spectrum. In order to see how robust the biexciton-exciton protocol is dipole detunings, we define the detunings of the XX transition lines from their cavities as $\delta_{XX1}$ and $\delta_{XX2}$. In Fig \[fig8\] we plot the dependence of fidelity on dipole detunings $\delta_{XX1}$ and $\delta_{XX2}$ for the above case. For both $\delta_{XX1}=0$ and $\delta_{XX2}=0$, fidelity is 1 as expected. When we increase $\delta_{XX1}$ and $\delta_{XX2}$, the transmission and reflection coefficients are modified due to the coupling of the X transition to the cavity. This lowers the fidelity of the output state. The drop is fidelity is gradual and for a cavity linewidth separation of the dipoles from the cavity resonance( 50 GHz), fidelity drops to only 0.96. As we further increase the detunings to 100 GHz, fidelity drops to 0.85. Thus, even for large detunings between the cavities and the dipoles, reasonable high fidelity(0.85) states of the QDs can be obtained. Thus, the exciton-biexciton scheme can be used to create entanglement between QDs even if the exciton line couples to the cavity. The performance of the protocol can be further improved by fabricating cavities with high quality factors. 7. Conclusions ============== In conclusion, we have shown that one can achieve high fidelity entangled states between two dipoles, even when their emission frequencies are different. The method is robust to dipole and cavity frequency mismatch. Efficiency loss for a cavity linewidth change in dipole detuning is about 50$\%$ for a constant fidelity. Therefore, relatively high fidelity can be obtained over a large range of dipole detunings without significant loss of efficiency. The development of protocols that are robust to these imperfections is extremely important for semiconductor based implementations of quantum networks. The authors acknowledge the support of Army Research Office under the Grant No. W911NF0710427. APPENDIX ======== The input coherent fields are ${\|\alpha\rangle}$ and ${\|\beta\rangle}$. Both the dipoles are initialized in a superposition of states ${\|g\rangle}$ and ${\|m\rangle}$. Thus, the initial state of the system is $${\|\Psi_i\rangle} = {\|\alpha\rangle}{\|\beta\rangle}({\|g\rangle} + {\|m\rangle})({\|g\rangle} + {\|m\rangle})/2$$ The coherent states can be replaced by their corresponding displacement operators to account for all order of photon numbers. Thus, $$\begin{split} {\|\Psi_i\rangle} = &\frac{1}{2}e^{(\alpha \hat{\textbf{a}}^{\dagger}_{in} - \alpha^{} \hat{\textbf{a}}_{in})}{\|0\rangle}_{a_{in}} e^{(\beta \hat{\textbf{c}}^{\dagger}_{in} - \beta^{} \hat{\textbf{c}}_{in})}{\|0\rangle}_{c_{in}}\\ &({\|gg\rangle} +{\|gm\rangle} + {\|mg\rangle} +{\|mm\rangle})\\ \end{split}$$ The input fields after interactions with the cavity-dipole systems are transformed according to Eq 2. Thus, when the dipoles are in state ${\|gg\rangle}$ $$\begin{split} &\hat{\textbf{a}}^{\dagger}_{in} \rightarrow r_1^g \hat{\textbf{a}}^{\dagger}_{out}+ t_1^g \hat{\textbf{b}}^{\dagger}_{out}\\ &\hat{\textbf{c}}^{\dagger}_{in} \rightarrow r_2^g \hat{\textbf{c}}^{\dagger}_{out}+ t_2^g \hat{\textbf{d}}^{\dagger}_{out}\\ \end{split}$$ Similar transformation equations apply when the dipoles are in the states ${\|gm\rangle}$, ${\|mg\rangle}$ and ${\|mm\rangle}$. The reflected field from the two cavities is mixed on a 50/50 beamsplitter that applies the transformation: $$\begin{split} &\hat{\textbf{a}}^{\dagger}_{out}\rightarrow(\hat{\textbf{d}}_{1}+\hat{\textbf{d}}_{2})/\sqrt{2}\\ &\hat{\textbf{c}}^{\dagger}_{out}\rightarrow(\hat{\textbf{d}}_{1}-\hat{\textbf{d}}_{2})/\sqrt{2}\\ \end{split}$$ Applying the cavity and beamsplitter transformations on the initial state ${\|\Psi_i\rangle}$ we get $$\begin{split} {\|\Psi_{{gg}}\rangle} = &D(\frac{\alpha r_1^g + \beta r_2^g}{\sqrt{2}})D(\frac{\alpha r_1^g - \beta r_2^g}{\sqrt{2}})D(\alpha t_1^g)D(\beta t_2^g)\\ &{\|0\rangle}_{d_1,d_2,b_{out},d_{out}}{\|gg\rangle}\\ \end{split}$$ ${\|\Psi_{{gg}}\rangle}$ is the state of the output modes for the dipoles in state ${\|gg\rangle}$. This state can be split up into the modes of detector $\hat{\textbf{d$_1$}}$, $\hat{\textbf{b}}_{out}$ and $\hat{\textbf{d}}_{out}$ and detector $\hat{\textbf{d$_2$}}$. Thus, $$\label{dipstate} \begin{split} &{\|\Psi_{{gg}}\rangle}={\|\psi_{{gg}}\rangle}{\|\mu_{gg}\rangle}{\|gg\rangle}\\ &{\|\psi_{{gg}}\rangle}= D(\frac{\alpha r_1^g + \beta r_2^g}{\sqrt{2}})D(\alpha t_1^g)D(\beta t_2^g){\|0\rangle}_{d_1,b_{out},d_{out}}\\ &{\|\mu_{gg}\rangle}=D(\frac{\alpha r_1^g - \beta r_2^g}{\sqrt{2}}){\|0\rangle}_{d_2}\\ \end{split}$$ ${\|\psi_{{gg}}\rangle}$ is the state of the output modes at detector $\hat{\textbf{d$_1$}}$ and the transmitted modes $\hat{\textbf{b}}_{out}$ and $\hat{\textbf{d}}_{out}$ when the dipoles are in state ${\|gg\rangle}$. ${\|\mu_{gg}\rangle}$ is the field amplitude at detector $\hat{\textbf{d$_2$}}$ when the dipoles are in state ${\|gg\rangle}$. Similarly, we can obtain the field amplitudes ${\|\Psi_{{gm}}\rangle}$, ${\|\Psi_{{mg}}\rangle}$ and ${\|\Psi_{{mm}}\rangle}$ when the dipoles are in states ${\|gm\rangle}$, ${\|mg\rangle}$ and ${\|mm\rangle}$. The final state of the system is given by $${\|\Psi_{{f}}\rangle}={\|\Psi_{{gg}}\rangle}+{\|\Psi_{{gm}}\rangle}+{\|\Psi_{{mg}}\rangle}+{\|\Psi_{{mm}}\rangle}$$ These states ${\|\Psi_{{gm}}\rangle}$, ${\|\Psi_{{mg}}\rangle}$ and ${\|\Psi_{{mm}}\rangle}$ can be further decomposed on similar lines to Eq \[dipstate\] to obtain the field amplitudes ${\|\psi_{{gm}}\rangle}$ and ${\|\mu_{gm}\rangle}$,${\|\psi_{{mg}}\rangle}$ and ${\|\mu_{mg}\rangle}$ and ${\|\psi_{{mm}}\rangle}$ and ${\|\mu_{mm}\rangle}$ respectively. We define the projection matrix M as $\sum_{n=1}^\infty{\|n\rangle}_{d_2}{\langle n\|}$. M can also be written as $$\begin{split} M &= \sum_{n=0}^\infty{\|n\rangle}_{d_2}{\langle n\|} - {\|0\rangle}_{d_2}{\langle 0\|}\\ &= I - {\|0\rangle}_{d_2}{\langle 0\|}\\ \end{split}$$ $$\rho_{dipoles}=\frac{tr_{\mathrm{(fields)}}\{{\langle M\|\Psi_f\rangle}{\langle \Psi_f\|M\rangle}\}}{tr_{\mathrm{(dipoles \& fields)}}\{{\langle M\|\Psi_f\rangle}{\langle \Psi_f\|M\rangle}\}}$$ $$\begin{split} F & = {\langle \Psi_-\|}{\rho_{dipoles}}{\|\Psi_-\rangle}\\ &= {\langle \Psi_-\|}{\frac{tr_{\mathrm{(fields)}}\{{\langle M\|\Psi_f\rangle}{\langle \Psi_f\|M\rangle}\}}{tr_{\mathrm{(dipoles \& fields)}}\{{\langle M\|\Psi_f\rangle}{\langle \Psi_f\|M\rangle}\}}}{\|\Psi_-\rangle}\\ \end{split}$$ The denominator is the probability of getting a detection at detector $\hat{\textbf{d$_2$}}$. We identify this as efficiency $\eta$. $$\begin{split} F & = {\langle \Psi_-\|}{\rho_{dipoles}}{\|\Psi_-\rangle}\\ &= {\langle \Psi_-\|}{\frac{tr_{\mathrm{(fields)}}\{{\|\Psi_f\rangle}{\langle \Psi_f\|}\}-tr_{\mathrm{(fields)}}\{{\langle 0\|\Psi_f\rangle}{\langle \Psi_f\|0\rangle}_{d_2}\}}{\eta}}{\|\Psi_-\rangle}\\ &=\frac{F_1-F_2}{\eta}\\ \end{split}$$ The individual terms can be evaluated to give $$\begin{split} &F_1 = \frac{1}{4}-\frac{{\langle \Psi_{{gm}}\|\Psi_{{mg}}\rangle}}{2} - \frac{{\langle \Psi_{{mg}}\|\Psi_{{gm}}\rangle}}{2}\\ F_2 =& \frac{1}{8}e^{-\|\mu_{gm}\|^2}+\frac{1}{8}e^{-\|\mu_{mg}\|^2}-\frac{1}{2}e^{-(\|\mu_{gm}\|^2+\|\mu_{mg}\|^2)/2}{\langle \psi_{mg}\|\psi_{gm}\rangle}\\ &-\frac{1}{2}e^{-(\|\mu_{gm}\|^2+\|\mu_{mg}\|^2)/2}{\langle \psi_{gm}\|\psi_{mg}\rangle}\\ &\eta = \frac{1}{4}[e^{-\mu_{gg}^2}+e^{-\mu_{gm}^2}+e^{-\mu_{mg}^2}+e^{-\mu_{mm}^2}]\\ \end{split}$$ Thus, the complete expression for fidelity and efficiency can be obtained. 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--- author: - François Boulogne - Sepideh Khodaparast - Christophe Poulard - 'Howard A. Stone' bibliography: - 'biblio.bib' title: Protocol to perform pressurized blister tests on thin elastic films --- Introduction {#sec:intro} ============ Characterization of adhesion energies between materials is of broad interest for scientific and engineering purposes. In practice, measuring the work of adhesion remains a difficult task both for the complex mechanical problems involved and for the technical barriers. Several measurement techniques have been developed in the past century. The most intuitive test is probably the peeling test [@Obreimoff1930; @Kendall1975] that aims to propagate an interfacial crack between two materials. However, certain materials are difficult to manipulate as required in the peeling test, especially when the film is particularly thin and brittle. In addition, the Johnson-Kendall-Roberts contact adhesion test relies on the deformation of a small sphere in contact with a surface of interest [@Johnson1971], which assumes thick materials to avoid finite-thickness effects [@Shull1997; @Shull2002; @Barthel2008]. Dannenberg developed the blister test in 1961 to measure the adhesion of paints on surfaces [@Dannenberg1961] which was further developed in particular by Jensen [@Jensen1991; @Jensen1998]. The so-called blister test consists of debonding a thin film by imposing a pressure via injecting a fluid between the film and the substrate to form a blister. Since then the blister test has been employed in different systems to quantify the adhesion energy between films of different properties and thicknesses and a large variety of substrates [@Hinkley:1983; @Briscoe1991; @Sizemore1995; @Dupeux:1998; @Cao:2014] including graphene membranes [@Koenig:2011; @Huang:2011; @Boddeti:2013; @Metten:2014]. Originally, the pressure and the volume of the injected liquid were measured to estimate the work required to detach the film [@Dannenberg1961]. However, recent studies have often employed more advanced optical techniques such as interferometry to quantify the shape of the blister [@Cao:2014; @Yahiaoui:2001]. Multiple studies have investigated the theoretical and empirical models to correlate the adhesion energy with the shape of the blister for the potential different deformation regimes in a blister test, including the bending plate, the stretching membrane and the transitional regime in between [@Guo:2005; @Komaragiri:2005; @Wang:2016]. Recently, Sofla et al. (2010) proposed a mechanical model, which was solved numerically, to relate the energy release rate to the morphology of the deformed film in a blister test for a wide range of physical parameters, which covers both the membrane and the plate regimes [@Sofla2010]. These authors quantified the adhesion energy of millimeter thickness polydimethylsiloxane PDMS films on glass. However, for thinner and more brittle materials, this measurement can be significantly more difficult to carry out. In particular, in the present study we show the preparation and the measurement protocols for polystyrene films with higher elastic modulus ($E = 3.4$ GPa) compared to less stiff materials such as PDMS ($E \approx 1.2$ MPa), which imposes additional challenges in the preparation and transfer of the thin film, and visualization of the deflection of the blister. Although preparation of the blister device and performing the tests are often described to be rather simple tasks, multiple practical challenges are faced during the process, which are rarely discussed in earlier works. In fact, most of the previous studies are dedicated more to the estimation of the work of adhesion using the blister test than to how to perform the test. Thus, in this paper, we aim to focus on how to prepare sample devices and perform the blister test itself, especially for thin elastic films. We illustrate our method on polystyrene films with thicknesses that range between $590$ to $1200$ nm. This protocol includes the preparation of a specific device to pressurize a film and the coating of this device with the film of interest. To validate our experimental approach and the numerical model used to derive the energy release rate, we compared our results to those obtained with a cleavage test on a polystyrene film of thickness about $100$ $\mu$m. Preparation of the blister test device {#sec:device} ====================================== In this section, we describe the preparation of a device to perform the blister test depicted in Fig. \[fig:blister\_test\], which aims to measure the energy release rate between a thin film and a substrate. The device must consist of a flat substrate made of the material of interest and pierced in its center for the inlet. The surface must be covered by the thin elastic film, which is then pressurized by injecting a liquid through the inlet to perform the blister test measurement. The shape of the blister formed by the thin film can be related to the energy release rate. To compensate for the high elastic modulus of materials such as polystyrene, adhesion energy measurements of films must be performed on small thicknesses to avoid crack propagation over large distances. Therefore, the inherent difficulty of the blister test is to make a thin film on a pierced surface, which prevents the use of spin-coating techniques. The principle of our protocol is to spin-coat a solution of polystyrene on a donor substrate, and the resulting thin film is then floated on a water bath. We prepare a receiver substrate of interest, pierced in its center, which is connected to an inlet for injecting the liquid. Then, the film is transferred to the latter surface with a method inspired by the Langmuir-Blodgett (LB) technique [@Blodgett1937]. A second difficulty is to withdraw the air trapped in the injection hole. Having a blister filled only with the liquid phase is of critical importance, especially if the shape of the blister is estimated according to the volume of the liquid [@Hohlfelder1996]. Here, we propose to add a microchannel to the back of the blister test device for this purpose, as shown in Fig. \[fig:blister\_test\]. Donor substrates {#subsec:donor} ---------------- Microscope glass slides (Dow Corning, 25 mm by 75 mm) are used as donor substrates in all experiments. To clean the surface of the glass slides, we plunge them in a bath of acetone for 30 minutes and then they are thoroughly rinsed with deionized (DI) water, and a solution of ethanol and acetone. Glass slides are dried with clear air and heated at $100^{\circ}$C for 30 minutes prior to the experiments. Preparation of PS films on donor substrates {#subsec:psfilm} ------------------------------------------- Polystyrene PS films are produced from a solution of PS (Sigma-Aldrich, $M_w \simeq 280$ kg/mol) in toluene. To achieve thin films of uniform thickness, the PS solution is spin-coated on solid glass substrates at 2000 rpm for 30 seconds. Solutions of different mass fractions of PS in toluene ($0.005 < \frac{m_{PS}}{m_t} < 0.14$) are used to achieve film thicknesses ranging from $t = 50$ nm to $t = 2$ $\mu$m. A frame is then cut out with a sharp blade around the spin-coated films before they were annealed at $130^{\circ}$C under vacuum for two hours to release any pre-stress in the films. After annealing, the thickness of the polystyrene films are measured by a Leica DCM 3D optical profilometer. To ensure the spatial uniformity of the film thickness for the polystyrene films, we measured the film thickness in four different locations of the film using a Woollam M2000 Spectroscopic Ellipsometer to confirm the values obtained with the optical profilometer. The average film thicknesses $t$ achieved at different polystyrene concentrations are presented in Fig. \[fig:PS-film\]. The resulting films have an elastic modulus $E = 3.4 $ GPa and Poisson’s ratio $\nu = 0.33$ [@Brandrup1989]. The measurement of the elastic modulus can be performed with different techniques such as indentation [@Oliver2004] or buckling instability [@Stafford2004]. Receiver substrates ------------------- Microscope glass slides (Dow Corning, 25 mm by 75 mm) are used as receiver substrates. A hole of approximately 1 mm diameter is drilled in the middle of the glass slide with a 1 mm diameter diamond drill bit mounted on a Dremel tool. The receiver glass slides are cleaned after the drilling process following the procedure described above for the donor substrates. A microchannel of length $L = 20$ mm and with a cross-section of $250 \times 250$ $\mu {\rm m}^{2}$ is fabricated in PDMS (Dow Corning, Sylgard 184 at a 1:10 wt ratio of crosslinking agent to prepolymer) using conventional soft lithography techniques [@Xia:1998]. The lithography mask used in our experiments is provided in Supplementary Materials. As shown in Fig. \[fig:blister\_test\]a, two vertical openings of 1 mm diameter are created in the PDMS slab at the inlet and the outlet of the channel with a biopsy punch (Miltex, 98PUN6-1). The PDMS microchannel and the receiver glass slide are then activated with a plasma gun (Electro-Technic Product, BD20-AC) [@Haubert2006]. Following this step, the inlet of the microchannel is aligned with the hole drilled in the receiver glass slide, and the two activated pieces are bonded together. In order to enhance the bonding, the assembly is heated at $90^{\circ}$C for 1 hour (Fig. \[fig:blister\_test\]b). Transferring the PS film ------------------------ The thin PS film is transferred from the donor substrate to the receiver substrate. The first step consists of detaching the PS film from the donor surface. To this end, the donor substrate is slowly dipped into a water bath until the entire PS film is floating on the free air-water interface. The water bath is then placed on a motorized translation stage (Thorlabs, NRT200) with the receiver glass slide dipped in it. The second step in the transfer process starts with a PS film floating at the water interface. The thin floating PS film is brought in contact with the glass side of the receiver substrate and the film slightly bends due to the curvature of the meniscus. Simultaneously, the water bath is displaced downward at a speed $U = 1$ $\mu$m/s (Fig. \[fig:transfer\]), to mimic the LB deposition technique [@Roberts:1981]. The air-water interface, which holds the thin PS film moves towards the hydrophilic glass slide and subsequently transfers the thin film onto the substrate [@Savelski:1995; @Cerro:2003]. The film transfer is easier for small water contact angles on the receiving glass slide. Optionally, a stripe of one centimeter at the top of the receiver substrate can be activated by a plasma treatment. During this treatment, the rest of the glass slide is covered with another glass slide to prevent any modification of the surface that would affect the adhesion energy. ![Schematic of the experimental setup for the transfer of the PS film (orange) onto the device.[]{data-label="fig:transfer"}](schematic2.pdf) Finally the prepared device is annealed at $130^{\circ}$C under vacuum overnight to release the eventual pre-stress in the films and remove water trapped underneath the film. The pre-stress could be caused by the mechanical vibrations of the translation stage occurring during the film transfer. At the end of this process, we obtain a device coated with a PS film ready for the blister test (Fig. \[fig:blister\_test\]b). Measurement of the adhesion energy in a blister test {#sec:blister_test} ==================================================== Devices are mounted with the PS film facing up on a 3-axis translational stage (Thorlabs, PT3) and a rotational stage (Thorlabs, PR01) equipped with micrometers to ensure the alignment of the device with a horizontal microscope. The inlet and outlet of the microchannel on the PDMS side of the device are connected to two syringes, which are mounted on two syringe pumps (Harvard Apparatus, PhD Ultra) functioning in infusion and withdrawal modes, respectively. Fig. \[fig:blister\_test\]c shows the schematic of the experimental setup used in the present blister test measurements. For visualization purposes, the inlet syringe is filled with a solution of a dye (Sigma-Aldrich, methylene blue) dissolved in DI water. The microchannel is slowly filled with water by pushing the liquid through the inlet while removing the trapped air in the channel through the outlet. At this point the blister test device is ready for measurements. A finite volume of liquid is pumped through the hole in the glass substrate until an interfacial crack is initiated between the two materials. The system is left to achieve equilibrium at this stage, especially by inspecting that the radius of the bulge does not further evolve. Once a blister is formed underneath the PS film, the radius and the height of the blister are measured by top-view and side-view visualization of the blister, respectively. Visualization of the blister shape ---------------------------------- The side-view visualization is performed using a horizontal home-made microscope. The microscope consists of a long working distance infinity corrected apochromatic objective (Mitutoyo 10$\times$, focal length $20$ mm) aligned to a tube lens (Mitutoyo, MT-1, $1\times$) with a $57$ mm long tube (2.5 cm diameter). The tube lens is mounted on a DSLR camera (Nikon, D7100) with a 161 mm long tube and a C-mount-to-Nikon lens mount adapter (Fotodiox). This optical setup provides a spatial resolution of 0.5 $\mu$m per pixel. A green LED light source (Thorlabs, M530L3) is positioned in line with the tube microscope and behind the blister test device. The LED light is collimated using a biconvex lens (Thorlabs, LB1761) located at its focal distance from the LED light source. The top-view visualization, which provides the diameter $2r$ of the blister, is performed with a macro lens (Nikon, 105 mm) mounted on a DSLR camera (Nikon, D5100). The spatial resolution in the top-view optical system is 10 $\mu$m per pixel. Sample snapshots of the top-view and the side-view visualizations are provide in Fig. \[fig:top\_view\] and Fig. \[fig:side\_view\], respectively. The diameter of the blister $2r$ is quantified by fitting a circle to the thresholded image of the blister from the top. A Python code performing this task with Scikit-image [@Vanderwalt2014] is provided in the Supplementary Materials. The height is simply measured in the side-view images by identifying the distance between initial level of the thin film and the highest level of the blister for every injection volume. ![Visualization of the deflection of the film. (a) Side-view visualization of the center of the blister. The dashed red line represents the zero level before the injection of liquid under the PS film. (b) The corresponding top-view images are included to facilitate visualizing the size of the blister.[]{data-label="fig:side_view"}](side_view.pdf) The reproducibility of the measurements is controlled by performing measurements on three different devices. As an example, Fig. \[fig:reproducibility\] shows the measurements of the height of the blister versus its radius obtained on three devices for a film thickness $t = 850$ nm. Estimation of the adhesion energy --------------------------------- Blister tests measurements are performed for three different film thicknesses, namely $t = 590, 850, 1200$ nm. To derive the energy release rate $G$ as a function of the blister height $h$, we use the expressions provided by Sofla et al. [@Sofla2010] $$\label{eq:adhesion_buldge} G = \frac{\bar E t^5}{2 r^4} \left( 12 \bar M(h)^2 + \bar N(h)^2 \right)$$ where $\bar M(h)$ and $ \bar N(h)$ are the dimensionless moment per unit length and the normal force, respectively and $\bar E = E / (1 - \nu^2)$. The expressions for $\bar M(h)$ and $\bar N(h)$ are given in the Appendix. The average values of adhesion energy $G$ obtained in the blister tests are presented in Table \[tbl:blister\_adhesion\]. The reported results and the corresponding errors indicate the average and standard deviation achieved for individual tests in three separate devices. The agreement of these values of $G$ for different thicknesses shows the self-consistency of the method. Film thickness, $t$ \[nm\] Adhesion energy, $G$ \[mJ/${\mbox m}^2$\] -------------------------------- ----------------------------------------------- [590 $\pm$ 10]{} [18.5 $\pm$ 2.5]{} [850 $\pm$ 30]{} [17.8 $\pm$ 2]{} [1200 $\pm$80]{} [18.6 $\pm$ 2]{} : Measurements of adhesion energy using blister test. The uncertainty on the film thickness is calculated from the standard deviation of the thicknesses measured on a batch of films obtained under the same experimental conditions. []{data-label="tbl:blister_adhesion"} Comparison with a cleavage test {#sec:result} =============================== To validate the analysis made on the blister test, we measured the adhesion energy with a second method, namely a cleavage test, on the same material. The cleavage test consists of propagating an interfacial crack between the film and the substrate with a wedge. As the wedge must be pushed between the film and the substrate, this test prevents us to measure the adhesion energy on thicknesses smaller than few tens of micrometers. To prepare these thicker polystyrene films, the spin-coating or the Landau-Levich coating [@Rio2017] from a solution of polystyrene in toluene causes thickness variations due to surface instabilities triggered during the solvent evaporation [@Bassou2009]. As a consequence, we prepare our polystyrene film with a solvent-free method by melting the polymer. Glass slides are prepared with the protocol described in Sec. \[subsec:donor\]. Polystyrene pellets (Sigma-Aldrich, $M_w \simeq 280$ kg/mol, as used before) are placed between two glass slides, themselves placed between two aluminum plates (1 cm $\times$ 10 cm $\times$ 10 cm) held together by three screws. This press is placed in a oven at an initial temperature of $130^\circ$C. Screws are regularly tightened to squeeze the pellets and the temperature is raised until $220^\circ$C and the desired thickness is obtained. The temperature is decreased to $130^\circ$C and maintained for 12 hours to relax the difference of thermal dilation between the film and the substrate and is then further decreased to the room temperature. Aluminum plates and one of the glass slides are removed and the film is annealed again with the same protocol as described in Section \[subsec:psfilm\]. Films are selected for their film thickness uniformity measured with a micrometer caliper. The film thickness is typically of $100$ $\mu$m in our experiments. A razor blade of thickness $\delta=385$ $\mu$m is used as a wedge. The wedge is placed parallel and in contact with the substrate and pushed by a translational stage. Once the crack is initiated, we stop the motor and measure the distance $a$ between the wedge and the crack tip (Fig. \[fig:cleavage\]). From the minimization of the sum of the bending and adhesion energies, the energy release rate is [@Kendall1994] $$\label{eq:adhesion_cleavage} G = \frac{3}{16} \frac{E t^3 \delta^2}{(1-\nu^2) a^4},$$ where $\delta$ is the height of the blade and $a$ the length of the crack. From our measurements, we obtain $G= 14\pm 2$ mJ/m$^2$, which is in good agreement with the values from the blister test presented in Table \[tbl:blister\_adhesion\] and thus validates our approach. Conclusion ========== In this work, a step-by-step guide is provided for preparation of a device to perform blister test for thin delicate materials. More specifically, we performed blister tests to determine the adhesion energy between glass substrates and polystyrene films of micron and sub-micron thicknesses. Side-view and top-view visualizations were performed to quantify the maximum deflection and the radius of the blister, respectively. The adhesion energy was estimated for the experimental measurements based on the closed-form expressions proposed by Sofla et al. (2010). The average adhesion energy per unit area $G$ measured using the present blister test was $G = 18 \pm 2$ $ \mbox{mJ}/{\mbox m}^2$, which is in good agreement with the results of independent measurements of $G = 14 \pm 2$ $ \mbox{mJ}/{\mbox m}^2$ obtained in our cleavage test. The current device can be used for blister test measurements in the fixed radius mode to estimate the material properties of ultra thin films. Moreover, the protocols described here can be tailored conveniently to be applied to less conventional material pairs such as multi-layers of soft films and biomaterials such as biofilms for which performing a peeling test may damage the substrate and/or the adhering substance. Acknowledgments =============== We thank F. Restagno and T. Salez for valuable discussions on the preparation of PS films. F.B. acknowledges that part of the research leading to these results received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement 623541. S.K. appreciates the early mobility funding from the Swiss National Science Foundation (P2ELP2-158896). Appendix ======== Sofla et al. (2010) provided the relationships for the characteristics of the film deformation [@Sofla2010]. Here, we recall the expressions of the moment and the normal force. The dimensionless moment $\bar M = \frac{(1-\nu^2) r^2 M}{E t^4}$ and the normal force $\bar N = \frac{(1-\nu^2) r^2 N}{E t^4}$ are given by $$\begin{aligned} \bar M &=& \frac{2}{3} \bar h + \left( \frac{m(\nu) \bar h^{1.25}}{2.2 + \bar h^{1.25}} \right) \bar h^2, \\ \bar N &=& n(\nu) \left( -0.255 \bar h^2 \exp\left(-0.16 \bar h^{1.3}\right) + 0.667 \bar h^2\right),\end{aligned}$$ where the dimensionless deflection is $\bar h = h/t$ and $$\begin{aligned} m(\nu) &=& 0.509+0.221\nu-0.263\nu^2,\\ n(\nu) &=& 0.809-1.073\nu -0.816\nu^2.\end{aligned}$$
--- abstract: 'We investigate the birational section conjecture for curves over function fields of characteristic zero and prove that the conjecture holds over finitely generated fields over $\operatorname{\mathbb{Q}}$ if it holds over number fields.' author: - Mohamed Saïdi and Michael Tyler title: '***' --- Introduction and statement of results ===================================== The birational section conjecture --------------------------------- For a smooth, geometrically connected, projective curve $X$ over a field $k$, we define the absolute Galois group of $X$* to be the group $$G_X:=\operatorname{\textup{Gal}}(k(X)^{\textup{sep}}/k(X))$$ where $k(X)$ denotes the function field of $X$ and $k(X)^{\operatorname{\textup{sep}}}$ is a separable closure of $k(X)$. The Galois group $G_X$ fits into an exact sequence $${\begin{CD}1 @>>> G_{X_{\bar k}} @>>> G_X @>>> G_k @>>> 1\end{CD}}$$ where $G_k := \operatorname{\textup{Gal}}(\bar k\|k)$, $\bar k$ being the algebraic closure of $k$ in $k(X)^{\operatorname{\textup{sep}}}$, and $G_{X_{\bar k}}:=\operatorname{\textup{Gal}}(K(X)^{\operatorname{\textup{sep}}}/K(X) \cdot \bar k)$. Let $x \in X(k)$ be a $k$-rational point, and let $\tilde x$ be a valuation of $k(X)^{\operatorname{\textup{sep}}}$ extending the valuation $\nu_x$ of $k(X)$ corresponding to $x$. We will refer to $\tilde x$ as an extension of $x$ to $k(X)^{\operatorname{\textup{sep}}}$. The decomposition group $D_{\tilde x}$ of $\tilde x$ fits into a commutative diagram of exact sequences $$\begin{tikzcd} 1 \arrow{r}{} &I_{\tilde x} \arrow{r}{}\arrow[hookrightarrow]{d}{} &D_{\tilde x} \arrow{r}{}\arrow[hookrightarrow]{d}{} &G_{k(x)} \arrow{r}{}\arrow[equals]{d}{} &1\\ 1 \arrow{r}{} &G_{X_{\bar k}} \arrow{r}{} &G_X \arrow{r}{} &G_k \arrow{r}{} &1 \end{tikzcd}$$ where $I_{\tilde x}$ is the inertia group at $\tilde x$. We will refer to a splitting of the lower sequence in the above diagram as a section of $G_X$. A splitting of the upper exact sequence naturally defines a section $s_{\tilde x} : G_k \to G_X$ of $G_X$, with image contained in $D_{\tilde x}$. \[geometricgaloissections\] We say a section $s$ of $G_X$ is geometric if its image $s(G_k)$ is contained in a decomposition group $D_{\tilde x}$ for some $k$-rational point $x\in X(k)$ and some extension $\tilde x$ of $x$ to $k(X)^{\operatorname{\textup{sep}}}$. In this case, we say that the section $s$ arises from the point $x$. The birational analogue of Grothendieck’s anabelian section conjecture for étale fundamental groups may be stated as follows. Let $k$ be a finitely generated field over $\operatorname{\mathbb{Q}}$, and let $X$ be a smooth, projective, geometrically connected curve over $k$. Then every section of $G_X$ is geometric and arises from a unique $k$-rational point $x\in X(k)$. We will refer to this as the birational section conjecture or BSC. One may consider the statement for more general fields $k$, so we establish the following terminology. \[bscholds\] 1. Let $X$ be a smooth, geometrically connected, projective curve over a field $k$. We say the birational section conjecture (or BSC) holds for $X$ if every section of $G_X$ is geometric and arises from a unique $k$-rational point $x\in X(k)$. 2. For a field $k$, we say the birational section conjecture (or BSC) holds over $k$ if the BSC holds for every smooth, geometrically connected, projective curve over $k$. \[bscuniqueness\] To prove that the BSC holds for $X$ it suffices to prove that every section of $G_X$ arises from a $k$-rational point $x \in X(k)$. This is necessarily the unique such point, since decomposition subgroups of $G_X$ associated to distinct rank $1$ valuations of $k(X)^{\operatorname{\textup{sep}}}$ intersect trivially [@CONF Corollary 12.1.3] (in loc. cit. $k$ is a global field but the same argument of proof works over any field). It is hoped that the birational section conjecture might be used to prove the Grothendieck anabelian section conjecture for étale fundamental groups, via the theory of “cuspidalisation” of sections of arithmetic fundamental groups [@saidicusp]. Conversely, the anabelian section conjecture for $\pi_1$ implies the birational section conjecture, as follows easily from the “limit argument” of Akio Tamagawa [@tamagawa Proposition 2.8 (iv)]. At present the BSC over finitely generated fields over $\operatorname{\mathbb{Q}}$, as well as the anabelian section conjecture for $\pi_1$, are quite open. Very few examples are known of curves over number fields for which the [BSC]{} holds (see [@stoll Remark 8.9] for some examples). Also, some conditional results on the birational section conjecture over number fields of small degree are known (see [@hoshi]). Note that a $p$-adic analog of the birational section conjecture holds by [@koenigsmann]. To the best of our knowledge no result on the birational section conjecture is known for curves over finitely generated fields over $\operatorname{\mathbb{Q}}$ of positive transcendence degree. Statement of the Main Theorems ------------------------------ In this paper we investigate the birational section conjecture over function fields. We prove that, for a certain class of fields $k$ of characteristic zero, and under the condition of finiteness of certain Shafarevich-Tate groups, proving that the BSC holds over function fields over $k$ reduces to proving that it holds over finite extensions of $k$. This class of fields contains, in particular, the finitely generated extensions of $\operatorname{\mathbb{Q}}$, and for such fields we show further that the statement is independent of finiteness of the Shafarevich-Tate groups. The result therefore reduces the BSC over finitely generated extensions of $\operatorname{\mathbb{Q}}$ unconditionally to the case of number fields. Our approach stems from the proof in [@saidiSCOFF] of a similar result for the section conjecture for étale fundamental groups. Let us start by describing the aforementioned class of fields, which was introduced in [@saidiSCOFF Definition 0.2]. \[conditions\] For a field $k'$ of characteristic zero, consider the following conditions on $k'$. 1. The [BSC]{} holds over $k'$. 2. For every prime integer $\ell$, the $\ell$-cyclotomic character $\chi_{\ell}:G_{k'}\rightarrow\operatorname{\mathbb{Z}}^{\times}_{\ell}$ is non-Tate, meaning that any $G_{k'}$-map $\operatorname{\mathbb{Z}}_{\ell}(1)\rightarrow T_{\ell} A$, for some abelian variety $A$ over $k'$ and its $\ell$-adic Tate module $T_{\ell}A$, vanishes. 3. Given an abelian variety $A$ over $k'$, any quotient $A(k')\twoheadrightarrow D$ of the group of $k'$-rational points $A(k')$ satisfies the following: 1. The natural map $D\rightarrow \widehat D$ is injective, where $\widehat D:=\varprojlim_{N \geq 1} D/ND$. 2. The $N$-torsion subgroup $D[N]$ of $D$ is finite for all $N\geq 1$, and the Tate module $TD$ is trivial (cf. Notation). 4. Given a separated, smooth, connected curve $C$ over $k'$ with function field $K=k'(C)$, $K$ admits the structure of a Hausdorff topological field, so that $X(K)$ is compact for any smooth, geometrically connected, projective, hyperbolic curve $X$ over $K$. 5. Given a separated, smooth, and connected (not necessarily projective) curve $C$ over $k'$ with function field $K=k'(C)$ and a finite morphism $\tilde C\rightarrow C$ with $\tilde C$ separated and smooth, then the following holds. If $\tilde C_c(k'(c))\ne\emptyset$ for all closed points $c\in C^{\textup{cl}}$, where $k'(c)$ denotes the residue field at $c$ and $\tilde C_c$ is the scheme-theoretic inverse image of $c$ in $\tilde C$, then $\tilde C(K)\ne\emptyset$. For a field $k$ of characteristic zero, we say that $k$ strongly satisfies one of the above conditions (i), (ii), (iii), (iv) and (v) if this condition is satisfied by any finite extension $k'\|k$ of $k$. Condition (i) is the strongest of the above conditions. Conditions (ii)-(v) are much milder conditions which in particular are satisfied by finitely generated fields over $\operatorname{\mathbb{Q}}$. Condition (v) is satisfied by Hilbertian fields (cf. [@saidiSCOFF Lemma 4.1.5]). \[sha\] Let $k$ be a field of characteristic zero and $C$ a smooth, separated, connected curve over $k$ with function field $K$. Let $\mathcal{A}\rightarrow C$ be an abelian scheme with generic fibre $A := \mathcal{A} \times_C K$. For each closed point $c\in C^{\textup{cl}}$ denote by $K_c$ the completion of $K$ at $c$, and write $A_c := A \times_K K_c$. We define the Shafarevich-Tate group $$\Sha(\mathcal{A}) := \ker(H^1(G_K,A)\rightarrow\prod_{c\in C^{\textup{cl}}}H^1(G_{K_c},A_c))$$ where the product is taken over all the closed points of $C$. We now state our first two main Theorems. Let $k$ be a field of characteristic zero that strongly satisfies conditions (i)-(v) of Definition \[conditions\]. Let $C$ be a smooth, separated, connected curve over $k$ with function field $K$. Let $\operatorname{\mathcal{X}}\rightarrow C$ be a flat, proper, smooth relative curve, with generic fibre $X:=\operatorname{\mathcal{X}}\times_C K$ which is a geometrically connected hyperbolic curve over $K$ such that $X(K)\ne\emptyset$. Denoting by $\mathcal{J}:=\textup{Pic}^0_{\operatorname{\mathcal{X}}/C}$ the relative Jacobian of $\operatorname{\mathcal{X}}$, assume $T\Sha(\mathcal{J})=0$. Then the [BSC]{} holds for $X$. Let $k$, $C$ and $K$ be as in Theorem A. For any finite extension $L$ of $K$, let $C^L$ denote the normalisation of $C$ in $L$. Assume that for any such finite extension $L$ and any flat, proper, smooth relative curve $\operatorname{\mathcal{Y}}\to C^L$ we have $T\Sha(\operatorname{\mathcal{J}}_{\!\operatorname{\mathcal{Y}}}) = 0$, where $\operatorname{\mathcal{J}}_{\!\operatorname{\mathcal{Y}}} := \operatorname{\textup{Pic}}^0_{\operatorname{\mathcal{Y}}/C^L}$ is the relative Jacobian of $\operatorname{\mathcal{Y}}$. Then the [BSC]{} holds over all finite extensions of $K$. In the context of $\operatorname{\mathbb{Q}}$, this means that if the birational section conjecture holds over all fields of some fixed transcendence degree over $\operatorname{\mathbb{Q}}$ then it holds over all fields of transcendence degree one higher, provided that finiteness of $\Sha$ holds. By induction this means that, under the assumption on the relevant Shafarevich-Tate groups, if the birational section conjecture holds over number fields then it holds over all finitely generated fields over $\operatorname{\mathbb{Q}}$. As a consequence of one of the main results in [@saiditamagawa], asserting the finiteness of $\Sha$ for isotrivial abelian varieties over finitely generated fields, and using Theorem A, we can prove the following. Assume that the [BSC]{} holds over all number fields. Then the [BSC]{} holds over all finitely generated fields over $\Bbb Q$. Thus Theorem C reduces the proof of the birational section conjecture to the case of number fields. Guide to the paper ------------------ Our approach to proving the above Theorems is inspired by the method in [@saidiSCOFF], and relies on a local-global argument. This requires studying the properties of sections of absolute Galois groups of curves over local fields of equal characteristic zero and over function fields of curves in characteristic zero. We relate these two settings by investigating “étale abelian sections”, from where arises the constraint on the Shafarevich-Tate groups. The layout of the paper is as follows. In §2 we will recall some properties of étale and geometrically abelian fundamental groups and their sections, which will be necessary for the proofs of our main Theorems. In §3 we work in a local setting. We consider a flat, proper, smooth relative curve over the spectrum of a complete discrete valuation ring with residue characteristic zero, and explain how to define a specialisation homomorphism of absolute Galois groups (Theorem \[galspec\]) using the specialisation homomorphism of fundamental groups of affine curves. We apply this in §\[ramificationofsections\] to study the specialisation of sections, and the phenomenon of ramification. In §4 we return to the global setting of Theorems A and B, and consider curves over function fields of characteristic zero. In §\[specialisationofsections\] we explain how to pass to the local setting and apply the results from §3. In §\[etaleabeliansections\] we consider étale abelian sections and show how to apply a local-global principle, during which we encounter the issue of finiteness of the Shafarevich-Tate group (Proposition \[sabjkhat\]). In §5 we use the results of §3 and §4 to prove Theorems A and B, and apply these, along with one of the main results of [@saiditamagawa], to prove Theorem C. Notation {#notation .unnumbered} -------- - For a scheme $X$, we will denote the set of closed points of $X$ by $X^{\operatorname{\textup{cl}}}$. - Given a ring $R$ and morphisms of schemes $X \to Y$ and $\operatorname{\textup{Spec}}R \to Y$, we will denote the fibre product $X \times_Y \operatorname{\textup{Spec}}R$ by $X_R$. - For a field $k$ and a given separable closure $k^{\operatorname{\textup{sep}}}$ of $k$, we will denote the absolute Galois group $\operatorname{\textup{Gal}}(k^{\operatorname{\textup{sep}}}\|k)$ by $G_k$. - For an abelian group $A$ and a positive integer $N$, we denote by $A[N]$ the kernel of the homomorphism $N : A \to A$, $a \mapsto N \cdot a$. - Given an abelian group $A$, we denote by $\widehat{A}$ the inverse limit $\widehat{A} := \varprojlim_{N \geq 1} A/NA$ and by $TA := \varprojlim_{N \geq 1} A[N]$ the Tate module of $A$. - Given an abelian variety $B$ over a field $k$, an algebraic closure $\bar k$ of $k$, we will denote by $TB$ the Tate module of the abelian group $B(\bar k)$. Sections of étale and geometrically abelian fundamental groups ============================================================== Sections of étale fundamental groups ------------------------------------ Let $k$ be a field of characteristic zero, $X$ a smooth, geometrically connected, projective curve over $k$, and $U\subset X$ an open subset with complement $S := X \setminus U$ (when considering an open $U\subset X$ we assume $U$ non-empty). Write $K=k(X)$ for the function field of $X$. Let $z$ be a geometric point of $U$ with image in the generic point. Thus $z$ determines algebraic closures ${\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}$, and $\bar k$, of $K$ and $k$ respectively, as well as a geometric point of $U_{\bar k}$ that we denote $\bar z$. The étale fundamental group $\pi_1(U, z)$ of $U$ fits into an exact sequence: $${\begin{CD}1 @>>> \pi_1(U_{\bar k},\bar z) @>>> \pi_1(U,z) @>>> G_k=\operatorname{\textup{Gal}}(\bar k/k) @>>> 1\end{CD}}$$ We will refer to this as the fundamental exact sequence of $U$. The fundamental group $\pi_1(U_{\bar k},\bar z)$ will be called the geometric fundamental group of $U$. \[universalcover\] A universal pro-étale cover $\tilde U\rightarrow U$ of $U$ is an inverse system of finite étale covers $\{V_i\rightarrow U\}_{i\in I}$, corresponding to open subgroups of $\pi_1(U,z)$, such that for any étale cover $V\rightarrow U$, corresponding to an open subgroup of $\pi_1(U,z)$, there is a $U$-morphism $V_i\rightarrow V$ for some $i\in I$. A (closed) point $\tilde x$ of the universal pro-étale cover $\tilde U$ is a compatible system of (closed) points $\{x_i\in V_i\}_{i\in I}$. Fix a universal pro-étale cover $\tilde U \to U$, and let $\{V_i \to U\}_{i\in I}$ be the inverse system of étale covers defining it. \[normalisation\] Let $Y_i \to X$ denote the unique extension of $V_i \to U$ to a finite morphism of smooth, connected, projective curves over $k$. The inverse system of morphisms $\{Y_i \to X\}_{i\in I}$ will be called the normalisation of $X$ in $\tilde U$, denoted $\tilde X_{\tilde U} \to X$. A (closed) point $\tilde x$ of $\tilde X_{\tilde U}$ is a compatible system of (closed) points $\{x_i\in Y_i\}_{i\in I}$. \[decompinertiapi1\] With the above notation, let $x\in X^{\operatorname{\textup{cl}}}$ be a closed point and $\tilde x$ a point above $x$ in the normalisation $\tilde X_{\tilde U}$ of $X$ in $\tilde U$. The decomposition group $D_{\tilde x}$ of $\tilde x$ is the stabiliser of $\tilde x$ under the action of $\pi_1(U,z)$ (which acts naturally on the points of $\tilde X_{\tilde U}$). The inertia group $I_{\tilde x}$ is the kernel of the natural projection $D_{\tilde x}\twoheadrightarrow G_{k(x)}= \operatorname{\textup{Aut}}(\bar k/k(x))$, where $k(x)$ is the residue field at $x$. It follows immediately from the definition that decomposition and inertia groups for different choices of $\tilde x$ above $x$ are conjugate, that is, for any $\sigma\in\pi_1(U,z)$ we have $I_{\sigma\tilde x}=\sigma I_{\tilde x}\sigma^{-1}$ and $D_{\sigma\tilde x}=\sigma D_{\tilde x}\sigma^{-1}$. The inertia group $I_{\tilde x}$ is trivial if $x \in U$, while if $x \in S$ it is isomorphic as a $G_{k(x)}$-module to the Tate twist $\hat\operatorname{\mathbb{Z}}(1)$. Recall that the curve $U$ is hyperbolic if the geometric fundamental group $\pi_1(U_{\bar k}, \bar z)$ is non-abelian. Denoting by $g$ the genus of $X$, this occurs exactly when $2 - 2g - \deg_k S < 0$, where $S$ is regarded as a reduced effective divisor on $X$. We refer to [@hoshimochizuki Lemma 1.5] for a proof of the following statement. \[inertiaproperties\] With the above notation, assume further that $U$ is hyperbolic. Then any two inertia subgroups $I_{\tilde x}, I_{\tilde x'} \subset \pi_1(U, z)$ corresponding to distinct points $\tilde x\ne\tilde x'$ of $\tilde X_{\tilde U}$ intersect trivially. Let $x \in X(k)$ be a $k$-rational point of $X$, and let $\tilde x$ be a point above $x$ in the normalisation $\tilde X_{\tilde U}$ of $X$ in $\tilde U$. There is a commutative diagram of exact sequences $$\label{decompinjectpi1} \begin{tikzcd} 1 \arrow{r}{} &I_{\tilde x} \arrow{r}{}\arrow[hookrightarrow]{d}{} &D_{\tilde x} \arrow{r}{}\arrow[hookrightarrow]{d}{} &G_{k(x)} \arrow{r}{}\arrow[equals]{d}{} &1\\ 1 \arrow{r}{} &\pi_1(U_{\bar k},\bar z) \arrow{r}{} &\pi_1(U,z) \arrow{r}{} &G_{k} \arrow{r}{} &1 \end{tikzcd}$$ where the middle vertical map is the natural inclusion, via which a section of the upper exact sequence naturally defines one of the (lower) fundamental exact sequence. When $x \in U$ the inertia group $I_{\tilde x}$ is trivial, hence there is an isomorphism $D_{\tilde x} \simeq G_{k(x)}=G_k$ (induced by the projection $\pi_1(U,z)\twoheadrightarrow G_k$) and thus $D_{\tilde x}$ gives rise naturally to a section of $\pi_1(U, z)$ in this case. Action by $\pi_1(U_{\bar k}, \bar z)$ permutes the points $\tilde x$ of $\tilde U$ above $x$, so $x$ in fact induces a conjugacy class of sections of the fundamental exact sequence. It is well known that the upper exact sequence in diagram (\[decompinjectpi1\]) also splits when $x$ is contained in the complement $S = X \setminus U$. \[cuspidal\] A section $s:G_k\rightarrow\pi_1(U,z)$ is called cuspidal if it factors through $D_{\tilde x}$ for some (necessarily $k$-rational) point $x \in S$ and some $\tilde x\in\tilde X_{\tilde U}$ above $x$. Now let $\xi : \operatorname{\textup{Spec}}\Omega \to X$ be a geometric point with image the generic point of $X$. The geometric point $\xi$ naturally determines a choice of a separable closure $K^{\operatorname{\textup{sep}}}$ of $K$ and, for each open subset $U \subset X$, a geometric point $\xi:\operatorname{\textup{Spec}}\Omega\to U$ with image the generic point of $U$. The following is well-known. \[Gallim\] With $X$, $K$ and $\xi$ as above, there is a canonical isomorphism $$G_X:=\operatorname{\textup{Gal}}(K^{\operatorname{\textup{sep}}}/K)\simeq\varprojlim_{U\subset X\textup{ open}}\pi_1(U,\xi)$$ where the limit is taken over all the open subsets of $X$, partially ordered by inclusion. In particular, for any open subset $U \subset X$, the fundamental group $\pi_1(U, \xi)$ is naturally a quotient of $G_X$. In fact we have a commutative diagram $$\begin{tikzcd} 1 \arrow{r}{} &G_{X_{\bar k}} \arrow{r}{}\arrow[twoheadrightarrow]{d}{} &G_X \arrow{r}{}\arrow[twoheadrightarrow]{d}{} &G_k \arrow{r}{}\arrow[equals]{d}{} &1\\ 1 \arrow{r}{} &\pi_1(U_{\bar k},\bar\xi) \arrow{r}{} &\pi_1(U,\xi) \arrow{r}{} &G_k \arrow{r}{} &1 \end{tikzcd}$$ where the middle vertical map is the natural projection, via which a section $s : G_k \to G_X$ of $G_X$ naturally induces a section $s_U : G_k \to \pi_1(U, \xi)$ of the projection $\pi_1(U, \xi)\twoheadrightarrow G_k$. Thus, by Lemma \[Gallim\], a section $s$ of $G_X$ determines, and is determined by, a compatible system of sections $s_U : G_k \to \pi_1(U, \xi)$ for the open subsets $U \subset X$. Geometrically abelian fundamental groups ---------------------------------------- Let $k$ be a field of characteristic zero, and let $X$ be a smooth, geometrically connected, projective curve over $k$ such that $X(k) \ne \emptyset$. Let $z:\operatorname{\textup{Spec}}\Omega\rightarrow X$ be a geometric point with value in the generic point, which determines an algebraic closure $\bar k$ of $k$ and a geometric point $\bar z$ of $X_{\bar k}$. Denote by $\pi_1(X_{\bar k}, \bar z)^{\operatorname{\textup{ab}}}$ the maximal abelian quotient of the geometric fundamental group of $X$. The geometrically abelian quotient of $\pi_1(X, z)$, denoted $\pi_1(X, z)^{(\operatorname{\textup{ab}})}$, is defined by the pushout diagram $$\begin{tikzcd} 1 \arrow{r}{} &\pi_1(X_{\bar k},\bar z) \arrow{r}{}\arrow[twoheadrightarrow]{d}{} &\pi_1(X,z) \arrow{r}{}\arrow[twoheadrightarrow]{d}{} &G_k \arrow{r}{}\arrow[equals]{d}{} &1\\ 1 \arrow{r}{} &\pi_1(X_{\bar k},\bar z)^{\operatorname{\textup{ab}}} \arrow{r}{} &\pi_1(X,z)^{(\operatorname{\textup{ab}})} \arrow{r}{} &G_k \arrow{r}{} &1 \end{tikzcd}$$ where the upper row is the fundamental exact sequence. By commutativity of this diagram, a section $s : G_k \to \pi_1(X,z)$ of the étale fundamental group of $X$ naturally induces a section $s^{\operatorname{\textup{ab}}} : G_k \to \pi_1(X, z)^{(\operatorname{\textup{ab}})}$ of the geometrically abelian quotient, which we will call the étale abelian section induced by $s$. Fix a $k$-rational point $x_0\in X(k)$. Let $J$ denote the Jacobian of $X$, and let $\iota:X\rightarrow J$ be the closed immersion mapping $x_0$ to the zero section of $J$. Note that $\iota$ maps an arbitrary $k$-rational point $x \in X(k)$ to the class of the degree zero divisor $[x]-[x_0]$. Moreover, it induces a commutative diagram of exact sequences $$\label{pi1xpi1j} \begin{tikzpicture}[descr/.style={fill=white}, baseline=(current bounding box.center)] \matrix(m)[matrix of math nodes, row sep=3em, column sep=2.5em, text height=1.5ex, text depth=0.25ex] {1 & \pi_1(X_{\bar k},\bar z)^{\operatorname{\textup{ab}}} & \pi_1(X,z)^{(\operatorname{\textup{ab}})} & G_k & 1\\ 1 & \pi_1(J_{\bar k},\bar z) & \pi_1(J,z) & G_k & 1\\}; \path[->] (m-1-1) edge (m-1-2); \path[->] (m-1-2) edge (m-1-3); \path[->,font=\scriptsize] (m-1-2) edge node[below,rotate=90]{$\sim$} (m-2-2); \path[->] (m-1-3) edge (m-1-4); \path[->,font=\scriptsize] (m-1-3) edge node[below,rotate=90]{$\sim$} (m-2-3); \path[->] (m-1-4) edge (m-1-5); \path[-] (m-1-4) edge[double distance=2pt] (m-2-4); \path[->] (m-2-1) edge (m-2-2); \path[->] (m-2-2) edge (m-2-3); \path[->] (m-2-3) edge (m-2-4); \path[->] (m-2-4) edge (m-2-5); \end{tikzpicture}$$ where the vertical maps are isomorphisms. Hence there is an identification of $G_k$-modules $\pi_1(X_{\bar k},\bar z)^{\operatorname{\textup{ab}}} \simeq \pi_1(J_{{\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu}}, \bar z) \simeq TJ$, where $TJ$ is the Tate module of $J$ (see Notation). We fix a base point of the torsor of splittings of the upper exact sequence in (\[pi1xpi1j\]) to be the splitting arising from the rational point $x_0$, and the corresponding base point of the torsor of splittings of the lower exact sequence in (\[pi1xpi1j\]), and identify the set of $\pi_1(X_{\bar k},\bar z)^{\operatorname{\textup{ab}}}$-conjugacy classes of sections of the upper exact sequence in (\[pi1xpi1j\]) with the Galois cohomology group $H^1(G_k, TJ)$. By functoriality of the fundamental group, any $k$-rational point $x\in X(k)$ induces a section $s_x:G_k\to\pi_1(X,z)$, which in turn induces an étale abelian section $s_x^{\operatorname{\textup{ab}}}:G_k\rightarrow\pi_1(X,z)^{(\operatorname{\textup{ab}})}$. Thus we have a map $X(k)\to H^1(G_k,TJ)$ defined by $x\mapsto [s_x^{\operatorname{\textup{ab}}}]$, where $[s_x^{\operatorname{\textup{ab}}}]$ denotes the $\pi_1(X_{\bar k}, \bar z)^{\operatorname{\textup{ab}}}$-conjugacy class of sections containing $s_x^{\operatorname{\textup{ab}}}$. This map factors through $J(k)$, that is, it coincides with the composite map $$\begin{tikzcd}[column sep=small]X(k)\arrow[hookrightarrow]{r}{\iota} &J(k) \arrow{r}{} &H^1(G_k,TJ).\end{tikzcd}$$ This is due to the isomorphism $\pi_1(X,z)^{(\operatorname{\textup{ab}})}\simeq\pi_1(J,z)$, via which $s_x^{\operatorname{\textup{ab}}}$ corresponds to the section of $\pi_1(J,z)$ induced by functoriality from the $k$-rational point $\iota(x)\in J(k)$. \[kummerexactseq\] There is an exact sequence $${\begin{CD}0 @>>> \widehat{J(k)} @>>> H^1(G_k,TJ) @>>> TH^1(G_k,J) @>>> 0\end{CD}}$$ where $\widehat{J(k)}:=\varprojlim_N J(k)/NJ(k)$ (see Notation) and $TH^1(G_k,J)$ is the Tate module of the Galois cohomology group $H^1(G_k,J)$. We shall refer to this sequence as the Kummer exact sequence. One can easily derive it from the Kummer exact sequences $\begin{tikzcd}[column sep=small]0 \arrow{r}{} &J(\bar k)[N] \arrow{r}{} &J(\bar k) \arrow{r}{N} &J(\bar k) \arrow{r}{} &0,\end{tikzcd}$ $N \geq 0$, during which one sees, in particular, that the map $J(k)\to H^1(G_k,TJ)$ factors through $\widehat{J(k)}$. Thus we have a sequence of maps: $$\label{sabptth} \begin{tikzcd}[column sep=small]X(k) \arrow[hookrightarrow]{r}{\iota} &J(k) \arrow{r}{} &\widehat{J(k)} \arrow[hookrightarrow]{r}{} &H^1(G_k,TJ).\end{tikzcd}$$ Specialisation of sections in a local setting ============================================= A specialisation homomorphism for absolute Galois groups {#spechom} -------------------------------------------------------- Let $R$ be a complete discrete valuation ring with uniformiser $\pi$, field of fractions $K$ and residue field $k:=R/\pi R$ of characteristic zero. Let $X$ be a flat, proper, smooth, geometrically connected relative curve over $\operatorname{\textup{Spec}}R$, and denote by $X_K:=X\times _{\operatorname{\textup{Spec}}R}\operatorname{\textup{Spec}}K$ its generic fibre and $X_k:=X\times _{\operatorname{\textup{Spec}}R}\operatorname{\textup{Spec}}k$ its special fibre. Fix an algebraic closure ${\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}$ of $K$, and denote by ${\mkern 1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern 1.5mu}$ the integral closure of $R$ in ${\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}$, and by $\bar k$ the residue field of ${\mkern 1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern 1.5mu}$, which is an algebraic closure of $k$. Let $\bar\xi_1:\operatorname{\textup{Spec}}\Omega_1\rightarrow X_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}$ be a geometric point with image the generic point of $X_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}$, and similarly let $\bar\xi_2:\operatorname{\textup{Spec}}\Omega_2\rightarrow X_{\bar k}$ be a geometric point with image the generic point of $X_{\bar k}$. These induce geometric points of $X_K$ and $X_k$, which we denote by $\xi_1$ and $\xi_2$ respectively. \[stilde\] For each closed point $x$ of $X_k^{\operatorname{\textup{cl}}}$, fix a choice of closed point $y\in X_K^{\operatorname{\textup{cl}}}$ which specialises to $x$ and whose residue field is the unique unramified extension of $K$ whose valuation ring has residue field $k(x)$ (such a point exists since $X \to \operatorname{\textup{Spec}}R$ is smooth). We define $\tilde S$ to be the set of these chosen closed points $y\in X_K^{\operatorname{\textup{cl}}}$. Thus, $\tilde S$ is a subset of $X^{\operatorname{\textup{cl}}}_K$ in bijection with $X_k^{\operatorname{\textup{cl}}}$. We denote by $\tilde S_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}$ the inverse image of $\tilde S$ via the map $X_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}^{\operatorname{\textup{cl}}}\to X^{\operatorname{\textup{cl}}}$. Thus, $\tilde S_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}$ is a subset of $X_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}^{\operatorname{\textup{cl}}}$ in bijection with $X_{\bar k}^{\operatorname{\textup{cl}}}$. \[pi1stilde\] With $\tilde S$ and $\tilde S_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}$ as in Definition \[stilde\], we define the group $\pi_1(X_K - \tilde S)$ to be the inverse limit $$\pi_1(X_K - \tilde S) := \varprojlim_{B\subset\tilde S\textup{ finite}}\pi_1(X_K \setminus B, \xi_1)$$ over the open subsets of $X_K$ whose complements are finite subsets of $\tilde S$, ordered by inclusion. Similarly, we define $\pi_1(X_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}} - \tilde S_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}) := \varprojlim_{B\subset\tilde S_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}\textup{ finite}}\pi_1(X_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}} \setminus B, \bar\xi_1)$. \[universalcovertilde\] A universal pro-étale cover $\tilde X_{\tilde S} \to X_K - \tilde S$ is an inverse system of finite morphisms $\{Y_i\to X_K\}_{i\in I}$, with $Y_i$ smooth, corresponding to open subgroups of $\pi_1(X_K - \tilde S)$, such that for any finite morphism $Y \to X_K$, with $Y$ smooth, corresponding to an open subgroup of $\pi_1(X_K - \tilde S)$, there is an $X_K$-morphism $Y_i\to Y$ for some $i\in I$. A (closed) point $\tilde x$ of $\tilde X_{\tilde S}$ is a compatible system of (closed) points $\{y_i \in Y_i\}_{i\in I}$. For a universal pro-étale cover $\tilde X_{\tilde S} \to X_K - \tilde S$ and any closed point $\tilde x$ of $\tilde X_{\tilde S}$, we define decomposition and inertia subgroups $D_{\tilde x}, I_{\tilde x} \subset \pi_1(X_K - \tilde S)$ exactly as in Definition \[decompinertiapi1\], and they satisfy analogous properties. \[galspec\] There exists a surjective (continuous) homomorphism $\operatorname{\textup{Sp}}: \pi_1(X_K-\tilde S) \twoheadrightarrow G_{X_k}$, an isomorphism ${\mkern 1.5mu\overline{\mkern-1.5mu\operatorname{\textup{Sp}}\mkern-1.5mu}\mkern 1.5mu} : \pi_1(X_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}} - \tilde S_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}) \simeq G_{X_{\bar k}}$, and a commutative diagram of exact sequences: $$\label{specialisationgalois} \begin{tikzpicture}[descr/.style={fill=white}, baseline=(current bounding box.center)] \matrix(m)[matrix of math nodes, row sep=3em, column sep=2.5em, text height=1.5ex, text depth=0.25ex] {1 & G_{X_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}} & G_{X_K} & G_K & 1\\ 1 & \pi_1(X_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}-\tilde S_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}) & \pi_1(X_K-\tilde S) & G_K & 1\\ 1 & G_{X_{\bar k}} & G_{X_k} & G_k & 1\\}; \path[->,font=\scriptsize] (m-1-1) edge node[above]{} (m-1-2); \path[->,font=\scriptsize] (m-1-2) edge node[above]{} (m-1-3); \path[->>,font=\scriptsize] (m-1-2) edge node[left]{} (m-2-2); \path[->,font=\scriptsize] (m-1-3) edge node[above]{} (m-1-4); \path[->>,font=\scriptsize] (m-1-3) edge node[left]{} (m-2-3); \path[->,font=\scriptsize] (m-1-4) edge node[above]{} (m-1-5); \path[-,font=\scriptsize] (m-1-4) edge[double distance=2pt] node[left]{} (m-2-4); \path[->,font=\scriptsize] (m-2-1) edge node[above]{} (m-2-2); \path[->,font=\scriptsize] (m-2-2) edge node[above]{} (m-2-3); \path[->>,font=\scriptsize] (m-2-2) edge node[left]{${\mkern 1.5mu\overline{\mkern-1.5mu\operatorname{\textup{Sp}}\mkern-1.5mu}\mkern 1.5mu}$} (m-3-2); \path[->,font=\scriptsize] (m-2-3) edge node[above]{} (m-2-4); \path[->>,font=\scriptsize] (m-2-3) edge node[left]{$\operatorname{\textup{Sp}}$} (m-3-3); \path[->,font=\scriptsize] (m-2-4) edge node[above]{} (m-2-5); \path[->>,font=\scriptsize] (m-2-4) edge node[left]{} (m-3-4); \path[->,font=\scriptsize] (m-3-1) edge node[above]{} (m-3-2); \path[->,font=\scriptsize] (m-3-2) edge node[above]{} (m-3-3); \path[->,font=\scriptsize] (m-3-3) edge node[above]{} (m-3-4); \path[->,font=\scriptsize] (m-3-4) edge node[above]{} (m-3-5); \end{tikzpicture}$$ The homomorphism $\operatorname{\textup{Sp}}$, resp. ${\mkern 1.5mu\overline{\mkern-1.5mu\operatorname{\textup{Sp}}\mkern-1.5mu}\mkern 1.5mu}$, is defined only up to conjugation. The homomorphisms ${\mkern 1.5mu\overline{\mkern-1.5mu\operatorname{\textup{Sp}}\mkern-1.5mu}\mkern 1.5mu}$ and $\operatorname{\textup{Sp}}$ will be referred to as specialisation homomorphisms. For the proof we need the following well-known result. \[specialisationaffine\] Let $S$ be a divisor on $X$ which is finite étale over $R$, and denote $U := X \setminus S$ which is an open sub-scheme of $X$. Then there exists a surjective (continuous) homomorphism $\operatorname{\textup{Sp}}_U : \pi_1(U_K, \xi_1) \twoheadrightarrow \pi_1(U_k, \xi_2)$ and an isomorphism ${\mkern 1.5mu\overline{\mkern-1.5mu\operatorname{\textup{Sp}}\mkern-1.5mu}\mkern 1.5mu}_U : \pi_1(U_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}, \bar\xi_1) \simeq \pi_1(U_{\bar k}, \bar\xi_1)$ making the following diagram commutative. $$\label{specialisationaffine2} \begin{tikzpicture}[descr/.style={fill=white}, baseline=(current bounding box.center)] \matrix(m)[matrix of math nodes, row sep=3em, column sep=2.5em, text height=1.5ex, text depth=0.25ex] {1 & \pi_1(U_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}},\bar \xi_1) & \pi_1(U_K,\xi_1) & G_K & 1\\ 1 & \pi_1(U_{\bar k},\bar \xi_2) & \pi_1(U_k,\xi_2) & G_k & 1\\}; \path[->,font=\scriptsize] (m-1-1) edge node[above]{} (m-1-2); \path[->,font=\scriptsize] (m-1-2) edge node[above]{} (m-1-3); \path[->,font=\scriptsize] (m-1-2) edge node[left]{${\mkern 1.5mu\overline{\mkern-1.5mu\operatorname{\textup{Sp}}\mkern-1.5mu}\mkern 1.5mu}_U$} node[below,rotate=90]{$\sim$} (m-2-2); \path[->,font=\scriptsize] (m-1-3) edge node[above]{} (m-1-4); \path[->>,font=\scriptsize] (m-1-3) edge node[left]{$\operatorname{\textup{Sp}}_U$} (m-2-3); \path[->,font=\scriptsize] (m-1-4) edge node[above]{} (m-1-5); \path[->>,font=\scriptsize] (m-1-4) edge node[left]{} (m-2-4); \path[->,font=\scriptsize] (m-2-1) edge node[above]{} (m-2-2); \path[->,font=\scriptsize] (m-2-2) edge node[above]{} (m-2-3); \path[->,font=\scriptsize] (m-2-3) edge node[above]{} (m-2-4); \path[->,font=\scriptsize] (m-2-4) edge node[above]{} (m-2-5); \end{tikzpicture}$$ The homomorphism ${\mkern 1.5mu\overline{\mkern-1.5mu\operatorname{\textup{Sp}}\mkern-1.5mu}\mkern 1.5mu}_U$, respectively $\operatorname{\textup{Sp}}_U$, is defined only up to inner automorphism of $\pi_1(U_{{\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu}},\bar\xi_2)$, resp. $\pi_1(U,\xi_2)$. The homomorphisms ${\mkern 1.5mu\overline{\mkern-1.5mu\operatorname{\textup{Sp}}\mkern-1.5mu}\mkern 1.5mu}_U$ and $\operatorname{\textup{Sp}}_U$ are called specialisation homomorphisms of fundamental groups. The homomorphism $\operatorname{\textup{Sp}}_U$ is defined in a natural way and induces the homomorphism ${\mkern 1.5mu\overline{\mkern-1.5mu\operatorname{\textup{Sp}}\mkern-1.5mu}\mkern 1.5mu}_U$. For an exposition, in particular the fact that ${\mkern 1.5mu\overline{\mkern-1.5mu\operatorname{\textup{Sp}}\mkern-1.5mu}\mkern 1.5mu}_U$ is an isomorphism, see [@Luminy]. Let $B\subset\tilde S$ be a finite subset of $\tilde S$, viewed as a reduced closed sub-scheme of $X_K$, and let $\operatorname{\mathbb{B}}$ denote its schematic closure in $X$. By construction, $\operatorname{\mathbb{B}}$ is a divisor on $X$ which is finite étale over $R$ such that $\operatorname{\mathbb{B}}_K=B$. Denoting $U := X - \operatorname{\mathbb{B}}$, by Lemma \[specialisationaffine\] there exist specialisation homomorphisms $\operatorname{\textup{Sp}}_U : \pi_1(U_K, \xi_1) \twoheadrightarrow \pi_1(U_k, \xi_2)$ and ${\mkern 1.5mu\overline{\mkern-1.5mu\operatorname{\textup{Sp}}\mkern-1.5mu}\mkern 1.5mu}_U : \pi_1(U_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}, \bar\xi_1) \simeq \pi_1(U_{\bar k}, \bar\xi_2)$. Since there exist such homomorphisms for every finite subset of $\tilde S$, we have a compatible system of surjective homomorphisms $\{ \operatorname{\textup{Sp}}_U \}$, resp. of isomorphisms $\{ {\mkern 1.5mu\overline{\mkern-1.5mu\operatorname{\textup{Sp}}\mkern-1.5mu}\mkern 1.5mu}_U \}$ (the compatibility follows from the construction of these homomorphisms). By Lemma \[Gallim\], taking inverse limits gives rise respectively to the surjective homomorphism $\operatorname{\textup{Sp}}: \pi_1(X_K-\tilde S)\twoheadrightarrow G_{X_k}$ and the isomorphism ${\mkern 1.5mu\overline{\mkern-1.5mu\operatorname{\textup{Sp}}\mkern-1.5mu}\mkern 1.5mu} : \pi_1(X_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}} - \tilde S_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}) \simeq G_{X_{\bar k}}$, and moreover $\pi_1(X_K-\tilde S)$ and $\pi_1(X_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}} - \tilde S_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}})$ are naturally quotients of $G_{X_K}$ and $G_{X_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}}$ respectively. Thus we have the required homomorphisms in diagram (\[specialisationgalois\]), and this diagram is clearly commutative. One could consider the composite homomorphism $G_{X_K} \twoheadrightarrow \pi_1(X_K-\tilde S) \twoheadrightarrow G_{X_k}$, and similarly for $G_{X_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}}$, to be a specialisation homomorphism for absolute Galois groups. However, we will reserve the label ‘Sp’ for the homomorphism $\pi_1(X_K-\tilde S) \twoheadrightarrow G_{X_k}$, since this will be important in the next section. Ramification of sections {#ramificationofsections} ------------------------ We use the notation of §\[spechom\], and assume further that the closed fibre $X_k$ is hyperbolic. With $S$ and $U$ as in Lemma \[specialisationaffine\], consider again diagram (\[specialisationaffine2\]), and recall that the kernel of the projection $G_K\twoheadrightarrow G_k$ is the inertia group $I_K$ associated to the discrete valuation on $K$. \[sppullback\] 1. The projection $\pi_1(U_K, \xi_1) \twoheadrightarrow G_K$ restricts to an isomorphism $\ker(\operatorname{\textup{Sp}}_U)\simeq I_K$. 2. The right square in diagram (\[specialisationaffine2\]) is cartesian. The isomorphism $\ker(\operatorname{\textup{Sp}})\simeq I_K$ follows from a simple diagram chase, and (ii) follows easily from (i). Fix universal pro-étale covers $\tilde U_K \to U_K$ and $\tilde U_k \to U_k$ (corresponding to the geometric points $\xi_1$ and $\xi_2$ respectively), and let $\tilde X_{\tilde U_K}$ denote the normalisation of $X_K$ in $\tilde U_K$, and likewise $\tilde X_{\tilde U_k}$ the normalisation of $X_k$ in $\tilde U_k$ (see Definitions \[universalcover\] and \[normalisation\]). \[cuspdecomppullback\] Let $x\in S(k)$, and let $y'$ be the unique ($K$-rational) point of $S_K$ which specialises to $x$. Let $\tilde y'$ be a point of $\tilde X_{\tilde U_K}$ above $y'$. There exists a unique $\tilde x$ in $\tilde X_{\tilde U_k}$ above $x$, so that we have the following commutative diagram: $$\begin{tikzpicture}[descr/.style={fill=white}, baseline=(current bounding box.center)] \matrix(m)[matrix of math nodes, row sep=3em, column sep=2.5em, text height=1.5ex, text depth=0.25ex] {1 & I_{\tilde y'} & D_{\tilde y'} & G_K & 1\\ 1 & I_{\tilde x} & D_{\tilde x} & G_k & 1\\}; \path[->,font=\scriptsize] (m-1-1) edge node[above]{} (m-1-2); \path[->,font=\scriptsize] (m-1-2) edge node[above]{} (m-1-3); \path[->,font=\scriptsize] (m-1-2) edge node[left]{${\mkern 1.5mu\overline{\mkern-1.5mu\operatorname{\textup{Sp}}\mkern-1.5mu}\mkern 1.5mu}_U$} node[below,rotate=90]{$\sim$} (m-2-2); \path[->,font=\scriptsize] (m-1-3) edge node[above]{} (m-1-4); \path[->>,font=\scriptsize] (m-1-3) edge node[left]{$\operatorname{\textup{Sp}}_U$} (m-2-3); \path[->,font=\scriptsize] (m-1-4) edge node[above]{} (m-1-5); \path[->>,font=\scriptsize] (m-1-4) edge node[left]{} (m-2-4); \path[->,font=\scriptsize] (m-2-1) edge node[above]{} (m-2-2); \path[->,font=\scriptsize] (m-2-2) edge node[above]{} (m-2-3); \path[->,font=\scriptsize] (m-2-3) edge node[above]{} (m-2-4); \path[->,font=\scriptsize] (m-2-4) edge node[above]{} (m-2-5); \end{tikzpicture}$$ where $D_{\tilde y'}$ (resp. $D_{\tilde x}$) is the decomposition group of $\tilde y'$ (resp. $\tilde x$) in $\pi_1(U_K,\xi_1)$ (resp. $\pi_1(U_k,\xi_2)$). Moreover, the right square in this diagram is cartesian. The image of $D_{\tilde y'}$ under $\operatorname{\textup{Sp}}_U$ is contained in $D_{\tilde x}$ for some $\tilde x$ in $\tilde X_{U_k}$ above $x$, as follows easily from the functoriality of fundamental groups and the specialisation of points on (coverings of) $R$-curves. Moreover, such $\tilde x$ is unique by Lemma \[inertiaproperties\], since ${\mkern 1.5mu\overline{\mkern-1.5mu\operatorname{\textup{Sp}}\mkern-1.5mu}\mkern 1.5mu}_U$ is an isomorphism, which implies the inertia subgroup $I_{\tilde y'}\subset D_{\tilde y'}$ maps isomorphically to $I_{\tilde x}\subset\pi_1(U_{\bar k},\bar\xi_2)$. Commutativity of diagram (\[specialisationaffine2\]) then implies that $D_{\tilde y'}$ maps surjectively onto $D_{\tilde x}$, whence the above diagram. As in the proof of Lemma \[sppullback\], the right square in this diagram is cartesian. \[cuspidalevenifyisnotacusp\] With the notation of Lemma \[cuspdecomppullback\], for any $K$-rational point $y$ of $U_K$ specialising to $x$ and any point $\tilde y$ of $\tilde U_K$ above $y$, $D_{\tilde y}$ is contained in $D_{\tilde y'}$ for some $\tilde y'$ in $\tilde X_{\tilde U_K}$ above $y'$. In particular, a section $s:G_K\to\pi_1(U_K,\xi_1)$ with image contained in (hence equal to) $D_{\tilde y}$ is cuspidal (Definition \[cuspidal\]), even though $y\not\in S_K$. By specialisation, the point $\tilde y$ determines a point $\tilde x\in \tilde X_{\tilde U_k}$ such that $\operatorname{\textup{Sp}}_U(D_{\tilde y}) \subset D_{\tilde x}$. The statement then follows from Lemma \[cuspdecomppullback\] and the universal property of cartesian squares. Note that for a point $\tilde x$ in $\tilde X_{\tilde U_k}$ above $x$ there exists a point $\tilde y'$ of $\tilde X_{\tilde U_K}$ above $y'$ such that the conclusion of Lemma \[cuspdecomppullback\] holds (follows easily from a limit argument). By Lemma \[sppullback\] (ii), any section of $\pi_1(U_k,\xi_2)$ naturally induces a section of $\pi_1(U_K,\xi_1)$, but the converse is not true in general. Given any section $s:G_K\rightarrow\pi_1(U_K,\xi_1)$, let us define a homomorphism $\varphi_s:G_K\rightarrow\pi_1(U_k,\xi_2)$ by the composition $\varphi_s:=\operatorname{\textup{Sp}}_U\circ s$. $$\label{defofvarphisandramified} \begin{tikzpicture}[descr/.style={fill=white}, baseline=(current bounding box.center)] \matrix(m)[matrix of math nodes, row sep=4em, column sep=3em, text height=1.5ex, text depth=0.25ex] {1 & \pi_1(U_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}},\bar\xi_1) & \pi_1(U_K, \xi_1) & G_K & 1\\ 1 & \pi_1(U_{\bar k},\bar\xi_2) & \pi_1(U_k, \xi_2) & G_k & 1\\}; \path[->] (m-1-1) edge (m-1-2); \path[->,font=\scriptsize] (m-1-2) edge (m-1-3) edge node[left]{${\mkern 1.5mu\overline{\mkern-1.5mu\operatorname{\textup{Sp}}\mkern-1.5mu}\mkern 1.5mu}_U$} node[below,rotate=90]{$\sim$} (m-2-2); \path[->] (m-1-3) edge (m-1-4); \path[->>,font=\scriptsize] (m-1-3) edge node[left]{$\operatorname{\textup{Sp}}_U$} (m-2-3); \path[->,font=\scriptsize] (m-1-4) edge (m-1-5) edge[out=160,in=20] node[above]{$s$} (m-1-3) edge[out=200,in=60] node[left]{$\varphi_s$} (m-2-3); \path[->>,font=\scriptsize] (m-1-4) edge node[left]{} (m-2-4); \path[->] (m-2-1) edge (m-2-2); \path[->] (m-2-2) edge (m-2-3); \path[->] (m-2-3) edge (m-2-4); \path[->] (m-2-4) edge (m-2-5); \end{tikzpicture}$$ \[ramified\] We say the section $s$ is unramified if $\varphi_s(I_K)=1$. Otherwise we say $s$ is ramified. For an unramified section $s:G_K\rightarrow\pi_1(U_K,\xi_1)$ the map $\varphi_s$ factors through the projection $G_K\twoheadrightarrow G_k$ and $s$ induces a section $\bar s:G_k\rightarrow\pi_1(U_k,\xi_1)$. This induced section will be called the specialisation of $s$ and denoted $\bar s$. We now investigate under what conditions we may conclude that a given section $s : G_K \to \pi_1(U_K, \xi_1)$ has image contained in a decomposition group. \[unramifiedcuspidal\] Let $s:G_K\rightarrow\pi_1(U_K,\xi_1)$ be an unramified section, and suppose $\bar s(G_k)\subset D_{\tilde x}$ for some $x \in S(k)$ and some $\tilde x$ in $\tilde X_{\tilde U_k}$ above $x$. Then $s(G_K)\subset D_{\tilde y'}$ for a unique $\tilde y'$ in $\tilde X_{\tilde U_K}$ above the unique point $y' \in S(K)$ specialising to $x$. There exists a point $\tilde y'$ in $\tilde X_{\tilde U_K}$ above $y'$ such that the image of $D_{\tilde y'} \subset \pi_1(U_K, \xi_1)$ under $\operatorname{\textup{Sp}}_U$ is contained in $D_{\tilde x}$ (cf. proof of Corollary \[cuspidalevenifyisnotacusp\]), and this $\tilde y'$ is unique as follows from Lemma \[inertiaproperties\]. The pullback of $\bar s$ via the natural projection $G_K\twoheadrightarrow G_k$ gives rise to the section $s$, so $s(G_K)$ must be contained in the pullback of $D_{\tilde x}$ via $G_K\twoheadrightarrow G_k$, which is $D_{\tilde y'}$ by Lemma \[cuspdecomppullback\]. \[weightargument\] Assume that $k$ satisfies condition (ii) in Definition \[conditions\]. Let $s:G_K\rightarrow\pi_1(U_K,\xi_1)$ be a section, and denote $\varphi_s:=\operatorname{\textup{Sp}}_U\circ s$ as above. If $\varphi_s(I_K)$ is non-trivial then it is contained in the inertia group $I_{\tilde x}$ of a unique point $\tilde x$ of $\tilde X_{\tilde U_k}$ above a point of $S_k$. This follows from [@hoshimochizuki Lemma 1.6]. Indeed, if $\varphi_s(I_K)$ is non-trivial then, by commutativity of diagram (\[defofvarphisandramified\]), it is contained in $\pi_1(U_{\bar k},\bar\xi_2)$, and it is a procyclic subgroup of $\pi_1(U_{\bar k},\bar\xi_2)$ because $I_K\simeq\hat\operatorname{\mathbb{Z}}(1)$ is procyclic. Since $k$ satisfies condition (ii) of Definition \[conditions\], the image of $\varphi_s(I_K)$ under the composite $G_k$-homomorphism $$\begin{tikzcd}[column sep=small] \pi_1(U_{\bar k},\bar\xi_2) \arrow[twoheadrightarrow]{r}{} &\pi_1(X_{\bar k},\bar\xi_2) \arrow[twoheadrightarrow]{r}{} &\pi_1(X_{\bar k},\bar\xi_2)^{\operatorname{\textup{ab}}} \end{tikzcd}$$ is trivial. It then follows from loc. cit. that $\varphi_s(I_K)$ must be contained in an inertia subgroup of $\pi_1(U_{\bar k}, \bar\xi_2)$, which is unique by Lemma \[inertiaproperties\]. \[properimpliesunramified\] Under the hypotheses of Proposition \[weightargument\], if $S=\emptyset$, so that $U=X$ is proper over $\operatorname{\textup{Spec}}R$, then it follows from the arguments in the proof of Proposition \[weightargument\] that any section $s:G_K\rightarrow\pi_1(X_K,\xi_1)$ is necessarily unramified. \[ramified=cuspidal\] Assume that $k$ satisfies condition (ii) in Definition \[conditions\], and let $s:G_K\to\pi_1(U_K,\xi_1)$ be a section and $\varphi_s:=\operatorname{\textup{Sp}}_U\circ s$. If $s$ is ramified then $\varphi_s(G_K)\subset D_{\tilde x}$ for a unique $k$-rational point $x$ of $S_k$ and a unique $\tilde x$ in $\tilde X_{\tilde U_k}$ above $x$, and $s(G_K)\subset D_{\tilde y'}$ for a unique point $\tilde y'$ of $\tilde X_{\tilde U_K}$ above the unique $K$-rational point $y'$ of $S_K$ specialising to $x$. In particular, $s$ is cuspidal. If $\varphi_s(I_K)$ is non-trivial then, by Proposition \[weightargument\], it must be contained in a unique inertia group $I_{\tilde x}$ for some $x\in S_k$ and some $\tilde x\in\tilde X_{\tilde U_k}$ above $x$. Since $\varphi_s(G_K)$ normalises $\varphi_s(I_K)$, for some $\sigma\in G_K$ we have $\varphi_s(I_K)=\varphi_s(\sigma)\cdot \varphi_s(I_K)\cdot \varphi_s(\sigma)^{-1}\subseteq \varphi_s(\sigma)\cdot I_{\tilde x}\cdot \varphi_s(\sigma)^{-1}=I_{\varphi_s(\sigma)\cdot\tilde x}$. But $\varphi_s(I_K)$ is contained in a unique inertia group, so $\varphi_s(\sigma)\cdot\tilde x=\tilde x$ and $\varphi_s(G_K)$ fixes $\tilde x$, i.e. $\varphi_s(G_K)\subseteq D_{\tilde x}$. Moreover, $x$ is necessarily a $k$-rational point since, by commutativity of diagram (\[defofvarphisandramified\]), $\varphi_s(G_K)$ maps surjectively onto $G_k$. A similar argument to that used in the proof of Corollary 3.8 implies that $s(G_K)\subseteq D_{\tilde y'}$ for some $\tilde y'$ in $\tilde X_{\tilde U_K}$ above the unique point $y'$ of $S_K$ specialising to $x$. Moreover, such point $\tilde y'$ is unique by Lemma \[inertiaproperties\]. Let $\tilde S$ be as in Definition \[stilde\]. For a section $s:G_K\rightarrow\pi_1(X_K-\tilde S)$ (see Definition \[pi1stilde\]), we write $\varphi_s:=\operatorname{\textup{Sp}}\circ s$ for the composition of $s$ with the specialisation homomorphism $\operatorname{\textup{Sp}}: \pi_1(X_K - \tilde S)\twoheadrightarrow G_{X_k}$ of Theorem \[galspec\]. $$\label{varphistilde} \begin{tikzpicture}[descr/.style={fill=white}, baseline=(current bounding box.center)] \matrix(m)[matrix of math nodes, row sep=4em, column sep=3em, text height=1.5ex, text depth=0.25ex] {\pi_1(X_K-\tilde S) & G_K\\ G_{X_k} & G_k\\}; \path[->] (m-1-1) edge (m-1-2); \path[->>,font=\scriptsize] (m-1-1) edge node[left]{$\operatorname{\textup{Sp}}$} (m-2-1); \path[->,font=\scriptsize] (m-1-2) edge[out=150,in=30] node[above]{$s$} (m-1-1) edge[out=200,in=60] node[left]{$\varphi_s$} (m-2-1); \path[->>,font=\scriptsize] (m-1-2) edge node[left]{} (m-2-2); \path[->,font=\scriptsize] (m-2-1) edge node[above]{} (m-2-2); \end{tikzpicture}$$ We define the ramification and specialisation of $s$ analogously to Definition \[ramified\]. For an open subset $U_k \subset X_k$ with complement $S_k$, we will denote by $S_K$ the set of points of $\tilde S$ which specialise to $S_k$, $D$ the schematic closure of $S_K$ in $X$, and $U := X \setminus D$, thus $U_K=X_K\setminus S_K$. We will denote by $s_U : G_K\rightarrow\pi_1(U_K,\xi_1)$ the section of $\pi_1(U_K, \xi_1)$ naturally induced by $s$, and by $\varphi_U$ the composition $\varphi_U := \operatorname{\textup{Sp}}_U \circ s_U : G_K \to \pi_1(U_k, \xi_2)$ of $s_U$ with the specialisation homomorphism $\operatorname{\textup{Sp}}_U : \pi_1(U_K,\xi_1) \twoheadrightarrow \pi_1(U_k,\xi_2)$ of Lemma \[specialisationaffine\]. Since $s$ induces such a section $s_U : G_K \to \pi_1(U_K, \xi_1)$ for every open subset $U_K\subset X_K$ as above, it determines a compatible system of sections $\{ s_U \}$, parameterised by the open subsets $U\subset X$ as above. Conversely, such a system determines a section of $\pi_1(X_K-\tilde S)$. Similarly, since $s$ induces a homomorphism $\varphi_U : G_K \to \pi_1(U_k, \xi_2)$ for every open subset of $X_k$, it determines a compatible system of homomorphisms $\{ \varphi_U \}$ whose inverse limit $\varprojlim_{U_k \subset X_k \textup{ open}}\varphi_U$ is exactly the homomorphism $\varphi_s$ of diagram (\[varphistilde\]). \[toramifiedsubsets\] A section $s:G_K\to\pi_1(X_K-\tilde S)$ is ramified if and only if there is a non empty open subset $U_k\subset X_k$ for which the section $s_U:G_K\to\pi_1(U_K,\xi_1)$ induced as above by $s$ is ramified. Since $\varphi_s(I_K) = \varprojlim_{U_k \subset X_k \textup{ open}} \varphi_U(I_K)$ (with surjective transition maps), $\varphi_s(I_K)$ is trivial if and only if $\varphi_U(I_K)$ is trivial for every open subset $U_k \subset X_k$. \[ramifiedsubsets\] Assume that $k$ satisfies condition (ii) of Definition \[conditions\], and let $s:G_K\to\pi_1(X_K-\tilde S)$ be a section. Let $U'_k \subset U_k \subset X_k$ be any two non-empty open subsets, and let $\tilde X_{\tilde U_k}$, resp. $\tilde X_{\tilde U'_k}$ be the normalisation of $X_k$ in some universal pro-étale cover $\tilde U_k \to U_k$, resp. $\tilde U'_k \to U'_k$. Suppose that $s_U$ is ramified, with $\varphi_U(I_K)$ contained in the inertia subgroup $I_{\tilde x_U} \subset \pi_1(U_k,\xi_2)$ for some $x\in (X_k\setminus U_k)(k)$ and some $\tilde x_U$ in $\tilde X_{\tilde U_k}$ above $x$ (see Lemma \[ramified=cuspidal\] and its proof). Then $s_{U'}$ is ramified, and $\varphi_{U'}(I_K)$ is contained in the inertia subgroup $I_{\tilde x_{U'}}\subset\pi_1(U'_k,\xi_2)$ of some $\tilde x_{U'}$ in $\tilde X_{\tilde U'_k}$ above the same point $x\in (X_k\setminus U_k)(k)\subset (X_k\setminus U_k')(k)$. The image of $\varphi_{U'}(I_K)$ under the homomorphism $\pi_1(U'_k,\xi_2)\twoheadrightarrow\pi_1(U_k,\xi_2)$ coincides with $\varphi_U(I_K)$, which is nontrivial by assumption. Thus $\varphi_{U'}(I_K)$ must also be non-trivial, and therefore it is contained in an inertia subgroup $I_{\tilde z_{U'}} \subset \pi_1(U'_k,\xi_2)$ for some $z\in (X_k\setminus U_k')(k)$ and some $\tilde z_{U'}$ in $\tilde X_{\tilde U'_k}$ above $z$ (see Lemma \[ramified=cuspidal\] and its proof). Let $\tilde z_U$ be the image of $\tilde z_{U'}$ in $\tilde X_{\tilde U_k}$. Suppose $\tilde z_U \ne \tilde x_U$. If $z \in X_k\setminus U_k$ then, by functoriality, the image of $I_{\tilde z_{U'}}$ under $\pi_1(U'_k,\xi_2)\twoheadrightarrow\pi_1(U_k,\xi_2)$ is the inertia subgroup $I_{\tilde z_U} \subset \pi_1(U_k,\xi_2)$ at $\tilde z_U$, which intersects trivially with $I_{\tilde x_U}$ by Lemma \[inertiaproperties\]. Meanwhile, if $z \not\in X_k\setminus U_k$ the image of $I_{\tilde z_{U'}}$ in $\pi_1(U_k,\xi_2)$ is trivial. Both of these contradict compatibility of $\varphi_U$ and $\varphi_{U'}$, so we must have $\tilde z_U=\tilde x_U$ and $z=x$. The ramification of a section $s:G_K\to\pi_1(X_K-\tilde S)$ is therefore characterised by the ramification of the system of sections $s_U$ it induces. Let us now fix a universal pro-étale cover $\tilde X_{\tilde S} \to X_K - \tilde S$ (see Definition \[universalcovertilde\]), and denote by $k(X_k)^{\operatorname{\textup{sep}}}$ the separable closure of the function field $k(X_k)$ determined by the geometric point $\xi_2$. \[galptthunramified\] Let $s:G_K\to\pi_1(X_K-\tilde S)$ be an unramified section, and suppose its specialisation $\bar s : G_k \to G_{X_k}$ is geometric with $\bar s(G_k)\subset D_{\tilde x}$ for some $x\in X(k)$ and some extension $\tilde x$ of $x$ to $k(X_k)^{\operatorname{\textup{sep}}}$. Then $s(G_K)\subset D_{\tilde y}$ for a unique $\tilde y$ in $\tilde X_{\tilde S}$ above the unique ($K$-rational) point $y$ of $\tilde S$ specialising to $x$. For any open subset $U_k \subset X_k$, the section $s_U$ is unramified by Lemma \[toramifiedsubsets\]. Choose $U_k$ so that $x \not\in U_k$, and let $\tilde X_{\tilde U_K}$, respectively $\tilde X_{\tilde U_k}$ denote the normalisation of $X_K$, resp. $X_k$ in some universal pro-étale cover of $U_K$, resp. $U_k$. By compatibility of the homomorphisms $\varphi_U$, we have $\bar s_U(G_k) \subset D_{\tilde x_U}$ for some $\tilde x_U$ in $\tilde X_{U_k}$ above $x$. Hence, by Lemma \[unramifiedcuspidal\], $s_U(G_K) \subset D_{\tilde y_U}$ for some unique $\tilde y_U$ in $\tilde X_{U_K}$ above the unique point $y$ of $S_K$ specialising to $x$. This is also true for every open subset of $X_k$ contained in $U_k$, so taking inverse limits yields the required statement. \[galptthramified\] Assume that $k$ satisfies condition (ii) of Definition \[conditions\], and let $s:G_K\to\pi_1(X_K-\tilde S)$ be a section. If $s$ is ramified then $\varphi_s(G_K)\subset D_{\tilde x}$ for a unique valuation $\tilde x$ on $k(X_k)^{\operatorname{\textup{sep}}}$ extending a $k$-rational point $x$ of $X_k$, and $s(G_K)\subset D_{\tilde y}$ for a unique $\tilde y$ in $\tilde X_{\tilde S}$ above the unique $K$-rational point $y$ in $\tilde S$ specialising to $x$. Lemma \[toramifiedsubsets\] implies that, for some open subset $U_k \subset X_k$, the section $s_U$ is ramified. Let $\tilde X_{\tilde U_K}$, respectively $\tilde X_{\tilde U_k}$ denote the normalisation of $X_K$, resp. $X_k$ in some universal pro-étale cover of $U_K$, resp. $U_k$. By Lemma \[ramified=cuspidal\] we have $\varphi_U(G_K) \subset D_{\tilde x_U}$ for some $x \in (X_k\setminus U_k)(k)$ and some $\tilde x_U$ in $\tilde X_{\tilde U_k}$ above $x$, and $s_U(G_K) \subset D_{\tilde y_{U}}$ for some unique $\tilde y_U$ in $\tilde X_{\tilde U_K}$ above the unique $K$-rational point $y$ of $\tilde S$ specialising to $x$. By Proposition \[ramifiedsubsets\], this is true for every open subset $U'_k\subset X_k$ contained in $U_k$, thus taking inverse limits over the open subsets of $X_k$ yields the required statement (the uniqueness of $\tilde x$ follows from the same argument as in the proof of [@CONF Corollary 12.1.3]). Sections of absolute Galois groups of curves over function fields ================================================================= Specialisation of sections of absolute Galois groups {#specialisationofsections} ---------------------------------------------------- Let $k$ be a field of characteristic zero, and let $C$ a smooth, separated, connected curve over $k$ with function field $K$. Let $\operatorname{\mathcal{X}}\rightarrow C$ be a flat, proper, smooth relative curve, with generic fibre $X := \operatorname{\mathcal{X}}\times_C K$ which is a geometrically connected curve over $K$. For some $c\in C^{\operatorname{\textup{cl}}}$, let $K_c$ denote the completion of $K$ with respect to the valuation corresponding to $c$, and let $X_c:=X \times_K K_c$ be the base change of $X$ to $K_c$. Let $\xi$ be a geometric point of $X$ with value in its generic point. This determines a separable closure $K(X)^{\operatorname{\textup{sep}}}$ of the function field of $X$ and an algebraic closure ${\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}$ of $K$, as well as a geometric point $\bar\xi$ of $X_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}$. Likewise let $\xi_c$ be a geometric point of $X_c$ with value in its generic point, which determines an algebraic closure ${\mkern 1.5mu\overline{\mkern-1.5muK_c\mkern-1.5mu}\mkern 1.5mu}$ of $K_c$ and a geometric point $\bar\xi_c$ of $X_{{\mkern 1.5mu\overline{\mkern-1.5muK_c\mkern-1.5mu}\mkern 1.5mu}}$. Fix an embedding ${\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}\hookrightarrow {\mkern 1.5mu\overline{\mkern-1.5muK_c\mkern-1.5mu}\mkern 1.5mu}$. This determines a natural inclusion $i : X({\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}) \hookrightarrow X_c({\mkern 1.5mu\overline{\mkern-1.5muK_c\mkern-1.5mu}\mkern 1.5mu})$. Set-theoretically, an element $y : \operatorname{\textup{Spec}}{\mkern 1.5mu\overline{\mkern-1.5muK_c\mkern-1.5mu}\mkern 1.5mu} \to X_c$ of $X_c({\mkern 1.5mu\overline{\mkern-1.5muK_c\mkern-1.5mu}\mkern 1.5mu})$ maps the unique point of $\operatorname{\textup{Spec}}{\mkern 1.5mu\overline{\mkern-1.5muK_c\mkern-1.5mu}\mkern 1.5mu}$ to a closed point of $X_c$. This closed point will be called an algebraic point of $X_{K_c}$ if $y \in i(X({\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}))$; otherwise, it will be called a transcendental point. Let us denote by $X_c^{\operatorname{\textup{tr}}}$ the complement in $X_c^{\operatorname{\textup{cl}}}$ of the set of all algebraic points of $X_c^{\operatorname{\textup{cl}}}$. With the above notation, define the group $\pi_1(X_c^{\operatorname{\textup{tr}}})$ to be the inverse limit $$\pi_1(X_c^{\operatorname{\textup{tr}}}) := \varprojlim_{U \subset X \textup{ open}} \pi_1(U_c, \xi_c)$$ over all open subsets $U \subset X$, where $U_c$ denotes the base change $U \times_K K_c$. Recall there is an exact sequence $1 \to G_{X_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}} \to G_X \to G_K := \operatorname{\textup{Gal}}({\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}/K) \to 1$, where $G_X$ is the Galois group of $X$ with base point $\xi$ (cf. Lemma \[Gallim\]). \[pullbackgx\] We have a commutative diagram of exact sequences $$\begin{tikzpicture}[descr/.style={fill=white}, baseline=(current bounding box.center)] \matrix(m)[matrix of math nodes, row sep=3em, column sep=2.5em, text height=1.5ex, text depth=0.25ex] {1 & \pi_1(X_{{\mkern 1.5mu\overline{\mkern-1.5muK_c\mkern-1.5mu}\mkern 1.5mu}}^{\operatorname{\textup{tr}}}) & \pi_1(X_c^{\operatorname{\textup{tr}}}) & G_{K_c} & 1\\ 1 & G_{X_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}} & G_X & G_K & 1\\}; \path[->] (m-1-1) edge (m-1-2); \path[->] (m-1-2) edge (m-1-3) edge (m-2-2); \path[->] (m-1-3) edge (m-1-4) edge (m-2-3); \path[->] (m-1-4) edge (m-1-5) edge (m-2-4); \path[->] (m-2-1) edge (m-2-2); \path[->] (m-2-2) edge (m-2-3); \path[->] (m-2-3) edge (m-2-4); \path[->] (m-2-4) edge (m-2-5); \end{tikzpicture}$$ where $G_{K_c}=\operatorname{\textup{Gal}}({\mkern 1.5mu\overline{\mkern-1.5muK_c\mkern-1.5mu}\mkern 1.5mu}/K_c)$, $\pi_1(X_{{\mkern 1.5mu\overline{\mkern-1.5muK_c\mkern-1.5mu}\mkern 1.5mu}}^{\operatorname{\textup{tr}}})$ is defined so that the upper horizontal sequence is exact, the middle vertical map is defined up to conjugation, the left vertical map is an isomorphism, and the right square is cartesian. For each open subset $U \subset X$, functoriality of the fundamental group yields a diagram $$\label{pullbackdiagrampi1u} \begin{tikzpicture}[descr/.style={fill=white}, baseline=(current bounding box.center)] \matrix(m)[matrix of math nodes, row sep=3em, column sep=2.5em, text height=1.5ex, text depth=0.25ex] {1 & \pi_1(U_{{\mkern 1.5mu\overline{\mkern-1.5muK_c\mkern-1.5mu}\mkern 1.5mu}},\bar\xi_c) & \pi_1(U_c,\xi_c) & G_{K_c} & 1\\ 1 & \pi_1(U_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}},\bar\xi) & \pi_1(U,\xi) & G_K & 1\\}; \path[->] (m-1-1) edge (m-1-2); \path[->] (m-1-2) edge (m-1-3) edge (m-2-2); \path[->] (m-1-3) edge (m-1-4) edge (m-2-3); \path[->] (m-1-4) edge (m-1-5) edge (m-2-4); \path[->] (m-2-1) edge (m-2-2); \path[->] (m-2-2) edge (m-2-3); \path[->] (m-2-3) edge (m-2-4); \path[->] (m-2-4) edge (m-2-5); \end{tikzpicture}$$ where the rows are the fundamental exact sequences, the middle vertical map is defined up to conjugation, the left vertical map is an isomorphism (see [@Luminy Théorème 1.6]), and the right square is cartesian (follows as in the proof of Lemma \[sppullback\]). The statement then follows by taking the projective limit of these diagrams over the open subsets of $X$. Denote by $\operatorname{\mathcal{X}}_c:=\operatorname{\mathcal{X}}\times_C k(c)$ the fibre of $\operatorname{\mathcal{X}}$ above $c$. \[liftingtoalgpts\] For each closed point $x$ of $\operatorname{\mathcal{X}}_c^{\operatorname{\textup{cl}}}$, there exists an algebraic point $y$ of $X_c$ specialising to $x$ whose residue field is the unique unramified extension $L$ of $K_c$ whose valuation ring $\operatorname{\mathcal{O}}_{L}$ has residue field $k(x)$. Let us write $X_{c,L} := X_c \times_{K_c} L$ and $\operatorname{\mathcal{X}}_{c,k(x)} := \operatorname{\mathcal{X}}_c \times_{k(c)} k(x)$. Let $x'$ be a $k(x)$-rational point of $\operatorname{\mathcal{X}}_{c,k(x)}$ which maps to $x$ under the projection $\operatorname{\mathcal{X}}_{c,k(x)} \to \operatorname{\mathcal{X}}_c$, and let ${\mathfrak{m}}_{L}$ denote the maximal ideal of $\operatorname{\mathcal{O}}_{L}$. The set of $L$-rational points of $X_{c,L}$ which specialise to $x'$ is in bijection with ${\mathfrak{m}}_{L}$ [@Liu Proposition 10.1.40]. Let $F$ be a finite extension of $K$ whose completion at a place above $c$ is $L$. Then an element of ${\mathfrak{m}}_{L} \cap F$ corresponds to an $L$-rational algebraic point $y'$ of $X_{c,L}$ which specialises to $x'$. The image $y$ of $y'$ under the projection $X_{c,L} \to X_c$ is then an $L$-rational algebraic point of $X_c$ ($k(y)=L$) which specialises to $x$. \[stildekc\] For each closed point $x$ of $\operatorname{\mathcal{X}}_c^{\operatorname{\textup{cl}}}$, fix a choice of algebraic point $y\in X_c^{\operatorname{\textup{cl}}}$ which specialises to $x$ and whose residue field is the unique unramified extension of $K_c$ whose valuation ring has residue field $k(x)$ (such a point exists by Lemma \[liftingtoalgpts\]). We define $\tilde S_c$ to be the set of these chosen algebraic points $y\in X_c^{\operatorname{\textup{cl}}}$. Thus, $\tilde S_c$ is a subset of $X_c^{\operatorname{\textup{cl}}}$ which consists of algebraic points and which is in bijection with $\operatorname{\mathcal{X}}_c^{\operatorname{\textup{cl}}}$. We denote by $\tilde S_{c,{\mkern 1.5mu\overline{\mkern-1.5muK_c\mkern-1.5mu}\mkern 1.5mu}}$ the set of points of $X_c\times _{K_c}{\mkern 1.5mu\overline{\mkern-1.5muK_c\mkern-1.5mu}\mkern 1.5mu}$ which map to points in $\tilde S_c$. Thus $\tilde S_{c,{\mkern 1.5mu\overline{\mkern-1.5muK_c\mkern-1.5mu}\mkern 1.5mu}}$ is a subset of $X_{{\mkern 1.5mu\overline{\mkern-1.5muK_c\mkern-1.5mu}\mkern 1.5mu}}^{\operatorname{\textup{cl}}}$ in bijection with $\operatorname{\mathcal{X}}_{{\mkern 1.5mu\overline{\mkern-1.5muk(c)\mkern-1.5mu}\mkern 1.5mu}}^{\operatorname{\textup{cl}}}$. Let $\xi'_c$ be a geometric point of $\operatorname{\mathcal{X}}_c$ with value in its generic point, which determines an algebraic closure ${\mkern 1.5mu\overline{\mkern-1.5muk(c)\mkern-1.5mu}\mkern 1.5mu}$ of $k(c)$ and a geometric point $\bar\xi'_c$ of $\operatorname{\mathcal{X}}_{{\mkern 1.5mu\overline{\mkern-1.5muk(c)\mkern-1.5mu}\mkern 1.5mu}}$. Let $1 \to G_{\operatorname{\mathcal{X}}_{{\mkern 1.5mu\overline{\mkern-1.5muk(c)\mkern-1.5mu}\mkern 1.5mu}}} \to G_{\operatorname{\mathcal{X}}_c} \to G_{k(c)} \to 1$ be the exact sequence of the absolute Galois group of $\operatorname{\mathcal{X}}_c$ with base point $\xi'_c$. \[galspeckc\] With $\tilde S_c$ and $\tilde S_{c,{\mkern 1.5mu\overline{\mkern-1.5muK_c\mkern-1.5mu}\mkern 1.5mu}}$ as in Definition \[stildekc\], there exists a surjective homomorphism $\operatorname{\textup{Sp}}: \pi_1(X_c - \tilde S_c) \twoheadrightarrow G_{\operatorname{\mathcal{X}}_c}$, an isomorphism ${\mkern 1.5mu\overline{\mkern-1.5mu\operatorname{\textup{Sp}}\mkern-1.5mu}\mkern 1.5mu} : \pi_1(X_{{\mkern 1.5mu\overline{\mkern-1.5muK_c\mkern-1.5mu}\mkern 1.5mu}} - \tilde S_{c,{\mkern 1.5mu\overline{\mkern-1.5muK_c\mkern-1.5mu}\mkern 1.5mu}}) \simeq G_{\operatorname{\mathcal{X}}_{{\mkern 1.5mu\overline{\mkern-1.5muk(c)\mkern-1.5mu}\mkern 1.5mu}}}$ and a commutative diagram of exact sequences: $$\begin{tikzpicture}[descr/.style={fill=white}, baseline=(current bounding box.center)] \matrix(m)[matrix of math nodes, row sep=3em, column sep=2.5em, text height=1.5ex, text depth=0.25ex] {1 & \pi_1(X_{{\mkern 1.5mu\overline{\mkern-1.5muK_c\mkern-1.5mu}\mkern 1.5mu}}^{\operatorname{\textup{tr}}}) & \pi_1(X_c^{\operatorname{\textup{tr}}}) & G_{K_c} & 1\\ 1 & \pi_1(X_{{\mkern 1.5mu\overline{\mkern-1.5muK_c\mkern-1.5mu}\mkern 1.5mu}}-\tilde S_{c,{\mkern 1.5mu\overline{\mkern-1.5muK_c\mkern-1.5mu}\mkern 1.5mu}}) & \pi_1(X_c-\tilde S_c) & G_{K_c} & 1\\ 1 & G_{\operatorname{\mathcal{X}}_{{\mkern 1.5mu\overline{\mkern-1.5muk(c)\mkern-1.5mu}\mkern 1.5mu}}} & G_{\operatorname{\mathcal{X}}_c} & G_{k(c)} & 1\\}; \path[->] (m-1-1) edge (m-1-2); \path[->] (m-1-2) edge (m-1-3); \path[->] (m-1-3) edge (m-1-4); \path[->] (m-1-4) edge (m-1-5); \path[->>] (m-1-2) edge (m-2-2); \path[->>] (m-1-3) edge (m-2-3); \path[-] (m-1-4) edge[double distance=2pt] (m-2-4); \path[->] (m-2-1) edge (m-2-2); \path[->] (m-2-2) edge (m-2-3); \path[->] (m-2-3) edge (m-2-4); \path[->] (m-2-4) edge (m-2-5); \path[->,font=\scriptsize] (m-2-2) edge node[left]{${\mkern 1.5mu\overline{\mkern-1.5mu\operatorname{\textup{Sp}}\mkern-1.5mu}\mkern 1.5mu}$} node[below,rotate=90]{$\sim$} (m-3-2); \path[->>,font=\scriptsize] (m-2-3) edge node[left]{$\operatorname{\textup{Sp}}$} (m-3-3); \path[->>,font=\scriptsize] (m-2-4) edge node[left]{} (m-3-4); \path[->] (m-3-1) edge (m-3-2); \path[->] (m-3-2) edge (m-3-3); \path[->] (m-3-3) edge (m-3-4); \path[->] (m-3-4) edge (m-3-5); \end{tikzpicture}$$ where the map $\operatorname{\textup{Sp}}$ is defined up to conjugation. The existence of ${\mkern 1.5mu\overline{\mkern-1.5mu\operatorname{\textup{Sp}}\mkern-1.5mu}\mkern 1.5mu}$ and $\operatorname{\textup{Sp}}$ follows from Theorem \[galspec\]. Since we have chosen $\tilde S_c$ to consist of algebraic points, $\pi_1(X_c-\tilde S_c)$, respectively $\pi_1(X_{{\mkern 1.5mu\overline{\mkern-1.5muK_c\mkern-1.5mu}\mkern 1.5mu}} - \tilde S_{c,{\mkern 1.5mu\overline{\mkern-1.5muK_c\mkern-1.5mu}\mkern 1.5mu}})$ is naturally a quotient of $\pi_1(X_c^{\operatorname{\textup{tr}}})$, resp. $\pi_1(X_{{\mkern 1.5mu\overline{\mkern-1.5muK_c\mkern-1.5mu}\mkern 1.5mu}}^{\operatorname{\textup{tr}}})$. Thus we have the required homomorphisms in the diagram, and it is clearly commutative. Let $s:G_K\rightarrow G_X$ be a section, and let $s_c:G_{K_c}\rightarrow\pi_1(X_c^{\operatorname{\textup{tr}}})$ be the section of $\pi_1(X_c^{\operatorname{\textup{tr}}})$ induced from $s$ (cf. Lemma \[pullbackgx\]). This naturally induces a section $\tilde s_c:G_{K_c}\rightarrow\pi_1(X_c-\tilde S_c)$ of the projection $\pi_1(X_c-\tilde S_c)\twoheadrightarrow G_{K_c}$. \[stildeptth\] With the notation of the above paragraph, assume further that $X$ is hyperbolic. Let $\tilde X_{c, \tilde S_c} \to X_c - \tilde S_c$ be a universal pro-étale cover (Definition \[universalcovertilde\]), and denote by $k(\operatorname{\mathcal{X}}_c)^{\operatorname{\textup{sep}}}$ the separable closure of the function field of $\operatorname{\mathcal{X}}_c$ determined by the geometric point $\xi'_c$. 1. Suppose $\tilde s_c$ is unramified and induces a section $\bar s_c:G_{k(c)}\rightarrow G_{\operatorname{\mathcal{X}}_c}$. If $\bar s_c$ is geometric with $\bar s_c(G_{k(c)})\subset D_{\tilde x}$ for some $x\in\operatorname{\mathcal{X}}_c(k(c))$ and some extension $\tilde x$ of $x$ to $k(\operatorname{\mathcal{X}}_c)^{\operatorname{\textup{sep}}}$, then $\tilde s_c(G_{K_c})\subset D_{\tilde y}$ for some unique $\tilde y$ in $\tilde X_{c, \tilde S_c}$ above the unique ($K_c$-rational) point $y$ of $\tilde S_c$ specialising to $x$. 2. Assume further that $k$ satisfies condition (ii) of Definition \[conditions\]. Suppose $\tilde s_c$ is ramified, and denote $\varphi_s:=\operatorname{\textup{Sp}}\circ\tilde s_c$. Then $\varphi_s(G_{K_c})\subset D_{\tilde x}$ for a unique valuation on $k(\operatorname{\mathcal{X}}_c)^{\operatorname{\textup{sep}}}$ extending a $k(c)$-rational point $x$ of $\operatorname{\mathcal{X}}_c$, and $\tilde s_c(G_{K_c})\subset D_{\tilde y}$ for some unique $\tilde y$ in $\tilde X_{c, \tilde S_c}$ above the unique ($K_c$-rational) point $y$ of $\tilde S_c$ specialising to $x$. Part (i) follows from Lemma \[galptthunramified\], and (ii) follows from Lemma \[galptthramified\]. Étale abelian sections {#etaleabeliansections} ---------------------- We use the notation of §\[specialisationofsections\], and hereafter we assume that $X$ is hyperbolic and that $X(K)\ne\emptyset$. For each $c \in C^{\operatorname{\textup{cl}}}$, there is a commutative diagram $$\label{specialisationofset} \begin{tikzpicture}[descr/.style={fill=white}, baseline=(current bounding box.center)] \matrix(m)[matrix of math nodes, row sep=3em, column sep=2.5em, text height=1.5ex, text depth=0.25ex] {1 & \pi_1(X_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}, \bar\xi) & \pi_1(X, \xi) & G_K & 1\\ 1 & \pi_1(X_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}, \bar\xi) & \pi_1(\operatorname{\mathcal{X}}, \xi) & \pi_1(C, \xi) & 1\\ 1 & \pi_1(\operatorname{\mathcal{X}}_{{\mkern 1.5mu\overline{\mkern-1.5muk(c)\mkern-1.5mu}\mkern 1.5mu}}, \bar\xi'_c) & \pi_1(\operatorname{\mathcal{X}}_c, \xi'_c) & G_{k(c)} & 1\\}; \path[->] (m-1-1) edge (m-1-2); \path[->] (m-1-2) edge (m-1-3); \path[->] (m-1-3) edge (m-1-4); \path[->, font = \scriptsize] (m-1-4) edge (m-1-5); \path[-] (m-1-2) edge[double distance = 2pt] (m-2-2); \path[->>] (m-1-3) edge (m-2-3); \path[->>] (m-1-4) edge (m-2-4); \path[->] (m-2-1) edge (m-2-2); \path[->] (m-2-2) edge (m-2-3); \path[->] (m-2-3) edge (m-2-4); \path[->, font = \scriptsize] (m-2-4) edge (m-2-5); \path[->] (m-3-1) edge (m-3-2); \path[->, font = \scriptsize] (m-3-2) edge (m-3-3) edge node[below, rotate = 90]{$\sim$} (m-2-2); \path[->] (m-3-3) edge (m-3-4); \path[right hook->] (m-3-3) edge (m-2-3); \path[->, font = \scriptsize] (m-3-4) edge (m-3-5); \path[right hook->] (m-3-4) edge (m-2-4); \end{tikzpicture}$$ where the lower vertical homomorphisms are defined up to conjugation, the middle and right of those maps are injective and the left one is an isomorphism. This follows from functoriality of the fundamental group and the fundamental exact sequences for $X$ and $\operatorname{\mathcal{X}}_c$. Exactness of the middle row follows from [@SGA Exposé XIII, Proposition 4.3]. Let us fix a section $s:G_K\rightarrow G_X$ of $G_X$, and denote by $s^{\operatorname{\textup{\'et}}}:G_K\rightarrow\pi_1(X,\xi)$ the section of the étale fundamental group of $X$ induced by $s$. By Lemma \[pullbackgx\], $s$ pulls back to a section $s_c:G_{K_c}\rightarrow\pi_1(X_c^{\operatorname{\textup{tr}}})$, which in turn induces a section $s_c^{\operatorname{\textup{\'et}}}:G_{K_c}\rightarrow\pi_1(X_c,\xi_c)$. By Proposition \[weightargument\] (see also Remark \[properimpliesunramified\]), $s_c^{\operatorname{\textup{\'et}}}$ specialises to a section $\bar s_c^{\operatorname{\textup{\'et}}}:G_{k(c)}\rightarrow\pi_1(\operatorname{\mathcal{X}}_c,\xi'_c)$. By the following, we may also consider $\bar s_c^{\operatorname{\textup{\'et}}}$ the specialisation of $s^{\operatorname{\textup{\'et}}}$. The section $s^{\operatorname{\textup{\'et}}}$ extends to a section $s_C^{\operatorname{\textup{\'et}}} : \pi_1(C, \xi) \to \pi_1(\operatorname{\mathcal{X}}, \xi)$ of the projection $\pi_1(\operatorname{\mathcal{X}}, \xi) \twoheadrightarrow \pi_1(C, \xi)$ which restricts to the section $\bar s_c^{\operatorname{\textup{\'et}}}$ for each $c \in C^{\operatorname{\textup{cl}}}$. The kernel of the homomorphism $G_K\twoheadrightarrow\pi_1(C, \xi)$ is the inertia group $I_C$ normally generated by the inertia subgroups associated to the closed points of $C$. Since $s_c^{\operatorname{\textup{\'et}}}$ is unramified for every $c \in C^{\operatorname{\textup{cl}}}$, the image of each of these inertia groups under the composition $\operatorname{\textup{Sp}}_X \circ s_c^{\operatorname{\textup{\'et}}} : G_{K_c} \to \pi_1(\operatorname{\mathcal{X}}_c, \xi'_c)$ of $s_c^{\operatorname{\textup{\'et}}}$ with the specialisation homomorphism $\operatorname{\textup{Sp}}_X : \pi_1(X_c, \xi_c) \twoheadrightarrow \pi_1(\operatorname{\mathcal{X}}_c, \xi'_c)$ is trivial, hence the image of $I_C$ under the composite $\begin{tikzcd}[column sep=small]G_C \arrow{r}{s^{\operatorname{\textup{\'et}}}} & \pi_1(X, \xi) \arrow[twoheadrightarrow]{r}{} & \pi_1(\operatorname{\mathcal{X}}, \xi)\end{tikzcd}$ is trivial. Thus $s^{\operatorname{\textup{\'et}}}$ extends to a section $s_C^{\operatorname{\textup{\'et}}}:\pi_1(C, \xi) \rightarrow \pi_1(\operatorname{\mathcal{X}}, \xi)$, which must restrict to $\bar s_c^{\operatorname{\textup{\'et}}}:G_{k(c)}\rightarrow\pi_1(\operatorname{\mathcal{X}}_c, \xi'_c)$. Let $\operatorname{\mathcal{J}}:=\textup{Pic}^0_{\operatorname{\mathcal{X}}/C}\rightarrow C$ denote the relative Jacobian of $\operatorname{\mathcal{X}}$, and $J:=\operatorname{\mathcal{J}}_K$ the Jacobian of $X$. For each closed point $c\in C^{\operatorname{\textup{cl}}}$, let $J_c:=J_{K_c}$ denote the Jacobian of $X_c$ and $\operatorname{\mathcal{J}}_c:=\operatorname{\mathcal{J}}_{k(c)}$ that of $\operatorname{\mathcal{X}}_c$. The above sections $s^{\operatorname{\textup{\'et}}}$, $s_c^{\operatorname{\textup{\'et}}}$ and $\bar s_c^{\operatorname{\textup{\'et}}}$ induce étale abelian sections $s^{\operatorname{\textup{ab}}}$, $s_c^{\operatorname{\textup{ab}}}$ and $\bar s_c^{\operatorname{\textup{ab}}}$ respectively, while diagram (\[specialisationofset\]) induces a commutative diagram of exact sequences of geometrically abelian fundamental groups $$\label{specialisationofsab} \begin{tikzpicture}[descr/.style={fill=white}, baseline=(current bounding box.center)] \matrix(m)[matrix of math nodes, row sep=3em, column sep=2.5em, text height=1.5ex, text depth=0.25ex] {1 & \pi_1(X_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}, \bar\xi)^{\operatorname{\textup{ab}}} & \pi_1(X, \xi)^{(\operatorname{\textup{ab}})} & G_K = G_C & 1\\ 1 & \pi_1(X_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}, \bar\xi)^{\operatorname{\textup{ab}}} & \pi_1(\operatorname{\mathcal{X}}, \xi)^{(\operatorname{\textup{ab}})} & \pi_1(C, \xi) & 1\\ 1 & \pi_1(\operatorname{\mathcal{X}}_{{\mkern 1.5mu\overline{\mkern-1.5muk(c)\mkern-1.5mu}\mkern 1.5mu}}, \bar\xi'_c)^{\operatorname{\textup{ab}}} & \pi_1(\operatorname{\mathcal{X}}_c, \xi'_c)^{(\operatorname{\textup{ab}})} & G_{k(c)} & 1\\}; \path[->] (m-1-1) edge (m-1-2); \path[->] (m-1-2) edge (m-1-3); \path[->] (m-1-3) edge (m-1-4); \path[->, font = \scriptsize] (m-1-4) edge (m-1-5) edge[out = 165, in = 15] node[above]{$s^{\operatorname{\textup{ab}}}$} (m-1-3); \path[-] (m-1-2) edge[double distance = 2pt] (m-2-2); \path[->>] (m-1-3) edge (m-2-3); \path[->>] (m-1-4) edge (m-2-4); \path[->] (m-2-1) edge (m-2-2); \path[->] (m-2-2) edge (m-2-3); \path[->] (m-2-3) edge (m-2-4); \path[->, font = \scriptsize] (m-2-4) edge (m-2-5) edge[out = 165, in = 15] node[above]{$s_C^{\operatorname{\textup{ab}}}$} (m-2-3); \path[->] (m-3-1) edge (m-3-2); \path[->, font = \scriptsize] (m-3-2) edge (m-3-3) edge node[below, rotate = 90]{$\sim$} (m-2-2); \path[->] (m-3-3) edge (m-3-4); \path[right hook->] (m-3-3) edge (m-2-3); \path[->, font = \scriptsize] (m-3-4) edge (m-3-5) edge[out = 160, in = 15] node[above]{$\bar s_c^{\operatorname{\textup{ab}}}$} (m-3-3); \path[right hook->] (m-3-4) edge (m-2-4); \end{tikzpicture}$$ where the middle horizontal row is obtained as the push-out of the middle horizontal row in diagram (9) by the projection $\pi_1(X_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}},\bar\xi)\twoheadrightarrow \pi_1(X_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}},\bar \xi)^{\operatorname{\textup{ab}}}$, and $s_C^{\operatorname{\textup{ab}}} : \pi_1(C, \xi) \to \pi_1(\operatorname{\mathcal{X}}, \xi)^{(\operatorname{\textup{ab}})}$ is induced by $s_C^{\operatorname{\textup{\'et}}}$. Since $X(K) \ne \emptyset$ by assumption, the étale abelian sections $s^{\operatorname{\textup{ab}}}$, $s_c^{\operatorname{\textup{ab}}}$, $s_C^{\operatorname{\textup{ab}}}$ and $\bar s_c^{\operatorname{\textup{ab}}}$ correspond to elements of the cohomology groups $H^1(G_K,TJ)$, $H^1(G_{K_c},TJ_c)$, $H^1(\pi_1(C, \xi), TJ)$ and $H^1(G_{k(c)},T\operatorname{\mathcal{J}}_c)$ respectively, which are related by the following restriction and inflation maps: $$\begin{tikzpicture}[descr/.style={fill=white}] \matrix(m)[matrix of math nodes, row sep=3em, column sep=2.5em, text height=2ex, text depth=0.25ex] {H^1(G_K,TJ) & H^1(G_{K_c},TJ_c)\\ H^1(\pi_1(C, \xi), TJ) & H^1(G_{k(c)},T\operatorname{\mathcal{J}}_c)\\}; \path[->,font=\scriptsize] (m-1-1) edge node[above]{$\operatorname{\textup{res}}_c$} (m-1-2); \path[->, font = \scriptsize] (m-2-1) edge node[left]{$\inf_C$} (m-1-1) edge node[above]{$\operatorname{\textup{res}}_{C, c}$} (m-2-2); \path[->, font=\scriptsize] (m-2-2) edge node[right]{$\inf_c$} (m-1-2); \end{tikzpicture}$$ \[sabresinf\] With the above notation, we have the following. 1. $\operatorname{\textup{res}}_c(s^{\operatorname{\textup{ab}}}) = \inf_c(\bar s_c^{\operatorname{\textup{ab}}}) = s_c^{\operatorname{\textup{ab}}}$; 2. $\inf_C(s_C^{\operatorname{\textup{ab}}}) = s^{\operatorname{\textup{ab}}}$ and $\operatorname{\textup{res}}_{C, c}(s_C^{\operatorname{\textup{ab}}}) = \bar s_c^{\operatorname{\textup{ab}}}$. Part (i) follows from the fact that $s^{\operatorname{\textup{\'et}}}$ and $\bar s^{\operatorname{\textup{\'et}}}_c$ both pull back to $s^{\operatorname{\textup{\'et}}}_c$ - see diagrams (\[pullbackdiagrampi1u\]) and (\[defofvarphisandramified\]), considering the case when $U=X$ is projective. Part (ii) follows from diagram (\[specialisationofsab\]). See also [@saidiSCOFF Lemma 3.4]. The image of $s^{\operatorname{\textup{ab}}}$ under the diagonal map $$\prod_{c\in C^{\operatorname{\textup{cl}}}}\operatorname{\textup{res}}_c:H^1(G_K,TJ)\longrightarrow\prod_{c\in C^{\operatorname{\textup{cl}}}}H^1(G_{K_c},TJ_c)$$ is therefore the family $(s_c^{\operatorname{\textup{ab}}})_{c\in C^{\operatorname{\textup{cl}}}}$. This diagonal map fits into the following commutative diagram of Kummer exact sequences (see Lemma \[kummerexactseq\]). $$\begin{tikzpicture} \matrix(m)[matrix of math nodes, row sep=3em, column sep=2em, text height=2ex, text depth=0.25ex] {0 & \widehat{J(K)} & H^1(G_K,TJ) & TH^1(G_K,J) & 0\\ 0 & \displaystyle\prod_{c\in C^{\operatorname{\textup{cl}}}}\widehat{J_c(K_c)} & \displaystyle\prod_{c\in C^{\operatorname{\textup{cl}}}}H^1(G_{K_c},TJ_c) & \displaystyle\prod_{c\in C^{\operatorname{\textup{cl}}}}TH^1(G_{K_c},J_c) & 0\\}; \path[->] (m-1-1) edge (m-1-2); \path[->] (m-1-2) edge (m-1-3) edge (m-2-2); \path[->] (m-1-3) edge (m-1-4) edge (m-2-3); \path[font=\scriptsize,->] (m-1-4) edge (m-1-5) edge node[left]{} (m-2-4); \path[->] (m-2-1) edge (m-2-2); \path[->] (m-2-2) edge (m-2-3); \path[->] (m-2-3) edge (m-2-4); \path[->] (m-2-4) edge (m-2-5); \end{tikzpicture}$$ Note that the kernel of the right vertical map is the Tate module $T\Sha(\operatorname{\mathcal{J}})$ of the Shafarevich-Tate group $\Sha(\operatorname{\mathcal{J}})$ (Definition \[sha\]). Commutativity of this diagram immediately implies the following. \[tshatrivial\] Suppose that, for every $c\in C^{\operatorname{\textup{cl}}}$, the section $s_c^{\operatorname{\textup{ab}}}\in H^1(G_{K_c},TJ_c)$ is contained in $\widehat{J_c(K_c)}$. Then if $T\Sha(\operatorname{\mathcal{J}})=0$, the section $s^{\operatorname{\textup{ab}}}\in H^1(G_K,TJ)$ is contained in $\widehat{J(K)}$. \[sabjkhat\] Assume that $T\Sha(\operatorname{\mathcal{J}})=0$, and that $k$ strongly satisfies conditions (i), (ii) and (iii)(a) of Definition \[conditions\]. Then we have the following. 1. For each $c \in C^{\operatorname{\textup{cl}}}$, $\bar s_c^{\operatorname{\textup{ab}}}$ is in the image of the injective map $\operatorname{\mathcal{X}}_c(k(c)) \hookrightarrow H^1(G_{k(c)},T\operatorname{\mathcal{J}}_c)$, and $s_c^{\operatorname{\textup{ab}}}$ is in the image of $X_c(K_c)\to H^1(G_{K_c},TJ_c)$. 2. $s^{\operatorname{\textup{ab}}}$ is contained in $\widehat{J(K)}$. Let $\tilde s_c : G_{K_c} \to \pi_1(X_c - \tilde S_c)$ denote the section of $\pi_1(X_c - \tilde S_c)$ induced by $s_c$, and write $\varphi_{s_c} := \operatorname{\textup{Sp}}\circ \tilde s_c$. Let $\tilde X_{c, \tilde S_c} \to X_c - \tilde S_c$ be a universal pro-étale cover, and recall $k(\operatorname{\mathcal{X}}_c)^{\operatorname{\textup{sep}}}$ the separable closure of the function field of $\operatorname{\mathcal{X}}_c$ determined by $\xi'_c$. By Theorem \[stildeptth\], for every $c\in C^{\operatorname{\textup{cl}}}$ we have $\varphi_{s_c}(G_{K_c}) \subset D_{\tilde x}$ for a unique $x \in \operatorname{\mathcal{X}}_c(k(c))$ and some unique extension $\tilde x$ of $x$ to $k(\operatorname{\mathcal{X}}_c)^{\operatorname{\textup{sep}}}$, and $\tilde s_c(G_{K_c}) \subset D_{\tilde y}$ for some unique $\tilde y$ in $\tilde X_{c, \tilde S_c}$ above the unique ($K_c$-rational) point $y$ of $\tilde S_c$ specialising to $x$. This implies that $s_c^{\operatorname{\textup{\'et}}}(G_{K_c})=D_{\tilde y'}$, for some $\tilde y'$ above $y$ in a universal pro-étale cover $\tilde X_c \to X_c$, and $\bar s_c^{\operatorname{\textup{\'et}}}(G_{k(c)}) = D_{\tilde x'}$ for some $\tilde x'$ above $x$ in a universal pro-étale cover $\tilde\operatorname{\mathcal{X}}_c \to \operatorname{\mathcal{X}}_c$. This means that $s_c^{\operatorname{\textup{\'et}}}$, respectively $\bar s_c^{\operatorname{\textup{\'et}}}$ arises from $y \in X_c(K_c)$, resp. $x \in \operatorname{\mathcal{X}}_c(k(c))$ by functoriality of the fundamental group, which proves (i) (see the discussion before Lemma \[kummerexactseq\]). The map $\operatorname{\mathcal{X}}_c(k(c))\to H^1(G_{k(c)},T\operatorname{\mathcal{J}}_c)$ is injective by condition (iii)(a) of Definition \[conditions\]. Since the map $X_c(K_c)\to H^1(G_{K_c},TJ_c)$ factors through the inclusion $\widehat{J_c(K_c)} \hookrightarrow H^1(G_{K_c}, TJ_c)$ (see sequence (\[sabptth\])), part (i) implies in particular that $s_c^{\operatorname{\textup{ab}}}$ is contained in $\widehat{J_c(K_c)}$, and since this is true for every $c\in C^{\operatorname{\textup{cl}}}$, Lemma \[tshatrivial\] implies that $s^{\operatorname{\textup{ab}}}\in\widehat{J(K)}$, which proves (ii). Proof of the Main Theorems ========================== Proof of Theorem A ------------------ Let $k$ be a field of characteristic zero that strongly satisfies the conditions of Definition \[conditions\]. Let $C$ be a smooth, separated, connected curve over $k$ with function field $K$. Let $\operatorname{\mathcal{X}}\rightarrow C$ be a flat, proper, smooth relative curve whose generic fibre $X:=\operatorname{\mathcal{X}}\times_C K$ is geometrically connected and hyperbolic, with $X(K)\ne\emptyset$. Let $\operatorname{\mathcal{J}}:=\operatorname{\textup{Pic}}^0_{\operatorname{\mathcal{X}}/C}$ denote the relative Jacobian of $\operatorname{\mathcal{X}}$, and $J := \operatorname{\mathcal{J}}_K$ the Jacobian of $X$. For a closed point $c \in C^{\operatorname{\textup{cl}}}$, denote by $\operatorname{\mathcal{J}}_c := \operatorname{\mathcal{J}}_{k(c)}$ the Jacobian of $\operatorname{\mathcal{X}}_c$. Assume that $T\Sha(\operatorname{\mathcal{J}})=0$. We show that the birational section conjecture holds for $X$ (see Definitions \[geometricgaloissections\] and \[bscholds\] and Remark \[bscuniqueness\]). Let $s:G_K\rightarrow G_X$ be a section. Under our assumptions, the étale abelian section $s^{\operatorname{\textup{ab}}}$ induced by $s$ is contained in $\widehat{J(K)}$, by Proposition \[sabjkhat\] (ii). \[sabinjk\] The homomorphism $J(K)\rightarrow\widehat{J(K)}$ is injective and $s^{\operatorname{\textup{ab}}}$ is contained in $J(K)$. There exist $c_1,c_2\in C^{\operatorname{\textup{cl}}}$ such that the natural specialisation homomorphism $J(K)\rightarrow\operatorname{\mathcal{J}}_{c_1}(k(c_1))\times\operatorname{\mathcal{J}}_{c_2}(k(c_2))$ is injective [@poonenvoloch Proposition 2.4]. Let $\ell $ be a finite extension of $k$ that contains $k(c_1)$ and $k(c_2)$. Then there is an injective homomorphism $\operatorname{\mathcal{J}}_{c_i}(k(c_i))\hookrightarrow\operatorname{\mathcal{J}}_{c_i}(\ell )$ for each $i=1,2$, hence an injective homomorphism $\operatorname{\mathcal{J}}_{c_1}(k(c_1))\times\operatorname{\mathcal{J}}_{c_2}(k(c_2))\hookrightarrow\operatorname{\mathcal{J}}_{c_1}(\ell)\times\operatorname{\mathcal{J}}_{c_2}(\ell)\simeq(\operatorname{\mathcal{J}}_{c_1}\times\operatorname{\mathcal{J}}_{c_2})(\ell)$. For ease of notation, let us write $\operatorname{\mathcal{J}}_{1,2}(\ell):= (\operatorname{\mathcal{J}}_{c_1}\times\operatorname{\mathcal{J}}_{c_2})(\ell)$. We have a commutative diagram of exact sequences: $$\begin{tikzpicture}[descr/.style={fill=white}] \matrix(m)[matrix of math nodes, row sep=3em, column sep=3em, text height=2.5ex, text depth=0.25ex] {0 & J(K) & \operatorname{\mathcal{J}}_{1,2}(\ell) & H & 0\\ 0 & \widehat{J(K)} & \widehat{\operatorname{\mathcal{J}}_{1,2}(\ell)} & \widehat{H} & 0\\}; \path[->] (m-1-1) edge (m-1-2); \path[->] (m-1-2) edge (m-1-3) edge (m-2-2); \path[->,font=\scriptsize] (m-1-3) edge (m-1-4) edge node[left]{$\phi$} (m-2-3); \path[->] (m-1-4) edge (m-1-5) edge (m-2-4); \path[->] (m-2-1) edge (m-2-2); \path[->,font=\scriptsize] (m-2-2) edge node[above]{$\psi$} (m-2-3); \path[->] (m-2-3) edge (m-2-4); \path[->] (m-2-4) edge (m-2-5); \end{tikzpicture}$$ where $H$ is defined so that the upper horizontal sequence is exact. Exactness of the lower sequence follows easily from condition (iii) (b) of Definition \[conditions\], while condition (iii) (a) implies that the middle and right vertical maps are injective. Therefore the left vertical map is also injective, and the equality $J(K)=\phi(\operatorname{\mathcal{J}}_{1,2}(\ell))\cap\psi(\widehat{J(K)})$ holds inside $\widehat{\operatorname{\mathcal{J}}_{1,2}(\ell)}$. For each $c_i$, $i=1,2$, the section $s$ induces an element $\bar s_{c_i}^{\operatorname{\textup{ab}}}\in H^1(G_{k(c_i)},T\operatorname{\mathcal{J}}_{c_i})$, which is contained in the image of the map $\operatorname{\mathcal{X}}_{c_i}(k(c_i)) \to H^1(G_{k(c_i)}, T\operatorname{\mathcal{J}}_{c_i})$ by Proposition \[sabjkhat\] (i). This map is injective by condition (iii)(a) of Definition \[conditions\], so we may consider $\bar s_{c_i}^{\operatorname{\textup{ab}}}$ to be contained in $\operatorname{\mathcal{X}}_{c_i}(k(c_i))$. Then $\bar s_{c_i}^{\operatorname{\textup{ab}}}$ is contained in $\operatorname{\mathcal{J}}_{c_i}(\ell)$ for each $i=1,2$, due to injectivity of the maps $\operatorname{\mathcal{X}}_{c_i}(k(c_i))\hookrightarrow\operatorname{\mathcal{J}}_{c_i}(k(c_i))\hookrightarrow\operatorname{\mathcal{J}}_{c_i}(\ell)$. Thus $(\bar s_{c_1}^{\operatorname{\textup{ab}}},\bar s_{c_2}^{\operatorname{\textup{ab}}})$ is contained in $\operatorname{\mathcal{J}}_{c_1}(\ell)\times\operatorname{\mathcal{J}}_{c_2}(\ell)$, hence in $\phi(\operatorname{\mathcal{J}}_{1,2}(\ell))$. By Lemma \[sabresinf\], the image of $s^{\operatorname{\textup{ab}}}\in\widehat{J(K)}$ in $\widehat{\operatorname{\mathcal{J}}_{1,2}(\ell)}$ under $\psi$ is the element $(\bar s_{c_1}^{\operatorname{\textup{ab}}},\bar s_{c_2}^{\operatorname{\textup{ab}}})$, and since this lies in $\phi(\operatorname{\mathcal{J}}_{1,2}(\ell))$ we have $s^{\operatorname{\textup{ab}}}\in\phi(\operatorname{\mathcal{J}}_{1,2}(\ell))\cap\psi(\widehat{J(K)})=J(K)$. Fix a $K$-rational point $x_0\in X(K)=\operatorname{\mathcal{X}}(C)$ (non-empty by assumption), and let $\iota:\operatorname{\mathcal{X}}\rightarrow\operatorname{\mathcal{J}}$ denote the closed immersion mapping $x_0$ to the zero section of $\operatorname{\mathcal{J}}$. \[sabinxk\] $s^{\operatorname{\textup{ab}}}$ is contained in $X(K)$. Since $s^{\operatorname{\textup{ab}}}$ is contained in $J(K) = \operatorname{\mathcal{J}}(C)$, it may be regarded as a morphism $s^{\operatorname{\textup{ab}}} : C \to \operatorname{\mathcal{J}}$. By Lemma \[sabresinf\], the pullback of this morphism to $\operatorname{\textup{Spec}}k(c)$ is precisely $\bar s_c^{\operatorname{\textup{ab}}}$, considered an element of $\operatorname{\mathcal{J}}_c(k(c)) \subset H^1(G_{k(c)}, T\operatorname{\mathcal{J}}_c)$. But $\bar s_c^{\operatorname{\textup{ab}}}$ is contained in $\operatorname{\mathcal{X}}_c(k(c))$ by Proposition \[sabjkhat\] (i), hence the morphism $\bar s_c^{\operatorname{\textup{ab}}} : \operatorname{\textup{Spec}}k(c) \to \operatorname{\mathcal{J}}_c$ must factor through $\operatorname{\mathcal{X}}_c$, where $\operatorname{\mathcal{X}}_c$ is considered a closed subscheme of $\operatorname{\mathcal{J}}_c$ via $\iota$. Thus, for each $c \in C^{\operatorname{\textup{cl}}}$, the image of $c$ under the morphism $s^{\operatorname{\textup{ab}}} : C \to \operatorname{\mathcal{J}}$ is a closed point of $\operatorname{\mathcal{X}}_c$. This implies that $s^{\operatorname{\textup{ab}}} : C \to \operatorname{\mathcal{J}}$ factors through $\iota:\operatorname{\mathcal{X}}\rightarrow\operatorname{\mathcal{J}}$, thus $s^{\operatorname{\textup{ab}}}$ is contained in the subset $\iota(X(K))\subseteq J(K)$. Let $z$ be the point in $X(K)$ such that $\iota(z)=s^{\operatorname{\textup{ab}}}$, and for each $c \in C^{\operatorname{\textup{cl}}}$ let $\bar z_c$ denote its specialisation to $\operatorname{\mathcal{X}}_c$. Let $\bar x_c\in\operatorname{\mathcal{X}}_c(k(c))$ be the point associated to $\bar s_c^{\operatorname{\textup{ab}}}$ by Proposition \[sabjkhat\] (i). \[zc=xc\] $\bar z_c=\bar x_c$ in $\operatorname{\mathcal{X}}_c(k(c))$ for all $c\in C^{\operatorname{\textup{cl}}}$. Lemma \[sabresinf\] implies that, for each $c \in C^{\operatorname{\textup{cl}}}$, $\bar s_c^{\operatorname{\textup{ab}}}$ is the image of both $\bar x_c$ and $\bar z_c$ under the map $\operatorname{\mathcal{X}}_c(k(c)) \to H^1(G_{k(c)},T\operatorname{\mathcal{J}}_c)$. This map is injective by condition (iii)(a) of Definition \[conditions\], hence $\bar z_c=\bar x_c$. \[sisgeometric\] $s$ is geometric. By the “limit argument” of Tamagawa [@tamagawa Proposition 2.8 (iv)], it suffices to prove that for any open subgroup $H \subset G_X$ which contains $s(G_K)$, if $Y \to X$ denotes the corresponding finite morphism with $Y$ smooth, we have $Y(K) \ne \emptyset$. By construction, we have $G_Y = H$, and $s$ defines a section $s_Y : G_K \to G_Y$ of $G_Y$. For $c\in C^{\operatorname{\textup{cl}}}$, let $K_c$ be the completion of $K$ at $c$, and write $X_c:=X\times_K K_c$ and $Y_c:=Y\times_K K_c$. By Lemma \[pullbackgx\], the section $s$ pulls back to a section $s_c:G_{K_c}\rightarrow\pi_1(X_c^{\operatorname{\textup{tr}}})$, and likewise $s_Y$ pulls back to a section $s_{Y_c}:G_{K_c}\rightarrow\pi_1(Y_c^{\operatorname{\textup{tr}}})$. We have the following commutative diagram. $$\begin{tikzpicture}[descr/.style = {fill = white}, baseline = (current bounding box.center)] \matrix(m)[matrix of math nodes, row sep=3em, column sep=3em, text height=2ex, text depth=0.25ex] {\pi_1(Y_c^{\operatorname{\textup{tr}}}) & \pi_1(X_c^{\operatorname{\textup{tr}}}) & G_{K_c}\\ G_Y & G_X & G_K\\}; \path[right hook->] (m-1-1) edge (m-1-2); \path[->] (m-1-1) edge (m-2-1); \path[->] (m-1-2) edge (m-1-3); \path[->] (m-1-2) edge (m-2-2); \path[->,font=\scriptsize] (m-1-3) edge[out=165,in=15] node[above left, near end]{$s_c$} (m-1-2) edge[out=145,in=25] node[below]{$s_{Y_c}$} (m-1-1); \path[->] (m-1-3) edge (m-2-3); \path[right hook->] (m-2-1) edge (m-2-2); \path[->] (m-2-2) edge (m-2-3); \path[->,font=\scriptsize] (m-2-3) edge[out=165,in=15] node[above left, near end]{$s$} (m-2-2) edge[out=150,in=25] node[above left,near end]{$s_Y$} (m-2-1); \end{tikzpicture}$$ Let $\operatorname{\mathcal{Y}}$ be the normalisation of $\operatorname{\mathcal{X}}$ in the function field of $Y$, and for each $c \in C^{\operatorname{\textup{cl}}}$ let $\operatorname{\mathcal{Y}}_c$ denote the closed fibre of $\operatorname{\mathcal{Y}}$ at $c$. After possibly removing finitely many points from $C$, we may assume that $\operatorname{\mathcal{Y}}$ is smooth over $C$. Indeed, the closed fibres $\operatorname{\mathcal{Y}}_c$ are smooth except possibly for finitely many closed points $c\in C^{\operatorname{\textup{cl}}}$ [@Liu Proposition 10.1.21]. So, if necessary, we may replace $C$ by the largest open sub-scheme $C'\subset C$ such that $\operatorname{\mathcal{Y}}_c$ is smooth for every $c\in(C')^{\operatorname{\textup{cl}}}$, and $\operatorname{\mathcal{Y}}\to \operatorname{\mathcal{X}}$ by the induced map of fibre products $\operatorname{\mathcal{Y}}\times_C C' \to \operatorname{\mathcal{X}}\times_C C'$. So we assume that the fibres $\operatorname{\mathcal{Y}}_c$ are smooth for all $c\in C^{\operatorname{\textup{cl}}}$. For each closed point $\bar x_c \in \operatorname{\mathcal{X}}_c^{\operatorname{\textup{cl}}}$, respectively $\bar y_c \in \operatorname{\mathcal{Y}}_c^{\operatorname{\textup{cl}}}$, choose an algebraic point $x_c \in X_c$, resp. $y_c \in Y_c$ specialising to $\bar x_c$, resp. $\bar y_c$ whose residue field is the unique unramified extension of $K_c$ whose valuation ring has residue field $k(\bar x_c)$, resp. $k(\bar y_c)$. Let $\tilde S_c$, respectively $\tilde T_c$ denote the set of these chosen algebraic points of $X_c$, resp. $Y_c$ (see Definition \[stildekc\]). The groups $\pi_1(Y_c-\tilde T_c)$ and $\pi_1(X_c-\tilde S_c)$ are naturally quotients of $\pi_1(Y_c^{\operatorname{\textup{tr}}})$ and $\pi_1(X_c^{\operatorname{\textup{tr}}})$ respectively, hence $s_c$ naturally induces a section $\tilde s_c:G_{K_c}\rightarrow\pi_1(X_c-\tilde S_c)$, and likewise $s_{Y_c}$ induces a section $\tilde s_{Y_c}:G_{K_c}\rightarrow\pi_1(Y_c-\tilde T_c)$. By Theorem \[galspeckc\], there exist specialisation homomorphisms $\operatorname{\textup{Sp}}_X : \pi_1(X_c - \tilde S_c) \twoheadrightarrow G_{\operatorname{\mathcal{X}}_c}$ and $\operatorname{\textup{Sp}}_Y : \pi_1(Y_c - \tilde T_c) \twoheadrightarrow G_{\operatorname{\mathcal{Y}}_c}$ and a commutative diagram $$\begin{tikzpicture}[descr/.style = {fill = white}, baseline = (current bounding box.center)] \matrix(m)[matrix of math nodes, column sep=3em, text height=2ex, text depth=0.25ex] {\pi_1(Y_c^{\operatorname{\textup{tr}}}) & \pi_1(X_c^{\operatorname{\textup{tr}}}) & G_{K_c}\\[2em] \pi_1(Y_c-\tilde T_c) & & \\[0em] & & G_{K_c}\\[0em] & \pi_1(X_c-\tilde S_c) & \\[3.5em] G_{\operatorname{\mathcal{Y}}_c} & G_{\operatorname{\mathcal{X}}_c} & G_{k(c)}\\}; \path[right hook->] (m-1-1) edge (m-1-2); \path[->>] (m-1-1) edge (m-2-1); \path[->] (m-1-2) edge (m-1-3); \path[->>] (m-1-2) edge (m-4-2); \path[->,font=\scriptsize] (m-1-3) edge[out=165,in=15] node[above left, near end]{$s_c$} (m-1-2) edge[out=150,in=25] node[below]{$s_{Y_c}$} (m-1-1); \path[-] (m-1-3) edge[double distance=2pt] (m-3-3); \path[font=\scriptsize,->] (m-2-1) edge[out=0, in=165] (m-3-3); \path[font=\scriptsize,->>] (m-2-1) edge node[left]{$\operatorname{\textup{Sp}}_Y$} (m-5-1); \path[->] (m-4-2) edge[out=5, in=205] (m-3-3); \path[font=\scriptsize,->>] (m-4-2) edge node[left]{$\operatorname{\textup{Sp}}_X$} (m-5-2); \path[->,font=\scriptsize] (m-3-3) edge[out=187,in=17] node[above left, near end]{$\tilde s_c$} (m-4-2) edge[out=150,in=10] node[above left,near end]{$\tilde s_{Y_c}$} (m-2-1) edge[out=220,in=70] node[below right]{$\varphi_X$} (m-5-2) edge[out=220,in=70] node[below right,near end]{$\varphi_Y$} (m-5-1); \path[font=\scriptsize,->>] (m-3-3) edge node[left]{} (m-5-3); \path[font=\scriptsize,right hook->] (m-5-1) edge node[above]{} (m-5-2); \path[->] (m-5-2) edge (m-5-3); \end{tikzpicture}$$ where we denote $\varphi_X:=\operatorname{\textup{Sp}}_X\circ\tilde s_c$ and $\varphi_Y:=\operatorname{\textup{Sp}}_Y\circ\tilde s_{Y_c}$. By Theorem \[stildeptth\], we have $\varphi_Y(G_{K_c}) \subset D_{\tilde y_c} \subset G_{\operatorname{\mathcal{Y}}_c}$ for a unique valuation $\tilde y_c$ on $k(\operatorname{\mathcal{X}}_c)^{\operatorname{\textup{sep}}}$ extending a unique $k(c)$-rational point $\bar y_c \in \operatorname{\mathcal{Y}}_c(k(c))$. By commutativity of the above diagram, this implies that $\varphi_X(G_{K_c}) \subset D_{\tilde y_c} \subset G_{\operatorname{\mathcal{X}}_c}$ for the same valuation $\tilde y_c$ on $k(\operatorname{\mathcal{X}}_c)^{\operatorname{\textup{sep}}}$, whose restriction to $k(\operatorname{\mathcal{X}}_c)$ corresponds to the image $\bar x'_c$ of $\bar y_c$ in $\operatorname{\mathcal{X}}_c$. Thus we have found, for every $c\in C^{\operatorname{\textup{cl}}}$, unique $k(c)$-rational points $\bar y_c\in\operatorname{\mathcal{Y}}_c(k(c))$ and $\bar x'_c\in\operatorname{\mathcal{X}}_c(k(c))$ such that $\bar y_c$ maps to $\bar x'_c$ via $\operatorname{\mathcal{Y}}_c\rightarrow\operatorname{\mathcal{X}}_c$. Moreover, $\bar x'_c$ must be the same as the point $\bar x_c$ associated to $\bar s_c^{\operatorname{\textup{ab}}}$ (see Lemma \[zc=xc\] and the paragraph before it). Recall the section $s^{\operatorname{\textup{ab}}}$ is associated to a $K$-rational point $z$ (see Lemma \[sabinxk\] and the paragraph after it). View $z\in X(K) = \operatorname{\mathcal{X}}(C)$ as a section $z:C\rightarrow\operatorname{\mathcal{X}}$, and denote by $\operatorname{\mathcal{Y}}_z$ the pullback of the image $z(C)$ via the map $\operatorname{\mathcal{Y}}\rightarrow\operatorname{\mathcal{X}}$. Then $\operatorname{\mathcal{Y}}_z\rightarrow z(C)$ is a finite morphism, and we can assume, after possibly shrinking $C$, that $\operatorname{\mathcal{Y}}_z$ is smooth. Since $z$ specialises to $\bar x_c\in\operatorname{\mathcal{X}}_c(k(c))$ (Lemma \[zc=xc\]), $\bar x_c\in z(C)$ and therefore $\bar y_c\in\operatorname{\mathcal{Y}}_z(k(c))$ for every $c\in C^{\operatorname{\textup{cl}}}$. Then condition (v) of Definition \[conditions\] implies that $\operatorname{\mathcal{Y}}_z(K)\ne\emptyset$. Thus $\operatorname{\mathcal{Y}}_z(K)\subseteq\operatorname{\mathcal{Y}}(K)=Y(K)\ne\emptyset$, which completes the proof of Proposition \[sisgeometric\]. Thus $s(G_k)$ is contained in a decomposition group associated to a $K$-rational point $x\in X(K)$, which is unique (cf. Remark \[bscuniqueness\]). This concludes the proof of Theorem A. Proof of Theorem B ------------------ In this section we explain how Theorem B is deduced from Theorem A. Let $k$ be a field of characteristic zero that strongly satisfies the conditions of Definition \[conditions\]. Let $C$ be a smooth, separated, connected curve over $k$ with function field $K$. For any finite extension $L$ of $K$, let $C^L$ denote the normalisation of $C$ in $L$, and for any flat, proper, smooth relative curve $\operatorname{\mathcal{Y}}\to C^L$, let $\operatorname{\mathcal{J}}_{\!\operatorname{\mathcal{Y}}} := \operatorname{\textup{Pic}}^0_{\operatorname{\mathcal{Y}}/C^L}$ denote the relative Jacobian of $\operatorname{\mathcal{Y}}$. Assume that for any such finite extension $L$ and any such relative curve $\operatorname{\mathcal{Y}}$ we have $T\Sha(\operatorname{\mathcal{J}}_{\!\operatorname{\mathcal{Y}}})=0$. We will show that for any finite extension $L$ of $K$ and any smooth, projective, geometrically connected (not necessarily hyperbolic) curve $X$ over $L$, the birational section conjecture holds for $X$. With the above notation and hypotheses, let $s:G_L\rightarrow G_X$ be a section of $G_X$. Then $s$ is geometric. By the Hurwitz formula, we may choose an open subgroup $H\subset G_X$ containing $s(G_L)$ such that, denoting by $Y\rightarrow X$ the corresponding finite morphism with $Y$ smooth, $Y$ is hyperbolic. We have an isomorphism $H\simeq G_Y$, and $s$ naturally defines a section $s_Y:G_L\rightarrow G_Y$ of the natural projection $G_Y\twoheadrightarrow G_L$. Let $L'\|L$ be a finite extension such that $Y(L')\ne\emptyset$, and let $M\|L$ be a Galois extension of $L$ containing $L'$. Then $Y_M(M)\ne\emptyset$, and $s_Y$ restricts to a section $s_{Y_M}:G_M\rightarrow G_{Y_M}$ of the absolute Galois group of $Y_M$. $$\begin{tikzpicture}[descr/.style = {fill = white}, baseline = (current bounding box.center)] \matrix(m)[matrix of math nodes, row sep=3em, column sep=3em, text height=1.5ex, text depth=0.25ex] {1 & G_{Y_{{\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu}}} & G_{Y_M} & G_M & 1\\ 1 & G_{Y_{{\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu}}} & G_Y & G_L & 1\\}; \path[->] (m-1-1) edge (m-1-2); \path[->,font=\scriptsize] (m-1-2) edge (m-1-3); \path[-] (m-1-2) edge[double distance=2pt] (m-2-2); \path[->] (m-1-3) edge (m-1-4); \path[right hook->,font=\scriptsize] (m-1-3) edge (m-2-3); \path[->,font=\scriptsize] (m-1-4) edge (m-1-5) edge[out=160,in=20] node[above]{$s_{Y_M}$} (m-1-3); \path[right hook->,font=\scriptsize] (m-1-4) edge (m-2-4); \path[->] (m-2-1) edge (m-2-2); \path[->] (m-2-2) edge (m-2-3); \path[->] (m-2-3) edge (m-2-4); \path[->,font=\scriptsize] (m-2-4) edge (m-2-5) edge[out=160,in=20] node[above]{$s_Y$} (m-2-3); \end{tikzpicture}$$ Let $C^M$ denote the normalisation of $C$ in $M$, and let $\operatorname{\mathcal{Y}}\to C^M$ be a flat and proper model of $Y_M$ over $C^M$. As in the proof of Proposition \[sisgeometric\], after possibly removing finitely many closed points from $C^M$ we may assume that the closed fibres $\operatorname{\mathcal{Y}}_c:=\operatorname{\mathcal{Y}}\times_{C^M} k(c)$ of $\operatorname{\mathcal{Y}}$ are smooth for all $c\in (C^M)^{\operatorname{\textup{cl}}}$. Then $\operatorname{\mathcal{Y}}\to C^M$ is a flat, proper, smooth relative curve whose generic fibre $Y_M$ is hyperbolic and has at least one $M$-rational point. Theorem A then implies that $s_{Y_M}(G_M)$ is contained in a decomposition subgroup $D^M_{\tilde y} \subset G_{Y_M}$ for a unique $M$-rational point $y$ of $Y_M$ and some extension $\tilde y$ of $y$ to $k(X)^{\operatorname{\textup{sep}}}$. Note we use a superscript $M$ to emphasise that $D^M_{\tilde y}$ is a subgroup of $G_{Y_M}$. Since $M\|L$ is a Galois extension, $G_M$ is a normal subgroup of $G_L$, hence $s_Y(G_L)$ normalises $s_{Y_M}(G_M)$ in $G_Y$. Therefore, for any $\sigma \in G_L$, $s_{Y_M}(G_M)$ is also contained in $s_Y(\sigma)^{-1}D^M_{\tilde y} s_Y(\sigma) = D^M_{s_Y(\sigma)\cdot\tilde y}$, which implies that $\tilde y=s_Y(\sigma)\cdot\tilde y$ [@CONF Corollary 12.1.3]. Thus, $s_Y(G_L)$ normalises $D^M_{\tilde y}$ in $G_Y$, so it is contained in the normaliser of $D^M_{\tilde y}$ in $G_Y$, which is precisely $D_{\tilde y} \subset G_Y$. This implies that $s(G_L)$ is contained in the decomposition subgroup $D_{\tilde y} \subset G_X$ of the same valuation $\tilde y$ of $k(X)^{\operatorname{\textup{sep}}}$, whose restriction to $k(X)$ corresponds to the image $x$ of $y$ in $X$. The point $x$ is then necessarily $L$-rational, since $D_{\tilde y}$ must map surjectively to $G_L$. This concludes the proof of Theorem B. Proof of Theorem C ------------------ In this section we prove Theorem C. Assume the [BSC]{} holds over all number fields. We prove that the [BSC]{} holds over all finitely generated fields over $\operatorname{\mathbb{Q}}$ of transcendence degree $n \ge 1$. We argue by induction on $n$ and assume that the [BSC]{} holds over all finitely generated fields over $\operatorname{\mathbb{Q}}$ of transcendence degree $<n$. Let $K$ be a finitely generated field over $\operatorname{\mathbb{Q}}$ of transcendence degree $n$ and $k\subset K$ a subfield which is algebraically closed in $K$ over which $K$ has transcendence degree $1$. We show the [BSC]{} holds over $K$. It is well-known that in order to prove that the [BSC]{} holds over $K$ it suffices to prove that the [BSC]{} holds for the projective line over $K$ (cf. [@saidiBASC Lemma 2.1]). Thus, we will show the following. With $K$ and $k$ as above, let $X=\Bbb P^1_K$, and let $s:G_K\to G_X$ a section of the projection $G_X\twoheadrightarrow G_K$. Then $s$ is geometric. Let ${\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}$ and ${\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu}$ be the algebraic closures of $K$ and $k$, respectively, induced by the geometric point $\xi$ defining $G_X$. We claim that there exists an open subgroup $H \subset G_X$ containing $s(G_K)$ such that, denoting by $Y \to X$ the corresponding finite morphism with $Y$ smooth, $Y$ is hyperbolic and isotrivial, meaning that $Y_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}$ descends to a smooth curve $Y_{\bar k}$ over $\bar k$. Indeed, let $U\subset \Bbb P^1_k$ be an open subset, $U_{\bar k}=U\times _k \bar k$, $U_K=U\times _k K$, and $U_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}=U\times _k {\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}$. We have a natural commutative diagram of exact sequences $$\CD 1@>>> \pi_1(U_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}},\bar \xi) @>>> \pi_1(U_K,\xi) @>>> G_{K} @>>> 1\\ @. @VVV @VVV @VVV @. \\ 1@>>> \pi_1(U_{\bar k},\bar \xi) @>>> \pi_1(U_k,\xi)@>>> G_{k}@>>> 1\\ \endCD$$ where the left vertical map is an isomorphism. Let $\tilde \Delta$ be a characteristic open subgroup of $\pi_1(U_{\bar k},\bar \xi)$ corresponding to a finite morphism $Y_{\bar k}\to \Bbb P^1_{\bar k}$ with $Y_{\bar k}$ smooth and hyperbolic. Such a subgroup exists by the Riemann-Hurwitz formula and the fact that $\pi_1(U_{\bar k},\bar \xi)$ is finitely generated. We write $\Delta$ for the corresponding subgroup of $\pi_1(U_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}},\bar \xi)$ and $Y_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}\to \Bbb P^1_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}$ the corresponding finite morphism with $Y_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}$ smooth. The section $s$ induces a section $s_U : G_K \to \pi_1(U_{K},\xi)$ of the projection $\pi_1(U_K,\xi) \twoheadrightarrow G_{K}$. Let $\widetilde H=\Delta \cdot s_U(G_K)$ and $H$ the inverse image of $\widetilde H$ in $G_X$. Then $H$ and the corresponding finite morphism $Y\to X$ are as claimed above. The section $s$ induces a section $s_Y : G_K \to G_Y = H$ of the natural projection $G_Y\twoheadrightarrow G_K$ with $s_Y(G_K) = s(G_K)$, and one easily verifies that the section $s$ is geometric if (and only if) the section $s_Y$ is geometric. Let $C$ be a separated, smooth and connected curve over $k$ with function field $k(C)=K$, and $\operatorname{\mathcal{Y}}\to C$ a flat, smooth and proper relative $C$-curve with generic fibre $\operatorname{\mathcal{Y}}_K=Y$. Without loss of generality, we can assume that $Y(K)\ne \emptyset$ (cf. proof of Theorem B). Let $\operatorname{\mathcal{J}}$ be the relative jacobian of $\operatorname{\mathcal{Y}}$. The Shafarevich-Tate group $\Sha (\operatorname{\mathcal{J}})$ is finite by [@saiditamagawa Theorem 4.1] since $\operatorname{\mathcal{J}}_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}$, being the jacobian of $Y_{{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}}$, is isotrivial (i.e. descends to an abelian variety over $\bar k$). Moreover, $k$ strongly satisfies the conditions in Definition \[conditions\], since it is finitely generated and by the above induction assumption. 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--- abstract: 'The decay of Taylor-Couette turbulence, i.e the flow between two coaxial and independently rotating cylinders, is numerically studied by instantaneously stopping the forcing from an initially statistically stationary flow field at a Reynolds number of $Re=3.5\times 10^4$. The effect of wall-friction is analysed by comparing three separate cases, in which the cylinders are either suddenly made no-slip or stress-free. Different life stages are observed during the decay. In the first stage, the decay is dominated by large-scale rolls. Counterintuitively, when these rolls fade away, if the flow inertia is small a redistribution of energy occurs, the energy of the azimuthal velocity behaves non-monotonically: first decreasing by almost two orders of magnitude, and then increasing during the redistribution. The second stage is dominated by non-normal transient growth of perturbations in the axial (spanwise) direction. Once this mechanism is exhausted, the flow enters the final life stage, viscous decay, which is dominated by wall-friction. We show that this stage can be modeled by a one-dimensional heat equation, and that self-similar velocity profiles collapse onto the theoretical solution.' author: - 'Rodolfo Ostilla-Mónico' - Xiaojue Zhu - Vamsi Spandan - Roberto Verzicco - Detlef Lohse bibliography: - 'literatur.bib' title: 'Life stages of wall-bounded decay of Taylor-Couette turbulence' --- Turbulence is a classic example of a non-equilibrium phenomenon: it requires a constant energy injection as energy is constantly dissipated by viscous effects. For statistically stationary turbulence, the classical picture by Richardson [@ric20] and Kolmogorov [@kol41a; @kol41b; @kol41c] is that energy is injected at the larger scales, cascades down to smaller scales and is dissipated by viscosity. In this picture, only the large scales are dependent on the boundary conditions, the forcing and the geometry of the flow. The intermediate and small scales are assumed to be homogeneous and isotropic, and thus the energy dissipation mechanisms are postulated to be universal and self-similar. Once the forcing is turned off, turbulence decays as energy is no longer injected into the system, but it is still dissipated. Decaying turbulence has been employed not only to study unforced systems, but also the mechanisms for energy dissipation in statistically stationary turbulence, including the dissipation anomaly, a cornerstone of every theory of turbulence [@vas15]. Historically, studies have focused on the decay of statistically homogeneous isotropic turbulence (HIT), performed both experimentally (using grid-induced turbulence) [@smi93a; @sta99] or numerically (using a triply periodic box forced randomly) [@geo09]. Some theoretical power-laws for the decay of vorticity and kinetic energy can be obtained both from dimensional analysis, or from a Navier-Stokes equation-based mean field theory [@loh94a], and are in good agreement with available data [@smi93a; @sta99; @ll87]. However, the dependence of decay on initial conditions [@lav07; @val07; @hur11; @tei09], and the degree of self-similarity of the decay have been debated in the literature [@saf67; @geo09; @bif03; @eyi00]. Real systems however are very far from homogeneous and isotropic. Most flows in technology and industry are wall-bounded and thus anisotropic. A considerable part of the energy dissipation occurs next to the walls. And even in geo- and astrophysical flows, where walls might be absent, strong anisotropies still exist as in accretion disks or mantle convection. Studies of turbulence decay outside HIT are more limited [@tou02]. Taylor-Couette (TC) flow, i.e. the turbulent flow between two concentric and independently rotating cylinders [@gro16], is an ideal system to study the decay of wall-bounded turbulence because it is a closed system and the cylindrical geometry allows for experiments with relatively small end-effects and more modest sizes unlike channel or pipe flow. While TC had already been employed to study decay of puffs at low Reynolds numbers [@bor10], or turbulence decay due to linear stabilization [@ost14b], the experiment by Verschoof et al. [@ver16] focused on the fully unstable turbulent regime, at Reynolds numbers of $Re\sim\mathcal{O}(10^6)$. The cylinders were stopped, and the turbulence was allowed to decay. The decay did not follow a pure power law like HIT, but instead decayed faster due to the viscous drag applied by the walls. Nonetheless, the decay was found to be self-similar. However, the considerable inertia of the massive metal cylinders causes the braking to take a substantial amount of time (approximately 3000 eddy turnover times), during which the decay could not be measured. The experiment also has secondary flows due to residual cooling effects, which become important at later times and cause deviations from the models in Ref. [@ver16]. In this manuscript, we numerically simulate Taylor-Couette turbulence, analogous to the experiments of Ref. [@ver16], but now we instantaneously stop the cylinders (which is of course possible in numerical simulations) to better understand the initial stages of the decay, and run the simulation until the fully viscous final stage dominates the system dynamics. In addition, we can remove the effect of wall-friction by making the walls stress-free, and thus we can clearly separate the role of the walls from other decay mechanisms in the flow. We find that the decay regime observed and discussed in Ref. [@ver16] corresponds to an intermediate stage of decay. We do not find the proposed self-similar regime until the last life-stage which can be described as a quenching problem. Prior to this last viscosity-dominated stage, two earlier life stages of decay are observed, which are dominated by linear instabilities and non-normal transient growth, respectively. We simulate the incompressible Navier-Stokes equations using a second-order energy-conserving centered finite difference scheme, which treats the viscous terms semi-implicitly [@ver96; @poe15]. The initial starting field corresponds to a TC system driven by pure inner cylinder rotation in an inertial reference frame. To reduce dispersive errors in the simulations, we simulate the system in a rotating frame such that the velocities at both cylinders are always equal and of opposite signs ($\pm U/2$). Due to this reference frame change, a Coriolis force naturally arises [@ost14d]. The initial shear Reynolds number was taken as $Re_s = Ud/\nu = 3.52\times 10^4$, where $r_i$ ($r_o$) is the inner (outer) cylinder radius, $d$ the gap width $d=r_o-r_i$, and $\nu$ the kinematic viscosity of the fluid. Periodic boundary conditions are employed in the axial direction. The system has a radius ratio of $\eta=r_i/r_o=0.909$ and an aspect ratio of $\Gamma=L_z/(r_o-r_i)=4$, where $L_z$ is the axial periodicity length. This geometry was chosen such that the system had two fixed roll pairs (i.e. four rolls) and curvature effects are limited. A rotational (azimuthal) symmetry of order $n_{sym}=10$ was forced on the system to reduce computational costs. This means that the system is $4\pi$ half-gaps long in the streamwise direction at its smallest extent. A resolution of $N_\theta \times N_r \times N_z = 1024\times512\times1024$ was used, the grid nodes were uniformly distributed in the axial and azimuthal directions, while a clipped Chebychev distribution was used for the radial direction to cluster points near the walls. This is equivalent to the resolutions used for previous studies [@ost16vlb]. The simulations were run until a statistically stationary state was reached, where two roll pairs could be seen in the velocity fields, and the torque at both cylinders was equal when temporally averaging over a sufficiently long time. We simulate three cases: for the first case, which we denote as sudden stop as in the experiment (SS-EXP), we instantaneously stop the inner cylinder at $t=0$, and allow the turbulence to decay. The main difference between the simulation and the experiment is that we stop the cylinders instantaneously, while the stopping time of the experiment by Verschoof et al. [@ver16] is around 12 s. Even if this is still orders of magnitude smaller than the viscous time ($d^2/\nu=6\cdot 10^3$ s) it corresponds to over three-thousand large eddy turn-over times as mentioned previously ($d/U=3.2\cdot10^{-3}$ s). To eliminate as much as possible the effects of the wall, we run a second case, denoted by sudden disengagement (SD) in which we suddenly change the boundary condition at the wall to stress-free. The flow is no longer forced by shear, and thus the turbulence decays. Finally, we run an intermediate, third case, denoted by (SS-V0) where we set the velocity of both cylinders to the mean velocity, i.e. zero in the rotating frame. While in the experiment, and in the SS-EXP simulation, the inner cylinder is brought to the same velocity as the outer cylinder, and there is a substantial mean momentum which must be transported from bulk to walls, this is not the case in the SS-V0 case because the mean momentum respect to the cylinders is close to zero. In both the SS-V0 case and the SD case, the main decay is that of velocity fluctuations while in the SS-EXP case it is of the mean azimuthal (streamwise) velocity. Thus, we expect the SS-V0 case to behave in an intermediate way between the SD case and the SS-V0 case. As turbulence decays, the computational method is switched for efficiency. At $\tilde{t} = tU/d \approx 125$ for the SS-V0 case, and $\tilde{t} \approx 325$ for the SD and SS-EXP cases, the flow fields were down-sampled to a resolution of $512^3$, and the treatment of the viscous terms in the homogeneous directions was made implicit. This allowed the time-step to increase to $\Delta \tilde{t} = 1$, dramatically reducing the computational cost. The simulations were then advanced in time up to $\tilde{t}=7500$ (SS-V0/SD) and $\tilde{t}=15000$ (SS-EXP), late enough to be in the asymptotic viscous stage. ![Top left: Temporal evolution of the kinetic energy in the azimuthal velocity. Bottom left: Temporal evolution of the wind kinetic energy. The figures show the same data in semilogarithmic scale (right) and double logarithmic scale (left). Symbols: SS-EXP (green), SD (red) and SS-V0 (blue). []{data-label="fi:urms"}](pdffigs/urms_bsbs_loglog.pdf "fig:"){width="48.00000%"} ![Top left: Temporal evolution of the kinetic energy in the azimuthal velocity. Bottom left: Temporal evolution of the wind kinetic energy. The figures show the same data in semilogarithmic scale (right) and double logarithmic scale (left). Symbols: SS-EXP (green), SD (red) and SS-V0 (blue). []{data-label="fi:urms"}](pdffigs/urms_bsbs_semilogy.pdf "fig:"){width="48.00000%"}\ ![Top left: Temporal evolution of the kinetic energy in the azimuthal velocity. Bottom left: Temporal evolution of the wind kinetic energy. The figures show the same data in semilogarithmic scale (right) and double logarithmic scale (left). Symbols: SS-EXP (green), SD (red) and SS-V0 (blue). []{data-label="fi:urms"}](pdffigs/krms_bsbs_loglog.pdf "fig:"){width="48.00000%"} ![Top left: Temporal evolution of the kinetic energy in the azimuthal velocity. Bottom left: Temporal evolution of the wind kinetic energy. The figures show the same data in semilogarithmic scale (right) and double logarithmic scale (left). Symbols: SS-EXP (green), SD (red) and SS-V0 (blue). []{data-label="fi:urms"}](pdffigs/krms_bsbs_semilogy.pdf "fig:"){width="48.00000%"}\ ![Temporal evolution of the energy dissipation. The figures show the same data in semilogarithmic scale (right) and double logarithmic scale (left). Symbols: SS-EXP (green), SD (red), SS-V0 (blue), experimental data from Ref. [@ver16] (right, data matches top temporal axis only). []{data-label="fi:diss"}](pdffigs/diss_bsbs_loglog.pdf "fig:"){width="48.00000%"} ![Temporal evolution of the energy dissipation. The figures show the same data in semilogarithmic scale (right) and double logarithmic scale (left). Symbols: SS-EXP (green), SD (red), SS-V0 (blue), experimental data from Ref. [@ver16] (right, data matches top temporal axis only). []{data-label="fi:diss"}](pdffigs/diss_bsbs_semilogy.pdf "fig:"){width="48.00000%"}\ ![image](pdffigs/2-eps-converted-to.pdf){width="80.00000%"} ![Pseudocolor visualization of the azimuthal velocity in a constant-azimuth cut for the SS-EXP case at five different times (left to right): $\tilde{t}=0$, $10$, $100$, $1000$ and $17000$. []{data-label="fi:vizthcutnsexp"}](pdffigs/NSEXP_q1thcut-eps-converted-to.pdf){width="98.00000%"} Figure \[fi:urms\] shows the temporal evolution of the average kinetic energy of the azimuthal velocity ($\frac{1}{2}\langle u_\theta^2\rangle_\Omega$) and the wind kinetic energy ($\frac{1}{2}\langle u_r^2 + u_z^2 \rangle_\Omega$) for all cases, while figure \[fi:diss\] shows the temporal evolution of the total kinetic energy dissipation rate $\epsilon=\frac{1}{2}\nu\langle \partial_i u_{j}\rangle^2_\Omega$, where $\langle \dots \rangle$ denotes the averaging operator and $\Omega$ the entire fluid volume. On the right panel of figure \[fi:diss\], experimental data from [@ver16] are also added for comparison. This data only matches the top temporal axis (viscid units). No overarching behavior or power law which describes either the dissipation or the evolution of the kinetic energy can be seen. Instead, several different life stages of the decay are revealed, which we detail here. In figure \[fi:urms\], the main difference between the SS-EXP and the other two cases can be clearly seen: the azimuthal kinetic energy (in the bulk) is several orders of magnitude larger than in the other cases. Throughout the entire decay, it is significantly larger than the wind kinetic energy, in line with what was observed by Ref.[@ver16]. As we will see, this dominates the physics of the decay. The black curve in the right panel of Fig. \[fi:diss\] shows a much faster decay in the dissipation rate (when measured in viscous time units) than any of the numerical cases simulated, the reasons for this will be explored later. The stages of decay are illustrated by Figures \[fi:vizthcutnsvzero\]-\[fi:vizthcutnsexp\], which shows a visualizations of the azimuthal velocity at different time instants for the SS-V0 and SS-EXP cases, from the start of the simulation. Aside from the magnitude of the velocity being different, we can also see wide variation in the flow topology. A similar flow topology as the one seen for SS-V0 is seen for the SD case, too. We briefly describe the stages for the SS-V0 and SD cases: during the first life stage, which takes place between $\tilde{t}=0$ to $\tilde{t}\approx10$ (corresponding to Figure \[fi:vizthcutnsvzero\]b), the large-scale rolls remain in motion, as they are still being fed from the perturbations inside the boundary layers. After this, the flow undergoes non-linear non-normal transient growth, as in an ordinary shear flow. Finally, in the last, asymptotic stage, (shown in the rightmost panels), viscous diffusion dominates and the flow becomes homogeneous in the azimuthal and axial directions. The SS-EXP case shows different behaviour, as the large scale roll remains being fed throughout the entire simulation by the outer cylinder, while the inner cylinder undergoes a rapid change between feeding the roll through a centrifugal instability for $\tilde{t}<0$, to being centrifugally stable for $\tilde{t}>0$. This is especially noticeable in the second panel of figure \[fi:vizthcutnsexp\], where remnants of ejection zones of the outer cylinder are clearly visible, but the flow topology at the inner cylinder has changed. After this phase, the flow loses the clear axial signature of the rolls (third panel), but the strong boundary layer asymmetry remains in place as evident in the fourth panel. At even later times, shown in the last panel, the flow also enters the asymptotic stage, where it becomes practically homogeneous in the azimuthal and axial direction, analogous to the right panel of figure \[fi:vizthcutnsvzero\]. No significant axial signature was observed in Ref. [@ver16] after the long stopping time of the cylinders, which provides some indication that their measurements start at a time where the flow approximately behaves like in the third panel of figure \[fi:vizthcutnsexp\]. Due to the less clean delimitation between stages in the SS-EXP case, we first focus on the three stages of the SS-V0 and SD cases. During the first life stage, which takes place between $\tilde{t}=0$ to $\tilde{t}\approx10$ (corresponding to Figure \[fi:vizthcutnsvzero\]b), the large-scale rolls remain in motion, as they are still fed from the perturbations inside the boundary layers. Thhe wind kinetic energy remains constant (horizontal) between $\tilde{t}=0$ and $\tilde{t}\approx 10$, while the kinetic energy of the azimuthal velocity decreases by more than an order of magnitude. The different behavior of velocity components highlights the anisotropic character of the decay. We note that the time-scale of this first stage is the same for both the SS-V0 and the SD case, highlighting that the rolls drain energy much more efficiently than the wall. A simple scaling estimate for the diffusive time across the boundary layer of the available kinetic energy to the wall is $t_{\nu,BL} = \lambda_\omega^2/\nu$. If we estimate the boundary layer thickness $\lambda_\omega$ as $d/(2Nu_\omega)$, with $Nu_\omega$, the generalized Nusselt number from the initial turbulent simulation and equal to $Nu_\omega \approx 25$ [@ost16vlb], we obtain an estimate for $t_{\nu,BL} U/d \approx 200$, an order of magnitude larger than the time-scale seen. This stage is not seen in Fig. \[fi:urms\] for the SS-EXP case, because the kinetic energy is dominated by the bulk, and not by the boundary layers. From this we would also not expect this stage to be present in the experiment. Further proof of this is seen in figure \[fi:kinen\_s1\], which shows the distribution of the azimuthal kinetic energy at the start and end of the first stage, and at similar times for the SS-EXP case. At the start of the decay, energy is concentrated near the boundary layers. Once this energy is drained by the rolls, they fade away. The boundary layers grow, and the fluctuation maximum moves away from the walls. Energy is redistributed between the existing structures, and the flow topology changes from Figure \[fi:vizthcutnsvzero\]b to \[fi:vizthcutnsvzero\]c. Counterintuitively, the decay of azimuthal kinetic energy is non-monotonic, and in Fig. \[fi:kinen\_s1\], we see an increase in azimuthal kinetic energy at around $\tilde{t}\approx 15$ which is drawn from the wind kinetic energy during the redistribution, up to a maximum at $\tilde{t}\approx 40$. Wall-friction plays a secondary role during this stage. The main player draining the available energy in the boundary layer are the rolls. The wall-friction comes in as a small correction, making the SS case undergo a transition from the first life stage to the second life stage slightly faster than the SD case. Non-monotonic behavior can be seen for both SS and SD cases. For the SS-EXP case, no redistribution is seen. The energy levels for the bulk are much higher and this dominates $\langle u^2_\theta \rangle_\Omega$. ![Left: Azimuthally and axially averaged azimuthal kinetic energy dissipation for the SS-V0 case, and $\tilde{t}=0$ (blue), $\tilde{t}=10$ (orange), $\tilde{t}=30$ (green) and $\tilde{t}=50$ (red). Right: same, for SS-EXP case and $\tilde{t}=0$ (blue), $\tilde{t}=10$ (orange) and $\tilde{t}=50$ (green) []{data-label="fi:kinen_s1"}](pdffigs/kineticenergy_stage1.pdf "fig:"){width="48.00000%"} ![Left: Azimuthally and axially averaged azimuthal kinetic energy dissipation for the SS-V0 case, and $\tilde{t}=0$ (blue), $\tilde{t}=10$ (orange), $\tilde{t}=30$ (green) and $\tilde{t}=50$ (red). Right: same, for SS-EXP case and $\tilde{t}=0$ (blue), $\tilde{t}=10$ (orange) and $\tilde{t}=50$ (green) []{data-label="fi:kinen_s1"}](pdffigs/kineticenergy_nsexp.pdf "fig:"){width="48.00000%"} Then the second life stage is entered which takes place between $\tilde{t}\approx 10 $ and $\tilde{t} \approx 500$. With the rolls faded away, perturbations are amplified non-normally. A detailed discussion of this non-normal-non-linear mechanism is found in Ref. [@gro00rmp]. Here it is sufficient to state that perturbations in the axial (spanwise) direction tend to grow faster than those in other directions. As typical for this type of instability, the growth of these perturbations is transient, and after several time units they decay. For the SS-V0 case, the wind kinetic energy drops to a minimum around $\langle u^2 \rangle_\Omega \approx 10^{-5}$ and it remains at that level until the end of this regime, while for the SD case, the wind kinetic energy does not reach a plateau but instead oscillations appear throughout the entire decay. The SS case appears to drain the available energy of the many non-normal modes in a shorter amount of time, while this draining occurs over a longer time-scale for the SD case. The signature of non-normal transient decay can be clearly seen for the wind kinetic energy in the SD and SS-EXP cases up to the end of the simulation. When one of the non-normal modes grows, amplifies transiently and decays, it imprints the oscillations seen in the wind kinetic energy for $\tilde{t}>100$. ![Pseudocolor visualization of the azimuthal velocity in a constant-radius cut at the mid-gap ($r=r_a$) for the SS-V0 case at two different times $\tilde{t}=75$ (top) and $1000$ (bottom). []{data-label="fi:vizrcut"}](pdffigs/4-eps-converted-to.pdf){width="98.00000%"} Due to the non-normal transient growth and dissipation of axially-oriented perturbations, the flow becomes practically homogeneous in the axial direction for $\tilde{t} > 500$ in the SS-V0 case. The large-scale flow structure is now columnar. This is shown in Figure \[fi:vizrcut\]. The SD and SS-EXP cases do not show this, and instead show both azimuthal and axial dependency even at later times. Unlike the SS-V0 case, the marks of non-linear transient growth seem to be present in the wind kinetic energy even up to $\tilde{t} \approx 2500$ for these two cases. The experimental data shows an even steeper decay of $\epsilon$ than the DNS in Fig. \[fi:diss\]. The instantaneous Reynolds number is still significantly higher in the experiment than in the simulations even for later times $Re\approx 500$. Significant non-linear non-normal transient growth can still happen, which means an overall faster draining of energy and an overall faster decay rate (when looking at the flow in viscid time units). This is reflected in a steeper decay of $\epsilon$. The role of wall-friction is minor in the transition between the second and third stage, as this happens when transient non-normal growth is exhausted. Once either the non-linear transient growth mechanism is drained, or the wind kinetic energy is sufficiently small that its effect cannot be felt, the decay enters its last life stage, in which the decay is dominated by viscosity. ![Left: Axially and azimuthally averaged normalized azimuthal velocity profiles for the SS-V0 at $\tilde{t}=500$ (blue), $\tilde{t}=1000$ (green) and $\tilde{t}=5000$ (orange), as well as the theoretical solution based on Bessel functions (black dashed). Right: same, for the SS-EXP case at times $\tilde{t}=10$ (blue), $\tilde{t}=100$ (orange), $\tilde{t}=1000$ (green), $\tilde{t}=5000$ (red) and $\tilde{t}=15000$ (purple). []{data-label="fi:selfsim"}](pdffigs/selfsim_nsns_2.pdf "fig:"){width="48.00000%"} ![Left: Axially and azimuthally averaged normalized azimuthal velocity profiles for the SS-V0 at $\tilde{t}=500$ (blue), $\tilde{t}=1000$ (green) and $\tilde{t}=5000$ (orange), as well as the theoretical solution based on Bessel functions (black dashed). Right: same, for the SS-EXP case at times $\tilde{t}=10$ (blue), $\tilde{t}=100$ (orange), $\tilde{t}=1000$ (green), $\tilde{t}=5000$ (red) and $\tilde{t}=15000$ (purple). []{data-label="fi:selfsim"}](pdffigs/selfsim_nsexp_new.pdf "fig:"){width="48.00000%"} In this last life stage the energy dissipation rate behaves in a quasi-exponential manner. The wind kinetic energy is negligible and we can expect the azimuthal velocity to behave like a passive scalar with diffusion, i.e. a heat equation. With the boundary and initial conditions, the problem is analogous to the quenching of a cylindrical annulus, whose solution is given by $u_\theta(r,t)=\sum_n A_n J_0(\lambda_n r)\exp(-\nu \lambda_n^2 t)$, where $J_0(r)$ are Bessel functions of the zeroth order and $\lambda_n$ their associated eigenvalues, and $A_n$ depend on the initial conditions [@prosbook]. If $t$ is large, only the slowest-decaying function with the smallest eigenvalue has significant energy. The energy dissipation behaves exponentially in time, and self-similarity comes out naturally. To demonstrate that for very large times (beyond $\tilde{t} > 5000$ for SS-V0 and $\tilde{t} > 15000$ for SS-EXP), the asymptotic self-similar decay regime is entered, in Figure \[fi:selfsim\] we show several azimuthal velocity profiles, normalized by their mean value. For the SS-V0 case, the normalized velocity profiles at times $\tilde{t}=5000$ to the end of our simulation $\tilde{t}=7500$ (corresponding to $t\nu/d^2\approx 0.2$) collapse, and agree with the theoretically calculated fundamental Bessel function, while the earlier normalized profiles between $\tilde{t}\approx1000$ and $\tilde{t}\approx5000$ show slight deviations as the asymptotic stage has not yet been reached. The normalized profile around $\tilde{t}\approx500$ is very different, as it is in the non-normal transient growth state. For the SS-EXP case, the asymptotic, self-similar stage is only reached for $\tilde{t}\approx 15000$, corresponding to $t\nu/d^2 \approx 0.25$. When comparing these profiles to the proposed self-similarity in Ref. [@ver16], we find that they are similar at intermediate times: a flat profile in the center, indicating strong turbulent mixing, and a steeper boundary layer at the outer cylinder, due to the centrifugal instability at the outer cylinder and the centrifigual stability at the inner cylinder. This intuition coincides with the fact that the largest reported profile at $t=100s$ is only $t\nu/d^2 \approx 0.03$ in viscous units. The proposed self-similarity is shown for a small interval of time ($0.001< t\nu/d^2 < 0.03$), and may be just be a product of the relatively small time interval. In summary, in this manuscript, we have studied the decay of Taylor-Couette turbulence, focusing also on the early and late stages of decay that could not be observed in Ref. [@ver16]. We have shown that three distinct life stages are seen where different decay mechanisms dominate. In the first life stage, the energy decays mainly through the linearly unstable modes, i.e. the rolls. After this, the available energy is directed towards non-normal modes which sustain transient growth. During the transition between stages, a redistribution of energy between structures occurs, and $\langle u_\theta \rangle^2_\Omega$ increases. While this can only be seen if the flow inertia is small, it highlights that the decay process is anisotropic and be temporally non-monotonic. Finally, the decay becomes purely viscous, in spite of the relatively high instantaneous Reynolds number, and self-similarity can be observed. We have shown that no simple power law can cover all three stages. The most unstable modes dominate the early decay; the modes decay progressively, from linearly-unstable modes with roll-like instabilities, to non-normal transient growth to self-similar decay, which behaves like a quenching problem. This progression of decaying modes from more to less unstable can shed light onto how real-world unforced turbulence decays. Our finding that anisotropic flows can decay in a viscid way even at moderate Reynolds numbers, that redistribution of energy between structures can lead to non-monotonic behavior, and that wall-friction is a secondary mechanism in the initial decay can provide insight into the decay of geo- and astrophysically relevant systems even if confinement of the Taylor-Couette system could still play a role. Further insights into decay and the cross-over between the first two stages can be provided by studying the decay in linearly stable Taylor-Couette flow [@ost16b], or in plane Couette flow, i.e. the flow between two parallel plates, which could make the first stage of decay less important. The TC geometries studied have very limited curvature, and its (de)stabilizing role in reducing the non-normal transient growth can be explored. Finally, studies of the decay of thermal turbulence [@ahl09], and further exploring the analogy between TC and Rayleigh-Bénard convection [@dub02; @eck07b], the flow in a layer heated from below and cooled from above, which have been called the “twins of turblence research” [@bus12], is another research line which can lead to increased understanding of decaying geo- and astrophysical turbulence. Acknowledgments: We thank B. Eckhardt for the fruitful and stimulating discussions, and R. A. Verschoof for providing the data from Ref. [@ver16]. This research was supported in part by the National Science Foundation under Grant No. PHY11-25915. We also gratefully acknowledge computational time for the simulations provided by SURFsara on resource Cartesius through a NWO grant.
--- abstract: 'The classical output theorem for the $M/M/1$ queue, due to Burke (1956), states that the departure process from a stationary $M/M/1$ queue, in equilibrium, has the same law as the arrivals process, that is, it is a Poisson process. In this paper we show that the associated measure-preserving transformation is metrically isomorphic to a two-sided Bernoulli shift. We also discuss some extensions of Burke’s theorem where it remains an open problem to determine if, or under what conditions, the analogue of this result holds.' address: - 'Department of Mathematics and Computer Science, Wesleyan University, Middletown, CT 06459, USA' - 'Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK' author: - Michael Keane - 'Neil O’Connell' title: 'The $M/M/1$ queue is Bernoulli' --- ł ø Ø Ł[[L]{}]{} ¶[[P]{}]{} [2000 MSC:]{} Primary 60K25, 37A50; Secondary 60J15, 60J65, 37H99. Introduction ============ The classical output theorem for the $M/M/1$ queue, due to Burke [@burke], states that the departure process from a stationary $M/M/1$ queue, in equilibrium, has the same law as the arrivals process, that is, it is a Poisson process. To be more precise, let $A$ and $S$ be Poisson processes on $\R$ with repective intensities $\lambda<\xi$ and define, for $t\in\R$, $$Q(t)=\sup_{s\le t} (A(s,t]-S(s,t]).$$ For each $t$, $Q(t)$ should be interpreted as the number of customers in the queue at time $t$. Customers arrive according to the Poisson process $A$ (the arrivals process) and at times given by the points of $S$, if the queue is non-empty, a customer is served and departs from the queue. The departure process $D$ is defined to be the point process of times at which customers depart from the queue. More precisely, we define, for $s<t$, $$D(s,t]=Q(s)+A(s,t]-Q(t).$$ Burke’s theorem states that $D$ is a Poisson process with intensity $\lambda$, and moreover that $(D(t,0],t<0)$ is independent of $Q(0)$. The standard proof of this fact, due to Reich [@reich], is a reversibility argument which exploits the dynamical symmetry of the queue and the fact that $Q$ is a stationary, reversible Markov chain. For more background on queueing theory, see, for example, Kelly [@kelly]. The nature of Burke’s theorem suggest that there may be a measure-preserving transformation somewhere nearby. It is not immediately obvious how to find it, since $D$ is not only a function of $A$, it also depends on $S$. However, it was shown in [@oy02] that, if we define $R=A+S-D$, then the pair $(D,R)$ has the same joint law as $(A,S)$, thus exhibiting a measure-preserving transformation; moreover, the restriction of $(D,R)$ to $(-\infty,0]^2$ is independent of $Q(0)$. We can restate this as follows. For $t\in\R$, set $$X(t)=\begin{cases} S(0,t]-A(0,t] & t>0\\ A(-t,0]-S(-t,0] & t\le 0.\end{cases}$$ and $$Y(t)=\begin{cases} R(0,t]-D(0,t] & t>0\\ D(-t,0]-R(-t,0] & t\le 0.\end{cases}$$ Note that we can write $$Y(t)=2M(t)-X(t)-2M(0),\qquad M(t)=\sup_{-\infty<s\le t} X(s).$$ Then $X$ is a two-sided continuous-time simple random walk with positive drift $\xi-\lambda$, and the transformation which maps $X$ to $Y$ is measure-preserving; moreover, $(Y(t),t\le 0)$ is independent of $Q(0)\equiv M(0)$. This statement can be further simplified by considering only the times at which events occur (i.e. the times at which the random walk $X$ jumps). Denote these times (which are almost surely distinct) by $$\cdots<\tau_{-2}<\tau_{-1}<0<\tau_1<\tau_2<\cdots$$ and set $x_n=X(\tau_n)$ and $y_n=Y(\tau_n)$, for $n\in\Z$. Note that, for $n\in\Z$, $$y_n=2s_n-x_n-2s_0,\qquad s_n=\sup_{m\le n}x_m.$$ Then $(x_n,n\in\Z)$ is a two-sided, discrete-time simple random walk, as is $(y_n,n\in\Z)$, and $(y_n,n\le 0)$ is independent of $s_0$. Finally, let $\O=\{-1,1\}^\Z$ be equipped with Bernoulli product measure with parameter $p=\xi/(\lambda+\xi)$. Set $\epsilon_n=x_n-x_{n-1}$ and $\sigma_n=y_n-y_{n-1}$. Then we can write $\sigma=T\epsilon$, where $T$, defined almost everywhere on $\O$, is a measure-preserving transformation. The fact that $(\sigma_n,n\le 0)$ is independent of $s_0$ can now be interpreted as saying that $T$ has a [factor]{} which is Bernoulli, that is, a factor which is metrically isomorphic to a two-sided Bernoulli shift (see section 2 for details). The main result of this paper is that $T$ is, in fact, Bernoulli. This will be presented in section 2. In section 3 we discuss the Brownian analogue of Burke’s theorem where it is only possible to show that the corresponding transformation has a Bernoulli factor. The difficulty here is similar to that encountered in the open question, posed by Marc Yor, of determining whether Lévy’s transformation of Brownian motion is ergodic. Dubins and Smorodinsky [@ds] proved that there is a discrete version of Lévy’s transformation which is isomorphic to a one-sided Bernoulli shift. In section 4 we describe a natural extension of Burke’s theorem to the more general setting of iterated random functions, and leave it as an open problem to determine under what conditions the corresponding transformation is Bernoulli. The main result =============== Let $\m$ be a Bernoulli product measure on $\O=\{-1,1\}^\Z$ with $$\m\{\o\in\O:\ \o_0=1\}=p>1/2.$$ Define a two-sided simple random walk $x=(x_n,n\in\Z)$ by $x_0=0$, $$x_n=\begin{cases}x_{n-1}+\o_n & n>0 ,\\ x_{n+1}-\o_{n+1} & n<0.\end{cases}$$ For $n\in\Z$, set $s_n=\sup_{m\le n} x_m$ and $\O'=\{s_0(\o)<\infty\}$. Note that $\m(\O')=1$. Write $y=2s-x$ and define $T:\O'\to\O$ by setting $(T\o)_n=y_n-y_{n-1}$ for each $n\in\Z$. In order to discuss the inverse transformation we further define $$\O''=\{\o\in\O':\ \liminf_n(s_n-x_n)=0\}$$ and note that $\m(\O'')=1$. Let $R:\O\to\O$ be the ‘time-reversal’ operator defined by $(R\o)_n=\o_{-n}$ for $n\in\Z$, and set $\rho=p^{-1}(1-p)$. We first recall the analogue of Burke’s theorem in this discrete setting. \[burke\] - $\m\circ T^{-1}=\mu$. - For $x\ge 0$, $\m\{\o:\ s_0(\o)=x\}=(1-\rho)\rho^x$. - The random variable $s_0$ is independent of $((T\o)_n,n\le 0)$. - If $\o\in\O''$ then $\o=(RTR)(T\o)$. [Proof.]{} The measure-preserving property (i) is essentially equivalent to the output theorem for the stationary $M/M/1$ queue, as discussed in section 1, which follows easily from the fact that the Markov chain $q=s-x$ is stationary and reversible. Property (ii) is well-known. Properties (iii) and (iv) follow from (i) and the fact that, for $\o\in\O''$, $s_n=\min_{l\ge n} y_l$, $\forall n$. $\Box$ An immediate consequence of (iv) is that there exists $\O^\subset\O$ with $\m(\O^)=1$ and on which $T^k$ is defined for all $k\in\Z$. Define a mapping $\p:\O^\to\N^\Z$ by putting $(\p\o)_k=s_0(T^k\o)$ for each $k\in\Z$. Denote the shift operator on $\N^\Z$ by $\theta$ and let $\g$ be the $\theta$-invariant product measure on $\N^\Z$ with $$\g\{\a\in\N^\Z: \a_0=x\}=(1-\rho)\rho^x\qquad x\ge 0.$$ \[main\] - $\m\circ \p^{-1}=\g$. - Almost every $\o\in\O^$ is uniquely determined by $\p\o$. - $T=\p^{-1}\circ\theta\circ\p$ almost everywhere. [Proof.]{} Claim (i) follows from Theorem \[burke\] (iii). To prove (ii) we first note that $\o_0=(-1)^N$ where $N=\min\{k\ge 0:\ s_0(T^k\o)=0\}$. Indeed, if $s_0(T^k\o)>0$, then $(T^{k+1}\o)_0=-(T^k\o)_0$ whereas, if $s_0(T^k\o)=0$, then $(T^k\o)_0=1$. By the same reasoning, for any $k\ge 0$, we have $(T^k\o)_0=(-1)^{N_k}$, where $N_k=\min\{l\ge 0:\ s_0(T^{k+l}\o)=0\}.$ Thus, we can recover $((T^k\o)_0,k\in\Z)$ from $\p\o$. In exactly the same way, for any $n\in\Z$, we can recover $((T^k\o)_n,k\in\Z)$ from the sequence $(q_n(T^k\o),k\in\Z)$, where $q=s-x$. Combining this observation with the identity $$q_{n-1}(T^k\o)=\max\{ q_n(T^k\o)+(T^{k+1}\o)_n,0\}$$ we see that, for any $n\le 0$ we can recover $((T^k\o)_n,k\in\Z)$ from $\p\o$. In particular, we recover $(\o_n,n\le 0)$, from $\p\o$. A similar argument works in the other direction, starting with the observation that, if $s_0(T^k\o)>0$, then $(T^{k+1}\o)_1=-(T^k\o)_1$ whereas, if $s_0(T^k\o)=0$, then $(T^{k+1}\o)_1=1$; this leads to the conclusion that $\{\o_n,n\ge 1\}$ can be recovered from $\p\o$, which completes the proof of (ii), and (iii) follows. $\Box$ Brownian version ================ Let $(X(t),t\in\R)$ be a two-sided standard Brownian motion with drift $\nu>0$ and with $X(0)=0$. For $t\in\R$, set $$Y(t)=2M(t)-X(t)-2M(0),\qquad M(t)=\sup_{-\infty<s\le t} X(s).$$ The continuous analogue of Burke’s theorem (see, for example, [@oy02b] and references therein) states that $Y$ has the same law as $X$ and, moreover, that $(Y(t),t\le 0)$ is independent of $M(0)$, which is exponentially distributed with parameter $2\nu$. It follows that the measure-preserving transformation $T$, which maps $X$ to $Y$, has a factor which is metrically isomorphic to the shift operator on $\R_+^\Z$, equipped with the product measure $\varepsilon^{\otimes\Z}$, where $\varepsilon$ is the exponential distribution on $\R_+$ with parameter $2\nu$. However, it is not clear in this setting whether or not $X$ can be recovered from the sequence $(\sup_{-\infty<s\le 0} (T^k X)(s),\ k\in\Z)$, so we cannot conclude that $T$ is Bernoulli. The recovery map for the discrete case, defined in the proof of Theorem \[main\], does not have an obvious continuous analogue. It is thus an open problem to determine whether or not this transformation is Bernoulli, or even ergodic. This is reminiscent of a (still open) question, originally posed by Marc Yor, in relation to the following transformation of Brownian motion. Let $(B_t,t\ge 0)$ be a standard one-dimensional Brownian motion. It is a classical result, due to Paul Lévy, that the process $$\left( \|B_t\|-L_t^0(\|B\|),\ t\ge 0\right)$$ is also a standard Brownian motion, where $L_t^0(\|B\|)$ denotes the local time at zero of $\|B\|$ up to time $t$. Is this an ergodic transformation? Dubins and Smorodinsky [@ds] proved that there is a discrete version which is metrically isomorphic to a (one-sided) Bernoulli shift. Iterated random functions ========================= The classical output theorem for the $M/M/1$ queue extends quite naturally to the more general setting of iterated random functions. Loosely following [@df], let $S$ be a topological space equipped with its Borel $\sigma$-algebra, $\{ f_\t,\ \t\in\T\}$ a family of continuous functions that map $S$ to itself and $\mu$ a probability distribution on $\T$. Let $(\t_n,n\in\Z)$ be a sequence of random variables with common law $\kappa$. Consider the markov chain $x=(x_n,n\ge 0)$ with state space $S$ defined by $x_0=s$ and $$\label{rec} x_n=f_{\t_{n}}(x_{n-1})=(f_{\t_n}\circ\cdots\circ f_{\t_1})(s) , \qquad n>0.$$ We will assume that this Markov chain has reversible transition probabilities with respect to a unique invariant probability measure. Now consider the [backward]{} iterations: $$u_m=(f_{\t_1}\circ\cdots\circ f_{\t_m})(s).$$ Under certain regularity conditions, as discussed in [@df], the sequence $u_m$ converges almost surely, as $m\to\infty$, to a limiting random variable $u_\infty$ which does not depend on $s$ and which realises the invariant distribution of $x$. We will assume that this property holds. It follows that, for each $n\in\Z$, the limit $$\label{Zdef} z_n=\lim_{m\to\infty} (f_{\t_{1+n}}\circ\cdots\circ f_{\t_{m}})(s)$$ exists almost surely and does not depend on $s$. By continuity, these random variables satisfy $$\label{brec} z_n=f_{\t_{n+1}}(z_{n+1}), \qquad n\in\Z,$$ from which it follows, recalling that $x$ is has reversible transition probabilities, that the sequence $z=(z_n,\ n\in\Z)$ is a two-sided stationary version of $x$. Now suppose that, for each $s \in S$, the map $\t\mapsto (s,f_\t(s))$ is injective, and define $F(r,s)=\t$ whenever $s=f_\t(r)$. Then we can write $$\label{z} \t_{n}=F(z_{n},z_{n-1}), \qquad n\in\Z.$$ Define a sequence of random variables $\eta=\{\eta_n,n\in\Z\}$ by setting $$\label{eta} \eta_{n}=F(z_{n-1},z_{n}), \qquad n\in\Z,$$ so that $$\label{brec2} z_{n}=f_{\eta_{n}}(z_{n-1}), \qquad n\in\Z .$$ Reversibility ensures that $\eta$ is well-defined. \[burke2\] In the above context, $\eta$ has the same distribution as $\t$ and the sequence $\eta_1,\eta_2,\ldots$ is independent of $z_0$. [Proof.]{} The first claim follows from (\[z\]) and (\[eta\]), and the fact that $z$ is stationary and reversible. By (\[brec2\]) we can write, almost surely, $$z_0=f_{\eta_0} ( f_{\eta_{-1}} (f_{\eta_{-2}}( \cdots$$ which is independent of $\eta_1,\eta_2,\ldots$ as required. $\Box$ This defines a measure-preserving transformation (mapping $\theta$ to $\eta$) which has a Bernoulli factor. When is it Bernoulli? The $M/M/1$ queue corresponds to the special case where $\T=\{-1,1\}$, $1-\kappa\{-1\}=\kappa\{1\}=q<1/2$ and $f_{\theta}(x)=\max\{x+\theta,0\}$. Examples of iterated random functions where Theorem \[burke2\] applies can be found in [@dyson] and [@oc03]. Further examples which arise from taking products of random matrices, and for which the invariant measure is known explicitly, are discussed in the paper [@mtw]; note however that not all of these are reversible. [Acknowledgements.]{} This research was supported by Science Foundation Ireland, grant number SFI 04-RP1-I512. [10]{} P. Burke. The output of a queueing system. 4 (1956) 699–704. Persi Diaconis and David Freedman. Iterated random functions. [SIAM Review]{} 41 (1999) 45–76. Lester E. Dubins and Meir Smorodinsky. The modified, discrete, LŽvy-transformation is Bernoulli. [Séminaire de Probabilités]{}, XXVI, 157–161, Lecture Notes in Math., 1526, Springer, Berlin, 1992. F. J. Dyson. The dynamics of a disordered linear chain. [Phys. Rev.]{} 92 (1953) 1331–1338. F.P. Kelly. . Wiley, 1979. J. Marklof, Y. Tourigny and L. Wolowski. Explicit invariant measures for products of random matrices. [Trans. Amer. Math. Soc.]{}, to appear. Neil O’Connell. Random matrices, non-colliding processes and queues. [Séminaire de Probabilités]{}, XXXVI, 165–182, Lecture Notes in Math., 1801, Springer, Berlin, 2003. Neil O’Connell and Marc Yor. A representation for non-colliding random walks. [Electron. Comm. Probab.]{} 7 (2002). Neil O’Connell and Marc Yor. Brownian analogues of [B]{}urke’s theorem. 96 (2001) 285–304. E. Reich. Waiting times when queues are in tandem. 28 (1957) 527–530.
--- abstract: 'Rapid advancement in the domain of quantum technologies have opened up researchers to the real possibility of experimenting with quantum circuits, and simulating small-scale quantum programs. Nevertheless, the quality of currently available qubits and environmental noise pose a challenge in smooth execution of the quantum circuits. Therefore, efficient design automation flows for mapping a given algorithm to the Noisy Intermediate Scale Quantum (NISQ) computer becomes of utmost importance. State-of-the-art quantum design automation tools are primarily focused on reducing logical depth, gate count and qubit counts with recent emphasis on topology-aware (nearest-neighbour compliance) mapping. In this work, we extend the technology mapping flows to simultaneously consider the topology and gate fidelity constraints while keeping logical depth and gate count as optimization objectives. We provide a comprehensive problem formulation and multi-tier approach towards solving it. The proposed automation flow is compatible with commercial quantum computers, such as IBM QX and Rigetti. Our simulation results over 10 quantum circuit benchmarks, show that the fidelity of the circuit can be improved up to 3.37$\times$ with an average improvement of 1.87$\times$.' author: - 'Email: [debjyoti001@ntu.edu.sg]{}' bibliography: - 'ref.bib' title: 'MUQUT: Multi-Constraint Quantum Circuit Mapping on Noisy Intermediate-Scale Quantum Computers' ---

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